﻿<s>
Recalling_VVG the_AT definitions_NN2 (_( 10_MC )_) and_CC (_( 11_MC )_) of_IO @S_FO and_CC @S_FO as_II31 well_II32 as_II33 @S_FO ,_, throughout_II our_APPGE analysis_NN1 we_PPIS2 make_VV0 the_AT following_JJ assumptions_NN2 ._. 
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Thus_RR w0_FO also_RR has_VHZ a_AT1 unique_JJ critical_JJ point_NN1 ._. 
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In_RR21 particular_RR22 ,_, they_PPHS2 can_VM be_VBI used_VVN to_TO deduce_VVI the_AT existence_NN1 of_IO (_( categorified_JJ )_) cluster_VV0 structures_NN2 ._. 
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For_IF completeness_NN1 the_AT proof_NN1 of_IO Lemma_NN1 2.4_MC is_VBZ given_VVN below_RL ._. 
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For_IF the_AT unknotted_JJ circle_NN1 ,_, the_AT homology_NN1 @S_FO has_VHZ rank_NN1 3_MC ._. 
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Set_VV0 @S_FO and_CC @S_FO ,_, where_CS r_ZZ1 is_VBZ a_AT1 positive_JJ integer_NN1 so_CS21 that_CS22 L_ZZ1 and_CC L_ZZ1 '_NULL are_VBR Cartier_NP1 ._. 
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<s>
Indeed_RR ,_, over_II the_AT next_MD few_DA2 decades_NNT2 research_NN1 has_VHZ confirmed_VVN the_AT idea_NN1 "_" that_CST reversal_NN1 error_NN1 is_VBZ not_XX an_AT1 issue_NN1 of_IO mere_JJ carelessness_NN1 ,_, but_CCB it_PPH1 is_VBZ a_AT1 more_RGR deeply_RR rooted_JJ problem_NN1 of_IO how_RRQ students_NN2 comprehend_VV0 the_AT problem_NN1 and_CC interact_VV0 with_IW mathematical_JJ notation_NN1 "_" (_( Kim_NP1 et_RA21 al._RA22 ,_, 2014_MC ,_, p._NN1 12_MC )_) ._. 
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If_CS participants_NN2 reported_VVD wanting_VVG to_TO teach_VVI for_IF conceptual_JJ understanding_NN1 or_CC to_TO promote_VVI productive_JJ struggle_NN1 (_( grappling_VVG )_) ,_, among_II their_APPGE students_NN2 ,_, they_PPHS2 were_VBDR also_RR likely_JJ to_TO report_VVI that_CST their_APPGE teacher_NN1 education_NN1 program_NN1 promoted_VVN these_DD2 practices_NN2 ._. 
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Let_VV0 us_PPIO2 define_VVI for_IF any_DD t_ZZ1 >_FO 0_MC @F_FO ._. 
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We_PPIS2 have_VH0 for_IF any_DD t_ZZ1 >_FO 0_MC @F_FO ,_, where_CS we_PPIS2 have_VH0 set_VVN pcm_NNU :_: =_FO pcm_NNU (_( 0_MC )_) ._. 
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In_II the_AT case_NN1 where_CS M_ZZ1 =_FO 0_MC ,_, then_RT we_PPIS2 have_VH0 @S_FO ._. 
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This_DD1 allows_VVZ us_PPIO2 to_TO define_VVI a_AT1 parameter_NN1 @S_FO ._. 
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The_AT two_MC parameters_NN2 @S_FO completely_RR define_VV0 the_AT system_NN1 because_CS @S_FO ._. 
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We_PPIS2 must_VM now_RT say_VVI something_PN1 about_II the_AT range_NN1 of_IO values_NN2 the_AT parameter_NN1 r_ZZ1 can_VM take_VVI on_RP ._. 
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The_AT shock_NN1 location_NN1 asymptotically_RR approaches_VVZ spacelike_JJ infinity_NN1 @S_FO if_CS and_CC only_RR if_CS the_AT shock_NN1 speed_NN1 is_VBZ initially_RR positive_JJ ._. 
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Practitioner_NN1 contributions_NN2 and_CC implications_NN2 We_PPIS2 found_VVD that_DD1 classroom_NN1 practices_NN2 of_IO our_APPGE expert_NN1 MTEs_NN2 differed_VVN from_II the_AT (_( documented_VVN )_) practices_NN2 of_IO mathematics_NN1 faculty/staff_FU in_II the_AT content_JJ courses_NN2 ._. 
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As_CSA with_IW some_DD of_IO the_AT Covariation_NN1 interpretations_NN2 ,_, Unit_NN1 rate_NN1 relied_VVN on_II "_" for_IF every_AT1 "_" language_NN1 ,_, but_CCB considered_VVD the_AT amount_NN1 per_II one_MC1 ,_, specifically_RR ._. 
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<s>
Since_CS @S_FO is_VBZ etale_NN1 above_II a_AT1 neighborhood_NN1 of_IO the_AT diagonal_JJ point_NN1 ,_, we_PPIS2 find_VV0 that_CST @F_FO ._. 
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<s>
Applying_VVG the_AT localization_NN1 exact_JJ sequence_NN1 &lsqb;_( Ful_NP1 ,_, Proposition_NN1 1.8_MC &rsqb;_) to_II the_AT inclusion_NN1 @S_FO ,_, we_PPIS2 then_RT conclude_VV0 from_II (_( 3_MC )_) ,_, (_( 4_MC )_) ,_, and_CC (_( 5_MC )_) that_CST @F_FO ,_, where_CS z_ZZ1 is_VBZ a_AT1 zero-cycle_NN1 on_II VK_NP1 whose_DDQGE support_NN1 is_VBZ contained_VVN in_II @S_FO ._. 
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In_II Schwarzschild_NP1 coordinates_NN2 ,_, denoted_VVN @S_FO (_( the_AT parameter_NN1 @S_FO being_VBG the_AT speed_NN1 )_) ,_, the_AT metric_JJ of_IO interest_NN1 reads_VVZ @F_FO with_IW @S_FO and_CC @S_FO ,_, where_CS @S_FO is_VBZ the_AT canonical_JJ metric_JJ on_II the_AT two-sphere_NN1 @S_FO (_( with_IW @S_FO and_CC @S_FO )_) ._. 
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This_DD1 particular_JJ activity_NN1 of_IO actually_RR making_VVG the_AT shapes_NN2 ,_, drawing_VVG where_RRQ they_PPHS2 are_VBR ,_, or_CC filling_VVG in_II the_AT shapes_NN2 in_II an_AT1 outline_NN1 with_IW the_AT Tangram_NN1 pieces_NN2 ,_, can_VM be_VBI done_VDN in_II kindergarten_NN1 ._. 
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<s>
Now_RT let_VV0 finite_JJ sets_NN2 Mi_NP1 ,_, M2_FO ,_, and_CC M3_FO be_VBI bases_NN2 of_IO the_AT linear_JJ spaces_NN2 V1_FO ,_, V2_FO ,_, and_CC V3_FO as_CSA above_RL ._. 
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An_AT1 approximation_NN1 error_NN1 on_II the_AT time_NNT1 interval_NN1 @S_FO is_VBZ @S_FO provided_CS21 that_CS22 @S_FO with_IW @S_FO ._. 
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This_DD1 error_NN1 tends_VVZ to_TO zero_VVI as_CSA @S_FO ,_, i.e._REX ,_, as_CSA @S_FO ._. 
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Rather_CS21 than_CS22 directly_RR resorting_VVG to_II the_AT optimality_NN1 condition_NN1 ,_, we_PPIS2 adopt_VV0 an_AT1 algorithmic_JJ perspective_NN1 to_TO analyze_VVI the_AT regularized_JJ MLE_NN1 0_MC ._. 
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By_II combining_VVG Lemma_NN1 6.3_MC with_IW Lemma_NN1 6.4_MC ,_, we_PPIS2 reach_VV0 the_AT following_JJ result_NN1 ._. 
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The_AT various_JJ points_NN2 @S_FO are_VBR all_DB distinct_JJ ._. 
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In_II Sect._NP1 5_MC ,_, we_PPIS2 refine_VV0 the_AT notion_NN1 of_IO Burnside_NN1 rings_NN2 which_DDQ insures_VVZ the_AT multiplicativity_NN1 of_IO the_AT specialization_NN1 map_NN1 in_II this_DD1 more_RGR general_JJ situation_NN1 ._. 
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The_AT Lipschitz_NP1 estimate_NN1 (_( in_II space_NN1 )_) under_II coercivity_NN1 condition_NN1 (_( 1.12_MC )_) for_IF equation_NN1 (_( A.23_FO )_) has_VHZ been_VBN established_VVN in_II &lsqb;_( 26_MC &rsqb;_) ,_, and_CC subsequently_RR discussed_VVN in_II &lsqb;_( 12_MC &rsqb;_) ._. 
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To_TO see_VVI the_AT efficiency_NN1 ,_, we_PPIS2 check_VV0 numerical_JJ error_NN1 versus_II computational_JJ cost_NN1 ._. 
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For_IF a_AT1 vertex_NN1 v_ZZ1 of_IO Ca(Sg)_NP1 ,_, we_PPIS2 will_VM now_RT define_VVI two_MC different_JJ join_NN1 decompositions_NN2 of_IO @S_FO ,_, one_MC1 topological_JJ and_CC one_PN1 combinatorial_JJ ._. 
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Clearly_RR ,_, @S_FO ._. 
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We_PPIS2 now_RT use_VV0 the_AT properties_NN2 of_IO Z_ZZ1 described_VVN in_II Assumption_NN1 2.1_MC ._. 
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Since_CS @S_FO is_VBZ self-adjoint_JJ (_( @S_FO is_VBZ symmetric_JJ )_) ,_, @F_FO ._. 
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Hence_RR ,_, w_ZZ1 is_VBZ a_AT1 distributional_JJ solution_NN1 of_IO (_( EP_NP1 )_) ._. 
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As_CSA @S_FO converges_VVZ C2_FO to_II S_ZZ1 as_CSA @S_FO ,_, and_CC @S_FO limits_VVZ to_II y(i)_NNU ,_, then_RT y(i)_NNU also_RR has_VHZ vertical_JJ flux_NN1 vector_NN1 ._. 
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This_DD1 is_VBZ equivalent_JJ to_II &lsqb;_( e_ZZ1 ,_, -_- &rsqb;_) -invariance_NN1 of_IO GRicydiv_NP1 ,_, i.e._REX to_II @F_FO ._. 
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This_DD1 is_VBZ essentially_RR a_AT1 consequence_NN1 of_IO the_AT local_JJ Lipschitz_NN1 character_NN1 of_IO the_AT velocity_NN1 field_NN1 (_( 2.9_MC )_) observed_VVD in_II Section_NN1 2_MC ._. 
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The_AT numerical_JJ realization_NN1 of_IO Algorithm_NN1 4.8_MC and_CC comparison_NN1 with_IW standard_JJ iterative_JJ hard_JJ thresholding_NN1 and_CC iterative_JJ hard_JJ weighted_JJ thresholding_NN1 will_VM be_VBI conducted_VVN in_II future_JJ work_NN1 ._. 
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Let_VV0 p_ZZ1 be_VBI a_AT1 quantum_NN1 state_NN1 on_II a_AT1 Gaussian_JJ quantum_NN1 system_NN1 with_IW finite_JJ average_JJ energy_NN1 ,_, and_CC let_VVN &;_NULL be_VBI the_AT thermal_JJ Gaussian_JJ quantum_NN1 state_NN1 wih_NN1 the_AT same_DA average_JJ energy_NN1 ._. 
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Let_VV0 6abm_FO be_VBI a_AT1 quantum_NN1 state_NN1 on_II ABM_NP1 such_CS21 that_CS22 @F_FO ,_, and_CC let_VV0 us_PPIO2 suppose_VVI that_CST A_ZZ1 and_CC B_ZZ1 are_VBR conditionally_RR independent_JJ given_JJ M_NN1 :_: @F_FO ._. 
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Then_RT ,_, for_IF any_DD 0_MC <_FO k_ZZ1 <_FO 1_MC1 the_AT quantum_NN1 linear_JJ conditional_JJ Entropy_NN1 Power_NN1 Inequality_NN1 holds_VVZ :_: @F_FO ._. 
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<s>
The_AT quantum_NN1 conditional_NN1 Entropy_NN1 Power_NN1 Inequality_NN1 follows_VVZ maximizing_VVG over_II k_ZZ1 the_AT righthand_JJ side_NN1 of_IO (_( 106_MC )_) :_: @F_FO ._. 
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Indeed_RR ,_, we_PPIS2 set_VV0 @F_FO where_RRQ @S_FO ._. 
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<s>
Then_RT (_( 4.1_MC )_) and_CC (_( 4.3_MC )_) imply_VV0 that_CST @L_FO ,_, where_RRQ ,_, as_RR21 usual_RR22 ,_, we_PPIS2 have_VH0 denoted_VVN Ba_NP1 =_FO Ba_NP1 (_( 0_MC )_) for_IF every_AT1 a_AT1 >_FO 0_MC ._. 
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New_JJ :_: The_AT algorithm_NN1 described_VVN in_II this_DD1 paper_NN1 ._. 
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Until_II now_RT ,_, we_PPIS2 discussed_VVD the_AT special_JJ polyhedral_JJ set_NN1 @S_FO ._. 
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<s>
Prediger_VV0 et_RA21 al_RA22 ._. 
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<s>
(_( 2015_MC ,_, p._NN1 880_MC )_) distinguish_VV0 between_II two_MC archetypes_NN2 of_IO design_NN1 research_NN1 ,_, with_IW one_PN1 focusing_VVG on_II curriculum_NN1 innovations_NN2 aiming_VVG at_II direct_JJ practical_JJ use_NN1 ,_, and_CC the_AT other_NN1 focusing_VVG on_II teaching_NN1 and_CC learning_VVG processes_NN2 and_CC developing_JJ theories_NN2 ._. 
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<s>
We_PPIS2 represent_VV0 a_AT1 point_NN1 in_II PE_NN1 as_CSA (_( x_ZZ1 ,_, &lsqb;_( v_ZZ1 &rsqb;_) )_) where_RRQ &lsqb;_( v_ZZ1 &rsqb;_) is_VBZ an_AT1 equivalence_NN1 class_NN1 of_IO non-zero_JJ vectors_NN2 in_II the_AT fiber_NN1 E(x)_NN1 ._. 
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The_AT stream_NN1 formulation_NN1 Because_II21 of_II22 the_AT incompressibility_NN1 of_IO the_AT flow_NN1 @S_FO ,_, we_PPIS2 write_VV0 th_NNU velocity_NN1 as_II the_AT gradient_NN1 perpendicular_NN1 of_IO a_AT1 stream_NN1 function_NN1 @S_FO ,_, i.e._REX @F_FO ,_, with_IW @S_FO ._. 
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Then_RT ,_, computing_VVG the_AT curl_NN1 of_IO the/e.olutionequation_FU of_IO the_AT velocity_NN1 ,_, we_PPIS2 get_VV0 the_AT following_JJ Poisson_NP1 equation_NN1 :_: @F_FO ._. 
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Taking_VVG in_II account_NN1 (_( 6.8_MC )_) and_CC the_AT no-slip_JJ condition_NN1 we_PPIS2 obtain_VV0 the_AT boundary_NN1 condition_NN1 @S_FO ._. 
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Thus_RR ,_, we_PPIS2 need_VV0 to_TO impose_VVI @S_FO where_RRQ @S_FO could_VM be_VBI ,_, in_II principle_NN1 ,_, different_JJ from_II b_ZZ1 ._. 
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The_AT term_NN1 pK_NNU is_VBZ related_VVN to_II the_AT oscillation_NN1 of_IO f_ZZ1 ,_, the_AT datum_NN1 of_IO the_AT problem_NN1 (_( 1_MC1 )_) ,_, and_CC is_VBZ typical_JJ also_RR in_II the_AT finite_JJ element_NN1 framework_NN1 ._. 
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The_AT result_NN1 follows_VVZ immediately_RR from_II the_AT fact_NN1 that_CST @S_FO is_VBZ a_AT1 norm_NN1 on_II H2(Q)_FO x_ZZ1 H_ZZ1 )_) (_( Q_ZZ1 )_) ,_, equivalent_JJ with_IW the_AT norm_NN1 @S_FO (_( see_VV0 Chapter_NN1 6_MC of_IO Ref._NN1 52_MC )_) ._. 
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The_AT algorithm_NN1 makes_VVZ use_NN1 of_IO the_AT fact_NN1 that_CST a_AT1 class_NN1 of_IO point_NN1 processes_NN2 is_VBZ represented_VVN as_II a_AT1 mixture_NN1 of_IO Poisson_NP1 processes_NN2 with_IW different_JJ event_NN1 rates_NN2 ._. 
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U(n)_NP1 is_VBZ the_AT random_JJ word_NN1 taking_VVG values_NN2 from_II n_ZZ1 according_II21 to_II22 p_ZZ1 ._. 
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Semiparametric_JJ efficiency_NN1 of_IO EL_NP1 with_IW estimating_VVG equations_NN2 is_VBZ shown_VVN in_II Qin_NP1 and_CC Lawless_JJ (_( 1994_MC )_) ._. 
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<s>
Roughly_RR ,_, this_DD1 will_VM tell_VVI us_PPIO2 that_CST at_II such_DA points_VVZ the_AT Rk_NP1 splitting_NN1 is_VBZ preserved_VVN at_II all_DB scales_NN2 ,_, which_DDQ is_VBZ the_AT result_NN1 required_VVN to_TO prove_VVI the_AT main_JJ theorem_NN1 ._. 
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<s>
Explicit_JJ formulas_NN2 for_IF En_FW and_CC en_FW are_VBR given_VVN by_II @F_FO ,_, @F_FO and_CC @F_FO ,_, where_CS r_ZZ1 is_VBZ the_AT stability_NN1 function_NN1 of_IO the_AT method_NN1 and_CC @S_FO ._. 
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While_CS Sena_NP1 was_VBDZ potentially_RR supporting_JJ student_NN1 understanding_NN1 by_II connecting_VVG to_II children_NN2 '_NULL s_ZZ1 experiences_NN2 in_II a_AT1 familiar_JJ community_NN1 location_NN1 ,_, she_PPHS1 was_VBDZ not_XX eliciting_VVG or_CC connecting_VVG to_II ways_NN2 that_CST children_NN2 and_CC families_NN2 might_VM engage_VVI in_II mathematics_NN1 outside_II21 of_II22 school_NN1 ._. 
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Let_VV0 P_ZZ1 ,_, V_ZZ1 and_CC R_ZZ1 e_ZZ1 P3_FO as_CSA above_RL be_VBI given_VVN and_CC form_VV0 the_AT functor_NN1 r_ZZ1 ._. 
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<s>
Our_APPGE results_NN2 show_VV0 the_AT robustness_NN1 of_IO the_AT evaluation_NN1 complexity_NN1 bounds_VVZ with_II31 respect_II32 to_II33 such_DA perturbations_NN2 ._. 
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A_AT1 turn_NN1 for_IF the_AT curriculum_NN1 document_NN1 begins_VVZ when_RRQ the_AT reader_NN1 begins_VVZ to_TO read_VVI ._. 
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<s>
Since_CS they_PPHS2 are_VBR also_RR pointwise_JJ monotone_NN1 limits_NN2 of_IO the_AT sequences_NN2 F2n+1_FO and_CC F2n_FO ,_, respectively_RR (_( see_VV0 Corollary_NN1 4.3_MC and_CC (_( 4.7_MC )_) )_) ,_, by_II Dini_NP1 '_NULL s_ZZ1 theorem_NN1 ,_, the_AT convergence_NN1 is_VBZ in_II fact_NN1 uniform_NN1 ._. 
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<s>
If_CS x0=0_FO ,_, @F_FO ,_, which_DDQ follows_VVZ from_II the_AT inequality_NN1 of_IO &lsqb;_( 51_MC ,_, Lemma_NN1 18_MC &rsqb;_) ,_, and_CC |_NULL v0_FO |_NULL <_FO ._. 
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<s>
Convergence_NN1 for_IF Burgers_NN2 '_NULL equation_NN1 with_IW energy-dissipative_JJ flux_NN1 ,_, using_VVG several_DA2 standard_JJ RK_NP1 methods_NN2 (_( solid_JJ lines_NN2 )_) and_CC their_APPGE energy-conservative_JJ modifications_NN2 (_( dashed_JJ lines_NN2 )_) ._. 
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<s>
These_DD2 were_VBDR also_RR supported_VVN by_II web-based_JJ discussions_NN2 that_CST allowed_VVD for_IF grade-level_JJ collaboration_NN1 across_II schools_NN2 ._. 
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In_II this_DD1 section_NN1 we_PPIS2 turn_VV0 to_II the_AT proof_NN1 of_IO Theorem_NN1 1.1_MC ._. 
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<s>
If_CS we_PPIS2 are_VBR willing_JJ to_TO focus_VVI on_II high_JJ dimensional_JJ linear_JJ models_NN2 ,_, however_RR ,_, it_PPH1 is_VBZ possible_JJ to_TO tighten_VVI the_AT connection_NN1 between_II the_AT estimation_NN1 strategy_NN1 and_CC the_AT objective_NN1 of_IO estimating_VVG the_AT average_JJ treatment_NN1 effect_NN1 and_CC ,_, in_II doing_VDG so_RR ,_, to_TO extend_VVI the_AT number_NN1 of_IO settings_NN2 where_RRQ n-consistent_JJ inference_NN1 is_VBZ possible_JJ ._. 
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<s>
When_CS asked_VVN in_II the_AT interview_NN1 if_CS she_PPHS1 wanted_VVD to_TO show_VVI her_APPGE finger_NN1 multiplication_NN1 method_NN1 to_II the_AT teacher_NN1 ,_, May_NPM1 said_VVD ,_, "_" No_UH ,_, I_PPIS1 don_VV0 '_NULL t_ZZ1 want_VV0 to_TO &lsqb;_( show_VV0 it_PPH1 to_II the_AT teacher_NN1 &rsqb;_) ._. 
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For_IF all_DB t2R_FO and_CC u2H2(M)_FO we_PPIS2 have_VH0 @F_FO ,_, where_CS @S_FO ,_, and_CC the_AT constant_JJ C_ZZ1 is_VBZ independent_JJ of_IO t_ZZ1 and_CC h_ZZ1 ._. 
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<s>
Tables_NN2 5_MC and_CC 6_MC highlight_VV0 the_AT smallest_JJT value_NN1 in_II each_DD1 row_NN1 and_CC also_RR show_VV0 results_NN2 for_IF the_AT backtracking_JJ version_NN1 ofFISTA_NN1 ,_, the_AT most_RGT efficient_JJ variant_NN1 of_IO ISTA/FISTA_NN1 that_CST we_PPIS2 could_VM identify_VVI for_IF this_DD1 class_NN1 of_IO problems_NN2 ._. 
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<s>
How_RGQ many_DA2 two-card_JJ hands_NN2 can_VM we_PPIS2 make_VVI that_CST have_VH0 one_MC1 spade_NN1 and_CC one_MC1 heart_NN1 ?_? )_) ._. 
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<s>
As_II a_AT1 consequence_NN1 ,_, the_AT horizontal_JJ lines_NN2 @S_FO have_VH0 empty_JJ intersection_NN1 with_IW @S_FO ._. 
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<s>
Consider_VV0 the_AT measures_NN2 @S_FO ,_, where_CS @S_FO is_VBZ the_AT Lebesgue_NN1 measure_NN1 restricted_VVN to_II the_AT interval_NN1 @S_FO is_VBZ the_AT unit_NN1 mass_NN1 supported_VVN at_II 0_MC ._. 
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<s>
As_II an_AT1 illustrative_JJ numerical_JJ experiment_NN1 ,_, we_PPIS2 consider_VV0 the_AT charged-particle_JJ motion_NN1 in_II the_AT magnetic_JJ field_NN1 @F_FO and_CC the_AT electric_JJ field_NN1 @S_FO with_IW the_AT potential_NN1 @F_FO ._. 
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<s>
The_AT initial_JJ values_NN2 are_VBR chosen_VVN as_CSA @S_FO and_CC @F_FO ._. 
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<s>
Zk_NP1 p(Zk)_NNU admits_VVZ a_AT1 conformal_JJ extension_NN1 through_II dZk_NNU (_( because_CS p(Zk)_NNU ci_MC )_) ._. 
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Then_RT there_EX exists_VVZ a_AT1 unique_JJ solution_NN1 @S_FO to_II the_AT problem_NN1 @F_FO ._. 
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<s>
By_II a_AT1 curve_NN1 ,_, we_PPIS2 shall_VM mean_VVI the_AT image_NN1 of_IO a_AT1 piecewise_JJ (_( regular_JJ )_) real-analytic_JJ function3_FO y_ZZ1 from_II a_AT1 closed_JJ interval_NN1 &lsqb;_( a_AT1 ,_, b_ZZ1 &rsqb;_) (_( where_CS a_AT1 <_FO b_ZZ1 )_) to_II either_RR the_AT plane_NN1 or_CC the_AT sphere_NN1 ._. 
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<s>
The_AT proof_NN1 that_CST A_ZZ1 is_VBZ isotopic_JJ to_II A_ZZ1 for_IF the_AT operations_NN2 of_IO Figures_NN2 5.6_MC c_ZZ1 i_ZZ1 )_) ,_, d_ZZ1 )_) ,_, e_ZZ1 i_ZZ1 )_) are_VBR all_RR31 the_RR32 same_RR33 ._. 
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<s>
However_RR ,_, the_AT resources_NN2 and_CC possibilities_NN2 of_IO the_AT schools_NN2 strongly_RR condition_VV0 the_AT practicum_NN1 that_CST can_VM be_VBI carried_VVN out_RP ._. 
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<s>
Tillema_NP1 (_( 2018_MC )_) identified_VVD that_CST MC1_FO students_NN2 create_VV0 pairs_NN2 as_II part_NN1 of_IO the_AT activity_NN1 they_PPHS2 produce_VV0 in_II a_AT1 situation_NN1 ._. 
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<s>
It_PPH1 follows_VVZ that_CST @S_FO is_VBZ simple_JJ ,_, @S_FO for_IF large_JJ T._NP1 This_DD1 gives_VVZ the_AT first_MD examples_NN2 of_IO non-analytic_JJ Anosov_NP1 flows_NN2 and_CC geodesic_JJ flows_NN2 in_II variable_JJ negative_JJ curvature_NN1 where_CS the_AT Fried_JJ conjecture_NN1 holds_VVZ true_JJ ._. 
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<s>
We_PPIS2 consider_VV0 the_AT following_JJ linearized_JJ K-approximate_JJ problem_NN1 @F_FO ._. 
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<s>
Consider_VV0 two_MC distinct_JJ ,_, immiscible_JJ ,_, incompressible_JJ MHD_NP1 fluids_NN2 evolving_VVG in_II a_AT1 moving_JJ domain_NN1 @S_FO for_IF time_NNT1 t_ZZ1 >_FO 0_MC ,_, where_CS the_AT upper_JJ fluid_NN1 fills_VVZ the_AT upper_JJ domain_NN1 @F_FO ,_, and_CC the_AT lower_JJR fluid_NN1 fills_VVZ the_AT lower_JJR domain_NN1 @F_FO ._. 
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<s>
We_PPIS2 assume_VV0 that_CST h_ZZ1 and_CC l_ZZ1 are_VBR given_VVN constants_NN2 satisfying_VVG h_ZZ1 >_FO l_ZZ1 ,_, but_CCB the_AT internal_JJ surface_NN1 function_NN1 @S_FO is_VBZ free_JJ and_CC unknown_JJ ._. 
</s>
<s>
Our_APPGE key_JJ observation_NN1 is_VBZ that_CST the_AT -curl_NN1 of_IO U_ZZ1 satisfies_VVZ a_AT1 transport_NN1 equation_NN1 which_DDQ at_II the_AT top_JJ order_NN1 decouples_VVZ from_II the_AT rest_NN1 of_IO the_AT dynamics_NN ,_, allowing_VVG us_PPIO2 to_TO obtain_VVI "_" good_JJ "_" estimates_NN2 for_IF the_AT (_( Lagrangian_JJ pull-back_NN1 of_IO )_) -curl_NN1 of_IO U._NP1 The_AT extended_JJ Tchebycheff_NN1 spaces_NN2 and_CC their_APPGE dimensions_NN2 are_VBR allowed_VVN to_TO change_VVI from_II interval_NN1 to_II interval_NN1 ._. 
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<s>
According_II21 to_II22 the_AT heuristic_JJ argument_NN1 in_II the_AT introduction_NN1 ,_, we_PPIS2 believe_VV0 that_CST our_APPGE estimates_NN2 are_VBR optimal_JJ as_II21 regards_II22 the_AT size_NN1 of_IO chaos_NN1 and_CC the_AT rate_NN1 of_IO convergence_NN1 &lsqb;_( see_VV0 also_RR the_AT classical_JJ estimate_NN1 on_II independent_JJ random_JJ variables_NN2 for_IF which_DDQ the_AT same_DA result_NN1 is_VBZ easily_RR obtained_VVN (_( for_REX21 example_REX22 &lsqb;_( 24,39_MC &rsqb;_) )_) &rsqb;_) ._. 
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<s>
Let_VV0 the_AT initial_JJ data_NN @S_FO ,_, and_CC assume_VV0 @F_FO ,_, by_II the_AT H1_FO theory_NN1 of_IO the_AT primitive_JJ equations_NN2 ,_, see_VV0 &lsqb;_( 12_MC &rsqb;_) ,_, there_EX is_VBZ a_AT1 unique_JJ global_JJ strong_JJ solution_NN1 (_( v_ZZ1 ,_, w_ZZ1 )_) to_II (_( PEs_NN2 )_) ,_, subject_II21 to_II22 the_AT boundary_NN1 and_CC initial_JJ conditions_NN2 (_( 1.2_MC )_) (_( 1.4_MC )_) ,_, such_CS21 that_CS22 @F_FO ._. 
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<s>
Then_RT ,_, using_VVG the_AT boundary_NN1 condition_NN1 (_( 1.2_MC )_) and_CC the_AT symmetry_NN1 condition_NN1 (_( 1.4_MC )_) ,_, the_AT vertical_JJ component_NN1 w_ZZ1 of_IO the_AT velocity_NN1 can_VM be_VBI uniquely_RR determined_VVN as_CSA @F_FO ._. 
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<s>
This_DD1 theorem_NN1 is_VBZ new_JJ even_RR in_II the_AT case_NN1 G=Z_FO ._. 
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<s>
The_AT result_NN1 now_RT follows_VVZ by_II Theorem_NN1 1.3_MC ._. 
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<s>
This_DD1 first_MD construction_NN1 allows_VVZ us_PPIO2 to_TO obtain_VVI only_JJ examples_NN2 where_RRQ the_AT velocity_NN1 field_NN1 is_VBZ neither_RR smooth_JJ nor_CC uniformly_RR bounded_VVN in_II W1_FO ,_, TO_II ._. 
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<s>
However_RR ,_, for_IF the_AT score_NN1 test_NN1 and_CC split_JJ Lasso_NN1 ,_, the_AT time_NNT1 becomes_VVZ increasing_VVG when_RRQ k_ZZ1 is_VBZ large_JJ ;_; this_DD1 is_VBZ because_CS the_AT computation_NN1 time_NNT1 to_TO aggregate_VVI results_NN2 from_II different_JJ splits_NN2 is_VBZ no_RR21 longer_RR22 negligible_JJ for_IF very_RG large_JJ k_ZZ1 '_NULL s_ZZ1 ._. 
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<s>
Finally_RR ,_, the_AT size_NN1 of_IO @S_FO is_VBZ certainly_RR bounded_VVN above_RL and_CC below_RL in_II31 terms_II32 of_II33 @S_FO ,_, by_II the_AT classification_NN1 of_IO semisimple_NN1 groups_NN2 ._. 
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<s>
We_PPIS2 may_VM assume_VVI furthermore_RR that_CST every_AT1 element_NN1 w_ZZ1 of_IO Q_ZZ1 almost_RR minimizes_VVZ the_AT length_NN1 of_IO its_APPGE T-orbit_NN1 ,_, for_REX21 example_REX22 that_CST it_PPH1 satisfies_VVZ @F_FO ._. 
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<s>
Indeed_RR ,_, if_CS @S_FO is_VBZ an_AT1 enumeration_NN1 of_IO T_ZZ1 (_( say_VV0 with_IW 70=e_FO )_) ,_, and_CC if_CS @S_FO for_IF the_AT first_MD n_ZZ1 satisfying_JJ @S_FO ,_, then_RT we_PPIS2 may_VM replace_VVI Q_ZZ1 by_II f_ZZ1 (_( Q_ZZ1 )_) ,_, which_DDQ remains_VVZ a_AT1 Borel_NN1 fundamental_JJ domain_NN1 for_IF T_ZZ1 ,_, which_DDQ still_RR satisfies_VVZ (_( 5.1_MC )_) and_CC which_DDQ moreover_RR satisfies_VVZ (_( 5.2_MC )_) ._. 
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<s>
In_II a_AT1 collaboration_NN1 network_NN1 a_AT1 c-author_JJ publication_NN1 induces_VVZ a_AT1 c-clique_NN1 in_II the_AT graph_NN1 ,_, because_CS every_AT1 pair_NN of_IO the_AT c_ZZ1 coauthors_NN2 will_VM share_VVI an_AT1 edge_NN1 ,_, c_ZZ1 (_( c_ZZ1 1_MC1 )_) /2_MF edges_NN2 in_II total_NN1 from_II a_AT1 c-author_JJ publication_NN1 ._. 
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<s>
The_AT use_NN1 of_IO negative_JJ norms_NN2 to_TO measure_VVI mixing_NN1 was_VBDZ proposed_VVN in_II &lsqb;_( 39_MC &rsqb;_) ,_, where_CS the_AT equivalence_NN1 between_II the_AT decay_NN1 of_IO the_AT H1/2_FU norm_NN1 and_CC mixing_VVG in_II the_AT ergodic_JJ sense_NN1 was_VBDZ established_VVN ._. 
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<s>
In_RR21 particular_RR22 ,_, a_AT1 lot_NN1 of_IO attention_NN1 was_VBDZ paid_VVN to_II convex_JJ risk_NN1 and_CC uncertainty_NN1 measures_NN2 ;_; see_VV0 ,_, e.g._REX ,_, &lsqb;_( 6_MC ,_, 23_MC &rsqb;_) ._. 
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<s>
By_II combining_VVG Theorem_NN1 1_MC1 with_IW Theorem_NN1 2_MC ,_, we_PPIS2 can_VM relate_VVI the_AT scaling_NN1 limit_NN1 of_IO the_AT total_JJ height_NN1 of_IO a_AT1 Boltzmann_NN1 triangulation_NN1 with_IW boundary_NN1 with_IW the_AT extinction_NN1 time_NNT1 @S_FO of_IO the_AT growth-fragmentation_JJ process_NN1 X_ZZ1 (_( which_DDQ is_VBZ known_VVN to_TO be_VBI almost_RR surely_RR finite_JJ &lsqb;_( 10_MC &rsqb;_) ,_, Corollary_NN1 3_MC )_) ._. 
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<s>
Overall_RR our_APPGE motivation_NN1 is_VBZ not_XX to_TO propose_VVI the_AT best_JJT method_NN1 for_IF (_( 78_MC )_) but_CCB to_TO demonstrate_VVI the_AT performance_NN1 of_IO aGRAAL_JJ on_II some_DD real-world_JJ problems_NN2 ._. 
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<s>
Proof_NN1 We_PPIS2 may_VM assume_VVI without_IW loss_NN1 of_IO generality_NN1 that_CST X_ZZ1 →_NULL T_ZZ1 has_VHZ a_AT1 section_NN1 ._. 
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<s>
A_AT1 score_NN1 of_IO 4_MC indicates_VVZ that_CST an_AT1 explicit_JJ rule_NN1 (_( not_XX recursive_JJ only_JJ )_) was_VBDZ given_VVN for_IF any_DD item_NN1 in_II words_NN2 (_( 4.1_MC )_) ,_, with_IW a_AT1 symbolic_JJ expression_NN1 (_( 4.2_MC )_) ,_, or_CC with_IW a_AT1 full_JJ equation_NN1 with_IW both_DB2 variables_NN2 symbolised_VVN (_( 4.3_MC )_) ._. 
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This_DD1 eventually_RR leads_VVZ to_II less_RGR competitive_JJ timing_NN1 performance_NN1 than_CSN PICASSO_NP1 ._. 
</s>
<s>
It_PPH1 may_VM add_VVI too_RG many_DA2 inactive_JJ coordinates_NN2 into_II the_AT active_JJ set_NN1 ,_, and_CC compromise_VV0 the_AT solution_NN1 sparsity_NN1 ._. 
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<s>
In_II the_AT following_JJ ,_, we_PPIS2 demonstrate_VV0 the_AT power_NN1 of_IO our_APPGE approach_NN1 ,_, illustrating_VVG how_RGQ new_JJ and_CC simple_JJ globally_RR convergent_JJ schemes_NN2 can_VM be_VBI derived_VVN for_IF the_AT broad_JJ class_NN1 of_IO problems_NN2 (_( QIP_NP1 )_) ._. 
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<s>
The_AT experimental_JJ study_NN1 by_II Harackiewicz_NP1 ,_, Rozek_NP1 ,_, Hulleman_NP1 ,_, and_CC Hyde_NP1 (_( 2012_MC )_) showed_VVD that_DD1 helping_NN1 parents_NN2 to_TO talk_VVI with_IW their_APPGE children_NN2 about_II the_AT value_NN1 and_CC importance_NN1 of_IO science_NN1 and_CC mathematics_NN1 in_II high_JJ school_NN1 increased_JJ enrolment_NN1 in_II advanced_JJ mathematics_NN1 among_II achieving_VVG male_JJ and_CC low_RR achieving_VVG female_JJ students_NN2 in_II the_AT treatment_NN1 group_NN1 ._. 
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<s>
The_AT fact_NN1 that_CST a_AT1 triangle_NN1 comprised_VVN of_IO line_NN1 segments_NN2 whose_DDQGE lengths_NN2 had_VHD long_RR decimal_JJ tails_NN2 could_VM still_RR produce_VVI an_AT1 area_NN1 with_IW a_AT1 round_JJ number_NN1 like_II 2.5_MC was_VBDZ unexpected_JJ and_CC initially_RR hard_JJ for_IF students_NN2 to_TO make_VVI sense_NN1 of_IO ._. 
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<s>
We_PPIS2 were_VBDR also_RR interested_JJ in_II students_NN2 '_NULL ability_NN1 to_TO interpret_VVI and_CC construct_VVI different_JJ representations_NN2 of_IO the_AT linear_JJ functional_JJ relationship_NN1 :_: ordered_JJ pairs_NN2 ,_, descriptive_JJ rules_NN2 ,_, and_CC symbolic_JJ equations_NN2 ._. 
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<s>
D_ZZ1 &C;_NULL restructures_VVZ the_AT SVD_NP1 algorithm_NN1 somewhat_RR ,_, as_CSA is_VBZ shown_VVN in_II Algorithm_NN1 4_MC ,_, compared_VVN with_IW the_AT QR_NP1 iteration_NN1 version_NN1 in_II Algorithm_NN1 2_MC ._. 
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<s>
Here_RL ,_, we_PPIS2 notice_VV0 that_CST ,_, due_II21 to_II22 the_AT facts_NN2 that_CST we_PPIS2 just_RR use_VV0 propagation_NN1 for_IF a_AT1 uniform_JJ finite_JJ time_NNT1 and_CC that_CST the_AT Hamiltonian_JJ flow_NN1 τ_NULL t_ZZ1 is_VBZ smooth_JJ in_II τ_NULL ,_, the_AT proof_NN1 of_IO &lsqb;_( 16_MC ,_, Prop_VV0 ._. 
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<s>
2.5_MC &rsqb;_) can_VM be_VBI repeated_VVN uniformly_RR for_IF τ_NULL close_JJ enough_RR to_II 0_MC ._. 
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<s>
This_DD1 property_NN1 is_VBZ fundamental_JJ in_II statistics_NN ,_, approximation_NN1 theory_NN1 ,_, and_CC data_NN interpolation_NN1 &lsqb;_( 40_MC &rsqb;_) ._. 
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<s>
In_II Figs._NN2 10_MC and_CC 11_MC we_PPIS2 give_VV0 the_AT comparison_NN1 of_IO the_AT local_JJ distribution_NN1 of_IO the_AT total_JJ error_NN1 @S_FO and_CC the_AT sum_NN1 @S_FO of_IO the_AT local_JJ indicators_NN2 ._. 
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<s>
The_AT focus_NN1 students_NN2 '_NULL utterances_NN2 were_VBDR further_RRR evaluated_VVN as_CSA having_VHG high_JJ epistemic_JJ quality_NN1 if_CS two_MC of_IO the_AT following_JJ categories_NN2 were_VBDR fulfilled_VVN (_( Erath_NP1 ,_, 2017b_FO ,_, pp_NN1 ._. 
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<s>
We_PPIS2 explain_VV0 in_II Appendix_NN1 A_ZZ1 how_RRQ to_TO modify_VVI the_AT arguments_NN2 for_IF the_AT nonperiodic_JJ case_NN1 in_BCL21 order_BCL22 to_TO prove_VVI Theorem_NN1 2.13_MC ._. 
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<s>
The_AT nonparametric_JJ rate_NN1 for_IF estimation_NN1 of_IO convex_JJ sequences_NN2 is_VBZ of_IO order_NN1 n-4/5_FU for_IF equispaced_JJ design_NN1 points_NN2 ._. 
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<s>
The_AT corresponding_JJ mass_JJ matrix_NN1 is_VBZ therefore_RR singular_JJ ,_, which_DDQ is_VBZ of_RR21 course_RR22 an_AT1 issue_NN1 when_CS considering_VVG explicit_JJ discretisations_NN2 of_IO time-dependent_JJ (_( even_RR linear_JJ )_) problems_NN2 ;_; solving_VVG this_DD1 issue_NN1 requires_VVZ the_AT usage_NN1 of_IO enriched_JJ @S_FO elements_NN2 &lsqb;_( 10_MC &rsqb;_) ._. 
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<s>
Throughout_II the_AT section_NN1 (_( X_ZZ1 ,_, d_ZZ1 ,_, m_ZZ1 )_) is_VBZ an_AT1 RCD*_FO (_( K_ZZ1 ,_, N_ZZ1 )_) -space_NN1 for_IF some_DD @S_FO and_CC @S_FO and_CC @S_FO are_VBR points_NN2 in_II X_ZZ1 satisfying_JJ @S_FO (_( of_RR21 course_RR22 ,_, by_II applying_VVG the_AT estimates_NN2 recursively_RR ,_, one_PN1 can_VM also_RR consider_VVI points_NN2 farther_RRR apart_RL )_) ._. 
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<s>
It_PPH1 is_VBZ well_RR known_VVN that_CST defining_VVG @S_FO as_II a_AT1 polynomial_NN1 interpolant_NN1 on_II a_AT1 fixed_JJ stencil_NN1 yields_VVZ oscillatory_JJ results_NN2 in_II any_DD high_JJ order_NN1 scheme_NN1 ._. 
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<s>
The_AT free_JJ energies_NN2 of_IO Gn_NNU are_VBR then_RT equal_JJ to_II the_AT free_JJ energies_NN2 of_IO the_AT 4-cycle_NN1 when_CS n_ZZ1 is_VBZ even_RR and_CC those_DD2 of_IO the_AT 6-cycle_NN1 when_CS n_ZZ1 is_VBZ odd_JJ ._. 
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<s>
In_II the_AT following_JJ two_MC lemmas_NN2 ,_, we_PPIS2 will_VM estimate_VVI the_AT left_JJ side_NN1 of_IO (_( 43_MC )_) and_CC the_AT first_MD term_NN1 on_II the_AT right_JJ side_NN1 respectively_RR ._. 
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<s>
Using_VVG the_AT Poincare_NN1 inequality_NN1 ,_, it_PPH1 is_VBZ not_XX difficult_JJ to_TO show_VVI that_CST E_ZZ1 is_VBZ V-elliptic_JJ if_CS the_AT diameter_NN1 of_IO D_ZZ1 is_VBZ small_JJ enough_RR (_( see_VV0 &lsqb;_( 16_MC &rsqb;_) ,_, pages_NN2 385-387_MCMC )_) ._. 
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<s>
Thus_RR (_( 15_MC )_) holds_VVZ for_IF @S_FO and_CC all_DB k_ZZ1 sufficiently_RR large_JJ ,_, @S_FO and_CC (_( 23_MC )_) finally_RR allows_VVZ us_PPIO2 to_TO deduce_VVI that_CST F(x*)_FO =_FO 0_MC ._. 
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<s>
Thus_RR these_DD2 complexity_NN1 bounds_NN2 remain_VV0 valid_JJ with_IW growingp_NN1 and_CC approach_NN1 O(e-1)_MC1 ._. 
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<s>
Naturality_NN1 with_II31 respect_II32 to_II33 the_AT morphism_NN1 pN_NNU pdN_NNU is_VBZ built_VVN into_II the_AT cocycle_NN1 computation_NN1 above_RL ._. 
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<s>
Here_RL and_CC below_RG ok_RR stands_VVZ for_IF the_AT column_NN1 vector_NN1 @S_FO ._. 
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<s>
Since_CS the_AT coefficients_NN2 are_VBR assumed_VVN to_TO vanish_VVI outside_II G+_FO ,_, the_AT flow_NN1 X_ZZ1 ,_, and_CC in_II fact_NN1 any_DD flow_NN1 appearing_VVG below_II that_DD1 is_VBZ built_VVN from_II the_AT coefficients_NN2 a_AT1 and_CC o_ZZ1 ,_, are_VBR trivial_JJ outside_II G+_FO ._. 
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<s>
Let_VV0 V_ZZ1 be_VBI the_AT cover_NN1 witnessing_NN1 shadowing_VVG ._. 
</s>
<s>
Our_APPGE approach_NN1 to_II systematic_JJ derivations_NN2 of_IO model_NN1 equations_NN2 is_VBZ from_II the_AT point_NN1 of_IO view_NN1 of_IO Hamiltonian_JJ perturbation_NN1 theory_NN1 ._. 
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<s>
Comparing_VVG these_DD2 values_NN2 for_IF some_DD parts_NN2 of_IO the_AT tank_NN1 helped_VVD him_PPHO1 to_TO draw_VVI the_AT true_JJ graph_NN1 and_CC to_TO be_VBI attentive_JJ to_II the_AT finer_JJR details_NN2 of_IO the_AT other_JJ graphs_NN2 ._. 
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<s>
We_PPIS2 report_VV0 the_AT thresholds_NN2 ofvarious_JJ testing_NN1 procedures_NN2 in_II Table_NN1 1_MC1 ._. 
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<s>
We_PPIS2 can_VM see_VVI that_CST more_DAR information_NN1 can_VM be_VBI harvested_VVN from_II the_AT data_NN by_II using_VVG auxiliary_JJ information_NN1 ._. 
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<s>
Finally_RR ,_, the_AT results_NN2 proven_VVN in_II the_AT paper_NN1 reveal_VV0 relations_NN2 among_II these_DD2 concepts_NN2 in_II a_AT1 general_JJ system_NN1 theoretic_JJ framework_NN1 ,_, which_DDQ we_PPIS2 believe_VV0 might_VM be_VBI suitable_JJ for_IF an_AT1 extension_NN1 of_IO our_APPGE results_NN2 to_II LQ_NP1 optimal_JJ control_NN1 problems_NN2 in_II an_AT1 infinite_JJ dimensional_JJ setting_NN1 ._. 
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<s>
The_AT dimension_NN1 @S_FO is_VBZ the_AT dimension_NN1 of_IO G_ZZ1 as_II a_AT1 p-adic_JJ manifold_NN1 ._. 
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<s>
Applying_VVG to_II the_AT case_NN1 of_IO @S_FO we_PPIS2 obtain_VV0 the_AT following_JJ ._. 
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<s>
Therefore_RR the_AT assertion_NN1 is_VBZ a_AT1 direct_JJ consequence_NN1 ofProposition_NN1 4.1_MC ._. 
</s>
<s>
To_II the_AT best_JJT of_IO our_APPGE knowledge_NN1 ,_, the_AT first_MD explicit_JJ mention_NN1 of_IO this_DD1 invariance_NN1 was_VBDZ made_VVN by_II Serre_NP1 &lsqb;_( 39_MC &rsqb;_) thus_RR establishing_VVG an_AT1 analogy_NN1 to_II the_AT mass-critical_JJ nonlinear_JJ Schr?dinger_NN1 equation_NN1 ,_, known_VVN to_TO possess_VVI a_AT1 pseudo-conformal_JJ invariance_NN1 ._. 
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<s>
Then_RT ,_, by_II Clarke_NP1 '_NULL s_ZZ1 inverse_JJ function_NN1 theorem_NN1 ,_, there_EX exists_VVZ a_AT1 Lipschitz_NP1 continuous_JJ solution_NN1 function_NN1 @S_FO such_CS21 that_CS22 @S_FO and_CC the_AT Lipschitz_NP1 constant_NN1 is_VBZ bounded_VVN by_II @S_FO for_IF all_DB @F_FO ._. 
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<s>
Then_RT Assumption_NN1 2.4_MC holds_VVZ ._. 
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<s>
We_PPIS2 assume_VV0 that_CST H_ZZ1 has_VHZ a_AT1 unique_JJ gapped_JJ ground_NN1 state_NN1 and_CC hence_RR local_JJ charge_NN1 fluctuations_NN2 ._. 
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<s>
After_CS this_DD1 ,_, one_MC1 of_IO the_AT researchers_NN2 again_RT watched_VVD all_DB the_AT video_NN1 documentations_NN2 and_CC categorized_VVD the_AT episodes_NN2 of_IO critical_JJ incidents_NN2 into_II the_AT four_MC categories_NN2 ._. 
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<s>
We_PPIS2 can_VM thus_RR Write_VV0 @F_FO where_RRQ @S_FO is_VBZ a_AT1 reduced_JJ centered_JJ randomariable_JJ ._. 
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<s>
Let_VV0 t_ZZ1 be_VBI a_AT1 Schroder_NP1 tree_NN1 ._. 
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<s>
The_AT distribution_NN1 of_IO the_AT understanding_NN1 levels_NN2 of_IO the_AT two_MC 4th-grade_JJ textbooks_NN2 is_VBZ roughly_RR similar_JJ ;_; there_EX is_VBZ wide_JJ discrepancy_NN1 in_II the_AT distributions_NN2 of_IO the_AT 8th-grade_JJ books_NN2 ._. 
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<s>
We_PPIS2 also_RR prove_VV0 a_AT1 corresponding_JJ universality_NN1 result_NN1 :_: the_AT above_JJ scaling_NN1 limit_NN1 holds_VVZ for_IF essentially_RR any_DD distribution_NN1 on_II face_NN1 degrees_NN2 (_( or_CC ,_, dually_RR ,_, vertex_VV0 degrees_NN2 )_) of_IO the_AT random_JJ map_NN1 ._. 
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<s>
This_DD1 understanding_NN1 of_IO Amanda_NP1 '_NULL s_ZZ1 noticing_VVG contrasts_NN2 with_IW images_NN2 in_II the_AT literature_NN1 of_IO a_AT1 lone_JJ teacher_NN1 creating_VVG meaning_NN1 from_II chaos_NN1 ,_, making_VVG sense_NN1 of_IO a_AT1 "_" blooming_JJ ,_, buzzing_JJ confusion_NN1 of_IO sensory_JJ data_NN "_" (_( Sherin_NP1 &;_NULL Star_NN1 ,_, 2011_MC ,_, p._NN1 69_MC )_) largely_RR on_II his_APPGE or_CC her_APPGE own_DA ,_, based_VVN on_II personal_JJ knowledge_NN1 ,_, skill_NN1 ,_, and_CC experience_NN1 ._. 
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<s>
Thus_RR Ak_NP1 is_VBZ obtained_VVN from_II Ak-1_MC1 by_II a_AT1 sequence_NN1 of_IO tube_NN1 sliding_VVG moves_NN2 and_CC a_AT1 tube_NN1 locus_NN1 free_JJ Whitney_NP1 move_NN1 ._. 
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<s>
Shulman_NP1 (_( 1987_MC )_) noted_VVD seven_MC forms_NN2 of_IO knowledge_NN1 as_II a_AT1 basis_NN1 for_IF teaching_NN1 :_: content_JJ knowledge_NN1 (_( MCK_NP1 )_) ,_, pedagogical_JJ knowledge_NN1 ,_, pedagogical_JJ content_JJ knowledge_NN1 (_( PCK_NP1 )_) ,_, curriculum_NN1 knowledge_NN1 ,_, knowledge_NN1 of_IO students_NN2 ,_, knowledge_NN1 of_IO educational_JJ contexts_NN2 ,_, and_CC knowledge_NN1 of_IO purposes_NN2 of_IO education_NN1 ._. 
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<s>
Those_DD2 results_NN2 are_VBR helpful_JJ for_IF understanding_VVG the_AT gap_NN1 between_II the_AT cardinality-_NN1 or_CC rank-constrained_JJ formulations_NN2 and_CC the_AT @S_FO approximation_NN1 formulations_NN2 ._. 
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<s>
Then_RT the_AT integrality_NN1 of_IO for_CS U_ZZ1 of_IO any_DD finite_JJ type_NN1 follows_VVZ by_II Theorem_NN1 5.3(2)_FO ._. 
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<s>
Interestingly_RR ,_, the_AT child_NN1 sees_VVZ the_AT berries_NN2 both_RR as_RG "_" threes_MC2 "_" (_( units_NN2 of_IO three_MC circles_NN2 )_) and_CC as_RG single_JJ units_NN2 (_( one-two-three_MC )_) and_CC acts_VVZ according_II21 to_II22 this_DD1 different_JJ way_NN1 of_IO seeing_VVG the_AT berries_NN2 simultaneously_RR ._. 
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<s>
First_MD of_IO all_DB ,_, to_TO avoid_VVI inessential_JJ difficulties_NN2 ,_, we_PPIS2 will_VM assume_VVI that_CST the_AT market_NN1 density_NN1 @S_FO has_VHZ a_AT1 certain_JJ number_NN1 of_IO moments_NN2 bounded_VVN ,_, more_RGR precisely_RR @F_FO ._. 
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<s>
Among_II observable_JJ quantities_NN2 ,_, by_II letting_VVG 1_MC1 in_II (_( 2.4_MC )_) one_MC1 shows_VVZ that_CST the_AT mass_NN1 is_VBZ consewed_VVN ._. 
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<s>
Finally_RR ,_, neither_DD1 set_NN1 is_VBZ a_AT1 subset_NN1 of_IO the_AT other_JJ ,_, and_CC ,_, although_CS for_IF N_ZZ1 =_FO 100_MC ,_, @S_FO has_VHZ smaller_JJR volume_NN1 than_CSN @S_FO ,_, the_AT reverse_NN1 holds_VVZ for_IF larger_JJR N._NNU Consequently_RR ,_, the_AT best_JJT choice_NN1 of_IO set_NN1 likely_RR depends_VVZ on_II N._NP1 Phages_NP2 ,_, viruses_NN2 infecting_VVG bacteria_NN2 ,_, are_VBR essential_JJ for_IF areas_NN2 as_RG diverse_JJ as_CSA the_AT ecosystem_NN1 of_IO the_AT oceans_NN2 and_CC gut_NN1 health_NN1 ,_, and_CC epidemics_NN2 caused_VVN by_II plant_NN1 and_CC animal_NN1 viruses_NN2 make_VV0 severe_JJ impacts_NN2 on_II agriculture_NN1 and_CC human_JJ health_NN1 ._. 
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<s>
Beckenbach_NP1 and_CC Rado_NP1 proved_VVD in_II &lsqb;_( 7_MC &rsqb;_) our_APPGE Theorem_NN1 1.1_MC for_IF smooth_JJ 2-dimensional_JJ Riemannian_JJ manifolds_NN2 ,_, finding_VVG a_AT1 connection_NN1 between_II log-subharmonicity_NN1 ,_, isoperimetric_JJ inequalities_NN2 and_CC curvature_NN1 bounds_NN2 ._. 
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<s>
We_PPIS2 are_VBR forth_RR to_TO do_VDI it_PPH1 in_BCL21 order_BCL22 to_TO control_VVI all_DB the_AT terms_NN2 ._. 
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<s>
For_IF the_AT measurement_NN1 of_IO strategy_NN1 use_NN1 ,_, up_RG21 to_RG22 three_MC points_NN2 per_II item_NN1 were_VBDR given_VVN according_II21 to_II22 the_AT total_JJ number_NN1 of_IO conceptually_RR different_JJ strategies_NN2 used_VVN with_IW a_AT1 correct_JJ solution_NN1 for_IF each_DD1 problem_NN1 ._. 
</s>
<s>
We_PPIS2 claim_VV0 that_CST @F_FO ._. 
</s>
<s>
Assume_VV0 that_CST this_DD1 was_VBDZ false_JJ ._. 
</s>
<s>
There_EX exists_VVZ some_DD @S_FO (_( depending_VVG only_RR on_II the_AT law_NN1 of_IO Z_ZZ1 )_) so_CS21 that_CS22 for_IF every_AT1 R_ZZ1 >_FO 0_MC the_AT following_JJ holds_NN2 ._. 
</s>
<s>
We_PPIS2 refer_VV0 to_II this_DD1 setting_NN1 as_CSA semi-discrete_JJ optimal_JJ transport_NN1 ._. 
</s>
<s>
The_AT only_JJ additional_JJ step_NN1 is_VBZ in_II bounding_VVG v._II Since_RR ,_, for_IF all_DB @S_FO ,_, it_PPH1 holds_VVZ that_CST &lsqb;_( 28_MC &rsqb;_) @F_FO ,_, it_PPH1 follows_VVZ that_CST @F_FO ._. 
</s>
<s>
Using_VVG this_DD1 inequality_NN1 gives_VVZ the_AT required_JJ bounds_NN2 ._. 
</s>
<s>
For_IF notational_JJ simplicity_NN1 ,_, we_PPIS2 denote_VV0 @S_FO for_IF @S_FO ._. 
</s>
<s>
The_AT argument_NN1 is_VBZ a_AT1 comparison_NN1 between_II m1_FO and_CC @S_FO ,_, where_CS @S_FO is_VBZ given_VVN by_II @F_FO ,_, and_CC the_AT constants_NN2 k_ZZ1 and_CC a_AT1 are_VBR defined_VVN by_II @F_FO ._. 
</s>
<s>
Her_APPGE laugh_NN1 at_II the_AT end_NN1 of_IO the_AT sentence_NN1 seems_VVZ to_TO suggest_VVI the_AT difficulty_NN1 entailed_VVN in_II her_PPHO1 taking_VVG this_DD1 stance_NN1 in_II31 front_II32 of_II33 her_APPGE colleague_NN1 ._. 
</s>
<s>
Proof_NN1 of_IO Proposition_NN1 5.3_MC Recall_VV0 that_CST @F_FO ._. 
</s>
<s>
From_II these_DD2 balls_NN2 ,_, we_PPIS2 add_VV0 to_II Fk+1_FO those_DD2 which_DDQ do_VD0 not_XX intersect_VVI @S_FO ._. 
</s>
<s>
The_AT class_NN1 of_IO generalized_JJ alpha_NN1 investing_VVG rules_NN2 is_VBZ quite_RG broad_JJ ._. 
</s>
<s>
The_AT right_JJ side_NN1 of_IO the_AT figure_NN1 ,_, the_AT last_MD phase_NN1 ,_, is_VBZ stepping_VVG down_RP to_II (_( teaching_NN1 )_) practice_NN1 ._. 
</s>
<s>
The_AT same_DA argument_NN1 applies_VVZ in_II the_AT second_MD case_NN1 when_CS g_ZZ1 has_VHZ one_MC1 more_DAR derivative_NN1 ,_, using_VVG Theorem_NN1 2.10_MC ._. 
</s>
<s>
Furthermore_RR ,_, a_AT1 fractional_JJ convergence_NN1 bound_NN1 holds_VVZ true_JJ for_IF less_RGR regular_JJ solutions_NN2 ,_, see_VV0 Theorem_NN1 13_MC ._. 
</s>
<s>
The_AT 95%_NNU credible_JJ interval_NN1 for_IF the_AT standardised_JJ effect_NN1 size_NN1 was_VBDZ &lsqb;_( -.326_MC ,_, .233_MC &rsqb;_) ._. 
</s>
<s>
There_EX are_VBR two_MC predominant_JJ approaches_NN2 for_IF approximating_VVG the_AT fractional_JJ derivative_NN1 :_: one_MC1 approach_NN1 is_VBZ by_II using_VVG Lubich_NP1 '_NULL s_ZZ1 convolution_NN1 quadrature_NN1 &lsqb;_( 27_MC &rsqb;_) ,_, &lsqb;_( 28_MC &rsqb;_) ,_, &lsqb;_( 29_MC &rsqb;_) and_CC another_DD1 approach_NN1 is_VBZ by_II using_VVG the_AT L1_FO scheme_NN1 (_( or_CC Diethelm_VV0 '_NULL s_ZZ1 finite_JJ difference_NN1 method_NN1 )_) ._. 
</s>
<s>
Of_RR21 course_RR22 ,_, it_PPH1 is_VBZ already_RR known_VVN that_CST any_DD Ito_NN1 process_NN1 @F_FO is_VBZ completely_RR characterized_VVN by_II @S_FO The_AT main_JJ message_NN1 of_IO Theorem_NN1 4.3_MC is_VBZ that_CST @F_FO ,_, where_CS @S_FO can_VM be_VBI intrinsically_RR constructed_VVN by_II31 means_II32 of_II33 any_DD stable_JJ imbedded_JJ discrete_JJ structure_NN1 Y_ZZ1 satisfying_JJ (_( 4.29_MC )_) and_CC (_( 4.30_MC )_) ._. 
</s>
<s>
In_BCL21 order_BCL22 to_TO establish_VVI (_( 6.5_MC )_) in_II the_AT general_JJ case_NN1 ,_, where_CS Fj_NP1 is_VBZ merely_RR continuous_JJ ,_, we_PPIS2 use_VV0 a_AT1 spherical_JJ approximate_JJ identity_NN1 @S_FO ,_, @S_FO ,_, (_( instead_II21 of_II22 repeating_VVG the_AT arguments_NN2 from_II the_AT proof_NN1 of_IO &lsqb;_( 62_MC ,_, Theorem_NN1 6.3_MC &rsqb;_) )_) to_TO define_VVI @S_FO for_IF every_AT1 @S_FO ._. 
</s>
<s>
Then_RT @S_FO is_VBZ SO(n)-equivariant_JJ and_CC smooth_JJ ,_, and_CC by_II what_DDQ we_PPIS2 have_VH0 already_RR shown_VVN and_CC the_AT fact_NN1 that_CST multiplier_NN1 transformations_NN2 commute_VV0 ,_, @F_FO ._. 
</s>
<s>
Letting_VVG now_RT @S_FO ,_, we_PPIS2 obtain_VV0 (_( 6.5_MC )_) from_II Lemmas_NN2 2.5_MC and_CC 2.6_MC ._. 
</s>
<s>
Moreover_RR ,_, CJ_NP1 contains_VVZ all_DB the_AT modules_NN2 of_IO the_AT form_NN1 V(i)_NP1 (_( q_ZZ1 )_) s_ZZ1 for_IF 1≤i≤N1_FO and_CC s∈i1+2Z_FO (_( see_VV0 Sect._NP1 4.7_MC )_) ._. 
</s>
<s>
These_DD2 are_VBR the_AT unstable_JJ fixed_JJ points_NN2 ._. 
</s>
<s>
It_PPH1 remains_VVZ to_TO consider_VVI the_AT case_NN1 w_ZZ1 =1_FO ._. 
</s>
<s>
A_AT1 thorough_JJ treatment_NN1 of_IO the_AT definition_NN1 of_IO linearization_NN1 and_CC strong_JJ linearization_NN1 and_CC their_APPGE implications_NN2 can_VM be_VBI found_VVN in_II &lsqb;_( 17_MC &rsqb;_) ._. 
</s>
<s>
While_CS the_AT order_NN1 of_IO accuracy_NN1 can_VM be_VBI arbitrarily_RR high_JJ ,_, and_CC the_AT individual_JJ integral_JJ equations_NN2 are_VBR well-conditioned_JJ ,_, the_AT stability_NN1 of_IO the_AT resulting_JJ scheme_NN1 remains_VVZ to_TO be_VBI studied_VVN ._. 
</s>
<s>
As_CSA explained_VVN earlier_RRR ,_, if_CS we_PPIS2 condition_VV0 on_II @S_FO and_CC @S_FO ,_, then_RT the_AT conditional_JJ law_NN1 of_IO b_ZZ1 in_II the_AT remaining_JJ domain_NN1 (_( up_II21 until_II22 it_PPH1 hits_VVZ @S_FO is_VBZ that_DD1 of_IO an_AT1 SLEk_NP1 (_( k_ZZ1 -_- 6_MC )_) process_VV0 ._. 
</s>
<s>
We_PPIS2 start_VV0 with_IW observations_NN2 about_II the_AT structure_NN1 of_IO Q_ZZ1 and_CC R._NP1 Since_CS the_AT first_MD r_ZZ1 columns_NN2 of_IO Q_ZZ1 are_VBR identically_RR those_DD2 of_IO U_ZZ1 ,_, we_PPIS2 let_VV0 Zr_NP1 be_VBI the_AT @S_FO matrix_NN1 such_CS21 that_CS22 @S_FO ._. 
</s>
<s>
For_IF the_AT same_DA reason_NN1 ,_, R_ZZ1 has_VHZ the_AT block_NN1 structure_NN1 @F_FO ,_, where_CS the_AT matrices_NN2 R12_FO and_CC R22_FO satisfy_VV0 @S_FO ,_, so_CS21 that_CS22 @F_FO ._. 
</s>
<s>
Since_CS Vr_NP1 has_VHZ orthogonal_JJ columns_NN2 ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
In_RR21 particular_RR22 ,_, we_PPIS2 proved_VVD that_CST Laplace-based_JJ importance_NN1 sampling_NN1 behaves_VVZ well_RR in_II the_AT small_JJ noise_NN1 or_CC large_JJ data_NN size_NN1 limit_NN1 ,_, respectively_RR ._. 
</s>
<s>
Proposition_NN1 5.2_MC suggests_VVZ that_CST if_CS the_AT kernel_NN1 Kexp_NN1 converges_VVZ to_II Kunder_NP1 some_DD scalings_NN2 ,_, then_RT the_AT Fredholm_NP1 Pfaffian_NN1 of_IO Kexp_NN1 should_VM converge_VVI to_TO FGSE_VVI ._. 
</s>
<s>
The_AT second_MD part_NN1 of_IO the_AT post-intervention_JJ test_NN1 had_VHD six_MC questions_NN2 of_IO extended_JJ answer_NN1 format_NN1 ._. 
</s>
<s>
In_II this_DD1 paper_NN1 ,_, we_PPIS2 develop_VV0 a_AT1 min–max_NN1 theory_NN1 for_IF the_AT construction_NN1 of_IO constant_JJ mean_JJ curvature_NN1 (_( CMC_NP1 )_) hypersurfaces_NN2 of_IO prescribed_JJ mean_JJ curvature_NN1 in_II an_AT1 arbitrary_JJ closed_JJ manifold_NN1 ._. 
</s>
<s>
When_CS finding_VVG the_AT area_NN1 of_IO a_AT1 rectangle_NN1 ,_, students_NN2 must_VM learn_VVI to_TO coordinate_VVI linear_JJ unit_NN1 counting_VVG with_IW area_NN1 unit_NN1 counting_NN1 ._. 
</s>
<s>
Since_CS T_ZZ1 is_VBZ essentially_RR supported_VVN in_II a_AT1 rectangle_NN1 of_IO dimensions_NN2 R1/2×1_VV0 ,_, with_IW the_AT long_JJ direction_NN1 parallel_NN1 to_II S1r_FO at_II points_NN2 in_II τ_NULL S1r_FO ,_, we_PPIS2 can_VM bound_VVI @F_FO ,_, where_CS j_ZZ1 ,_, τ_NULL ,_, r_ZZ1 is_VBZ again_RT rapidly_RR decaying_VVG outside_II21 of_II22 @S_FO ,_, but_CCB a_RR21 bit_RR22 more_RGR slowly_RR than_CSN j_ZZ1 ,_, τ_NULL ,_, r_ZZ1 ._. 
</s>
<s>
The_AT advantage_NN1 of_IO working_VVG with_IW smooth_JJ translation-invariant_JJ valuations_NN2 instead_II21 of_II22 merely_RR continuous_JJ ones_NN2 is_VBZ that_CST the_AT space_NN1 @S_FO admits_VVZ more_RGR algebraic_JJ structure_NN1 ._. 
</s>
<s>
On_II the_AT other_JJ hand_NN1 ,_, for_IF non-local_JJ operators_NN2 ,_, it_PPH1 is_VBZ not_XX known_VVN whether_CSW EHI_NP1 implies_VVZ a_JJ21 priori_JJ22 elliptic_JJ Holder_NN1 regularity_NN1 (_( EHR_NP1 )_) and_CC we_PPIS2 suspect_VV0 it_PPH1 is_VBZ not_XX (_( although_CS parabolic_JJ Harnack_NN1 inequality_NN1 (_( PHI_NN1 )_) does_VDZ imply_VVI parabolic_JJ Holder_NN1 regularity_NN1 and_CC hence_RR EHR_VV0 ,_, see_VV0 Theorem_NN1 1.19_MC below_RL )_) ._. 
</s>
<s>
By_II Example_NN1 5.13_MC ,_, the_AT functor_NN1 @S._FO is_VBZ coherent_JJ ._. 
</s>
<s>
Participants_NN2 ofthe_VV0 study_NN1 were_VBDR 935_MC public_JJ middle_JJ school_NN1 students_NN2 between_II grades_NN2 6_MC and_CC 8_MC (_( between_II ages_NN2 11_MC and_CC 14_MC )_) ._. 
</s>
<s>
The_AT following_JJ result_NN1 provides_VVZ verifiable_JJ sufficient_JJ conditions_NN2 which_DDQ ensure_VV0 that_CST a_AT1 computation_NN1 composed_VVN of_IO individual_JJ BPNMs_NP1 is_VBZ a_AT1 Bayesian_JJ computation_NN1 ._. 
</s>
<s>
The_AT lengths_NN2 of_IO the_AT lines_NN2 are_VBR in_II31 proportion_II32 to_II33 the_AT recipe_NN1 :_: one_MC1 line_NN1 for_IF lemon_NN1 juice_NN1 ,_, two_MC for_IF water_NN1 ,_, and_CC a_AT1 line_NN1 half_RR as_RG long_RR for_IF sugar_NN1 ,_, adding_VVG @S_FO under_II the_AT last_MD line_NN1 for_IF clarification_NN1 (_( Fig._NN1 2_MC )_) ._. 
</s>
<s>
Remark_VV0 7_MC (_( characterization_NN1 (_( v_ZZ1 )_) and_CC related_JJ notions_NN2 )_) Note_VV0 that_CST ,_, unlike_II the_AT other_JJ characterizations_NN2 ,_, (_( v_ZZ1 )_) provides_VVZ only_RR a_AT1 qualitative_JJ criterion_NN1 of_IO transversality_NN1 ._. 
</s>
<s>
The_AT proofs_NN2 are_VBR based_VVN on_II exact_JJ identities_NN2 for_IF LCFT_NP1 correlation_NN1 functions_NN2 which_DDQ rely_VV0 on_II the_AT underlying_JJ Gaussian_JJ structure_NN1 of_IO LCFT_NP1 combined_VVD with_IW estimates_NN2 from_II the_AT theory_NN1 of_IO critical_JJ Gaussian_JJ Multiplicative_JJ Chaos_NN1 and_CC a_AT1 careful_JJ analysis_NN1 of_IO singular_JJ integrals_NN2 (_( Beurling_NP1 transforms_VVZ and_CC generalizations_NN2 )_) ._. 
</s>
<s>
Since_CS on_II a_AT1 technical_JJ level_NN1 the_AT rest_NN1 of_IO Section_NN1 3_MC is_VBZ independent_JJ from_II those_DD2 that_CST follow_VV0 ,_, we_PPIS2 emphasize_VV0 that_CST the_AT reader_NN1 who_PNQS wishes_VVZ to_TO skip_VVI to_II Section_NN1 4_MC may_VM safely_RR do_VDI so_RR ._. 
</s>
<s>
Roughly_RR ,_, the_AT benefit_NN1 of_IO US_NP1 is_VBZ due_II21 to_II22 the_AT facts_NN2 that_CST averages_NN2 with_II31 respect_II32 to_II33 the_AT @S_FO are_VBR often_RR sufficient_JJ to_TO solve_VVI for_IF all_DB desired_JJ quantities_NN2 ,_, and_CC that_CST one_PN1 can_VM choose_VVI @S_FO so_CS21 that_CS22 averages_NN2 with_II31 respect_II32 to_II33 the_AT @S_FO converge_VV0 much_RR more_RGR quickly_RR than_CSN averages_NN2 with_II31 respect_II32 to_II33 itself_PPX1 ._. 
</s>
<s>
Even_CS21 though_CS22 all_DB teachers_NN2 and_CC researchers_NN2 in_II mathematics_NN1 education_NN1 recognize_VV0 the_AT pedagogic_JJ value_NN1 of_IO visual_JJ language_NN1 ,_, they_PPHS2 do_VD0 not_XX all_DB give_VVI visual_JJ language_NN1 the_AT same_DA importance_NN1 as_CSA other_JJ languages_NN2 ,_, despite_II it_PPH1 being_VBG the_AT object_NN1 of_IO many_DA2 studies_NN2 (_( e.g._REX ,_, David_NP1 &;_NULL Tomaz_NP1 ,_, 2012_MC ;_; Stylianou_NP1 &;_NULL Silver_NP1 ,_, 2004_MC )_) ._. 
</s>
<s>
Next_MD ,_, I_PPIS1 analyzed_VVD the_AT PSTs_NP1 '_NULL responses_NN2 using_VVG these_DD2 summaries_NN2 and_CC considering_VVG the_AT research_NN1 questions_NN2 ._. 
</s>
<s>
One_PN1 would_VM typically_RR choose_VVI @S_FO in_II this_DD1 deterministic_JJ iteration_NN1 ;_; we_PPIS2 consider_VV0 arbitrary_JJ @S_FO to_TO motivate_VVI the_AT stochastic_JJ approximation_NN1 algorithm_NN1 developed_VVN in_II subsection_NN1 3.3_MC ._. 
</s>
<s>
Moreover_RR ,_, the_AT method_NN1 converges_VVZ with_IW larger_JJR orders_NN2 for_IF the_AT rest_NN1 of_IO the_AT transmission_NN1 eigenvalues_NN2 ._. 
</s>
<s>
Second_MD ,_, we_PPIS2 exploit_VV0 the_AT (_( hidden_VVN )_) submodularity_NN1 of_IO the_AT 0-1_MCMC SOC_NN1 constraints_NN2 and_CC employ_VV0 extended_JJ polymatroid_JJ valid_JJ inequalities_NN2 to_TO accelerate_VVI solving_VVG DCBP_NP1 ._. 
</s>
<s>
After_CS a_AT1 rearrangement_NN1 this_DD1 gives_VVZ @F_FO ,_, with_IW @F_FO ._. 
</s>
<s>
Note_VV0 that_CST ,_, using_VVG our_APPGE assumption_NN1 (_( 3.3_MC )_) @F_FO ._. 
</s>
<s>
Contingency_NN1 The_AT teachers_NN2 indicated_VVN as_CSA they_PPHS2 walked_VVD around_RP observing_VVG and_CC talking_VVG to_II the_AT students_NN2 during_II the_AT lessons_NN2 ,_, they_PPHS2 had_VHD to_TO spontaneously_RR respond_VVI to_II students_NN2 '_NULL tool-based_JJ strategies_NN2 which_DDQ were_VBDR changing_VVG and_CC developing_JJ ,_, for_REX21 example_REX22 in_II Group_NN1 Case_NN1 2_MC ,_, the_AT intended_JJ plan_NN1 and_CC the_AT implemented_JJ plan_NN1 were_VBDR different_JJ ._. 
</s>
<s>
Chance-constrained_JJ programs_NN2 are_VBR difficult_JJ to_TO solve_VVI ,_, mainly_RR because_CS the_AT feasible_JJ region_NN1 described_VVN by_II constraints_NN2 (_( 1_MC1 )_) is_VBZ nonconvex_NN1 in_RR21 general_RR22 &lsqb;_( 26_MC &rsqb;_) ._. 
</s>
<s>
These_DD2 examples_NN2 show_VV0 that_CST the_AT nesting_NN1 strategy_NN1 for_IF research-based_JJ design_NN1 imposes_VVZ research_NN1 questions_NN2 :_: When_RRQ teachers_NN2 '_NULL learning_VVG pathways_NN2 toward_II digital_JJ tools_NN2 in_II algebra_NN1 classrooms_NN2 are_VBR considered_VVN to_TO be_VBI a_AT1 relevant_JJ part_NN1 of_IO the_AT nested_JJ FPD_NP1 content_NN1 ,_, then_RT they_PPHS2 should_VM be_VBI investigated_VVN ._. 
</s>
<s>
In_II Peter_NP1 '_NULL s_ZZ1 discussion_NN1 ,_, he_PPHS1 described_VVD the_AT constant_JJ difference_NN1 between_II the_AT thermometers_NN2 as_CSA justifying_VVG that_DD1 scenario_NN1 1_MC1 is_VBZ a_AT1 proportion_NN1 ;_; however_RR ,_, scenario_NN1 1_MC1 is_VBZ an_AT1 additive_JJ linear_JJ (_( affine_JJ )_) situation_NN1 ._. 
</s>
<s>
In_II the_AT purely_RR infinite_JJ case_NN1 ,_, groupoid_JJ models_NN2 and_CC hence_RR Cartan_JJ subalgebras_NN2 have_VH0 been_VBN constructed_VVN in_II &lsqb;_( 41_MC &rsqb;_) (_( see_VV0 also_RR &lsqb;_( 29_MC ,_, §_FO 5_MC &rsqb;_) )_) ._. 
</s>
<s>
The_AT assumption_NN1 that_CST there_EX is_VBZ a_AT1 single_JJ quantum_NN1 register_NN1 is_VBZ without_IW loss_NN1 of_IO generality_NN1 ,_, as_CSA one_PN1 can_VM think_VVI of_IO multiple_JJ registers_NN2 as_CSA being_VBG placed_VVN "_" side-by-side_RR "_" to_TO form_VVI a_AT1 single_JJ register_NN1 (_( of_RR21 course_RR22 ,_, one_PN1 would_VM then_RT need_VVI to_TO specify_VVI what_DDQ operations_NN2 are_VBR allowed_VVN on_II the_AT resulting_JJ register_NN1 )_) ._. 
</s>
<s>
The_AT claim_NN1 that_CST (_( ii_MC )_) implies_VVZ (_( iii_MC )_) with_IW the_AT limit_NN1 expressed_VVN as_CSA described_VVN in_II Remark_NN1 5.4(ii)is_FO essentially_RR identical_JJ to_II the_AT proof_NN1 of_IO Theorem_NN1 3.3_MC from_II Section_NN1 4.2_MC and_CC is_VBZ left_VVN to_II the_AT reader_NN1 ._. 
</s>
<s>
During_II the_AT @S_FO iteration_NN1 of_IO the_AT NEUS_NN algorithm_NN1 ,_, we_PPIS2 update_VV0 the_AT current_JJ approximations_NN2 @S_FO and_CC @S_FO based_VVN on_II statistics_NN gathered_VVN from_II @S_FO independent_JJ excursions_NN2 @S_FO defined_JJ accor_NN1 ing_VVG to_II the_AT rules_NN2 governing_VVG @S_FO enumerated_VVN above_RL with_IW @S_FO drawn_VVN from_II @S_FO ,_, the_AT current_JJ (_( at_II the_AT mth_NNU iteration_NN1 of_IO the_AT scheme_NN1 )_) estimate_NN1 of_IO the_AT flux_NN1 distribution_NN1 @S_FO ._. 
</s>
<s>
The_AT surface_NN1 tension_NN1 of_IO a_AT1 measured_JJ foliation_NN1 a_AT1 is_VBZ @F_FO ,_, where_RRQ ,_, writing_VVG @S_FO ,_, @S_FO is_VBZ minus_II the_AT free_JJ energy_NN1 of_IO @S_FO ;_; see_VV0 (_( 114_MC )_) ._. 
</s>
<s>
Jacobi_JJ Polynomials_NN2 ,_, Gn(p.q.x)_NP1 ,_, on_II (_( 0_MC ._. 
</s>
<s>
1_MC1 )_) ._. 
</s>
<s>
We_PPIS2 also_RR use_VV0 them_PPHO2 to_TO show_VVI ,_, via_II the_AT pullback_NN1 argument_NN1 ,_, that_CST any_DD two_MC combinatorially_RR equivalent_JJ Siegel_NP1 pacmen_NN2 are_VBR hybrid_JJ equivalent_NN1 (_( Theorem_NN1 3.11_MC )_) ,_, i.e._REX ,_, there_EX is_VBZ a_AT1 qc_NN1 conjugacy_NN1 between_II them_PPHO2 which_DDQ is_VBZ conformal_JJ on_II the_AT Siegel_NP1 disk_NN1 ._. 
</s>
<s>
Hence_RR ,_, if_CS we_PPIS2 write_VV0 the_AT solution_NN1 as_CSA @F_FO and_CC the_AT pressure_NN1 term_NN1 is_VBZ written_VVN as_CSA @F_FO ._. 
</s>
<s>
Then_RT ,_, the_AT exact_JJ evolution_NN1 equations_NN2 for_IF the_AT perturbation_NN1 become_VV0 @F_FO ,_, besides_II the_AT no-slip_JJ condition_NN1 @S_FO on_II @S_FO ._. 
</s>
<s>
For_IF hydrodynamic_JJ stability_NN1 questions_NN2 ,_, naturally_RR p(0)_FO is_VBZ assumed_VVN initially_RR small_JJ in_II certain_JJ norm_NN1 ._. 
</s>
<s>
We_PPIS2 prove_VV0 that_CST the_AT Boltzmann_NP1 equation_NN1 without_IW cut-off_NN1 can_VM be_VBI written_VVN in_II this_DD1 form_NN1 and_CC satisfies_VVZ our_APPGE assumptions_NN2 provided_CS21 that_CS22 the_AT mass_JJ density_NN1 is_VBZ bounded_VVN away_II21 from_II22 vacuum_NN1 ,_, and_CC mass_NN1 ,_, energy_NN1 and_CC entropy_NN1 densities_NN2 are_VBR bounded_VVN above_RL ._. 
</s>
<s>
In_II fact_NN1 ,_, the_AT formulation_NN1 gives_VVZ surprisingly_RR accurate_JJ and_CC smooth_JJ prediction_NN1 of_IO pressure_NN1 pressure_NN1 distribution_NN1 ,_, which_DDQ is_VBZ unexpected_JJ from_II unresolved_JJ or_CC marginally_RR resolved_VVN simulations_NN2 of_IO flows_NN2 on_II non-body-fitted_JJ meshes_NN2 ._. 
</s>
<s>
We_PPIS2 set_VV0 @S_FO ._. 
</s>
<s>
For_IF 0≤k≤R_FO we_PPIS2 define_VV0 @S_FO as_II the_AT set_NN1 of_IO (_( xn_FO )_) Rn=0_FO satisfying_JJ @S_FO except_II21 for_II22 at_RR21 most_RR22 k_ZZ1 entries_NN2 ._. 
</s>
<s>
We_PPIS2 see_VV0 the_AT use_NN1 of_IO the_AT sole_JJ and_CC combined_JJ proof_NN1 schemes_NN2 proposed_VVN in_II this_DD1 paper_NN1 as_CSA being_VBG of_IO assistance_NN1 to_II teachers_NN2 and_CC researchers_NN2 interested_JJ in_II noticing_VVG the_AT signals_NN2 of_IO where_RRQ this_DD1 change_NN1 of_IO classroom_NN1 culture_NN1 may_VM be_VBI on_II the_AT cusp_NN1 of_IO having_VHG the_AT capacity_NN1 to_TO occur_VVI ._. 
</s>
<s>
Conversely_RR ,_, it_PPH1 appears_VVZ that_CST accelerated_JJ gradient_NN1 descent_NN1 is_VBZ more_RGR central_JJ to_II our_APPGE approach_NN1 ._. 
</s>
<s>
It_PPH1 remains_VVZ to_TO investigate_VVI the_AT quantity_NN1 @F_FO ,_, where_RRQ recall_VV0 that_CST @S_FO ._. 
</s>
<s>
Using_VVG again_RT the_AT law_NN1 of_IO M_ZZ1 ,_, which_DDQ is_VBZ exponential_NN1 with_IW parameter_NN1 @S_FO ,_, and_CC making_VVG the_AT change_NN1 of_IO variables_NN2 @S_FO ,_, we_PPIS2 get_VV0 @F_FO ._. 
</s>
<s>
In_CS21 case_CS22 one_MC1 of_IO them_PPHO2 is_VBZ invertible_JJ ,_, it_PPH1 gives_VVZ a_AT1 quasi-isomorphism_NN1 Aa_NP1 Aa_NP1 and_CC hence_RR ,_, by_II Corollary_NN1 3.3.4_MC ,_, Ma_NP1 =_FO 0_MC ._. 
</s>
<s>
Sometimes_RT this_DD1 might_VM be_VBI the_AT "_" domain_NN1 ,_, "_" or_CC set_NN1 of_IO objects_NN2 ,_, for_IF which_DDQ a_AT1 statement_NN1 is_VBZ ,_, and_CC is_VBZ not_XX ,_, true_JJ ;_; at_II other_JJ times_NNT2 ,_, it_PPH1 might_VM be_VBI about_II the_AT underlying_JJ "_" argument_NN1 ,_, "_" and_CC whether_CSW the_AT argument_NN1 would_VM hold_VVI in_II a_AT1 different_JJ setting_NN1 ._. 
</s>
<s>
It_PPH1 shows_VVZ what_DDQ can_VM be_VBI done_VDN with_IW almost_RR no_AT preparation_NN1 ,_, and_CC it_PPH1 has_VHZ proven_VVN to_TO be_VBI a_AT1 reliable_JJ way_NN1 of_IO scaffolding_NN1 the_AT "_" rediscovery_NN1 "_" of_IO one_MC1 of_IO Galileo_NP1 '_NULL s_ZZ1 fundamental_JJ accomplishments_NN2 :_: the_AT description_NN1 of_IO "_" falling_VVG as_II a_AT1 motion_NN1 with_IW constant_JJ acceleration_NN1 ._. 
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<s>
Other_JJ validating_JJ activities_NN2 might_VM be_VBI observed_VVN during_II tasks_NN2 that_CST encourage_VV0 students_NN2 to_TO create_VVI new_JJ mathematics_NN1 ,_, repurpose_VV0 already-known_JJ mathematics_NN1 ,_, reference_VV0 a_AT1 domain_NN1 the_AT modeler_NN1 has_VHZ little_DA1 experience_NN1 thinking_VVG about_II mathematically_RR ,_, or_CC in_II authentic_JJ ,_, open-ended_JJ modeling_NN1 projects_NN2 ._. 
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<s>
However_RR ,_, these_DD2 works_NN left_VVD wide_RR open_JJ the_AT question_NN1 of_IO which_DDQ values_NN2 may_VM be_VBI prescribedthat_NN1 is_VBZ ,_, for_IF which_DDQ constants_NN2 c_ZZ1 does_VDZ there_RL exist_VVI a_AT1 closed_JJ hypersurface_NN1 of_IO constant_JJ mean_JJ curvature_NN1 c_ZZ1 ?_? 
</s>
<s>
We_PPIS2 claim_VV0 that_CST solutions_NN2 to_II (_( 2.9_MC )_) convergetoasolutionof_NN1 (_( 3.10_MC )_) ._. 
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<s>
By_II assumption_NN1 ,_, we_PPIS2 have_VH0 Wij_NP1 >_FO 0_MC on_II the_AT edges_NN2 ,_, and_CC for_IF all_DB spanning_VVG trees_NN2 T_ZZ1 ,_, since_CS edges_NN2 appear_VV0 at_RR21 most_RR22 once_RR ,_, @F_FO ,_, which_DDQ implies_VVZ @S_FO ._. 
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<s>
From_II the_AT expression_NN1 of_IO QW(du)_NP1 ,_, we_PPIS2 deduce_VV0 that_CST @F_FO ._. 
</s>
<s>
These_DD2 formulas_NN2 may_VM prove_VVI useful_JJ for_IF the_AT analysis_NN1 of_IO other_JJ statistical_JJ methods_NN2 under_II high-dimensional_JJ asymptotics_NN2 ,_, such_II21 as_II22 principal_JJ component_NN1 regression_NN1 ._. 
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<s>
This_DD1 is_VBZ the_AT first_MD paper_NN1 that_CST takes_VVZ the_AT different_JJ structures_NN2 in_II different_JJ modes_NN2 into_II account_NN1 to_TO develop_VVI a_AT1 new_JJ theory_NN1 on_II structured_JJ robust_JJ stability_NN1 and_CC boundedness_NN1 for_IF highly_RR nonlinear_JJ hybrid_JJ SDDEs_NN2 ._. 
</s>
<s>
A_AT1 completely_RR analogous_JJ argument_NN1 yields_VVZ the_AT remaining_JJ bound_NN1 on_II |_NULL |_NULL ≤N&lsqb;_FO ,_, CurlA_NP1 &rsqb;_) θ_NULL 21++a,1_FO and_CC this_DD1 concludes_VVZ the_AT proof_NN1 of_IO the_AT proposition_NN1 ._. 
</s>
<s>
Let_VV0 H_ZZ1 denote_VVI the_AT closed_JJ upper_JJ half_NN1 of_IO the_AT unit_NN1 disk_NN1 and_CC @S_FO the_AT supremum_NN1 norm_NN1 on_II H._NP1 Together_RL with_IW (_( 4.45_MC )_) ,_, this_DD1 remark_NN1 implies_VVZ that_CST @F_FO ._. 
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<s>
We_PPIS2 now_RT extend_VV0 the_AT consistency_NN1 analysis_NN1 to_II the_AT practical_JJ two-scale_JJ operator_NN1 @S_FO ._. 
</s>
<s>
Consider_VV0 @S_FO ._. 
</s>
<s>
By_II (_( 2.8b_FO )_) ,_, (_( 4.2a_FO )_) ,_, (_( 4.4a_FO )_) ,_, (_( 4.6a_FO )_) ,_, and_CC (_( 4.8a_FO )_) ,_, we_PPIS2 have_VH0 @F_FO ._. 
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<s>
The_AT above_JJ with_IW the_AT fact_NN1 that_CST @S_FO for_IF all_DB @S_FO implies_VVZ (_( 4.10a_FO )_) ._. 
</s>
<s>
For_IF each_DD1 @S_FO ,_, there_EX exists_VVZ a_AT1 set_NN1 @S_FO such_CS21 that_CS22 @S_FO and_CC @S_FO for_IF uncountably_RR many_DA2 @S_FO ._. 
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<s>
For_IF N_ZZ1 <_FO n_ZZ1 ,_, let_VV0 @S_FO be_VBI the_AT set_NN1 of_IO words_NN2 of_IO length_NN1 N_ZZ1 that_CST arise_VV0 as_II a_AT1 subword_NN1 of_IO a_AT1 word_NN1 in_II Fn_NP1 ._. 
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<s>
By_II the_AT definition_NN1 in_II (_( 7.24_MC )_) and_CC (_( 2.4_MC )_) it_PPH1 follows_VVZ that_CST @F_FO ._. 
</s>
<s>
Next_MD ,_, notice_VV0 that_CST for_CS @S_FO ,_, the_AT vector_NN1 @S_FO is_VBZ the_AT projection_NN1 of_IO @S_FO onto_II the_AT one-dimensional_JJ subspace_NN1 generated_VVN by_II v._NNU Hence_RR @S_FO and_CC using_VVG (_( 2.7_MC )_) we_PPIS2 have_VH0 @F_FO ,_, so_CS21 that_CS22 (_( 7.28_MC )_) follows_VVZ by_II combining_VVG the_AT last_MD two_MC displays_NN2 ._. 
</s>
<s>
In_II the_AT limit_NN1 of_IO vanishing_JJ effective_JJ springs_NN2 ,_, however_RR ,_, the_AT mean_JJ squared_JJ displacement_NN1 (_( 3.16_MC )_) reduces_VVZ to_II (_( 3.4_MC )_) ,_, since_CS @S_FO ._. 
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<s>
In_II the_AT early_JJ 1960s_MC2 ,_, Polya_NP1 (_( 1961_MC )_) asserted_VVD that_CST solving_VVG a_AT1 problem_NN1 is_VBZ finding_VVG a_AT1 way_NN1 out_II21 of_II22 a_AT1 difficulty_NN1 ,_, a_AT1 way_NN1 around_II an_AT1 obstacle_NN1 or_CC attaining_VVG an_AT1 aim_NN1 which_DDQ was_VBDZ not_XX immediately_RR attainable_JJ ._. 
</s>
<s>
Since_CS is_VBZ a_AT1 unipotent_JJ character_NN1 and_CC the_AT Harish-Chandra_NP1 restriction_NN1 preserves_VVZ rational_JJ series_NN ,_, every_AT1 irreducible_JJ constituent_NN1 of_IO @S_FO is_VBZ a_AT1 unipotent_JJ character_NN1 of_IO L_ZZ1 ,_, and_CC so_RR contains_VVZ @S_FO in_II its_APPGE kernel_NN1 ._. 
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<s>
We_PPIS2 study_VV0 the_AT case_NN1 @S_FO in_II detail_NN1 and_CC prove_VV0 that_CST the_AT @S-equivariant_FO coherent_JJ Satake_NN1 category_NN1 of_IO @S_FO is_VBZ a_AT1 monoidal_JJ categorification_NN1 of_IO an_AT1 explicit_JJ quantum_NN1 cluster_NN1 algebra_NN1 ._. 
</s>
<s>
Note_VV0 that_CST the_AT multiparameter_JJR version_NN1 of_IO this_DD1 last_MD result_NN1 fails_VVZ spectacularly_RR :_: for_IF simple_JJ eigenvalues_NN2 ,_, a_AT1 locally_RR Lipschitz-dependence_JJ holds_NN2 but_CCB differentiability_NN1 might_VM fail_VVI and_CC it_PPH1 may_VM not_XX even_RR be_VBI possible_JJ to_TO choose_VVI eigenvectors_NN2 in_II a_AT1 continuous_JJ way_NN1 (_( see_VV0 the_AT discussion_NN1 in_II &lsqb;_( 24_MC &rsqb;_) )_) ._. 
</s>
<s>
These_DD2 smallness_NN1 conditions_NN2 are_VBR the_AT same_DA as_CSA in_II &lsqb;_( 28_MC &rsqb;_) ._. 
</s>
<s>
Indeed_RR ,_, standard_JJ methods_NN2 do_VD0 not_XX work_VVI in_II a_AT1 linearized_JJ framework_NN1 ,_, being_VBG these_DD2 based_VVN on_II the_AT identification_NN1 of_IO bad_JJ parts_NN2 of_IO the_AT function_NN1 via_II coarea_NN1 formula_NN1 and_CC their_APPGE removal_NN1 via_II truncation_NN1 &lsqb;_( 6_MC &rsqb;_) ._. 
</s>
<s>
To_TO bound_VVI the_AT second_MD factor_NN1 ,_, we_PPIS2 note_VV0 that_CST 2_MC of_IO a_AT1 ball_NN1 of_IO radius_NN1 r1_FO is_VBZ at_RR21 most_RR22 r_ZZ1 ._. 
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<s>
The_AT above_JJ analysis_NN1 can_VM be_VBI extended_VVN to_II nondifferentiable_JJ convex_JJ functions_NN2 ._. 
</s>
<s>
It_PPH1 is_VBZ obvious_JJ that_CST in_BCL21 order_BCL22 to_TO establish_VVI (_( 1.11_MC )_) it_PPH1 is_VBZ sufficient_JJ to_TO prove_VVI (_( 1.12_MC )_) for_IF every_AT1 ball_NN1 @S_FO ._. 
</s>
<s>
Here_RL we_PPIS2 show_VV0 how_RRQ to_TO reduce_VVI the_AT proof_NN1 of_IO (_( 1.12_MC )_) to_II the_AT case_NN1 when_CS both_DB2 the_AT solution_NN1 u_ZZ1 and_CC datum_NN1 are_VBR more_RGR regular_JJ ,_, that_REX21 is_REX22 ,_, (_( 3.1_MC )_) holds_VVZ ._. 
</s>
<s>
Alternatively_RR ,_, a_AT1 gradient_NN1 descent_NN1 on_II the_AT low-rank_NN1 manifold_NN1 M_NN1 can_VM be_VBI used_VVN to_TO find_VVI the_AT correction_NN1 that_CST needs_VVZ to_TO be_VBI added_VVN to_II modes_NN2 Un_NP1 and_CC coefficients_NN2 Zn_NP1 to_TO evaluate_VVI the_AT SVD_NP1 truncation_NN1 @S_FO (_( (_( 41_MC )_) and_CC (_( 42_MC )_) )_) ._. 
</s>
<s>
To_TO do_VDI so_RR ,_, we_PPIS2 start_VV0 by_II establishing_VVG consistency_NN1 of_IO 0(x)_FO for_IF 0(x)_FO given_VVN our_APPGE assumptions_NN2 ;_; we_PPIS2 note_VV0 that_CST this_DD1 is_VBZ the_AT only_JJ point_NN1 in_II the_AT paper_NN1 where_CS we_PPIS2 use_VV0 the_AT fact_NN1 that_CST W_ZZ1 is_VBZ the_AT negative_JJ gradient_NN1 of_IO a_AT1 convex_JJ loss_NN1 as_CSA in_II Assumption_NN1 6_MC ._. 
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<s>
If_CS Sq(Gn)_NP1 converges_VVZ to_II a_AT1 closed_JJ set_NN1 @S_FO in_II the_AT Hausdorff_NN1 metric_JJ ,_, then_RT the_AT limit_NN1 (_( 2.14_MC )_) exists_VVZ for_IF all_DB a_AT1 e_ZZ1 Aq_NP1 and_CC all_DB symmetric_JJ J_ZZ1 e_ZZ1 Rqxq_NP1 and_CC can_VM be_VBI expressed_VVN as_CSA @F._FO (_( ii_MC )_) Let_VV0 @S_FO and_CC the_AT limit_NN1 (_( 2.15_MC )_) exists_VVZ for_IF all_DB symmetric_JJ @S_FO ,_, then_RT the_AT limit_NN1 (_( 2.14_MC )_) exists_VVZ for_IF all_DB such_DA J_ZZ1 and_CC @F_FO ._. 
</s>
<s>
We_PPIS2 see_VV0 diversity_NN1 and_CC individualism_NN1 as_CSA principles_NN2 of_IO the_AT inclusive_JJ model_NN1 primarily_RR because_CS these_DD2 teachers_NN2 do_VD0 not_XX support_VVI the_AT exclusion_NN1 or_CC stigmatization_NN1 of_IO "_" weak_JJ "_" students_NN2 ;_; rather_RR ,_, they_PPHS2 assert_VV0 the_AT need_NN1 "_" to_TO educate_VVI everyone_PN1 "_" :_: "_" We_PPIS2 need_VV0 to_TO work_VVI with_IW all_DB who_PNQS come_VV0 ,_, those_DD2 ready_JJ to_TO take_VVI it_PPH1 all_DB ,_, and_CC those_DD2 not_XX ready_JJ ,_, sick_JJ children_NN2 ,_, with_IW weak_JJ health_NN1 ,_, and_CC healthy_JJ children_NN2 "_" (_( Interview_NN1 97_MC )_) ._. 
</s>
<s>
Again_RT ,_, the_AT underlying_JJ class_NN1 is_VBZ F_ZZ1 =_FO @S_FO ,_, and_CC to_TO define_VVI the_AT regularization_NN1 function_NN1 let_VV0 @S_FO and_CC set_VVI @F_FO ,_, where_RRQ (_( tf_NNU )_) d=1_FO denotes_VVZ the_AT nonincreasing_JJ re-arrangement_NN1 of_IO @S_FO ._. 
</s>
<s>
Thus_RR ,_, the_AT SLOPE_NN1 norm_NN1 is_VBZ a_AT1 sorted_JJ ,_, weighted_JJ @S-norm_FO ,_, and_CC for_IF @S_FO ,_, SLOPE_NN1 regularization_NN1 coincides_VVZ with_IW the_AT LASSO_NN1 ._. 
</s>
<s>
How_RGQ many_DA2 more_DAR marbles_NN2 does_VDZ John_NP1 have_VHI ?_? 
</s>
<s>
We_PPIS2 also_RR require_VV0 that_CST every_AT1 pair_NN of_IO vertices_NN2 on_II the_AT right_JJ boundary_NN1 of_IO the_AT closure_NN1 of_IO the_AT structure_NN1 have_VH0 a_AT1 "_" line-of-sight_NN1 "_" to_II each_PPX221 other_PPX222 ,_, unless_CS the_AT structure_NN1 contains_VVZ an_AT1 edge_NN1 (_( missing_VVG or_CC nonmissing_VVG )_) connecting_VVG them_PPHO2 ._. 
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<s>
The_AT set_NN1 ofgaps_VVZ @S_FO is_VBZ the_AT set_NN1 of_IO positive_JJ integers_NN2 which_DDQ are_VBR not_XX contained_VVN in_II R._NP1 Equivalently_NP1 ,_, the_AT gaps_NN2 are_VBR defined_VVN as_II all_DB natural_JJ numbers_NN2 which_DDQ can_VM not_XX be_VBI written_VVN as_CSA non-negative_JJ integer_NN1 linear_JJ combination_NN1 of_IO the_AT generators_NN2 @S_FO ._. 
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<s>
Let_VV0 @S_FO be_VBI a_AT1 metric_JJ measure_NN1 space_NN1 ,_, @S_FO Lipschitz_NP1 and_CC d<m_FO and_CC integer_NN1 ._. 
</s>
<s>
More_RGR precisely_RR ,_, we_PPIS2 consider_VV0 the_AT liquids_NN2 for_IF which_DDQ the_AT density_NN1 p_ZZ1 is_VBZ supposed_JJ to_TO be_VBI uniformly_RR bounded_VVN from_II below_RL by_II a_AT1 positive_JJ constant_JJ ,_, so_CS21 that_CS22 the_AT equations_NN2 (_( 1.1_MC )_) can_VM be_VBI rewritten_VVN equivalently_RR as_II a_AT1 symmetric_JJ hyperbolic_JJ system_NN1 for_IF sufficiently_RR smooth_JJ solutions_NN2 ._. 
</s>
<s>
We_PPIS2 have_VH0 attempted_VVN to_TO review_VVI various_JJ primal-dual_JJ methods_NN2 which_DDQ are_VBR most_RRT related_VVN to_II our_APPGE work_NN1 ._. 
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<s>
The_AT bivariate_JJ density_NN1 @S_FO can_VM be_VBI well_RR estimated_VVN by_II using_VVG a_AT1 standard_JJ kernel_NN1 method_NN1 (_( Silverman_NP1 ,_, 1986_MC ;_; Wand_NN1 and_CC Jones_NP1 ,_, 1995_MC )_) ._. 
</s>
<s>
Hence_RR we_PPIS2 shall_VM focus_VVI on_II the_AT quantity_NN1 @S_FO ._. 
</s>
<s>
Suppose_VV0 that_CST we_PPIS2 are_VBR interested_JJ in_II counting_VVG howfrequently_RR @S_FO from_II the_AT null_JJ distribution_NN1 @S_FO would_VM fall_VVI into_II an_AT1 interval_NN1 in_II the_AT neighbourhood_NN1 of_IO @S_FO The_AT quantity_NN1 is_VBZ relevant_JJ because_CS @S_FO The_AT counting_NN1 task_NN1 is_VBZ difficult_JJ as_CSA we_PPIS2 do_VD0 notknow_VVI the_AT value_NN1 of_IO @S_FO ._. 
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<s>
Our_APPGE idea_NN1 is_VBZ ?_? first_MD to_TO apply_VVI a_AT1 screening_NN1 method_NN1 to_TO select_VVI the_AT nulls_NN2 @S_FO ,_, and_CC then_RT to_TO construct_VVI an_AT1 estimator_NN1 based_VVN on_II selected_JJ cases_NN2 ._. 
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<s>
Figure_VV0 1_MC1 :_: First_MD and_CC third_MD approximate_JJ eigenfunctions_NN2 with_IW hexagonal_JJ meshes_NN2 ._. 
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<s>
It_PPH1 is_VBZ possible_JJ that_CST graduates_NN2 with_IW Profile_NN1 1_MC1 foregrounded_VVD a_AT1 commitment_NN1 toward_II considerations_NN2 of_IO doing_VDG mathematics_NN1 compared_VVN to_II the_AT other_JJ commitments_NN2 ,_, manifesting_VVG as_CSA desiring_VVG to_TO provide_VVI conceptual_JJ explanations_NN2 when_CS teaching_VVG ._. 
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<s>
One_MC1 indicator_NN1 that_CST MC2_FO students_NN2 have_VH0 interiorized_VVN the_AT multiplicative_JJ relationship_NN1 shown_VVN in_II Figure_NN1 3_MC is_VBZ that_CST they_PPHS2 can_VM reason_VVI with_IW and_CC about_II pairs_NN2 even_CS21 if_CS22 they_PPHS2 have_VH0 not_XX created_VVN them_PPHO2 in_II immediate_JJ past_JJ experience_NN1 ._. 
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<s>
One_MC1 of_IO the_AT signed_JJ trees_NN2 associated_VVN with_IW a_ZZ1 =_FO 3214576_MC ._. 
</s>
<s>
Work_VV0 with_IW your_APPGE partner_NN1 and_CC use_VV0 the_AT toothpicks_NN2 to_TO create_VVI a_AT1 new_JJ rectangular_JJ area_NN1 for_IF the_AT cows_NN2 ._. "_" 
</s>
<s>
The_AT student_NN1 who_PNQS solved_VVD APs_NP2 using_VVG multiplication_NN1 learned_VVD to_TO create_VVI a_AT1 second_MD set_NN1 (_( e.g._REX ,_, eight_MC colors_NN2 )_) from_II the_AT first_MD set_NN1 (_( e.g._REX ,_, eight_MC colors_NN2 )_) where_RRQ she_PPHS1 considered_VVD the_AT second_MD set_NN1 to_TO be_VBI identical_JJ to_II the_AT first_MD ,_, and_CC established_VVD the_AT entire_JJ second_NNT1 set_VVN prior_II21 to_II22 making_VVG any_DD outcomes_NN2 ._. 
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<s>
So_RR ,_, having_VHG them_PPHO2 come_VV0 up_RP with_IW it_PPH1 ,_, but_CCB also_RR having_VHG them_PPHO2 think_VV0 of_IO all_DB these_DD2 cases_NN2 are_VBR not_XX only_RR possible_JJ ,_, but_CCB then_RT you_PPY can_VM kind_RR21 of_RR22 like_VVI create_VV0 the_AT function_NN1 to_TO match_VVI it_PPH1 ._. 
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<s>
Let_VV0 @S_FO be_VBI a_AT1 quantum_NN1 seed_NN1 in_II A._NP1 The_AT quantum_NN1 cluster_NN1 algebra_NN1 Aqi/2(S)_NN2 associated_VVN to_II the_AT quantum_NN1 seed_NN1 S_ZZ1 is_VBZ the_AT of_IO the_AT skew_NN1 field_NN1 K_ZZ1 generated_VVN by_II all_DB the_AT quantum_NN1 cluster_NN1 variables_NN2 in_II the_AT quantum_NN1 seeds_NN2 obtained_VVN from_II S_ZZ1 by_II any_DD sequence_NN1 of_IO mutations_NN2 ._. 
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<s>
Nonlinear_JJ parabolic_JJ equations_NN2 of_IO the_AT form_NN1 @F_FO ,_, equipped_VVN with_IW suitable_JJ boundary_NN1 and_CC initial_JJ conditions_NN2 ,_, are_VBR frequently_RR encountered_VVN in_II applications_NN2 ._. 
</s>
<s>
In_II Tables_NN2 1_MC1 and_CC 2_MC ,_, the_AT upper_JJ rows_NN2 describe_VV0 more_RGR rudimentary_JJ strategies_NN2 ,_, whereas_CS the_AT more_RGR sophisticated_JJ strategies_NN2 are_VBR delineated_VVN in_II the_AT lower_JJR rows_NN2 ._. 
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<s>
Note_VV0 that_CST @S_FO ,_, @S_FO ,_, and_CC @S_FO ._. 
</s>
<s>
For_IF a_AT1 subring_JJ A_ZZ1 of_IO Q(q)_NP1 ,_, we_PPIS2 say_VV0 that_CST L_ZZ1 is_VBZ an_AT1 A-lattice_NN1 of_IO a_AT1 Q(q)-vector_JJ space_NN1 V_ZZ1 if_CS L_ZZ1 is_VBZ a_AT1 free_JJ A-submodule_NN1 of_IO V_ZZ1 such_CS21 that_CS22 V_ZZ1 =_FO Q(q)_NP1 L._NP1 Rh_FO ,_, N_ZZ1 U_ZZ1 Rh_FO n_ZZ1 ,_, with_IW t_ZZ1 >_FO 0_MC small_JJ ,_, @S_FO the_AT generalised_JJ outer_JJ normal_JJ to_II Q_ZZ1 atXh_NN1 ,_, N_ZZ1 ,_, and_CC @S_FO an_AT1 orthonormal_JJ basis_NN1 of_IO @S_FO ._. 
</s>
<s>
Moreover_RR ,_, let_VV0 u_ZZ1 '_NULL hN_NNU ?_NNU GSBDP_NP1 (_( Rh_FO N_ZZ1 )_) be_VBI the_AT functions_NN2 provided_VVN by_II Lemma_NN1 2.8_MC for_IF which_DDQ the_AT analogous_JJ of(4.10)_FO hold_VV0 ._. 
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<s>
Let_VV0 the_AT RK_NP1 method_NN1 with_IW coefficients_NN2 (_( A_ZZ1 ,_, b_ZZ1 )_) be_VBI absolutely_RR monotonic_JJ at_II @S_FO ._. 
</s>
<s>
Then_RT the_AT method_NN1 @S_FO with_IW @S_FO is_VBZ also_RR absolutely_RR monotonic_JJ at_II @S_FO ._. 
</s>
<s>
Since_CS G1_FO is_VBZ embedded_VVN in_II T2_FO there_EX is_VBZ an_AT1 induced_JJ linear_JJ map_NN1 @S_FO ,_, and_CC the_AT image_NN1 of_IO X_ZZ1 under_II this_DD1 map_NN1 is_VBZ a_AT1 convex_JJ polygon_NN1 @S_FO with_IW integer_NN1 vertices_NN2 ,_, the_AT unit_NN1 flow_NN1 polygon_NN1 ._. 
</s>
<s>
The_AT conjecture_NN1 is_VBZ true_JJ in_II dimensions_NN2 @S_FO ,_, as_CSA pointed_VVN out_RP earlier_RRR ,_, and_CC in_II the_AT case_NN1 in_II which_DDQ @S_FO is_VBZ a_AT1 Kahler-Ricci_JJ flow_NN1 (_( see_VV0 &lsqb;_( Zha10_FO &rsqb;_) )_) ._. 
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<s>
Tanaya_NP1 :_: Because_CS the_AT person_NN1 riding_NN1 is_VBZ going_VVG at_II a_AT1 constant_JJ speed_NN1 ,_, obviously_RR I_PPIS1 don_VV0 '_NULL t_ZZ1 really_RR know_VV0 how_RGQ fast_RR they_PPHS2 '_NULL re_II going_VVG ,_, to_TO like_VVI increase_VV0 the_AT distance_NN1 from_II the_AT ground_NN1 &lsqb;_( gestures_NN2 along_II lower-left_NN1 quadrant_NN1 of_IO wheel_NN1 &rsqb;_) ,_, so_CS it_PPH1 '_NULL s_ZZ1 probably_RR not_XX actually_RR a_AT1 straight_JJ line_NN1 ._. 
</s>
<s>
The_AT difficulty_NN1 comes_VVZ from_II the_AT sign_NN1 ,_, and_CC hence_RR we_PPIS2 focus_VV0 on_II the_AT case_NN1 |_NULL x_ZZ1 |_NULL =_FO 2_MC ,_, the_AT general_JJ one_MC1 following_VVG easily_RR ._. 
</s>
<s>
Let_VV0 @S_FO and_CC @S_FO ,_, uo_NNU >_FO 0_MC ,_, be_VBI given_VVN ._. 
</s>
<s>
We_PPIS2 next_MD use_NN1 Lemma_NN1 4.5_MC to_TO show_VVI that_CST the_AT expectation_NN1 of_IO the_AT solution_NN1 of_IO the_AT planar_JJ metric_JJ problem_NN1 is_VBZ approximately_RR affine_JJ far_JJ from_II the_AT boundary_NN1 plane_NN1 ,_, thereby_RR obtaining_VVG the_AT first_MD statement_NN1 of_IO Proposition_NN1 4.1_MC ._. 
</s>
<s>
To_II the_AT best_JJT of_IO our_APPGE knowledge_NN1 ,_, Theorem_NN1 1.1_MC and_CC its_APPGE generalization_NN1 in_II Corollary_NN1 3.8_MC below_RG respectively_RR are_VBR the_AT first_MD results_NN2 in_II the_AT literature_NN1 which_DDQ imply_VV0 exponential_NN1 integrability_NN1 properties_NN2 for_IF numerical_JJ approximations_NN2 of_IO the_AT stochastic_JJ Ginzburg-Landau_NP1 equation_NN1 in_II Subsection_NN1 4.2_MC ,_, for_IF numerical_JJ approximations_NN2 of_IO the_AT stochastic_JJ Lorenz_NP1 equation_NN1 with_IW additive_JJ noise_NN1 in_II Subsection_NN1 4.3_MC ,_, for_IF numerical_JJ approximations_NN2 of_IO the_AT stochastic_JJ van_NP1 der_NP1 Pol_NP1 oscillator_NN1 in_II Subsection_NN1 4.4_MC ,_, for_IF numerical_JJ approximations_NN2 of_IO the_AT stochastic_JJ Duffing-van_JJ der_FU Pol_NP1 oscillator_NN1 in_II Subsection_NN1 4.5_MC ,_, for_IF numerical_JJ approximations_NN2 of_IO the_AT model_NN1 from_II experimental_JJ psychology_NN1 in_II Subsection_NN1 4.6_MC ,_, for_IF numerical_JJ approximations_NN2 of_IO the_AT stochastic_JJ SIR_NN1 model_NN1 in_II Subsection_NN1 4.7_MC ,_, or_CC -_- under_II additional_JJ assumptions_NN2 on_II the_AT model_NN1 -_- for_IF numerical_JJ approximations_NN2 of_IO the_AT Langevin_NP1 dynamics_NN in_II Subsection_NN1 4.8_MC ._. 
</s>
<s>
Similarly_RR ,_, from_II Lemma_NN1 5.5_MC we_PPIS2 obtain_VV0 @F_FO ._. 
</s>
<s>
Moreover_RR ,_, @F_FO and_CC for_IF @S_FO denoting_VVG the_AT entry_NN1 of_IO An_AT1 in_II row_NN1 j_ZZ1 and_CC column_NN1 k_ZZ1 @F_FO ._. 
</s>
<s>
We_PPIS2 also_RR perform_VV0 a_AT1 convergence_NN1 analysis_NN1 at_II the_AT discrete_JJ level_NN1 and_CC the_AT effect_NN1 of_IO temporal_JJ discretizations_NN2 is_VBZ explored_VVN ._. 
</s>
<s>
Introducing_VVG @S_FO (_( @S_FO )_) and_CC using_VVG a_AT1 triangle_NN1 inequality_NN1 ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Similarly_RR ,_, introducing_VVG @S_FO and_CC using_VVG the_AT triangle_NN1 inequality_NN1 and_CC the_AT definition_NN1 of_IO @S_FO ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
An_AT1 estimate_NN1 on_II @S_FO as_CSA in_II Theorem_NN1 2.12_MC therefore_RR also_RR yields_VVZ an_AT1 estimate_NN1 on_II @S_FO and_CC @S_FO ,_, modulo_NN1 the_AT additional_JJ interpolation_NN1 errors_NN2 @S_FO and_CC @S_FO ._. 
</s>
<s>
If_CS @S_FO has_VHZ function_NN1 and_CC gradient_NN1 reconstructions_NN2 that_CST are_VBR piecewise_JJ polynomial_NN1 of_IO high-order_NN1 ,_, these_DD2 interpolation_NN1 errors_NN2 can_VM be_VBI expected_VVN to_TO have_VHI a_AT1 high_JJ rate_NN1 of_IO convergence_NN1 with_II31 respect_II32 to_II33 the_AT mesh_NN1 size_NN1 ._. 
</s>
<s>
Anyway_RR ,_, these_DD2 values_NN2 will_VM finally_RR depend_VVI only_RR on_II n_ZZ1 ,_, p_ZZ1 ,_, v_ZZ1 ,_, L_ZZ1 ;_; see_VV0 Remark_NN1 4_MC below_RL ._. 
</s>
<s>
We_PPIS2 start_VV0 with_IW the_AT definition_NN1 of_IO the_AT optimal_JJ value_NN1 of_IO X._NP1 This_DD1 is_VBZ done_VDN via_II the_AT mutual_JJ information_NN1 :_: We_PPIS2 start_VV0 by_II computing_VVG the_AT two-orbital_JJ entropy_NN1 s_ZZ1 (_( i_ZZ1 ,_, j_ZZ1 )_) ._. 
</s>
<s>
At_RR21 least_RR22 four_MC competing_JJ explanation_NN1 hypotheses_NN2 can_VM be_VBI formulated_VVN to_TO explain_VVI the_AT non-significance_NN1 of_IO differences_NN2 and_CC need_VVI further_JJR investigation_NN1 :_: (_( 1_MC1 )_) Sample_NN1 sizes_NN2 too_RG small_JJ :_: There_EX is_VBZ a_AT1 difference_NN1 with_IW low_JJ effect_NN1 size_NN1 which_DDQ may_VM become_VVI more_RGR visible_JJ with_IW larger_JJR sample_NN1 sizes_NN2 ._. 
</s>
<s>
We_PPIS2 obtain_VV0 a_AT1 good_JJ match_NN1 to_II the_AT theoretical_JJ results_NN2 ,_, i.e._REX ,_, the_AT RMSE_NN1 for_IF choosing_VVG the_AT prior_JJ measure_NN1 as_CSA importance_NN1 distribution_NN1 behaves_VVZ like_II @S_FO in_II accordance_NN1 to_II Lemma_NN1 2_MC ._. 
</s>
<s>
When_CS f_ZZ1 is_VBZ instead_RR lower_JJR semicontinuous_JJ ,_, the_AT existence_NN1 of_IO the_AT optimal_JJ equilibrium_NN1 still_RR holds_VVZ ._. 
</s>
<s>
Proof_VV0 It_PPH1 is_VBZ clear_JJ that_CST the_AT continuities_NN2 (_( 41_MC )_) hold_VV0 ._. 
</s>
<s>
We_PPIS2 notice_VV0 in_II our_APPGE simulation_NN1 that_CST if_CS the_AT obstacle_NN1 is_VBZ sufficiently_RR small_JJ it_PPH1 does_VDZ not_XX visibly_RR affect_VVI the_AT flux_NN1 ,_, so_CS we_PPIS2 consider_VV0 now_RT the_AT case_NN1 of_IO a_AT1 squared_JJ obstacle_NN1 with_IW side_NN1 41_MC ._. 
</s>
<s>
If_CS the_AT optimal_JJ decay_NN1 were_VBDR indeed_RR exponential_NN1 ,_, we_PPIS2 would_VM then_RT deduce_VVI that_DD1 for_IF s_ZZ1 >_FO 1_MC1 self-similarity_NN1 is_VBZ too_RG restrictive_JJ and_CC only_RR allows_VVZ for_IF suboptimal_JJ decay_NN1 rates_NN2 ._. 
</s>
<s>
The_AT spacing_NN1 between_II horizontal_JJ slats_NN2 is_VBZ R/2_FU ,_, and_CC so_RR the_AT number_NN1 of_IO horizontal_JJ slats_NN2 that_CST intersect_VV0 the_AT ball_NN1 B_ZZ1 (_( x_ZZ1 ,_, r_ZZ1 )_) is_VBZ at_RR21 most_RR22 @F_FO ._. 
</s>
<s>
Each_DD1 horizontal_JJ slat_NN1 intersects_VVZ B_ZZ1 (_( x_ZZ1 ,_, r_ZZ1 )_) in_RP at_II most_DAT AR1-boxes_NN2 ._. 
</s>
<s>
Let_VV0 y_ZZ1 '_NULL be_VBI the_AT point_NN1 on_II this_DD1 geodesic_JJ ray_NN1 such_CS21 that_CS22 @S_FO ._. 
</s>
<s>
We_PPIS2 first_MD claim_NN1 that_CST @F_FO ._. 
</s>
<s>
Indeed_RR ,_, it_PPH1 is_VBZ clear_JJ that_CST @F_FO ._. 
</s>
<s>
Le@S_FO ,_, and_CC le@S_FO ._. 
</s>
<s>
As_CSA @S_FO is_VBZ adjacent_II21 to_II22 u_ZZ1 ,_, the_AT sector_NN1 @S_FO contains_VVZ the_AT geodesic_JJ ra@S_FO ._. 
</s>
<s>
The_AT quantity_NN1 @F_FO makes_VVZ perfect_JJ sense_NN1 because_CS all_DB terms_NN2 of_IO the_AT form_NN1 @S_FO are_VBR integrated_VVN against_II smooth_JJ functions_NN2 of_IO xi_NN1 ,_, xj_NNU with_IW sufficient_JJ decay_NN1 ._. 
</s>
<s>
Certainly_RR n_ZZ1 o_ZZ1 f_ZZ1 sends_VVZ the_AT set_NN1 Cai_NN1 into_II the_AT open_JJ set_NN1 Zai_NN1 ,_, but_CCB in_II fact_NN1 the_AT following_JJ stronger_JJR property_NN1 is_VBZ true_JJ ._. 
</s>
<s>
The_AT function_NN1 @S_FO is_VBZ continuous_JJ in_II t_ZZ1 ._. 
</s>
<s>
Many_DA2 available_JJ curricula_NN2 reflect_VV0 years_NNT2 of_IO efforts_NN2 from_II a_AT1 consortium_NN1 of_IO teachers_NN2 ,_, mathematics_NN1 educators_NN2 ,_, and_CC mathematicians_NN2 ;_; how_RRQ is_VBZ it_PPH1 that_DD1 prospective_JJ teachers_NN2 draw_VV0 upon_II the_AT result_NN1 of_IO those_DD2 efforts_NN2 mathematics_NN1 curriculumas_NN2 they_PPHS2 begin_VV0 to_TO design_VVI their_APPGE own_DA instruction_NN1 to_TO support_VVI children_NN2 '_NULL s_ZZ1 mathematical_JJ understanding_NN1 ?_? 
</s>
<s>
The_AT main_JJ goals_NN2 of_IO this_DD1 section_NN1 are_VBR twofold_RR :_: on_II the_AT one_MC1 hand_NN1 ,_, we_PPIS2 compare_VV0 the_AT finite_JJ sample_NN1 behavior_NN1 of_IO the_AT ML_NNU estimation_NN1 of_IO the_AT microergodic_JJ parameter_NN1 of_IO the_AT GW_NP1 model_NN1 with_IW the_AT asymptotic_JJ distributions_NN2 given_VVN in_II Theorems_NN2 8_MC and_CC 9_MC ._. 
</s>
<s>
For_IF all_DB extended_VVD polymatroid_JJ inequalities_NN2 @S_FO with_II31 regard_II32 to_II33 bin_NN1 i_MC1 for_IF all_DB @S_FO ,_, inequality_NN1 @F_FO is_VBZ valid_JJ for_IF the_AT DCBP_NP1 formulation_NN1 ._. 
</s>
<s>
While_CS our_APPGE focus_NN1 in_II this_DD1 work_NN1 is_VBZ on_II undirected_JJ (_( as_II31 opposed_II32 to_II33 directed_JJ )_) configuration_NN1 models_NN2 ,_, directed_VVD configuration_NN1 models_NN2 are_VBR discussed_VVN briefly_RR in_II section_NN1 3.2_MC ._. 
</s>
<s>
For_IF any_DD composition_NN1 a_AT1 we_PPIS2 have_VH0 @F_FO ,_, where_CS @S_FO is_VBZ the_AT right-hand_JJ side_NN1 of_IO (_( 2.2_MC )_) ._. 
</s>
<s>
Insight_NN1 of_IO this_DD1 sort_NN1 may_VM lead_VVI to_II improved_JJ generation_NN1 or_CC detection_NN1 of_IO synthetic_JJ fingerprints_NN2 ._. 
</s>
<s>
On_II the_AT last_MD tab_NN1 ,_, there_EX is_VBZ one_MC1 button_NN1 ,_, Pump_NN1 Oil_NN1 ._. 
</s>
<s>
Some_DD of_IO the_AT univariate_NN1 change_NN1 point_NN1 methodologies_NN2 have_VH0 been_VBN extended_VVN to_II multivariate_JJ set_NN1 tings_NN2 ._. 
</s>
<s>
Let_VV0 n_ZZ1 be_VBI the_AT smallest_JJT natural_JJ number_NN1 such_CS21 that_CS22 @S_FO ._. 
</s>
<s>
Consider_VV0 the_AT two_MC projections_NN2 @S_FO and_CC the_AT associated_JJ fibre_NN1 sequence_NN1 of_IO relative_JJ K-theory_JJ spectra_NN2 @F_FO ._. 
</s>
<s>
In_II the_AT kinetic_JJ context_NN1 ,_, the_AT distribution_NN1 function_NN1 for_IF the_AT tumor_NN1 cells_NN2 is_VBZ a_AT1 mesoscopic_JJ quantity_NN1 depending_VVG not_XX only_RR on_II time_NNT1 and_CC position_NN1 ,_, but_CCB also_RR on_II the_AT cell_NN1 velocity_NN1 and_CC the_AT activity_NN1 variables_NN2 mentioned_VVN above_RL ._. 
</s>
<s>
Replacing_VVG Q_ZZ1 by_II some_DD power_NN1 allows_VVZ to_TO remove_VVI the_AT factor_NN1 2C_FO ._. 
</s>
<s>
Group_NN1 IV_MC started_VVD with_IW a_AT1 visual_JJ representation_NN1 in_II a_AT1 table_NN1 (_( Figure_NN1 5_MC ,_, left_JJ )_) ,_, carrying_VVG out_RP a_AT1 treatment_NN1 without_IW leaving_VVG the_AT visual_JJ system_NN1 ._. 
</s>
<s>
Then_RT they_PPHS2 satisfy_VV0 Assumptions_NN2 2.1_MC and_CC 2.3_MC ,_, respectively_RR ._. 
</s>
<s>
In_II that_DD1 case_NN1 the_AT product_NN1 on_II the_AT right_JJ hand_NN1 side_NN1 of_IO (_( 2.9_MC )_) vanishes_VVZ as_CSA @S_FO with_IW increasing_JJ volume_NN1 |_NULL A_ZZ1 |_NULL ,_, and_CC we_PPIS2 get_VV0 @F_FO ._. 
</s>
<s>
See_VV0 also_RR the_AT discussion_NN1 in_II the_AT paragraph_NN1 preceding_VVG &lsqb;_( 22_MC ,_, Theorem_NN1 9_MC &rsqb;_) ._. 
</s>
<s>
We_PPIS2 revisit_VV0 the_AT proof_NN1 of_IO sublinear_JJ convergence_NN1 in_II section_NN1 D.2_FO ,_, noting_VVG that_CST if_CS @S_FO ,_, the_AT corollary_NN1 is_VBZ immediate_JJ ,_, so_CS we_PPIS2 need_VM consider_VVI only_RR the_AT case_NN1 that_CST @S_FO ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI the_AT vector_NN1 (_( 41_MC )_) that_DD1 Lemma_NN1 D.2_FO guarantees_VVZ ,_, and_CC let_VV0 @S_FO be_VBI the_AT vector_NN1 (_( 40_MC )_) that_DD1 Lemma_NN1 D.3_FO guarantees_VVZ ._. 
</s>
<s>
Finally_RR ,_, for_IF a_AT1 general_JJ function_NN1 @S_FO ,_, we_PPIS2 define_VV0 :_: @F_FO ._. 
</s>
<s>
Acknowledgments_NN2 The_AT first_MD author_NN1 was_VBDZ partially_RR supported_VVN through_II NSF_NP1 grant_NN1 1551514_MC ._. 
</s>
<s>
The_AT family_NN1 of_IO quasi-modes_NN2 @S_FO and_CC the_AT family_NN1 of_IO quasi-modes_NN2 @S_FO are_VBR defined_VVN by_II :_: for_IF @S_FO ,_, and_CC for_IF @S_FO :_: @F_FO ._. 
</s>
<s>
Here_RL ,_, we_PPIS2 assume_VV0 there_EX is_VBZ a_AT1 lower_JJR bound_NN1 Ln_NP1 in_II (_( 3.2a_FO )_) to_TO avoid_VVI unboundedness_NN1 of_IO the_AT forward_JJ problem_NN1 ._. 
</s>
<s>
We_PPIS2 equip_VV0 X_ZZ1 in_II (_( H1_FO )_) with_IW the_AT partial_JJ order_NN1 <_FO for_IF which_DDQ x_ZZ1 <_FO y_ZZ1 if_CS and_CC only_RR if_CS x(z)_NNU <_FO y(z)_VV0 for_IF all_DB @S_FO ._. 
</s>
<s>
For_IF existence_NN1 theorems_NN2 ,_, which_DDQ can_VM be_VBI used_VVN to_TO verify_VVI (_( H1_FO )_) ,_, we_PPIS2 refer_VV0 to_II &lsqb;_( 12_MC ,_, Chapters_NN2 3_MC ,_, 7_MC &rsqb;_) ._. 
</s>
<s>
Finally_RR ,_, the_AT estimator_NN1 of_IO @S_FO does_VDZ not_XX require_VVI selection_NN1 of_IO tuning_VVG parameters_NN2 ,_, which_DDQ is_VBZ in_II contrast_NN1 to_II other_JJ variable_NN1 selection_NN1 procedures_NN2 like_II the_AT Lasso_NP1 (_( Tibshirani_NP1 ,_, 1996_MC )_) which_DDQ typically_RR uses_VVZ cross-validation_JJ to_TO select_VVI the_AT tuning_NN1 parameters_NN2 (_( Hastie_NP1 et_RA21 al._RA22 ,_, 2016_MC )_) ;_; all_DB the_AT components_NN2 of_IO our_APPGE threshold_NN1 in_II equation_NN1 (_( 5_MC )_) arepre-determined_NN1 from_II the_AT inputs_NN2 provided_VVN in_II Section_NN1 3.1_MC ._. 
</s>
<s>
Choose_VV0 an_AT1 integer_NN1 i_ZZ1 uniformly_RR at_RR21 random_RR22 from_II @S_FO ._. 
</s>
<s>
Let_VV0 U_ZZ1 be_VBI a_AT1 (_( disk_NN1 )_) neighborhood_NN1 of_IO x_ZZ1 which_DDQ contains_VVZ no_AT other_JJ point_NN1 of_IO F._NP1 According_II21 to_II22 Lemma_NN1 3.9_MC ,_, f_ZZ1 fixes_VVZ x_ZZ1 ,_, so_CS we_PPIS2 can_VM choose_VVI a_AT1 smaller_JJR disk_NN1 neighborhood_NN1 V_ZZ1 of_IO x_ZZ1 such_CS21 that_CS22 @S_FO ._. 
</s>
<s>
Let_VV0 @S_FO ._. 
</s>
<s>
We_PPIS2 are_VBR looking_VVG for_IF a_AT1 curve_NN1 belonging_VVG to_II Cz_NP1 and_CC included_VVN in_II U._NP1 Let_VV0 a_AT1 be_VBI a_AT1 simple_JJ closed_JJ curve_NN1 around_RG x_MC in_II V_ZZ1 x_ZZ1 based_VVN at_II z_ZZ1 ,_, let_VV0 @S_FO be_VBI a_AT1 curve_NN1 from_II z_ZZ1 to_II f_ZZ1 (_( z_ZZ1 )_) in_II U_ZZ1 ,_, and_CC y_ZZ1 be_VBI any_DD curve_NN1 in_II Cz_NP1 ._. 
</s>
<s>
We_PPIS2 would_VM like_VVI to_TO thank_VVI Alexander_NP1 Stolyar_NP1 for_IF helpful_JJ discussions_NN2 on_II stochastic_JJ processes_NN2 ._. 
</s>
<s>
The_AT former_DA "_" refers_VVZ to_II using_VVG representations_NN2 of_IO mathematics_NN1 to_TO communicate_VVI mathematical_JJ concepts_NN2 or_CC ideas_NN2 ,_, "_" whereas_CS the_AT latter_DA "_" links_NN2 mathematics_NN1 and_CC authentic_JJ real-world_JJ questions_NN2 "_" (_( Cirillo_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
2016_MC )_) ._. 
</s>
<s>
It_PPH1 follows_VVZ as_CSA in_CS21 Case_CS22 1(b)_FO that_DD1 is_VBZ invariant_JJ under_II U2_NP1 and_CC hence_RR invariant_JJ under_II U_JJ β_NULL for_IF every_AT1 positive_JJ root_NN1 β_NULL ∈+_FO ._. 
</s>
<s>
There_RL exist_VV0 several_DA2 methods_NN2 for_IF the_AT Monge-Ampeere_JJ equation_NN1 ._. 
</s>
<s>
Error_NN1 bars_NN2 for_IF the_AT EM_FU and_CC limit_VV0 methods_NN2 are_VBR comparable_JJ to_II symbol_NN1 sizes_NN2 and_CC thus_RR not_XX presented_VVN ._. 
</s>
<s>
Compared_VVN with_IW the_AT symmetric_JJ Nitsche_NN1 method_NN1 ,_, the_AT non-symmetric_JJ Nitsche_NN1 method_NN1 does_VDZ not_XX require_VVI additional_JJ stabilization_NN1 and_CC therefore_RR does_VDZ not_XX depend_VVI on_II the_AT penalty_NN1 stabilization_NN1 parameter_NN1 ._. 
</s>
<s>
Moreover_RR ,_, the_AT growth_NN1 rates_NN2 of_IO the_AT penalty_NN1 parameters_NN2 induced_VVN by_II this_DD1 scheme_NN1 are_VBR adaptive_JJ ,_, without_IW involving_VVG any_DD prior_JJ information_NN1 on_II the_AT active_JJ set_NN1 S._NP1 Moreover_RR ,_, @F_FO ._. 
</s>
<s>
Consequently_RR ,_, we_PPIS2 derive_VV0 from_II the_AT SMHD_NP1 model_NN1 that_CST (_( n_ZZ1 u_ZZ1 ,_, B_ZZ1 ,_, q_ZZ1 )_) satisfies_VVZ the_AT following_JJ initial-boundary_JJ value_NN1 problem_NN1 with_IW an_AT1 internal_JJ interface_NN1 :_: @F_FO ._. 
</s>
<s>
Given_VVN a_AT1 filtration_NN1 F_ZZ1 of_IO R_ZZ1 ,_, set_VV0 @F_FO for_IF @S_FO ._. 
</s>
<s>
We_PPIS2 set_VV0 @S_FO for_IF @S_FO ._. 
</s>
<s>
The_AT ideal_JJ bp(F)_NNU is_VBZ well_RR defined_VVN ,_, and_CC b_ZZ1 ._. 
</s>
<s>
(_( F_ZZ1 )_) is_VBZ a_AT1 graded_JJ sequence_NN1 of_IO ideals_NN2 assuming_VVG F_ZZ1 is_VBZ non-trivial_JJ &lsqb;_( BJ17_FO ,_, 3.17-3.18_MCMC &rsqb;_) ._. 
</s>
<s>
By_II the_AT same_DA argument_NN1 as_CSA in_II the_AT first_MD comment_NN1 below_II the_AT proof_NN1 of_IO Theorem_NN1 3_MC ,_, we_PPIS2 may_VM assume_VVI that_CST both_RR Fn_NP1 and_CC Gn_NNU are_VBR orientationpreserving_VVG for_IF every_AT1 n_ZZ1 ._. 
</s>
<s>
To_TO overcome_VVI this_DD1 issue_NN1 ,_, we_PPIS2 consider_VV0 to_TO be_VBI the_AT FE_NP1 approximation_NN1 of_IO degree_NN1 @S_FO of_IO problem_NN1 (_( 5.4_MC )_) for_IF a_AT1 given_JJ triangulation_NN1 ._. 
</s>
<s>
For_IF future_JJ use_NN1 let_VV0 us_PPIO2 state_VVI an_AT1 auxiliary_JJ lemma_NN1 ._. 
</s>
<s>
Then_RT for_IF every_AT1 >0_FO and_CC for_IF all_DB sufficiently_RR large_JJ n_ZZ1 we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Extensions_NN2 of_IO the_AT randomized_JJ primal-dual_JJ gradient_NN1 method_NN1 to_II non-strongly_RR convex_JJ ,_, nonsmooth_NN1 ,_, and_CC unbounded_JJ problems_NN2 are_VBR also_RR discussed_VVN in_II this_DD1 paper_NN1 ._. 
</s>
<s>
Using_VVG open_JJ coding_NN1 on_II his_APPGE Redback_NN1 to_II 28_MC PTs_NN2 on_II narrative_JJ portfolios_NN2 and_CC critical_JJ incident_NN1 portfolios_NN2 ._. 
</s>
<s>
The_AT inequalities_NN2 (_( 37b_FO )_) and_CC (_( 37c_FO )_) follow_VV0 from_II Proposition_NN1 13_MC ._. 
</s>
<s>
In_II this_DD1 study_NN1 ,_, we_PPIS2 were_VBDR unable_JK to_TO provide_VVI partial_JJ credits_NN2 for_IF the_AT open-ended_JJ items_NN2 ._. 
</s>
<s>
For_IF any_DD i_ZZ1 =_FO 1_MC1 ,_, ..._... ,_, 
</s>
<s>
M_ZZ1 ,_, @S_FO and_CC we_PPIS2 can_VM also_RR suppose_VVI that_DD1 ,_, as_CSA @S_FO converges_VVZ to_II @S_FO uniformly_RR in_II @S_FO ._. 
</s>
<s>
Clearly_RR ,_, for_IF any_DD i_ZZ1 =_FO 1_MC1 ,_, ..._... ,_, 
</s>
<s>
M_ZZ1 ,_, @S_FO and_CC @S_FO belong_VV0 to_II @S_FO and_CC are_VBR Lipschitz_NP1 with_IW Lipschitz_NP1 constant_NN1 bounded_VVN by_II L._NP1 In_II this_DD1 section_NN1 we_PPIS2 perform_VV0 some_DD numerical_JJ simulations_NN2 of_IO system_NN1 (_( 1_MC1 )_) using_VVG our_APPGE scheme_NN1 ,_, Eqs_NN2 ._. 
</s>
<s>
With_IW (_( 3.5_MC )_) ,_, for_IF r_ZZ1 =_FO 1_MC1 the_AT corresponding_JJ steady-state_JJ diffusion_NN1 equation_NN1 with_IW homogeneous_JJ boundary_NN1 conditions_NN2 becomes_VVZ @F_FO ._. 
</s>
<s>
We_PPIS2 need_VV0 to_TO show_VVI that_CST @S_FO for_IF any_DD partition_NN1 A._NP1 We_PPIS2 will_VM do_VDI it_PPH1 by_II showing_VVG that_CST @F_FO ,_, where_CS @S_FO is_VBZ the_AT partition_NN1 obtained_VVN by_II prepending_VVG to_II A_ZZ1 a_AT1 first_MD row_NN1 of_IO length_NN1 N_ZZ1 (_( for_IF sufficiently_RR large_JJ N_ZZ1 )_) ._. 
</s>
<s>
Proof_VV0 The_AT construction_NN1 of_IO Section_NN1 4.1_MC yields_VVZ the_AT bounds_NN2 (_( 4.34_MC )_) and_CC the_AT uniqueness_NN1 up_II21 to_II22 @S_FO of_IO the_AT coefficient_NN1 functions_NN2 ._. 
</s>
<s>
Moreover_RR ,_, in_II each_DD1 of_IO B+_FO and_CC B-_NN1 the_AT function_NN1 @S_FO is_VBZ either_RR constant_JJ or_CC we_PPIS2 have_VH0 @S_FO ._. 
</s>
<s>
The_AT following_JJ example_NN1 is_VBZ a_AT1 trivial_JJ example_NN1 for_IF this_DD1 phenomenon_NN1 ._. 
</s>
<s>
As_CSA @S_FO is_VBZ submodular_JJ in_II y_ZZ1 ,_, it_PPH1 suffices_VVZ to_TO prove_VVI that_CST @S_FO is_VBZ submodular_JJ ._. 
</s>
<s>
Our_APPGE assumption_NN1 is_VBZ more_RGR strict_JJ ,_, but_CCB does_VDZ not_XX influence_VVI anything_PN1 in_II the_AT analysis_NN1 of_IO the_AT equations_NN2 we_PPIS2 consider_VV0 ._. 
</s>
<s>
As_II the_AT measures_NN2 are_VBR derived_VVN from_II examining_VVG the_AT residuals_NN2 following_VVG for_REX21 example_REX22 a_AT1 lasso_NN1 ?_? 
</s>
<s>
The_AT statement_NN1 of_IO Theorem_NN1 4.13_MC remains_VVZ valid_JJ in_II the_AT setting_NN1 of_IO Theorem_NN1 4.4_MC if_CS one_MC1 replaces_VVZ a_AT1 and_CC s_ZZ1 by_II @S_FO and_CC @S_FO ,_, respectively_RR ._. 
</s>
<s>
This_DD1 is_VBZ why_RRQ it_PPH1 is_VBZ useful_JJ that_CST these_DD2 misconceptions_NN2 are_VBR identified_VVN by_II trainers_NN2 ,_, in_BCL21 order_BCL22 to_TO intervene_VVI effectively_RR in_II the_AT area_NN1 of_IO the_AT pre-service_JJ teachers_NN2 '_NULL proximal_JJ professional_JJ development_NN1 ._. 
</s>
<s>
We_PPIS2 start_VV0 with_IW the_AT case_NN1 @S_FO ._. 
</s>
<s>
First_MD note_NN1 that_CST (_( 2.2_MC )_) implies_VVZ @S_FO ,_, almost_RR everywhere_RL on_II @S_FO ._. 
</s>
<s>
Now_RT we_PPIS2 can_VM write_VVI B_ZZ1 as_II A/Ti_FU for_IF some_DD ideal_JJ I_ZZ1 c_ZZ1 Oe_NP1 ,_, where_CS Ti_NP1 is_VBZ the_AT kernel_NN1 of_IO I_ZZ1 acting_VVG on_II A._NNU Since_CS A_ZZ1 with_IW its_APPGE endomorphisms_NN2 is_VBZ defined_VVN over_II Q(A)0_FO ,_, it_PPH1 follows_VVZ that_CST B_ZZ1 can_VM be_VBI defined_VVN over_II Q(A)0_FO as_RR21 well_RR22 ,_, so_CS Q(B)_FO c_ZZ1 Q(A)0_FO ._. 
</s>
<s>
We_PPIS2 note_VV0 that_CST it_PPH1 holds_VVZ that_CST for_IF all_DB @S_FO ,_, p_ZZ1 >_FO 1_MC1 ,_, @S_FO since_CS the_AT weight_NN1 @S_FO (_( Theorem_NN1 3.9_MC )_) is_VBZ constructed_VVN by_II "_" 1+polynomials_FO of_IO Brownian_JJ motions_NN2 ._. "_" 
</s>
<s>
More_RGR precisely_RR ,_, for_IF every_AT1 v_ZZ1 >_FO 0_MC and_CC every_AT1 k_ZZ1 G_ZZ1 N_ZZ1 ,_, there_EX exists_VVZ a_AT1 map_NN1 @S_FO ,_, such_CS21 that_CS22 for_IF all_DB @S_FO ,_, @F._FO @S_FO such_CS21 that_CS22 for_IF @S_FO one_MC1 has_VHZ @F_FO ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI a_AT1 bijection_NN1 ._. 
</s>
<s>
These_DD2 are_VBR listed_VVN in_II Table_NN1 1_MC1 ._. 
</s>
<s>
Here_RL ,_, @S_FO denotes_VVZ a_AT1 @S_FO matrix_NN1 of_IO Is_VBZ ,_, ,_, the_AT functions_NN2 Sum_NN1 and_CC ColumnSum_NP1 take_VV0 the_AT sums_NN2 of_IO all_DB elements_NN2 in_II a_AT1 matrix_NN1 and_CC its_APPGE columns_NN2 respectively_RR and_CC an_AT1 asterisk_NN1 denotes_VVZ the_AT elementwise_JJ mul_NN1 tiplication_NN1 operator_NN1 ._. 
</s>
<s>
Consider_VV0 the_AT following_JJ contrast_NN1 in_II Dan_NP1 '_NULL s_ZZ1 thinking_VVG on_II the_AT pre-test_NN1 and_CC pre-interview_JJ assessment_NN1 ._. 
</s>
<s>
We_PPIS2 can_VM write_VVI @S._FO @F_FO ._. 
</s>
<s>
For_IF finite_JJ element_NN1 nodes_NN2 which_DDQ lie_VV0 inside_II an_AT1 element_NN1 T_ZZ1 ,_, i.e._REX ,_, @S_FO ,_, we_PPIS2 have_VH0 @S_FO ._. 
</s>
<s>
For_IF @S_FO we_PPIS2 use_VV0 the_AT definition_NN1 of_IO @S_FO and_CC Lemma_NN1 3.4_MC and_CC thus_RR obtain_VV0 @F_FO ._. 
</s>
<s>
Let_VV0 z_ZZ1 N_ZZ1 (_( 0_MC ,_, Ipxp_NP1 )_) and_CC let_VV0 X_ZZ1 g_ZZ1 Rpxp_NP1 be_VBI a_AT1 positive_JJ semidefinite_NN1 matrix_NN1 with_IW maxj=1_FO ,_, p_ZZ1 ?jj_FO <_FO 1_MC1 ._. 
</s>
<s>
The_AT reduced_JJ basis_NN1 is_VBZ then_RT derived_VVN from_II the_AT generators_NN2 of_IO the_AT Voronoi_JJ clustering_NN1 ._. 
</s>
<s>
A_AT1 standard_JJ argument_NN1 (_( see_VV0 ,_, e.g._REX ,_, Nualart_NP1 and_CC Pardoux_NP1 &lsqb;_( 24_MC &rsqb;_) )_) shows_VVZ that_CST every_AT1 process_NN1 @S_FO is_VBZ a_AT1 strong_JJ Markov_NP1 process_NN1 with_II31 respect_II32 to_II33 F._NP1 In_RR21 addition_RR22 ,_, Ssnal_NP1 can_VM solve_VVI the_AT instance_NN1 pyrim5_FO in_II 9_MC seconds_NNT2 while_CS ADMM_NP1 reaches_VVZ the_AT maximum_NN1 of_IO 20000_MC iterations_NN2 and_CC consumes_VVZ about_RG 2_MC hours_NNT2 but_CCB only_RR produces_VVZ a_AT1 rather_RG inaccurate_JJ solution_NN1 ._. 
</s>
<s>
It_PPH1 is_VBZ known_VVN that_CST on_II a_AT1 bounded_JJ ,_, smooth_JJ ,_, strictly_RR pseudoconvex_VV0 domain_NN1 in_II Cn_NP1 ,_, all_DB four_MC classical_JJ invariant_JJ metrics_NN2 are_VBR uniformly_RR equivalent_JJ to_II each_PPX221 other_PPX222 (_( see_VV0 ,_, for_REX21 example_REX22 ,_, &lsqb;_( Die70_FO ,_, Gra75_FO ,_, CY80_FO ,_, Lem81_FO ,_, BFG83_FO ,_, Wu93_FO &rsqb;_) and_CC the_AT references_NN2 therein_RR )_) ._. 
</s>
<s>
The_AT multifidelity_NN1 importance_NN1 sampling_NN1 approach_NN1 introduced_VVN in_II &lsqb;_( 160_MC &rsqb;_) uses_VVZ a_AT1 low-fidelity_JJ model_NN1 to_TO construct_VVI the_AT biasing_JJ distribution_NN1 in_II the_AT first_MD step_NN1 of_IO importance_NN1 sampling_NN1 and_CC derives_VVZ the_AT statistics_NN using_VVG high-fidelity_JJ model_NN1 evaluations_NN2 in_II step_NN1 2_MC ._. 
</s>
<s>
This_DD1 problem_NN1 has_VHZ been_VBN actively_RR studied_VVN ._. 
</s>
<s>
This_DD1 slightly_RR surprising_JJ condition_NN1 is_VBZ equivalent_JJ to_II the_AT complete_JJ interaction_NN1 being_VBG maximally_RR single-trace_JJ ._. 
</s>
<s>
The_AT other_JJ step_NN1 size_NN1 parameters_NN2 are_VBR chosen_VVN as_CSA @L_FO ._. 
</s>
<s>
Let_VV0 u0_FO be_VBI the_AT initial_JJ datum_NN1 in_II (_( 59_MC )_) and_CC let_VV0 uj_NN1 be_VBI the_AT solution_NN1 to_II the_AT linear_JJ parabolic_JJ equation_NN1 @F_FO ,_, where_CS @S_FO ._. 
</s>
<s>
Note_VV0 that_CST the_AT existence_NN1 of_IO solutions_NN2 to_II the_AT above_JJ linear_JJ initial-boundary_JJ problem_NN1 is_VBZ guaranteed_VVN by_II using_VVG the_AT heat_NN1 kernel_NN1 @F_FO ._. 
</s>
<s>
Indeed_RR ,_, define_VV0 two_MC heat_NN1 operators_NN2 M1_FO and_CC M2_FO by_II @F_FO ._. 
</s>
<s>
Then_RT @F_FO ._. 
</s>
<s>
Step_VV0 (_( ii_MC )_) ._. 
</s>
<s>
Consequently_RR ,_, T_ZZ1 (_( f_ZZ1 )_) g_ZZ1 belongs_VVZ to_II L2(R)_FO ._. 
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<s>
The_AT proof_NN1 also_RR makes_VVZ use_NN1 of_IO a_AT1 precise_JJ quantitative_JJ form_NN1 of_IO non-divergence_NN1 of_IO unipotent_JJ orbits_NN2 by_II Kleinbock_NP1 and_CC Margulis_NP1 ,_, and_CC an_AT1 extension_NN1 by_II de_FW la_FW Salle_NP1 of_IO strong_JJ property_NN1 (_( T_ZZ1 )_) to_II representations_NN2 of_IO nonuniform_JJ lattices_NN2 ._. 
</s>
<s>
We_PPIS2 henceforth_RT assume_VV0 that_CST XY_FO is_VBZ smooth_JJ over_II Zp_NP1 ._. 
</s>
<s>
We_PPIS2 claim_VV0 that_CST ,_, for_IF @S_FO ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
We_PPIS2 are_VBR now_RT ready_JJ to_II upper_JJ bound_VVD the_AT number_NN1 of_IO successful_JJ iterations_NN2 of_IO Algorithm_NN1 2.1_MC until_CS termination_NN1 ._. 
</s>
<s>
In_II fact_NN1 @F_FO ._. 
</s>
<s>
The_AT first_MD term_NN1 of_RR21 course_RR22 vanishes_VVZ for_IF @S_FO ._. 
</s>
<s>
The_AT second_MD term_NN1 satisfies_VVZ the_AT inequality_NN1 @F_FO ._. 
</s>
<s>
Averaged_JJ outgoing_JJ flux_NN1 vs._II number_NN1 of_IO pedestrians_NN2 ._. 
</s>
<s>
It_PPH1 has_VHZ been_VBN proposed_VVN ,_, based_VVN on_II heuristic_JJ arguments_NN2 and_CC simulations_NN2 ,_, that_DD1 "_" activity-weighted_NN1 "_" spanning_VVG trees_NN2 should_VM have_VHI SLE_NN1 scaling_NN1 limits_VVZ with_IW k_ZZ1 anywhere_RL in_II the_AT range_NN1 &lsqb;_( 4/3,4_MF )_) and_CC k_ZZ1 '_NULL anywhere_RL in_II the_AT range_NN1 (_( 4_MC ,_, 12_MC &rsqb;_) &lsqb;_( 36_MC &rsqb;_) ._. 
</s>
<s>
In_RR21 general_RR22 ,_, an_AT1 organic_JJ theory_NN1 of_IO transport_NN1 and_CC Jacobi_JJ fields_NN2 along_II abnormal_JJ geodesics_NN1 is_VBZ still_RR lacking_VVG ._. 
</s>
<s>
In_II §2.3_FO we_PPIS2 discuss_VV0 the_AT important_JJ notions_NN2 of_IO wall_NN1 trees_NN2 and_CC panel_NN1 trees_NN2 ._. 
</s>
<s>
Thus_RR ,_, our_APPGE main_JJ contribution_NN1 is_VBZ to_II (_( a_ZZ1 )_) generalize_VV0 the_AT error_NN1 estimate_NN1 (_( 1.8_MC )_) to_II the_AT case_NN1 of_IO the_AT fractional_JJ order_NN1 evolution_NN1 model_NN1 (_( 1.1_MC )_) and_CC (_( b_ZZ1 )_) provide_VV0 pointwise-in-time_JJ optimal_JJ L2(Q)-error_JJ estimate_NN1 for_IF the_AT time-stepping_JJ schemes_NN2 (_( 1.6_MC )_) and_CC (_( 1.7_MC )_) for_IF initial_JJ data_NN @S_FO with_IW @S_FO ;_; see_VV0 the_AT definition_NN1 of_IO the_AT dotted_JJ space_NN1 @S_FO below_RL ._. 
</s>
<s>
The_AT bar_NN1 involution_NN1 on_II U_ZZ1 extends_VVZ to_II a_AT1 bar_NN1 involution_NN1 ,_, again_RT denoted_VVD by_RP ,_, on_II U._NP1 While_CS observing_VVG and_CC analyzing_VVG processes_NN2 of_IO learning_NN1 we_PPIS2 do_VD0 keep_VVI in_II mind_NN1 that_CST what_DDQ is_VBZ happening_VVG covertly_RR inside_II a_AT1 person_NN1 is_VBZ of_IO utmost_JJ significance_NN1 to_II what_DDQ is_VBZ overtly_RR happening_VVG between_II people_NN ._. 
</s>
<s>
The_AT function_NN1 @S_FO is_VBZ subharmonic_JJ and_CC bounded_VVN above_RL on_II S._NP1 Then_RT (_( 2.24_MC )_) follows_VVZ from_II Perron_NP1 '_NULL s_ZZ1 construction_NN1 of_IO solutions_NN2 to_II the_AT Dirichlet_NN1 problem_NN1 via_II subharmonic_JJ functions_NN2 ;_; see_VV0 ,_, for_REX21 instance_REX22 ,_, &lsqb;_( Con95_FO ,_, §19.7_FO &rsqb;_) ._. 
</s>
<s>
In_RR21 addition_RR22 ,_, it_PPH1 is_VBZ easy_JJ to_TO check_VVI that_CST the_AT result_NN1 in_II Lemma_NN1 2_MC implies_VVZ that_CST f_ZZ1 lies_VVZ in_II the_AT subset_NN1 Fs_NP2 s_ZZ1 defined_VVN in_II (_( 3.7_MC )_) with_IW high_JJ probability_NN1 ._. 
</s>
<s>
Division_NN1 by_II the_AT SD_NP1 makes_VVZ the_AT measure_NN1 standardized_VVN (_( independent_NN1 of_IO a_AT1 unit_NN1 )_) ,_, which_DDQ allows_VVZ comparison_NN1 across_II studies_NN2 ._. 
</s>
<s>
In_RR21 addition_RR22 ,_, for_IF numbers_NN2 @S_FO ,_, @S_FO ,_, @S_FO ,_, a_AT1 set_NN1 @S_FO ,_, and_CC an_AT1 open_JJ and_CC convex_JJ set_NN1 @S_FO we_PPIS2 denote_VV0 by_II @S_FO in_II &lsqb;_( 18_MC &rsqb;_) )_) the_AT set_NN1 given_VVN by_II @F_FO ._. 
</s>
<s>
Next_MD let_VV0 @S_FO and_CC @S_FO be_VBI the_AT mappings_NN2 with_IW the_AT property_NN1 that_CST for_IF all_DB @S_FO it_PPH1 holds_VVZ that_CST @S_FO and_CC @S_FO ._. 
</s>
<s>
The_AT Chi-square_JJ test_NN1 confirmed_VVD that_CST the_AT findings_NN2 are_VBR not_XX statistically_RR significant_JJ (_( @S_FO ,_, p_ZZ1 =_FO 0.071_MC )_) ._. 
</s>
<s>
On_II the_AT other_JJ hand_NN1 ,_, a_AT1 frame_NN1 guarantees_VVZ at_RR21 least_RR22 one_MC1 approximation_NN1 with_IW small-norm_JJ coefficientsnamely_RR ,_, the_AT truncated_JJ canonical_JJ dual_JJ frame_NN1 expansionalthough_NN1 ,_, as_CSA seen_VVN above_RL ,_, better_JJR approximations_NN2 often_RR exist_VV0 ._. 
</s>
<s>
In_II31 line_II32 with_II33 this_DD1 expectation_NN1 ,_, for_IF low_JJ population_NN1 size_NN1 ,_, the_AT variation_NN1 in_II interaction_NN1 with_IW the_AT wall_NN1 makes_VVZ no_AT substantial_JJ difference_NN1 in_II the_AT outgoing_JJ flux_NN1 ,_, supporting_VVG the_AT observation_NN1 that_CST in_II simple_JJ environments_NN2 ,_, heuristics_NN2 are_VBR not_XX particularly_RR useful_JJ ._. 
</s>
<s>
The_AT teachers_NN2 took_VVD running_JJ notes_NN2 of_IO the_AT students_NN2 '_NULL strategies_NN2 ,_, what_DDQ they_PPHS2 thought_VVD each_DD1 student_NN1 was_VBDZ thinking_VVG ,_, and_CC ranked_VVD the_AT students_NN2 '_NULL responses_NN2 in_II31 terms_II32 of_II33 sophistication_NN1 of_IO understanding_NN1 ,_, using_VVG the_AT LT_NNB if_CSW possible_JJ ._. 
</s>
<s>
Moreover_RR ,_, we_PPIS2 improve_VV0 the_AT existing_JJ estimates_NN2 of_IO the_AT controllability_NN1 time_NNT1 and_CC we_PPIS2 show_VV0 that_CST our_APPGE estimates_NN2 are_VBR sharp_JJ ,_, at_RR21 least_RR22 when_CS the_AT control_NN1 is_VBZ active_JJ for_IF very_RG low_JJ ages_NN2 ._. 
</s>
<s>
Derivation_NN1 of_IO kineic-type_JJ models_NN2 The_AT derivation_NN1 of_IO kinetic_JJ models_NN2 moves_VVZ from_II the_AT structure_NN1 defined_VVN by_II Eq_NN1 ._. 
</s>
<s>
(_( 2.6_MC )_) and_CC is_VBZ carried_VVN out_RP by_II extending_VVG the_AT rationale_NN1 proposed_VVN at_II the_AT microscopic_JJ scale_NN1 to_TO model_VVI the_AT terms_NN2 n_ZZ1 and_CC A._NNU First_MD ,_, Lemma_NN1 25_MC together_RL with_IW (_( 79_MC )_) and_CC (_( 94_MC )_) implies_VVZ the_AT existence_NN1 of_IO some_DD c_ZZ1 >_FO 0_MC such_CS21 that_CS22 for_IF all_DB @S_FO and_CC h_ZZ1 small_JJ enough_RR ,_, @F_FO ._. 
</s>
<s>
Then_RT An_AT1 is_VBZ a_AT1 univalent_JJ Nqn-wall_NN1 (_( see_VV0 Lemma_NN1 B.13_FO )_) enclosing_VVG an_AT1 open_JJ topological_JJ disk_NN1 On_II 3_MC a_AT1 such_DA that_CST An_AT1 respects_NN2 Hn_NP1 as_CSA above_RL (_( see_VV0 (_( a_ZZ1 )_) -(c)_ZZ1 )_) ._. 
</s>
<s>
Our_APPGE measured_JJ tone_NN1 notwithstanding_RR ,_, we_PPIS2 stress_VV0 the_AT importance_NN1 of_IO these_DD2 concepts_NN2 relative_II21 to_II22 today_RT '_NULL s_ZZ1 pressing_JJ issues_NN2 ._. 
</s>
<s>
The_AT arguments_NN2 provided_VVN here_RL are_VBR due_II21 to_II22 Guillaume_NP1 Rond_NP1 ._. 
</s>
<s>
Let_VV0 us_PPIO2 note_VVI that_CST this_DD1 bound_NN1 can_VM be_VBI improved_VVN for_IF particular_JJ density_NN1 functions_NN2 ._. 
</s>
<s>
The_AT construction_NN1 is_VBZ uniform_JJ with_II31 respect_II32 to_II33 the_AT Planck_NP1 constant_JJ ._. 
</s>
<s>
A_AT1 sequence_NN1 S=i∈_FO is_VBZ called_VVN a_AT1 critical_JJ sequence_NN1 if_CS Lc(S)=Lc()_FO ._. 
</s>
<s>
The_AT first_MD empirical_JJ evidence_NN1 from_II a_AT1 variety_NN1 of_IO studies_NN2 that_CST this_DD1 unequal_JJ participation_NN1 is_VBZ critical_JJ with_II31 respect_II32 to_II33 equity_NN1 (_( e.g._REX ,_, Bailey_NP1 ,_, 2007_MC ;_; Barwell_NP1 ,_, 2012_MC ;_; Gresalfi_NP1 ,_, Martin_NP1 ,_, Hand_NN1 ,_, &;_NULL Greeno_NP1 ,_, 2009_MC ;_; Krummheuer_NP1 ,_, 2011_MC )_) can_VM be_VBI combined_VVN into_II a_AT1 hypothesized_JJ chain_NN1 of_IO connections_NN2 (_( Fig._NN1 1_MC1 )_) ._. 
</s>
<s>
This_DD1 contrasts_VVZ with_IW so-called_JJ dressed-up_JJ word_NN1 problems_NN2 ,_, where_CS a_AT1 mathematical_JJ content_NN1 is_VBZ merely_RR embellished_VVN by_II a_AT1 context_NN1 :_: "_" Students_NN2 just_RR have_VH0 to_TO "_" undress_VVI "_" the_AT problem_NN1 by_II picking_VVG out_RP the_AT simplified_JJ real_JJ model_NN1 ,_, which_DDQ is_VBZ already_RR provided_VVN in_II the_AT situational_JJ description_NN1 ._. 
</s>
<s>
For_IF every_AT1 fixed_JJ @S_FO ,_, with_IW probability_NN1 at_RR21 least_RR22 @S_FO ,_, @F_FO ._. 
</s>
<s>
We_PPIS2 define_VV0 g(x)∈_FO &lsqb;_( 0,1_MC &rsqb;_) Z_ZZ1 by_II @F_FO ,_, where_CS n_ZZ1 is_VBZ the_AT integer_NN1 in_RP E(x)_II satisfying_JJ x_ZZ1 ,_, n≤t<_FO β_NULL x_ZZ1 ,_, n_ZZ1 ._. 
</s>
<s>
Roughly_RR speaking_VVG ,_, we_PPIS2 have_VH0 attached_VVN the_AT "_" perturbation_NN1 map_NN1 "_" G#I_NN2 (_( x_ZZ1 ,_, n_ZZ1 )_) 1_MC1 to_II each_DD1 interval_NN1 I_ZZ1 (_( x_ZZ1 ,_, n_ZZ1 )_) ._. 
</s>
<s>
The_AT curves_NN2 a_AT1 ,_, b_ZZ1 ,_, c_ZZ1 ,_, and_CC d3_FO ._. 
</s>
<s>
The_AT shape_NN1 of_IO the_AT optimal_JJ shrinker_NN1 is_VBZ determined_VVN by_II the_AT choice_NN1 of_IO loss_NN1 function_NN1 and_CC ,_, crucially_RR ,_, by_II inconsistency_NN1 of_IO both_RR eigenvalues_NN2 and_CC eigenvectors_NN2 of_IO the_AT sample_NN1 covariance_NN1 matrix_NN1 ._. 
</s>
<s>
Of_RR21 course_RR22 ,_, such_DA apparent_JJ contradictions_NN2 can_VM only_RR be_VBI explained_VVN through_II considering_VVG the_AT underlying_JJ problem_NN1 :_: following_VVG a_AT1 migration_NN1 ,_, T_ZZ1 cells_NN2 must_VM spend_VVI a_AT1 certain_JJ time_NNT1 controlling_VVG their_APPGE local_JJ environment_NN1 for_IF any_DD antigen_NN1 presenting_VVG (_( i.e._REX infected_JJ )_) cells_NN2 ,_, often_RR detected_VVN through_II direct_JJ cellcell_NN1 contact_NN1 ,_, and_CC hence_RR '_NULL waiting_VVG '_NULL is_VBZ an_AT1 intrinsic_JJ component_NN1 of_IO the_AT search/detection_NN1 process_NN1 ._. 
</s>
<s>
From_II a_AT1 mathematical_JJ point_NN1 of_IO view_NN1 ,_, consensus_NN1 is_VBZ then_RT a_AT1 pattern_NN1 to_II which_DDQ the_AT system_NN1 tends_VVZ naturally_RR to_TO be_VBI attracted_VVN ._. 
</s>
<s>
We_PPIS2 formulate_VV0 our_APPGE general_JJ model_NN1 ,_, our_APPGE assumptions_NN2 on_II the_AT random_JJ potential_NN1 ,_, and_CC the_AT definitions_NN2 of_IO the_AT various_JJ notions_NN2 of_IO BEC_NP1 used_VVD in_II this_DD1 work_NN1 in_II Section_NN1 2_MC ._. 
</s>
<s>
We_PPIS2 will_VM also_RR abuse_VVI notation_NN1 by_II denoting_VVG e_ZZ1 as_II an_AT1 arbitrary_JJ constant_JJ with_IW different_JJ values_NN2 at_II different_JJ occurrences_NN2 ,_, arising_VVG from_II the_AT usage_NN1 of_IO inequality_NN1 (_( 4.3_MC )_) ._. 
</s>
<s>
It_PPH1 is_VBZ a_AT1 problem-based_JJ teaching_NN1 and_CC learning_VVG approach_NN1 ._. 
</s>
<s>
She_PPHS1 labeled_VVD the_AT lowest_JJT level_NN1 as_CSA level_NN1 1_MC1 where_CS the_AT major_JJ aim_NN1 is_VBZ to_TO organize_VVI instructional_JJ setting_NN1 to_TO support_VVI students_NN2 '_NULL learning_VVG and_CC teacher-student_JJ interaction_NN1 is_VBZ minimum_JJ ._. 
</s>
<s>
In_II MBI_JJ computations_NN2 the_AT DSD/SST_NN1 functions_NN2 as_II a_AT1 moving-mesh_JJ method_NN1 ._. 
</s>
<s>
We_PPIS2 say_VV0 that_CST @S_FO is_VBZ a_AT1 mirror_NN1 map_NN1 if_CS it_PPH1 satisfies_VVZ the_AT following_JJ properties_NN2 :_: @L_FO ._. 
</s>
<s>
These_DD2 are_VBR pertinent_JJ questions_NN2 since_CS sofic_JJ entropy_NN1 is_VBZ easier_JJR to_TO define_VVI ,_, compute_VVI ,_, and_CC understand_VVI when_RRQ there_EX exists_VVZ a_AT1 finite_JJ generating_JJ partition_NN1 ._. 
</s>
<s>
Analogously_RR ,_, it_PPH1 could_VM be_VBI argued_VVN that_CST MTEs_NN2 need_VV0 to_TO know_VVI about_II (_( 1_MC1 )_) SMTPCK_NP1 (_( the_AT knowledge_NN1 they_PPHS2 want_VV0 PSTs_NP1 to_TO acquire_VVI )_) ,_, (_( 2_MC )_) their_APPGE students_NN2 (_( PSTs_NP1 )_) and_CC the_AT PSTs_NP1 '_NULL relationship_NN1 to_II SMTPCK_NP1 ,_, (_( 3_MC )_) teaching_VVG PSTs_NP1 ,_, and_CC (_( 4_MC )_) the_AT curriculum_NN1 for_IF teaching_VVG PSTs_NP1 how_RRQ to_TO teach_VVI school_NN1 mathematics_NN1 ._. 
</s>
<s>
After_CS these_DD2 preparations_NN2 ,_, the_AT terms_NN2 in_II (_( 10.10_MC )_) are_VBR now_RT estimated_VVN separately_RR ._. 
</s>
<s>
Let_VV0 S_ZZ1 be_VBI a_AT1 numerical_JJ semigroup_NN1 with_IW profile_NN1 (_( p1_FO ,_, p2_FO )_) ._. 
</s>
<s>
Using_VVG the_AT notation_NN1 for_IF the_AT gauge_NN1 transformations_NN2 Cx_MC introduced_VVN in_II (_( 29_MC )_) ,_, we_PPIS2 define_VV0 the_AT quantities_NN2 @F_FO by_II the_AT requirements_NN2 that_CST @S_FO and_CC ,_, for_IF all_DB @S_FO ,_, @F_FO ._. 
</s>
<s>
The_AT Polya_NP1 conjecture_NN1 for_IF the_AT Neumann_NP1 eigenvalues_NN2 holds_VVZ for_IF k=2_FO in_II any_DD dimension_NN1 of_IO the_AT space_NN1 ._. 
</s>
<s>
Table_NN1 2_MC Integrated_JJ autocorrelation_NN1 times_NNT2 of_IO the_AT auxiliary_JJ chains_NN2 @S_FO on_II levels_NN2 =_FO 0_MC ,_, ..._... ,_, 4_MC ._. 
</s>
<s>
Based_VVN on_II the_AT findings_NN2 within_II three_MC large-scale_JJ random_JJ assignment_NN1 studies_NN2 of_IO teacher_NN1 PD_NP1 ,_, they_PPHS2 further_VV0 underlined_VVN that_CST "_" doing_VDG so_RR will_VM better_RRR enable_VVI developers_NN2 to_TO design_VVI PD_NP1 that_CST focuses_VVZ on_II improving_VVG those_DD2 aspects_NN2 of_IO knowledge_NN1 or_CC practice_NN1 that_CST will_VM most_RGT likely_RR translate_VVI into_II improvements_NN2 in_II student_NN1 achievement_NN1 "_" (_( p._NN1 1_MC1 )_) ._. 
</s>
<s>
Therefore_RR ,_, a_AT1 mixed_JJ method_NN1 research_NN1 was_VBDZ performed_VVN to_TO address_VVI the_AT research_NN1 questions_NN2 of_IO the_AT study_NN1 ._. 
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<s>
Lastly_RR ,_, A-level_NN1 Chemistry_NN1 grade_NN1 A_ZZ1 has_VHZ a_AT1 strong_JJ effect_NN1 with_IW a_AT1 mean_NN1 of_IO 6.1%_FO &lsqb;_( -8.4_MC :_: 20.6_MC &rsqb;_) but_CCB the_AT interval_NN1 clearly_RR shows_VVZ substantial_JJ uncertainty_NN1 ._. 
</s>
<s>
As_CSA these_DD2 authors_NN2 make_VV0 clear_JJ ,_, problem_NN1 context_NN1 shapes_NN2 the_AT process_NN1 in_II which_DDQ data_NN are_VBR generated_VVN and_CC reasons_NN2 for_IF which_DDQ data_NN are_VBR analyzed_VVN ,_, and_CC thus_RR also_RR the_AT types_NN2 of_IO inferences_NN2 that_CST can_VM be_VBI made_VVN from_II the_AT data_NN ._. 
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<s>
Particularly_RR ,_, let_VV0 E_ZZ1 be_VBI an_AT1 event_NN1 ,_, we_PPIS2 define_VV0 1E_FO as_II an_AT1 indicator_NN1 function_NN1 with_IW @S_FO if_CS E_ZZ1 holds_VVZ and_CC @S_FO otherwise_RR ._. 
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<s>
Once_RR these_DD2 coefficients_NN2 cqn_NNU ,_, m_ZZ1 are_VBR introduced_VVN we_PPIS2 can_VM give_VVI explicit_JJ formulae_NN2 for_IF the_AT Fourier_NP1 coefficients_NN2 of_IO the_AT Melnikov_NP1 potential_JJ L._NP1 According_II21 to_II22 Castillo-Garsow_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
(_( 2013_MC )_) ,_, the_AT ways_NN2 of_IO reasoning_NN1 while_CS coordinating_VVG the_AT covarying_JJ quantities_NN2 (_( i.e._REX ,_, continuous_JJ covariational_JJ reasoning_NN1 )_) ,_, may_VM appear_VVI in_II "_" chunky_JJ "_" or_CC "_" smooth_JJ "_" forms_NN2 ._. 
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<s>
As_II an_AT1 example_NN1 of_IO properties_NN2 which_DDQ are_VBR very_RG simple_JJ to_TO prove_VVI using_VVG the_AT definition_NN1 of_IO Besov_NP1 semi-norms_NN2 in_II31 terms_II32 of_II33 finite_JJ differences_NN2 ,_, let_VV0 us_PPIO2 prove_VVI the_AT first_MD point_NN1 in_II Theorem_NN1 1.2_MC ._. 
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<s>
Summary_NN1 and_CC interpretation_NN1 Theorem_NN1 2.1_MC and_CC the_AT subsequent_JJ analysis_NN1 tells_VVZ us_PPIO2 that_CST increasing_VVG the_AT parameter_NN1 @S_FO in_II Pa_NP1 leads_VVZ to_II more_RGR clustered_JJ eigenvalues_NN2 of_IO @S_FO for_IF a_AT1 range_NN1 of_IO Stokes_NP1 problems_NN2 ,_, and_CC should_VM result_VVI in_II more_RGR rapid_JJ convergence_NN1 of_IO the_AT MINRES_NP2 algorithm_NN1 ._. 
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<s>
Assume_VV0 that_CST the_AT following_NN1 two_MC conditions_NN2 are_VBR satisfied_JJ ._. 
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<s>
Additionally_RR ,_, the_AT model_NN1 suggests_VVZ that_CST reformed_JJ beliefs_NN2 about_II the_AT nature_NN1 of_IO mathematics_NN1 and_CC its_APPGE learning_NN1 directly_RR affect_VV0 teachers_NN2 '_NULL beliefs_NN2 about_II the_AT value_NN1 of_IO MPS_NNU ._. 
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<s>
Remark_VV0 4.2_MC While_CS a_AT1 set_NN1 generating_VVG a_AT1 Gabor_NP1 Riesz_NP1 sequence_NN1 is_VBZ necessarily_RR separated_VVN ,_, a_AT1 set_NN1 generating_VVG a_AT1 frame_NN1 may_VM be_VBI only_RR relatively_RR separated_VVN ,_, and_CC Theorem_NN1 4.1_MC (_( a_ZZ1 )_) does_VDZ not_XX directly_RR apply_VVI to_II non-separated_JJ sets_NN2 ._. 
</s>
<s>
In_II some_DD cases_NN2 ,_, these_DD2 new_JJ models_NN2 show_VV0 interesting_JJ qualitative_JJ features_NN2 consistent_JJ with_IW physical_JJ reality_NN1 ,_, that_CST are_VBR not_XX shown_VVN by_II purely_RR phenomenological_JJ models_NN2 ._. 
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<s>
However_RR ,_, it_PPH1 is_VBZ conceivable_JJ that_CST a_AT1 large_JJ positive_JJ clique_NN1 exists_VVZ even_CS21 when_CS22 <_FO (_( log1n_FO )_) ,_, in_II which_DDQ case_VV0 our_APPGE methods_NN2 would_VM continue_VVI to_TO be_VBI effective_JJ ._. 
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<s>
Namely_REX ,_, we_PPIS2 recall_VV0 the_AT following_JJ result_NN1 proved_VVN in_II &lsqb;_( 17_MC &rsqb;_) ._. 
</s>
<s>
We_PPIS2 now_RT consider_VV0 other_JJ phase_NN1 functions_NN2 @S_FO ,_, which_DDQ are_VBR small_JJ C1_FO perturbations_NN2 of_IO Tpar_NP1 @S_FO on_II @S_FO ._. 
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<s>
For_IF each_DD1 such_DA phase_NN1 function_NN1 T_ZZ1 ,_, and_CC each_DD1 scale_NN1 R_ZZ1 ,_, we_PPIS2 define_VV0 an_AT1 operator_NN1 @F_FO ._. 
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<s>
Hormander_NP1 introduced_VVD this_DD1 type_NN1 of_IO operator_NN1 in_II &lsqb;_( H_ZZ1 &rsqb;_) ._. 
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<s>
For_IF an_AT1 arbitrary_JJ open_JJ set_NN1 @S_FO containing_VVG g_ZZ1 ,_, we_PPIS2 can_VM choose_VVI @S_FO such_CS21 that_CS22 @S_FO holds_VVZ ._. 
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<s>
If_CS PSTs_NP1 use_NN1 procedures_NN2 to_TO solve_VVI the_AT tasks_NN2 ,_, in_II the_AT interview_NN1 setting_NN1 we_PPIS2 can_VM readily_RR probe_VVI their_APPGE reasoning_NN1 to_TO determine_VVI whether_CSW they_PPHS2 are_VBR able_JK to_TO unpack_VVI the_AT procedures_NN2 to_TO provide_VVI more_RGR explicit_JJ evidence_NN1 or_CC counterevidence_NN1 of_IO coordination_NN1 of_IO three_MC levels_NN2 of_IO units_NN2 ._. 
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<s>
Namely_REX F_ZZ1 is_VBZ Hermitean_JJ and_CC @F_FO ._. 
</s>
<s>
The_AT two_MC form_VV0 H_ZZ1 is_VBZ an_AT1 auxiliary_JJ field_NN1 and_CC its_APPGE integration_NN1 contour_NN1 is_VBZ chosen_VVN so_CS21 that_CS22 the_AT corresponding_JJ gaussian_JJ integral_JJ converges_VVZ ._. 
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<s>
By_II coning_VVG off_II g_ZZ1 the_AT situation_NN1 changes_NN2 drastically_RR ._. 
</s>
<s>
The_AT numerical_JJ method_NN1 based_VVN on_II the_AT discretization_NN1 of_IO evolution_NN1 equations_NN2 of_IO geometric_JJ quantities_NN2 ,_, as_CSA presented_VVN here_RL ,_, is_VBZ computationally_RR more_RGR expensive_JJ than_CSN Dziuk_NP1 '_NULL s_ZZ1 method_NN1 (_( roughly_RR by_II about_II a_AT1 factor_NN1 2_MC )_) ,_, but_CCB on_II the_AT other_JJ hand_NN1 it_PPH1 provides_VVZ full-order_JJ approximations_NN2 to_II basic_JJ geometric_JJ quantitiesthe_NN1 normal_JJ vector_NN1 and_CC curvaturein_NN1 addition_NN1 to_II the_AT position_NN1 and_CC velocity_NN1 of_IO the_AT surface_NN1 ._. 
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<s>
The_AT main_JJ idea_NN1 of_IO Tukey_NP1 '_NULL s_ZZ1 median_NN1 is_VBZ to_TO project_VVI multivariate_JJ data_NN onto_II all_DB one-dimensional_JJ subspaces_NN2 and_CC obtain_VV0 the_AT deepest_JJT point_NN1 by_II evaluating_VVG depths_NN2 in_II those_DD2 one-dimensional_JJ subspaces_NN2 ._. 
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<s>
We_PPIS2 refer_VV0 the_AT reader_NN1 to_II &lsqb;_( 19_MC ,_, p._NN1 1331_MC &rsqb;_) for_IF a_AT1 (_( more_RRR )_) comprehensive_JJ list_NN1 of_IO such_DA costs_NN2 ._. 
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<s>
The_AT initial_JJ ideas_NN2 for_IF this_DD1 paper_NN1 were_VBDR first_MD discussed_VVN during_II the_AT Research_NN1 Cluster_NN1 on_II "_" Computational_JJ Challenges_NN2 in_II Sparse_JJ and_CC Redundant_JJ Representations_NN2 "_" at_II ICERM_NP1 in_II November_NPM1 2014_MC ._. 
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<s>
This_DD1 is_VBZ a_AT1 very_RG efficient_JJ method_NN1 with_IW computational_JJ cost_NN1 comparable_JJ to_II the_AT MPSA_NN1 method_NN1 ._. 
</s>
<s>
The_AT set_NN1 of_IO all_DB Borel_NN1 probability_NN1 measures_NN2 on_II a_AT1 metric_JJ space_NN1 X_ZZ1 will_VM be_VBI denoted_VVN by_II P_ZZ1 (_( X_ZZ1 )_) ._. 
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<s>
When_RRQ data_NN are_VBR collected_VVN over_II time_NNT1 ,_, heterogeneity_NN1 often_RR manifests_VVZ itself_PPX1 through_II non-stationa-rity_NN1 ,_, where_CS the_AT data-generating_JJ mechanism_NN1 varies_VVZ with_IW time_NNT1 ._. 
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<s>
A_AT1 subtlety_NN1 arises_VVZ for_IF the_AT canonical_JJ ensemble_NN1 as_II the_AT Fermi_NP1 level_NN1 for_IF the_AT finite_JJ system_NN1 depends_VVZ globally_RR on_II the_AT atom_NN1 configuration_NN1 ,_, which_DDQ would_VM destroy_VVI the_AT locality_NN1 ._. 
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<s>
It_PPH1 is_VBZ up_II21 to_II22 the_AT reader_NN1 to_TO decide_VVI if_CSW the_AT data_NN and_CC subsequent_JJ inferences_NN2 are_VBR transferable_JJ to_II their_APPGE situations_NN2 ._. 
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<s>
In_II31 terms_II32 of_II33 the_AT sub-goals_NN2 ,_, the_AT majority_NN1 of_IO students_NN2 '_NULL responses_NN2 were_VBDR related_VVN to_II their_APPGE self-efficacy_JJ beliefs_NN2 in_II doing_VDG mathematics_NN1 (_( 7.87%_FO in_II year_NNT1 8_MC ;_; 8.45%_FO in_II year_NNT1 9_MC )_) and_CC their_APPGE hope_NN1 to_TO achieve_VVI a_AT1 better_JJR grade_NN1 (_( 3.64%_FO in_II year_NNT1 8_MC ;_; 3.52%_FO in_II year_NNT1 9_MC )_) ._. 
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<s>
We_PPIS2 conducted_VVD Monte_NP1 Carlo_NP1 studies_NN2 for_IF the_AT following_JJ i.i.d_NNU ._. 
</s>
<s>
We_PPIS2 assume_VV0 that_CST the_AT direction_NN1 of_IO M_ZZ1 is_VBZ parallel_RR to_II the_AT vector_NN1 y2y1_FO ._. 
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<s>
Both_DB2 of_IO them_PPHO2 are_VBR proportional_JJ because_CS the_AT increase_NN1 and_CC decrease_NN1 are_VBR constant_JJ ._. 
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<s>
There_EX were_VBDR students_NN2 whose_DDQGE responses_NN2 were_VBDR at_II the_AT levels_NN2 of_IO extending_VVG the_AT pattern_NN1 (_( level_NN1 1_MC1 )_) or_CC recursive_JJ thinking_NN1 (_( level_VV0 2/3_MF )_) ,_, but_CCB not_XX algebraic_JJ generalisation_NN1 ,_, who_PNQS nonetheless_RR evidenced_VVN covariational_JJ reasoning_NN1 that_CST coordinates_NN2 direction_NN1 and_CC amount_NN1 of_IO change_NN1 in_II one_MC1 variable_NN1 with_IW changes_NN2 in_II the_AT other_JJ variable_NN1 (_( level_NN1 2.2_MC ;_; fifth_MD column_NN1 )_) ._. 
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<s>
In_II the_AT first_MD case_NN1 Theorem_NN1 2_MC applies_VVZ directly_RR ._. 
</s>
<s>
We_PPIS2 suggest_VV0 that_CST a_AT1 child_NN1 who_PNQS views_VVZ the_AT equal_JJ sign_NN1 relationally_RR would_VM not_XX need_VVI to_TO "_" put_VVI &lsqb;_( the_AT equation_NN1 &rsqb;_) around_RP ,_, "_" but_CCB could_VM start_VVI with_IW 6_MC and_CC might_VM reason_VVI that_CST "_" the_AT equation_NN1 is_VBZ false_JJ because_CS 6_MC is_VBZ not_XX the_AT same_DA as_CSA 3_MC +_FO 2_MC ,_, since_II 3_MC +_FO 2_MC is_VBZ 5_MC ._. "_" 
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<s>
Here_RL ,_, a_AT1 stable_JJ region_NN1 appears_VVZ for_IF k_ZZ1 from_II 150_MC up_RG21 to_RG22 500_MC ,_, leading_VVG to_TO anestimate_VVI between_II 3.73_MC million_NNO and_CC 4.12_MC million_NNO ._. 
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<s>
Since_CS the_AT differential_NN1 @S_FO is_VBZ zero_NN1 for_IF fields_NN2 in_II which_DDQ @S_FO ,_, also_RR @S_FO ,_, which_DDQ concludes_VVZ the_AT proof_NN1 ._. 
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<s>
Since_CS @S_FO is_VBZ reflexive_NN1 ,_, then_RT we_PPIS2 shall_VM apply_VVI Theorem_NN1 2.1_MC and_CC Corollary_NN1 3.3_MC in_II &lsqb;_( 18_MC &rsqb;_) to_TO state_VVI that_CST @S_FO is_VBZ weakly-relatively_RR compact_JJ w.r.t_NNU ._. 
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<s>
The_AT involutions_NN2 da_NN1 on_II TS_ZZ2 and_CC @S_FO on_II @S_FO then_RT induce_VV0 involutions_NN2 on_II TC_NP1 and_CC @S_FO ;_; we_PPIS2 denote_VV0 the_AT induced_JJ involutions_NN2 also_RR by_II da_FU and_CC da*_FO ._. 
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<s>
Figure_NN1 3_MC depicts_VVZ the_AT simulation_NN1 with_IW Gaussian_JJ kernels_NN2 as_CSA interaction_NN1 potentials_NN2 ._. 
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<s>
Teacher_NN1 noticing_VVG of_IO student_NN1 thinking_NN1 imbedded_VVD within_II video_NN1 club_NN1 meetings_NN2 provided_VVD a_AT1 lens_NN1 for_IF interpreting_VVG curricula_NN2 materials_NN2 and_CC instructional_JJ decision_NN1 making_VVG ._. 
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<s>
Also_RR ,_, the_AT language_NN1 we_PPIS2 were_VBDR taught_VVN to_II uselike_JJ "_" groups_NN2 of_IO "_" and_CC "_" copies_NN2 of_IO "_" helped_VVN me_PPIO1 to_TO understand_VVI more_RGR as_RR21 well_RR22 ._. 
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<s>
Finally_RR ,_, (_( 4.30_MC )_) is_VBZ obtained_VVN choosing_VVG @F_FO ,_, in_II (_( 4.31_MC )_) ,_, using_VVG (_( 4.29_MC )_) and_CC recalling_VVG that_CST @S_FO is_VBZ an_AT1 orthonormal_JJ basis_NN1 in_II @S_FO ._. 
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<s>
Remark_VV0 4.6_MC ._. 
</s>
<s>
The_AT Andre-Oort_NP1 Conjecture_NN1 holds_VVZ for_IF Ag_FO for_IF any_DD @S_FO ._. 
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<s>
Consequently_RR ,_, with_IW X*_FO globally_RR stable_JJ ,_, there_EX exists_VVZ some_DD @S_FO such_CS21 that_CS22 @F_FO ._. 
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<s>
If_CS @S_FO ,_, (_( 4.12_MC )_) yields_VVZ @S_FO ._. 
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<s>
Hence_RR ,_, minimizing_VVG over_II @S_FO ,_, we_PPIS2 get_VV0 @S_FO ,_, so_CS @S_FO ._. 
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<s>
However_RR ,_, it_PPH1 is_VBZ unclear_JJ whether_CSW this_DD1 is_VBZ the_AT case_NN1 for_IF every_AT1 initial_JJ datum_NN1 ._. 
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<s>
We_PPIS2 use_VV0 the_AT notion_NN1 of_IO real_JJ orientation_NN1 introduced_VVN in_II this_DD1 paper_NN1 to_TO obtain_VVI isomorphisms_NN2 of_IO real_JJ bundle_NN1 pairs_NN2 over_II families_NN2 of_IO symmetric_JJ surfaces_NN2 and_CC then_RT apply_VV0 the_AT determinant_NN1 functor_NN1 to_II these_DD2 isomorphisms_NN2 ._. 
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<s>
However_RR ,_, @S_FO ,_, which_DDQ is_VBZ related_VVN to_II the_AT stopping_VVG criterion_NN1 of_IO the_AT homotopy-smoothing_JJ method_NN1 ._. 
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<s>
These_DD2 are_VBR clearly_RR outstanding_JJ research_NN1 directions_NN2 for_IF the_AT future_NN1 ._. 
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<s>
We_PPIS2 choose_VV0 again_RT a_AT1 nowhere_RL dense_JJ closed_JJ subscheme_NN1 YX_NN1 such_CS21 that_CS22 p_ZZ1 is_VBZ an_AT1 isomorphism_NN1 outside_II Y._NP1 Theorem_NN1 A_ZZ1 shows_VVZ that_CST the_AT horizontal_JJ sequences_NN2 of_IO the_AT diagram_NN1 of_IO pro-groups_NN2 @T_FO are_VBR exact_JJ ._. 
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<s>
The_AT student-invented_JJ dissections_NN2 to_TO compare_VVI the_AT space_NN1 covered_VVN by_II different_JJ figures_NN2 ,_, and_CC to_II structure_NN1 areas_NN2 by_II unit_NN1 dissection_NN1 ,_, supported_VVD change_NN1 in_II students_NN2 '_NULL initial_JJ images_NN2 of_IO area_NN1 measure_NN1 as_CSA simply_RR a_AT1 matter_NN1 of_IO multiplying_VVG "_" length_NN1 x_II width_NN1 ,_, "_" but_CCB stopped_VVD short_JJ of_IO assisting_VVG students_NN2 to_TO think_VVI about_II area_NN1 dynamically_RR -_- as_CSA generated_VVN ._. 
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<s>
Moreover_RR ,_, by_II assumption_NN1 @S_FO is_VBZ an_AT1 Fp-algebra_NN1 ._. 
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<s>
By_II &lsqb;_( Wri14_FO &rsqb;_) ,_, the_AT monodromy_NN1 of_IO p(TM)_NNU is_VBZ totally_RR irreducible_JJ ._. 
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<s>
Such_DA a_AT1 basis_NN1 of_IO V_ZZ1 is_VBZ @S_FO ._. 
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<s>
To_TO describe_VVI the_AT special_JJ fibre_NN1 of_IO this_DD1 model_NN1 ,_, note_VV0 that_CST the_AT order_NN1 of_IO @S_FO is_VBZ invertible_JJ on_II OX_NN1 ,_, and_CC we_PPIS2 therefore_RR have_VH0 @F_FO ._. 
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<s>
This_DD1 implies_VVZ that_CST @S_FO ,_, and_CC hence_RR @S_FO ._. 
</s>
<s>
The_AT description_NN1 of_IO the_AT special_JJ fibre_NN1 follows_VVZ from_II that_DD1 of_IO the_AT action_NN1 of_IO @S_FO in_II Theorem_NN1 6.5_MC ._. 
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<s>
Now_RT we_PPIS2 can_VM prove_VVI the_AT compactness_NN1 of_IO the_AT set_NN1 of_IO quotients_NN2 ._. 
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<s>
Jacobi_NN2 Elliptic_JJ functions_NN2 are_VBR obtained_VVN by_II inverting_VVG incomplete_JJ elliptic_JJ integrals_NN2 of_IO the_AT first_MD kind_NN1 ._. 
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<s>
By_II property_NN1 (_( 5_MC )_) ,_, the_AT vertices_NN2 @S_FO give_VV0 rise_NN1 to_II a_AT1 pair_NN of_IO non-separating_JJ arcs_NN2 in_II R_ZZ1 based_VVN at_II the_AT marked_JJ point_NN1 ,_, and_CC these_DD2 arcs_NN2 fill_VV0 a_AT1 subsurface_NN1 Qx_NP1 of_IO R_ZZ1 homeomorphic_JJ to_II a_AT1 surface_NN1 of_IO genus_NN1 one_MC1 with_IW one_MC1 boundary_NN1 component_NN1 and_CC one_MC1 marked_JJ point_NN1 ._. 
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<s>
However_RR ,_, change_NN1 occurred_VVD in_II respect_NN1 to_II her_APPGE confidence_NN1 in_II enactment_NN1 at_II the_AT classroom_NN1 level_NN1 and_CC how_RRQ one_PN1 adapted_VVD materials_NN2 ._. 
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<s>
The_AT normal_JJ equations_NN2 are_VBR given_VVN by_II @F_FO ,_, which_DDQ define_VV0 the_AT vector_NN1 c_ZZ1 that_CST minimizes_VVZ the_AT @S_FO residual_JJ of_IO (_( 5.4_MC )_) ._. 
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<s>
Then_RT for_IF any_DD fixed_JJ x_ZZ1 e_ZZ1 Sn-1_MC1 ,_, IE_REX fL(x)_NNU (_( A_ZZ1 )_) |_NULL <_FO (_( a1_FO +_FO a2_FO )_) &lsqb;_( d_ZZ1 ,_, with_IW a1_FO ,_, a2_FO as_CSA in_II Proposition_NN1 2.4_MC ._. 
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<s>
For_IF this_DD1 ,_, observe_VV0 that_CST @F_FO ._. 
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<s>
Note_VV0 that_CST @S_FO is_VBZ bounded_VVN from_II above_RL by_II a_AT1 constant_JJ that_CST only_RR depends_VVZ on_II A_ZZ1 and_CC D_ZZ1 (_( see_VV0 Proposition_NN1 3.1_MC )_) ._. 
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<s>
We_PPIS2 first_MD discretize_VV0 the_AT operator_NN1 @S_FO by_II the_AT centered_JJ finite_JJ difference_NN1 formula_NN1 with_IW @S_FO ,_, @F_FO ,_, @F_FO ,_, where_CS Ix_MC e_ZZ1 Rmxm_NP1 is_VBZ the_AT identity_NN1 matrix_NN1 ._. 
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<s>
In_BCL21 order_BCL22 to_TO show_VVI that_CST this_DD1 function_NN1 is_VBZ constant_JJ ,_, it_PPH1 suffices_VVZ to_TO prove_VVI that_CST the_AT Prym_NP1 PGa_NP1 (_( which_DDQ acts_VVZ through_II automorphisms_NN2 of_IO Mg_NNU )_) ,_, acts_VVZ transitively_RR on_II n0_FO (_( M.HE_NP1 a_ZZ1 )_) ._. 
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<s>
First_MD ,_, the_AT power_NN1 is_VBZ much_RR higher_JJR ,_, with_IW WTCCC_NP1 (_( 2007_MC )_) making_VVG 9_MC discoveries_NN2 ,_, while_CS knockoffs_NN2 made_VVD 18_MC discoveries_NN2 on_II average_NN1 ,_, doubling_VVG the_AT power_NN1 ._. 
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<s>
We_PPIS2 will_VM see_VVI that_CST these_DD2 subsets_NN2 are_VBR in_II fact_NN1 subvarieties_NN2 of_IO Z_ZZ1 ,_, and_CC that_DD1 ,_, in_II generic_JJ cases_NN2 ,_, they_PPHS2 are_VBR transverse_JJ complete_JJ intersections_NN2 ._. 
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<s>
Candes_NN2 ,_, Sing-Long_JJ and_CC Trzasko_NP1 (_( 20i3_FO )_) ,_, Donoho_NP1 and_CC Gavish_NN1 (_( 20i4_FO )_) ,_, Gavish_JJ and_CC Donoho_NP1 (_( 20i4_FO )_) studied_VVD the_AT algorithm_NN1 for_IF recovering_VVG X_ZZ1 ,_, where_CS singular_JJ value_NN1 thresholding_VVG (_( SVT_NP1 )_) and_CC hard_RR singular_JJ value_NN1 thresholding_VVG (_( HSVT_NP1 )_) ,_, stated_VVN as_CSA @F_FO were_VBDR proposed_VVN ._. 
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<s>
Then_RT ,_, for_IF any_DD @S_FO there_EX holds_VVZ for_IF the_AT error_NN1 @F_FO ,_, of_IO the_AT Laplace-based_JJ importance_NN1 sampling_VVG with_IW @S_FO samples_VVZ that_CST @F_FO ,_, where_CS @S_FO with_IW @S_FO ._. 
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<s>
This_DD1 is_VBZ different_JJ to_II the_AT variance-reduced_JJ stochastic_JJ quasi-Newton_NP1 methods_NN2 in_II &lsqb;_( 19_MC ,_, 24_MC &rsqb;_) that_CST attempt_VV0 to_TO reduce_VVI only_RR the_AT noise_NN1 of_IO gradient_NN1 approximations_NN2 ._. 
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<s>
GT-spline_JJ spaces_NN2 with_IW pieces_NN2 drawn_VVN from_II (_( different_JJ )_) generalized_JJ polynomial_NN1 spaces_NN2 containing_VVG polynomial_NN1 ,_, exponential_NN1 ,_, or_CC trigonometric_JJ functions_NN2 (_( see_VV0 ,_, e.g._REX ,_, Examples_NN2 7.4--7.6_MCMC )_) are_VBR of_IO particular_JJ interest_NN1 both_RR in_II geometric_JJ design_NN1 and_CC numerical_JJ simulation_NN1 because_CS they_PPHS2 offer_VV0 a_AT1 valid_JJ alternative_NN1 to_II NURBS_NN2 ._. 
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<s>
Wigfield_NN1 and_CC Meece_NP1 (_( 1988_MC )_) distinguish_VV0 between_II a_AT1 predominantly_RR cognitive_JJ "_" worry_NN1 "_" component_NN1 ,_, involving_VVG mathematics_NN1 performance_NN1 anxiety_NN1 and_CC a_AT1 predominantly_RR affective_JJ component_NN1 ,_, involving_VVG negative_JJ emotions_NN2 in_II the_AT presence_NN1 of_IO mathematical_JJ stimuli_NN2 ._. 
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<s>
To_II this_DD1 end_NN1 ,_, let_VV0 @S_FO be_VBI a_AT1 local_JJ maximum_JJ point_NN1 of_IO @S_FO for_IF some_DD @S_FO ._. 
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<s>
In_II Fig._NN1 3_MC ,_, we_PPIS2 show_VV0 the_AT convergence_NN1 lines_NN2 of_IO the_AT @S_FO error_NN1 on_II the_AT velocity_NN1 and_CC the_AT @S_FO error_NN1 on_II the_AT pressure_NN1 ,_, respectively_RR ._. 
</s>
<s>
Linear_JJ convergence_NN1 occurs_VVZ when_RRQ @S_FO ,_, where_CS @S_FO is_VBZ the_AT condition_NN1 number_NN1 for_IF the_AT problem_NN1 ,_, and_CC the_AT error_NN1 falls_VVZ beneath_II ?_NNU in_II roughly_RR @S_FO Lanczos_NP1 iterations_NN2 ._. 
</s>
<s>
Suppose_VV0 @S_FO satisfies_VVZ A2_FO ,_, and_CC A3_FO holds_VVZ ._. 
</s>
<s>
We_PPIS2 have_VH0 already_RR established_VVN in_II (_( 2.15_MC )_) that_CST @S_FO ._. 
</s>
<s>
The_AT antisymmetric_JJ part_NN1 of_IO the_AT gradient_NN1 tensor_NN1 then_RT ,_, can_VM be_VBI reconstructed_VVN applying_VVG a_AT1 zeroth_MD order_NN1 pseudo-differential_NN1 operator_NN1 to_II S._NP1 We_PPIS2 find_VV0 that_CST @F_FO ._. 
</s>
<s>
Because_CS this_DD1 is_VBZ a_AT1 zeroth_MD order_NN1 operator_NN1 related_VVN to_II the_AT Riesz_NP1 transform_VV0 ,_, it_PPH1 is_VBZ bounded_VVN from_II Lp_NN1 to_II Lp_NN1 for_IF 1_MC1 <_FO p_ZZ1 <_FO +8_MC ,_, but_CCB we_PPIS2 will_VM only_RR have_VHI Calderon-Zygmund_NP1 type_NN1 estimates_NN2 ,_, so_CS our_APPGE control_NN1 will_VM be_VBI very_RG bad_JJ ._. 
</s>
<s>
Assume_VV0 that_CST S_ZZ1 has_VHZ two_MC continuous_JJ derivatives_NN2 in_II D_ZZ1 ,_, a_AT1 neighborhood_NN1 of_IO x*_FO ,_, and_CC @F_FO ._. 
</s>
<s>
Then_RT functional_JJ iteration_NN1 (_( A.2_FO )_) is_VBZ locally_RR q-quadratically_RR convergent_JJ to_II x*_FO ._. 
</s>
<s>
Let_VV0 B_ZZ1 be_VBI as_CSA in_II Lemma_NN1 27_MC and_CC fix_VV0 some_DD n_ZZ1 G_ZZ1 B._NP1 Since_CS @F_FO ,_, we_PPIS2 can_VM apply_VVI Proposition_NN1 21_MC with_IW @S_FO and_CC write_VVI @F_FO ,_, where_CS each_DD1 ni_NN2 is_VBZ a_AT1 Bernoulli_JJ measure_NN1 supported_VVN at_II a_AT1 pair_NN of_IO points_NN2 of_IO distance_NN1 between_II 2n-1_MC1 and_CC 2n+1_FO ,_, vo_NN1 is_VBZ a_AT1 non-negative_JJ measure_NN1 ,_, and_CC @F_FO ,_, where_CS c_ZZ1 is_VBZ an_AT1 absolute_JJ constant_JJ ._. 
</s>
<s>
We_PPIS2 now_RT define_VV0 the_AT set_NN1 X_ZZ1 as_CSA follows_VVZ :_: @F_FO ._. 
</s>
<s>
Note_VV0 that_CST ,_, for_IF each_DD1 k>1_FO ,_, there_EX exists_VVZ a_AT1 set_NN1 @S_FO such_CS21 that_CS22 @F_FO ._. 
</s>
<s>
We_PPIS2 set_VV0 @S_FO ._. 
</s>
<s>
Then_RT ,_, for_IF all_DB @S_FO and_CC m_ZZ1 ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Lemma_NN1 6.6_MC then_RT shows_VVZ that_CST if_CS the_AT discrepancy_NN1 property_NN1 holds_VVZ for_IF A_ZZ1 ,_, then_RT deterministically_RR the_AT heavy_JJ couples_NN2 give_VV0 a_AT1 small_JJ contribution_NN1 to_TO xTAx_VVI for_IF any_DD vector_NN1 x_ZZ1 ._. 
</s>
<s>
As_CSA in_II &lsqb;_( 11_MC ,_, Problem_NN1 (_( DQ_NP1 '_NULL )_) &rsqb;_) ,_, problem_NN1 (_( P1_FO )_) has_VHZ a_AT1 dual_JJ formulation_NN1 that_CST one_PN1 can_VM solve_VVI numerically_RR for_IF any_DD given_JJ Nusing_VVG a_AT1 semidefinite_NN1 program_NN1 (_( SDP_NP1 )_) to_TO determine_VVI an_AT1 upper_JJ bound_NN1 on_II the_AT cost_NN1 function_NN1 worst-case_NN1 bound_VVN for_IF any_DD FSFOM_NN1 :_: @F_FO ,_, where_CS @S_FO ,_, and_CC @F_FO ._. 
</s>
<s>
This_DD1 means_VVZ that_CST one_PN1 can_VM compute_VVI a_AT1 valid_JJ upper_JJ bound_NN1 (_( D_ZZ1 )_) of_IO (_( P_ZZ1 )_) for_IF given_JJ step_NN1 coefficients_NN2 h_ZZ1 using_VVG a_AT1 SDP_NP1 ._. 
</s>
<s>
Mortar_VV0 methods_NN2 ,_, as_CSA introduced_VVN in_II &lsqb;_( 8_MC &rsqb;_) ,_, form_VV0 an_AT1 appealing_JJ framework_NN1 for_IF fracture_NN1 modeling_NN1 ,_, since_CS both_DB2 nonmatching_VVG grids_NN2 and_CC intersections_NN2 are_VBR naturally_RR handled_VVN ._. 
</s>
<s>
The_AT first_MD two_MC sets_NN2 (_( Di_NP1 ,_, D2_FO )_) are_VBR provided_VVN by_II linear_JJ operators_NN2 from_II W_ZZ1 ,_, into_II R_ZZ1 and_CC the_AT set_NN1 D3_FO by_II linear_JJ operators_NN2 from_II @S_FO into_II R._NP1 For_IF all_DB @S_FO they_PPHS2 are_VBR defined_VVN as_CSA follows_VVZ :_: D1_FO contains_VVZ linear_JJ operators_NN2 evaluating_VVG v_ZZ1 at_II the_AT N_ZZ1 vertices_NN2 of_IO K_ZZ1 ,_, D2_FO contains_VVZ linear_JJ operators_NN2 evaluating_VVG Vvh_NP1 at_II the_AT Nk_NP1 vertices_NN2 of_IO K_ZZ1 ,_, D3_FO contains_VVZ linear_JJ operators_NN2 evaluating_VVG @S_FO at_II the_AT Nk_NP1 vertices_NN2 of_IO K._NP1 In_II the_AT context_NN1 of_IO the_AT standard_JJ parabolic_JJ counterpart_NN1 with_IW @S_FO data_NN and_CC zero_NN1 forcing_VVG term_NN1 ,_, Luskin_NP1 and_CC Rannacher_NP1 &lsqb;_( 26_MC &rsqb;_) analyzed_VVD a_AT1 fully_RR discrete_JJ scheme_NN1 based_VVN on_II Galerkin_NP1 FEM_NP1 in_II space_NN1 and_CC the_AT backward_JJ Euler_NN1 method_NN1 in_II time_NNT1 ,_, and_CC proved_VVD a_AT1 first-order_JJ temporal_JJ convergence_NN1 ._. 
</s>
<s>
Under_II the_AT conditions_NN2 and_CC scaling_NN1 of_IO Theorem_NN1 3.8_MC ,_, @S_FO ._. 
</s>
<s>
In_II Section_NN1 4_MC ,_, optimal_JJ error_NN1 estimates_NN2 (_( with_II31 respect_II32 to_II33 both_DB2 the_AT convergence_NN1 order_NN1 and_CC the_AT regularity_NN1 of_IO u0_FO )_) in_II the_AT @S_FO will_VM be_VBI proved_VVN using_VVG novel_JJ energy_NN1 arguments_NN2 ,_, see_VV0 Theorem_NN1 4.3_MC ._. 
</s>
<s>
We_PPIS2 now_RT aim_VV0 at_II constructing_VVG random_JJ variables_NN2 a_AT1 '_NULL n_ZZ1 such_CS21 that_CS22 @F_FO ,_, so_CS21 that_CS22 we_PPIS2 can_VM apply_VVI Theorem_NN1 1.6(i)_FO of_IO &lsqb;_( 23_MC &rsqb;_) in_II each_DD1 m_ZZ1 g_ZZ1 Q._NP1 The_AT corresponding_JJ result_NN1 for_IF sound-soft_JJ scatterers_NN2 ,_, that_REX21 is_REX22 ,_, in_II the_AT Dirichlet_NN1 case_NN1 ,_, was_VBDZ considered_VVN earlier_RRR in_II &lsqb;_( 32_MC &rsqb;_) ._. 
</s>
<s>
Proof_VV0 Our_APPGE assumptions_NN2 imply_VV0 that_CST almost-every_AT1 orbit_NN1 is_VBZ infinite_JJ ._. 
</s>
<s>
In_II this_DD1 respect_NN1 ,_, it_PPH1 is_VBZ also_RR worth_II pointing_VVG out_RP that_CST an_AT1 exponential_NN1 bound_VVN in_II the_AT error_NN1 term_NN1 in_II Theorem_NN1 2_MC ,_, say_VV0 of_IO the_AT form_NN1 exp(cd)_NN1 for_IF some_DD c>0_FO ,_, would_VM easily_RR imply_VVI the_AT Lehmer_NP1 conjecture_NN1 (_( arguing_VVG ,_, say_VV0 ,_, as_CSA in_II &lsqb;_( 10_MC ,_, Lemma_NN1 16_MC &rsqb;_) )_) ._. 
</s>
<s>
In_II our_APPGE case_NN1 the_AT relevant_JJ groups_NN2 are_VBR not_XX discrete_JJ ,_, so_CS we_PPIS2 use_VV0 a_AT1 semisimplicial_JJ space_NN1 instead_RR ._. 
</s>
<s>
If_CS f_ZZ1 g_ZZ1 (_( 2_MC ,_, 2_MC &rsqb;_) ,_, assume_VV0 further_RRR that_DD1 V_ZZ1 can_VM be_VBI enhanced_VVN to_II a_AT1 rough_JJ distribution_NN1 V._II Then_RT there_EX exists_VVZ a_AT1 unique_JJ probability_NN1 measure_NN1 P_ZZ1 ,_, which_DDQ solves_VVZ the_AT martingale_NN1 problem_NN1 with_IW generator_NN1 GV_NP1 starting_VVG at_II x_ZZ1 (_( as_CSA described_VVN above_RL )_) ,_, for_IF every_AT1 x_ZZ1 g_ZZ1 Rd_NN1 ._. 
</s>
<s>
This_DD1 suggests_VVZ that_DD1 convexity_NN1 alone_RR may_VM not_XX be_VBI sufficient_JJ for_IF the_AT trajectories_NN2 of_IO (_( 1_MC1 )_) to_TO converge_VVI strongly_RR ,_, but_CCB one_PN1 can_VM reasonably_RR expect_VVI it_PPH1 to_TO be_VBI the_AT case_NN1 under_II some_DD additional_JJ conditions_NN2 ._. 
</s>
<s>
The_AT desired_JJ assertion_NN1 (_( 5.4_MC )_) must_VM therefore_RR hold_VVI ._. 
</s>
<s>
Consider_VV0 the_AT discrete_JJ Schrodinger_NN1 operator_NN1 @S_FO in_II @S_FO ,_, where_CS @F_FO ,_, is_VBZ the_AT discrete_JJ Laplace_NP1 operator_NN1 ,_, and_CC V_ZZ1 is_VBZ the_AT operator_NN1 of_IO multiplication_NN1 by_II the_AT potential_NN1 given_VVN by_II @F_FO ,_, where_CS the_AT real_JJ numbers_NN2 v0_FO and_CC v1_FO are_VBR fixed_VVN ._. 
</s>
<s>
Let_VV0 us_PPIO2 prove_VVI the_AT energy-dissipation_JJ inequality_NN1 (_( EI_NP1 )_) k_ZZ1 ._. 
</s>
<s>
Noss_VV0 and_CC Hoyles_NP2 (_( 1992_MC )_) note_VV0 that_CST "_" there_EX are_VBR tasks_NN2 that_CST children_NN2 can_VM do_VDI with_IW a_AT1 computer_NN1 that_CST would_VM be_VBI impossible_JJ without_IW one_MC1 "_" (_( p._NNU 455_MC )_) ._. 
</s>
<s>
Academic_JJ self-efficacy_NN1 ,_, as_CSA a_AT1 context-related_JJ construct_NN1 refers_VVZ to_II people_NN '_NULL s_ZZ1 beliefs_NN2 about_II their_APPGE own_DA capabilities_NN2 for_IF successfully_RR executing_VVG a_AT1 course_NN1 of_IO action_NN1 that_CST leads_VVZ to_II a_AT1 desired_JJ outcome_NN1 "_" (_( Vasile_NP1 ,_, Marhan_NP1 ,_, Singer_NP1 ,_, &;_NULL Stoicescu_NP1 ,_, 2011_MC ,_, p._NN1 479_MC )_) ._. 
</s>
<s>
On_II a_AT1 Hilbert_NP1 space_NN1 H_ZZ1 we_PPIS2 will_VM denote_VVI by_II @S_FO the_AT inner_JJ product_NN1 generating_VVG the_AT norm_NN1 @S_FO ._. 
</s>
<s>
For_IF an_AT1 arbitrary_JJ normed_JJ linear_JJ space_NN1 @S_FO ,_, the_AT topological_JJ dual_JJ space_NN1 is_VBZ @F_FO ,_, which_DDQ is_VBZ a_AT1 Banach_NN1 space_NN1 for_IF the_AT norm_NN1 @S_FO ._. 
</s>
<s>
Yet_RR ,_, we_PPIS2 can_VM fix_VVI a_AT1 closed_JJ one_PN1 form_VV0 0∈H1_NN1 (_( M_ZZ1 ,_, R_ZZ1 )_) such_CS21 that_CS22 γ_NULL 000_MC ._. 
</s>
<s>
Our_APPGE goal_NN1 is_VBZ to_TO describe_VVI the_AT limit_NN1 of_IO an_AT1 when_CS n_ZZ1 goes_VVZ to_II infinity_NN1 ._. 
</s>
<s>
In_II this_DD1 section_NN1 we_PPIS2 will_VM need_VVI notation_NN1 for_IF new_JJ types_NN2 of_IO crossing_VVG distances_NN2 ._. 
</s>
<s>
Section_NN1 4_MC is_VBZ devoted_JJ to_II presenting_VVG the_AT innovative_JJ theoretical_JJ results_NN2 and_CC analysis_NN1 for_IF polynomial_NN1 approximation_NN1 using_VVG our_APPGE versions_NN2 of_IO weighted_JJ @S_FO minimization_NN1 and_CC iterative_JJ hard_JJ thresholding_JJ algorithms_NN2 ._. 
</s>
<s>
As_II a_AT1 result_NN1 of_IO the_AT corrections_NN2 above_RL ,_, equations_NN2 (_( 4.14_MC )_) and_CC (_( 4.19_MC )_) were_VBDR removed_VVN ,_, so_RR these_DD2 equation_NN1 numbers_NN2 no_RR21 longer_RR22 exist_VV0 in_II this_DD1 reprint_NN1 ._. 
</s>
<s>
The_AT tip_NN1 displacement_NN1 fluctuation_NN1 has_VHZ a_AT1 frequency_NN1 of_IO 13.6_MC Hz_NNU ,_, and_CC 0.2_MC mm_NNU amplitude_NN1 around_II a_AT1 mean_JJ displacement_NN1 of_IO 2.66_MC mm_NNU ._. 
</s>
<s>
Finally_RR ,_, we_PPIS2 also_RR note_VV0 the_AT work_NN1 of_IO Hsu_NP1 ,_, Kakade_NP1 and_CC Zhang_NP1 (_( 2014_MC )_) ,_, who_PNQS provide_VV0 finite-sample_JJ concentration_NN1 inequalities_NN2 on_II the_AT prediction_NN1 error_NN1 of_IO random-design_JJ ridge_NN1 regression_NN1 ,_, without_IW obtaining_VVG limiting_JJ formulas_NN2 ._. 
</s>
<s>
Let_VV0 X1_FO ,_, X2_FO ,_, ..._... ,_, 
</s>
<s>
Xk_FO be_VBI k_ZZ1 uniform_JJ and_CC independent_JJ points_NN2 in_II &lsqb;_( 0_MC ,_, 1_MC1 &rsqb;_) ,_, independent_JJ from_II (_( e_ZZ1 ,_, S_ZZ1 )_) and_CC set_VV0 X_ZZ1 =_FO X1_FO ,_, X2_FO ,_, ..._... ,_, 
</s>
<s>
Xk_FO ._. 
</s>
<s>
SF_NN1 '_NULL s_ZZ1 work_NN1 is_VBZ partially_RR supported_VVN by_II NSF_NP1 DMS-1216393_MC ._. 
</s>
<s>
However_RR ,_, test_VV0 statistic(1.8)_FO makes_VVZ sense_NN1 even_CS21 when_CS22 K_ZZ1 =_FO KN_NP1 →_NULL ∞_FO ,_, which_DDQ we_PPIS2 consider_VV0 in_II Section_NN1 4.3_MC ._. 
</s>
<s>
Given_CS21 that_CS22 the_AT students_NN2 took_VVD two_MC different_JJ approaches_NN2 to_II the_AT application_NN1 of_IO the_AT discount_NN1 in_II the_AT revenue_NN1 maximization_NN1 task_NN1 suggests_VVZ that_CST the_AT wording_NN1 of_IO the_AT task_NN1 may_VM not_XX have_VHI been_VBN clear_JJ enough_RR to_II the_AT participants_NN2 ._. 
</s>
<s>
This_DD1 result_NN1 together_RL with_IW some_DD of_IO Oleinik_NP1 '_NULL s_ZZ1 other_JJ works_NN is_VBZ well_RR presented_VVN in_II the_AT monograph_NN1 &lsqb;_( 17_MC &rsqb;_) ._. 
</s>
<s>
Also_RR we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
The_AT lemma_NN1 is_VBZ proved_VVN ._. 
</s>
<s>
As_CSA outlined_VVN in_II Section_NN1 2.3_MC ,_, we_PPIS2 approximate_VV0 the_AT entropy_NN1 density_NN1 by_II (_( 22_MC )_) and_CC the_AT diffusion_NN1 matrix_NN1 by_II (_( 23_MC )_) ._. 
</s>
<s>
We_PPIS2 will_VM show_VVI a_AT1 negative_JJ answer_NN1 in_II section_NN1 3_MC ._. 
</s>
<s>
On_II the_AT other_JJ hand_NN1 ,_, if_CS (_( 2b_FO )_) holds_VVZ ,_, then_RT the_AT sequence_NN1 (_( 5.3_MC )_) splits_VVZ after_II reparametrizing_VVG the_AT action_NN1 by_II @S_FO for_IF sufficiently_RR divisible_JJ integers_NN2 d_ZZ1 (_( see_VV0 proof_NN1 of_IO Theorem_NN1 4.3_MC )_) ._. 
</s>
<s>
Given_VVN a_AT1 holomorphic_JJ deformation_NN1 B_ZZ1 for_IF @S_FO at_II p_ZZ1 as_CSA in_II Definition_NN1 2.9_MC ,_, one_MC1 may_VM also_RR define_VVI the_AT rank_NN1 of_IO the_AT deformation_NN1 to_TO be_VBI @F_FO ._. 
</s>
<s>
The_AT metric_JJ spaces_NN2 @S_FO ,_, are_VBR all_DB coarsely_RR embeddable_JJ ,_, and_CC the_AT sequence_NN1 @S_FO is_VBZ equi-coarsely_RR embeddable_JJ into_II a_AT1 Banach_NN1 space_NN1 that_CST has_VHZ a_AT1 spreading_JJ model_NN1 E_ZZ1 generated_VVN by_II a_AT1 normalized_JJ weakly_RR null_JJ sequence_NN1 ,_, which_DDQ is_VBZ not_XX isomorphic_JJ to_II co_NN1 ._. 
</s>
<s>
In_RR21 particular_RR22 ,_, the_AT variables_NN2 zi_NN2 are_VBR updated_VVN as_CSA @F_FO ._. 
</s>
<s>
Observe_VV0 in_II the_AT update_NN1 in_II (_( 12_MC )_) that_CST the_AT variable_NN1 associated_VVN with_IW the_AT function_NN1 fit_NN1 is_VBZ set_VVN to_TO be_VBI the_AT updated_JJ variable_NN1 xt+1_FO while_CS the_AT other_JJ iterates_NN2 are_VBR simply_RR kept_VVN as_CSA their_APPGE previous_JJ value_NN1 ._. 
</s>
<s>
Also_RR ,_, an_AT1 initial_JJ Lagrangian_JJ bound_NN1 can_VM be_VBI computed_VVN from_II this_DD1 initialization_NN1 ._. 
</s>
<s>
BA_NP1 ?_? '_NULL ;_; 
</s>
<s>
"_" Numbers_NN2 -_- quantities_NN2 ._. '_NULL ;_; 
</s>
<s>
"_" 1C_FO "_" ._. 
</s>
<s>
A_AT1 sequence_NN1 of_IO 1-spheres_NN2 in_II R2_FO ,_, with_IW north_ND1 poles_NN2 spaced_VVN at_II distance_NN1 @S_FO ._. 
</s>
<s>
We_PPIS2 will_VM adaptively_RR select_VVI the_AT primal_JJ constraints_NN2 to_TO deal_VVI with_IW such_DA coefficients_NN2 as_CSA pioneered_VVN in_II &lsqb;_( 62_MC &rsqb;_) ._. 
</s>
<s>
If_CS @S_FO is_VBZ an_AT1 even_JJ integer_NN1 ,_, the_AT interaction_NN1 V_ZZ1 is_VBZ called_VVN even_RR ._. 
</s>
<s>
Also_RR ,_, say_VV0 we_PPIS2 remove_VV0 an_AT1 edge_NN1 from_II the_AT middle_JJ n-path_NN1 so_CS21 that_CS22 the_AT middle_JJ n-path_NN1 contributes_VVZ j_ZZ1 vertices_VVZ to_II the_AT top_JJ connected_JJ piece_NN1 ._. 
</s>
<s>
By_II the_AT assumption_NN1 on_II E_NP1 this_DD1 is_VBZ also_RR a_AT1 weak_JJ equivalence_NN1 ._. 
</s>
<s>
Denote_VV0 E∞_FO the_AT open_JJ domain_NN1 enclosed_VVN by_II and_CC Fj_NP1 ,_, ∞_FO ,_, j=1_FO ,_, ,_, k_ZZ1 ._. 
</s>
<s>
Similarly_RR ,_, as_CSA with_IW Lemma_NN1 2.4_MC ,_, we_PPIS2 may_VM show_VVI that_CST @F_FO ._. 
</s>
<s>
This_DD1 is_VBZ a_AT1 modification_NN1 of_IO &lsqb;_( Hai14b_FO ,_, Theorem_NN1 5.12_MC &rsqb;_) ._. 
</s>
<s>
The_AT compensation_NN1 strategy_NN1 was_VBDZ not_XX explicitly_RR taught_VVN and_CC did_VDD not_XX spontaneously_RR occur_VVI in_II kindergarten_NN1 ._. 
</s>
<s>
Typically_RR ,_, @S_FO evaluates_VVZ the_AT loss_NN1 of_IO the_AT decision_NN1 rule_NN1 parametrized_VVD by_II x_ZZ1 on_II a_AT1 data_NN point_VV0 g_ZZ1 ._. 
</s>
<s>
In_II many_DA2 common_JJ settings_NN2 ,_, null_JJ p-values_NN2 are_VBR conservative_JJ but_CCB not_XX necessarily_RR exactly_RR uniform_JJ ._. 
</s>
<s>
Computations_NN2 with_IW @S_FO and_CC m_ZZ1 >_FO 14_MC are_VBR not_XX sufficiently_RR convergent_JJ and_CC hence_RR are_VBR not_XX reported_VVN ._. 
</s>
<s>
This_DD1 is_VBZ a_AT1 condition_NN1 that_CST we_PPIS2 introduced_VVD in_II &lsqb;_( GRW14b_FO ,_, Section_NN1 5.1_MC &rsqb;_) ,_, and_CC we_PPIS2 will_VM refer_VVI there_RL for_IF some_DD of_IO its_APPGE basic_JJ properties_NN2 ._. 
</s>
<s>
Under_II this_DD1 condition_NN1 ,_, making_VVG use_NN1 of_IO the_AT property_NN1 @F_FO ,_, we_PPIS2 find_VV0 that_CST the_AT variation_NN1 of_IO the_AT cohomological_JJ observable_JJ @S_FO determined_VVN by_RP is_VBZ S-exact_JJ ._. 
</s>
<s>
We_PPIS2 consider_VV0 a_AT1 sequence_NN1 Bj_NP1 of_IO nested_JJ balls_NN2 ,_, with_IW radii_NN2 rj_NNU ,_, shrinking_VVG to_II x0_FO ,_, and_CC the_AT related_JJ gradient_NN1 averages_NN2 @F_FO ._. 
</s>
<s>
Theorem_NN1 1.2_MC (_( or_CC Theorem_NN1 1.4_MC for_IF abelian_JJ coefficients_NN2 )_) therefore_RR implies_VVZ the_AT following_JJ ._. 
</s>
<s>
We_PPIS2 now_RT apply_VV0 Theorem_NN1 1.1_MC to_TO find_VVI a_AT1 pointed_JJ affine_JJ etale_NN1 k-morphism_NN1 @S_FO that_DD1 induces_VVZ an_AT1 isomorphism_NN1 of_IO stabilizers_NN2 at_II w0_FO ._. 
</s>
<s>
In_CS21 case_CS22 (_( a_ZZ1 )_) @S_FO ,_, so_CS there_EX is_VBZ no_AT middle_JJ degree_NN1 ,_, and_CC this_DD1 is_VBZ the_AT simplest_JJT case_NN1 ,_, which_DDQ we_PPIS2 handle_VV0 in_II Example_NN1 5.8_MC ._. 
</s>
<s>
As_RG rely_VV0 on_II Lemma_NN1 8.4_MC in_II choosing_VVG to_II >_FO 0_MC suitably_RR large_JJ such_CS21 that_CS22 @F_FO and_CC conclude_VV0 ._. 
</s>
<s>
Let_VV0 @S_FO denote_VVI the_AT Jacobi_JJ orthonormal_JJ polynomials_NN2 ._. 
</s>
<s>
This_DD1 advantage_NN1 is_VBZ established_VVN through_II the_AT introduction_NN1 of_IO lower_JJR RIP_NP1 ,_, a_AT1 weaker_JJR version_NN1 of_IO RIP_NP1 that_DD1 is_VBZ associated_VVN with_IW lower_JJR sets_NN2 ,_, and_CC an_AT1 optimal_JJ choice_NN1 of_IO polynomial_NN1 subspace_NN1 ._. 
</s>
<s>
Pick_VV0 a_AT1 point_NN1 p_ZZ1 in_II @S_FO and_CC consider_VV0 the_AT holonomy_NN1 group_NN1 of_IO @S_FO at_II this_DD1 point_NN1 ,_, as_CSA a_AT1 subgroup_NN1 of_IO SQ(3)_FO (_( the_AT automorphisms_NN2 of_IO the_AT fiber_NN1 of_IO @S_FO at_II p_ZZ1 )_) ._. 
</s>
<s>
Note_VV0 that_CST this_DD1 right_JJ derivative_NN1 is_VBZ also_RR continuous_JJ in_II A._NP1 It_PPH1 is_VBZ well_RR known_VVN (_( see_VV0 ,_, e.g._REX ,_, &lsqb;_( 9_MC &rsqb;_) )_) that_CST a_AT1 function_NN1 with_IW continuous_JJ right_JJ derivative_NN1 on_II an_AT1 open_JJ interval_NN1 is_VBZ continuously_RR differentiable_JJ on_II this_DD1 interval_NN1 ._. 
</s>
<s>
Suppose_VV0 that_CST A_ZZ1 is_VBZ a_AT1 DPT_NP1 over_II @S_FO ,_, with_IW @S_FO ._. 
</s>
<s>
Assume_VV0 also_RR that_CST its_APPGE normal_JJ traces_NN2 on_II the_AT top/bottom_JJ boundaries_NN2 are_VBR bounded_VVN measures_NN2 ._. 
</s>
<s>
Suppose_VV0 that_CST there_EX exist_VV0 constants_NN2 @S_FO and_CC a_AT1 E_ZZ1 (_( 0_MC ,_, 1_MC1 &rsqb;_) such_CS21 that_CS22 @F_FO ,_, for_IF some_DD r_ZZ1 with_IW @S_FO ,_, where_CS @S_FO is_VBZ the_AT constant_JJ in_II PI(?)_NN1 ._. 
</s>
<s>
Despite_II having_VHG been_VBN consistently_RR mentioned_VVN and_CC studied_VVN during_II the_AT last_MD four_MC decades_NNT2 by_II many_DA2 researchers_NN2 representing_VVG different_JJ branches_NN2 of_IO mathematics_NN1 ,_, including_II computer_NN1 science_NN1 ,_, algebraic_JJ geometry_NN1 ,_, and_CC combinatorics_NN2 ,_, both_DB2 Conjecture_VV0 1_MC1 and_CC its_APPGE strong_JJ form_NN1 remained_VVD completely_RR open_VV0 to_II this_DD1 date_NN1 ._. 
</s>
<s>
On_II the_AT other_JJ hand_NN1 ,_, Gabbi_NP1 coordinated_VVD incremental_JJ increases_NN2 in_II one_MC1 distance_NN1 with_IW incremental_JJ decreases_NN2 in_II the_AT other_JJ on_II the_AT Pace_NN1 car_NN1 task_NN1 ,_, and_CC thus_RR thought_VVN about_II the_AT relationship_NN1 between_II these_DD2 two_MC distances_NN2 in_II a_AT1 way_NN1 that_CST reflected_VVD its_APPGE invariance_NN1 with_II31 respect_II32 to_II33 speed_NN1 ._. 
</s>
<s>
The_AT Poincare_NN1 series_NN of_IO M_ZZ1 is_VBZ the_AT generating_JJ series_NN @F_FO ._. 
</s>
<s>
Specifically_RR ,_, these_DD2 participants_NN2 had_VHD a_AT1 higher_JJR probability_NN1 of_IO reporting_VVG that_CST they_PPHS2 would_VM teach_VVI for_IF conceptual_JJ understanding_NN1 (_( 58.5%_FO )_) compared_VVN to_II those_DD2 in_II Profile_NN1 1_MC1 ._. 
</s>
<s>
Proof_NN1 First_MD write_VV0 the_AT definition_NN1 of_IO f_ZZ1 as_CSA follows_VVZ :_: @F_FO ._. 
</s>
<s>
Note_VV0 that_CST 1zizj_FO does_VDZ not_XX contribute_VVI to_II the_AT denominator_NN1 because_II21 of_II22 symmetrization_NN1 ._. 
</s>
<s>
We_PPIS2 next_MD look_NN1 at_II the_AT consequences_NN2 of_IO Theorems_NN2 4.3_MC and_CC 4.4_MC ._. 
</s>
<s>
Since_CS Ho_NP1 is_VBZ respected_VVN by_II A_AT1 '_NULL ew_NN1 ,_, the_AT flower_NN1 H0_FO also_RR admits_VVZ a_AT1 full_JJ lift_NN1 Hn_NP1 to_II the_AT dynamical_JJ plane_NN1 of_IO fn_NNU such_CS21 that_CS22 Hn_NP1 is_VBZ respected_VVN by_II n_ZZ1 ._. 
</s>
<s>
Given_CS21 that_CS22 a_AT1 significant_JJ number_NN1 of_IO students_NN2 were_VBDR missing_JJ information_NN1 on_II the_AT dependent_JJ variable_NN1 or_CC a_AT1 key_JJ control_NN1 variable_NN1 ,_, we_PPIS2 deemed_VVD it_PPH1 more_RGR appropriate_JJ to_TO exclude_VVI them_PPHO2 from_II our_APPGE analytic_JJ sample_NN1 rather_CS21 than_CS22 impute_VVI data_NN for_IF them_PPHO2 (_( for_IF a_AT1 similar_JJ approach_NN1 ,_, see_VV0 Chetty_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
2014_MC )_) ._. 
</s>
<s>
This_DD1 method_NN1 is_VBZ able_JK to_TO control_VVI the_AT condition_NN1 number_NN1 of_IO the_AT stiffness_NN1 matrix_NN1 also_RR for_IF the_AT case_NN1 of_IO higher-order_JJ discretizations_NN2 ._. 
</s>
<s>
In_II summary_NN1 ,_, for_IF Chemistry_NN1 undergraduate_NN1 ,_, grades_NN2 A_ZZ1 and_CC B_ZZ1 in_II A-Level_NN1 Mathematics_NN1 do_VD0 not_XX have_VHI clear_JJ positive_JJ effects_NN2 ,_, but_CCB grades_NN2 C/D/E_NP1 do_VD0 have_VHI negative_JJ effects_NN2 ._. 
</s>
<s>
Thus_RR the_AT estimate_NN1 in_II Lemma_NN1 6.4_MC follows_VVZ by_II the_AT standard_JJ energy_NN1 method_NN1 if_CS we_PPIS2 can_VM show_VVI that_DD1 ,_, for_IF any_DD k_ZZ1 >_FO 0_MC ,_, 1_MC1 <_FO j_ZZ1 <_FO 4_MC and_CC i_ZZ1 >_FO 0_MC with_IW @S_FO ,_, @F_FO ._. 
</s>
<s>
Note_VV0 that_CST (_( 55_MC )_) obviously_RR holds_VVZ for_IF j_ZZ1 1_MC1 because_CS @S_FO ._. 
</s>
<s>
It_PPH1 remains_VVZ to_TO consider_VVI the_AT cases_NN2 when_RRQ j_ZZ1 2_MC ,_, 3_MC ,_, 4_MC ._. 
</s>
<s>
Under_II the_AT assumptions_NN2 (_( EXP_NP1 )_) ,_, (_( BSCT_NP1 )_) ,_, and_CC (_( Green_NP1 )_) ,_, if_CS @S_FO has_VHZ the_AT form_NN1 @S_FO for_IF the_AT operators_NN2 F7_FO in_II Corollary_NN1 10.3_MC ,_, then_RT lim_VV0 lim_NN1 @S_FO ._. 
</s>
<s>
Thus_RR ,_, the_AT effect_NN1 of_IO covariance_NN1 structure_NN1 depends_VVZ on_II the_AT loss_NN1 function_NN1 ._. 
</s>
<s>
As_CSA with_IW the_AT smoothed_JJ dual_JJ formulation_NN1 ,_, we_PPIS2 can_VM also_RR obtain_VVI a_AT1 feasible_JJ primal_JJ solution_NN1 by_II averaging_VVG subgradients_NN2 ._. 
</s>
<s>
We_PPIS2 assume_VV0 that_CST X_ZZ1 ,_, is_VBZ parametrized_VVN by_II &lsqb;_( 0,1_MC &rsqb;_) and_CC framed_VVN so_CS21 that_CS22 C(0)_FO is_VBZ in_II the_AT plane_NN1 spanned_VVD by_II @S_FO as_CSA in_II Definition_NN1 5.4_MC ._. 
</s>
<s>
We_PPIS2 refer_VV0 the_AT reader_NN1 to_II those_DD2 papers_NN2 for_IF further_JJR details_NN2 ._. 
</s>
<s>
Lemma_NN1 1_MC1 implies_VVZ that_CST the_AT optimality_NN1 system_NN1 (_( 2.17_MC )_) has_VHZ a_AT1 unique_JJ solution_NN1 ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI the_AT new_JJ 2-sphere_NN1 where_CS the_AT two_MC surfaces_NN2 are_VBR identified_VVN ._. 
</s>
<s>
Let_VV0 @S_FO denote_VVI the_AT tube_NN1 in_II A_ZZ1 that_CST parallels_VVZ @S_FO ._. 
</s>
<s>
Sliding_VVG T(k)_NP1 over_II f_ZZ1 (_( p_ZZ1 )_) entangles_VVZ T(t)_UH with_IW T(k)_NP1 ._. 
</s>
<s>
Let_VV0 a_AT1 <_FO b_ZZ1 ,_, and_CC p_ZZ1 ?_NNU C_ZZ1 1&lsqb;a_FO ,_, b_ZZ1 &rsqb;_) satisfy_VV0 @F_FO ._. 
</s>
<s>
If_CS S_ZZ1 =_FO 0_MC ,_, then_RT clearly_RR @S_FO ._. 
</s>
<s>
Our_APPGE system_NN1 of_IO equations_NN2 really_RR only_RR has_VHZ two_MC degrees_NN2 of_IO freedom_NN1 ,_, because_II21 of_II22 the_AT condition_NN1 @S_FO ,_, but_CCB because_CS we_PPIS2 are_VBR interested_JJ in_II the_AT ratios_NN2 of_IO the_AT eigenvalues_NN2 asymptotically_RR ,_, we_PPIS2 will_VM reduce_VVI the_AT system_NN1 to_II the_AT two_MC parameters_NN2 @S_FO ._. 
</s>
<s>
These_DD2 two_MC parameters_NN2 completely_RR determine_VV0 our_APPGE system_NN1 because_CS @S_FO ._. 
</s>
<s>
We_PPIS2 now_RT will_VM rewrite_VVI our_APPGE system_NN1 of_IO ODEs_NN2 as_CSA follows_VVZ :_: @F_FO ._. 
</s>
<s>
For_IF i_ZZ1 ,_, j∈J_FO ,_, let_VV0 =RL(i)z_FO ,_, L(j)z_NP1 :_: @S_FO ;_; i.e._REX ,_, the_AT RJ(i+j)-module_FO homomorphism_NN1 given_VVN by_II @F_FO ,_, where_CS 1_MC1 is_VBZ the_AT intertwiner_NN1 defined_VVN in_II Sect._NP1 1.6_MC ._. 
</s>
<s>
Next_MD ,_, we_PPIS2 show_VV0 that_DD1 system_NN1 (_( 3.30_MC )_) coupled_VVD with_IW "_" S-part_NN1 "_" of_IO (_( 3.29_MC )_) is_VBZ Mittag-Leffler_NP1 stable_JJ ._. 
</s>
<s>
If_CS @S_FO is_VBZ a_AT1 generic_JJ regular_JJ homotopy_NN1 with_IW @S_FO an_AT1 embedded_JJ surface_NN1 ,_, M_ZZ1 a_AT1 smooth_JJ 4-manifold_NN1 ,_, G_ZZ1 a_AT1 transverse_JJ embedded_JJ sphere_NN1 to_II @S_FO ,_, and_CC ft_NNU is_VBZ supported_VVN away_II21 from_II22 G_ZZ1 ,_, then_RT ft_NNU is_VBZ shadowed_VVN by_II tubed_JJ surfaces_NN2 ._. 
</s>
<s>
He_PPHS1 also_RR reported_VVD that_CST the_AT errors_NN2 are_VBR mostly_RR of_IO overestimation_NN1 ._. 
</s>
<s>
Participants_NN2 experienced_VVD metacognitive_JJ blindness_NN1 when_CS they_PPHS2 did_VDD not_XX notice_VVI that_CST an_AT1 assumption_NN1 was_VBDZ insufficiently_RR mathematized_VVN and_CC continued_VVN on_RP ._. 
</s>
<s>
This_DD1 result_NN1 ,_, that_DD1 was_VBDZ only_RR known_VVN to_TO be_VBI true_JJ for_IF @S_FO ,_, is_VBZ optimal_JJ :_: @S_FO is_VBZ a_AT1 W1,2W1,2_FO singular_JJ stable_JJ solution_NN1 for_IF @S_FO ._. 
</s>
<s>
Then_RT with_IW the_AT above_JJ notation_NN1 ,_, for_IF almost_RR all_DB @S_FO ,_, the_AT function_NN1 @S_FO is_VBZ the_AT probabilistic_JJ solution_NN1 of_IO (_( 17_MC )_) on_II Q_ZZ1 (_( w_ZZ1 )_) ,_, with_IW initial_JJ condition_NN1 f_ZZ1 and_CC boundary_NN1 condition_NN1 0_MC ._. 
</s>
<s>
Strictly_RR speaking_VVG ,_, this_DD1 mechanism_NN1 given_VVN by_II the_AT application_NN1 of_IO scattering_VVG maps_NN2 produce_VV0 indeed_RR pseudo-orbits_NN2 ,_, that_REX21 is_REX22 ,_, heteroclinic_JJ connections_NN2 between_II different_JJ periodic_JJ orbits_NN2 in_II the_AT infinity_NN1 manifold_NN1 which_DDQ are_VBR commonly_RR known_VVN as_II transition_NN1 chains_NN2 after_II Arnold_NP1 '_NULL s_ZZ1 pioneering_JJ work_NN1 &lsqb;_( Arn64_FO &rsqb;_) ._. 
</s>
<s>
Consider_VV0 an_AT1 elevation_NN1 @S_FO of_IO a_AT1 surface_NN1 in_II Sh_UH and_CC suppose_VV0 that_CST the_AT stabilizer_NN1 of_IO S_ZZ1 intersects_VVZ a_AT1 stabilizer_NN1 of_IO a_AT1 boundary_NN1 plane_NN1 T_ZZ1 along_II an_AT1 infinite_JJ cyclic_JJ group_NN1 ._. 
</s>
<s>
Proof_NN1 Recall_VV0 that_CST the_AT N-orbit_NN1 of_IO any_DD @S_FO is_VBZ a_AT1 closed_JJ torus_NN1 ._. 
</s>
<s>
Note_VV0 that_CST hm_UH solves_VVZ the_AT equation_NN1 (_( see_VV0 Lemma_NN1 B.2_FO )_) @F_FO ,_, where_CS @S_FO and_CC @F_FO ._. 
</s>
<s>
Clearly_RR ,_, @F_FO ._. 
</s>
<s>
It_PPH1 remains_VVZ to_TO estimate_VVI the_AT terms_NN2 on_II the_AT right_JJ hand_NN1 side_NN1 ,_, which_DDQ will_VM be_VBI done_VDN in_II the_AT next_MD two_MC lemmas_NN2 ._. 
</s>
<s>
In_RR21 particular_RR22 ,_, for_IF fixed_JJ g_ZZ1 ,_, itis_NN1 enough_RR to_TO have_VHI @S_FO for_IF @S_FO (_( and_CC hence_RR @S_FO )_) to_TO be_VBI strongly_RR consistent_JJ ._. 
</s>
<s>
We_PPIS2 now_RT move_VV0 to_II the_AT second_MD remainder_NN1 term_NN1 R2_FO (_( f_ZZ1 )_) g_ZZ1 ._. 
</s>
<s>
Throughout_II its_APPGE history_NN1 in_II research_NN1 ,_, it_PPH1 stubbornly_RR dodged_JJ operationalization_NN1 ._. 
</s>
<s>
From_II (_( 3.26_MC )_) we_PPIS2 can_VM consider_VVI the_AT vector_NN1 @S_FO ._. 
</s>
<s>
Following_VVG the_AT same_DA process_NN1 as_CSA before_RT ,_, we_PPIS2 have_VH0 that_DD1 @F_FO ._. 
</s>
<s>
Grouping_NN1 terms_NN2 we_PPIS2 obtain_VV0 @F_FO ._. 
</s>
<s>
And_CC due_II21 to_II22 the_AT fact_NN1 that_CST @S_FO ,_, @F_FO ._. 
</s>
<s>
From_II (_( 3.26_MC )_) and_CC (_( 3.27_MC )_) it_PPH1 is_VBZ clear_JJ that_CST m_ZZ1 =_FO 2t_FO for_IF kinks_NN2 and_CC m_ZZ1 =_FO 4t_FO for_IF jumps_NN2 ._. 
</s>
<s>
More_RGR generally_RR ,_, we_PPIS2 could_VM replace_VVI the_AT fitting_JJ procedure_NN1 in_II the_AT M-step_NN1 by_II penalized_JJ GLM_NP1 (_( glmnet_NN1 package_NN1 )_) ,_, a_AT1 generalized_JJ additive_JJ model_NN1 (_( gam_NN1 or_CC mgcv_NNU package_NN1 )_) or_CC generalized_JJ boostingregression_NN1 (_( gbm_NNU package_NN1 )_) ._. 
</s>
<s>
We_PPIS2 are_VBR now_RT ready_JJ to_TO state_VVI our_APPGE sufficient_JJ conditions_NN2 for_IF assumption_NN1 (_( A5_FO )_) as_II the_AT following_JJ immediate_JJ corollaries_NN2 of_IO Lemma_NN1 5.3_MC and_CC Theorem_NN1 5.3_MC ._. 
</s>
<s>
Let_VV0 (_( 0_MC ,_, f_ZZ1 (_( 0_MC )_) )_) in_II the_AT physical_JJ plane_NN1 be_VBI transformed_VVN into_II (_( 0_MC ,_, M_ZZ1 )_) in_II the_AT potential_JJ plane_NN1 for_IF Q._NN1 It_PPH1 follows_VVZ from_II &lsqb;_( 32_MC ,_, Proposition_NN1 48_MC and_CC Theorems_NN2 24_MC ,_, 39_MC &rsqb;_) ,_, where_CS we_PPIS2 obtained_VVD the_AT structure_NN1 of_IO sonic_JJ curves_NN2 and_CC the_AT properties_NN2 of_IO the_AT characteristics_NN2 from_II the_AT sonic_JJ points_NN2 for_IF smooth_JJ transonic_JJ flows_NN2 ,_, that_CST in_II the_AT potential_JJ plane_NN1 ,_, the_AT positive_JJ characteristic_NN1 from_II (_( 0_MC ,_, M_ZZ1 )_) intersects_VVZ the_AT sonic_JJ curve_NN1 at_II a_AT1 nonexceptional_JJ point_NN1 ,_, while_CS the_AT negative_JJ characteristic_NN1 S_ZZ1 from_II (_( 0_MC ,_, M_ZZ1 )_) is_VBZ located_VVN in_II the_AT supersonic_JJ region_NN1 ._. 
</s>
<s>
In_II a_AT1 similar_JJ way_NN1 @S_FO ,_, and_CC the_AT result_NN1 follows_VVZ ._. 
</s>
<s>
We_PPIS2 set_VV0 @F_FO ._. 
</s>
<s>
Note_VV0 that_CST we_PPIS2 will_VM only_RR ever_RR consider_VVI moduli_NN2 stacks_NN2 of_IO modules_NN2 supported_VVN on_II the_AT principal_JJ parts_NN2 of_IO quivers_NN2 ._. 
</s>
<s>
We_PPIS2 illustrate_VV0 applications_NN2 of_IO the_AT framework_NN1 developed_VVN in_II Section_NN1 3_MC on_II convergence_NN1 and_CC error_NN1 estimation_NN1 for_IF numerical_JJ solutions_NN2 of_IO a_AT1 number_NN1 of_IO static_JJ contact_NN1 problems_NN2 with_IW elastic_JJ materials_NN2 ._. 
</s>
<s>
The_AT limit_NN1 problem_NN1 The_AT limits_NN2 of_IO the_AT previous_JJ section_NN1 allow_VV0 to_TO investigate_VVI the_AT limit_NN1 of_IO the_AT elastic_JJ problem_NN1 ._. 
</s>
<s>
Mathematics_NN1 education_NN1 should_VM offer_VVI youth_NN1 opportunities_NN2 to_TO quantify_VVI ,_, measure_VV0 ,_, project_NN1 ,_, and_CC model_NN1 migration_NN1 ;_; to_TO understand_VVI and_CC evaluate_VVI models_NN2 currently_RR in_II use_NN1 ;_; and_CC to_TO engage_VVI with_IW considering_VVG how_RRQ mathematics_NN1 might_VM be_VBI insufficient_JJ or_CC misguided_JJ relative_II21 to_II22 human_JJ rights_NN2 ._. 
</s>
<s>
This_DD1 finding_NN1 highlights_VVZ the_AT critical_JJ role_NN1 of_IO mathematical_JJ fluency_NN1 and_CC understanding_VVG for_IF the_AT planning_NN1 of_IO pedagogy_NN1 ._. 
</s>
<s>
In_II (_( 6.8_MC )_) ,_, @S_FO contains_VVZ the_AT boundary_NN1 data_NN @S_FO on_II boundary_NN1 b_ZZ1 at_II time_NNT1 level_NN1 k_ZZ1 ,_, @S_FO is_VBZ from_II (_( 5.11_MC )_) and_CC @S_FO ,_, where_CS each_DD1 @S_FO is_VBZ a_AT1 bounded_JJ positive_JJ definite_JJ matrix_NN1 ._. 
</s>
<s>
In_II the_AT proof_NN1 of_IO Theorem_NN1 3.1_MC we_PPIS2 will_VM have_VHI the_AT following_JJ dichotomy_NN1 :_: Take_VV0 p∈Z_FO ._. 
</s>
<s>
The_AT point_NN1 p_ZZ1 is_VBZ said_VVN to_TO be_VBI wild_JJ if_CS p∈_FO (_( x_ZZ1 ,_, L04_FO )_) ._. 
</s>
<s>
SDP-1_MC1 ,_, in_II its_APPGE basic_JJ form_NN1 ,_, tends_VVZ to_TO partition_VVI the_AT network_NN1 into_II blocks_NN2 of_IO similar_JJ sizes_NN2 ._. 
</s>
<s>
We_PPIS2 construct_VV0 the_AT extensive_JJ formulation_NN1 on_II the_AT scenario_NN1 tree_NN1 and_CC use_VV0 CPLEX_VV0 to_TO solve_VVI the_AT problem_NN1 as_CSA one_MC1 large_JJ MIP_NN1 ._. 
</s>
<s>
Lemma_NN1 4.1_MC Fix_VV0 n_ZZ1 ,_, N∈N_FO ._. 
</s>
<s>
Then_RT for_IF any_DD ε>0_FO there_EX is_VBZ ε0>0_FO such_CS21 that_CS22 for_IF any_DD simplicial_JJ complex_NN1 E_ZZ1 of_IO dimension_NN1 at_II most_DAT N_ZZ1 we_PPIS2 have_VH0 for_IF all_DB @S_FO @F_FO ._. 
</s>
<s>
This_DD1 includes_VVZ different_JJ valley_NN1 formations_NN2 ,_, namely_REX cuboids_NN2 of_IO varying_JJ side_NN1 lengths_NN2 ,_, cf._VV0 Situational_JJ interest_NN1 is_VBZ seen_VVN as_II a_AT1 certain_JJ motivational_JJ state_NN1 (_( Hidi_NP1 &;_NULL Renninger_NP1 ,_, 2006_MC )_) ._. 
</s>
<s>
In_II their_APPGE current_JJ instantiation_NN1 ,_, it_PPH1 does_VDZ not_XX seem_VVI possible_JJ to_TO solve_VVI quasilinear_JJ SPDEs_NN2 via_II regularity_NN1 structures_NN2 ._. 
</s>
<s>
However_RR ,_, Mance_NP1 was_VBDZ not_XX attending_VVG to_II the_AT differential_JJ equation_NN1 itself_PPX1 ._. 
</s>
<s>
Assume_VV0 further_RRR that_DD1 @S_FO ,_, for_IF every_AT1 k_ZZ1 ._. 
</s>
<s>
Then_RT ,_, @F_FO ._. 
</s>
<s>
Also_RR ,_, as_CSA researchers_NN2 partaking_VVG in_II this_DD1 study_NN1 ,_, we_PPIS2 both_DB2 assemble_VV0 with_IW the_AT data_NN and_CC the_AT theoretical_JJ perspectives_NN2 that_CST we_PPIS2 are_VBR investigating_VVG ._. 
</s>
<s>
Recall_VV0 that_CST @S_FO is_VBZ the_AT ergodic_JJ decomposition_NN1 of_IO Q._NP1 The_AT values_NN2 of_IO the_AT Euler_NN1 form_NN1 are_VBR determined_VVN from_II the_AT exact_JJ sequences_NN2 above_RL ._. 
</s>
<s>
For_IF this_DD1 we_PPIS2 need_VV0 that_DD1 sin_VVI 2p_NNU goes_VVZ to_TO zero_VVI (_( at_RR21 least_RR22 linearly_RR )_) where_RRQ v_ZZ1 =_FO 0_MC ._. 
</s>
<s>
Here_RL ,_, heavy-tailed_NN1 is_VBZ understood_VVN in_II the_AT sense_NN1 that_CST only_RR a_AT1 finite_JJ number_NN1 of_IO moments_NN2 exist_VV0 ._. 
</s>
<s>
For_IF @S_FO ,_, write_VV0 @S_FO if_CS i_ZZ1 is_VBZ a_AT1 neighbor_NN1 of_IO j_ZZ1 ._. 
</s>
<s>
We_PPIS2 write_VV0 dg_NNU for_IF the_AT graph_NN1 distance_NN1 in_II G_ZZ1 ,_, and_CC for_IF two_MC subsets_NN2 U_ZZ1 ,_, U_ZZ1 '_NULL of_IO V_ZZ1 ,_, we_PPIS2 define_VV0 @S_FO ._. 
</s>
<s>
While_CS it_PPH1 is_VBZ true_JJ that_CST the_AT conspicuous_JJ manifestations_NN2 of_IO complete_JJ integrability_NN1 of_IO KdV_NP1 played_VVD no_AT particular_JJ role_NN1 in_II the_AT series_NN of_IO works_NN we_PPIS2 have_VH0 just_RR described_VVN ,_, it_PPH1 is_VBZ difficult_JJ to_TO completely_RR decouple_VVI these_DD2 successes_NN2 from_II the_AT exact_JJ structure_NN1 of_IO the_AT KdV_NP1 equation_NN1 ._. 
</s>
<s>
Consider_VV0 the_AT function_NN1 f_ZZ1 presented_VVN in_II Example_NN1 3.2_MC ._. 
</s>
<s>
Finally_RR ,_, when_CS @S_FO and_CC @S_FO ,_, (_( 3.1_MC )_) shows_VVZ that_CST @S_FO ._. 
</s>
<s>
Teachers_NN2 used_VVD particular_JJ strategies_NN2 to_TO increase_VVI student-directed_JJ learning_NN1 ._. 
</s>
<s>
By_II the_AT discussion_NN1 in_II Section_NN1 4.2_MC ,_, it_PPH1 suffices_VVZ to_TO show_VVI that_CST one_PN1 has_VHZ the_AT following_JJ expansion_NN1 as_CSA z_ZZ1 goes_VVZ to_II 0_MC :_: @F_FO ._. 
</s>
<s>
Note_VV0 that_CST since_II now_RT @S_FO ,_, we_PPIS2 need_VV0 a_AT1 Taylor_NP1 expansion_NN1 only_RR to_II 0-th_JJ order_NN1 ._. 
</s>
<s>
By_II doubling_VVG this_DD1 standard_JJ model_NN1 ,_, we_PPIS2 obtain_VV0 a_AT1 foam_NN1 in_II @S_FO with_IW two_MC seams_NN2 of_IO the_AT same_DA type_NN1 ._. 
</s>
<s>
Both_RR series_NN are_VBR related_VVN through_II @S_FO ._. 
</s>
<s>
In_II the_AT next_MD Lemma_NN1 we_PPIS2 see_VV0 that_CST the_AT terms_NN2 @F_FO of_IO second_MD order_NN1 with_II31 respect_II32 to_II33 s_ZZ1 satisfy_VV0 a_AT1 very_RG exponentially_RR small_JJ bound_NN1 for_IF large_JJ G._NP1 They_PPHS2 rated_VVD the_AT relevance_NN1 of_IO each_DD1 misconception_NN1 in_II the_AT participants_NN2 '_NULL memories_NN2 of_IO their_APPGE own_DA school_NN1 day_NNT1 (_( nine-point_JJ scale_NN1 ,_, with_IW responses_NN2 ranging_VVG from_II "_" extremely_RR relevant_JJ "_" to_II "_" not_XX at_RR21 all_RR22 relevant_JJ "_" )_) (_( one_MC1 screen_NN1 page_NN1 )_) ._. 
</s>
<s>
Let_VV0 @S_FO satisfy_VVI @S_FO ._. 
</s>
<s>
Following_VVG our_APPGE conventions_NN2 ,_, we_PPIS2 define_VV0 @F_FO and_CC @F_FO ._. 
</s>
<s>
By_II Corollary_NN1 4.14_MC ,_, @F_FO is_VBZ a_AT1 union_NN1 of_IO connected_JJ components_NN2 of_IO @S_FO ,_, and_CC so_RR there_EX are_VBR natural_JJ isomorphisms_NN2 @F_FO and_CC @F_FO ._. 
</s>
<s>
Schur_VV0 nonnegative_JJ specializations_NN2 of_IO Sym_NP1 are_VBR specializations_NN2 taking_VVG values_NN2 in_II R>0_FO when_CS applied_VVN to_TO skew_VVI Schur_NN1 functions_NN2 @S_FO for_IF any_DD partitions_NN2 k_ZZ1 and_CC @S_FO ._. 
</s>
<s>
Thoma_NN1 '_NULL s_ZZ1 theorem_NN1 (_( see_VV0 &lsqb;_( 30_MC &rsqb;_) and_CC references_NN2 therein_RR )_) provides_VVZ a_AT1 classification_NN1 of_IO such_DA specializations_NN2 ._. 
</s>
<s>
These_DD2 are_VBR constructed_VVN as_CSA suspensions_NN2 over_II Axiom_NN1 A_ZZ1 maps_NN2 with_IW piecewise_JJ constant_NN1 (_( but_CCB not_XX constant_JJ )_) return_VV0 time_NNT1 and_CC consequently_RR are_VBR not_XX Anosov_NP1 flows_VVZ ._. 
</s>
<s>
Now_RT we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Hence_RR ,_, we_PPIS2 deduce_VV0 the_AT equation_NN1 @F_FO ._. 
</s>
<s>
From_II (_( 9.7_MC )_) ,_, we_PPIS2 see_VV0 that_CST @S_FO ._. 
</s>
<s>
To_TO get_VVI the_AT result_NN1 ,_, we_PPIS2 only_RR need_VV0 to_TO prove_VVI that_CST @S_FO ._. 
</s>
<s>
If_CS β_NULL =B0_FO ,_, B1_FO is_VBZ a_AT1 pre-partition_JJ satisfying_JJ @F_FO ,_, then_RT R∈-algredG_JJ (_( β_NULL )_) ._. 
</s>
<s>
In_II 1945_MC Polya_NP1 anticipated_VVD a_AT1 lot_NN1 of_IO what_DDQ Wing_NN1 is_VBZ putting_VVG forward_RL ,_, and_CC more_RRR ._. 
</s>
<s>
The_AT Z/2Z-graded_FU k-linear_JJ category_NN1 K_ZZ1 is_VBZ freely_RR generated_VVN by_II the_AT Kronecker_NN1 quiver_NN1 with_IW two_MC vertices_NN2 and_CC two_MC parallel_JJ arrows_NN2 ._. 
</s>
<s>
Remark_VV0 3.2_MC (_( Market-implied_NN1 vs._II innate_JJ discounting_NN1 )_) ._. 
</s>
<s>
Examples_NN2 include_VV0 devising_VVG a_AT1 mathematical_JJ proof_NN1 of_IO a_AT1 statement_NN1 ,_, finding_VVG and_CC implementing_VVG a_AT1 way_NN1 to_TO solve_VVI a_AT1 mathematical_JJ problem_NN1 ,_, reducing_VVG a_AT1 Fig._NN1 1_MC1 A_ZZ1 visual_JJ representation_NN1 of_IO the_AT eight_MC mathematical_JJ competencies_NN2 (_( adapted_VVN from_II Niss_NP1 &;_NULL Jensen_NP1 ,_, 2002_MC ,_, p._NN1 45_MC )_) @T_FO complex_JJ symbolic_JJ expression_NN1 to_II an_AT1 as_RG simple_JJ as_CSA possible_JJ equivalent_JJ expression_NN1 ,_, or_CC actively_RR communicating_VVG one_MC1 '_NULL s_ZZ1 building_NN1 of_IO a_AT1 mathematical_JJ model_NN1 ._. 
</s>
<s>
This_DD1 was_VBDZ not_XX the_AT end_NN1 of_IO the_AT students_NN2 '_NULL discussion_NN1 of_IO the_AT task_NN1 ;_; as_CSA captured_VVN in_II the_AT next_MD section_NN1 ,_, their_APPGE discussion_NN1 next_MD turned_VVN to_II the_AT issue_NN1 of_IO the_AT definition_NN1 of_IO a_AT1 mathematical_JJ object_NN1 ._. 
</s>
<s>
For_IF this_DD1 reason_NN1 ,_, in_II the_AT sequel_NN1 pd_NNU in_II (_( 16_MC )_) will_VM be_VBI called_VVN a_AT1 Christoffel_NN1 polynomial_NN1 ._. 
</s>
<s>
In_II this_DD1 section_NN1 ,_, we_PPIS2 prove_VV0 Theorem_NN1 2.1_MC starting_VVG from_II Corollary_NN1 4.7_MC ._. 
</s>
<s>
In_II this_DD1 case_NN1 ,_, we_PPIS2 aim_VV0 at_II constructing_VVG a_AT1 legal_JJ canonical-path_NN1 that_CST empties_VVZ @S_FO using_VVG only_RR flips_VVZ there_RL ._. 
</s>
<s>
Therefore_RR ,_, it_PPH1 follows_VVZ from_II (_( 3.14_MC )_) that_CST @F_FO ._. 
</s>
<s>
Observe_VV0 that_CST Mq_NP1 <_FO M_ZZ1 1_MC1 ._. 
</s>
<s>
We_PPIS2 prove_VV0 directional_JJ FDR_NP1 control_NN1 by_II conditioning_NN1 on_II @S_FO ._. 
</s>
<s>
For_IF any_DD y(°)such_FO that_CST the_AT event_NN1 E_ZZ1 holds_VVZ ,_, we_PPIS2 will_VM show_VVI that_DD1 for_IF the_AT knockoff_NN1 procedure_NN1 ,_, @F_FO ._. 
</s>
<s>
This_DD1 will_VM not_XX create_VVI any_DD issues_NN2 for_IF us_PPIO2 ,_, as_CSA the_AT results_NN2 in_II &lsqb;_( Bam17_FO &rsqb;_) surprisingly_RR do_VD0 not_XX depend_VVI on_II such_DA a_AT1 curvature_NN1 characterization_NN1 ._. 
</s>
<s>
The_AT case_NN1 when_CS @S_FO ._. 
</s>
<s>
Note_VV0 that_CST @S_FO imposes_VVZ that_CST @S_FO in_II this_DD1 case_NN1 ._. 
</s>
<s>
Specifically_RR ,_, we_PPIS2 consider_VV0 two_MC coupled_JJ copies_NN2 @S_FO ,_, where_CS @S_FO is_VBZ a_AT1 fixed_JJ parameter_NN1 indicating_VVG some_DD fraction_NN1 of_IO common_JJ edges_NN2 in_II the_AT two_MC graphs_NN2 ._. 
</s>
<s>
Indeed_RR ,_, for_IF the_AT function_NN1 concerned_JJ in_II Figure_NN1 1_MC1 ,_, consisting_VVG of_IO trigonometric_JJ and_CC rational_JJ univariate_NN1 functions_NN2 ,_, The_AT case_NN1 when_CS Z1_FO =_FO Z2_FO was_VBDZ obtained_VVN in_II &lsqb;_( 4_MC &rsqb;_) :_: take_VV0 the_AT square_NN1 of_IO (_( 8.17_MC )_) in_II Lemma_NN1 8.2_MC ,_, whose_DDQGE proof_NN1 was_VBDZ given_VVN in_II Section_NN1 9.2_MC ._. 
</s>
<s>
Then_RT we_PPIS2 have_VH0 @S_FO if_CS one_MC1 of_IO the_AT conditions_NN2 in_II the_AT lemma_NN1 holds_VVZ ._. 
</s>
<s>
The_AT following_JJ setting_NN1 will_VM be_VBI kept_VVN in_II the_AT rest_NN1 of_IO the_AT paper_NN1 ._. 
</s>
<s>
We_PPIS2 begin_VV0 with_IW the_AT proof_NN1 of_IO the_AT global_JJ existence_NN1 ._. 
</s>
<s>
As_CSA we_PPIS2 will_VM see_VVI later_RRR ,_, this_DD1 is_VBZ also_RR true_JJ for_IF general_JJ radial_JJ solutions_NN2 ._. 
</s>
<s>
Besides_RR ,_, their_APPGE analysis_NN1 for_IF general_JJ loss_NN1 functions_NN2 also_RR requires_VVZ the_AT restrictive_JJ assumption_NN1 :_: @S_FO ,_, where_CS R_ZZ1 is_VBZ a_AT1 constant_JJ and_CC does_VDZ not_XX scale_VVI with_IW (_( n_ZZ1 ,_, s*_FO ,_, d_ZZ1 )_) ._. 
</s>
<s>
Mathematical_JJ inquiry_NN1 presents_VVZ evident_JJ similarities_NN2 with_IW scientific_JJ inquiry_NN1 ._. 
</s>
<s>
Remark_VV0 4_MC In_II (_( 41_MC )_) ,_, the_AT resolvents_NN2 are_VBR assumed_VVN to_TO be_VBI computed_VVN exactly_RR to_TO simplify_VVI the_AT presentation_NN1 ._. 
</s>
<s>
The_AT Bethe/gauge_NN1 correspondence_NN1 between_II the_AT supersymmetric_JJ gauge_NN1 theories_NN2 and_CC quantum_NN1 integrable_JJ systems_NN2 is_VBZ a_AT1 subject_NN1 of_IO research_NN1 spanning_VVG over_RP a_AT1 decade_NNT1 &lsqb;_( 1–9_MCMC &rsqb;_) and_CC even_RR longer_RRR in_II the_AT context_NN1 of_IO topological_JJ gauge_NN1 theories_NN2 &lsqb;_( 10–13_MCMC &rsqb;_) ._. 
</s>
<s>
There_EX is_VBZ a_AT1 subtlety_NN1 :_: for_REX21 example_REX22 ,_, the_AT natural_JJ composition_NN1 is_VBZ @S_FO ;_; to_TO define_VVI an_AT1 @Sstructure_FO on_II @S_FO itself_PPX1 requires_VVZ a_AT1 careful_JJ choice_NN1 of_IO Hamiltonian_JJ perturbations_NN2 depending_II21 on_II22 the_AT domain_NN1 and_CC the_AT use_NN1 of_IO the_AT Liouville_NP1 flow_NN1 to_II "_" rescale_NN1 "_" from_II kH_NNU to_II H_ZZ1 ;_; see_VV0 &lsqb;_( 1_MC1 &rsqb;_) ._. 
</s>
<s>
If_CS X_ZZ1 has_VHZ the_AT resolution_NN1 property_NN1 (_( e.g._REX ,_, @S_FO ,_, where_CS @S_FO )_) ,_, then_RT X_ZZ1 is_VBZ coherently_RR complete_VV0 along_II @S_FO &lsqb;_( GZB15_FO ,_, Th._NP1 1.1_MC &rsqb;_) ._. 
</s>
<s>
Erickson_NP1 (_( 2011_MC )_) describes_VVZ pedagogical_JJ commitments_NN2 as_CSA "_" basic_JJ ontological_JJ assumptions_NN2 ,_, both_RR tacit_JJ and_CC explicit_JJ ,_, concerning_VVG manifold_JJ aspects_NN2 of_IO teaching_NN1 and_CC learning_VVG activity_NN1 ..._... "_" 
</s>
<s>
A_AT1 stability_NN1 property_NN1 of_IO type_NN1 (_( 1.2_MC )_) for_IF Alikhanov-type_JJ semidiscretizations_NN2 will_VM be_VBI obtained_VVN in_II section_NN1 4.4_MC ,_, which_DDQ will_VM allow_VVI to_TO extend_VVI our_APPGE error_NN1 analysis_NN1 to_II this_DD1 case_NN1 ._. 
</s>
<s>
Esma_NP1 :_: I_PPIS1 considered_VVD the_AT gears_NN2 as_II the_AT wheels_NN2 ofa_NN1 tractor_NN1 ._. 
</s>
<s>
The_AT first_MD relates_VVZ to_II seeking_VVG similarities_NN2 in_II different_JJ examples_NN2 :_: in_II lesson_NN1 US3_FO ritual-enabling_JJ OTLs_NN2 were_VBDR used_VVN to_TO produce_VVI examples_NN2 (_( e.g._REX ,_, @S_FO ;_; @S_FO ;_; @S_FO )_) ._. 
</s>
<s>
The_AT set_NN1 of_IO @S_FO that_CST satisfy_VV0 the_AT assumptions_NN2 of_IO Theorem_NN1 2.3_MC is_VBZ non_FU empty_JJ for_IF y_ZZ1 v_ZZ1 V2_FO ._. 
</s>
<s>
Equivalence_NN1 of_IO Gaussian_JJ measures_NN2 &lsqb;_( Ibragimov_NP1 and_CC Rozanov_NP1 (_( 1978_MC )_) ,_, Skorokhod_NP1 and_CC Yadrenko_NP1 (_( 1973_MC )_) &rsqb;_) represents_VVZ an_AT1 essential_JJ tool_NN1 to_TO establish_VVI the_AT asymptotic_JJ properties_NN2 of_IO both_RR prediction_NN1 and_CC estimation_NN1 of_IO Gaussian_JJ fields_NN2 under_II fixed_JJ domain_NN1 asymptotics_NN2 ._. 
</s>
<s>
Set_VV0 @S_FO and_CC go_VV0 to_II 2._MC ,_, unless_CS only_RR one_MC1 detail_NN1 coefficient_NN1 was_VBDZ extracted_VVN in_II step_NN1 4_MC ,_, for_IF which_DDQ ,_, necessarily_RR ,_, @S_FO ._. 
</s>
<s>
At_II this_DD1 point_NN1 ,_, the_AT TGUH_NN1 transform_VV0 is_VBZ completed_VVN ._. 
</s>
<s>
To_II our_APPGE knowledge_NN1 ,_, this_DD1 connection_NN1 between_II GES_NP2 and_CC the_AT Chow-Liu_NP1 algorithm_NN1 can_VM not_XX be_VBI found_VVN in_II the_AT literature_NN1 on_II the_AT Chow-Liu_NP1 algorithm_NN1 and_CC extensions_NN2 thereof_RR for_IF learning_VVG polytrees_NN2 &lsqb;_( Rebane_NP1 and_CC Pearl_NP1 (_( 1987_MC )_) ,_, Huete_NP1 and_CC de_NP1 Campos_NP2 (_( 1993_MC )_) ,_, de_NP1 Campos_NP2 (_( 1998_MC )_) ,_, Ouerd_NP1 ,_, Oommen_NP1 and_CC Matwin_NP1 (_( 2004_MC )_) &rsqb;_) ._. 
</s>
<s>
However_RR ,_, we_PPIS2 make_VV0 here_RL a_AT1 few_DA2 brief_JJ comments_NN2 on_II what_DDQ is_VBZ known_VVN concerning_II these_DD2 conjectures_NN2 and_CC which_DDQ will_VM be_VBI used_VVN in_II the_AT proof_NN1 of_IO the_AT main_JJ theorem_NN1 of_IO this_DD1 paper_NN1 ._. 
</s>
<s>
This_DD1 corollary_NN1 is_VBZ an_AT1 immediate_JJ consequence_NN1 of_IO previous_JJ results_NN2 ._. 
</s>
<s>
It_PPH1 comprises_VVZ 0.2%_FO of_IO the_AT U.S._NP1 labor_NN1 force_NN1 ,_, and_CC is_VBZ projected_VVN to_TO decline_VVI 8%_NNU over_II the_AT next_MD decade_NNT1 ,_, owing_VVG mainly_RR to_II foreign_JJ competition_NN1 ._. 
</s>
<s>
We_PPIS2 now_RT consider_VV0 the_AT uniqueness_NN1 question_NN1 in_II Conjecture_NN1 2_MC ,_, and_CC whether_CSW it_PPH1 could_VM hold_VVI for_IF the_AT maximization_NN1 problem_NN1 ._. 
</s>
<s>
The_AT narrative_NN1 consisted_VVN of_IO two_MC essential_JJ parts_NN2 :_: (_( 1_MC1 )_) A_ZZ1 summary_NN1 with_IW the_AT presentation_NN1 and_CC contextualization_NN1 that_CST identifies_VVZ the_AT theme_NN1 ,_, the_AT objectives_NN2 ,_, the_AT spatial_JJ and_CC temporal_JJ organization_NN1 and_CC other_JJ relevant_JJ data_NN ._. 
</s>
<s>
We_PPIS2 now_RT treat_VV0 the_AT case_NN1 @S_FO ._. 
</s>
<s>
We_PPIS2 set_VV0 @S_FO ,_, and_CC write_VV0 @S_FO as_CSA @F_FO ._. 
</s>
<s>
The_AT first_MD and_CC second_MD terms_NN2 can_VM be_VBI treated_VVN easily_RR using_VVG (_( 4.5_MC )_) and_CC (_( W-3_FO )_) respectively_RR ._. 
</s>
<s>
In_II this_DD1 model_NN1 ,_, mathematics_NN1 teachers_NN2 are_VBR strongly_RR convinced_JJ that_CST students_NN2 '_NULL abilities_NN2 in_II mathematics_NN1 have_VH0 been_VBN distributed_VVN unequally_RR ._. 
</s>
<s>
For_REX21 example_REX22 ,_, extending_VVG Delpit_NP1 '_NULL s_ZZ1 idea_NN1 of_IO how_RRQ making_NN1 "_" rules_VVZ "_" explicit_JJ facilitates_VVZ the_AT acquisition_NN1 of_IO power_NN1 ,_, we_PPIS2 wondered_VVD whether_CSW instances_NN2 in_II which_DDQ teachers_NN2 chose_VVD high-level_JJ tasks_NN2 but_CCB did_VDD not_XX maintain_VVI their_APPGE potential_JJ cognitive_JJ demand_NN1 throughout_II the_AT lesson_NN1 might_VM be_VBI indicative_JJ of_IO a_AT1 reduction_NN1 in_II not_XX only_RR opportunity_NN1 to_TO learn_VVI but_CCB also_RR in_II coherence_NN1 and_CC explicitness_NN1 of_IO the_AT "_" rules_NN2 ,_, "_" or_CC rather_RR ,_, explicitness_NN1 of_IO what_DDQ counts_VVZ as_RG successful_JJ participation_NN1 ._. 
</s>
<s>
Theory_NN1 Related_JJ Fields_NN2 @S_FO established_VVN that_CST the_AT nonparametric_JJ rate_NN1 of_IO convex_JJ regression_NN1 is_VBZ of_IO order_NN1 n-4/5_FU for_IF equispaced_JJ design_NN1 points_NN2 ,_, we_PPIS2 show_VV0 that_CST the_AT nonparametric_JJ rate_NN1 of_IO convex_JJ regression_NN1 can_VM be_VBI as_RG slow_JJ as_CSA @S_FO for_IF some_DD worst-case_JJ design_NN1 points_NN2 ._. 
</s>
<s>
The_AT simplest_JJT method_NN1 for_IF obtaining_VVG a_AT1 guarantee_NN1 of_IO the_AT form_NN1 (_( 2_MC )_) is_VBZ gradient_NN1 descent_NN1 (_( GD_JJ )_) ._. 
</s>
<s>
More_RGR recently_RR ,_, &lsqb;_( 15_MC &rsqb;_) showed_VVD that_CST valid_JJ Lagrangian_JJ lower_JJR bounds_NN2 can_VM be_VBI calculated_VVN from_II the_AT iterates_NN2 of_IO the_AT PH_NN1 algorithm_NN1 when_CS the_AT sets_NN2 Ks_NP2 are_VBR not_XX convex_JJ ._. 
</s>
<s>
The_AT following_JJ lemma_NN1 is_VBZ the_AT heart_NN1 of_IO the_AT proof_NN1 of_IO Theorem_NN1 8(b)_FO ._. 
</s>
<s>
However_RR ,_, we_PPIS2 may_VM consider_VVI a_AT1 more_RGR general_JJ space_NN1 which_DDQ contains_VVZ @S_FO for_IF some_DD absolute_JJ constant_JJ a2_FO bounded_VVN in_II some_DD interval_NN1 &lsqb;_( M1_FO ,_, M_ZZ1 &rsqb;_) ._. 
</s>
<s>
Now_RT ,_, in_II the_AT inverse_JJ case_NN1 ,_, when_CS one_PN1 cools_VVZ the_AT bottom_NN1 and_CC heats_NN2 the_AT top_NN1 ,_, it_PPH1 is_VBZ expected_VVN that_CST the_AT system_NN1 remains_VVZ stable_JJ ._. 
</s>
<s>
We_PPIS2 review_VV0 the_AT literature_NN1 on_II continuum_NN1 stability_NN1 estimates_NN2 and_CC give_VV0 the_AT proofs_NN2 bridging_VVG the_AT gap_NN1 between_II the_AT existing_JJ estimates_NN2 and_CC those_DD2 in_II Theorems_NN2 1_MC1 and_CC 2_MC in_II an_AT1 "_" Appendix_NN1 "_" ._. 
</s>
<s>
Observe_VV0 that_CST @F_FO where_RRQ the_AT last_MD equality_NN1 is_VBZ by_II definition_NN1 ._. 
</s>
<s>
As_II a_AT1 consequence_NN1 of_IO Theorem_NN1 2.3_MC ,_, we_PPIS2 obtain_VV0 the_AT following_JJ statement_NN1 ,_, which_DDQ implies_VVZ Theorem_NN1 1.3_MC (_( announced_VVN in_II the_AT Introduction_NN1 )_) ._. 
</s>
<s>
Let_VV0 L1_FO =_FO 1.0093_MC ,_, L2_FO =_FO 1_MC1 ,_, and_CC X_ZZ1 e_ZZ1 0.013459_MC ,_, 0.25346_MC ,_, 0.50346_MC andconsiderthe_VV0 corresponding_JJ numerical_JJ approximation_NN1 u_ZZ1 (_( t_ZZ1 ,_, y_ZZ1 )_) depicted_VVD Figure2_FO ._. 
</s>
<s>
In_II contrast_NN1 ,_, the_AT Chinese_JJ teachers_NN2 showed_VVD strength_NN1 in_II professional_JJ noticing_VVG on_II aspects_NN2 like_II using_VVG knowledge_NN1 to_TO make_VVI relevant_JJ judgments_NN2 on_II students_NN2 '_NULL work_VV0 or_CC to_TO identify_VVI critical_JJ characteristics_NN2 of_IO students_NN2 '_NULL activities_NN2 ,_, evaluating_VVG students_NN2 '_NULL mistakes_NN2 ,_, and_CC developing_JJ alternative_JJ ways_NN2 of_IO teaching_NN1 ._. 
</s>
<s>
This_DD1 is_VBZ mirrored_VVN by_II eigenvalue_NN1 computations_NN2 (_( not_XX shown_VVN )_) which_DDQ ,_, in_II both_DB2 cases_NN2 ,_, display_VV0 qualitatively_RR similar_JJ behavior_NN1 to_II the_AT 2D_NNU model_NN1 problems_NN2 ._. 
</s>
<s>
Finally_RR we_PPIS2 observe_VV0 that_CST for_IF all_DB @S_FO ,_, we_PPIS2 have_VH0 that_DD1 @F_FO ._. 
</s>
<s>
This_DD1 completes_VVZ the_AT proof_NN1 ._. 
</s>
<s>
This_DD1 is_VBZ not_XX necessarily_RR the_AT case_NN1 with_IW Proposition_NN1 4.9_MC simply_RR because_CS the_AT right_JJ scalar_JJ multiplication_NN1 does_VDZ not_XX necessarily_RR produce_VVI a_AT1 closed_JJ convex_JJ function_NN1 ._. 
</s>
<s>
We_PPIS2 introduce_VV0 the_AT stopping_VVG time_NNT1 @F_FO ._. 
</s>
<s>
Lemma_NN1 1_MC1 Assume_VV0 all_DB conditions_NN2 in_II Theorem_NN1 1_MC1 ._. 
</s>
<s>
Here_RL ,_, we_PPIS2 briefly_RR describe_VV0 the_AT method_NN1 ._. 
</s>
<s>
In_II &lsqb;_( 23_MC &rsqb;_) itwas_NN2 found_VVD that_CST as_CSA crosses_VVZ a_AT1 classical_JJ cut_NN1 of_IO Yi_NP1 (_( )_) ,_, the_AT Yi_NP1 (_( )_) is_VBZ transformed_VVN by_II the_AT reflection_NN1 in_II Wg_NP1 generated_VVD by_II i-th_MD simple_JJ root_NN1 &lsqb;_( 23_MC &rsqb;_) ._. 
</s>
<s>
At_II the_AT beginning_NN1 of_IO this_DD1 section_NN1 we_PPIS2 required_VVD the_AT manifold_JJ M_NN1 to_TO be_VBI spin_NN1 ._. 
</s>
<s>
We_PPIS2 consider_VV0 vectors_NN2 as_CSA column_NN1 vectors_NN2 ,_, so_CS21 that_CS22 the_AT bilinear_JJ form_NN1 associated_VVN to_II a_AT1 square_JJ matrix_NN1 C_ZZ1 is_VBZ defined_VVN by_II @S_FO ._. 
</s>
<s>
The_AT students_NN2 had_VHD to_TO deal_VVI substantially_RR with_IW the_AT extra-mathematical_JJ context_NN1 by_II determining_VVG which_DDQ information_NN1 is_VBZ superfluous_JJ (_( e.g._REX ,_, 8_MC weeks_NNT2 ago_RA )_) as_II31 well_II32 as_II33 by_II making_VVG an_AT1 assumption_NN1 about_II the_AT missing_JJ number_NN1 (_( amount_NN1 of_IO the_AT needed_VVN riding_VVG hours_NNT2 )_) ._. 
</s>
<s>
Referring_VVG to_II the_AT example_NN1 of_IO the_AT weight_NN1 ,_, this_DD1 behavior_NN1 is_VBZ clearly_RR satisfied_JJ ,_, since_CS it_PPH1 is_VBZ more_DAR pleasnt_NN1 to_TO eat_VVI (_( and_CC to_TO gain_VVI weight_NN1 )_) than_CSN to_II fast_JJ (_( and_CC to_TO loose_VVI it_PPH1 )_) ._. 
</s>
<s>
The_AT numerical_JJ invariants_NN2 @S_FO called_VVN parabolic_JJ jumps_NN2 for_IF @S_FO and_CC @S_FO are_VBR defined_VVN by_II @F_FO ._. 
</s>
<s>
Note_VV0 that_CST for_IF all_DB i_ZZ1 ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
The_AT rank_NN1 ,_, degree_NN1 and_CC @S_FO are_VBR all_DB additive_JJ for_IF short_JJ exact_JJ sequences_NN2 ._. 
</s>
<s>
Fig._NN1 3.2_MC Relative_JJ error_NN1 @S_FO of_IO the_AT modified_JJ magnetic_JJ moment_NN1 @S_FO as_II a_AT1 function_NN1 of_IO time_NNT1 ,_, along_II the_AT numerical_JJ solution_NN1 of_IO the_AT variational_JJ integrator_NN1 (_( 1.7_MC )_) ,_, obtained_VVN with_IW @S_FO ,_, where_CS @S_FO (_( black_JJ )_) and_CC @S_FO (_( grey_JJ )_) ._. 
</s>
<s>
Recalling_VVG that_CST a_AT1 sequence_NN1 is_VBZ exact_JJ if_CS the_AT image_NN1 of_IO each_DD1 operator_NN1 coincides_VVZ with_IW the_AT kernel_NN1 of_IO the_AT following_JJ one_PN1 ,_, and_CC that_CST P_ZZ1 is_VBZ contractible_JJ ,_, from_II (_( 2.2_MC )_) and_CC (_( 2.3_MC )_) it_PPH1 is_VBZ easy_JJ to_TO check_VVI that_CST the_AT following_JJ sequence_NN1 is_VBZ exact_JJ :_: @F_FO ,_, where_CS i_ZZ1 denotes_VVZ the_AT mapping_NN1 that_CST to_II every_AT1 real_JJ number_NN1 r_ZZ1 associates_VVZ the_AT constant_JJ function_NN1 identically_RR equal_JJ to_II r_ZZ1 and_CC 0_MC is_VBZ the_AT mapping_NN1 that_CST to_II every_AT1 function_NN1 associates_VVZ the_AT number_NN1 0_MC ._. 
</s>
<s>
By_II Corollary_NN1 5.7(i)_FO ,_, the_AT errors_NN2 are_VBR expected_VVN to_TO be_VBI @S_FO ._. 
</s>
<s>
Our_APPGE numerical_JJ results_NN2 clearly_RR confirm_VV0 the_AT sharpness_NN1 of_IO this_DD1 corollary_NN1 for_IF the_AT considered_JJ case_NN1 ._. 
</s>
<s>
Over_II &lsqb;_( hgDS/Hg_FU &rsqb;_) ,_, all_DB the_AT groups_NN2 appearing_VVG in_II (_( 21_MC )_) are_VBR tori_NN2 which_DDQ are_VBR canonically_RR equivalent_JJ to_II the_AT torus_NN1 of_IO regular_JJ semisimple_NN1 centralizers_NN2 on_II &lsqb;_( h_ZZ1 sd/Hs_FU &rsqb;_) ._. 
</s>
<s>
In_II the_AT same_DA way_NN1 as_CSA algebraic_JJ or_CC Lie_VV0 groups_NN2 are_VBR important_JJ in_II algebraic_JJ or_CC differential_JJ geometry_NN1 ,_, the_AT understanding_NN1 of_IO groups_NN2 definable_JJ in_II a_AT1 given_JJ first-order_JJ structure_NN1 (_( or_CC in_II certain_JJ classes_NN2 of_IO first-order_JJ structures_NN2 )_) is_VBZ important_JJ for_IF model_NN1 theory_NN1 as_II31 well_II32 as_II33 its_APPGE applications_NN2 ._. 
</s>
<s>
In_II what_DDQ follows_VVZ ,_, Roman_JJ indices_NN2 take_VV0 on_RP values_NN2 @S_FO ,_, where_CS @S_FO is_VBZ the_AT spatial_JJ dimension_NN1 ,_, and_CC summation_NN1 convention_NN1 on_II repeated_JJ indices_NN2 is_VBZ applied_VVN ._. 
</s>
<s>
Future_JJ research_NN1 should_VM examine_VVI these_DD2 dispositions_NN2 more_RGR closely_RR ._. 
</s>
<s>
Consider_VV0 the_AT copy_NN1 of_IO Yi_NP1 in_II Ji_NN1 that_CST contains_VVZ a_AT1 lift_NN1 of_IO aC_NN1 to_II @S_FO ._. 
</s>
<s>
If_CS Al_NP1 is_VBZ disjoint_JJ from_II Yi_NP1 ,_, then_RT aC_NN1 is_VBZ a_AT1 piece_NN1 and_CC dC_NN1 is_VBZ a_AT1 concatenation_NN1 of_IO at_RR21 most_RR22 7_MC pieces_NN2 ,_, which_DDQ contradicts_VVZ @S_FO ._. 
</s>
<s>
Otherwise_RR let_VV0 @S_FO ._. 
</s>
<s>
Hence_RR aC_NN1 projects_NN2 into_II the_AT quotient_NN1 @S_FO of_IO @S_FO in_II Y_ZZ1 ,_, ._. 
</s>
<s>
These_DD2 ideas_NN2 are_VBR fundamental_JJ for_IF the_AT error_NN1 analysis_NN1 in_II Sect._NP1 5_MC ._. 
</s>
<s>
By_II Claim_NN1 3(B)_FO and_CC the_AT same_DA argument_NN1 in_II Claim_NN1 4(A)_FO ,_, we_PPIS2 know_VV0 that_CST @F_FO ,_, where_RRQ &lsqb;_( Tz2_FO &rsqb;_) and_CC &lsqb;_( Tx1_FO &rsqb;_) denote_VV0 the_AT un-oriented_JJ tangent_JJ planes_NN2 of_IO 2_MC and_CC 1_MC1 (_( without_IW counting_VVG multiplicity_NN1 )_) respectively_RR ._. 
</s>
<s>
The_AT items_NN2 assessing_VVG mastery_NN1 goals_NN2 correspond_VV0 to_II those_DD2 assessing_VVG mastery-approach_JJ goals_NN2 ._. 
</s>
<s>
Anyhow_RR ,_, the_AT crucial_JJ issue_NN1 is_VBZ that_CST the_AT bound_NN1 degenerates_VVZ as_II the_AT spectrum_NN1 enlarges_VVZ ._. 
</s>
<s>
See_VV0 the_AT supplementary_JJ materials_NN2 (_( &lsqb;_( 29_MC &rsqb;_) ,_, Section_NN1 1.1.1_MC )_) ._. 
</s>
<s>
We_PPIS2 note_VV0 that_CST ,_, by_II using_VVG OLS_NN2 applied_VVN to_II a_AT1 larger_JJR set_NN1 of_IO variables_NN2 as_II the_AT RP_NP1 method_NN1 ,_, and_CC the_AT RSS_NP1 fromthe_NN1 resulting_VVG ?_? fit_NN1 as_II the_AT estimate_NN1 of_IO prediction_NN1 error_NN1 ,_, the_AT overall_JJ test_NN1 is_VBZ equivalent_JJ to_II apartial_JJ F_ZZ1 -test_JJ for_IF the_AT significance_NN1 of_IO the_AT additional_JJ group_NN1 of_IO variables_NN2 ._. 
</s>
<s>
It_PPH1 appears_VVZ particularly_RR significant_JJ to_TO analyse_VVI this_DD1 phenomenon_NN1 in_II the_AT context_NN1 of_IO the_AT degree_NN1 course_NN1 in_II Mathematics_NN1 ,_, studying_VVG students_NN2 '_NULL cognitive_JJ and_CC affective_JJ reactions_NN2 to_II the_AT (_( often_RR unexpected_JJ and_CC severe_JJ )_) difficulties_NN2 encountered_VVN in_II the_AT tertiary_JJ transition_NN1 ._. 
</s>
<s>
My_APPGE idea_NN1 is_VBZ to_TO change_VVI that_DD1 ,_, obviously_RR ._. 
</s>
<s>
From_II a_AT1 methodological_JJ point_NN1 of_IO view_NN1 ,_, it_PPH1 is_VBZ possible_JJ in_II the_AT state_NN1 space_NN1 context_NN1 to_TO use_VVI alternativesto_NN1 the_AT Hilbert_NP1 resampling_VVG sort_NN1 to_TO implement_VVI the_AT correlated_JJ pseudomarginal_JJ algorithm_NN1 (_( Lee_NP1 ,_, 2008_MC ;_; Malik_NP1 and_CC Pitt_NP1 ,_, 2011_MC ;_; LEcuyer_NP1 et_RA21 al._RA22 ,_, 2018_MC )_) and_CC several_DA2 such_DA methods_NN2 have_VH0 been_VBN proposed_VVN following_II the_AT first_MD version_NN1 of_IO this_DD1 work_NN1 (_( Doucet_NP1 et_RA21 al._RA22 ,_, 2015_MC )_) ;_; see_VV0 ,_, for_REX21 example_REX22 Jacob_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
(_( 2016_MC )_) and_CC Senet_NP1 al_NP1 ._. 
</s>
<s>
(_( 2018_MC )_) ._. 
</s>
<s>
By_II definition_NN1 ,_, @F_FO ,_, where_CS X_ZZ1 is_VBZ a_AT1 set_NN1 of_IO k_ZZ1 independent_JJ uniform_JJ variables_NN2 in_II &lsqb;_( 0_MC ,_, 1_MC1 &rsqb;_) ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI an_AT1 orthonormal_JJ basis_NN1 of_IO @S_FO consisting_VVG of_IO eigenvectors_NN2 of_IO the_AT operator_NN1 @S_FO and_CC let_VV0 @S_FO be_VBI the_AT corresponding_JJ eigenvalues_NN2 ._. 
</s>
<s>
It_PPH1 holds_VVZ that_CST @S_FO and_CC @S_FO ._. 
</s>
<s>
Additionally_RR ,_, if_CS @S_FO for_IF all_DB @S_FO ,_, then_RT @S_FO ._. 
</s>
<s>
Open_JJ access_NN1 funding_NN1 provided_VVN by_II University_NN1 of_IO Vienna_NP1 ._. 
</s>
<s>
Moreover_RR ,_, if_CS @S_FO in_II @S_FO ,_, then_RT by_II Remark_NN1 3.2_MC we_PPIS2 have_VH0 @S_FO in_II @S_FO ._. 
</s>
<s>
In_RR21 Part_RR22 II_MC ,_, we_PPIS2 first_MD "_" prewhiten_VV0 "_" the_AT data_NN by_II fitting_JJ univariate_NN1 time_NNT1 series_NN models_NN2 to_II each_DD1 time_NNT1 series_NN separately_RR ,_, and_CC then_RT apply_VV0 the_AT TGUH_NN1 method_NN1 with_IW default_NN1 parameters_NN2 to_II the_AT residual_JJ sequences_NN2 from_II these_DD2 fits_NN2 ._. 
</s>
<s>
Theorem_NN1 2.2_MC (_( Burton_NP1 and_CC Pemantle_NP1 (_( 1993_MC )_) )_) ._. 
</s>
<s>
By_II &lsqb;_( AL2_FO ,_, Proposition_NN1 4.3_MC &rsqb;_) any_DD Siegel_NP1 maps_NN2 f_ZZ1 ,_, g_ZZ1 can_VM be_VBI obtained_VVN by_II performing_VVG the_AT Douady-Ghys_NP1 surgery_NN1 on_II quasicritical_JJ circle_NN1 maps_NN2 f_ZZ1 ,_, g_ZZ1 ._. 
</s>
<s>
Circles_NN2 show_VV0 the_AT percentage_NN1 of_IO instances_NN2 where_RRQ the_AT five_MC most_RGT likely_JJ classifications_NN2 from_II the_AT network_NN1 do_VD0 not_XX include_VVI the_AT correct_JJ category_NN1 ,_, over_II the_AT training_NN1 data_NN images_NN2 ;_; crosses_NN2 show_VV0 the_AT same_DA measure_NN1 computed_VVN over_II the_AT validation_NN1 data_NN ._. 
</s>
<s>
I_PPIS1 have_VH0 two_MC aims_NN2 in_II writing_VVG this_DD1 article_NN1 ._. 
</s>
<s>
Thus_RR ,_, itis_NN1 natural_JJ to_TO apply_VVI this_DD1 modification_NN1 ._. 
</s>
<s>
The_AT exception_NN1 here_RL is_VBZ the_AT cluster_NN1 ,_, with_IW one_MC1 singular_NN1 @S_FO and_CC all_DB other_JJ @S_FO ._. 
</s>
<s>
In_II that_DD1 case_NN1 ,_, computing_VVG any_DD k_ZZ1 >_FO 1_MC1 vectors_NN2 was_VBDZ as_RG slow_JJ as_CSA computing_VVG all_DB vectors_NN2 with_IW bisection_NN1 ._. 
</s>
<s>
Section_NN1 6_MC and_CC "_" Appendix_NN1 "_" present_NN1 the_AT analysis_NN1 for_IF the_AT fully_RR discrete_JJ scheme_NN1 ._. 
</s>
<s>
Consider_VV0 ,_, too_RR ,_, a_AT1 scenario_NN1 where_RRQ students_NN2 spend_VV0 significant_JJ time_NNT1 comparing_VVG equivalent_JJ and_CC nonequivalent_JJ quantities_NN2 among_II sets_NN2 and_CC using_VVG this_DD1 as_II a_AT1 context_NN1 for_IF qualifying_JJ relationships_NN2 as_CSA "_" the_AT same_DA amount_NN1 as_CSA ,_, "_" "_" more_RRR than_CSN ,_, "_" or_CC "_" less_DAR than_CSN "_" before_II symbolizing_VVG these_DD2 relationships_NN2 with_IW "_" <_FO "_" ,_, "_" >_FO "_" ,_, and_CC "_" =_FO "_" ._. 
</s>
<s>
Suppose_VV0 (_( S_ZZ1 ,_, A_ZZ1 )_) is_VBZ a_AT1 compatible_JJ pair_NN and_CC the_AT base_NN1 scheme_NN1 S_ZZ1 contains_VVZ no_AT point_NN1 whose_DDQGE residue_NN1 characteristic_NN1 equals_VVZ two_MC ._. 
</s>
<s>
By_II Theorem_NN1 2.3_MC ,_, determining_VVG if_CSW a_AT1 binary_JJ linear_JJ system_NN1 game_NN1 has_VHZ a_AT1 perfect_JJ strategy_NN1 is_VBZ equivalent_JJ to_II determining_VVG if_CS J_ZZ1 =_FO 1_MC1 in_II a_AT1 solution_NN1 group_NN1 ._. 
</s>
<s>
Let_VV0 @S_FO with_IW @S_FO ,_, and_CC let_VV0 ni_NN2 ,_, no_UH ,_, ai_NNU ,_, ao_NNU ,_, Ap_NP1 ,_, An_AT1 be_VBI positive_JJ real_JJ numbers_NN2 ._. 
</s>
<s>
For_IF a_AT1 inequality_NN1 of_IO a_AT1 general_JJ form_NN1 ,_, we_PPIS2 prove_VV0 convergence_NN1 of_IO numerical_JJ solutions_NN2 ._. 
</s>
<s>
Furthermore_RR ,_, BO_NP1 and_CC BH_NP1 are_VBR strictly_RR upper_JJ triangular_JJ matrices_NN2 and_CC @S_FO is_VBZ a_AT1 diagonal_JJ matrix_NN1 ._. 
</s>
<s>
Suppose_VV0 m_ZZ1 ,_, n_ZZ1 ,_, and_CC l_ZZ1 are_VBR positive_JJ integers_NN2 and_CC @S_FO satisfying_JJ @F_FO ._. 
</s>
<s>
Then_RT ,_, C6_FO =_FO G_ZZ1 almost_RR uniformly_RR pointwise_RR as_CSA @S_FO ,_, and_CC @S_FO 63_MC ._. 
</s>
<s>
Hence_RR ,_, we_PPIS2 argue_VV0 that_CST the_AT consideration_NN1 of_IO a_AT1 multi-well_JJ potential_NN1 as_CSA in_II (_( 1.2_MC )_) ,_, although_CS it_PPH1 leads_VVZ to_II a_AT1 vectorial_JJ Cahn-Hilliard_NP1 system_NN1 ,_, may_VM yield_VVI a_AT1 mathematical_JJ model_NN1 that_CST is_VBZ further_RRR amenable_JJ to_II analytical_JJ and_CC numerical_JJ investigations_NN2 ,_, see_VV0 for_REX21 example_REX22 Ref._NN1 24_MC ._. 
</s>
<s>
This_DD1 study_NN1 aims_VVZ to_TO broaden_VVI the_AT understanding_NN1 of_IO how_RRQ the_AT content_NN1 in_II a_AT1 designed_JJ picture_NN1 book_NN1 directs_VVZ children_NN2 '_NULL s_ZZ1 attention_NN1 to_II numbers_NN2 ,_, and_CC what_DDQ kind_NN1 of_IO numerical_JJ reasoning_NN1 the_AT book_NN1 reading_NN1 entails_VVZ ._. 
</s>
<s>
CPU_NN1 time_NNT1 and_CC number_NN1 of_IO branching_JJ nodes_NN2 under_II different_JJ Omega-values_NN2 in_II constraint_NN1 (_( 9_MC )_) ._. 
</s>
<s>
Step_NN1 2_MC Let_VV0 us_PPIO2 show_VVI that_CST @F_FO ._. 
</s>
<s>
By_II taking_VVG the_AT scalar_JJ product_NN1 of_IO (_( 29_MC )_) by_II t2x(t)_FO ,_, we_PPIS2 get_VV0 @F_FO ._. 
</s>
<s>
Using_VVG the_AT Chain_NN1 Rule_NN1 and_CC the_AT Cauchy-Schwarz_NP1 inequality_NN1 ,_, we_PPIS2 obtain_VV0 @F_FO ._. 
</s>
<s>
Integration_NN1 by_II parts_NN2 yields_VVZ @F_FO ._. 
</s>
<s>
As_II a_AT1 consequence_NN1 ,_, @F_FO for_IF some_DD constant_JJ C0_FO depending_VVG only_RR on_II the_AT Cauchy_JJ data_NN ._. 
</s>
<s>
In_II this_DD1 section_NN1 we_PPIS2 provide_VV0 simulation_NN1 studies_NN2 illustrating_VVG the_AT performances_NN2 of_IO GS_NP1 ,_, TGS_NP1 and_CC WTGS_NP1 inthe_VV0 BVS_NP1 context_NN1 that_CST was_VBDZ described_VVN in_II Section_NN1 4_MC ._. 
</s>
<s>
This_DD1 data_NN is_VBZ represented_VVN in_II Table_NN1 1_MC1 ,_, column_NN1 titled_VVN %_NNU of_IO teachers_NN2 using_VVG this_DD1 strategy_NN1 ._. 
</s>
<s>
The_AT uncertainty_NN1 principle_NN1 with_IW variable_JJ amplitude_NN1 ._. 
</s>
<s>
Theorem_NN1 1.1_MC gives_VVZ the_AT range_NN1 p>2.8_FO ,_, and_CC the_AT best_JJT previous_JJ estimate_NN1 was_VBDZ p>3_FO ._. 
</s>
<s>
Let_VV0 O_ZZ1 be_VBI an_AT1 open_JJ subset_NN1 of_IO Rn_NP1 with_IW Lipschitz_NP1 boundary_NN1 ._. 
</s>
<s>
For_IF BPNMs_NP1 ,_, one_PN1 can_VM ask_VVI for_IF optimal_JJ information_NN1 ,_, @F_FO ,_, where_CS we_PPIS2 have_VH0 made_VVN explicit_JJ the_AT fact_NN1 that_CST the_AT optimal_JJ information_NN1 depends_VVZ on_II the_AT choice_NN1 of_IO prior_JJ @S_FO ._. 
</s>
<s>
Because_CS the_AT intended_JJ lesson_NN1 design_NN1 ,_, though_RR seems_VVZ to_TO be_VBI rigidly_RR structured_VVN ,_, allows_VVZ flexible_JJ student_NN1 self-directed_NN1 learning_NN1 elements_NN2 ,_, the_AT implemented_JJ lessons_NN2 still_RR capture_VV0 the_AT flow_NN1 of_IO the_AT inquiry-based_JJ modeling_NN1 cycle_NN1 ._. 
</s>
<s>
The_AT SIPG_NN1 stabilization_NN1 term_NN1 is_VBZ accounted_VVN for_IF in_II the_AT design_NN1 of_IO the_AT gradient_NN1 reconstruction_NN1 @S_FO through_II a_AT1 penalization_NN1 parameter_NN1 (_( denoted_VVN by_II @S_FO in_II &lsqb;_( 14_MC ,_, Chapter_NN1 11_MC &rsqb;_) )_) which_DDQ is_VBZ fixed_VVN at_II 0.6_MC in_II all_DB the_AT tests_NN2 ._. 
</s>
<s>
Assumption_NN1 2.4(iii)_FO is_VBZ also_RR satisfied_VVN with_IW @S_FO and_CC @S_FO being_VBG a_AT1 vector-valued_JJ version_NN1 of_IO the_AT continuous_JJ piecewise_JJ linear_JJ element_NN1 subspace_VV0 that_CST approximates_VVZ @S_FO as_CSA @S_FO ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI eigenvalues_NN2 of_IO @S_FO ._. 
</s>
<s>
To_TO define_VVI the_AT linear_JJ operator_NN1 A_ZZ1 ,_, consider_VV0 first_MD @S_FO jeFm_VV0 to_TO be_VBI an_AT1 approximate_JJ inverse_NN1 of_IO f_ZZ1 ,_, that_REX21 is_REX22 ,_, ni_NN2 @S_FO ._. 
</s>
<s>
And_CC then_RT it_PPH1 '_NULL s_ZZ1 like_JJ ,_, "_" Have_VH0 you_PPY done_VDN it_PPH1 ?_? 
</s>
<s>
BDDC_NP1 methods_NN2 for_IF vector_NN1 field_NN1 problems_NN2 discretized_VVD with_IW Raviart-Thomas_NP1 finite_JJ elements_NN2 are_VBR introduced_VVN ._. 
</s>
<s>
Hence_RR ,_, we_PPIS2 propose_VV0 to_TO plot_VVI other_JJ extreme_JJ scenarios_NN2 ,_, as_CSA shown_VVN in_II Fig._NN1 3_MC ,_, where_CS we_PPIS2 consider_VV0 different_JJ values_NN2 of_IO the_AT partial_JJ @S_FO of_IO the_AT unobserved_JJ confounder_NN1 with_IW the_AT outcome_NN1 ,_, including_II 75%_NNU and_CC 50%_NNU ._. 
</s>
<s>
Proceeding_VVG analogously_RR when_CS the_AT entries_NN2 of_IO (_( Gk_NP1 )_) ii_MC are_VBR negative_JJ ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Hence_RR ,_, for_IF sufficiently_RR large_JJ k_ZZ1 the_AT scalars_NN2 @S_FO have_VH0 constant_JJ sign_NN1 and_CC @F_FO ._. 
</s>
<s>
The_AT initial_JJ problem_NN1 corresponds_VVZ in_II this_DD1 case_NN1 to_II the_AT following_JJ Stokes_NP1 system_NN1 @F_FO which_DDQ admits_VVZ a_AT1 unique_JJ solution_NN1 (_( un_FW ,_, rn_NNU )_) ._. 
</s>
<s>
For_IF @S_FO a_AT1 domain_NN1 ,_, let_VV0 @S_FO be_VBI a_AT1 weight_NN1 function_NN1 ,_, and_CC consider_VV0 the_AT following_JJ space_NN1 of_IO w-weighted_JJ square-integrable_JJ real-valued_JJ functions_NN2 over_II D_ZZ1 :_: @F_FO ._. 
</s>
<s>
The_AT space_NN1 @S_FO is_VBZ a_AT1 Hilbert_NP1 space_NN1 with_IW an_AT1 inner_JJ product_NN1 and_CC norm_NN1 defined_VVD ,_, respectively_RR ,_, as_CSA @F_FO ._. 
</s>
<s>
To_TO simplify_VVI some_DD notation_NN1 later_RRR ,_, we_PPIS2 will_VM assume_VVI that_CST w_ZZ1 is_VBZ a_AT1 probability_NN1 density_NN1 function_NN1 ,_, i.e._REX ,_, that_CST @S_FO ._. 
</s>
<s>
This_DD1 is_VBZ not_XX a_AT1 particularly_RR strong_JJ assumption_NN1 since_CS it_PPH1 is_VBZ essentially_RR equivalent_JJ to_II requiring_VVG that_DD1 constant_JJ functions_NN2 are_VBR in_II @S_FO ._. 
</s>
<s>
The_AT following_JJ examples_NN2 illustrate_VV0 common_JJ choices_NN2 for_IF w_ZZ1 and_CC D._NP1 Given_VVN a_AT1 set_NN1 A_ZZ1 ,_, the_AT next_MD lemma_NN1 extracts_VVZ a_AT1 large_JJ subset_NN1 A0_FO of_IO a_AT1 suitable_JJ translation_NN1 of_IO A_ZZ1 ,_, such_CS21 that_CS22 points_NN2 in_II A0_FO are_VBR "_" not_XX too_RG close_RR to_II the_AT boundary_NN1 "_" of_IO 2d-adic_JJ intervals_NN2 ._. 
</s>
<s>
Consider_VV0 the_AT bubble_NN1 Kr+i_FO for_IF R_ZZ1 prime_JJ and_CC R_ZZ1 >_FO 3_MC ._. 
</s>
<s>
We_PPIS2 will_VM denote_VVI these_DD2 quantities_NN2 by_II @S_FO ,_, @S_FO and_CC @S_FO to_TO emphasize_VVI this_DD1 dependence_NN1 ._. 
</s>
<s>
Do_VD0 existing_JJ theories_NN2 explain_VV0 the_AT behavior_NN1 ofRDA_NN1 ?_? ._. 
</s>
<s>
Kermack_VV0 and_CC McKendrick_NP1 proposed_VVD and_CC developed_VVD the_AT first_MD "_" compartmental_JJ "_" model_NN1 of_IO disease_NN1 dynamics_NN and_CC disease_NN1 spread_VVN by_II considering_VVG a_AT1 population_NN1 of_IO individuals_NN2 sub-divided_VVN into_II three_MC separate_JJ classes_NN2 -_- susceptible_JJ ,_, infectious_JJ and_CC recovered_VVD -_- with_IW the_AT transmission_NN1 of_IO the_AT disease/infection_NN1 being_VBG dependent_JJ upon_II the_AT number_NN1 of_IO interactions_NN2 between_II individuals_NN2 and_CC the_AT u_ZZ1 ._. derlying_JJ rate_NN1 of_IO infection_NN1 ._. 
</s>
<s>
Thus_RR ,_, Year_NNT1 9_MC is_VBZ a_AT1 turning_JJ point_NN1 for_IF the_AT students_NN2 ,_, since_CS empirical_JJ verification_NN1 of_IO mathematical_JJ statements_NN2 leads_VVZ gradually_RR to_II logical_JJ reasoning_NN1 and_CC structured_JJ argumentation_NN1 based_VVN on_II definitions_NN2 and_CC properties_NN2 of_IO mathematical_JJ objects_NN2 ._. 
</s>
<s>
We_PPIS2 begin_VV0 by_II discussing_VVG the_AT category_NN1 of_IO perverse_JJ coherent_JJ sheaves_NN2 on_II the_AT affine_JJ Grassmannian_JJ following_NN1 &lsqb;_( AB10_FO ,_, BFM05_FO &rsqb;_) ._. 
</s>
<s>
For_IF every_AT1 Schroder_NP1 tree_NN1 t_ZZ1 ,_, 2t_FO |_NULL >_FO #t_FO +_FO 1_MC1 ._. 
</s>
<s>
In_II a_AT1 fashion_NN1 similar_JJ to_II Lemma_NN1 2.1_MC ,_, the_AT constants_NN2 in_II Lemma_NN1 3.1_MC depend_VV0 only_RR on_II d_ZZ1 ,_, m_ZZ1 ,_, Rc_NP1 ,_, hhop_VV0 ,_, hons_NN2 ,_, o_ZZ1 and_CC t_ZZ1 ,_, but_CCB Cj_NP1 are_VBR bounded_VVN above_RL and_CC nj_NNU are_VBR bounded_VVN away_II21 from_II22 0_MC on_II bounded_JJ intervals_NN2 for_IF t_ZZ1 ._. 
</s>
<s>
These_DD2 interactional_JJ contexts_NN2 also_RR have_VH0 implications_NN2 for_IF teacher_NN1 subject_NN1 knowledge_NN1 ._. 
</s>
<s>
Consider_VV0 an_AT1 open_JJ set_NN1 @S_FO and_CC define_VV0 the_AT first_MD eigenvalue_NN1 of_IO the_AT operator_NN1 δ+ew_FO in_II H10()_FO by_II @F_FO ,_, and_CC suppose_VV0 1_MC1 ,_, w()≤0_FO ._. 
</s>
<s>
The_AT other_JJ implication_NN1 may_VM be_VBI obtained_VVN as_CSA in_II Hulanicki_NP1 '_NULL s_ZZ1 characterization_NN1 of_IO amenability_NN1 (_( see_VV0 &lsqb;_( 49_MC ,_, Theorems_NN2 3.1.5_MC and_CC 4.3.2_MC &rsqb;_) )_) ._. 
</s>
<s>
Thus_RR ,_, they_PPHS2 are_VBR representations_NN2 or_CC descriptions_NN2 of_IO reality_NN1 that_CST move_VV0 beyond_II the_AT real-life_JJ situation_NN1 or_CC external_JJ world_NN1 and_CC examine_VV0 its_APPGE structural_JJ features_NN2 through_II mathematics_NN1 ._. 
</s>
<s>
First_MD we_PPIS2 establish_VV0 a_AT1 hyperbolic_JJ analogue_NN1 of_IO Proposition_NN1 3.1_MC ._. 
</s>
<s>
For_IF the_AT other_JJ direction_NN1 ,_, i.e._REX ,_, to_TO show_VVI (_( (_( 2_MC )_) or_CC (_( 3_MC )_) )_) (_( 1_MC1 )_) ,_, we_PPIS2 use_VV0 Proposition_NN1 2.7_MC and_CC consider_VV0 the_AT functors_NN2 @S_FO (_( or_CC the_AT telescopic_JJ analog_NN1 )_) ._. 
</s>
<s>
For_IF the_AT former_DA ,_, Let_VV0 @S_FO be_VBI given_VVN by_II @S_FO ._. 
</s>
<s>
Note_VV0 that_CST ,_, by_II construction_NN1 ,_, @S_FO for_IF all_DB k_ZZ1 and_CC i_ZZ1 ,_, so_RG in_RR21 particular_RR22 @S_FO is_VBZ a_AT1 nested_JJ intersection_NN1 of_IO the_AT closures_NN2 of_IO elements_NN2 of_IO the_AT open_JJ covers_NN2 ,_, and_CC hence_RR is_VBZ well-defined_JJ ._. 
</s>
<s>
Finally_RR ,_, by_II Theorem_NN1 2.9(ii)_FO ,_, in_II Theorem_NN1 2._MC i5_FO statement_NN1 (_( iv_MC )_) implies_VVZ statement_NN1 (_( v_ZZ1 )_) ,_, Our_APPGE goal_NN1 was_VBDZ not_XX to_TO evaluate_VVI curricula_NN2 ,_, and_CC as_CSA such_DA we_PPIS2 did_VDD not_XX make_VVI judgments_NN2 about_II how_RGQ curricular_JJ elements_NN2 may_VM or_CC may_VM not_XX support_VVI teacher_NN1 implementation_NN1 ;_; rather_RR ,_, we_PPIS2 sought_VVD to_TO identify_VVI and_CC bound_VVD sections_NN2 we_PPIS2 might_VM reasonably_RR expect_VVI teachers_NN2 to_TO make_VVI use_NN1 of_IO in_II planning_NN1 and_CC visualizing_VVG a_AT1 lesson_NN1 ._. 
</s>
<s>
We_PPIS2 require_VV0 that_CST at_II level_NN1 n_ZZ1 the_AT decomposition_NN1 satisfies_VVZ the_AT following_JJ grading_NN1 condition_NN1 ._. 
</s>
<s>
As_CSA expected_VVN ,_, the_AT temporal_JJ convergence_NN1 order_NN1 improves_VVZ when_RRQ the_AT grading_NN1 parameter_NN1 7_MC increases_NN2 ._. 
</s>
<s>
In_RR21 addition_RR22 ,_, we_PPIS2 note_VV0 that_CST g_ZZ1 has_VHZ stable_JJ integer_NN1 shifts_NN2 ._. 
</s>
<s>
We_PPIS2 take_VV0 k_ZZ1 points_VVZ X1_FO ,_, ..._... ,_, 
</s>
<s>
Xk_FO in_II &lsqb;_( 0,1_MC &rsqb;_) uniformly_RR at_RR21 random_RR22 ,_, independently_RR from_II each_PPX221 other_PPX222 and_CC from_II (_( e_ZZ1 ,_, S_ZZ1 )_) ,_, and_CC we_PPIS2 let_VV0 X_ZZ1 =_FO (_( X1_FO ,_, ..._... ,_, 
</s>
<s>
Xk_FO )_) ._. 
</s>
<s>
These_DD2 parallel_RR some_DD of_IO our_APPGE current_JJ discussion_NN1 ._. 
</s>
<s>
In_II &lsqb;_( RW_NP1 &rsqb;_) the_AT third_MD and_CC fourth_MD authors_NN2 ,_, inspired_VVN by_II &lsqb;_( AB_FO &rsqb;_) ,_, conjectured_VVD that_CST characters_NN2 of_IO tilting_VVG modules_NN2 in_II @S_FO can_VM be_VBI expressed_VVN in_II31 terms_II32 of_II33 the_AT -canonical_JJ basis_NN1 of_IO the_AT antispherical_JJ module_NN1 ._. 
</s>
<s>
Nevertheless_RR ,_, adopting_VVG for_REX21 instance_REX22 a_AT1 reduced_JJ integration_NN1 or_CC a_AT1 mixed_JJ interpolation_NN1 technique_NN1 ,_, this_DD1 phenomenon_NN1 can_VM be_VBI avoided_VVN ._. 
</s>
<s>
Using_VVG Lemma_NN1 2.8_MC ,_, construct_VV0 a_AT1 nonoverlapping_JJ collection_NN1 Jn_NP1 of_IO Nn_NP1 intervals_NN2 of_IO size_NN1 pn_NNU each_DD1 such_CS21 that_CS22 all_DB elements_NN2 of_IO Jn_NP1 intersect_VV0 An_AT1 and_CC @F_FO ._. 
</s>
<s>
Fix_VV0 a_AT1 cutoff_NN1 function_NN1 @F_FO ._. 
</s>
<s>
For_IF an_AT1 interval_NN1 J_ZZ1 with_IW center_NN1 J_ZZ1 ,_, define_VV0 the_AT function_NN1 @S_FO by_II @F_FO ,_, so_CS21 that_CS22 @F_FO ._. 
</s>
<s>
Now_RT ,_, define_VV0 the_AT weight_NN1 @S_FO by_II (_( see_VV0 Figure_NN1 3_MC )_) @F_FO ._. 
</s>
<s>
For_IF each_DD1 ,_, @S_FO ,_, there_RL exist_VV0 @S_FO and_CC @S_FO such_CS21 that_CS22 @S_FO ._. 
</s>
<s>
Also_RR ,_, @S_FO for_IF @S_FO ._. 
</s>
<s>
Therefore_RR ,_, @F_FO ._. 
</s>
<s>
Since_CS each_DD1 ,_, lies_VVZ in_RP at_RR21 most_RR22 500_MC intervals_NN2 in_II @S_FO ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Next_MD ,_, @F_FO ,_, where_CS C0_FO depends_VVZ only_RR on_II @S_FO ._. 
</s>
<s>
Here_RL we_PPIS2 use_VV0 the_AT formula_NN1 for_IF pn_NNU and_CC the_AT inequality_NN1 @S_FO valid_JJ for_IF all_DB @S_FO ._. 
</s>
<s>
Two_MC diffusion_NN1 processes_NN2 are_VBR mutually_RR absolutely_RR continuous_JJ if_CS and_CC only_RR if_CS they_PPHS2 differ_VV0 by_II a_AT1 pure_JJ drift_NN1 term_NN1 ._. 
</s>
<s>
Consider_VV0 three_MC independent_JJ Poisson_NP1 point_NN1 processes_NN2 :_: a_AT1 loop-soup_NN1 @S_FO ,_, a_AT1 P.p.p_NP1 ._. 
</s>
<s>
of_IO excursions_NN2 of_IO intensity_NN1 @S_FO ,_, a_AT1 P.p.p_NP1 ._. 
</s>
<s>
of_IO excursions_NN2 of_IO intensity_NN1 @S_FO ._. 
</s>
<s>
The_AT probability_NN1 that_CST the_AT two_MC P.p.p_NP1 ._. 
</s>
<s>
of_IO excursions_NN2 are_VBR connected_VVN either_RR directly_RR or_CC through_II a_AT1 cluster_NN1 of_IO @S_FO equals_VVZ ,_, according_II21 to_II22 Lemma_NN1 2.3_MC ,_, the_AT probability_NN1 that_CST @S_FO and_CC @S_FO are_VBR in_II the_AT same_DA cluster_NN1 of_IO @S_FO conditional_NN1 on_II @S_FO and_CC @S_FO ._. 
</s>
<s>
According_II21 to_II22 Lemma_NN1 2.1_MC this_DD1 probability_NN1 equals_VVZ @F_FO ._. 
</s>
<s>
Letting_VVG @S_FO we_PPIS2 get_VVI (_( 2.7_MC )_) ._. 
</s>
<s>
We_PPIS2 conduct_VV0 three_MC error_NN1 tests_NN2 (_( denoted_VVN by_II A_ZZ1 ,_, B_ZZ1 ,_, and_CC C_ZZ1 )_) ._. 
</s>
<s>
The_AT coefficients_NN2 @S_FO are_VBR associated_VVN with_IW the_AT function_NN1 @S_FO (_( see_VV0 Definition_NN1 42_MC and_CC (_( 65_MC )_) )_) and_CC with_IW 1-forms_NN2 @S_FO for_IF @S_FO (_( or_CC equivalently_RR ,_, @S_FO such_CS21 that_CS22 @S._FO @S_FO is_VBZ a_AT1 matrix_NN1 of_IO size_NN1 @S_FO ._. 
</s>
<s>
The_AT coefficients_NN2 @S_FO are_VBR associated_VVN with_IW the_AT function_NN1 and_CC with_IW 1-forms_NN2 @S_FO for_IF @S_FO (_( or_CC equivalently_RR ,_, @S_FO such_CS21 that_CS22 @S._FO @S_FO is_VBZ a_AT1 matrix_NN1 of_IO size_NN1 @S_FO ._. 
</s>
<s>
The_AT coefficients_NN2 @S_FO are_VBR associated_VVN with_IW 0-forms_NN2 @S_FO ,_, for_IF @S_FO and_CC with_IW 1-forms_NN2 @S_FO for_IF @S_FO (_( or_CC equivalently_RR ,_, @S_FO such_CS21 that_CS22 @S_FO ._. 
</s>
<s>
Remark_VV0 56_MC ._. 
</s>
<s>
This_DD1 is_VBZ quite_RG a_AT1 narrow_JJ focus_NN1 compared_VVN to_II the_AT spread_NN1 examined_VVN by_II Tatto_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
(_( 2008_MC )_) who_PNQS included_VVD greater_JJR detail_NN1 in_II number_NN1 and_CC algebra_NN1 and_CC a_RR31 great_RR32 deal_RR33 more_DAR evaluation_NN1 of_IO geometry_NN1 and_CC data_NN ._. 
</s>
<s>
While_CS Nesterov_NP1 '_NULL s_ZZ1 1983_MC development_NN1 of_IO acceleration_NN1 schemes_NN2 may_VM appear_VVI mysterious_JJ at_RR21 first_RR22 ,_, there_EX are_VBR multiple_JJ interpretations_NN2 of_IO AGD_NN1 as_II the_AT careful_JJ combination_NN1 of_IO different_JJ routines_NN2 for_IF function_NN1 minimization_NN1 ._. 
</s>
<s>
Section_NN1 3_MC presents_VVZ the_AT model_NN1 derivation_NN1 and_CC the_AT weak_JJ formulation_NN1 of_IO (_( 1.3_MC )_) ._. 
</s>
<s>
In_II those_DD2 examples_NN2 ,_, the_AT lattices_NN2 act_VV0 regularly_RR on_II the_AT set_NN1 of_IO vertices_NN2 of_IO their_APPGE builing_NN1 ;_; in_RR21 particular_RR22 ,_, the_AT types_NN2 of_IO vertices_NN2 are_VBR permuted_VVN cyclically_RR ._. 
</s>
<s>
Indeed_RR ,_, later_RRR on_RP in_II Section_NN1 5_MC we_PPIS2 will_VM see_VVI empirically_RR that_CST the_AT power_NN1 gain_NN1 can_VM be_VBI quite_RG substantial_JJ ._. 
</s>
<s>
Next_MD we_PPIS2 show_VV0 that_CST u1_FO is_VBZ tangent_JJ to_II the_AT segment_NN1 @S_FO ._. 
</s>
<s>
These_DD2 differences_NN2 in_II the_AT use_NN1 and_CC types_NN2 of_IO tables_NN2 of_IO values_NN2 in_II each_DD1 context_NN1 suggest_VV0 a_AT1 possible_JJ explanation_NN1 for_IF the_AT differences_NN2 found_VVN in_II some_DD students_NN2 '_NULL tendency_NN1 to_TO use_VVI an_AT1 habitual_JJ but_CCB invalid_JJ approach_NN1 for_IF the_AT Task_NN1 1_MC1 (_( Cleeremans_NP2 &;_NULL Jimenez_NP1 ,_, 2002_MC )_) ._. 
</s>
<s>
For_IF a_AT1 tall-skinny_JJ @S_FO matrix_NN1 ,_, we_PPIS2 accelerate_VV0 the_AT initial_JJ QR_NP1 factorization_NN1 &lsqb;_( 110_MC &rsqb;_) ._. 
</s>
<s>
The_AT R-module_JJ M_NN1 is_VBZ locally_RR free_JJ on_II the_AT punctured_JJ spectrum_NN1 ,_, and_CC hence_RR the_AT same_DA is_VBZ true_JJ of_IO its_APPGE @S_FO modules_NN2 ,_, @S_FO ._. 
</s>
<s>
The_AT Poincare_NN1 series_NN of_IO M_ZZ1 is_VBZ @S_FO (_( see_VV0 (_( 4.4_MC )_) )_) ,_, so_CS the_AT ranks_NN2 of_IO its_APPGE @S_FO modules_NN2 are_VBR @F_FO ._. 
</s>
<s>
The_AT projective_JJ dimension_NN1 of_IO (_( M_ZZ1 )_) is_VBZ 2n_FO d_ZZ1 and_CC its_APPGE depth_NN1 is_VBZ d_ZZ1 ._. 
</s>
<s>
Moreover_RR ,_, more_DAR can_VM be_VBI said_VVN about_II the_AT twist_NN1 ,_, and_CC also_RR about_II the_AT dimension_NN1 of_IO the_AT spectra_NN2 of_IO our_APPGE Cartan_JJ subalgebras_NN2 ._. 
</s>
<s>
The_AT mathematical_JJ model_NN1 and_CC computational_JJ method_NN1 parts_NN2 of_IO the_AT work_NN1 were_VBDR also_RR supported_VVN in_RR21 part_RR22 by_II Grant-in-Aid_NP1 for_IF Challenging_VVG Exploratory_JJ Research_NN1 16K13779_FO from_II JSPS_NP1 and_CC Grant-in-Aid_NP1 for_IF Scientific_JJ Research_NN1 (_( S_ZZ1 )_) 26220002_MC from_II the_AT Ministry_NN1 of_IO Education_NN1 ,_, Culture_NN1 ,_, Sports_NN2 ,_, Science_NN1 and_CC Technology_NN1 of_IO Japan_NP1 (_( MEXT_NP1 )_) (_( for_IF the_AT 5th_MD author_NN1 )_) and_CC ARO_NP1 Grant_NP1 W911NF-17-1-0046_NP1 and_CC Top_JJ Global_JJ University_NN1 Project_NN1 of_IO Waseda_NP1 University_NN1 (_( for_IF the_AT last_MD author_NN1 )_) ._. 
</s>
<s>
Since_CS y_ZZ1 is_VBZ a_AT1 classical_JJ subsolution_NN1 to_II the_AT initial-boundary_JJ problem_NN1 solved_VVN by_II @S_FO ,_, by_II comparison_NN1 this_DD1 yields_VVZ (_( 4.1_MC )_) ._. 
</s>
<s>
Thus_RR in_II what_DDQ follows_VVZ we_PPIS2 assume_VV0 that_CST G_ZZ1 is_VBZ a_AT1 free_JJ group_NN1 of_IO rank_NN1 at_RR21 least_RR22 2_MC ._. 
</s>
<s>
The_AT multiplier_NN1 substitution_NN1 equation_NN1 (_( 7.12_MC )_) for_IF the_AT eigenvalue_NN1 optimization_NN1 problem_NN1 (_( 7.1_MC )_) is_VBZ written_VVN @F_FO ,_, where_CS @F_FO with_IW @F_FO ._. 
</s>
<s>
It_PPH1 is_VBZ worth_II noting_VVG that_CST for_IF the_AT eigenvalue_NN1 optimization_NN1 problem_NN1 (_( 7.1_MC )_) the_AT least-squares_NN2 multiplier_NN1 approximation_NN1 formula_NN1 (_( 7.11_MC )_) becomes_VVZ the_AT Rayleigh_NP1 quotient_NN1 ._. 
</s>
<s>
Section_NN1 10.3_MC :_: We_PPIS2 prove_VV0 the_AT so-called_JJ gluing_JJ lemma_NN1 ,_, Lemma_NN1 10.5_MC ,_, that_DD1 uses_VVZ R_NP1 to_TO extend_VVI the_AT three_MC point_NN1 structure_NN1 constant_NN1 to_II a_AT1 holomorphic_JJ function_NN1 in_II a_AT1 neighborhood_NN1 of_IO Q._NN1 In_II conclusion_NN1 ,_, this_DD1 example_NN1 shows_VVZ that_CST having_VHG a_AT1 dense_JJ network_NN1 underneath_II the_AT model_NN1 (_( 1.1_MC )_) may_VM not_XX be_VBI enough_DD to_TO have_VHI uniform_NN1 (_( with_II31 respect_II32 to_II33 N_ZZ1 )_) controllability_NN1 properties_NN2 for_IF the_AT system_NN1 ,_, which_DDQ are_VBR then_RT transferred_VVN to_II the_AT corresponding_JJ infinite_JJ agent_NN1 equation_NN1 ._. 
</s>
<s>
Thus_RR ,_, in_II these_DD2 limiting_JJ cases_NN2 ,_, we_PPIS2 can_VM directly_RR apply_VVI Theorem_NN1 5.18_MC to_II @S_FO ._. 
</s>
<s>
After_II that_DD1 ,_, it_PPH1 only_RR remains_VVZ to_TO prove_VVI the_AT bound_NN1 on_II @S_FO ._. 
</s>
<s>
For_IF this_DD1 purpose_NN1 ,_, we_PPIS2 combine_VV0 Lemma_NN1 5.20_MC and_CC the_AT bound_NN1 on_II @S_FO in_II Theorem_NN1 5.18_MC for_IF obtaining_VVG @F_FO ._. 
</s>
<s>
This_DD1 ends_VVZ the_AT proof_NN1 ._. 
</s>
<s>
If_CS @S_FO is_VBZ an_AT1 SLn(Z)_NP1 contravariant_NN1 and_CC translation_NN1 invariant_JJ Minkowski_JJ valuation_NN1 ,_, then_RT @S_FO ._. 
</s>
<s>
Proof_NN1 ._. 
</s>
<s>
As_CSA explained_VVN above_RL ,_, we_PPIS2 will_VM start_VVI by_II specifying_VVG @S_FO ,_, but_CCB the_AT restriction_NN1 (_( 29_MC )_) means_VVZ that_CST the_AT true_JJ free_JJ function_NN1 is_VBZ its_APPGE derivative_JJ urf(r)_NN1 ,_, which_DDQ we_PPIS2 will_VM prescribe_VVI as_II a_AT1 continuous_JJ integrable_JJ function_NN1 @S_FO ,_, so_CS21 that_CS22 @F_FO ._. 
</s>
<s>
In_II this_DD1 sense_NN1 ,_, it_PPH1 would_VM be_VBI very_RG interesting_JJ to_TO compare_VVI the_AT students_NN2 '_NULL voices_NN2 collected_VVN in_II two_MC different_JJ moments_NN2 :_: during_II the_AT transition_NN1 and_CC at_II the_AT end_NN1 of_IO the_AT university_NN1 experience_NN1 ._. 
</s>
<s>
The_AT reason_NN1 is_VBZ that_CST the_AT '_NULL overlap_NN1 '_NULL between_II the_AT target_NN1 distribution_NN1 f_ZZ1 and_CC a_AT1 tempered_JJ version_NN1 ,_, such_II21 as_II22 @S_FO ,_, can_VM beextremely_RR low_JJ if_CS f_ZZ1 is_VBZ a_AT1 high_JJ dimensional_JJ distribution_NN1 ._. 
</s>
<s>
The_AT Z2-graded_JJ algebra_NN1 Rv_NP1 is_VBZ concentrated_VVN in_II degrees_NN2 in_II @S_FO ,_, so_CS it_PPH1 makes_VVZ sense_NN1 to_TO regard_VVI it_PPH1 as_CSA just_RR a_AT1 Z-graded_JJ algebra_NN1 ._. 
</s>
<s>
We_PPIS2 also_RR use_VV0 @S_FO (_( see_VV0 (_( 4.40_MC )_) )_) ._. 
</s>
<s>
If_CS the_AT operator_NN1 F_ZZ1 fulfills_VVZ the_AT Lipschitz_NP1 estimate_NN1 @F_FO for_IF all_DB @F_FO ,_, then_RT (_( 3.2_MC )_) is_VBZ fulfilled_VVN with_IW @S_FO ,_, and_CC @S_FO ._. 
</s>
<s>
From_II Proposition_NN1 3.1_MC ,_, we_PPIS2 know_VV0 that_CST each_DD1 imbedded_VVD discrete_JJ structure_NN1 @S_FO carries_VVZ a_AT1 sequence_NN1 of_IO Fk-special_JJ semimartingale_NN1 decompositions_NN2 @F_FO ,_, We_PPIS2 also_RR call_VV0 Ggjpj_NP1 the_AT pressure_NN1 difference_NN1 caused_VVN by_II gravity_NN1 ,_, since_CS ,_, similarly_RR to_II Ggp_NP1 ,_, it_PPH1 drives_VVZ the_AT growth_NN1 of_IO the_AT RT_NN1 instability_NN1 by_II acting_VVG on_II the_AT particles_NN2 on_II the_AT interface_NN1 ._. 
</s>
<s>
Let_VV0 P_ZZ1 be_VBI a_AT1 finite-degree_JJ polynomial_NN1 functor_NN1 over_II an_AT1 infinite_JJ field_NN1 K._NP1 Does_VDZ any_DD sequence_NN1 @S_FO of_IO ideals_NN2 in_II @S_FO eventually_RR become_VV0 constant_JJ ?_? 
</s>
<s>
For_IF the_AT proof_NN1 of_IO the_AT following_JJ proposition_NN1 ,_, we_PPIS2 refer_VV0 to_II Ref._NN1 44_MC ,_, Proposition_NN1 6_MC ._. 
</s>
<s>
One_MC1 may_VM use_VVI quasi-Monte_NP1 Carlo_NP1 low-discrepancy_JJ point_NN1 sets_NN2 &lsqb;_( 37_MC ,_, 18_MC &rsqb;_) ,_, which_DDQ have_VH0 been_VBN investigated_VVN in_II &lsqb;_( 39_MC &rsqb;_) ._. 
</s>
<s>
Standardized_JJ math_NN1 test_VV0 We_PPIS2 used_VVD an_AT1 achievement_NN1 test_NN1 consisting_VVG of_IO 14_MC reality-based_JJ in_II the_AT "_" linear_JJ functions_NN2 "_" subject_NN1 area_NN1 to_TO assess_VVI standardized_JJ mathematical_JJ competence_NN1 as_II a_AT1 central_JJ element_NN1 of_IO previous_JJ knowledge_NN1 for_IF a_AT1 competent_JJ solution_NN1 process_NN1 (_( Leiss_NP1 et_RA21 al._RA22 ,_, 2010_MC )_) ._. 
</s>
<s>
Finally_RR ,_, since_CS W_ZZ1 satisfies_VVZ (_( 2.3_MC )_) ,_, @F_FO holds_VVZ for_IF Ns(T)_NP1 sufficiently_RR small_JJ ._. 
</s>
<s>
Since_CS this_DD1 algebra_NN1 has_VHZ a_AT1 unique_JJ tracial_JJ state_NN1 ,_, and_CC this_DD1 trace_NN1 takes_VVZ rational_JJ values_NN2 on_II all_DB projections_NN2 ,_, we_PPIS2 see_VV0 that_CST this_DD1 value_NN1 of_IO k_ZZ1 must_VM be_VBI rational_JJ ._. 
</s>
<s>
The_AT sub-Riemannian_JJ distance_NN1 is_VBZ defined_VVN by_II :_: @_II ._. 
</s>
<s>
Assume_VV0 that_CST m_ZZ1 :_: =dim(L)≤n2_FO ,_, and_CC that_CST there_EX is_VBZ a_AT1 sequence_NN1 of_IO singular_JJ points_NN2 xk_FO →_NULL 0_MC and_CC radii_NN2 rk0_FO with_IW |_NULL xk_FO |_NULL ≤rk/2_FU such_CS21 that_CS22 @F_FO ,_, and_CC yk_VV0 :_: =xkrk_FO →_NULL y∞_FO ._. 
</s>
<s>
Then_RT y∞∈L_FO and_CC q(y∞)=0_FO ._. 
</s>
<s>
Afterwards_RT ,_, we_PPIS2 articulate_VV0 and_CC draw_VV0 a_AT1 contrast_NN1 with_IW alternative_JJ viewpoints_NN2 that_CST provide_VV0 a_AT1 critical_JJ stance_NN1 toward_II previous_JJ accounts_NN2 but_CCB also_RR provide_VV0 new_JJ ways_NN2 to_TO think_VVI about_II the_AT issues_NN2 under_II consideration_NN1 ._. 
</s>
<s>
The_AT bounds_NN2 on_II the_AT individual_JJ terms_NN2 in_II the_AT right-hand_JJ side_NN1 of_IO (_( 3.3_MC )_) are_VBR then_RT as_CSA follows_VVZ ._. 
</s>
<s>
We_PPIS2 show_VV0 that_CST for_IF arbitrarily_RR small_JJ @S_FO ,_, we_PPIS2 have_VH0 @F_FO and_CC ,_, under_II the_AT additional_JJ hypothesis_NN1 of_IO a_AT1 generic_JJ measure_NN1 and_CC associated_JJ generic_JJ point_NN1 whose_DDQGE orbit_NN1 closure_NN1 is_VBZ not_XX uniquely_RR ergodic_JJ ,_, @F_FO ._. 
</s>
<s>
The_AT theorem_NN1 follows_VVZ immediately_RR from_II these_DD2 estimates_NN2 ._. 
</s>
<s>
Nandini_NN2 taught_VVD 12_MC and_CC 21_MC decimal_JJ lessons_NN2 in_II Y1_FO and_CC Y2_FO ,_, respectively_RR ._. 
</s>
<s>
Proof_NN1 See_VV0 for_REX21 example_REX22 &lsqb;_( 5_MC ,_, Thm._NP1 2.10_MC &rsqb;_) ._. 
</s>
<s>
The_AT proof_NN1 is_VBZ analogous_JJ to_II the_AT one_PN1 of_IO Lemma_NN1 2.3_MC ,_, where_CS we_PPIS2 use_VV0 &lsqb;_( BCN_NP1 ,_, Theorem_NN1 1.3_MC &rsqb;_) and_CC &lsqb;_( GT_NP1 ,_, Corollary_NN1 8.36_MC &rsqb;_) in_II31 place_II32 of_II33 &lsqb;_( GT_NP1 ,_, Theorems_NN2 8.17_MC ,_, 8.18_MC and_CC 8.32_MC &rsqb;_) ._. 
</s>
<s>
In_II the_AT limit_NN1 problem_NN1 considered_VVN in_II the_AT present_JJ paper_NN1 such_DA a_AT1 recirculation_NN1 near_II the_AT junction_NN1 is_VBZ emphasized_VVN and_CC its_APPGE effect_NN1 produces_VVZ the_AT decoupling_NN1 of_IO the_AT flow_NN1 in_II the_AT two_MC branches_NN2 ._. 
</s>
<s>
Then_RT ,_, for_IF every_AT1 @S_FO and_CC for_IF every_AT1 @S_FO ,_, there_RL exist_VV0 a_AT1 lifting_NN1 @S_FO ,_, such_CS21 that_CS22 @L_FO ,_, where_CS c_ZZ1 is_VBZ a_AT1 positive_JJ constant_NN1 depending_VVG only_RR on_II and_CC on_II the_AT shape_NN1 of_IO T._NP1 We_PPIS2 thank_VV0 the_AT referees_NN2 for_IF interesting_JJ suggestions_NN2 and_CC for_IF the_AT accurate_JJ review_NN1 ,_, which_DDQ allowed_VVD us_PPIO2 to_TO improve_VVI this_DD1 paper_NN1 ._. 
</s>
<s>
Peaks_NN2 of_IO (_( pk_NNU )_) ,_, as_II31 well_II32 as_II33 small_JJ transition_NN1 probabilities_NN2 ,_, tend_VV0 to_TO "_" disconnect_VVI "_" the_AT graph_NN1 and_CC are_VBR bad_JJ for_IF mixing_NN1 ._. 
</s>
<s>
It_PPH1 is_VBZ common_JJ in_II the_AT classrooms_NN2 we_PPIS2 work_VV0 in_RP to_TO observe_VVI that_CST the_AT use_NN1 of_IO CT_NN1 creates_VVZ the_AT opportunity_NN1 for_IF children_NN2 to_TO exercise_VVI agency_NN1 ._. 
</s>
<s>
In_BCL21 order_BCL22 to_TO make_VVI the_AT presentation_NN1 clearer_JJR and_CC easier_RRR to_TO follow_VVI ,_, we_PPIS2 deferred_VVD several_DA2 proofs_NN2 of_IO technical_JJ claims_NN2 and_CC some_DD complementary_JJ material_NN1 to_II Appendices_NN2 A-E_FO ._. 
</s>
<s>
Finally_RR ,_, in_II Appendix_NN1 F_ZZ1 we_PPIS2 outline_VV0 a_AT1 possible_JJ strategy_NN1 to_TO approach_VVI the_AT global_JJ Birkhoff_NN1 conjecture_NN1 ,_, by_II31 means_II32 of_II33 the_AT affine_JJ length_NN1 shortening_VVG flow_NN1 ._. 
</s>
<s>
Notice_VV0 that_CST Crandall_NP1 '_NULL s_ZZ1 algorithm_NN1 (_( 4.4_MC )_) is_VBZ exactly_RR Ostrowski_JJ '_NULL s_ZZ1 algorithm_NN1 (_( 4.6_MC )_) ._. 
</s>
<s>
Next_MD consider_VV0 the_AT functions_NN2 @S_FO defined_VVN for_IF all_DB @S_FO by_II @F_FO ._. 
</s>
<s>
We_PPIS2 first_MD explain_VV0 a_AT1 version_NN1 of_IO this_DD1 construction_NN1 in_II which_DDQ @S_FO and_CC the_AT surface_NN1 is_VBZ a_AT1 unit-area_JJ LQG_NP1 sphere_NN1 decorated_VVN with_IW an_AT1 independent_JJ SLE8_FO ._. 
</s>
<s>
The_AT proof_NN1 of_IO the_AT following_JJ theorem_NN1 is_VBZ identical_JJ to_II that_DD1 of_IO Theorem_NN1 6.5_MC ._. 
</s>
<s>
For_IF ?_NNU small_JJ enough_RR ,_, the_AT derivative_NN1 of_IO any_DD @S_FO is_VBZ uniformly_RR bounded_VVN and_CC non-vanishing_NN1 on_II a_AT1 slightly_RR shrunk_VVN A_ZZ1 ;_; in_RR21 particular_RR22 g_ZZ1 has_VHZ no_AT critical_JJ points_NN2 in_II A._NNU In_II31 response_II32 to_II33 task_NN1 10_MC ,_, the_AT PST_NP1 did_VDD not_XX simply_RR shade_VVI fractional_JJ parts_NN2 within_II the_AT whole_NN1 ,_, but_CCB drew_VVD the_AT fractional_JJ part_NN1 from_II the_AT whole_NN1 ._. 
</s>
<s>
We_PPIS2 first_MD give_VV0 a_AT1 simple_JJ lemma_NN1 that_CST allows_VVZ us_PPIO2 to_TO decide_VVI when_RRQ a_AT1 permutation_NN1 group_NN1 is_VBZ m-transitive_JJ ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI a_AT1 set_NN1 of_IO Schwartz_NP1 functions_VVZ that_CST is_VBZ equicontinuous_JJ in_II H1(R)_FO ._. 
</s>
<s>
Denoting_VVG @S_FO ,_, we_PPIS2 have_VH0 @F_FO ,_, for_IF some_DD constants_NN2 @S_FO depending_VVG only_RR on_II q_ZZ1 ._. 
</s>
<s>
Since_CS study_NN1 satisfaction_NN1 predicts_VVZ actual_JJ drop-out_NN1 (_( Brandstatter_NP1 et_RA21 al._RA22 ,_, 2006_MC ;_; Schiefele_NP1 et_RA21 al._RA22 ,_, 2007_MC )_) ,_, our_APPGE results_NN2 indicate_VV0 that_CST individual_JJ interests_NN2 which_DDQ are_VBR congruent_JJ to_II the_AT contents_NN2 of_IO the_AT program_NN1 may_VM support_VVI students_NN2 to_TO retain_VVI in_II the_AT program_NN1 during_II the_AT transition_NN1 to_II university_NN1 mathematics_NN1 ._. 
</s>
<s>
By_II Lemma_NN1 2.1_MC ,_, one_MC1 has_VHZ @F_FO ._. 
</s>
<s>
The_AT cyclic_JJ object_NN1 E*_FO in_II (_( dgcat(2)_FO ,_, Mo_NP1 )_) is_VBZ 2-Segal_NP1 ._. 
</s>
<s>
The_AT main_JJ result_NN1 of_IO the_AT paper_NN1 then_RT reads_VVZ as_CSA follows_VVZ ._. 
</s>
<s>
Indeed_RR ,_, for_IF each_DD1 f_ZZ1 G_ZZ1 H_ZZ1 with_IW @S_FO on_II B(r)_JJ ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Now_RT for_IF any_DD N_ZZ1 >_FO 0_MC and_CC any_DD @S_FO ,_, substituting_VVG @F_FO yields_VVZ @F_FO ._. 
</s>
<s>
This_DD1 verifies_VVZ (_( 5.2_MC )_) ;_; hence_RR ,_, the_AT claim_NN1 is_VBZ proved_VVN ._. 
</s>
<s>
How_RRQ did_VDD the_AT student_NN1 think_VVI to_TO find_VVI the_AT response_NN1 ?_? 
</s>
<s>
However_RR ,_, the_AT difference_NN1 is_VBZ bounded_VVN from_II below_RL ,_, and_CC so_RR we_PPIS2 should_VM eventually_RR see_VVI the_AT slower_JJR convergence_NN1 rate_NN1 for_IF the_AT variance_NN1 as_CSA predicted_VVN by_II our_APPGE theory_NN1 ._. 
</s>
<s>
In_II this_DD1 case_NN1 ,_, we_PPIS2 are_VBR looking_VVG at_II the_AT feasible_JJ set_NN1 @F_FO ._. 
</s>
<s>
For_IF X_ZZ1 g_ZZ1 Xfree_NP1 ,_, we_PPIS2 still_RR have_VH0 X_MC >_FO 0_MC and_CC @S_FO ._. 
</s>
<s>
Thus_RR ,_, one_PN1 can_VM simply_RR relax_VVI to_II the_AT problem_NN1 denoted_VVN by_II SDP-3_MC in_II Table_NN1 1_MC1 ._. 
</s>
<s>
Otherwise_RR ,_, whether_CSW m_ZZ1 +_FO n_ZZ1 is_VBZ even_RR or_CC there_EX is_VBZ an_AT1 odd_JJ face_NN1 size_NN1 ,_, we_PPIS2 can_VM first_MD choose_VVI face_NN1 moves_VVZ to_TO change_VVI the_AT @S_FO coordinate_VV0 from_II m_ZZ1 to_II n_ZZ1 ,_, and_CC then_RT follow_VV0 them_PPHO2 by_II some_DD number_NN1 of_IO me_PPIO1 moves_VVZ to_TO change_VVI the_AT @S_FO coordinate_VV0 to_II n_ZZ1 ._. 
</s>
<s>
We_PPIS2 may_VM then_RT follow_VVI these_DD2 moves_NN2 by_II a_AT1 path_NN1 from_II (_( n_ZZ1 ,_, 0_MC )_) to_II itself_PPX1 with_IW length_NN1 given_VVN by_II any_DD sufficiently_RR large_JJ multiple_NN1 of_IO b_ZZ1 ._. 
</s>
<s>
We_PPIS2 choose_VV0 @S_FO to_TO be_VBI a_AT1 hyperplane_NN1 in_II Rn_NP1 and_CC @S_FO to_TO be_VBI a_AT1 half-plane_NN1 in_II Rn_NP1 ._. 
</s>
<s>
For_IF any_DD @S_FO and_CC for_IF any_DD @S_FO ,_, let_VV0 @S_FO ._. 
</s>
<s>
Since_CS the_AT holonomy_NN1 along_II F_ZZ1 is_VBZ asymptotically_RR conformal_JJ &lsqb;_( L1_FO ,_, Lemma_NN1 7.3_MC &rsqb;_) ,_, we_PPIS2 obtain_VV0 the_AT scaling_NN1 result_NN1 for_IF gnt_NNU ._. 
</s>
<s>
In_II this_DD1 section_NN1 ,_, we_PPIS2 analyze_VV0 the_AT pressure-robust_JJ method_NN1 given_VVN by_II equation_NN1 (_( PR_NP1 )_) ._. 
</s>
<s>
The_AT quantity_NN1 EK_NN1 will_VM signify_VVI the_AT translation_NN1 of_IO the_AT contours_NN2 @S_FO and_CC @S_FO ._. 
</s>
<s>
Specifically_RR ,_, we_PPIS2 make_VV0 the_AT following_JJ definitions_NN2 ,_, which_DDQ are_VBR analogous_JJ to_II Definition_NN1 6.3_MC ,_, Definition_NN1 6.4_MC and_CC Definition_NN1 6.5_MC ._. 
</s>
<s>
Bounds_VVZ similar_JJ to_II (_( 1.7_MC )_) have_VH0 been_VBN obtained_VVN in_II &lsqb;_( 5,12,20,21_MC &rsqb;_) for_IF N-boson_JJ systems_NN2 in_II the_AT mean_JJ field_NN1 limit_NN1 ,_, described_VVN by_II the_AT Hamilton_NP1 operator_NN1 @F_FO acting_VVG again_RT on_II L2(A3N)_FO ._. 
</s>
<s>
Since_CS a_AT1 similar_JJ procedure_NN1 is_VBZ used_VVN for_IF the_AT two-phase_JJ setting_NN1 and_CC is_VBZ presented_VVN in_II great_JJ detail_NN1 in_II Ref._NN1 27_MC ,_, we_PPIS2 only_RR briefly_RR outline_VV0 the_AT procedure_NN1 ._. 
</s>
<s>
Obviously_RR ,_, the_AT new_JJ terms_NN2 containing_VVG (_( 2.31_MC )_) and_CC (_( 2.32_MC )_) satisfy_VV0 (_( i_ZZ1 )_) ._. 
</s>
<s>
Assume_VV0 that_CST @S_FO ,_, as_CSA a_AT1 function_NN1 of_IO f_ZZ1 ,_, is_VBZ Schwartz_NP1 class_NN1 for_IF every_AT1 n_ZZ1 =_FO 0_MC ._. 
</s>
<s>
First_MD ,_, let_VV0 '_NULL s_ZZ1 suppose_VVI that_CST (_( Y_ZZ1 ,_, )_) is_VBZ non-atomic_JJ ._. 
</s>
<s>
Let_VV0 us_PPIO2 consider_VVI the_AT spaces_NN2 :_: @F_FO with_IW the_AT usual_JJ norms_NN2 ,_, and_CC let_VV0 @S_FO be_VBI the_AT bilinear_JJ form_NN1 defined_VVN by_II :_: @F_FO ,_, where_CS @F_FO and_CC @S_FO is_VBZ the_AT bilinear_JJ form_NN1 defined_VVN in_II (_( 2.3_MC )_) ._. 
</s>
<s>
If_CS @S_FO has_VHZ positive_JJ upper_JJ density_NN1 ,_, i.e._REX ,_, @F_FO ,_, then_RT A_ZZ1 contains_VVZ @S_FO ,_, where_CS B_ZZ1 and_CC C_ZZ1 are_VBR infinite_JJ subsets_NN2 of_IO N._NNU In_II31 view_II32 of_II33 the_AT subsequent_JJ numerical_JJ discretization_NN1 ,_, it_PPH1 is_VBZ convenient_JJ to_TO think_VVI of_IO @S_FO as_II the_AT position_NN1 at_II time_NNT1 t_ZZ1 of_IO a_AT1 moving_JJ particle_NN1 with_IW label_NN1 p_ZZ1 ,_, and_CC of_IO @S_FO as_II a_AT1 collection_NN1 of_IO such_DA particles_NN2 ._. 
</s>
<s>
Let_VV0 P(0)_FO and_CC P(1)_FO be_VBI Borel_NN1 probability_NN1 measures_NN2 on_II Rn-1_MC1 ._. 
</s>
<s>
The_AT proof_NN1 of_IO this_DD1 last_MD property_NN1 is_VBZ more_RGR involved_JJ ,_, and_CC it_PPH1 is_VBZ organized_VVN into_II three_MC steps_NN2 ._. 
</s>
<s>
Our_APPGE analysis_NN1 follows_VVZ the_AT well-established_JJ practice_NN1 of_IO appealing_VVG to_II uniform_JJ polynomial_NN1 approximations_NN2 &lsqb;_( 47_MC ,_, 33_MC &rsqb;_) to_TO construct_VVI "_" good_JJ "_" elements_NN2 in_II @S_FO achieving_VVG the_AT desired_JJ convergence_NN1 ,_, and_CC ,_, complementing_VVG this_DD1 approach_NN1 ,_, we_PPIS2 also_RR construct_VV0 reference_NN1 elements_NN2 in_II @S_FO based_VVN on_II Nesterov_NP1 '_NULL s_ZZ1 accelerated_JJ gradient_NN1 method_NN1 &lsqb;_( 35_MC ,_, 36_MC ,_, 49_MC &rsqb;_) ._. 
</s>
<s>
However_RR ,_, there_EX are_VBR some_DD significant_JJ differences_NN2 :_: We_PPIS2 use_VV0 the_AT functions_NN2 fj_NNU and_CC fj*_FO instead_II21 of_II22 the_AT original_JJ @S_FO ._. 
</s>
<s>
They_PPHS2 are_VBR (_( quasi_JJ )_) -eigenfunctions_NN2 ,_, respectively_RR of_IO the_AT non-selfadjoint_JJ operators_NN2 @S_FO and_CC @S_FO ._. 
</s>
<s>
In_RR21 general_RR22 VarJibli_JJ @S_FO ._. 
</s>
<s>
Actually_RR ,_, in_II the_AT special_JJ case_NN1 of_IO regular_JJ graphs_NN2 with_IW @S_FO ,_, one_PN1 has_VHZ @S_FO ._. 
</s>
<s>
We_PPIS2 did_VDD not_XX take_VVI the_AT square_NN1 of_IO @S_FO in_II the_AT definition_NN1 ._. 
</s>
<s>
Each_DD1 method_NN1 requires_VVZ different_JJ structure_NN1 assumptions_NN2 and_CC achieves_VVZ different_JJ guarantees_NN2 mostly_RR in_II an_AT1 ergodic_JJ or_CC averaging_VVG sense_NN1 ._. 
</s>
<s>
Then_RT fix_VV0 n_ZZ1 such_CS21 that_CS22 @S_FO for_IF all_DB @S_FO ._. 
</s>
<s>
They_PPHS2 might_VM even_RR go_VVI on_RP to_TO explore_VVI what_DDQ sets_VVZ of_IO six_MC numbers_NN2 may_VM be_VBI the_AT solutions_NN2 of_IO an_AT1 expression_NN1 polygon_NN1 ,_, or_CC experiment_NN1 with_IW having_VHG five_MC expressions_NN2 rather_II21 than_II22 four_MC ,_, for_REX21 instance_REX22 ._. 
</s>
<s>
So_RR (_( by_II Lemma_NN1 7.4_MC )_) its_APPGE projection_NN1 to_II Yv_NP1 is_VBZ also_RR of_IO full_JJ measure_NN1 ._. 
</s>
<s>
If_CS r_ZZ1 is_VBZ sufficiently_RR small_JJ ,_, this_DD1 includes_VVZ all_DB scales_NN2 we_PPIS2 need_VV0 to_TO consider_VVI in_II the_AT definition_NN1 of_IO 0-HE_PPHS1 ._. 
</s>
<s>
If_CS @S_FO is_VBZ an_AT1 SLn(Z)_NP1 equivariant_NN1 and_CC translation_NN1 invariant_JJ Minkowski_JJ valuation_NN1 ,_, then_RT Z_ZZ1 P_ZZ1 is_VBZ contained_VVN in_II a_AT1 subspace_NN1 parallel_NN1 to_TO aff_VVI P._NP1 Proof_NN1 ._. 
</s>
<s>
We_PPIS2 write_VV0 @S_FO with_IW @F_FO ._. 
</s>
<s>
Estimate_VV0 on_II J1_FO and_CC J2_FO ._. 
</s>
<s>
Group_NN1 coordinate_VV0 descent_NN1 has_VHZ been_VBN used_VVN previously_RR in_II &lsqb;_( 9_MC ,_, 50_MC &rsqb;_) without_IW quadratic_JJ approximation_NN1 ._. 
</s>
<s>
Teachers_NN2 do_VD0 seem_VVI to_TO be_VBI relatively_RR successful_JJ in_II choosing_VVG appropriate_JJ learning_NN1 goals_NN2 in_II31 relation_II32 to_II33 LTs_NP1 ._. 
</s>
<s>
For_IF a_AT1 pair_NN of_IO edges_NN2 (_( u_ZZ1 ,_, v_ZZ1 )_) and_CC (_( x_ZZ1 ,_, y_ZZ1 )_) there_EX are_VBR two_MC possible_JJ swaps_NN2 ,_, as_CSA shown_VVN in_II Figure_NN1 3_MC ._. 
</s>
<s>
Finally_RR ,_, note_VV0 that_CST @S_FO for_IF any_RR (_( compactly_RR supported_VVN )_) s_ZZ1 e_ZZ1 f(E)_NN1 ,_, so_CS the_AT first_MD term_NN1 of_IO (_( 5_MC )_) does_VDZ not_XX contribute_VVI to_II the_AT integral_JJ ._. 
</s>
<s>
We_PPIS2 define_VV0 the_AT conditional_JJ entropy_NN1 of_IO X_ZZ1 relative_II21 to_II22 Y_ZZ1 as_CSA @F_FO ._. 
</s>
<s>
We_PPIS2 recall_VV0 some_DD well-known_JJ properties_NN2 ._. 
</s>
<s>
The_AT space-time_JJ periodic_JJ solutions_NN2 of_IO (_( 1_MC1 )_) can_VM be_VBI expanded_VVN using_VVG the_AT Fourier_NP1 expansion_NN1 ,_, which_DDQ corresponds_VVZ to_TO let_VVI the_AT length_NN1 of_IO the_AT spatial_JJ interval_NN1 be_VBI @S_FO ._. 
</s>
<s>
Denote_VV0 by_II @S_FO the_AT sequence_NN1 of_IO Fourier_NP1 coefficients_NN2 of_IO u_ZZ1 (_( t_ZZ1 ,_, y_ZZ1 )_) ._. 
</s>
<s>
Theorem_NN1 4.2_MC For_IF any_DD i∈I_FO and_CC e=±1_FO ,_, the_AT braid_NN1 group_NN1 operators_NN2 Ti_NN1 ,_, e_ZZ1 and_CC Ti_NP1 ,_, e_ZZ1 restrict_VV0 to_II isomorphisms_NN2 of_IO U._NP1 Once_RR local_JJ existence_NN1 and_CC uniqueness_NN1 for_IF the_AT previous_JJ Stochastic_JJ Partial_JJ Differential_JJ Equation_NN1 (_( SPDE_NP1 )_) is_VBZ established_VVN and_CC one_PN1 has_VHZ shown_VVN that_CST ,_, in_II @S_FO can_VM be_VBI enhanced_VVN to_II a_AT1 rough_JJ distribution_NN1 ,_, we_PPIS2 obtain_VV0 the_AT following_JJ result_NN1 ._. 
</s>
<s>
Two_MC models_NN2 of_IO teachers_NN2 '_NULL beliefs_NN2 are_VBR revealed_VVN through_II our_APPGE interviews_NN2 :_: exclusive_JJ and_CC inclusive_JJ models_NN2 ._. 
</s>
<s>
The_AT replacement_NN1 of_IO constructible_JJ sheaves_NN2 by_II equivariant_JJ sheaves_NN2 has_VHZ a_AT1 straightforward_JJ analogue_NN1 in_II our_APPGE setting_NN1 ,_, and_CC it_PPH1 leads_VVZ to_II the_AT monoidal_JJ category_NN1 (_( @S_FO )_) of_IO B-equivariant_JJ parity_NN1 complexes_NN2 on_II @S_FO ._. 
</s>
<s>
Note_VV0 that_CST (_( 7.3_MC )_) says_VVZ that_CST if_CS we_PPIS2 have_VH0 control_NN1 on_II @S_FO ,_, then_RT we_PPIS2 have_VH0 control_NN1 on_II @S_FO for_IF all_DB @S.In_FO particular_JJ ,_, Lemma_NN1 7.4_MC implies_VVZ that_CST if_CS @S_FO (_( we_PPIS2 can_VM choose_VVI such_DA @S_FO without_IW loss_NN1 of_IO generality_NN1 )_) then_RT @S_FO on_II @S_FO for_IF all_DB @S_FO ,_, thus_RR @S_FO for_IF all_DB @S_FO and_CC @S_FO at_II each_DD1 @S_FO ._. 
</s>
<s>
Lemma_NN1 6.10_MC &lsqb;_( 71_MC ,_, Theorem_NN1 1.5_MC &rsqb;_) For_IF every_AT1 M_NN1 there_EX is_VBZ I_ZZ1 such_CS21 that_CS22 the_AT following_JJ holds_NN2 ._. 
</s>
<s>
In_RR21 addition_RR22 we_PPIS2 claim_VV0 the_AT estimates_NN2 @F_FO ._. 
</s>
<s>
We_PPIS2 note_VV0 that_CST the_AT exceptional_JJ zeros_NN2 are_VBR not_XX present_JJ by_II the_AT assumptions_NN2 of_IO the_AT theorem_NN1 ,_, so_CS the_AT right-hand_JJ side_NN1 of_IO the_AT displayed_JJ formula_NN1 in_II Proposition_NN1 20_MC becomes_VVZ @F_FO ._. 
</s>
<s>
Given_VVN the_AT assumptions_NN2 on_II the_AT approximate_JJ spaces_NN2 and_CC the_AT approximate_JJ bilinear_JJ forms_NN2 ,_, there_EX exists_VVZ @S_FO constant_JJ @S_FO ,_, independent_JJ of_IO @S_FO ,_, such_CS21 that_CS22 @F_FO ._. 
</s>
<s>
Such_DA a_AT1 behaviour_NN1 holds_VVZ regardless_RR ofwhether_VV0 the_AT original_JJ correlation_NN1 between_II regressors_NN2 is_VBZ positive_JJ or_CC negative_JJ ._. 
</s>
<s>
This_DD1 shows_VVZ (_( 5_MC )_) and_CC completes_VVZ the_AT proof_NN1 ._. 
</s>
<s>
Unfortunately_RR ,_, the_AT product_NN1 Hs(u)A(u)_NN1 ,_, where_CS @S_FO ,_, is_VBZ generally_RR not_XX positive_JJ definite_JJ and_CC we_PPIS2 need_VV0 to_TO approximate_VVI A(u)_FO ._. 
</s>
<s>
In_II other_JJ words_NN2 ,_, the_AT quantum_NN1 algorithm_NN1 is_VBZ applied_VVN to_II a_AT1 problem_NN1 for_IF which_DDQ it_PPH1 is_VBZ difficult_JJ to_TO classically_RR compute_VVI the_AT solution_NN1 ,_, but_CCB once_RR the_AT solution_NN1 (_( or_CC some_DD information_NN1 about_II it_PPH1 )_) is_VBZ obtained_VVN ,_, it_PPH1 is_VBZ easy_JJ to_TO classically_RR verify_VVI that_CST we_PPIS2 have_VH0 the_AT right_JJ answer_NN1 ._. 
</s>
<s>
This_DD1 is_VBZ a_AT1 Pfaffian_JJ analogue_NN1 of_IO the_AT determinantal_JJ Schur_NN1 process_NN1 introduced_VVN in_II &lsqb;_( 35_MC &rsqb;_) ._. 
</s>
<s>
As_CSA we_PPIS2 vary_VV0 ,_, the_AT Mv_NNU away_II21 from_II22 the_AT constant_JJ M0_FO ,_, the_AT roots_NN2 kj_NNU and_CC kj+1_FO can_VM not_XX merge_VVI ,_, since_CS that_DD1 would_VM lead_VVI to_II a_AT1 function_NN1 fj_NNU with_IW a_AT1 zero_NN1 boundary_NN1 value_NN1 ._. 
</s>
<s>
We_PPIS2 shall_VM also_RR need_VVI a_AT1 core_NN1 C_ZZ1 C_ZZ1 H_ZZ1 ,_, defined_VVN as_CSA follows_VVZ ._. 
</s>
<s>
We_PPIS2 will_VM show_VVI in_II the_AT next_MD section_NN1 ,_, Lemma_NN1 4.3_MC that_CST @F_FO ._. 
</s>
<s>
In_BCL21 order_BCL22 to_TO ensure_VVI (_( 2.5_MC )_) is_VBZ satisfied_JJ for_IF q_ZZ1 +_FO 1_MC1 ,_, we_PPIS2 design_VV0 po_NN1 such_CS21 that_CS22 @F_FO ._. 
</s>
<s>
We_PPIS2 thus_RR define_VV0 the_AT auxiliary_JJ function_NN1 @F_FO ._. 
</s>
<s>
The_AT term_NN1 9+2/2_FU is_VBZ added_VVN to_TO ensure_VVI that_CST we_PPIS2 leave_VV0 room_NN1 for_IF future_JJ corrections_NN2 and_CC the_AT max_NN1 is_VBZ in_II place_NN1 to_TO ensure_VVI that_CST we_PPIS2 do_VD0 not_XX correct_VVI the_AT energy_NN1 when_CS the_AT energy_NN1 of_IO vq_NNU is_VBZ already_RR sufficiently_RR close_JJ to_II the_AT prescribed_JJ energy_NN1 profile_NN1 ._. 
</s>
<s>
Instead_RR she_PPHS1 started_VVD the_AT discussion_NN1 by_II presenting_VVG a_AT1 solution_NN1 that_CST she_PPHS1 talked_VVD about_II as_CSA "_" having_VHG seen_VVN a_AT1 student_NN1 in_II another_DD1 class_NN1 doing_VDG ._. "_" 
</s>
<s>
In_II the_AT fall_NN1 of_IO 2017_MC Emmanuel_NP1 Letellier_NP1 visited_VVD 1ST_MD Austria_NP1 ._. 
</s>
<s>
The_AT following_JJ lemma_NN1 shows_VVZ that_CST B_ZZ1 (_( t_ZZ1 ,_, y_ZZ1 )_) satisfies_VVZ estimate_NN1 (_( 3.1_MC )_) in_II &lsqb;_( 3_MC &rsqb;_) on_II a_AT1 suitable_JJ ball_NN1 B2_FO for_IF every_AT1 @S_FO ;_; namely_REX there_EX exist_VV0 two_MC constants_NN2 a_AT1 >_FO 0_MC and_CC p_ZZ1 >_FO 0_MC ,_, independent_JJ of_IO the_AT diffeomorphisms_NN2 ,_, such_CS21 that_CS22 @F_FO for_IF every_AT1 @S_FO ,_, where_CS @S_FO ._. 
</s>
<s>
The_AT proof_NN1 is_VBZ based_VVN on_II the_AT results_NN2 of_IO Lemma_NN1 2.8_MC and_CC on_II a_AT1 careful_JJ estimate_NN1 of_IO the_AT constants_NN2 in_II the_AT second_MD Korn_NP1 inequality_NN1 ._. 
</s>
<s>
This_DD1 implies_VVZ that_CST there_EX exists_VVZ R1_FO >R0_FO large_JJ such_CS21 that_CS22 for_IF R_ZZ1 sufficiently_RR large_JJ ,_, @S_FO is_VBZ transverse_JJ to_II @F_FO ._. 
</s>
<s>
Finally_RR ,_, transversality_NN1 of_IO @S_FO to_II @S_FO for_IF R_ZZ1 large_JJ follows_VVZ from_II the_AT fact_NN1 that_CST under_II any_DD divergent_JJ sequence_NN1 of_IO vertical_JJ translations_NN2 of_IO S_ZZ1 ,_, a_AT1 subsequence_NN1 converges_VVZ to_II some_DD vertical_JJ translation_NN1 of_IO H_ZZ1 and_CC ,_, as_CSA R>_FO 1_MC1 ,_, the_AT unit_NN1 normal_JJ vectors_NN2 to_II @S_FO at_II points_NN2 of_IO @S_FO are_VBR converging_VVG to_II vertical_JJ unit_NN1 vectors_NN2 ._. 
</s>
<s>
In_II the_AT case_NN1 of_IO triangular_JJ meshes_NN2 ,_, E_ZZ1 is_VBZ the_AT reference_NN1 right_NN1 triangle_NN1 with_IW vertices_NN2 @S_FO ,_, @S_FO ,_, and_CC @S_FO ._. 
</s>
<s>
Let_VV0 r1_FO ,_, r2_FO ,_, and_CC r3_FO be_VBI the_AT corresponding_JJ vertices_NN2 of_IO E_ZZ1 ,_, oriented_VVD counterclockwise_RR ._. 
</s>
<s>
Observe_VV0 that_CST using_VVG (_( 2.2_MC )_) ,_, the_AT definition_NN1 of_IO V_ZZ1 '_NULL ,_, and_CC (_( 4.7_MC )_) we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Further_RRR ,_, using_VVG also_RR (_( 3.13_MC )_) ,_, (_( 4.8_MC )_) ,_, (_( 4.38_MC )_) ,_, and_CC integration_NN1 by_II parts_NN2 ,_, we_PPIS2 obtain_VV0 @F_FO ._. 
</s>
<s>
They_PPHS2 ,_, too_RR ,_, were_VBDR divided_VVN with_II31 respect_II32 to_II33 how_RRQ to_TO improve_VVI education_NN1 ._. 
</s>
<s>
Likewise_RR ,_, in_II the_AT case_NN1 in_II which_DDQ the_AT consumer_NN1 realizes_VVZ that_CST he_PPHS1 did_VDD not_XX use_VVI the_AT whole_JJ amount_NN1 of_IO money_NN1 in_II expenditure_NN1 ,_, @S_FO that_DD1 @S_FO ,_, he_PPHS1 will_VM leave_VVI leisurely_RR ,_, having_VHG the_AT possibility_NN1 to_TO spend_VVI more_DAR money_NN1 in_II the_AT next_MD occasion_NN1 ._. 
</s>
<s>
Multilevel_VV0 methods_NN2 reduce_VV0 the_AT total_JJ simulation_NN1 cost_NN1 by_II utilizing_VVG different_JJ discretizations_NN2 of_IO the_AT underlying_JJ model_NN1 ._. 
</s>
<s>
The_AT set_NN1 of_IO chambers_NN2 which_DDQ are_VBR opposite_JJ to_II C_ZZ1 is_VBZ an_AT1 open_JJ subset_NN1 @S_FO ._. 
</s>
<s>
This_DD1 is_VBZ equivalent_JJ to_II constructing_VVG a_AT1 spherical_JJ 13,13-code_FO C_ZZ1 of_IO size_NN1 2n2_FO ._. 
</s>
<s>
If_CS @S_FO is_VBZ an_AT1 SL2(Z)_FO contravariant_NN1 and_CC translation_NN1 invariant_JJ Minkowski_JJ valuation_NN1 ,_, then_RT there_EX exist_VV0 a_AT1 ,_, b_ZZ1 >_FO 0_MC such_CS21 that_CS22 @F_FO ._. 
</s>
<s>
Proof_NN1 ._. 
</s>
<s>
It_PPH1 is_VBZ clear_JJ that_CST estimates_NN2 (_( 5.9_MC )_) ,_, (_( 5.10_MC )_) ,_, (_( 5.11_MC )_) are_VBR ISS_NP1 estimates_VVZ w.r.t_NNU ._. 
</s>
<s>
Over_II the_AT regular_JJ semisimple_NN1 locus_NN1 ,_, this_DD1 group_NN1 scheme_NN1 can_VM be_VBI identified_VVN with_IW the_AT pullback_NN1 from_II gD/G_FU of_IO the_AT group_NN1 scheme_NN1 J_ZZ1 from_II &lsqb;_( Ngo10_FO &rsqb;_) ._. 
</s>
<s>
Consider_VV0 a_AT1 quantity_NN1 of_IO interest_NN1 of_IO the_AT form_NN1 @F_FO ._. 
</s>
<s>
Then_RT ,_, formally_RR (_( we_PPIS2 will_VM justify_VVI this_DD1 computation_NN1 case_NN1 by_II case_NN1 for_IF various_JJ @S_FO later_RRR )_) @F_FO ._. 
</s>
<s>
Fix_VV0 @S_FO ._. 
</s>
<s>
Then_RT X_ZZ1 is_VBZ S-regular_JJ with_IW constant_JJ @S_FO on_II scales_NN2 o0_FO to_II To1_FO ._. 
</s>
<s>
After_CS r_ZZ1 applications_NN2 of_IO the_AT next_MD lemma_NN1 we_PPIS2 can_VM assume_VVI that_CST r_ZZ1 =_FO 0_MC and_CC that_CST the_AT remaining_JJ data_NN defining_VVG A_ZZ1 is_VBZ unchanged_JJ ._. 
</s>
<s>
A_AT1 first_MD explanation_NN1 of_IO why_RRQ this_DD1 technique_NN1 works_NN is_VBZ the_AT following_JJ :_: If_CS all_DB the_AT substencils_NN2 @S_FO (_( three_MC of_IO four_MC points_NN2 ,_, two_MC of_IO five_MC points_NN2 ,_, and_CC one_MC1 of_IO six_MC points_NN2 )_) ,_, are_VBR smooth_JJ and_CC @S_FO ,_, all_DB of_IO them_PPHO2 are_VBR @S_FO ,_, n_ZZ1 =_FO 3,4_MC ,_, 5_MC (_( as_CSA shown_VVN in_II Theorem_NN1 3.1_MC )_) ._. 
</s>
<s>
Yeah_UH but_CCB i_MC1 want_VV0 you_PPY think_VV0 about_II ..._... 
</s>
<s>
The_AT really_RR important_JJ remaining_JJ datum_NN1 is_VBZ purely_RR 2-dimensional_JJ :_: it_PPH1 is_VBZ the_AT orientation_NN1 of_IO the_AT geometric_JJ triangle_NN1 itself_PPX1 ,_, which_DDQ determines_VVZ the_AT directions_NN2 of_IO the_AT morphisms_NN2 between_II the_AT objects_NN2 on_II its_APPGE edges_NN2 ._. 
</s>
<s>
We_PPIS2 start_VV0 by_II showing_VVG that_CST a_AT1 corrector_NN1 exists_VVZ ._. 
</s>
<s>
We_PPIS2 have_VH0 an_AT1 obvious_JJ @S-graded_FO analogue_NN1 of_IO the_AT concept_NN1 of_IO a_AT1 dg-category_NN1 :_: a_AT1 small_JJ category_NN1 enriched_VVN over_II @S_FO ._. 
</s>
<s>
We_PPIS2 refer_VV0 to_II these_DD2 structures_NN2 as_CSA 2-periodic_JJ ,_, or_CC @S-graded_FO ,_, dg-categories_NN2 and_CC will_VM leave_VVI out_RP the_AT extra_JJ adjective_NN1 when_CS it_PPH1 is_VBZ obvious_JJ from_II the_AT context_NN1 ._. 
</s>
<s>
We_PPIS2 claim_VV0 that_CST we_PPIS2 may_VM assume_VVI without_IW loss_NN1 of_IO generality_NN1 that_CST the_AT algebraic_JJ group_NN1 Holx(E)_NN1 is_VBZ connected_VVN ._. 
</s>
<s>
But_CCB since_II x_ZZ1 has_VHZ two_MC edges_NN2 contained_VVN in_II E1_FO E2_FO and_CC x_ZZ1 has_VHZ the_AT same_DA degree_NN1 in_II both_DB2 the_AT graph_NN1 G2_FO and_CC Gi_NN1 ,_, there_EX must_VM exist_VVI at_RR21 least_RR22 one_MC1 edge_NN1 @S_FO ,_, where_CS z_ZZ1 =_FO u_ZZ1 ,_, z_ZZ1 =_FO v._II Rewiring_NP1 (_( u_ZZ1 ,_, v_ZZ1 )_) and_CC (_( x_ZZ1 ,_, z_ZZ1 )_) in_II G2_FO produces_VVZ a_AT1 neighboring_JJ graph_NN1 G3_FO with_IW edge_NN1 (_( u_ZZ1 ,_, x_ZZ1 )_) and_CC thus_RR @S_FO ._. 
</s>
<s>
Note_VV0 that_CST gray_JJ and_CC black_JJ circles_NN2 and_CC squares_NN2 are_VBR overlapping_VVG ._. 
</s>
<s>
We_PPIS2 can_VM first_MD determine_VVI the_AT explicit_JJ solution_NN1 of_IO the_AT above_JJ equation_NN1 from_II (_( 44_MC )_) as_CSA follows_VVZ ._. 
</s>
<s>
Notice_VV0 that_CST the_AT powers_NN2 of_IO 2_NN1 (_( Nr1/2_FU (_( YM_NP1 )_) )_) cancel_VV0 ,_, leaving_VVG @F_FO ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI a_AT1 Kahler-Ricci_JJ flow_NN1 on_II a_AT1 Fano_NN1 manifold_NN1 M2n_FO ._. 
</s>
<s>
Fig._NN1 5.2_MC Pointwise_JJ error_NN1 @S_FO with_IW @S_FO and_CC @S_FO ._. 
</s>
<s>
The_AT proof_NN1 of_IO Proposition_NN1 8.4_MC below_RL follows_VVZ the_AT line_NN1 of_IO argument_NN1 given_VVN in_II &lsqb;_( 15_MC ,_, 4.1_MC &rsqb;_) (_( see_VV0 also_RR &lsqb;_( 12_MC ,_, Corollary_NN1 3.8_MC &rsqb;_) )_) ._. 
</s>
<s>
We_PPIS2 found_VVD that_CST @S_FO is_VBZ aconsistently_RR good_JJ choice_NN1 ._. 
</s>
<s>
By_II Theorem_NN1 4.4_MC (_( applied_VVN for_IF Tr_JJ in_II31 place_II32 of_II33 T_ZZ1 )_) ,_, we_PPIS2 get_VV0 the_AT identity_NN1 @F_FO ,_, where_CS the_AT average_NN1 @S_FO is_VBZ taken_VVN over_II those_DD2 @S_FO such_CS21 that_CS22 @S_FO ._. 
</s>
<s>
Using_VVG Theorem_NN1 5.4_MC ,_, we_PPIS2 get_VV0 that_CST the_AT average_NN1 on_II the_AT right-hand_JJ side_NN1 is_VBZ equal_JJ to_II @F_FO ,_, where_CS the_AT first_MD identity_NN1 follows_VVZ since_CS @S_FO and_CC the_AT second_NNT1 from_II Theorem_NN1 4.4_MC ._. 
</s>
<s>
Take_VV0 a_AT1 partition_NN1 @S_FO of_IO Q_ZZ1 into_II pairwise_RR disjoint_JJ cubes_NN2 of_IO side_NN1 length_NN1 @S_FO and_CC centred_VVN at_II @S_FO ._. 
</s>
<s>
Then_RT ,_, for_IF all_DB m_ZZ1 =_FO 3_MC ,_, ..._... ,_, d_ZZ1 ,_, define_VV0 recursively_RR a_AT1 subpartition_NN1 Qm_FO ,_, j_ZZ1 into_II pairwise_RR disjoint_JJ cubes_NN2 of_IO side_NN1 length_NN1 @S_FO and_CC centred_VVN at_II @S_FO in_II such_DA a_AT1 way_NN1 that_CST for_IF every_AT1 Qm_FO ,_, j_ZZ1 there_EX exists_VVZ a_AT1 @S_FO that_DD1 contains_VVZ it_PPH1 ._. 
</s>
<s>
If_CS r<t_FO ,_, then_RT we_PPIS2 can_VM directly_RR apply_VVI Proposition_NN1 7.2_MC ,_, noticing_VVG that_CST @F_FO ._. 
</s>
<s>
Helen_NP1 ,_, through_II Boris_NP1 ,_, has_VHZ introduced_VVN yet_RR another_DD1 potential_JJ student_NN1 misconception_NN1 ._. 
</s>
<s>
In_II this_DD1 section_NN1 we_PPIS2 further_RRR extend_VV0 the_AT results_NN2 on_II the_AT magnetic_JJ inhibition_NN1 in_II the_AT NMRT_NP1 problem_NN1 to_II the_AT magnetic_JJ Benard_NN1 problem_NN1 without_IW heat_NN1 conduction_NN1 ._. 
</s>
<s>
The_AT latter_DA are_VBR relevant_JJ because_CS ,_, as_CSA an_AT1 example_NN1 ,_, teachers_NN2 hold_VV0 beliefs_NN2 about_II mathematics_NN1 as_II a_AT1 discipline_NN1 and_CC mathematics_NN1 as_II a_AT1 school_NN1 subject_NN1 (_( Beswick_NP1 2012_MC )_) that_CST may_VM impact_NN1 on_II the_AT actions_NN2 they_PPHS2 take_VV0 as_CSA they_PPHS2 teach_VV0 ._. 
</s>
<s>
Future_JJ studies_NN2 may_VM analyze_VVI whether_CSW teaching_VVG sequences_NN2 that_CST combine_VV0 both_RR procedural_JJ and_CC conceptual_JJ approaches_NN2 and_CC ,_, as_II a_AT1 result_NN1 ,_, provide_VV0 students_NN2 with_IW tasks_NN2 that_CST illuminate_VV0 the_AT limitations_NN2 of_IO direct_JJ translation_NN1 strategies_NN2 ,_, have_VH0 the_AT pedagogic_JJ potential_NN1 to_TO improve_VVI students_NN2 '_NULL proficiency_NN1 to_TO solve_VVI algebraic_JJ word_NN1 problems_NN2 ._. 
</s>
<s>
Proof_NN1 By_II the_AT regularity_NN1 for_IF local_JJ minimizers_NN2 of_IO the_AT Ac_NN1 functional_JJ (_( Theorem_NN1 2.14_MC )_) ,_, we_PPIS2 know_VV0 that_CST each_DD1 i_ZZ1 is_VBZ a_AT1 smooth_JJ ,_, embedded_VVN ,_, stable_JJ c-boundary_NN1 in_II int(K)_NN1 by_II Proposition_NN1 5.8(iv)_FO ._. 
</s>
<s>
When_CS @S_FO and_CC @S_FO ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Letting_VVG @S_FO ,_, we_PPIS2 infer@S_FO ,_, that_DD1 is_VBZ @S_FO ,_, in_II the_AT distributional_JJ sense_NN1 ._. 
</s>
<s>
Recall_VV0 that_CST @S_FO denotes_VVZ the_AT group_NN1 of_IO diffeomorphisms_NN2 properly_RR homotopic_JJ to_II the_AT identity_NN1 ._. 
</s>
<s>
This_DD1 leads_VVZ to_II the_AT expected_JJ payoff_NN1 @S_FO ._. 
</s>
<s>
Thus_RR ,_, by_II comparing_VVG the_AT two_MC payoffs_VVZ f(x)_NNU and_CC @S_FO ,_, we_PPIS2 obtain_VV0 the_AT best_RRT stopping_VVG strategy_NN1 for_IF today_RT ,_, whose_DDQGE stopping_VVG region_NN1 is_VBZ given_VVN by_II @F_FO ._. 
</s>
<s>
Lemma_NN1 4.3_MC (_( recursive_JJ bound_NN1 for_IF global_JJ error_NN1 )_) ._. 
</s>
<s>
Suppose_VV0 that_CST the_AT poi@S_FO represented_VVN by_II Do_VD0 is_VBZ a_AT1 vertex_NN1 ._. 
</s>
<s>
As_CSA explained_VVN in_II the_AT introduction_NN1 ,_, we_PPIS2 are_VBR going_VVGK to_TO use_VVI Proposition_NN1 3.3_MC along_II the_AT characteristics_NN2 ._. 
</s>
<s>
The_AT Z_ZZ1 =_FO ±1_FO cases_NN2 are_VBR Fermi-Bose_NP1 sectors_NN2 (_( or_CC spin_VV0 up/down_RP sectors_NN2 )_) of_IO supersymmetric_JJ quantum_NN1 mechanics_NN2 ,_, where_CS W(x)_NP1 is_VBZ called_VVN the_AT super-potential_NN1 ._. 
</s>
<s>
We_PPIS2 do_VD0 not_XX analyze_VVI those_DD2 changes_NN2 here_RL because_CS the_AT reasoning_NN1 that_CST Terionna_NP1 produced_VVD largely_RR occurred_VVD prior_II21 to_II22 her_APPGE consideration_NN1 of_IO the_AT implications_NN2 it_PPH1 had_VHD for_IF her_APPGE array_NN1 representation_NN1 ._. 
</s>
<s>
Thus_RR ,_, the_AT value_NN1 of_IO problem_NN1 (_( 30_MC )_) is_VBZ zero_NN1 for_IF all_DB r_ZZ1 >_FO 8_MC ._. 
</s>
<s>
Suppose_VV0 that_CST @S_FO ._. 
</s>
<s>
Then_RT since_CS H_ZZ1 is_VBZ simply_RR connected_VVN and_CC the_AT simply_RR connected_JJ cover_NN1 is_VBZ unique_JJ up_II21 to_II22 isomorphism_NN1 ,_, the_AT automorphism_NN1 @S_FO of_IO i(H)_NN1 may_VM be_VBI lifted_VVN to_II an_AT1 F_ZZ1 -automorphism_NN1 of_IO H_ZZ1 ,_, and_CC in_RR21 particular_RR22 preserves_VVZ adelic_JJ points_NN2 ;_; so_CS @F_FO ._. 
</s>
<s>
Also_RR note_VV0 that_CST the_AT Haar_NN1 measure_NN1 on_II @S_FO is_VBZ not_XX changed_VVN by_II conjugation_NN1 by_II h_ZZ1 as_CSA H_ZZ1 is_VBZ semisimple_NN1 ._. 
</s>
<s>
However_RR ,_, if_CS we_PPIS2 restrict_VV0 to_II a_AT1 case_NN1 where_RRQ @F_FO ,_, we_PPIS2 can_VM prove_VVI an_AT1 alternative_JJ formula_NN1 describing_VVG the_AT index_NN1 as_II a_AT1 finite_JJ sum_NN1 over_II the_AT set_NN1 of_IO solutions_NN2 to_II certain_JJ transcendental_JJ equations_NN2 ,_, which_DDQ we_PPIS2 call_VV0 Bethe_NP1 Ansatz_NP1 Equations_NN2 (_( BAEs_NP2 )_) ._. 
</s>
<s>
The_AT filtration_NN1 @S_FO of_IO F&lsqb;G&rsqb;_NP1 gives_VVZ rise_NN1 to_II a_AT1 filtration_NN1 @S_FO of_IO the_AT local_JJ system_NN1 @S_FO on_II S_ZZ1 ,_, so_CS there_EX is_VBZ also_RR an_AT1 associated_JJ graded_JJ object_NN1 @S_FO ._. 
</s>
<s>
The_AT local_JJ system_NN1 @S_FO on_II S_ZZ1 is_VBZ trivial_JJ ,_, because_CS the_AT monodromy_JJ elements_NN2 @S_FO act_VV0 trivially_RR on_II Gr_NP1 ._. 
</s>
<s>
Construction_NN1 of_IO the_AT modulated_JJ Fourier_NP1 expansion_NN1 Our_APPGE construction_NN1 of_IO @S_FO and_CC the_AT coefficient_NN1 functions_NN2 @S_FO in_II (_( 4.1_MC )_) is_VBZ based_VVN on_II asymptotic_JJ expansions_NN2 ,_, which_DDQ are_VBR typically_RR divergent_JJ ._. 
</s>
<s>
While_CS the_AT research_NN1 on_II designing_VVG innovative_JJ aspects_NN2 of_IO content_JJ courses_NN2 has_VHZ directly_RR served_VVN the_AT purpose_NN1 of_IO improving_VVG the_AT learning_NN1 of_IO preservice_NN1 teachers_NN2 ,_, it_PPH1 also_RR shed_VV0 lights_NN2 on_II MTEs_NN2 '_NULL learning_VVG ._. 
</s>
<s>
Most_DAT PSTs_NP1 were_VBDR only_RR one_MC1 semester_NN1 away_II21 from_II22 their_APPGE final_JJ internship_NN1 where_CS they_PPHS2 would_VM take_VVI full_JJ classroom_NN1 responsibility_NN1 for_IF mathematics_NN1 instruction_NN1 ._. 
</s>
<s>
Moreover_RR ,_, the_AT Gauss_NN1 theorem_NN1 ensures_VVZ the_AT following_JJ rot-tangent_JJ component_NN1 relation_NN1 @F_FO ._. 
</s>
<s>
Finally_RR ,_, for_IF any_DD scalar_JJ function_NN1 v_ZZ1 defined_VVN in_II P_ZZ1 ,_, we_PPIS2 denote_VV0 with_IW @S_FO the_AT scalar_JJ function_NN1 defined_VVN in_II @S_FO such_CS21 that_CS22 @F_FO ._. 
</s>
<s>
Suppose_VV0 that_CST P_ZZ1 is_VBZ an_AT1 H-picture_JJ satisfying_JJ (_( p.1_FO )_) ,_, and_CC such_CS21 that_CS22 all_DB -cycles_NN2 in_II P_ZZ1 are_VBR facial_JJ covers_NN2 ._. 
</s>
<s>
Our_APPGE main_JJ result_NN1 is_VBZ that_CST all_DB generalized_JJ alpha_NN1 investing_VVG rules_NN2 control_VV0 FDR_NP1 ,_, provided_CS they_PPHS2 satisfy_VV0 a_AT1 natural_JJ monotonicity_NN1 condition_NN1 ._. 
</s>
<s>
All_DB were_VBDR referring_VVG to_II either_RR the_AT page_NN1 numbers_NN2 or_CC the_AT sign_NN1 on_II the_AT house_NN1 ._. 
</s>
<s>
Proof_NN1 of_IO the_AT Property_NN1 (_( P_ZZ1 )_) :_: Let_VV0 g_ZZ1 be_VBI a_AT1 smooth_JJ Riemannian_JJ metric_JJ ,_, 1>0_FO and_CC K>0_FO be_VBI constants_NN2 ,_, and_CC choose_VV0 @F_FO ._. 
</s>
<s>
We_PPIS2 deal_VV0 with_IW the_AT first_MD case_NN1 ,_, the_AT other_NN1 being_VBG similar_JJ ._. 
</s>
<s>
Now_RT we_PPIS2 choose_VV0 inductively_RR on_II n_ZZ1 unitaries_NN2 vn∈U(Cn)_FO and_CC un+1∈U(Cn+1)_FO such_CS21 that_CS22 ,_, for_IF all_DB n_ZZ1 ,_, vn(s)=1_FO for_IF all_DB @S_FO ,_, un+1(t)=un+1_FO ,_, t_ZZ1 for_IF t∈0,1_FO ,_, and_CC @S_FO :_: Simply_RR start_VV0 with_IW v1_FO :_: =1_FO ,_, and_CC if_CS vn_NNU and_CC un_FW have_VH0 been_VBN chosen_VVN ,_, choose_VV0 un+1∈U(Cn+1)_FO such_CS21 that_CS22 @S_FO for_IF all_DB t∈0,1_FO and_CC @S_FO for_IF all_DB @S_FO ,_, and_CC set_VV0 @S_FO ._. 
</s>
<s>
If_CS we_PPIS2 now_RT take_VV0 this_DD1 un+1_FO for_IF the_AT map_NN1 in_II (_( 14_MC )_) giving_VVG rise_NN1 to_II n_ZZ1 and_CC n_ZZ1 ,_, then_RT we_PPIS2 obtain_VV0 a_AT1 commutative_JJ diagram_NN1 @T_FO which_DDQ restricts_VVZ to_II @T_FO where_RRQ the_AT unitary_JJ u_ZZ1 ?_NULL n+1_FO for_IF the_AT map_NN1 in_II (_( 14_MC )_) for_IF ?_NULL n_ZZ1 is_VBZ now_RT trivial_JJ ,_, u_ZZ1 ?_NULL n+1=1_FO ,_, and_CC A_ZZ1 ?_NULL n_ZZ1 is_VBZ of_IO the_AT same_DA form_NN1 (_( 12_MC )_) as_CSA An_AT1 ,_, with_IW @S_FO of_IO the_AT same_DA form_NN1 as_CSA β_NULL t_ZZ1 for_IF t=0,1_FO (_( the_AT point_NN1 being_VBG that_RG vn(t)_NNU is_VBZ a_AT1 permutation_NN1 matrix_NN1 )_) ._. 
</s>
<s>
We_PPIS2 refer_VV0 the_AT reader_NN1 to_II &lsqb;_( Ngo10_FO ,_, Proposition_NN1 4.18.1_MC &rsqb;_) for_IF details_NN2 on_II the_AT construction_NN1 below_RL ._. 
</s>
<s>
By_II analogy_NN1 ,_, it_PPH1 would_VM be_VBI tempting_JJ to_TO think_VVI of_IO @S_FO as_II the_AT estimated_JJ marginal_JJ impact_NN1 of_IO the_AT confounder_NN1 on_II the_AT treatment_NN1 ._. 
</s>
<s>
For_IF k_ZZ1 =_FO 3_MC and_CC 4_MC ,_, Theorem_NN1 1.4_MC is_VBZ a_AT1 sharp_JJ upper_JJ bound_NN1 ,_, but_CCB we_PPIS2 do_VD0 not_XX know_VVI if_CSW it_PPH1 is_VBZ sharp_JJ for_IF k_ZZ1 >_FO 5_MC ._. 
</s>
<s>
Let_VV0 @S_FO ,_, be_VBI random_JJ vectors_NN2 that_CST are_VBR independent_JJ and_CC identically_RR distributed_VVN as_CSA (_( x_ZZ1 ,_, y_ZZ1 )_) ,_, where_CS y_ZZ1 is_VBZ aresponse_NN1 variable_NN1 and_CC @S_FO is_VBZ a_AT1 d-dimensional_JJ covariate_NN1 vector_NN1 ._. 
</s>
<s>
This_DD1 fact_NN1 yields_VVZ the_AT following_JJ result_NN1 ._. 
</s>
<s>
Finally_RR (_( 5.7_MC )_) follows_VVZ from_II (_( 5.8_MC )_) ,_, (_( 5.9_MC )_) ,_, (_( 5.10_MC )_) ,_, (_( 5.11_MC )_) ,_, (_( 5.12_MC )_) ,_, and_CC (_( 5.14_MC )_) ._. 
</s>
<s>
In_II31 view_II32 of_II33 the_AT structure_NN1 of_IO the_AT interaction_NN1 matrix_NN1 (_( 4.1_MC )_) ,_, if_CS we_PPIS2 denote_VV0 @S_FO ,_, we_PPIS2 can_VM easily_RR see_VVI that_DD1 system_NN1 (_( 1.1_MC )_) may_VM be_VBI written_VVN in_II matrix_NN1 notation_NN1 as_CSA @F_FO where_RRQ the_AT Laplacian_JJ matrix_NN1 L_ZZ1 is_VBZ given_VVN by_II @F_FO ,_, uniformly_RR bounded_VVN on_II N._NNU In_II this_DD1 regard_NN1 ,_, Prediger_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
(_( 2008_MC )_) systematized_VVD various_JJ possibilities_NN2 and_CC degrees_NN2 of_IO bringing_VVG in_II dialog_NN1 theories_NN2 ,_, which_DDQ led_VVD to_II the_AT development_NN1 of_IO a_AT1 conceptual_JJ framework_NN1 of_IO networking_VVG strategies_NN2 ._. 
</s>
<s>
Our_APPGE result_NN1 can_VM also_RR be_VBI used_VVN in_II studying_VVG Green_JJ '_NULL s_ZZ1 function_NN1 asymptotics_NN2 near_II spectral_JJ gap_NN1 edges_NN2 (_( see_VV0 &lsqb;_( 10_MC &rsqb;_) ,_, &lsqb;_( 19_MC &rsqb;_) and_CC &lsqb;_( 9_MC &rsqb;_) )_) and_CC to_TO obtain_VVI a_AT1 "_" variable_JJ period_NN1 "_" version_NN1 of_IO the_AT non-degeneracy_JJ conjecture_NN1 in_II two_MC dimensions_NN2 &lsqb;_( 20_MC &rsqb;_) ._. 
</s>
<s>
In_II Section_NN1 4_MC we_PPIS2 prove_VV0 Theorem_NN1 1.1_MC ._. 
</s>
<s>
Notice_VV0 that_CST such_DA a_AT1 surface_NN1 is_VBZ isometric_JJ to_II an_AT1 flat_JJ k-polygon_NN1 in_II R2_FO ._. 
</s>
<s>
We_PPIS2 have_VH0 the_AT following_JJ proposition_NN1 on_II the_AT properties_NN2 of_IO the_AT penalized_JJ EL_NP1 estimator_NN1 0n_FO as_CSA in_II (_( 2.7_MC )_) ._. 
</s>
<s>
If_CS SCT_NP1 consists_VVZ of_IO more_DAR than_CSN one_MC1 circle_NN1 ,_, &lsqb;_( 5_MC ,_, Cor._NP1 2.4_MC &rsqb;_) shows_VVZ that_CST this_DD1 conclusion_NN1 holds_VVZ after_II pulling_VVG back_RP to_II a_AT1 cover_NN1 of_IO @S_FO ._. 
</s>
<s>
The_AT orientability_NN1 of_IO the_AT compactified_JJ moduli_NN2 spaces_NN2 of_IO real_JJ maps_NN2 necessary_JJ for_IF defining_VVG real_JJ GW-invariants_NN2 is_VBZ not_XX considered_VVN in_II &lsqb;_( 5_MC &rsqb;_) ._. 
</s>
<s>
Section_NN1 5_MC establishes_VVZ the_AT necessary_JJ exponential_NN1 separation_NN1 of_IO projections_NN2 of_IO cylinders_NN2 ._. 
</s>
<s>
Let_VV0 M_ZZ1 and_CC N_ZZ1 be_VBI simple_JJ modules_NN2 ._. 
</s>
<s>
Thus_RR we_PPIS2 have_VH0 an_AT1 inclusion_NN1 @S_FO and_CC it_PPH1 remains_VVZ to_TO show_VVI that_CST the_AT image_NN1 of_IO @S_FO in_II @S_FO is_VBZ the_AT entire_JJ space_NN1 ._. 
</s>
<s>
The_AT coefficients_NN2 @S_FO ,_, @S_FO ,_, are_VBR the_AT aerodynamic_JJ admittance_NN1 functions_NN2 ,_, varying_VVG with_IW the_AT frequency_NN1 of_IO the_AT turbulent_JJ fluctuations.98_FO These_DD2 can_VM be_VBI interpreted_VVN as_CSA transfer_NN1 functions_NN2 between_II the_AT fluctuating_JJ wind_NN1 velocities_NN2 and_CC aerodynamic_JJ forces_NN2 ._. 
</s>
<s>
Let_VV0 X_ZZ1 and_CC U_ZZ1 be_VBI two_MC Hilbert_NP1 spaces_NN2 ,_, which_DDQ are_VBR accordingly_RR identified_VVN with_IW their_APPGE duals_NN2 ._. 
</s>
<s>
If_CS d_ZZ1 is_VBZ spherical_JJ ,_, any_DD standard_JJ d-structure_NN1 on_II Wip_NP1 extends_VVZ to_II the_AT closed_JJ manifold_NN1 @S_FO ._. 
</s>
<s>
In_II contrast_NN1 ,_, our_APPGE dual_JJ method_NN1 can_VM cope_VVI with_IW a_AT1 much_RR larger_JJR set_NN1 of_IO functions_NN2 ,_, and_CC in_RR21 particular_RR22 those_DD2 of_IO the_AT form_NN1 @S_FO ,_, i.e._REX ,_, obtained_VVN by_II precomposition_NN1 with_IW a_AT1 linear_JJ operator_NN1 ._. 
</s>
<s>
For_IF simplicity_NN1 we_PPIS2 say_VV0 that_CST a_AT1 homeomorphism_NN1 f_ZZ1 :_: U_ZZ1 C_ZZ1 is_VBZ conformal_JJ if_CS @S_FO is_VBZ conformal_JJ ._. 
</s>
<s>
Counter-maps_NN2 ,_, which_DDQ take_VV0 the_AT perspective_NN1 of_IO a_AT1 social_JJ group_NN1 that_CST is_VBZ typically_RR marginalized_VVN ,_, can_VM be_VBI used_VVN to_TO challenge_VVI dominant_JJ narratives_NN2 about_II places_NN2 ,_, social_JJ relations_NN2 ,_, and_CC power_NN1 (_( Mitchell_NP1 &;_NULL Elwood_NP1 ,_, 2012_MC ;_; Taylor_NP1 &;_NULL Hall_NN1 ,_, 2013_MC )_) ._. 
</s>
<s>
Let_VV0 γ_NULL :_: &lsqb;_( 0,1_MC &rsqb;_) →_NULL M_ZZ1 be_VBI the_AT unique_JJ minimizing_JJ geodesic_NN1 from_II x_ZZ1 to_II y_ZZ1 ,_, with_IW extremal_JJ :_: &lsqb;_( 0,1_MC &rsqb;_) →_NULL TM_NP1 ._. 
</s>
<s>
Of_RR21 course_RR22 ,_, the_AT unique_JJ minimizing_JJ geodesic_NN1 from_II y_ZZ1 to_II x_ZZ1 is_VBZ γ_NULL (_( t_ZZ1 )_) =_FO γ_NULL (_( 1t_FO )_) ._. 
</s>
<s>
For_IF this_DD1 reason_NN1 ,_, rather_II21 than_II22 using_VVG we_PPIS2 prefer_VV0 to_TO use_VVI W_ZZ1 and_CC W1_FO ._. 
</s>
<s>
Figure_NN1 19_MC shows_VVZ instantaneous_JJ pressure_NN1 distribution_NN1 on_II the_AT fuselage_NN1 surface_NN1 at_II two_MC different_JJ rotor_NN1 positions_NN2 ._. 
</s>
<s>
Unfortunately_RR ,_, the_AT intricate_JJ dependence_NN1 between_II W_ZZ1 and_CC x_ZZ1 has_VHZ not_XX been_VBN satisfactorily_RR resolved_VVN in_II previous_JJ work_NN1 ,_, where_CS only_RR suboptimal_JJ bounds_NN2 have_VH0 been_VBN produced_VVN for_IF @S_FO ,_, eventually_RR leading_VVG to_II suboptimal_JJ bounds_NN2 on_II the_AT acceptable_JJ noise_NN1 levels_NN2 a_AT1 &lsqb;_( 2_MC ,_, eq_VV0 ._. 
</s>
<s>
(_( 4.11_MC )_) &rsqb;_) &lsqb;_( 8_MC ,_, Lemma_NN1 12_MC &rsqb;_) &lsqb;_( 23_MC ,_, Proof_NN1 of_IO Thm._NP1 2_MC &rsqb;_) ._. 
</s>
<s>
Analogous_JJ estimates_NN2 have_VH0 been_VBN proved_VVN in_II &lsqb;_( 5_MC ,_, Theorem_NN1 2.5_MC &rsqb;_) for_IF the_AT case_NN1 of_IO C1_FO -domains_NN2 ._. 
</s>
<s>
The_AT teachers_NN2 in_II the_AT intervention_NN1 group_NN1 as_II31 well_II32 as_II33 the_AT control_NN1 group_NN1 most_RGT likely_RR learned_VVN from_II participating_VVG in_II the_AT collaborative_JJ groups_NN2 together_RL with_IW researchers_NN2 ._. 
</s>
<s>
There_EX has_VHZ been_VBN a_AT1 recent_JJ surge_NN1 of_IO work_NN1 on_II conducting_VVG formally_RR valid_JJ inference_NN1 in_II a_AT1 regression_NN1 setting_VVG after_II a_AT1 model_NN1 selection_NN1 event_NN1 has_VHZ occurred_VVN ;_; see_VV0 Bachoc_NP1 ,_, Leeb_NP1 and_CC Potscher_NP1 (_( 2014_MC )_) ,_, Berk_NN1 et_RA21 al_RA22 ._. 
</s>
<s>
(_( 2013_MC )_) ,_, Fithian_JJ ,_, Sun_NN1 and_CC Taylor_NP1 (_( 2014_MC )_) ,_, Lee_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
(_( 2016_MC )_) ,_, Lockhart_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
(_( 2014_MC )_) ,_, Tibshirani_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
(_( 2016_MC )_) ,_, just_RR to_TO name_VVI a_AT1 few_DA2 ._. 
</s>
<s>
Since_CS (_( 59_MC )_) is_VBZ a_AT1 parabolic_JJ equation_NN1 ,_, we_PPIS2 can_VM then_RT apply_VVI the_AT standard_JJ energy_NN1 estimate_NN1 in_II Gevrey_NP1 norms_NN2 ._. 
</s>
<s>
First_MD of_IO all_DB ,_, B_ZZ1 splits_VVZ into_II a_AT1 direct_JJ product_NN1 of_IO its_APPGE p-completions_NN2 for_IF p_ZZ1 |_NULL N_ZZ1 ,_, and_CC the_AT p-completion_NN1 @S_FO has_VHZ the_AT property_NN1 that_CST it_PPH1 is_VBZ annihilated_VVN by_II a_AT1 power_NN1 of_IO p_ZZ1 ._. 
</s>
<s>
Of_RR21 course_RR22 ,_, we_PPIS2 find_VV0 that_CST L_ZZ1 derives_VVZ from_II a_AT1 potential_JJ R_ZZ1 ,_, with_IW @F_FO ._. 
</s>
<s>
Let_VV0 us_PPIO2 denote_VVI @S_FO ._. 
</s>
<s>
Then_RT @S_FO ,_, where_CS @S_FO denotes_VVZ the_AT convex_JJ conjugate_NN1 of_IO @S_FO ._. 
</s>
<s>
We_PPIS2 have_VH0 @S_FO and_CC we_PPIS2 find_VV0 @S_FO where_RRQ @S_FO ._. 
</s>
<s>
With_IW @S_FO ,_, we_PPIS2 have_VH0 @F_FO and_CC the_AT convexity_NN1 of_IO R_ZZ1 in_II31 terms_II32 of_II33 q_ZZ1 (_( the_AT positivity_NN1 of_IO T_ZZ1 )_) amounts_VVZ to_II the_AT convexity_NN1 and_CC monotonicity_NN1 of_IO R_ZZ1 as_II a_AT1 function_NN1 of_IO @S_FO ._. 
</s>
<s>
We_PPIS2 observe_VV0 a_AT1 convergence_NN1 order_NN1 of_IO h2_FO as_CSA predicted_VVN ._. 
</s>
<s>
As_RR21 usual_RR22 ,_, by_II passing_VVG to_II a_AT1 finite_JJ cover_NN1 (_( which_DDQ is_VBZ a_AT1 homotopy_NN1 equivalence_NN1 away_II21 from_II22 torsion_NN1 primes_VVZ for_IF G_ZZ1 )_) ,_, we_PPIS2 can_VM restrict_VVI our_APPGE attention_NN1 to_II the_AT case_NN1 of_IO a_AT1 product_NN1 of_IO a_AT1 simple_JJ nonabelian_JJ Lie_NN1 group_NN1 and_CC a_AT1 torus_NN1 ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI as_CSA in_II (_( 4.5_MC )_) ._. 
</s>
<s>
If_CS @S_FO with_IW @S_FO ,_, we_PPIS2 have_VH0 @S_FO ._. 
</s>
<s>
By_II Case_NN1 2_MC in_II Theorem_NN1 6.3_MC (_( by_II Remark_NN1 6.4_MC ,_, the_AT assumption_NN1 that_CST 0_MC is_VBZ n-Diophantine_NN1 is_VBZ enough_RR )_) again_RT (_( with_IW @S_FO )_) ,_, we_PPIS2 obtain_VV0 (_( 97_MC )_) ._. 
</s>
<s>
The_AT prior_JJ (_( 17_MC )_) defines_VVZ a_AT1 Borel_NN1 probability_NN1 measure_NN1 on_II @S_FO and_CC thus_RR on_II the_AT class_NN1 @S_FO from_II Theorem_NN1 13_MC ._. 
</s>
<s>
What_DDQ do_VD0 I_PPIS1 do_VDI with_IW a_AT1 textbook_NN1 ?_? 
</s>
<s>
The_AT proof_NN1 is_VBZ ,_, to_II a_AT1 large_JJ degree_NN1 ,_, similar_JJ to_II the_AT arguments_NN2 in_II section_NN1 3.1_MC ,_, with_IW slight_JJ modifications_NN2 in_II the_AT truncation_NN1 error_NN1 estimation_NN1 ._. 
</s>
<s>
How_RGQ many_DA2 different_JJ combinations_NN2 of_IO people_NN will_VM fit_VVI in_II each_DD1 carriage_NN1 ?_? 
</s>
<s>
We_PPIS2 can_VM close_VVI the_AT induction_NN1 as_CS31 long_CS32 as_CS33 the_AT exponent_NN1 of_IO K_ZZ1 is_VBZ negative_JJ ._. 
</s>
<s>
At_II the_AT same_DA time_NNT1 ,_, they_PPHS2 dissected_VVD the_AT resulting_JJ area_NN1 as_CSA units_NN2 of_IO measure_NN1 (_( Kobiela_NP1 ,_, Lehrer_NP1 ,_, &;_NULL Pfaff_NP1 ,_, 2010_MC ;_; Lehrer_NP1 &;_NULL Slovin_NP1 ,_, 2014_MC ;_; Smith_NP1 ,_, 2016_MC ;_; Thompson_NP1 ,_, 2000_MC ;_; Vishnubhotla_NP1 &;_NULL Panorkou_NP1 ,_, 2017_MC )_) ._. 
</s>
<s>
Let_VV0 @S_FO ,_, then_RT K_ZZ1 is_VBZ a_AT1 connected_JJ compact_JJ set_NN1 ._. 
</s>
<s>
Observe_VV0 that_CST the_AT Stirling-type_JJ formula_NN1 in_II Robbins_NP1 &lsqb;_( 18_MC ,_, Displays_NN2 (_( 1_MC1 )_) -(2)_NN1 &rsqb;_) proves_VVZ for_IF all_DB @S_FO that_DD1 @F_FO ._. 
</s>
<s>
This_DD1 together_RL with_IW the_AT fact_NN1 that_CST @S_FO and_CC the_AT fact_NN1 that_CST for_IF all_DB @S_FO show_VV0 for_IF all_DB @S_FO that_DD1 @F_FO ._. 
</s>
<s>
Theorem_NN1 4.4_MC and_CC Lemma_NN1 4.5_MC ensure_VV0 that_CST @F_FO ._. 
</s>
<s>
It_PPH1 follows_VVZ that_CST @F_FO ._. 
</s>
<s>
This_DD1 proves_VVZ (_( 94_MC )_) ._. 
</s>
<s>
The_AT following_JJ two_MC immediate_JJ corollaries_NN2 are_VBR companion_NN1 results_NN2 to_II Corollaries_NN2 4.1_MC and_CC 4.2_MC ._. 
</s>
<s>
Unfortunately_RR ,_, many_DA2 practical_JJ applications_NN2 do_VD0 not_XX have_VHI strongly_RR convex_JJ objective_JJ functions_NN2 ._. 
</s>
<s>
Y+_FO and_CC f-_JJ maps_NN2 Y-_NN1 two-to-one_MC to_II S-_NN1 and_CC it_PPH1 maps_VVZ Y+_FO to_II S+_FO ._. 
</s>
<s>
However_RR ,_, the_AT modeling_NN1 activity_NN1 introduced_VVD him_PPHO1 to_II non-standard_JJ ways_NN2 to_TO write_VVI equations_NN2 ,_, even_CS21 if_CS22 they_PPHS2 were_VBDR not_XX in_II his_APPGE original_JJ thinking_NN1 ._. 
</s>
<s>
The_AT purpose_NN1 of_IO this_DD1 level_NN1 of_IO analysis_NN1 is_VBZ to_TO describe_VVI the_AT complexity_NN1 of_IO objects_NN2 and_CC meanings_NN2 that_CST form_VV0 part_NN1 of_IO mathematical_JJ and_CC didactic_JJ practices_NN2 ._. 
</s>
<s>
Next_MD ,_, we_PPIS2 have_VH0 to_TO mollify_VVI the_AT new_JJ function_NN1 along_II the_AT boundaries_NN2 of_IO Xi_NN1 ._. 
</s>
<s>
We_PPIS2 note_VV0 that_CST @S_FO ,_, since_CS the_AT @S_FO ._. 
</s>
<s>
Indeed_RR ,_, if_CS @S_FO ._. 
</s>
<s>
In_RR21 particular_RR22 ,_, the_AT recent_JJ papers_NN2 &lsqb;_( 13_MC &rsqb;_) ,_, &lsqb;_( 11_MC &rsqb;_) ,_, &lsqb;_( 12_MC &rsqb;_) show_VV0 "_" quantum_NN1 unique_JJ ergodic-ity_NN1 "_" for_IF the_AT adjacency_NN1 matrix_NN1 of_IO random_JJ regular_JJ graphs_NN2 :_: given_VVN an_AT1 observable_JJ @S_FO ,_, for_IF most_DAT @S-regular_FO graphs_NN2 on_II the_AT vertices_NN2 @S_FO ,_, we_PPIS2 have_VH0 that_DD1 @S_FO is_VBZ close_JJ to_II @S_FO for_IF all_DB indices_NN2 j_ZZ1 ._. 
</s>
<s>
If_CS @S_FO is_VBZ a_AT1 marked_JJ cobordism_NN1 from_II @S_FO to_II @S_FO ,_, we_PPIS2 say_VV0 that_CST the_AT marking_NN1 data_NN v_ZZ1 is_VBZ right-proper_JJ if_CS the_AT map_NN1 @F_FO is_VBZ surjective_JJ for_IF i_ZZ1 =_FO 0_MC and_CC 1_MC1 ._. 
</s>
<s>
Kierra_NN1 and_CC Isaac_NP1 recognized_VVD that_CST graphically_RR the_AT effect_NN1 of_IO applying_VVG the_AT discount_NN1 would_VM be_VBI a_AT1 slight_JJ change_NN1 in_II the_AT slope_NN1 of_IO the_AT graph_NN1 of_IO the_AT total_JJ revenue_NN1 function_NN1 after_II 300_MC computers_NN2 as_CSA shown_VVN in_II Figure_NN1 2_MC ._. 
</s>
<s>
It_PPH1 is_VBZ also_RR easily_RR seen_VVN from_II the_AT above_JJ formula_NN1 that_CST @F_FO ._. 
</s>
<s>
In_RR21 addition_RR22 ,_, the_AT students_NN2 had_VHD not_XX realised_VVN that_CST it_PPH1 was_VBDZ possible_JJ for_IF the_AT shape_NN1 of_IO cross-sections_NN2 to_TO be_VBI irregular_JJ ,_, and_CC they_PPHS2 ended_VVD up_RP constructing_VVG artefacts_NN2 that_CST looked_VVD somewhat_RR like_II triangles_NN2 (_( Fig._NN1 8b_FO )_) after_II engaging_VVG in_II the_AT task_NN1 ._. 
</s>
<s>
It_PPH1 is_VBZ convenient_JJ to_TO illustrate_VVI our_APPGE approach_NN1 to_II the_AT estimation_NN1 of_IO the_AT temporaldiscretization_NN1 error_NN1 using_VVG a_AT1 very_RG simple_JJ example_NN1 ._. 
</s>
<s>
Moreover_RR ,_, we_PPIS2 equip_VV0 this_DD1 space_NN1 with_IW the_AT following_JJ metric_JJ :_: @F_FO for_IF @S_FO ,_, '_NULL 1_MC1 '_NULL 2_MC e_ZZ1 DT_NP1 x_ZZ1 and_CC the_AT quantity_NN1 @F_FO ._. 
</s>
<s>
This_DD1 is_VBZ related_VVN to_II other_JJ differentiability_NN1 results_VVZ that_CST can_VM be_VBI for_REX21 instance_REX22 found_VV0 in_RP &lsqb;_( 10,31_MC &rsqb;_) ,_, and_CC that_CST have_VH0 been_VBN obtained_VVN under_II special_JJ structural_JJ assumptions_NN2 of_IO the_AT vector_NN1 field_NN1 a()_AT1 ;_; see_VV0 Theorem_NN1 4.2_MC below_RL ._. 
</s>
<s>
Let_VV0 us_PPIO2 assume_VVI that_CST (_( A0_FO )_) is_VBZ satisfied_JJ ._. 
</s>
<s>
Step_NN1 1_MC1 :_: solution_NN1 of_IO an_AT1 approximated_JJ problem_NN1 ._. 
</s>
<s>
Since_II then_RT @S_FO and_CC hence_RR @S_FO are_VBR deterministic_JJ ,_, the_AT situation_NN1 is_VBZ very_RG easy_JJ here_RL and_CC the_AT results_NN2 are_VBR not_XX surprising_JJ (_( see_VV0 Remark_NN1 2_MC )_) ._. 
</s>
<s>
For_IF @S_FO there_EX is_VBZ an_AT1 exact_JJ sequence_NN1 @F_FO ,_, where_CS @S_FO ._. 
</s>
<s>
From_II the_AT above_JJ description_NN1 of_IO @S_FO we_PPIS2 obtain_VV0 @F_FO ._. 
</s>
<s>
The_AT two_MC and_CC ,_, among_II other_JJ things_NN2 ,_, about_II the_AT degree_NN1 of_IO interconnectedness_NN1 of_IO a_AT1 knowledge_NN1 system_NN1 and_CC whether_CSW these_DD2 connections_NN2 already_RR exist_VV0 or_CC emerge_VV0 over_II time_NNT1 (_( see_VV0 Table_NN1 3_MC )_) ._. 
</s>
<s>
Questions_NN2 can_VM be_VBI explored_VVN ,_, at_II various_JJ scales_NN2 ,_, around_II how_RRQ industries_NN2 take_VV0 advantage_NN1 of_IO racialized_JJ geographies_NN2 when_CS selecting_VVG areas_NN2 for_IF manufacturing_NN1 or_CC waste_VV0 disposal_NN1 ,_, and_CC further_RRR ,_, how_RRQ boundary-making_JJ practices_NN2 of_IO who_PNQS lives_VVZ where_RRQ ,_, or_CC who_PNQS can_VM pollute_VVI here_RL ,_, are_VBR supported_VVN by_II policy_NN1 and_CC law_NN1 ._. 
</s>
<s>
The_AT first_MD two_MC are_VBR about_II content_NN1 ;_; the_AT third_MD is_VBZ about_II teaching_NN1 ._. 
</s>
<s>
The_AT next_MD theorem_NN1 provides_VVZ an_AT1 O(n-1)_MC1 approximation_NN1 to_II the_AT value_NN1 function_NN1 in_II the_AT n-player_JJ game_NN1 ._. 
</s>
<s>
First_MD consider_VV0 the_AT simpler_JJR case_NN1 of_IO @S_FO and_CC the_AT map_NN1 @S_FO ._. 
</s>
<s>
We_PPIS2 claim_VV0 this_DD1 map_NN1 is_VBZ an_AT1 isomorphism_NN1 ._. 
</s>
<s>
The_AT goal_NN1 of_IO this_DD1 section_NN1 is_VBZ to_TO establish_VVI the_AT first_MD parts_NN2 of_IO Proposition_NN1 5.2_MC and_CC Proposition_NN1 5.3_MC ._. 
</s>
<s>
This_DD1 approach_NN1 resembles_VVZ the_AT well-known_JJ mass-lumping_JJ procedure_NN1 ._. 
</s>
<s>
The_AT specific_JJ splitting_NN1 of_IO the_AT problem_NN1 @S_FO makes_VVZ it_PPH1 suitable_JJ for_IF the_AT well-known_JJ Dykstra_NP1 '_NULL s_ZZ1 algorithm_NN1 (_( see_VV0 Section_NN1 1.1_MC for_IF more_DAR background_NN1 on_II this_DD1 algorithm_NN1 )_) ._. 
</s>
<s>
Next_MD ,_, we_PPIS2 introduce_VV0 some_DD notation_NN1 used_VVN throughout_II these_DD2 notes_NN2 ._. 
</s>
<s>
A_AT1 total_NN1 of_IO 424_MC students_NN2 completed_VVD the_AT 2015_MC pre-_JJ and_CC post-_JJ surveys_NN2 (_( 118_MC students_NN2 in_II noniPad_NN1 classes_NN2 ,_, 150_MC students_NN2 in_II iPad_NN1 classes_NN2 in_II 2014_MC and_CC 2015_MC ,_, and_CC 156_MC students_NN2 from_II iPad_NN1 classes_NN2 in_II 2015_MC only_JJ )_) ._. 
</s>
<s>
To_II this_DD1 end_NN1 ,_, let_VV0 y_ZZ1 ,_, z2X_FO ,_, and_CC suppose_VV0 that_CST 16i_FO ,_, j6M_FO are_VBR such_CS21 that_CS22 @F_FO ._. 
</s>
<s>
There_RL exist_VV0 at_RR21 most_RR22 one_MC1 choice_NN1 for_IF each_DD1 of_IO i_ZZ1 and_CC j_ZZ1 ._. 
</s>
<s>
The_AT current_JJ study_NN1 adds_VVZ to_II the_AT multiple-goal_JJ literature_NN1 by_II exploring_VVG how_RRQ students_NN2 '_NULL achievement_NN1 goals_NN2 interact_VV0 with_IW each_PPX221 other_PPX222 to_TO moderate_VVI students_NN2 '_NULL perceptions_NN2 of_IO classroom_NN1 goal_NN1 structures_NN2 in_II learning_VVG mathematics_NN1 ._. 
</s>
<s>
Arlinghaus_NN2 and_CC Kerski_NP1 (_( 2014_MC )_) demonstrate_VV0 the_AT use_NN1 of_IO maps_NN2 to_TO teach_VVI mathematics_NN1 (_( coordinate_VV0 systems_NN2 ,_, measurement_NN1 ,_, trigonometry_NN1 ,_, transformations_NN2 ,_, vectors_NN2 ,_, scale_NN1 ,_, data_NN analysis_NN1 ,_, sampling_VVG )_) ._. 
</s>
<s>
We_PPIS2 consider_VV0 a_AT1 quantum_NN1 Hall_NN1 sample_NN1 of_IO an_AT1 area_NN1 L2_FO in_II an_AT1 external_JJ magnetic_JJ field_NN1 B._NP1 We_PPIS2 next_MD consider_VV0 the_AT equation_NN1 (_( 5.13_MC )_) ._. 
</s>
<s>
This_DD1 inequality_NN1 is_VBZ obtained_VVN along_II the_AT lines_NN2 of_IO the_AT proof_NN1 of_IO the_AT pioneering_JJ paper_NN1 by_II Otto_NP1 and_CC Villani_NP1 ._. 
</s>
<s>
Despite_II these_DD2 sophisticated_JJ drawing_NN1 behaviors_NN2 ,_, Charlotte_NP1 said_VVD her_APPGE drawing_NN1 (_( see_VV0 Figure_NN1 3a_FO )_) "_" bothered_VVD "_" her_PPHO1 because_CS "_" they_PPHS2 &lsqb;_( some_DD of_IO the_AT drawn_VVN squares_NN2 &rsqb;_) look_VV0 smaller_JJR ..._... like_CS they_PPHS2 look_VV0 like_CS they_PPHS2 were_VBDR three_MC across_RL ,_, but_CCB they_PPHS2 don_VV0 '_NULL t_ZZ1 look_VV0 as_RG much_DA1 as_CSA five_MC down_RP ._. "_" 
</s>
<s>
So_RR let_VV0 us_PPIO2 assume_VVI I._NP1 Now_RT choose_VV0 a_AT1 reduced_JJ expression_NN1 si1sil_FO of_IO w0_FO such_CS21 that_CS22 si1sik=w_FO (_( in_RR21 particular_RR22 i1∈I_FO )_) and_CC sk+1sil=ww0_FO ._. 
</s>
<s>
Define_VV0 an_AT1 extension_NN1 @S_FO of_IO a_AT1 by_II @F_FO ._. 
</s>
<s>
Clearly_RR ,_, @S_FO is_VBZ a_AT1 cocycle_NN1 ._. 
</s>
<s>
Assume_VV0 that_CST X_ZZ1 is_VBZ a_AT1 Banach_NN1 space_NN1 such_CS21 that_CS22 (_( 1_MC1 )_) @S_FO equi-coarsely_RR embeds_VVZ into_II X_ZZ1 ,_, (_( 2_MC )_) X_ZZ1 coarsely_RR embeds_VVZ into_II a_AT1 Banach_NN1 space_NN1 with_IW nontrivial_JJ type_NN1 ._. 
</s>
<s>
The_AT exact_JJ meaning_NN1 is_VBZ given_VVN below_RL ._. 
</s>
<s>
We_PPIS2 will_VM show_VVI that_DD1 for_IF every_AT1 @S_FO ,_, @F_FO ,_, that_REX21 is_REX22 ,_, @S_FO to_II in_II distribution_NN1 ._. 
</s>
<s>
This_DD1 process_NN1 was_VBDZ repeated_VVN multiple_JJ times_NNT2 throughout_II the_AT coding_NN1 of_IO data_NN ._. 
</s>
<s>
If_CS E_ZZ1 t_ZZ1 M_ZZ1 is_VBZ a_AT1 vector_NN1 bundle_NN1 with_IW a_AT1 non-degenerate_JJ symmetric_JJ pairing_NN1 @S_FO and_CC t_ZZ1 e_ZZ1 R_ZZ1 ,_, we_PPIS2 shall_VM define_VVI an_AT1 algebra_NN1 At(E)_VVD containing_VVG @S_FO ._. 
</s>
<s>
Looking_VVG at_II the_AT proof_NN1 and_CC having_VHG in_II mind_NN1 that_CST L1_FO equals_VVZ the_AT union_NN1 of_IO all_DB spaces_NN2 E_ZZ1 where_RRQ 4_MC satisfies_VVZ the_AT @S_FO ,_, this_DD1 could_VM be_VBI expected_VVN ._. 
</s>
<s>
The_AT local_JJ finiteness_NN1 of_IO CLEk_NP1 ,_, i.e._REX the_AT fact_NN1 that_CST the_AT number_NN1 of_IO loops_NN2 with_IW diameter_NN1 at_RR21 least_RR22 @S_FO is_VBZ for_IF each_DD1 @S_FO almost_RR surely_RR finite_JJ ,_, was_VBDZ established_VVN in_II &lsqb;_( 13_MC &rsqb;_) as_II a_AT1 consequence_NN1 of_IO the_AT almost_RR sure_JJ continuity_NN1 of_IO the_AT so-called_JJ space-filling_JJ SLE_NN1 ._. 
</s>
<s>
In_II the_AT spatially_RR homogeneous_JJ case_NN1 ,_, by_II iterating_VVG this_DD1 gain_NN1 of_IO regularity_NN1 ,_, it_PPH1 is_VBZ found_VVN that_CST solutions_NN2 belong_VV0 to_II the_AT Schwartz_NP1 class_NN1 for_IF all_DB positive_JJ times_NNT2 ._. 
</s>
<s>
Now_CS21 that_CS22 we_PPIS2 have_VH0 a_AT1 bound_NN1 on_II the_AT y-derivatives_NN2 of_IO @S_FO ,_, we_PPIS2 can_VM derive_VVI error_NN1 bounds_VVZ on_II a_AT1 Taylor_NP1 expansion_NN1 of_IO @S_FO in_II the_AT y-variable_NN1 on_II the_AT domain_NN1 @S_FO ._. 
</s>
<s>
Scalars_NP1 are_VBR marked_VVN in_II regular_JJ font_NN1 ._. 
</s>
<s>
Virtual_JJ learning_NN1 environments_NN2 (_( VLE_NP1 )_) have_VH0 been_VBN developed_VVN as_II a_AT1 locus_NN1 to_TO make_VVI online_JJ teacher_NN1 education_NN1 possible_JJ ._. 
</s>
<s>
Now_RT it_PPH1 is_VBZ easy_JJ to_TO see_VVI that_CST the_AT second_MD term_NN1 on_II the_AT above_JJ right-hand_JJ side_NN1 tends_VVZ to_TO zero_VVI as_CSA @S_FO ._. 
</s>
<s>
For_IF the_AT first_MD term_NN1 ,_, following_VVG the_AT proof_NN1 of_IO &lsqb;_( 23_MC ,_, Theorem_NN1 3.7_MC &rsqb;_) and_CC in_RR21 particular_RR22 &lsqb;_( 23_MC ,_, (_( 3.35_MC )_) &rsqb;_) ,_, we_PPIS2 have_VH0 @F_FO ,_, which_DDQ is_VBZ bounded_VVN uniformly_RR in_II @S_FO under_II the_AT assumption_NN1 on_II the_AT kernel_NN1 @S_FO ._. 
</s>
<s>
Assume_VV0 lastly_RR that_CST @S_FO ,_, F_ZZ1 is_VBZ Cm_NNU in_II a_AT1 neighborhood_NN1 of_IO @S_FO ,_, and_CC @S_FO ._. 
</s>
<s>
From_II (_( 203_MC )_) ,_, Let_VV0 @F_FO ._. 
</s>
<s>
The_AT second_MD condition_NN1 on_II w_ZZ1 is_VBZ the_AT "_" no_NN1 attack_NN1 "_" condition_NN1 as_CSA before_RT ._. 
</s>
<s>
Properties_NN2 (_( 1_MC1 )_) and_CC (_( 2_MC )_) follows_VVZ by_II direct_JJ computation_NN1 ._. 
</s>
<s>
Computing_VVG the_AT dimension_NN1 of_IO self-affine_JJ setsattractors_NN2 of_IO systems_NN2 of_IO affine_JJ contractions_NN2 of_IO Rdis_NP1 one_MC1 of_IO the_AT major_JJ open_JJ problems_NN2 in_II fractal_JJ geometry_NN1 ._. 
</s>
<s>
Actually_RR ,_, the_AT below_II argument_NN1 works_VVZ in_II all_DB dimensions_NN2 and_CC codimensions_NN2 since_CS the_AT same_DA is_VBZ true_JJ of_IO &lsqb;_( Ce1_FO &rsqb;_) ,_, &lsqb;_( Pa_NP1 &rsqb;_) ,_, and_CC the_AT Alexander_NP1 isotopy_NN1 ._. 
</s>
<s>
In_II section_NN1 3_MC ,_, we_PPIS2 propose_VV0 an_AT1 arbitrary_JJ high_JJ order_NN1 weak_JJ approximation_NN1 using_VVG Malliavin_NP1 calculus_NN1 ._. 
</s>
<s>
Let_VV0 @S_FO and_CC @S_FO ._. 
</s>
<s>
Then_RT @S_FO is_VBZ a_AT1 distributional_JJ (_( or_CC very_RG weak_JJ )_) solution_NN1 of_IO (_( 1.1_MC )_) if_CS for_IF all_DB @S_FO ,_, @S_FO and_CC @F_FO ._. 
</s>
<s>
Note_VV0 that_CST @S_FO if_CS ,_, e.g._REX ,_, @S_FO and_CC @S_FO continuous_JJ ._. 
</s>
<s>
Consequently_RR ,_, the_AT result_NN1 is_VBZ expected_VVN to_TO be_VBI a_AT1 checkerboard_NN1 reconstruction_NN1 ._. 
</s>
<s>
Although_CS we_PPIS2 shall_VM focus_VVI on_II mathematical_JJ matters_NN2 here_RL ,_, (_( KdV_NP1 )_) continues_VVZ to_TO be_VBI an_AT1 important_JJ effective_JJ model_NN1 for_IF a_AT1 diverse_JJ range_NN1 of_IO physical_JJ phenomena_NN2 ;_; see_VV0 ,_, for_REX21 example_REX22 ,_, the_AT review_NN1 &lsqb;_( 15_MC &rsqb;_) occasioned_VVN by_II the_AT centenary_NN1 of_IO &lsqb;_( 44_MC &rsqb;_) ._. 
</s>
<s>
In_II this_DD1 circuit_NN1 we_PPIS2 also_RR use_VV0 the_AT conjugate_NN1 transpose_VV0 @S_FO of_IO the_AT T_NP1 gate_NN1 ,_, but_CCB it_PPH1 is_VBZ easy_JJ to_TO see_VVI that_CST if_CS we_PPIS2 really_RR want_VV0 to_TO stick_VVI to_II the_AT gates_NN2 H_ZZ1 ,_, T_ZZ1 ,_, and_CC CNOT_VV0 only_RR ,_, @S_FO can_VM be_VBI constructed_VVN from_II T_ZZ1 because_CS @S_FO ._. 
</s>
<s>
Verifying_VVG correctness_NN1 of_IO the_AT construction_NN1 in_II Figure_NN1 13_MC requires_VVZ a_AT1 few_DA2 calculations_NN2 that_CST we_PPIS2 leave_VV0 as_II an_AT1 exercise_NN1 ._. 
</s>
<s>
As_CSA expected_VVN from_II the_AT 1D_NNU example_NN1 (_( Figure_NN1 3_MC )_) ,_, the_AT fully_RR linear_JJ scheme_NN1 induces_VVZ very_RG small_JJ numerical_JJ errors_NN2 near_II shocks_NN2 ,_, by_II either_RR smearing_VVG or_CC overshooting_VVG small_JJ details_NN2 ._. 
</s>
<s>
We_PPIS2 attempt_VV0 to_TO integrate_VVI the_AT vast_JJ literature_NN1 to_TO gain_VVI secure_JJ grounding_NN1 on_II which_DDQ to_TO develop_VVI a_AT1 theoretical_JJ framework_NN1 for_IF the_AT study_NN1 ._. 
</s>
<s>
Dividing_VVG (_( 2.2a_FO )_) by_II pi_NN1 and_CC using_VVG the_AT relation_NN1 @S_FO lead_VV0 to_II an_AT1 equation_NN1 for_IF @S_FO involving_VVG the_AT individual_JJ velocities_NN2 ._. 
</s>
<s>
The_AT representation_NN1 of_IO Figure_NN1 3_MC is_VBZ still_RR valid_JJ ,_, but_CCB progression_NN1 of_IO the_AT immune_JJ system_NN1 and_CC regression_NN1 of_IO the_AT virus_NN1 should_VM be_VBI related_VVN to_II well-defined_JJ medical_JJ actions_NN2 ._. 
</s>
<s>
Then_RT ,_, we_PPIS2 derive_VV0 from_II (_( 39_MC )_) and_CC (_( 40_MC )_) that_CST @F_FO and_CC ,_, therefore_RR ,_, that_CST @S_FO ._. 
</s>
<s>
Using_VVG the_AT fact_NN1 that_CST @S_FO ,_, we_PPIS2 then_RT deduce_VV0 that_CST @S_FO ._. 
</s>
<s>
Similarly_RR ,_, since_CS @S_FO ,_, we_PPIS2 have_VH0 @S_FO ._. 
</s>
<s>
Next_MD ,_, let_VV0 us_PPIO2 set_VVI @F_FO and_CC @F_FO ._. 
</s>
<s>
First_MD ,_, we_PPIS2 show_VV0 that_CST @S_FO for_IF every_AT1 @S_FO and_CC @S_FO for_IF every_AT1 @S_FO ._. 
</s>
<s>
The_AT first_MD assertion_NN1 is_VBZ obtained_VVN by_II invoking_VVG (_( 46_MC )_) ,_, (_( a_ZZ1 )_) ,_, (_( ii_MC )_) ,_, (_( 49_MC )_) ,_, and_CC (_( 50_MC )_) ,_, which_DDQ imply_VV0 that_CST @F_FO ._. 
</s>
<s>
Similarly_RR ,_, it_PPH1 follows_VVZ from_II (_( 52_MC )_) ,_, (_( a_ZZ1 )_) ,_, (_( ii_MC )_) ,_, (_( 55_MC )_) ,_, and_CC (_( 56_MC )_) that_CST @F_FO ._. 
</s>
<s>
We_PPIS2 next_MD perform_VV0 some_DD analysis_NN1 of_IO @S_FO and_CC @S_FO ._. 
</s>
<s>
We_PPIS2 derive_VV0 from_II (_( 57_MC )_) ,_, (_( 46_MC )_) ,_, and_CC (_( 41_MC )_) that_CST @F_FO ,_, which_DDQ yields_VVZ @F_FO and_CC @F_FO ._. 
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<s>
We_PPIS2 let_VV0 x0_FO be_VBI a_AT1 density_NN1 point_NN1 of_IO @S_FO ._. 
</s>
<s>
In_II other_JJ words_NN2 ,_, any_DD @S_FO admits_VVZ a_AT1 unique_JJ decomposition_NN1 @F_FO ,_, where_CS @S_FO and_CC @S_FO ._. 
</s>
<s>
Here_RL we_PPIS2 sketch_VV0 a_AT1 proof_NN1 (_( close_RR to_II what_DDQ we_PPIS2 do_VD0 for_IF mapping_VVG class_NN1 groups_NN2 )_) that_CST the_AT standard_JJ action_NN1 of_IO G_ZZ1 on_II the_AT circle_NN1 at_II infinity_NN1 δ_FO is_VBZ finitely_RR F-amenable_JJ for_IF the_AT family_NN1 F_ZZ1 of_IO cyclic_JJ subgroups_NN2 ._. 
</s>
<s>
Now_RT we_PPIS2 exploit_VV0 the_AT regularity_NN1 result_NN1 for_IF Vn_NP1 given_VVN by_II (_( 5.9_MC )_) ._. 
</s>
<s>
In_II other_JJ words_NN2 ,_, research_NN1 informs_VVZ us_PPIO2 that_CST teachers_NN2 need_VV0 to_TO be_VBI competent_JJ and_CC confident_JJ in_II their_APPGE competence_NN1 to_TO teach_VVI effectively_RR ._. 
</s>
<s>
John_NP1 and_CC Fred_NP1 concluded_VVD by_II advising_VVG the_AT family_NN1 to_TO sign_VVI the_AT contract_NN1 if_CS the_AT school_NN1 orders_NN2 "_" 330_MC computers_NN2 or_CC less_RRR ._. "_" 
</s>
<s>
This_DD1 discriminant_JJ validity_NN1 is_VBZ a_AT1 necessary_JJ prerequisite_NN1 for_IF the_AT further_JJR analysis_NN1 of_IO reasoning_VVG abilities_NN2 as_II a_AT1 possible_JJ foundation_NN1 of_IO successful_JJ strategy_NN1 use_NN1 ,_, according_II21 to_II22 research_NN1 question_NN1 2_MC ._. 
</s>
<s>
Again_RT ,_, Farrell_NP1 and_CC Jones_NP1 '_NULL original_JJ formulation_NN1 &lsqb;_( 30_MC &rsqb;_) for_IF the_AT group_NN1 ring_NN1 Z&lsqb;G&rsqb;_NP1 is_VBZ a_AT1 special_JJ case_NN1 of_IO this_DD1 more_RGR general_JJ formulation_NN1 ._. 
</s>
<s>
The_AT argument_NN1 is_VBZ standard_JJ and_CC mostly_RR borrowed_VVN from_II &lsqb;_( 1_MC1 ,_, proof_NN1 of_IO Proposition_NN1 2.2_MC &rsqb;_) ._. 
</s>
<s>
This_DD1 is_VBZ expected_VVN ,_, as_CSA in_II Corollary_NN1 15_MC we_PPIS2 need_VV0 to_TO use_VVI the_AT worst_JJT case_NN1 @S_FO to_TO determine_VVI the_AT trunction_NN1 parameter_NN1 A._NNU This_DD1 is_VBZ consistent_JJ with_IW the_AT prediction_NN1 given_VVN by_II the_AT ratio_NN1 @S_FO between_II the_AT dimension_NN1 of_IO the_AT ambient_JJ space_NN1 and_CC that_DD1 of_IO the_AT manifold_JJ M._NN1 After_II several_DA2 rounds_NN2 of_IO experiments_NN2 ,_, we_PPIS2 finally_RR opted_VVN for_IF the_AT third_MD choice_NN1 based_VVN on_II the_AT following_JJ observations_NN2 ._. 
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<s>
Instead_RR we_PPIS2 give_VV0 a_AT1 brief_JJ derivation_NN1 of_IO a_AT1 structured-crowd_JJ model_NN1 suitable_JJ for_IF dynamics_NN where_CS visibility_NN1 is_VBZ obscured_VVN ._. 
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<s>
Extension_NN1 We_PPIS2 note_VV0 that_CST there_EX is_VBZ an_AT1 extension_NN1 operator_NN1 @S_FO such_CS21 that_CS22 @F_FO ._. 
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<s>
This_DD1 result_NN1 follows_VVZ by_II mapping_VVG to_II a_AT1 reference_NN1 neighborhood_NN1 in_II R2_FO using_VVG a_AT1 smooth_JJ local_JJ chart_NN1 and_CC then_RT applying_VVG the_AT extension_NN1 theorem_NN1 ,_, see_VV0 &lsqb;_( 13_MC &rsqb;_) ,_, and_CC finally_RR mapping_VVG back_RP to_II the_AT surface_NN1 ._. 
</s>
<s>
Additionally_RR the_AT main_JJ diffusion_NN1 direction_NN1 ,_, i.e._REX the_AT largest_JJT eigenvector_NN1 of_IO Dw_NP1 ,_, is_VBZ shown_VVN in_II Fig._NN1 4(b)_FO as_II a_AT1 four-channel_JJ color-coded_JJ image_NN1 ._. 
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<s>
Caratheodory_NN1 '_NULL s_ZZ1 Theorem_NN1 gives_VVZ vectors_NN2 @S_FO such_CS21 that_CS22 0∈Conv_NN1 (_( v1_FO ,_, ,_, vN+1_FO )_) ._. 
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<s>
The_AT goal_NN1 of_IO this_DD1 paper_NN1 is_VBZ to_TO identify_VVI numerical_JJ approximations_NN2 @S_FO ,_, N_ZZ1 G_ZZ1 N_ZZ1 ,_, that_CST converge_VV0 in_II the_AT strong_JJ sense_NN1 to_II the_AT exact_JJ solution_NN1 of_IO the_AT SDE_NP1 (_( 1_MC1 )_) and_CC that_CST preserve_VV0 exponential_NN1 integrability_NN1 properties_NN2 in_II the_AT sense_NN1 that_CST for_IF all_DB suffciently_RR regular_JJ functions_NN2 @S_FO with_IW @S_FO it_PPH1 holds_VVZ that_CST @S_FO ._. 
</s>
<s>
Our_APPGE main_JJ motivation_NN1 for_IF this_DD1 is_VBZ that_CST such_DA exponential_NN1 integrability_NN1 properties_NN2 are_VBR a_AT1 key_JJ tool_NN1 for_IF establishing_VVG rates_NN2 of_IO strong_JJ and_CC numerically_RR weak_JJ convergence_NN1 for_IF a_AT1 large_JJ class_NN1 of_IO nonlinear_JJ SDEs_NN2 ._. 
</s>
<s>
Then_RT (_( 2.7_MC )_) is_VBZ violated_VVN ,_, which_DDQ contradicts_VVZ with_IW the_AT assumption_NN1 on_II semi-convexity_NN1 ._. 
</s>
<s>
For_IF consistency_NN1 with_IW the_AT continuous_JJ Stokes_NP1 system_NN1 the_AT matrix_NN1 B_ZZ1 should_VM satisfy_VVI @S_FO in_II the_AT case_NN1 of_IO enclosed_JJ flow_NN1 (_( see_VV0 ,_, e.g._REX ,_, &lsqb;_( 8_MC ,_, Chapter_NN1 3_MC &rsqb;_) )_) ._. 
</s>
<s>
Note_VV0 that_CST @S_FO ,_, and_CC @S_FO ._. 
</s>
<s>
By_II Lemma_NN1 2.5_MC ,_, we_PPIS2 conclude_VV0 that_CST @S_FO and_CC @S_FO are_VBR the_AT renormalized_JJ solutions_NN2 of_IO (_( 4.28_MC )_) j_ZZ1 and_CC (_( 5.5_MC )_) j_ZZ1 for_IF @S_FO ,_, respectively_RR ._. 
</s>
<s>
By_II construction_NN1 ,_, @S_FO forms_VVZ a_AT1 global_JJ ,_, locally_RR supported_VVN basis_NN1 for_IF the_AT space_NN1 @S_FO that_DD1 has_VHZ the_AT properties_NN2 listed_VVN in_II Theorem_NN1 3.7_MC ._. 
</s>
<s>
Clearly_RR ,_, there_EX is_VBZ not_XX a_AT1 unique_JJ way_NN1 to_TO measure_VVI the_AT relative_JJ strength_NN1 of_IO variables_NN2 (_( Kruskal_NP1 and_CC Majors_NP2 ,_, 1989_MC )_) ._. 
</s>
<s>
S_ZZ1 ;_; U_ZZ1 (_( Bki_NP1 (_( 0_MC )_) x_ZZ1 -bpr2_JJ )_) by_II the_AT weak_JJ maximum_JJ principle_NN1 (_( 2.8_MC )_) (_( recall_VV0 from_II Sect._NP1 3_MC that_CST @S_FO ._. 
</s>
<s>
Let_VV0 @S_FO and_CC consider_VVI the_AT function_NN1 Ue_NP1 defined_VVD in_II (_( 4.1_MC )_) with_IW the_AT choice_NN1 of_IO @S_FO as_CSA in_II Sect._NP1 4.1_MC (_( recall_VV0 (_( 4.1o_FO )_) )_) ._. 
</s>
<s>
In_RR21 particular_RR22 ,_, to_II the_AT best_JJT of_IO our_APPGE knowledge_NN1 ,_, the_AT numerical_JJ scheme_NN1 proposed_VVN in_II this_DD1 article_NN1 (_( see_VV0 (_( 6_MC )_) below_RL )_) is_VBZ the_AT first_MD approximation_NN1 method_NN1 for_IF which_DDQ temporal_JJ strong_JJ convergence_NN1 rates_NN2 have_VH0 been_VBN proved_VVN (_( see_VV0 Theorem_NN1 1.3_MC in_II &lsqb;_( 17_MC &rsqb;_) )_) for_IF at_RR21 least_RR22 one_MC1 multi-dimensional_JJ SDE_NN1 with_IW non-globally_RR monotone_NN1 coefficients_NN2 (_( see_VV0 Section_NN1 3.1_MC in_II &lsqb;_( 17_MC &rsqb;_) for_IF a_AT1 list_NN1 of_IO example_NN1 SDEs_NN2 for_IF which_DDQ temporal_JJ strong_JJ convergence_NN1 rates_NN2 for_IF the_AT numerical_JJ method_NN1 (_( 6_MC )_) below_RL have_VH0 been_VBN proved_VVN )_) ._. 
</s>
<s>
In_II contrast_NN1 ,_, in_II Extract_NN1 4_MC ,_, the_AT teacher_NN1 does_VDZ not_XX prevent_VVI Scott_NP1 from_II explaining_VVG and_CC does_VDZ not_XX treat_VVI this_DD1 turn_NN1 as_CSA dispreferred_VVN even_CS21 though_CS22 it_PPH1 deviates_VVZ from_II the_AT usual_JJ rules_NN2 of_IO turn-taking_NN1 (_( Ingram_NP1 &;_NULL Elliott_NP1 ,_, 2014_MC ;_; Mchoul_NP1 ,_, 1978_MC )_) ._. 
</s>
<s>
Also_RR ,_, we_PPIS2 emphasize_VV0 that_CST ,_, to_II our_APPGE knowledge_NN1 ,_, this_DD1 is_VBZ the_AT first_MD work_NN1 in_II the_AT literature_NN1 addressing_VVG the_AT asymptotics_NN1 of_IO the_AT high_JJ frequencies_NN2 for_IF a_AT1 thin_JJ T-like_JJ shaped_JJ structure_NN1 ._. 
</s>
<s>
We_PPIS2 do_VD0 this_DD1 by_II first_MD choosing_NN1 @S_FO ,_, which_DDQ ensures_VVZ @S_FO ._. 
</s>
<s>
Let_VV0 @S_FO ._. 
</s>
<s>
There_RL now_RT remains_VVZ to_TO estimate_VVI @S_FO by_II the_AT energy_NN1 of_IO @S_FO ._. 
</s>
<s>
For_IF this_DD1 ,_, we_PPIS2 will_VM need_VVI the_AT energy-minimizing_JJ ectension_NN1 of_IO any_DD finite_JJ element_NN1 function_NN1 defined_VVN on_II F._NP1 The_AT relevant_JJ matrix_NN1 is_VBZ @S_FO defined_VVN by_II @F_FO ._. 
</s>
<s>
Here_RL @S_FO is_VBZ principal_JJ minor_NN1 of_IO @S_FO with_II31 respect_II32 to_II33 @S_FO and_CC @S_FO an_AT1 off-diagonal_JJ block_NN1 of_IO @S_FO ._. 
</s>
<s>
We_PPIS2 need_VV0 to_TO establish_VVI a_AT1 bound_NN1 for_IF @F_FO and_CC to_TO show_VVI that_CST @F_FO ,_, where_CS @S_FO is_VBZ an_AT1 arbitrary_JJ extension_NN1 of_IO the_AT values_NN2 of_IO @S_FO on_II the_AT face_NN1 F_ZZ1 to_II the_AT rest_NN1 of_IO @S_FO ._. 
</s>
<s>
Employing_VVG now_RT the_AT arguments_NN2 similar_JJ to_II &lsqb;_( 8_MC ,_, Theorem_NN1 2_MC &rsqb;_) shows_VVZ that_CST the_AT latter_DA inequality_NN1 amounts_VVZ to_II the_AT existence_NN1 of_IO numbers_NN2 θ_NULL >_FO 0_MC and_CC M_ZZ1 >_FO 0_MC so_CS21 that_CS22 for_IF any_DD @S_FO the_AT estimate_NN1 @F_FO holds_VVZ ,_, which_DDQ is_VBZ the_AT well-known_JJ error_NN1 bound_VVD property_NN1 for_IF KKT_NP1 systems_NN2 with_IW inequality_NN1 constraints_NN2 ;_; see_VV0 &lsqb;_( 19_MC ,_, Theorem_NN1 1.43_MC &rsqb;_) and_CC &lsqb;_( 7_MC ,_, Proposition_NN1 6.2.7_MC &rsqb;_) for_IF more_DAR details_NN2 ._. 
</s>
<s>
We_PPIS2 expect_VV0 that_CST the_AT obstacle_NN1 can_VM influence_VVI the_AT resulting_JJ outgoing_JJ flux_NN1 ,_, because_II21 of_II22 excluded_JJ volume_NN1 ,_, in_II two_MC different_JJ ways_NN2 ._. 
</s>
<s>
In_II &lsqb;_( 23_MC &rsqb;_) ,_, the_AT authors_NN2 verified_VVD that_CST the_AT emergence_NN1 of_IO multi-cluster_JJ flocking_NN1 can_VM occur_VVI for_IF small_JJ coupling_NN1 strengths_NN2 ._. 
</s>
<s>
In_II this_DD1 section_NN1 ,_, we_PPIS2 provide_VV0 theoretical_JJ results_NN2 characterizing_VVG the_AT statistical_JJ properties_NN2 of_IO our_APPGE algorithm_NN1 ._. 
</s>
<s>
So_RR in_II the_AT same_DA way_NN1 we_PPIS2 obtain_VV0 a_AT1 standard_JJ form_NN1 model_NN1 C_ZZ1 for_IF @S_FO that_DD1 is_VBZ free_JJ over_II C(0)_FO and_CC minimal_JJ at_II r_ZZ1 =_FO (_( p_ZZ1 ,_, q_ZZ1 )_) with_IW etale_NN1 maps_NN2 c_ZZ1 ,_, d_ZZ1 that_CST are_VBR induced_VVN by_II actual_JJ maps_NN2 of_IO cdgas_NN2 ._. 
</s>
<s>
Adopting_VVG the_AT notation_NN1 from_II the_AT previous_JJ proof_NN1 and_CC using_VVG (_( 26_MC )_) we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Arguing_VVG indirectly_RR ,_, we_PPIS2 assume_VV0 that_CST there_EX exists_VVZ some_DD @S_FO such_CS21 that_CS22 @S_FO ._. 
</s>
<s>
Then_RT there_EX exists_VVZ @S_FO so_CS21 that_CS22 @F_FO ._. 
</s>
<s>
By_II (_( 2.12_MC )_) and_CC (_( 2.14_MC )_) ,_, we_PPIS2 then_RT obtain_VV0 @F_FO and_CC hence_RR @S_FO ._. 
</s>
<s>
This_DD1 contradiction_NN1 proves_VVZ the_AT desired_JJ fact_NN1 ._. 
</s>
<s>
While_CS Falkner_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
(_( 1999_MC )_) provide_VV0 some_DD evidence_NN1 that_CST students_NN2 as_RG early_RR as_CSA kindergarten_NN1 already_RR exhibit_VV0 operational_JJ thinking_NN1 ,_, studies_NN2 are_VBR needed_VVN that_CST systematically_RR unpack_VV0 the_AT nature_NN1 of_IO students_NN2 '_NULL thinking_VVG about_II the_AT equal_JJ sign_NN1 at_II this_DD1 early_RR ,_, transitional_JJ stage_NN1 ._. 
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<s>
For_IF higher_JJR k_ZZ1 ,_, using_VVG @S_FO by_II (_( 6.13_MC )_) ,_, we_PPIS2 get_VV0 @F_FO ._. 
</s>
<s>
These_DD2 findings_NN2 serve_VV0 to_TO justify_VVI the_AT use_NN1 of_IO teacher_NN1 noticing_VVG as_II an_AT1 analytic_JJ lens_NN1 by_II teacher-educators_NN2 regardless_RR of_IO grade_NN1 level_NN1 or_CC setting_NN1 ._. 
</s>
<s>
We_PPIS2 will_VM do_VDI this_DD1 by_II31 way_II32 of_II33 two_MC auxiliary_JJ lemmas_NN2 ._. 
</s>
<s>
Choosing_VVG c_ZZ1 β_NULL ,_, k>_FO (_( k+1_FO )_) (_( k+1_FO )_) t2kc_FO β_NULL ,_, ?1c_FO β_NULL ,_, k?+1_FO we_PPIS2 are_VBR done_VDN by_II the_AT second_MD case_NN1 considered_VVN above_RL ._. 
</s>
<s>
Erin_NP1 :_: I_PPIS1 don_VV0 '_NULL t_ZZ1 know_VV0 (_( looks_VVZ at_II the_AT triangle_NN1 in_II the_AT book_NN1 as_CSA she_PPHS1 walks_VVZ )_) ._. 
</s>
<s>
Furthermore_RR ,_, if_CS @S_FO forsome_JJ @S_FO with_IW @S_FO then_RT ,_, forany_NN1 1_MC1 <_FO q_ZZ1 <_FO 2_MC ,_, @F_FO ._. 
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<s>
Also_RR the_AT question_NN1 of_IO scaling_VVG the_AT Gaussian_JJ process_NN1 methodology_NN1 to_II high_JJ dimensional_JJ input_NN1 spaces_NN2 remains_VVZ open_JJ ._. 
</s>
<s>
Since_CS K_ZZ1 is_VBZ a_AT1 Koszul_NN1 twisted_JJ complex_NN1 ,_, @S_FO has_VHZ the_AT @F_FO ,_, This_DD1 differential_NN1 is_VBZ lower_JJR triangular_JJ ,_, hence_RR there_EX is_VBZ a_AT1 two-step_JJ spectral_JJ sequence_NN1 whose_DDQGE E1-page_NN1 is_VBZ @S_FO and_CC which_DDQ converges_VVZ to_II @S_FO ._. 
</s>
<s>
The_AT map_NN1 @S_FO has_VHZ the_AT form_NN1 @F_FO and_CC therefore_RR induces_VVZ a_AT1 map_NN1 of_IO filtered_JJ complexes_NN2 between_II @S_FO and_CC @S_FO ._. 
</s>
<s>
This_DD1 map_NN1 is_VBZ an_AT1 isomorphism_NN1 on_II the_AT Ei-page_NN1 since_CS aij_NN1 is_VBZ a_AT1 quasiisomorphism_NN1 ._. 
</s>
<s>
Therefore_RR ,_, we_PPIS2 can_VM see_VVI the_AT value_NN1 of_IO details_NN2 in_II the_AT mathematical_JJ discourse_NN1 "_" as_II a_AT1 major_JJ learning_NN1 outcome_NN1 in_II its_APPGE own_DA right_NN1 "_" (_( Clarke_NP1 ,_, 2013_MC ,_, p._NN1 22_MC )_) since_CS "_" the_AT more_RGR sensitive_JJ you_PPY are_VBR to_II noticing_VVG details_NN2 ,_, the_AT more_RRR tempted_VVD you_PPY are_VBR likely_JJ to_TO be_VBI to_TO act_VVI responsively_RR "_" (_( Mason_NP1 ,_, 2002_MC ,_, p._NN1 248_MC )_) ._. 
</s>
<s>
For_IF this_DD1 reason_NN1 ,_, we_PPIS2 make_VV0 several_DA2 simplifying_JJ assumptions_NN2 that_CST allow_VV0 us_PPIO2 to_TO focus_VVI on_II the_AT main_JJ ideas_NN2 ._. 
</s>
<s>
Lifting_VVG a_AT1 research_NN1 practice_NN1 means_VVZ that_CST certain_JJ types_NN2 of_IO research_NN1 questions_NN2 and/or_CC methods_NN2 from_II the_AT classroom_NN1 level_NN1 are_VBR implicitly_RR or_CC explicitly_RR transferred_VVN (_( and_CC adapted_VVN )_) to_II the_AT TPD_NP1 level_NN1 (_( or_CC from_II the_AT TPD_NP1 to_II FPD_NP1 level_NN1 )_) and_CC applied_VVN in_II an_AT1 analogous_JJ way_NN1 ._. 
</s>
<s>
It_PPH1 is_VBZ commonly_RR believed_VVN that_CST ,_, for_IF d>2_FO ,_, the_AT spectral_JJ gap_NN1 edges_NN2 are_VBR non-degenerate_JJ for_IF "_" generic_JJ "_" potentials_NN2 ;_; see_VV0 ,_, for_REX21 example_REX22 ,_, &lsqb;_( 18_MC ,_, Conjecture_NN1 5.1_MC &rsqb;_) and_CC the_AT recent_JJ review_NN1 &lsqb;_( 15_MC ,_, §5.9.2_FO &rsqb;_) ._. 
</s>
<s>
For_IF @S_FO ,_, we_PPIS2 shall_VM denote_VVI by_II @F_FO ._. 
</s>
<s>
Then_RT for_IF any_DD @S_FO ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
In_RR21 particular_RR22 ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
In_II the_AT next_MD proposition_NN1 ,_, we_PPIS2 extend_VV0 this_DD1 result_NN1 in_BCL21 order_BCL22 to_TO obtain_VVI a_AT1 velocity_NN1 field_NN1 and_CC a_AT1 solution_NN1 defined_VVN on_II I_ZZ1 x_ZZ1 R2_FO ,_, rather_CS21 than_CS22 on_II Q._NP1 The_AT approach_NN1 described_VVN in_II the_AT present_JJ work_NN1 could_VM be_VBI viewed_VVN as_II a_AT1 variant_NN1 in_II which_DDQ the_AT "_" embedded_JJ "_" method_NN1 is_VBZ simply_RR the_AT identity_NN1 map_NN1 ,_, but_CCB with_IW an_AT1 additional_JJ twist_NN1 that_CST requires_VVZ reinterpretation_NN1 of_IO the_AT new_JJ step_NN1 solution_NN1 as_II an_AT1 approximation_NN1 at_II a_AT1 slightly_RR different_JJ time_NNT1 ._. 
</s>
<s>
On_II the_AT Pace_NN1 car_NN1 task_NN1 ,_, Gabbi_NP1 used_VVD chunky_JJ covariational_JJ reasoning_NN1 (_( "_" increases_VVZ by_II a_AT1 certain_JJ amount_NN1 &lsqb;_( ..._... &rsqb;_) decreases_VVZ by_II the_AT same_DA "_" )_) to_TO create_VVI a_AT1 cogent_JJ argument_NN1 for_IF linearity_NN1 ._. 
</s>
<s>
Finally_RR ,_, Mrs._NNB Purl_NP1 was_VBDZ asked_VVN to_TO describe_VVI the_AT overall_JJ understanding_NN1 of_IO the_AT students_NN2 in_II her_APPGE classroom_NN1 and_CC to_TO discuss_VVI her_APPGE teaching_NN1 plans_NN2 for_IF the_AT next_MD day_NNT1 ._. 
</s>
<s>
If_CS in_II the_AT setting_NN1 of_IO Proposition_NN1 3.2_MC F(s)_NP2 has_VHZ a_AT1 finite_JJ chaos_NN1 expansion_NN1 of_IO length_NN1 n_ZZ1 for_IF all_DB @S_FO (_( see_VV0 section_NN1 3.2_MC for_IF the_AT definition_NN1 )_) ,_, then_RT also_RR E_ZZ1 (_( F_ZZ1 (_( s_ZZ1 )_) )_) has_VHZ a_AT1 chaos_NN1 expansion_NN1 of_IO length_NN1 @S_FO and_CC therefore_RR Gaussian_JJ hypercontractivity_NN1 shows_VVZ that_CST for_IF all_DB @S_FO ,_, @F_FO ._. 
</s>
<s>
In_II the_AT dynamical_JJ argument_NN1 we_PPIS2 will_VM use_VVI spectral_JJ gap_NN1 properties_NN2 and_CC dynamics_NN of_IO a_AT1 unipotent_JJ flow_NN1 ._. 
</s>
<s>
We_PPIS2 also_RR show_VV0 a_AT1 sharp_JJ integral_JJ Abresch-Gromoll_NP1 type_NN1 inequality_NN1 for_IF the_AT excess_JJ function_NN1 and_CC an_AT1 Abresch-Gromoll-type_NP1 inequality_NN1 for_IF the_AT gradient_NN1 of_IO the_AT excess_NN1 ._. 
</s>
<s>
The_AT reason_NN1 for_IF this_DD1 robustness_NN1 is_VBZ again_RT that_CST the_AT "_" divergence-free_JJ "_" property_NN1 of_IO VEM_NP1 yields_VVZ velocity_NN1 errors_NN2 that_CST do_VD0 not_XX depend_VVI directly_RR on_II the_AT pressure_NN1 (_( but_CCB only_RR indirectly_RR through_II the_AT higher_JJR order_NN1 load_NN1 approximation_NN1 term_NN1 ;_; see_VV0 Theorem_NN1 4.6_MC )_) ._. 
</s>
<s>
Moreover_RR ,_, if_CS there_EX is_VBZ a_AT1 global_JJ generic_JJ type_NN1 ,_, then_RT every_AT1 weakly_RR generic_JJ type_NN1 is_VBZ generic_JJ ,_, and_CC the_AT set_NN1 of_IO generic_JJ types_NN2 is_VBZ the_AT unique_JJ minimal_JJ flow_NN1 in_II @S._FO (_( 5_MC )_) A_ZZ1 type_NN1 @S_FO is_VBZ almost_RR periodic_JJ if_CS and_CC only_RR if_CS for_IF every_AT1 @S_FO ,_, the_AT set_NN1 @S_FO is_VBZ covered_VVN by_II finitely_RR many_DA2 left_JJ translates_VVZ of_IO @S_FO ._. 
</s>
<s>
Informally_RR ,_, in_BCL21 order_BCL22 for_IF p_ZZ1 to_TO have_VHI geometric_JJ mixing_NN1 scale_NN1 e_ZZ1 ,_, the_AT average_NN1 of_IO the_AT solution_NN1 on_II every_AT1 ball_NN1 of_IO radius_NN1 e_ZZ1 is_VBZ essentially_RR zero_MC ._. 
</s>
<s>
Note_VV0 that_CST since_CS the_AT integral_JJ vanishes_VVZ in_II zero_MC ,_, its_APPGE supremum_NN1 norm_NN1 can_VM be_VBI controlled_VVN by_II its_APPGE p-variation_NN1 ._. 
</s>
<s>
Proposition_NN1 @S_FO ._. 
</s>
<s>
Let_VV0 @S_FO and_CC set_VVI @S_FO ._. 
</s>
<s>
For_IF any_DD @S_FO ,_, the_AT Lasso_NN1 estimator_NN1 (_( 2.1_MC )_) with_IW tuning_VVG parameter_NN1 @F_FO ._. 
</s>
<s>
An_AT1 oracle_NN1 inequality_NN1 of_IO the_AT same_DA kind_NN1 as_CSA (_( 3.4_MC )_) was_VBDZ first_MD obtained_VVN in_II &lsqb;_( 14_MC &rsqb;_) ,_, Theorem_NN1 6.1_MC ,_, and_CC in_II a_AT1 slightly_RR less_RGR general_JJ form_NN1 ,_, with_IW some_DD factor_NN1 C_ZZ1 >_FO 1_MC1 in_II31 front_II32 of_II33 @S_FO ._. 
</s>
<s>
The_AT numerical_JJ constants_NN2 in_II Proposition_NN1 3.1_MC are_VBR taken_VVN from_II the_AT proof_NN1 of_IO Theorem_NN1 3_MC in_II &lsqb;_( 10_MC &rsqb;_) (_( cf._VV0 The_AT theoretical_JJ developments_NN2 presented_VVN below_RL appear_VV0 to_TO be_VBI generalisable_JJ ,_, at_RR21 least_RR22 in_II principle_NN1 ,_, to_II a_AT1 cut-cell-type_JJ setting_NN1 ,_, whereby_RRQ a_AT1 mesh_NN1 is_VBZ not_XX subordinate_JJ to_II the_AT interface_NN1 location_NN1 a_JJ21 priori_JJ22 ._. 
</s>
<s>
Now_RT ,_, the_AT proof_NN1 of_IO Lemma_NN1 3.12_MC is_VBZ complete_JJ ._. 
</s>
<s>
We_PPIS2 proceed_VV0 as_CSA in_II the_AT proof_NN1 of_IO Lemma_NN1 24_MC :_: for_IF every_AT1 c1_FO >_FO 1_MC1 ,_, we_PPIS2 have_VH0 @S_FO ,_, where_CS we_PPIS2 recall_VV0 that_CST Tn_NP1 is_VBZ a_AT1 uniform_JJ triangulation_NN1 of_IO the_AT sphere_NN1 with_IW n_ZZ1 vertices_VVZ ._. 
</s>
<s>
Then_RT @S_FO is_VBZ in_II @S_FO ._. 
</s>
<s>
Likewise_RR ,_, for_IF every_AT1 k_ZZ1 >_FO 0_MC ,_, the_AT chain_NN1 @S_FO belongs_VVZ to_II @S_FO ._. 
</s>
<s>
We_PPIS2 notice_VV0 that_CST the_AT sequence_NN1 @S_FO is_VBZ the_AT concatenation_NN1 of_IO the_AT chain_NN1 @S_FO ._. 
</s>
<s>
We_PPIS2 denote_VV0 the_AT concatenation_NN1 by_II @F_FO ._. 
</s>
<s>
By_II construction_NN1 ,_, lim_VV0 →_NULL An_AT1 ;_; n_ZZ1 has_VHZ the_AT desired_JJ Elliott_NP1 invariant_JJ (_( the_AT details_NN2 are_VBR as_CSA in_II the_AT unital_JJ case_NN1 ,_, see_VV0 &lsqb;_( 20_MC ,_, §_FO 13_MC &rsqb;_) )_) ._. 
</s>
<s>
Since_CS @S_FO by_II Lemma_NN1 4.2_MC ,_, we_PPIS2 have_VH0 @S_FO in_II D._NP1 We_PPIS2 now_RT want_VV0 to_TO compare_VVI v_ZZ1 and_CC u_ZZ1 on_II the_AT portion_NN1 of_IO d_ZZ1 D_ZZ1 required_JJ to_TO apply_VVI the_AT weak_JJ maximum_JJ principle_NN1 ._. 
</s>
<s>
To_TO obtain_VVI the_AT full_JJ extension_NN1 we_PPIS2 reassemble_VV0 the_AT structure_NN1 ._. 
</s>
<s>
However_RR with_IW only_RR bounded_VVN (_( which_DDQ is_VBZ critical_JJ if_CS we_PPIS2 want_VV0 to_TO apply_VVI this_DD1 to_II the_AT Biot–Savart_NN1 law_NN1 )_) ,_, we_PPIS2 are_VBR not_XX aware_JJ of_IO any_DD existing_JJ results_NN2 in_II the_AT literature_NN1 ._. 
</s>
<s>
The_AT proof_NN1 of_IO this_DD1 theorem_NN1 is_VBZ based_VVN on_II the_AT result_NN1 of_IO Theorem_NN1 7.5_MC ,_, and_CC the_AT two_MC proofs_NN2 have_VH0 a_RR21 lot_RR22 in_II common_JJ ._. 
</s>
<s>
However_RR ,_, this_DD1 event_NN1 is_VBZ definitely_RR not_XX a_AT1 black_JJ swan_NN1 ,_, although_CS it_PPH1 has_VHZ already_RR had_VHN a_AT1 great_JJ impact_NN1 all_RR over_II the_AT world_NN1 ._. 
</s>
<s>
The_AT me_PPIO1 moves_NN2 will_VM sew_VVI an_AT1 edge_NN1 to_II the_AT current_JJ marked_JJ bipolar_JJ map_NN1 upwards_RL from_II the_AT active_JJ vertex_NN1 and_CC move_VV0 the_AT active_JJ vertex_NN1 to_II the_AT upper_JJ endpoint_NN1 of_IO the_AT new_JJ edge_NN1 ._. 
</s>
<s>
Then_RT ,_, the_AT function_NN1 @S_FO is_VBZ of_IO class_NN1 @S_FO on_II @S_FO ,_, and_CC we_PPIS2 have_VH0 ,_, for_IF all_DB @S_FO such_CS21 that_CS22 @S_FO ,_, @F_FO ,_, if_CS @S_FO ,_, If_CS a_AT1 group_NN1 G_ZZ1 acting_VVG freely_RR on_II a_AT1 @S_FO cube_NN1 complex_NN1 @S_FO has_VHZ a_AT1 finite_JJ index_NN1 subgroup_NN1 G_ZZ1 '_NULL such_CS21 that_CS22 @S_FO is_VBZ special_JJ ,_, then_RT we_PPIS2 say_VV0 that_CST the_AT action_NN1 of_IO G_ZZ1 on_II X_ZZ1 is_VBZ virtually_RR special_JJ ._. 
</s>
<s>
The_AT protocol_NN1 provides_VVZ very_RG detailed_JJ and_CC explicit_JJ instructions_NN2 for_IF how_RRQ to_TO score_VVI individual_JJ classroom_NN1 events_NN2 and_CC uses_VVZ a_AT1 1-3_MCMC scale_NN1 to_TO measure_VVI a_AT1 stepwise_JJ progression_NN1 toward_II highly_RR valued_VVN events_NN2 ._. 
</s>
<s>
Then_RT @S_FO ,_, and_CC hence_RR we_PPIS2 have_VH0 for_IF every_AT1 character_NN1 @S_FO an_AT1 equality_NN1 @F_FO ._. 
</s>
<s>
Let_VV0 K_ZZ1 ,_, L_ZZ1 >_FO 2_MC and_CC let_VV0 S_ZZ1 =_FO 2s_MC2 ._. 
</s>
<s>
It_PPH1 is_VBZ thus_RR enough_RR (_( via_II contradiction_NN1 )_) to_TO show_VVI that_CST @S_FO is_VBZ almost_RR zero_MC with_II31 respect_II32 to_II33 (_( pg_NNU )_) 1p∞_FO ._. 
</s>
<s>
The_AT outline_NN1 of_IO the_AT paper_NN1 is_VBZ the_AT following_JJ ._. 
</s>
<s>
There_EX exists_VVZ a_AT1 natural_JJ averaging_NN1 triangulated_VVD functor_NN1 from_II @S_FO to_II @S_FO ,_, defined_VVN as_CSA convolution_NN1 on_II the_AT left_JJ with_IW the_AT object_NN1 AWJ1_FO ._. 
</s>
<s>
Remark_VV0 5.1_MC (_( The_AT origin_NN1 of_IO the_AT condition_NN1 (_( 3.1_MC )_) )_) ._. 
</s>
<s>
In_II the_AT following_JJ analysis_NN1 we_PPIS2 will_VM also_RR require_VVI the_AT following_JJ localized_JJ H2_FO stability_NN1 estimates_NN2 ._. 
</s>
<s>
We_PPIS2 see_VV0 a_AT1 clear_JJ clustering_NN1 structure_NN1 in_II at_RR21 least_RR22 the_AT first_MD eight_MC principal_JJ components_NN2 ._. 
</s>
<s>
A_AT1 highly_RR nontrivial_JJ step_NN1 is_VBZ to_TO obtain_VVI an_AT1 appropriate_JJ formulation_NN1 of_IO a_AT1 Poincare_NN1 inequality_NN1 adapted_VVN to_II the_AT Lie_NN1 group_NN1 action_NN1 related_VVN to_II the_AT equation_NN1 ._. 
</s>
<s>
A_AT1 version_NN1 of_IO Theorem_NN1 1.3_MC for_IF M_ZZ1 =_FO 0_MC was_VBDZ proved_VVN independently_RR by_II W._NP1 Sun_NN1 in_II Sun_NN1 (_( 2016_MC )_) using_VVG Temperley_NP1 '_NULL s_ZZ1 bijection_NN1 (_( Kenyon_NP1 ,_, Propp_NP1 and_CC Wilson_NP1 (_( 2000_MC )_) )_) between_II dimers_NN2 and_CC trees_NN2 ._. 
</s>
<s>
Therefore_RR the_AT continuous_JJ isomorphism_NN1 @S_FO is_VBZ an_AT1 isomorphism_NN1 of_IO topological_JJ groups_NN2 by_II Baire_NP1 ,_, since_CS @S_FO is_VBZ second_MD countable_JJ ._. 
</s>
<s>
Assume_VV0 that_DD1 VD_NN1 ,_, (_( 1.5_MC )_) ,_, (_( 1.6_MC )_) ,_, @S_FO hold_VV0 ._. 
</s>
<s>
For_REX21 instance_REX22 ,_, a_AT1 basic_JJ model_NN1 introducing_VVG the_AT three_MC effects_NN2 combines_VVZ the_AT repulsive-attractive_JJ forces_NN2 modeled_VVD by_II an_AT1 effective_JJ pairwise_RR interaction_NN1 potential_NN1 '_NULL (_( x_ZZ1 )_) and_CC the_AT gregariousness_NN1 behavior_NN1 of_IO individuals_NN2 by_II locally_RR averaging_VVG their_APPGE relative_JJ velocities_NN2 with_IW weights_NN2 depending_II21 on_II22 the_AT interindividual_JJ distances_NN2 (_( see_VV0 &lsqb;_( 12_MC ,_, 3_MC &rsqb;_) for_REX21 instance_REX22 )_) ._. 
</s>
<s>
Then_RT @F_FO ,_, For_IF the_AT first_MD term_NN1 in_II the_AT right_JJ hand_NN1 side_NN1 above_RL we_PPIS2 have_VH0 @F_FO ,_, by_II (_( 3.16_MC )_) ,_, (_( 3.17_MC )_) ,_, and_CC (_( 3.21_MC )_) (_( that_DD1 control_NN1 @S_FO ,_, see_VV0 the_AT first_MD inequality_NN1 in_II (_( 3.17_MC )_) ,_, and_CC then_RT @S_FO ._. 
</s>
<s>
As_II21 for_II22 the_AT second_MD term_NN1 in_II the_AT right_JJ hand_NN1 side_NN1 of_IO (_( 3.23_MC )_) ,_, it_PPH1 holds_VVZ that_CST @F.by_FO (_( 3.16_MC )_) ._. 
</s>
<s>
Participants_NN2 '_NULL post-test_JJ MCK_NP1 increased_VVD 1.153_MC for_IF each_DD1 mark_NN1 of_IO the_AT pre-test_NN1 MCK_NP1 ._. 
</s>
<s>
Before_II proceeding_VVG to_II the_AT discretization_NN1 ,_, it_PPH1 is_VBZ important_JJ to_TO analyze_VVI the_AT variational_JJ problem_NN1 (_( 2.23_MC )_) in_II the_AT continuous_JJ sense_NN1 ._. 
</s>
<s>
Intentionally_RR misleading_JJ problems_NN2 helped_VVD PSTs_NP1 see_VVI consequences_NN2 of_IO their_APPGE mathematical_JJ habits_NN2 and_CC highlighted_VVD the_AT importance_NN1 of_IO sense_NN1 making_VVG and_CC precision_NN1 when_CS creating_VVG problem_NN1 models_NN2 ._. 
</s>
<s>
Because_CS the_AT PSTs_NP1 did_VDD not_XX know_VVI what_DDQ the_AT students_NN2 learned_VVN in_II their_APPGE regular_JJ classroom_NN1 ,_, they_PPHS2 were_VBDR unsure_JJ about_II students_NN2 '_NULL knowledge_NN1 of_IO algebra_NN1 and_CC fractions_NN2 while_CS implementing_VVG the_AT first_MD tasks_NN2 of_IO those_DD2 content_JJ areas_NN2 ._. 
</s>
<s>
The_AT fourth_MD author_NN1 was_VBDZ partially_RR supported_VVN by_II the_AT DFG_NP1 grants_VVZ :_: NO_AT 1175/1-1_MCMC and_CC SFB_NP1 1085_MC -_- Higher_JJR Invariants_NN2 ,_, Regensburg_NP1 ._. 
</s>
<s>
Each_DD1 group_NN1 goes_VVZ around_RP to_II other_JJ groups_NN2 to_TO present_VVI to_II others_NN2 and_CC to_TO solicit_VVI comments_NN2 ._. 
</s>
<s>
We_PPIS2 use_VV0 the_AT method_NN1 of_IO characteristics_NN2 to_TO provide_VVI insight_NN1 into_II the_AT fascinating_JJ process_NN1 of_IO wave_NN1 breaking_VVG ._. 
</s>
<s>
Brennan_NP1 and_CC Resnick_VV0 also_RR identify_VV0 seven_MC programming_NN1 concepts_NN2 typically_RR used_VVN in_II Scratch_NN1 projects_NN2 ,_, but_CCB that_DD1 they_PPHS2 argue_VV0 are_VBR relevant_JJ to_II other_JJ programming_NN1 contexts_NN2 :_: sequences_NN2 ,_, loops_NN2 ,_, parallelism_NN1 ,_, events_NN2 ,_, conditionals_NN2 ,_, operators_NN2 ,_, and_CC data_NN ._. 
</s>
<s>
We_PPIS2 first_MD claim_NN1 that_CST for_IF every_AT1 n_ZZ1 >_FO 0_MC ,_, and_CC e_ZZ1 >_FO 0_MC there_EX is_VBZ @S_FO such_CS21 that_CS22 @F_FO ,_, which_DDQ implies_VVZ that_CST for_IF any_DD @S_FO there_EX is_VBZ @S_FO such_CS21 that_CS22 (_( 1.10_MC )_) holds_VVZ ._. 
</s>
<s>
Here_RL ,_, Dens(Rn)_NP1 denotes_VVZ the_AT 1-dimensional_JJ space_NN1 of_IO densities_NN2 on_II Rn_NP1 ._. 
</s>
<s>
Future_JJ studies_NN2 could_VM investigate_VVI other_JJ criteria_NN2 regarding_II this_DD1 SMN_NP1 such_DA as_CSA realistic_JJ ,_, complex_JJ ,_, or_CC original_JJ because_CS posing_VVG such_DA problems_NN2 are_VBR crucial_JJ for_IF gifted_JJ and_CC talented_JJ classrooms_NN2 ._. 
</s>
<s>
The_AT pay-out_NN1 is_VBZ not_XX tightly_RR determined_VVN by_II the_AT testing_NN1 level_NN1 a_AT1 ._. 
</s>
<s>
For_REX21 instance_REX22 if_CS N_ZZ1 ≡_NULL ,_, then_RT M_ZZ1 ?_NULL 2_MC does_VDZ not_XX depend_VVI on_II 1NdN0Nlog0N_FO ,_, divFL∞_FO and_CC log_NN1 ?_NULL W1_FO ,_, ∞_FO ._. 
</s>
<s>
See_VV0 the_AT proof_NN1 of_IO Theorem_NN1 2_MC in_II Sect._NP1 2.7_MC for_IF more_DAR details_NN2 ._. 
</s>
<s>
The_AT convergence_NN1 properties_NN2 of_IO @S_FO and_CC @S_FO toward_II @S_FO and_CC @S_FO ,_, together_RL with_IW the_AT convergence_NN1 @S_FO in_II @S_FO stated_VVN in_II (_( 2.10_MC )_) ,_, enable_VV0 us_PPIO2 to_TO take_VVI the_AT limit@S_FO of_IO the_AT scheme_NN1 to_TO see_VVI that_CST u_ZZ1 is_VBZ a_AT1 solution_NN1 to_II (_( 1.2_MC )_) ._. 
</s>
<s>
Note_VV0 that_CST by_II the_AT choice_NN1 of_IO ,_, the_AT above_JJ also_RR holds_VVZ for_IF 1_MC1 ._. 
</s>
<s>
For_IF a_AT1 more_RGR general_JJ case_NN1 @S_FO ,_, as_CS31 long_CS32 as_CS33 @S_FO <_FO M_ZZ1 with_IW some_DD absolute_JJ constant_JJ M_ZZ1 >_FO 0_MC ,_, the_AT result_NN1 remains_VVZ valid_JJ ._. 
</s>
<s>
Section_NN1 2_MC provides_VVZ preliminary_JJ definitions_NN2 ,_, notations_NN2 ,_, and_CC well-known_JJ results_NN2 concerning_II orthogonal_JJ polynomials_NN2 ,_, and_CC Gauss_NP1 and_CC anti-Gauss_JJ quadrature_NN1 formulae_NN2 ._. 
</s>
<s>
On_II the_AT other_JJ hand_NN1 ,_, @F_FO ,_, and_CC hence_RR h0_FO (_( X_ZZ1 ,_, S2E_FO )_) 0_MC ._. 
</s>
<s>
Since_CS local_JJ quadratic_JJ approximation_NN1 is_VBZ applied_VVN in_II the_AT algorithms_NN2 ,_, the_AT convexity_NN1 requirements_NN2 of_IO the_AT results_NN2 in_II Sections_NN2 2_MC and_CC 3_MC are_VBR met_VVN ._. 
</s>
<s>
The_AT converge_VV0 convergence_NN1 of_IO and_CC equalities_NN2 are_VBR easy_JJ consequences_NN2 of_IO the_AT above_JJ estimates_NN2 (_( 6.3_MC )_) ._. 
</s>
<s>
By_II the_AT limiting_JJ argument_NN1 as_CSA above_RL f_ZZ1 has_VHZ a_AT1 log-subharmonic_JJ representative_NN1 as_RR21 well_RR22 ._. 
</s>
<s>
This_DD1 completes_VVZ the_AT proof_NN1 of_IO the_AT lemma_NN1 ._. 
</s>
<s>
For_IF n=4_FO ,_, the_AT conjecture_NN1 is_VBZ that_CST (_( 1.2_MC )_) holds_VVZ for_IF p>22_FO ._. 
</s>
<s>
Thus_RR ,_, the_AT observation_NN1 of_IO eye_NN1 movements_NN2 can_VM offer_VVI insights_NN2 into_II cognitive_JJ processing_NN1 (_( Holmqvist_NN1 et_RA21 al._RA22 ,_, 2011_MC ;_; Just_RR &;_NULL Carpenter_NP1 ,_, 1976_MC )_) ._. 
</s>
<s>
Note_VV0 that_CST this_DD1 type_NN1 of_IO computing_NN1 device_NN1 is_VBZ similar_JJ to_II a_AT1 Turing_JJ machine_NN1 ,_, except_II21 for_II22 the_AT presence_NN1 of_IO a_AT1 tape_NN1 ._. 
</s>
<s>
In_II that_DD1 paper_NN1 it_PPH1 is_VBZ crucial_JJ that_CST the_AT surface_NN1 is_VBZ given_VVN as_II the_AT zero_NN1 level_NN1 of_IO a_AT1 smooth_JJ signed_JJ distance_NN1 function_NN1 which_DDQ is_VBZ explicitly_RR known_VVN ._. 
</s>
<s>
This_DD1 again_RT suggests_VVZ that_CST sharing_JJ authority_NN1 is_VBZ a_AT1 generative_JJ practice_NN1 in_II increasing_JJ access_NN1 to_II both_RR recognition_NN1 and_CC realization_NN1 rules_NN2 ._. 
</s>
<s>
Such_DA all-purpose_JJ preparation_NN1 has_VHZ led_VVN to_II a_AT1 corpus_NN1 of_IO elementary_JJ teachers_NN2 needing_VVG improved_JJ knowledge_NN1 and_CC capabilities_NN2 for_IF effectively_RR teaching_VVG mathematics_NN1 with_IW understanding_NN1 and_CC proficiency_NN1 in_II mathematical_JJ practices_NN2 at_II the_AT level_NN1 of_IO rigor_NN1 and_CC depth_NN1 depicted_VVN by_II the_AT Common_JJ Core_NN1 State_NN1 Standards_NN2 for_IF Mathematics_NN1 (_( CCSS-M_NNU ,_, NGACBP_NP1 and_CC CCSSO_NP1 2010_MC )_) ._. 
</s>
<s>
Chebyshev_NP1 expansions_NN2 are_VBR used_VVN because_II21 of_II22 their_APPGE near-optimal_JJ approximation_NN1 properties_NN2 and_CC associated_VVN fast_RR transforms_VVZ &lsqb;_( 11_MC ,_, 23_MC ,_, 38_MC &rsqb;_) and_CC Legendre_NN1 expansions_NN2 for_IF their_APPGE L2_FO orthogonality_NN1 &lsqb;_( 28_MC ,_, Table_NN1 18.3.1_MC &rsqb;_) as_II31 well_II32 as_II33 other_JJ recurrence_NN1 relations_NN2 that_CST they_PPHS2 satisfy_VV0 &lsqb;_( 15_MC &rsqb;_) ._. 
</s>
<s>
Similarly_RR ,_, Kleickmann_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
(_( 2015_MC )_) reported_VVD that_CST secondary_JJ mathematics_NN1 teachers_NN2 from_II Taiwan_NP1 outscored_VVD secondary_JJ mathematics_NN1 teachers_NN2 from_II Germany_NP1 on_II the_AT aspects_NN2 of_IO MCK_NP1 and_CC MPCK_NP1 as_RR21 well_RR22 ._. 
</s>
<s>
The_AT error_NN1 term_NN1 relies_VVZ on_II the_AT high-fidelity_JJ model_NN1 and_CC is_VBZ used_VVN to_TO correct_VVI for_IF the_AT error_NN1 introduced_VVN by_II the_AT low-fidelity_JJ model_NN1 ._. 
</s>
<s>
We_PPIS2 use_VV0 the_AT map_NN1 @F_FO ,_, to_TO identify_VVI the_AT elements_NN2 of_IO Z/_FU (_( n_ZZ1 +_FO 1_MC1 )_) with_IW the_AT set_NN1 of_IO (_( n_ZZ1 +_FO 1_MC1 )_) st_NNU roots_NN2 of_IO unity_NN1 contained_VVN in_II the_AT unit_NN1 circle_NN1 @S_FO ._. 
</s>
<s>
A_AT1 map_NN1 @S_FO in_II A_ZZ1 is_VBZ given_VVN by_II a_AT1 homotopy_NN1 class_NN1 of_IO continuous_JJ monotone_NN1 maps_NN2 @S_FO of_IO degree_NN1 1_MC1 ,_, mapping_VVG Z/_FU (_( m_ZZ1 +1_MC )_) into_II Z/_FU (_( n_ZZ1 +1_MC )_) ._. 
</s>
<s>
Briefly_RR ,_, the_AT reason_NN1 is_VBZ that_CST expectations_NN2 of_IO the_AT type_NN1 (_( 2.13_MC )_) are_VBR analytic_JJ in_II s_ZZ1 over_II a_AT1 region_NN1 determined_VVN by_II the_AT tail_NN1 asymptotics_NN1 of_IO the_AT random_JJ variable_NN1 @S_FO ,_, which_DDQ is_VBZ in_II turn_NN1 completely_RR determined_VVN by_II the_AT behavior_NN1 of_IO this_DD1 integral_JJ close_JJ to_II the_AT "_" worst_JJT "_" singularity_NN1 of_IO @S_FO ._. 
</s>
<s>
The_AT reflection_NN1 coefficient_NN1 enters_VVZ in_II the_AT description_NN1 of_IO the_AT tail_NN1 of_IO such_DA random_JJ variables_NN2 ._. 
</s>
<s>
Eq_NN1 ._. 
</s>
<s>
(_( 2.19_MC )_) does_VDZ not_XX fully_RR characterize_VVI the_AT error_NN1 '_NULL s_ZZ1 dependence_NN1 on_II X1_FO ,_, X2_FO ._. 
</s>
<s>
Note_VV0 that_CST the_AT values_NN2 of_IO uncertainty_NN1 are_VBR relative_II21 to_II22 the_AT nominal_JJ arc_NN1 length_NN1 ._. 
</s>
<s>
The_AT sessions_NNT2 took_VVD place_NN1 under_II laboratory_NN1 conditions_NN2 in_II rooms_NN2 that_CST were_VBDR specially_RR set_VVN aside_RL in_II the_AT school_NN1 ._. 
</s>
<s>
This_DD1 fact_NN1 goes_VVZ back_RP at_RR21 least_RR22 to_II &lsqb;_( 5_MC &rsqb;_) ;_; we_PPIS2 provide_VV0 details_NN2 for_IF completeness_NN1 ._. 
</s>
<s>
Observing_VVG these_DD2 four_MC strands_NN2 ,_, we_PPIS2 are_VBR only_RR missing_VVG the_AT pieces_NN2 in_II @S_FO ,_, so_CS21 that_CS22 we_PPIS2 can_VM already_RR see_VVI what_DDQ happened_VVD near_RL v._II In_RR21 particular_RR22 ,_, we_PPIS2 can_VM seemodulo_NN1 whether_CSW the_AT paths_NN2 y1_FO ,_, ..._... ,_, y4_FO hook_VV0 up_RP in_II the_AT right_JJ way_NN1 near_II uif_NN1 Ek(v)_NN1 can_VM hold_VVI or_CC not_XX ._. 
</s>
<s>
The_AT cardinality_NN1 of_IO the_AT automorphism_NN1 group_NN1 is_VBZ an_AT1 upper_JJ semi-continuous_JJ function_NN1 on_II the_AT compact_JJ moduli_NN2 space_NN1 ._. 
</s>
<s>
We_PPIS2 present_VV0 one_PN1 more_RGR technical_JJ lemma_NN1 we_PPIS2 will_VM need_VVI ._. 
</s>
<s>
We_PPIS2 will_VM use_VVI also_RR the_AT Wasserstein_NP1 distances_NN2 @S_FO ,_, with_IW @S_FO ,_, between_II two_MC probabiliy_JJ measures_NN2 @S_FO ,_, which_DDQ are_VBR defined_VVN as_CSA @F_FO ,_, being_NN1 @S_FO the_AT collection_NN1 of_IO all_DB probability_NN1 measures_NN2 on_II @S_FO with_IW marginal_JJ measures_NN2 v_ZZ1 and_CC @S_FO on_II the_AT first_MD and_CC second_MD factor_NN1 ,_, rspectively_RR ._. 
</s>
<s>
Our_APPGE design_NN1 ideas_NN2 came_VVD from_II three_MC sources_NN2 :_: the_AT work_NN1 of_IO mathematician_NN1 Peter_NP1 Taylor_NP1 of_IO Queen_NN1 '_NULL s_ZZ1 University_NN1 ,_, Canada_NP1 ,_, who_PNQS has_VHZ been_VBN using_VVG transformations_NN2 in_II his_APPGE work_NN1 with_IW grades_NN2 10-11_MCMC students_NN2 (_( )_) ;_; from_II NRICH_NP1 online_JJ resources_NN2 offered_VVN by_II the_AT University_NN1 of_IO Cambridge_NP1 ,_, United_NP1 Kingdom_NP1 :_: specifically_RR ,_, the_AT transformations_NN2 activity_NN1 available_JJ Figure_NN1 4_MC ._. 
</s>
<s>
On_II31 top_II32 of_II33 this_DD1 ,_, we_PPIS2 have_VH0 shown_VVN an_AT1 additional_JJ two_MC orders-of-magnitude_NN2 improvement_NN1 going_VVG from_II EISPACK_NN1 to_II PLASMA_NN1 (_( 146_MC Gflop/s_FU )_) on_II a_AT1 multicore_NN1 architecture_NN1 ,_, and_CC four_MC orders_NN2 of_IO magnitude_NN1 to_II DPLASMA_NP1 (_( 6.8_MC Tflop/s_FU )_) on_II a_AT1 distributed-memory_JJ machinewhile_NN1 moving_VVG from_II solving_VVG systems_NN2 of_IO dimension_NN1 100_MC to_II over_II 100,000yielding_FO over_RG six_MC orders-of-magnitude_NN2 performance_NN1 improvement_NN1 in_II 40_MC years_NNT2 ._. 
</s>
<s>
For_IF all_DB a_AT1 ,_, b_ZZ1 >_FO 2_MC the_AT graph_NN1 @S_FO is_VBZ claw-contractible-free_JJ ._. 
</s>
<s>
There_EX should_VM be_VBI no_AT corners_NN2 ,_, it_PPH1 should_VM continue_VVI smoothly_RR ..._... 
</s>
<s>
As_II a_AT1 reference_NN1 ,_, we_PPIS2 also_RR give_VV0 the_AT errors_NN2 for_IF the_AT implicit_JJ Euler_NN1 (_( IE_REX )_) scheme_NN1 ._. 
</s>
<s>
Implicit_JJ in_II this_DD1 extension_NN1 was_VBDZ the_AT observation_NN1 that_CST any_DD algorithm_NN1 that_CST can_VM efficiently_RR compute_VVI averages_NN2 with_II31 respect_II32 to_II33 the_AT stationary_JJ distribution_NN1 of_IO a_AT1 time-homogeneous_JJ Markov_NP1 process_NN1 can_VM be_VBI applied_VVN to_II computing_VVG dynamic_JJ averages_NN2 more_RGR generally_RR by_II an_AT1 enlargement_NN1 of_IO the_AT state_NN1 space_NN1 ,_, i.e._REX ,_, by_II applying_VVG the_AT scheme_NN1 to_II computing_VVG stationary_JJ averages_NN2 for_IF a_AT1 higher_JJR dimensional_JJ time-homogeneous_JJ Markov_NP1 process_NN1 ._. 
</s>
<s>
As_II a_AT1 matter_JJ31 of_JJ32 fact_JJ33 ,_, by_II self-similarity_NN1 ,_, it_PPH1 is_VBZ enough_DD to_TO deal_VVI with_IW all_DB x_ZZ1 in_II some_DD nonempty_JJ open_JJ set_NN1 ,_, but_CCB it_PPH1 is_VBZ not_XX enough_RR to_TO gain_VVI information_NN1 for_IF almost_RR all_DB values_NN2 of_IO x_ZZ1 ._. 
</s>
<s>
Note_VV0 that_CST the_AT underlying_JJ dynamical_JJ system_NN1 (_( X_ZZ1 ,_, T_ZZ1 )_) is_VBZ an_AT1 irrational_JJ rotation_NN1 on_II the_AT circle_NN1 (_( thanks_NN2 to_II p_ZZ1 and_CC q_ZZ1 being_VBG multiplicatively_RR independent_JJ )_) while_CS ,_, in_II the_AT case_NN1 of_IO Bernoulli_JJ convolutions_NN2 ,_, (_( X_ZZ1 ,_, T_ZZ1 )_) is_VBZ the_AT trivial_JJ one-point_JJ system_NN1 ._. 
</s>
<s>
Let_VV0 H_ZZ1 denote_VVI the_AT manifold_NN1 obtained_VVN from_II @S_FO by_II gluing_VVG @S_FO to_II dW1t1_FO along_II an_AT1 orientation_NN1 preserving_NN1 embedding_NN1 @F_FO ,_, which_DDQ we_PPIS2 also_RR choose_VV0 once_RR41 and_RR42 for_RR43 all_RR44 ._. 
</s>
<s>
From_II here_RL the_AT general_JJ strategy_NN1 is_VBZ to_TO build_VVI a_AT1 homotopy_NN1 by_II induction_NN1 on_II the_AT skeleta_NN1 of_IO @S_FO with_IW a_AT1 product_NN1 cell_NN1 structure_NN1 ._. 
</s>
<s>
The_AT last_MD condition_NN1 means_VVZ that_CST @F_FO ,_, where_CS I_ZZ1 is_VBZ a_AT1 finite_JJ set_NN1 ,_, di∈Z≥1_FO and_CC Di_NP1 are_VBR smooth_JJ irreducible_JJ divisors_NN2 in_II X_ZZ1 ,_, with_IW transversal_NN1 intersections_NN2 ._. 
</s>
<s>
Thus_RR we_PPIS2 have_VH0 the_AT next_MD definition_NN1 which_DDQ enables_VVZ us_PPIO2 to_TO keep_VVI track_NN1 of_IO how_RRQ to_TO attach_VVI tubes_NN2 to_II pairs_NN2 of_IO circles_NN2 and_CC more_RGR generally_RR to_TO attach_VVI pairs_NN2 of_IO tubes_NN2 to_II pairs_NN2 of_IO Hopf_NP1 bands_VVZ ._. 
</s>
<s>
Assume_VV0 that_CST in_II the_AT local_JJ case_NN1 (_( under_II assumptions_NN2 (_( 5.1_MC )_) or_CC (_( 6.1_MC )_) )_) the_AT map_NN1 @S_FO is_VBZ increasing_VVG for_IF all_DB @S_FO or_CC in_II the_AT nonlocal_JJ case_NN1 (_( under_II assumption_NN1 (_( 4.3_MC )_) )_) @F_FO ._. 
</s>
<s>
The_AT rate_NN1 of_IO @S_FO in_II theorem_NN1 2_MC is_VBZ due_II21 to_II22 the_AT recent_JJ results_NN2 that_CST were_VBDR established_VVN by_II Wegkamp_NP1 and_CC Zhao_NP1 (_( 2016_MC )_) and_CC Han_NP1 and_CC Liu_NP1 (_( 2017_MC )_) for_IF the_AT convergence_NN1 in_II spectral_JJ norm_NN1 of_IO the_AT modified_JJ Kendall_NP1 correlation_NN1 matrix_NN1 ._. 
</s>
<s>
Now_RT let_VV0 ?_ZZ1 be_VBI the_AT largest_JJT integer_NN1 such_CS21 that_CS22 ?<0_FO and_CC observe_VV0 that_CST C_ZZ1 is_VBZ ,_, in_RR21 particular_RR22 ,_, a_AT1 @S-code_FO ._. 
</s>
<s>
The_AT task_NN1 itself_PPX1 is_VBZ unsolvable_JJ unless_CS modeled_VVN with_IW assumptions_NN2 incompatible_JJ with_IW experience_NN1 ._. 
</s>
<s>
We_PPIS2 set_VV0 @S_FO ,_, and_CC the_AT rest_NN1 of_IO the_AT parameters_NN2 as_CSA in_II the_AT above_JJ section_NN1 ._. 
</s>
<s>
Fix_VV0 @S_FO and_CC @S_FO ._. 
</s>
<s>
For_IF every_AT1 @S_FO there_EX exists_VVZ @S_FO such_CS21 that_CS22 if_CS :_: (_( i_ZZ1 )_) (_( X_ZZ1 ,_, d_ZZ1 ,_, m_ZZ1 )_) is_VBZ an_AT1 @S-space_FO ,_, (_( ii_MC )_) there_RL exist_VV0 points_NN2 @S_FO of_IO X_ZZ1 ,_, for_IF some_DD k_ZZ1 <_FO N_ZZ1 ,_, such_CS21 that_CS22 @S_FO ,_, and_CC for_IF every_AT1 @S_FO ,_, @F_FO ,_, then_RT there_EX exists_VVZ a_AT1 p.m.m.s_NNU ._. 
</s>
<s>
Section_NN1 2_MC proves_VVZ several_DA2 formulas_NN2 for_IF efficient_JJ calculation_NN1 of_IO px(X)_NNU ._. 
</s>
<s>
Therefore_RR ,_, by_II applying_VVG the_AT conclusion_NN1 of_IO Lemma_NN1 3.2_MC to_II e_ZZ1 ,_, @F_FO ._. 
</s>
<s>
This_DD1 implies_VVZ that_CST @S_FO is_VBZ right-continuous_JJ at_II q_ZZ1 ._. 
</s>
<s>
From_II the_AT continuity_NN1 of_IO @S_FO it_PPH1 follows_VVZ that_CST there_EX is_VBZ @S_FO such_CS21 that_CS22 @F_FO ._. 
</s>
<s>
Let_VV0 t_ZZ1 >_FO 0_MC and_CC let_VV0 q_ZZ1 be_VBI the_AT initial_JJ value_NN1 q0_FO of_IO @S_FO ._. 
</s>
<s>
From_II (_( 3.7_MC )_) we_PPIS2 get_VV0 @F_FO ._. 
</s>
<s>
The_AT study_NN1 of_IO the_AT quasilinear_JJ ,_, degenerate_JJ case_NN1 has_VHZ been_VBN started_VVN in_II the_AT fundamental_JJ papers_NN2 &lsqb;_( 3,4_MC &rsqb;_) ofBoccardo_NN1 and_CC Gallouet_NP1 ,_, where_CS existence_NN1 and_CC regularity_NN1 estimates_NN2 have_VH0 been_VBN proved_VVN ._. 
</s>
<s>
This_DD1 multi-crowd_JJ scenario_NN1 is_VBZ treated_VVN as_II a_AT1 mean-field_JJ type_NN1 game_NN1 and_CC is_VBZ linked_VVN to_II an_AT1 optimal_JJ control_NN1 problem_NN1 ,_, for_IF which_DDQ we_PPIS2 prove_VV0 a_AT1 sufficient_JJ maximum_JJ principle_NN1 ._. 
</s>
<s>
Suppose_VV0 that_CST the_AT maximum_NN1 of_IO m_ZZ1 (_( S_ZZ1 (_( X_ZZ1 )_) )_) is_VBZ attained_VVN at_II @F_FO ,_, in_II a_AT1 position_NN1 belonging_VVG to_II the_AT sequence_NN1 70_MC ._. 
</s>
<s>
As_CSA shown_VVN in_II Figure_NN1 2_MC ,_, the_AT student_NN1 was_VBDZ presented_VVN with_IW a_AT1 twodimensional_JJ number_NN1 line_NN1 and_CC told_VVD ,_, for_REX21 example_REX22 ,_, "_" On_II the_AT number-line_NN1 below_RL ,_, mark_VV0 an_AT1 X_ZZ1 or_CC draw_VV0 a_AT1 dot_NN1 to_TO show_VVI where_RRQ @S_FO is_VBZ ._. "_" 
</s>
<s>
The_AT RO-DDU_JJ problem_NN1 performs_VVZ better_JJR than_CSN SO-DDU_JJ for_IF the_AT worst-case_JJ scenario_NN1 ._. 
</s>
<s>
Let_VV0 H_ZZ1 be_VBI a_AT1 basic_JJ subgroup_NN1 of_IO G._NP1 Since_CS H_ZZ1 is_VBZ not_XX abelian_JJ ,_, the_AT support_NN1 of_IO H_ZZ1 is_VBZ clearly_RR not_XX empty_JJ and_CC not_XX an_AT1 annulus_NN1 ._. 
</s>
<s>
Knot_NN1 positions_NN2 are_VBR visualized_VVN by_II vertical_JJ dotted_JJ lines_NN2 ._. 
</s>
<s>
We_PPIS2 could_VM replace_VVI in_II (_( 27_MC )_) the_AT @S_FO integral_JJ on_II the_AT boundary_NN1 with_IW a_AT1 piecewise_JJ GauB-Lobatto_NP1 combination_NN1 ,_, mapping_VVG each_DD1 edge_NN1 on_II the_AT reference_NN1 interval_NN1 @S_FO and_CC using_VVG (_( 40_MC )_) ;_; the_AT advantage_NN1 of_IO such_DA a_AT1 choice_NN1 is_VBZ that_CST we_PPIS2 can_VM automatically_RR use_VVI the_AT nodal_JJ degrees_NN2 of_IO freedom_NN1 on_II the_AT skeleton_NN1 ,_, assuming_VVG that_CST they_PPHS2 have_VH0 a_AT1 GauB-Lobatto_NP1 distribution_NN1 on_II each_DD1 edge_NN1 ._. 
</s>
<s>
This_DD1 paper_NN1 is_VBZ essentially_RR selfcontained_VVN and_CC does_VDZ not_XX rely_VVI on_II general_JJ results_NN2 from_II paradifferential_JJ calculus_NN1 ._. 
</s>
<s>
On_II the_AT other_JJ hand_NN1 ,_, Claim_VV0 3(A)_FO and_CC (_( 6.8_MC )_) imply_VV0 that_DD1 xi_NN1 ,_, ri_NN2 (_( 2Bri/2_FU (_( zi_NN2 )_) )_) converges_VVZ in_II the_AT Hausdorff_NN1 topology_NN1 to_II a_AT1 domain_NN1 in_II Tx1_FO ._. 
</s>
<s>
But_CCB the_AT action_NN1 of_IO h_ZZ1 (_( 0_MC )_) is_VBZ not_XX diagonalizable_JJ ._. 
</s>
<s>
To_TO conclude_VVI ,_, we_PPIS2 need_VV0 to_TO remark_VVI that_CST the_AT sum_NN1 pe_NN1 &K;_NULL c_ZZ1 '_NULL fnr6cc_NNU (_( p_ZZ1 ,_, a_ZZ1 )_) is_VBZ bounded_VVN independently_RR of_IO n_ZZ1 ;_; indeed_RR ,_, each_DD1 term_NN1 c?1t.nrocc_NNU (_( p_ZZ1 ,_, a_ZZ1 )_) is_VBZ bounded_VVN by_II 1_MC1 ,_, and_CC there_EX are_VBR K_ZZ1 !_! 
</s>
<s>
It_PPH1 is_VBZ clear_JJ from_II this_DD1 that_CST when_CS frame_NN1 elements_NN2 are_VBR close_JJ to_II being_VBG linearly_RR dependent_JJ ,_, the_AT constant_JJ AN_AT1 can_VM be_VBI quite_RG small_JJ ._. 
</s>
<s>
An_AT1 unconditional_JJ or_CC marginal_JJ point_NN1 of_IO view_NN1 is_VBZ also_RR possible_JJ ,_, which_DDQ we_PPIS2 now_RT describe_VV0 ._. 
</s>
<s>
Proving_VVG this_DD1 key_JJ proposition_NN1 amounts_VVZ to_II showing_VVG that_CST the_AT restriction_NN1 map_NN1 @F_FO is_VBZ surjective_JJ ._. 
</s>
<s>
We_PPIS2 define_VV0 @F_FO ._. 
</s>
<s>
For_IF each_DD1 ∈J_FO β_NULL and_CC a=1_FO ,_, ,_, n1_FO ,_, there_EX exists_VVZ a_AT1 Uq(g)-module_JJ homomorphism_NN1 @F_FO which_DDQ is_VBZ given_VVN by_II @F_FO for_IF vk∈_FO (_( VS(k)_NP1 )_) aff_NN1 (_( 1≤k≤n_FO )_) ._. 
</s>
<s>
Since_CS the_AT xj_NNU -arcs_NN2 fill_VV0 Qx_NP1 and_CC since_CS the_AT xj_NNU are_VBR disjoint_JJ from_II the_AT yj_NN1 it_PPH1 follows_VVZ from_II the_AT previous_JJ paragraph_NN1 that_CST if_CS we_PPIS2 take_VV0 the_AT y_ZZ1 ,_, -arcs_NN2 in_II R_ZZ1 and_CC intersect_VV0 them_PPHO2 all_DB with_IW Qx_NP1 we_PPIS2 obtain_VV0 a_AT1 set_NN1 of_IO four_MC parallel_JJ arcs_NN2 connecting_VVG the_AT marked_JJ point_NN1 to_II the_AT boundary_NN1 ._. 
</s>
<s>
Hine_NP1 2015_MC ;_; Kotzee_NP1 2012_MC ;_; Meyers_NP2 2012_MC ;_; Muller_NP1 2000_MC ;_; Stipek_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
2001_MC )_) ._. 
</s>
<s>
We_PPIS2 assume_VV0 that_CST Z-1_FO is_VBZ not_XX continuous_JJ for_IF some_DD @S_FO Then_RT ,_, there_EX exists_VVZ a_AT1 constant_JJ s_ZZ1 >_FO 0_MC ,_, such_CS21 that_CS22 ,_, for_IF any_DD i_MC1 >_FO 0_MC ,_, there_EX exists_VVZ a_AT1 point_NN1 x1_FO e_ZZ1 Q?_FO satisfying_JJ @S_FO and_CC @F_FO ._. 
</s>
<s>
Let_VV0 i_ZZ1 =_FO 1/n_FU ,_, we_PPIS2 denote_VV0 @S_FO ._. 
</s>
<s>
Since_CS @S_FO is_VBZ a_AT1 bounded_JJ sequence_NN1 ._. 
</s>
<s>
Indeed_RR ,_, it_PPH1 turns_VVZ out_RP that_DD1 for_IF higher-dimensional_JJR sticky_JJ diffusions_NN2 one_PN1 can_VM impose_VVI different_JJ dynamics_NN in_II the_AT interior_NN1 of_IO a_AT1 domain_NN1 and_CC on_II the_AT boundary_NN1 ,_, and_CC these_DD2 dynamics_NN don_VV0 '_NULL t_ZZ1 have_VH0 to_TO bear_VVI any_DD relation_NN1 to_II each_PPX221 other_PPX222 &lsqb;_( 45_MC &rsqb;_) ._. 
</s>
<s>
Proof_NN1 This_DD1 is_VBZ a_AT1 double_JJ counting_NN1 argument_NN1 ._. 
</s>
<s>
Before_CS we_PPIS2 can_VM proceed_VVI and_CC tackle_VVI the_AT general_JJ existence_NN1 theory_NN1 ,_, it_PPH1 turns_VVZ out_RP that_CST a_AT1 multiple_JJ version_NN1 of_IO the_AT generalized_JJ Riemann_NN1 problem_NN1 must_VM also_RR be_VBI solved_VVN and_CC this_DD1 is_VBZ done_VDN in_II Section_NN1 6_MC when_RRQ the_AT initial_JJ problem_NN1 with_IW three_MC steady_JJ states_NN2 separated_VVN with_IW two_MC discontinuities_NN2 is_VBZ analyzed_VVN ._. 
</s>
<s>
For_IF @S_FO ,_, let_VV0 W_ZZ1 denote_VVI the_AT product_NN1 of_IO the_AT first_MD w_ZZ1 primes_VVZ that_CST are_VBR relatively_RR prime_JJ to_II d_ZZ1 ._. 
</s>
<s>
The_AT given_JJ problems_NN2 were_VBDR assessed_VVN as_CSA "_" simple_JJ ,_, "_" "_" accessible_JJ ,_, "_" or_CC "_" human_NN1 "_" by_II some_DD of_IO the_AT students_NN2 ._. 
</s>
<s>
We_PPIS2 have_VH0 :_: @F_FO ._. 
</s>
<s>
In_II this_DD1 way_NN1 we_PPIS2 have_VH0 checked_VVN relations_NN2 (_( 45_MC )_) -(47)_NN1 ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI the_AT standard_JJ Sobolev_NP1 space_NN1 of_IO real-valued_JJ functions_NN2 that_CST are_VBR square_JJ integrable_JJ and_CC have_VH0 square-integrable_JJ derivatives_NN2 on_II @S_FO with_IW trace_NN1 zero_NN1 on_II the_AT boundary_NN1 @S_FO ._. 
</s>
<s>
It_PPH1 serves_VVZ as_II the_AT natural_JJ energy_NN1 space_NN1 of_IO (_( 3.6_MC )_) equipped_VVN with_IW an_AT1 inner_JJ product_NN1 and_CC norm_NN1 @F_FO ._. 
</s>
<s>
The_AT natural_JJ energy_NN1 space_NN1 associated_VVN with_IW (_( 3.5_MC )_) is_VBZ @F_FO for_IF @S_FO ._. 
</s>
<s>
The_AT higher_JJR discretization_NN1 accuracy_NN1 is_VBZ obtained_VVN by_II using_VVG an_AT1 isoparametric_JJ mapping_NN1 of_IO the_AT volume_NN1 mesh_NN1 based_VVN on_II a_AT1 high-order_JJ approximation_NN1 of_IO the_AT level_JJ set_NN1 function_NN1 ._. 
</s>
<s>
While_CS these_DD2 methods_NN2 differ_VV0 in_II the_AT specifics_NN2 of_IO how_RRQ they_PPHS2 address_VV0 (_( 1_MC1 )_) ,_, the_AT critical_JJ subroutine_NN1 in_II each_DD1 method_NN1 is_VBZ "_" solving_VVG the_AT inner_JJ problem_NN1 ._. "_" 
</s>
<s>
Within_II each_DD1 of_IO these_DD2 large_JJ rectangles_NN2 ,_, the_AT set_NN1 E_ZZ1 consists_VVZ of_IO evenly_RR spaced_VVN parallel_JJ rectangles_NN2 with_IW dimensions_NN2 R1/2×R1_VV0 ._. 
</s>
<s>
We_PPIS2 notice_VV0 that_CST @S_FO ._. 
</s>
<s>
Further_RRR ,_, by_II the_AT monotonicity_NN1 property_NN1 of_IO y_ZZ1 ,_, we_PPIS2 have_VH0 G_ZZ1 @S_FO for_CS I_PPIS1 <_FO m_ZZ1 ._. 
</s>
<s>
Thus_RR ,_, @F_FO ._. 
</s>
<s>
In_BCL21 order_BCL22 to_TO choose_VVI y_ZZ1 ,_, we_PPIS2 use_VV0 the_AT lower_JJR bound_NN1 A_ZZ1 (_( G_ZZ1 ,_, y_ZZ1 )_) as_II a_AT1 surrogate_NN1 objective_NN1 function_NN1 ._. 
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<s>
To_TO verify_VVI the_AT final_JJ statements_NN2 ,_, we_PPIS2 may_VM assume_VVI that_CST k_ZZ1 is_VBZ algebraically_RR closed_VVN ._. 
</s>
<s>
Fix_VV0 @S_FO and_CC @S_FO ._. 
</s>
<s>
Let_VV0 a_AT1 be_VBI a_AT1 trigonometric_JJ polynomial_NN1 with_IW @S_FO ._. 
</s>
<s>
We_PPIS2 highlight_VV0 that_CST this_DD1 feature_NN1 is_VBZ not_XX shared_VVN by_II the_AT method_NN1 defined_VVN in_II Ref._NN1 6_MC or_CC by_II most_DAT of_IO the_AT standard_NN1 mixed_VVD finite_JJ element_NN1 methods_NN2 ,_, where_CS the_AT divergence-free_JJ constraint_NN1 is_VBZ imposed_VVN only_RR in_II a_AT1 weak_JJ (_( relaxed_JJ )_) sense_NN1 ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI the_AT renormalised_JJ lattice_NN1 @S_FO and_CC let_VV0 ,_, for_IF @S_FO ,_, @S_FO ._. 
</s>
<s>
We_PPIS2 will_VM write_VVI @S_FO ,_, y_ZZ1 are_VBR nearest_JJT neighbors_NN2 in_II the_AT renormalised_JJ lattice_NN1 @S_FO ._. 
</s>
<s>
The_AT old_JJ "_" block_NN1 "_" variable_NN1 ax_NN1 g_ZZ1 S_ZZ1 associated_VVN to_II @S_FO is_VBZ renamed_VVN as_CSA @S_FO with_IW now_RT @S_FO for_IF all_DB @S_FO ._. 
</s>
<s>
In_RR21 particular_RR22 ,_, the_AT local_JJ variance_NN1 term_NN1 Varx_NN1 (_( f_ZZ1 )_) appearing_VVG in_II the_AT right-hand_JJ side_NN1 of_IO (_( 3.3_MC )_) becomes_VVZ @S_FO ._. 
</s>
<s>
Accordingly_RR ,_, we_PPIS2 rewrite_VV0 the_AT mapping_NN1 @S_FO ,_, as_CSA @S_FO ._. 
</s>
<s>
Let_VV0 A_ZZ1 ,_, B_ZZ1 be_VBI quantum_NN1 Gaussian_JJ systems_NN2 with_IW m_ZZ1 and_CC n_ZZ1 modes_NN2 ,_, respectively_RR ,_, and_CC :_: A_ZZ1 t_ZZ1 Bq_NP1 quantum_NN1 Gaussian_JJ channel_NN1 ._. 
</s>
<s>
Let_VV0 ,_, for_IF A_ZZ1 >_FO 0_MC ,_, be_VBI the_AT interaction_NN1 defined_VVN by_II @S_FO ,_, where_CS xa_NN1 is_VBZ the_AT characteristic_JJ function_NN1 of_IO a_AT1 ball_NN1 with_IW radius_NN1 A._NP1 Then_RT let_VV0 @F_FO ._. 
</s>
<s>
Since_II va_FW e_ZZ1 L2_FO (_( R_ZZ1 "_" )_) ,_, this_DD1 operator_NN1 is_VBZ self-adjoint_JJ on_II the_AT domain_NN1 D(Ha)_NP1 =_FO D(L)_MC ._. 
</s>
<s>
The_AT main_JJ purpose_NN1 of_IO this_DD1 section_NN1 is_VBZ to_TO introduce_VVI the_AT local_JJ minima_NN2 and_CC the_AT generalized_JJ saddle_NN1 points_NN2 of_IO f_ZZ1 ._. 
</s>
<s>
Dealing_VVG with_IW this_DD1 topic_NN1 generates_VVZ the_AT need_VM of_IO advanced_VVN computational_JJ methods_NN2 ._. 
</s>
<s>
The_AT rest_NN1 of_IO the_AT paper_NN1 is_VBZ organized_VVN as_CSA follows_VVZ ._. 
</s>
<s>
Let_VV0 @S_FO denote_VVI the_AT topological_JJ Markov_NP1 flow_NN1 with_IW roof_NN1 function_NN1 r_ZZ1 and_CC base_NN1 map_NN1 @S_FO (_( see_VV0 page_NN1 200_MC for_IF definition_NN1 )_) ._. 
</s>
<s>
We_PPIS2 aim_VV0 to_TO obtain_VVI upper_JJ bounds_NN2 on_II them_PPHO2 so_CS21 that_CS22 the_AT perturbed_JJ system_NN1 (_( 6.21_MC )_) remains_VVZ asymptotically_RR bounded_VVN ._. 
</s>
<s>
We_PPIS2 construct_VV0 a_AT1 sequence_NN1 of_IO approximate_JJ solutions_NN2 and_CC we_PPIS2 prove_VV0 that_CST the_AT weighted_JJ total_JJ variation_NN1 of_IO these_DD2 solutions_NN2 is_VBZ non-increasing_JJ in_II time_NNT1 ._. 
</s>
<s>
This_DD1 gives_VVZ a_AT1 tiling_NN1 of_IO R_ZZ1 ,_, which_DDQ has_VHZ several_DA2 nice_JJ properties_NN2 revealed_VVN in_II this_DD1 section_NN1 ._. 
</s>
<s>
The_AT former_DA have_VH0 determinant_NN1 12_MC ,_, the_AT latter_DA 1_MC1 ._. 
</s>
<s>
In_II fact_NN1 ,_, they_PPHS2 are_VBR not_XX contradictory_JJ ,_, and_CC (_( C2_FO )_) under_II (_( C1_FO )_) implies_VVZ (_( C40_FO )_) (_( see_VV0 &lsqb;_( 21_MC &rsqb;_) )_) ._. 
</s>
<s>
Nevertheless_RR ,_, the_AT interested_JJ reader_NN1 can_VM consult_VVI &lsqb;_( 5_MC &rsqb;_) for_IF a_AT1 more_RGR complete_JJ version_NN1 ._. 
</s>
<s>
As_II the_AT corresponding_JJ instance_NN1 of_IO (_( 3.7_MC )_) ,_, we_PPIS2 are_VBR looking_VVG for_IF a_AT1 pair_NN as_CSA in_II (_( 4.2_MC )_) that_CST satisfies_VVZ ,_, for_IF every_AT1 @S_FO ,_, the_AT conditions_NN2 @F_FO ,_, where_CS the_AT first_MD condition_NN1 is_VBZ equivalent_JJ to_II @F_FO ._. 
</s>
<s>
Level-rank_NN1 duality_NN1 of_IO type_NN1 D_ZZ1 also_RR gives_VVZ rise_NN1 to_II an_AT1 interesting_JJ coset_NN1 vertex_NN1 superalgebra_NN1 ._. 
</s>
<s>
Hence_RR ,_, in_II the_AT favorable_JJ cases_NN2 ,_, we_PPIS2 will_VM benefit_VVI from_II a_AT1 smaller_JJR distance_NN1 between_II the_AT new_JJ center_NN1 point_NN1 and_CC @S_FO ,_, while_CS in_II the_AT unfavorable_JJ cases_NN2 ,_, restarting_VVG should_VM not_XX affect_VVI too_RG much_RR the_AT convergence_NN1 ._. 
</s>
<s>
In_II Lemma_NN1 B.1_FO of_IO the_AT supplement_NN1 ,_, we_PPIS2 show_VV0 that_CST @S_FO almost_RR surely_RR and_CC in_II expectation_NN1 ._. 
</s>
<s>
In_II Figure_NN1 32_MC one_MC1 can_VM check_VVI that_CST the_AT weight_NN1 converges_VVZ ,_, as_CSA required_VVN ,_, to_II the_AT initial_JJ weight_NN1 at_II the_AT end_NN1 of_IO the_AT optimization_NN1 process_NN1 ._. 
</s>
<s>
Then_RT a_AT1 very_RG general_JJ hypersurface_NN1 @S_FO of_IO degree_NN1 d_ZZ1 and_CC with_IW multiplicity_NN1 d_ZZ1 '_NULL along_II L_ZZ1 does_VDZ not_XX admit_VVI an_AT1 integral_JJ decomposition_NN1 of_IO the_AT diagonal_JJ over_II the_AT algebraic_JJ closure_NN1 k_ZZ1 ._. 
</s>
<s>
There_EX are_VBR many_DA2 aspects_NN2 of_IO Lemma_NN1 5.1_MC that_CST deserve_VV0 our_APPGE attention_NN1 ._. 
</s>
<s>
For_REX21 example_REX22 ,_, a_AT1 common_JJ purpose_NN1 called_VVN "_" PTs_NN2 must_VM work_VVI in_II small_JJ groups_NN2 "_" was_VBDZ identified_VVN across_II all_DB MTEs_NN2 '_NULL initial_JJ interviews_NN2 ._. 
</s>
<s>
Again_RT ,_, the_AT same_DA holds_VVZ with_IW primes_NN2 ._. 
</s>
<s>
Let_VV0 @S_FO ,_, and_CC suppose_VVI @F_FO is_VBZ a_AT1 Qp-linear_JJ combination_NN1 of_IO the_AT Fi_NP1 ,_, their_APPGE single_JJ integrals_NN2 ,_, and_CC their_APPGE double_JJ integrals_NN2 ._. 
</s>
<s>
There_EX are_VBR parallels_NN2 between_II logic_NN1 programming_NN1 languages_NN2 such_II21 as_II22 Prolog_NP1 ,_, in_II which_DDQ relationships_NN2 between_II objects_NN2 are_VBR logical_JJ ones_NN2 ,_, and_CC the_AT spatial_JJ programming_NN1 in_II Sketchpad_NP1 ,_, where_RRQ the_AT relationships_NN2 are_VBR spatial_JJ ._. 
</s>
<s>
The_AT version_NN1 of_IO PA_NN1 by_II Buja_NP1 and_CC Eyuboglu_NP1 (_( 1992_MC )_) replaces_VVZ Gaussian_JJ simulations_NN2 by_II independent_JJ random_JJ permutations_NN2 within_II the_AT columns_NN2 of_IO the_AT data_NN matrix_NN1 (_( see_VV0 algorithm_NN1 1_MC1 in_II Table_NN1 1_MC1 )_) ._. 
</s>
<s>
She_PPHS1 had_VHD completed_VVN a_AT1 Master_NN1 '_NULL s_ZZ1 degree_NN1 and_CC was_VBDZ considered_VVN by_II the_AT school_NN1 district_NN1 to_TO be_VBI a_AT1 highly_RR qualified_JJ teacher_NN1 ._. 
</s>
<s>
The_AT last_MD curve_NN1 is_VBZ taken_VVN shortly_RR after_II shock_NN1 interaction_NN1 (_( T_ZZ1 =_FO 1_MC1 )_) ,_, and_CC a_AT1 zoom_NN1 of_IO the_AT interaction_NN1 region_NN1 is_VBZ shown_VVN in_II the_AT right_JJ panel_NN1 (_( zoom_VV0 2_MC )_) ._. 
</s>
<s>
From_II the_AT above_JJ theorizing_NN1 of_IO the_AT ritual_NN1 and_CC exploration_NN1 routines_NN2 ,_, we_PPIS2 suggest_VV0 two_MC teaching_NN1 goals_NN2 for_IF which_DDQ ritual-enabling_JJ OTLs_NN2 are_VBR a_AT1 necessary_JJ starting_NN1 point_NN1 ._. 
</s>
<s>
Damian_NP1 Knopoff_NP1 :_: Support_NN1 of_IO CONICET_NP1 (_( Grant_NP1 Number_NN1 PIP_NN1 11220150100500CO_FO )_) and_CC Secyt_NP1 UNC_NP1 (_( Grant_NP1 Number_NN1 33620180100326CB_FO )_) ._. 
</s>
<s>
In_II other_JJ words_NN2 ,_, the_AT profinite_NN1 completion_NN1 of_IO @S_FO coincides_VVZ with_IW a_AT1 finite-index_JJ subgroup_NN1 of_IO @S_FO ._. 
</s>
<s>
However_RR ,_, these_DD2 cuts_NN2 are_VBR tight_RR at_II the_AT proposed_JJ binary_JJ solution_NN1 @S_FO but_CCB could_VM be_VBI very_RG loose_JJ at_II other_JJ solutions_NN2 ,_, and_CC thus_RR lead_VV0 to_TO slow_VVI convergence_NN1 ._. 
</s>
<s>
There_EX exists_VVZ a_AT1 remarkable_JJ class_NN1 of_IO contact_NN1 structures_NN2 ,_, which_DDQ has_VHZ been_VBN introduced_VVN in_II &lsqb;_( 4_MC ,_, Definition_NN1 3.6_MC &rsqb;_) in_II any_DD dimension_NN1 ,_, called_VVN the_AT overtwisted_JJ contact_NN1 structures_NN2 ._. 
</s>
<s>
Otherwise_RR ,_, we_PPIS2 pick_VV0 Y_ZZ1 among_II the_AT Z_ZZ1 (_( s_ZZ1 ,_, )_) to_II which_DDQ the_AT first_MD case_NN1 applies_VVZ of_IO minimal_JJ distance_NN1 to_II XY_FO ._. 
</s>
<s>
Kolmogorov_NP1 and_CC researchers_NN2 around_II him_PPHO1 were_VBDR interested_JJ in_II the_AT area_NN1 even_RR earlier_RRR (_( Section_NN1 3.4_MC )_) ._. 
</s>
<s>
Coding_VVG smartness_NN1 The_AT social_JJ ,_, cultural_JJ ,_, and_CC political_JJ natures_NN2 of_IO mathematics_NN1 teacher_NN1 noticing_VVG are_VBR evident_JJ in_II the_AT ways_NN2 that_CST Amanda_NP1 '_NULL s_ZZ1 efforts_NN2 to_TO notice_VVI students_NN2 '_NULL "_" smartnesses_NN2 "_" intersected_VVD with_IW various_JJ coding_NN1 schemes_NN2 ._. 
</s>
<s>
We_PPIS2 refer_VV0 to_II them_PPHO2 as_CSA approximately_RR balancing_VVG weights_NN2 since_CS they_PPHS2 seek_VV0 to_TO make_VVI the_AT mean_NN1 of_IO the_AT reweighted_JJ control_NN1 sample_NN1 ,_, namely_REX @S_FO ,_, match_VV0 the_AT treated_JJ sample_NN1 mean_VV0 @S_FO as_RG closely_RR as_CSA possible_JJ ._. 
</s>
<s>
In_II the_AT discretization_NN1 we_PPIS2 search_VV0 for_IF @S_FO with_IW @S_FO ._. 
</s>
<s>
For_IF the_AT vector_NN1 representation_NN1 of_IO the_AT latter_DA constraint_NN1 we_PPIS2 introduce_VV0 @S_FO with_IW @S_FO ,_, @S_FO ,_, and_CC define_VV0 @F_FO ._. 
</s>
<s>
Hence_RR ,_, @S_FO ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI such_CS21 that_CS22 @F_FO ._. 
</s>
<s>
The_AT estimates_NN2 in_II (_( 6.1_MC )_) and_CC (_( 6.3_MC )_) imply_VV0 @F_FO ._. 
</s>
<s>
We_PPIS2 mention_VV0 recent_JJ work_NN1 using_VVG the_AT Reynolds_NP1 lubrication_NN1 equation_NN1 &lsqb;_( 32,40_MC &rsqb;_) as_II31 well_II32 as_II33 an_AT1 averaged_JJ Brinkman_NN1 equation_NN1 &lsqb;_( 12_MC &rsqb;_) ._. 
</s>
<s>
Now_RT ,_, we_PPIS2 are_VBR ready_JJ to_TO present_VVI our_APPGE first_MD result_NN1 on_II the_AT critical_JJ coupling_NN1 strength_NN1 for_IF the_AT emergence_NN1 of_IO mono-cluster_NN1 flocking_VVG ._. 
</s>
<s>
Due_II21 to_II22 the_AT relaxation_NN1 of_IO the_AT normal_JJ continuity_NN1 velocity_NN1 ,_, solutions_NN2 will_VM (_( in_II contrast_NN1 to_II the_AT discretization_NN1 in_II &lsqb;_( 25_MC &rsqb;_) with_IW @S_FO )_) be_VBI neither_RR exactly_RR solenoidal_JJ nor_CC pressure_NN1 robust_JJ ;_; i.e._REX ,_, the_AT error_NN1 in_II the_AT velocity_NN1 solution_NN1 will_VM depend_VVI on_II the_AT approximation_NN1 of_IO the_AT pressure_NN1 ._. 
</s>
<s>
As_CSA mentioned_VVN in_II &lsqb;_( 35_MC ,_, Section_NN1 7.1_MC &rsqb;_) ,_, the_AT statement_NN1 of_IO Lemma_NN1 23_MC still_JJ holds_NN2 under_II this_DD1 weaker_JJR assumption_NN1 ._. 
</s>
<s>
The_AT numerical_JJ values_NN2 of_IO the_AT support_NN1 points_NN2 and_CC their_APPGE weights_NN2 computed_VVN in_II the_AT above_JJ procedure_NN1 (_( and_CC displayed_VVN in_II Figure_NN1 2_MC )_) are_VBR listed_VVN in_II the_AT Supplementary_JJ Material_NN1 &lsqb;_( 2_MC &rsqb;_) ,_, Table_NN1 1_MC1 ._. 
</s>
<s>
In_II the_AT Gear_NN1 II_MC problem_NN1 ,_, she_PPHS1 used_VVD the_AT across-multiplication_JJ strategy_NN1 to_TO solve_VVI the_AT problem_NN1 (_( Fig._NN1 8a_FO )_) ._. 
</s>
<s>
A_ZZ1 contained_VVN in_II the_AT interior_NN1 of_IO the_AT positive_JJ Weyl_NN1 chamber_NN1 ._. 
</s>
<s>
Consider_VV0 the_AT first_MD of_IO the_AT stated_JJ mutations_NN2 ,_, the_AT other_NN1 being_VBG similar_JJ ._. 
</s>
<s>
They_PPHS2 frequently_RR arise_VV0 in_II algebraic_JJ geometry_NN1 as_CSA cohomology_NN1 and_CC Ext-functors_NN2 &lsqb;_( Har98_FO &rsqb;_) ._. 
</s>
<s>
Consider_VV0 an_AT1 input_NN1 @S_FO which_DDQ is_VBZ the_AT trace_NN1 of_IO a_AT1 function_NN1 @S_FO ._. 
</s>
<s>
Defining_VVG the_AT function_NN1 @F_FO ,_, and_CC using_VVG the_AT transformation_NN1 @F_FO ,_, we_PPIS2 are_VBR in_II a_AT1 position_NN1 to_TO study_VVI an_AT1 equivalent_JJ to_II (_( 5.1_MC )_) initial_JJ boundary_NN1 value_NN1 problem_NN1 @F_FO ._. 
</s>
<s>
Let_VV0 m_ZZ1 >_FO 0_MC be_VBI the_AT smallest_JJT integer_NN1 for_IF which_DDQ @S_FO ._. 
</s>
<s>
Then_RT &lsqb;_( 10_MC ,_, Theorem_NN1 6_MC ,_, page_NN1 365_MC &rsqb;_) in_II31 conjunction_II32 with_II33 &lsqb;_( 10_MC ,_, Theorem_NN1 4_MC ,_, page_NN1 288_MC &rsqb;_) and_CC &lsqb;_( 10_MC ,_, Theorem_NN1 6_MC ,_, page_NN1 270_MC &rsqb;_) guarantees_VVZ that_CST if_CS @S_FO for_IF every_AT1 @S_FO and_CC T_ZZ1 >_FO 0_MC ,_, @S_FO for_IF @S_FO ,_, where_CS @S_FO ,_, then_RT the_AT initial_JJ boundary_NN1 value_NN1 problem_NN1 (_( 5.4_MC )_) has_VHZ a_AT1 unique_JJ solution_NN1 @S_FO ._. 
</s>
<s>
Computational_JJ thinking_VVG The_AT recent_JJ push_NN1 for_IF computational_JJ thinking_NN1 as_II a_AT1 focus_NN1 of_IO educational_JJ efforts_NN2 stems_VVZ from_II the_AT idea_NN1 that_CST knowledge_NN1 and_CC skills_NN2 from_II the_AT field_NN1 of_IO computer_NN1 science_NN1 have_VH0 far-reaching_JJ applications_NN2 from_II which_DDQ everyone_PN1 can_VM benefit_VVI :_: "_" &lsqb;_( Computational_JJ thinking_NN1 &rsqb;_) represents_VVZ a_AT1 universally_RR applicable_JJ attitude_NN1 and_CC skill_NN1 set_VVD everyone_PN1 ,_, not_XX just_RR computer_NN1 scientists_NN2 ,_, would_VM be_VBI eager_JJ to_TO learn_VVI and_CC use_NN1 "_" (_( Wing_NN1 ,_, 2006_MC ,_, p._NN1 33_MC )_) ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI some_DD global_JJ almost_RR periodic_JJ types_NN2 given_VVN by_II Proposition_NN1 3.33_MC ,_, which_DDQ are_VBR also_RR @S-generic_FO by_II Proposition_NN1 3.31_MC ._. 
</s>
<s>
Table_NN1 1_MC1 Domains_NN2 and_CC topics_NN2 of_IO mathematics_NN1 education_NN1 addressed_VVN by_II the_AT studies_NN2 included_VVD m_MC the_AT review_NN1 @T_FO Table_NN1 1_MC1 (_( continued_JJ )_) @T_FO Table_NN1 1_MC1 (_( continued_JJ )_) @T_FO Table_NN1 1_MC1 (_( continued_JJ )_) @T_FO Reading_NP1 and_CC word_NN1 problem_NN1 solving_NN1 Studies_NN2 on_II mathematical_JJ reading_NN1 and_CC word_NN1 problem_NN1 solving_NN1 comprised_VVD 21_MC (_( 13%_NNU )_) of_IO the_AT studies_NN2 reviewed_VVN here_RL ._. 
</s>
<s>
The_AT memos_NN2 focused_VVN on_II the_AT degree_NN1 of_IO artifact_NN1 appropriation_NN1 of_IO the_AT teacher_NN1 and_CC typically_RR included_VVN a_AT1 degree_NN1 of_IO curriculum_NN1 use_NN1 :_: offloading_NN1 ,_, adapting_VVG ,_, or_CC improvising_VVG ._. 
</s>
<s>
Whilst_CS many_DA2 student_NN1 explanations_NN2 may_VM follow_VVI a_AT1 teacher_NN1 '_NULL s_ZZ1 why_RRQ or_CC how_RRQ question_NN1 ,_, not_XX all_RR do_VD0 ._. 
</s>
<s>
Recall_VV0 that_CST Idem_FW :_: @S_FO Set_NN1 commutes_VVZ with_IW limits_NN2 ,_, so_CS we_PPIS2 obtain_VV0 sheaves_NN2 of_IO sets_NN2 @S_FO ._. 
</s>
<s>
Moreover_RR ,_, S_ZZ1 is_VBZ a_AT1 cyclic_JJ normal_JJ subgroup_NN1 @S_FO acting_VVG as_II integer_NN1 translations_NN2 along_II the_AT line_NN1 factor_NN1 of_IO Y._NP1 Suppose_VV0 @S_FO is_VBZ a_AT1 weak_JJ equivalence_NN1 ._. 
</s>
<s>
Since_CS the_AT isomorphism_NN1 @S_FO identifies_VVZ the_AT initial_JJ quantum_NN1 cluster_NN1 variables_NN2 with_IW classes_NN2 of_IO sheave_NN1 of_IO the_AT form_NN1 @S_FO ,_, we_PPIS2 further_RRR obtain_VV0 the_AT following_JJ result_NN1 ._. 
</s>
<s>
By_II the_AT pointwise_JJ Schauder_NN1 estimates_VVZ for_IF linear_JJ equations_NN2 we_PPIS2 obtain_VV0 that_CST @S_FO is_VBZ pointwise_RR C3+a_FO at_II x°_FO ._. 
</s>
<s>
Taking_VVG @S_FO ,_, we_PPIS2 can_VM formally_RR decompose_VVI the_AT product_NN1 as_CSA @F._FO where_RRQ @F_FO ._. 
</s>
<s>
With_IW this_DD1 notation_NN1 ,_, the_AT following_JJ results_NN2 hold_VV0 ._. 
</s>
<s>
The_AT best_RRT result_VV0 available_JJ is_VBZ perhaps_RR from_II the_AT work_NN1 of_IO Mouhot_NP1 &lsqb;_( 53_MC &rsqb;_) ._. 
</s>
<s>
Fixing_VVG the_AT order_NN1 2_MC infinitesimal_JJ neighborhood_NN1 ._. 
</s>
<s>
Then_RT for_IF all_DB v1_FO ,_, v2∈span(Y)_FO we_PPIS2 have_VH0 @F_FO ,_, where_CS @S_FO for_IF i=1,2_FO are_VBR the_AT vectors_NN2 of_IO inner_JJ products_NN2 between_II vi_MC and_CC Y._NP1 Typically_RR ,_, the_AT lecture_NN1 or_CC workshop_NN1 activity_NN1 commenced_VVN with_IW the_AT analysis_NN1 of_IO a_AT1 school_NN1 student_NN1 '_NULL s_ZZ1 work_VV0 such_II21 as_II22 exemplified_VVD below_RL ._. 
</s>
<s>
This_DD1 paper_NN1 is_VBZ mainly_RR concerned_JJ on_II CWENO_NP1 reconstructions_NN2 in_II one_MC1 space_NN1 dimension_NN1 ._. 
</s>
<s>
The_AT proof_NN1 is_VBZ similar_JJ to_II the_AT proof_NN1 of_IO Proposition_NN1 7.2_MC ,_, so_CS we_PPIS2 describe_VV0 a_AT1 more_RGR general_JJ construction_NN1 which_DDQ applies_VVZ in_II the_AT case_NN1 r<t_FO as_RR21 well_RR22 ._. 
</s>
<s>
We_PPIS2 also_RR plan_VV0 to_TO address_VVI this_DD1 problem_NN1 in_II a_AT1 future_JJ work_NN1 ._. 
</s>
<s>
For_IF every_AT1 @S_FO ,_, let_VV0 @S_FO be_VBI the_AT set_NN1 of_IO @S-chains_FO g_ZZ1 which_DDQ are_VBR suitable_JJ from_II St_NP1 and_NP1 such_CS21 that_CS22 @S_FO belongs_VVZ to_II @S_FO :_: @F_FO ._. 
</s>
<s>
It_PPH1 will_VM be_VBI useful_JJ to_TO bound_VVI from_II above_RL @S_FO ._. 
</s>
<s>
From_II &lsqb;_( 23_MC ,_, Section_NN1 4.2_MC &rsqb;_) ,_, there_EX exists_VVZ a_AT1 @S_FO function_NN1 @S_FO with_IW @S_FO such_CS21 that_CS22 the_AT 1-form_NN1 @F_FO ,_, satisfies_VVZ @F_FO ._. 
</s>
<s>
Moreover_RR ,_, one_PN1 has_VHZ in_II the_AT limit_NN1 @S_FO (_( see_VV0 &lsqb;_( 23_MC ,_, Section_NN1 4.2_MC &rsqb;_) )_) :_: @F_FO ,_, where_CS the_AT remainder_NN1 term_NN1 O(h)_UH admits_VVZ a_AT1 full_JJ asymptotic_JJ expansion_NN1 in_II h_ZZ1 ._. 
</s>
<s>
The_AT radial_JJ basis_NN1 functions_NN2 often_RR depend_VV0 on_II hyper-parameters_NN2 ,_, e.g._REX ,_, the_AT bandwidth_NN1 of_IO the_AT Gaussian_JJ density_NN1 function_NN1 ._. 
</s>
<s>
Notice_VV0 that_CST here_RL we_PPIS2 start_VV0 from_II the_AT extension_NN1 of_IO u_ZZ1 with_IW value_NN1 0_MC outside_II Q_ZZ1 :_: so_RR we_PPIS2 have_VH0 to_TO include_VVI in_II the_AT analogue_NN1 of_IO (_( 4.12c_FO )_) also_RR the_AT contribution_NN1 due_II21 to_II22 dQ_NNU ,_, and_CC then_RT (_( 4.12c_FO )_) reads_VVZ in_II this_DD1 case_NN1 as_CSA @F_FO ._. 
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<s>
We_PPIS2 restrict_VV0 x_ZZ1 to_II a_AT1 ball_NN1 BR_NP1 of_IO radius_NN1 R._NP1 The_AT validity_NN1 of_IO the_AT Euclidean_JJ isoperimetric_JJ inequality_NN1 for_IF curves_NN2 with_II31 respect_II32 to_II33 the_AT Holmes-Thompson_NP1 definition_NN1 of_IO area_NN1 might_VM be_VBI related_VVN to_II other_JJ forms_NN2 of_IO convexity_NN1 beyond_II @S_FO ._. 
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<s>
In_II &lsqb;_( 16_MC &rsqb;_) ,_, the_AT authors_NN2 focus_VV0 on_II the_AT relationship_NN1 between_II vorticity_NN1 and_CC the_AT strain_NN1 tensor_NN1 in_II en-strophy_JJ production_NN1 ,_, as_CSA the_AT strain_NN1 tensor_NN1 and_CC vorticity_NN1 are_VBR related_VVN by_II a_AT1 linear_JJ zero_NN1 order_NN1 pseudo-differential_NN1 operator_NN1 ,_, @S_FO ._. 
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<s>
However_RR ,_, the_AT consistency_NN1 condition_NN1 is_VBZ actually_RR very_RG useful_JJ in_II dealing_VVG with_IW the_AT evolution_NN1 of_IO the_AT strain_NN1 tensor_NN1 ,_, because_CS a_AT1 number_NN1 of_IO the_AT terms_NN2 in_II the_AT evolution_NN1 equation_NN1 (_( 1.11_MC )_) are_VBR actually_RR in_II the_AT orthogonal_JJ compliment_NN1 of_IO L2_FO with_II31 respect_II32 to_II33 the_AT L2_FO inner_JJ product_NN1 ._. 
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<s>
The_AT asymptotic_JJ (_( Pitman_NP1 )_) efficiency_NN1 of_IO a_AT1 general_JJ graph-based_JJ test_NN1 is_VBZ derived_VVN ,_, which_DDQ includes_VVZ tests_NN2 based_VVN on_II geometricgraphs_NN2 ,_, such_II21 as_II22 the_AT Friedman–Rafsky_JJ test_NN1 ,_, the_AT test_NN1 based_VVN on_II the_AT K_ZZ1 -nearest-neighbour_JJ graph_NN1 ,_, thecross-match_JJ test_NN1 and_CC the_AT generalized_JJ edge_NN1 count_NN1 test_NN1 ,_, as_II31 well_II32 as_II33 tests_NN2 based_VVN on_II multivariate_JJ depthfunctions_NN2 (_( the_AT Liu–Singh_NN1 rank_NN1 sum_NN1 statistic_NN1 )_) ._. 
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<s>
The_AT second_MD condition_NN1 is_VBZ an_AT1 approximation_NN1 property_NN1 ._. 
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<s>
The_AT proof_NN1 follows_VVZ from_II the_AT robustness_NN1 of_IO the_AT renormalization_NN1 change_NN1 of_IO variables_NN2 in_II a_AT1 neighborhood_NN1 of_IO d_ZZ1 '_NULL Z_ZZ1 ,_, ._. 
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<s>
Note_VV0 that_CST since_CS this_DD1 paper_NN1 focuses_VVZ on_II a_AT1 part_NN1 of_IO our_APPGE larger_JJR study_NN1 ,_, not_XX all_DB of_IO the_AT codes_NN2 we_PPIS2 used_VVD in_II the_AT larger_JJR study_NN1 are_VBR related_VVN to_II the_AT findings_NN2 discussed_VVN here_RL ._. 
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<s>
Once_RR a_AT1 model_NN1 is_VBZ built_VVN ,_, regression_NN1 is_VBZ often_RR concerned_JJ with_IW topics_NN2 such_II21 as_II22 understanding_NN1 biasing_VVG properties_NN2 of_IO the_AT model_NN1 parameters_NN2 ,_, determining_VVG variable_JJ influence_NN1 ,_, and_CC understanding_VVG asymptotic_JJ convergence_NN1 properties_NN2 of_IO the_AT estimator_NN1 ._. 
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<s>
We_PPIS2 now_RT consider_VV0 the_AT subring_NN1 SH(M)c@S_FO consisting_VVG of_IO those_DD2 power_NN1 series_NN of_IO the_AT form_NN1 SA_NP1 for_IF some_DD @S_FO ,_, and_CC let_VV0 KM_NNU denote_VVI the_AT quotient_NN1 field_NN1 of_IO @S_FO ._. 
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<s>
Using_VVG @F_FO that_DD1 follows_VVZ from_II the_AT Galerkin_NN1 orthogonality_NN1 (_( 2.4_MC )_) and_CC the_AT results_NN2 of_IO the_AT two_MC previous_JJ sections_NN2 ,_, we_PPIS2 give_VV0 upper_JJ and_CC lower_JJR bounds_NN2 on_II the_AT discretization_NN1 error_NN1 ._. 
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<s>
In_II the_AT computations_NN2 below_RL ,_, we_PPIS2 explicitly_RR determine_VV0 @S_FO via_II this_DD1 diagram_NN1 ,_, using_VVG Lemma_NN1 5.2_MC to_TO characterise_VVI the_AT Frobenius_NP1 structure_NN1 Tn_NP1 uniquely_RR by_II its_APPGE universal_JJ property_NN1 ._. 
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<s>
So_RR the_AT sum_NN1 @S_FO reduces_VVZ to_II @S_FO ,_, where_CS depending_II21 on_II22 the_AT operator_NN1 ,_, @S_FO means_VVZ @S_FO (_( origin_NN1 and_CC terminus_NN1 of_IO w_ZZ1 )_) ,_, @S_FO ._. 
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In_II any_DD case_NN1 ,_, @S_FO ._. 
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<s>
So_RR @F_FO and_CC thus_RR @F_FO ._. 
</s>
<s>
It_PPH1 follows_VVZ that_CST @F_FO ._. 
</s>
<s>
Using_VVG Holder_NP1 '_NULL s_ZZ1 inequality_NN1 ,_, if_CS @S_FO ,_, using_VVG Remark_NN1 A.3_FO we_PPIS2 get_VV0 that_DD1 @F_FO uniformly_RR in_II A._NNU Here_RL ,_, @S_FO may_VM depend_VVI on_II T._NP1 The_AT result_NN1 of_IO performing_VVG these_DD2 mutations_NN2 in_II any_DD order_NN1 on_II Bn_NNU is_VBZ an_AT1 exchange_NN1 matrix_NN1 whose_DDQGE principal_JJ part_NN1 is_VBZ identified_VVN with_IW that_DD1 of_IO Bn_NNU upon_II permuting_VVG the_AT index_NN1 set_NN1 Iex_NN1 by_II the_AT involution_NN1 @S_FO ._. 
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<s>
For_IF @S_FO we_PPIS2 inductively_RR define_VV0 @S_FO as_II the_AT variable_NN1 introduced_VVN by_II mutating_VVG the_AT cluster_NN1 @S_FO in_II direction_NN1 @S_FO ._. 
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<s>
For_IF @S_FO we_PPIS2 define_VV0 @S_FO similarly_RR ,_, starting_VVG with_IW @S_FO ._. 
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<s>
We_PPIS2 now_RT turn_VV0 to_II the_AT problem_NN1 with_IW homogeneous_JJ boundary_NN1 conditions_NN2 in_II the_AT strictly_RR hyperbolic_JJ case_NN1 ._. 
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In_II the_AT multivariate_JJ Gaussian_JJ setting_NN1 ,_, conditional_JJ independence_NN1 tests_NN2 are_VBR equivalent_JJ to_II tests_NN2 for_IF zero_MC partial_JJ correlations_NN2 ._. 
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<s>
Then_RT the_AT ray_NN1 R(g)_NP1 lands_NN2 at_II x_ZZ1 for_IF all_DB g_ZZ1 in_II a_AT1 small_JJ neighborhood_NN1 of_IO f_ZZ1 ._. 
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<s>
While_CS at_II first_MD glance_NN1 this_DD1 might_VM appear_VVI to_TO involve_VVI a_AT1 compact_JJ perturbation_NN1 of_IO the_AT identity_NN1 ,_, that_DD1 is_VBZ not_XX the_AT case_NN1 ._. 
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<s>
Note_VV0 first_MD that_DD1 for_IF all_DB |_NULL v_ZZ1 |_NULL <_FO 1_MC1 and_CC @S_FO ,_, @F_FO ._. 
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<s>
Since_CS for_IF any_DD sets_NN2 @S_FO ,_, @S_FO ,_, in_BCL21 order_BCL22 to_TO check_VVI (_( H2_FO )_) (_( ii_MC )_) it_PPH1 is_VBZ enough_DD to_TO prove_VVI that_CST @F_FO ._. 
</s>
<s>
In_II this_DD1 appendix_NN1 we_PPIS2 collect_VV0 various_JJ results_NN2 ,_, some_DD of_IO which_DDQ we_PPIS2 referred_VVD to_II in_II the_AT previous_JJ text_NN1 ._. 
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<s>
If_CS @S_FO is_VBZ a_AT1 sequence_NN1 indexed_VVN by_II the_AT primes_NN2 ,_, we_PPIS2 write_VV0 @F_FO ,_, where_CS w(N)_NNU denotes_VVZ the_AT number_NN1 of_IO prime_JJ numbers_NN2 less_DAR than_CSN N_ZZ1 ,_, if_CS this_DD1 limit_NN1 exists_VVZ ._. 
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<s>
We_PPIS2 also_RR observe_VV0 that_CST the_AT Muskat_NN1 equation_NN1 is_VBZ parabolic_JJ as_CS31 long_CS32 as_CS33 one_PN1 controls_VVZ the_AT L-norm_NN1 of_IO fx_NNU only_RR ._. 
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<s>
This_DD1 observation_NN1 illustrates_VVZ ,_, however_RR ,_, that_CST because_CS it_PPH1 is_VBZ defined_VVN as_II the_AT argmin_NN1 of_IO a_AT1 linear_JJ program_NN1 ,_, the_AT true_JJ Wasserstein_NP1 barycenter_VV0 may_VM be_VBI unstable_JJ (_( when_CS viewed_VVN as_II an_AT1 histogram_NN1 ,_, and_CC not_XX in_II the_AT sense_NN1 of_IO the_AT @S_FO topology_NN1 of_IO measures_NN2 )_) ,_, even_RR for_IF such_DA a_AT1 simple_JJ problem_NN1 and_CC for_IF large_JJ n_ZZ1 as_CSA illustrated_VVN in_II Figure_NN1 4_MC ._. 
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<s>
A_AT1 proof_NN1 that_CST @S_FO is_VBZ locally_RR compact_JJ and_CC second_MD countable_JJ may_VM be_VBI found_VVN in_II &lsqb;_( 25_MC ,_, Prop_VV0 ._. 
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<s>
4.10_MC &rsqb;_) (_( formally_RR ,_, the_AT latter_DA statement_NN1 applies_VVZ to_II @@S_FO ;_; the_AT latter_DA coincides_VVZ with_IW @S_FO by_II (_( i_ZZ1 )_) )_) ._. 
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<s>
We_PPIS2 define_VV0 the_AT operators_NN2 @F_FO by_II @F_FO for_IF each_DD1 t_ZZ1 e_ZZ1 R_ZZ1 and_CC all_DB @S_FO ._. 
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<s>
Let_VV0 f_ZZ1 be_VBI a_AT1 nonnegative_JJ supersolution_NN1 of_IO the_AT Boltzmann_NP1 equation_NN1 in_II &lsqb;_( 0_MC ,_, T_ZZ1 &rsqb;_) x_ZZ1 BR_NP1 x_ZZ1 BR_NP1 ._. 
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<s>
If_CS the_AT ith_MD process_NN1 is_VBZ running_VVG in_II isolation_NN1 ,_, the_AT waiting_JJ time_NNT1 t_ZZ1 until_CS the_AT next_MD event_NN1 is_VBZ distributed_VVN according_II21 to_II22 @F_FO ,_, where_CS @S_FO is_VBZ the_AT survival_NN1 function_NN1 of_IO the_AT ith_MD process_NN1 given_VVN by_II (_( 3_MC )_) ._. 
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<s>
We_PPIS2 assume_VV0 that_CST w_ZZ1 factors_NN2 through_II the_AT projectivization_NN1 of_IO a_AT1 rank-4_MC vector_NN1 bundle_NN1 on_II S_ZZ1 such_CS21 that_CS22 the_AT fibers_NN2 are_VBR (_( possibly_RR singular_JJ )_) quadric_JJ surfaces_NN2 ;_; see_VV0 §3_FO for_IF relevant_JJ background_NN1 ._. 
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<s>
In_II Appendix_NN1 A_ZZ1 we_PPIS2 provide_VV0 an_AT1 example_NN1 of_IO a_AT1 radial_JJ measure_NN1 on_II Rd_NN1 whose_DDQGE support_NN1 is_VBZ an_AT1 annulus_NN1 (_( hence_RR is_VBZ not_XX simply_RR connected_VVN )_) but_CCB whose_DDQGE Poincare-Wirtinger_NP1 constant_JJ Cpw_NP1 is_VBZ nonetheless_RR positive_JJ ._. 
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<s>
Again_RT the_AT composition_NN1 γ_NULL is_VBZ an_AT1 isomorphism_NN1 and_CC by_II (_( i_ZZ1 )_) we_PPIS2 know_VV0 that_DD1 β_NULL is_VBZ a_AT1 monomorphism_NN1 ,_, so_RR γ_NULL 1_MC1 β_NULL is_VBZ an_AT1 inverse_NN1 to_II ._. 
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<s>
The_AT virus_NN1 ,_, Corona-virus_JJ disease_NN1 2019_MC (_( CO_FO ViD-19_MC )_) is_VBZ caused_VVN by_II the_AT infection_NN1 of_IO the_AT SARS-CoV-2_MC virus_NN1 ,_, a_AT1 corona-virus_NN1 ._. 
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<s>
Fix_VV0 a_AT1 point_NN1 @S_FO and_CC let_VV0 @S_FO be_VBI its_APPGE projection_NN1 onto_II @S_FO ._. 
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<s>
Let_VV0 X_ZZ1 be_VBI a_AT1 local_JJ positive_JJ oriented_JJ parametrization_NN1 of_IO @S_FO around_II q_ZZ1 and_CC @S_FO be_VBI the_AT induced_JJ parametrization_NN1 of_IO @S_FO around_RG pr(q)_NNU ._. 
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<s>
Second_MD ,_, a_AT1 group_NN1 G_ZZ1 of_IO continuous_JJ linear_JJ transformations_NN2 of_IO T_NP1 whose_DDQGE elements_NN2 0_MC e_ZZ1 G_ZZ1 fulfil_VV0 the_AT following_JJ condition_NN1 :_: @F_FO ,_, where_CS we_PPIS2 write_VV0 @S_FO as_II a_AT1 shorthand_NN1 for_IF @S_FO ._. 
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<s>
A_AT1 simple_JJ example_NN1 of_IO regularity_NN1 structure_NN1 is_VBZ given_VVN by_II the_AT polynomials_NN2 in_II d_ZZ1 +_FO 1_MC1 indeterminates_VVZ @S_FO ._. 
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<s>
For_IF every_AT1 @S_FO ,_, let_VV0 @S_FO be_VBI the_AT set_NN1 of_IO all_DB formal_JJ polynomials_NN2 in_II @S_FO with_IW s-scaled_JJ degree_NN1 equal_JJ to_II @S_FO ._. 
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<s>
Let_VV0 us_PPIO2 recall_VVI that_CST the_AT s-scaled_JJ degree_NN1 of_IO @S_FO ,_, for_IF any_DD given_JJ @S_FO ,_, is_VBZ equal_JJ to_II @S_FO ._. 
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<s>
The_AT set_NN1 of_IO homogeneities_NN2 in_II this_DD1 example_NN1 is_VBZ A_ZZ1 =_FO N_ZZ1 ,_, while_CS a_AT1 natural_JJ structure_NN1 group_NN1 is_VBZ the_AT group_NN1 of_IO translations_NN2 on_II Rd+1_FO ._. 
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<s>
For_IF PP_NNU2 (_( p_ZZ1 ,_, q_ZZ1 )_) ,_, the_AT likelihood_NN1 greatly_RR simplifies_VVZ ,_, since_CS @S_FO and_CC go_VV0 take_VV0 only_RR two_MC values_NN2 ,_, @F_FO and_CC similarly_RR for_IF @S_FO ._. 
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<s>
Since_CS @S_FO becomes_VVZ @F_FO ,_, where_CS the_AT constant_JJ term_NN1 does_VDZ not_XX depend_VVI on_II Z._NP1 With_IW the_AT condition_NN1 p_ZZ1 >_FO q_ZZ1 ,_, we_PPIS2 have_VH0 f(p)_NNU >_FO f(q)_NNU and_CC g(p)_NNU <_FO g(q)_NNU ._. 
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<s>
They_PPHS2 are_VBR not_XX assumed_VVN to_TO be_VBI the_AT Lebesgue_NN1 measure_NN1 (_( but_CCB most_DAT of_IO the_AT time_NNT1 ,_, they_PPHS2 are_VBR probability_NN1 measures_NN2 )_) ._. 
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<s>
Suppose_VV0 that_CST E_ZZ1 M_ZZ1 is_VBZ a_AT1 CA_NP1 and_CC that_DD1 ea_NNU is_VBZ a_AT1 local_JJ basis_NN1 of_IO E_ZZ1 such_CS21 that_CS22 @S_FO are_VBR constant_JJ functions_NN2 ._. 
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<s>
Let_VV0 @S_FO be_VBI an_AT1 arbitrary_JJ vector_NN1 ;_; let_VV0 us_PPIO2 partition_NN1 z_ZZ1 as_CSA @S_FO ,_, with_IW each_DD1 @S_FO ._. 
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<s>
Then_RT ,_, @F_FO ,_, where_CS in_II (_( a_ZZ1 )_) we_PPIS2 used_VVD @S_FO ._. 
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<s>
The_AT result_NN1 is_VBZ described_VVN in_II31 terms_II32 of_II33 two_MC functions_NN2 C(z)_NP1 and_CC D(z)_NP1 ._. 
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<s>
Consider_VV0 now_RT the_AT sequence_NN1 =0_FO ,_, 1_MC1 ,_, ,_, m=j_FO ,_, m+1=_FO in_II C(M)_NNU ;_; it_PPH1 trivially_RR satisfies_VVZ conditions_NN2 (_( a_ZZ1 )_) and_CC (_( b_ZZ1 )_) ._. 
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<s>
Namely_REX ,_, for_IF each_DD1 computed_JJ eigenvalue_NN1 @S_FO ,_, it_PPH1 first_MD computes_VVZ the_AT LU_NN1 factorization_NN1 of_IO the_AT shifted_JJ matrix_NN1 @S_FO with_IW partial_JJ pivoting_JJ ._. 
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<s>
The_AT design_NN1 of_IO computational_JJ codes_NN2 towards_II the_AT simulation_NN1 of_IO the_AT spatial_JJ dynamics_NN of_IO @S_FO can_VM be_VBI developed_VVN by_II different_JJ techniques_NN2 depending_II21 on_II22 the_AT mathematical_JJ structure_NN1 of_IO the_AT specific_JJ model_NN1 used_VVN for_IF the_AT simulations_NN2 ._. 
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<s>
We_PPIS2 extend_VV0 our_APPGE analysis_NN1 of_IO divergence-free_JJ positive_JJ symmetric_JJ tensors_NN2 (_( DPT_NP1 )_) begun_VVN in_II a_AT1 previous_JJ paper_NN1 ._. 
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<s>
Therefore_RR we_PPIS2 obtain_VV0 an_AT1 exact_JJ sequence_NN1 @F_FO ._. 
</s>
<s>
Our_APPGE paper_NN1 adds_VVZ a_AT1 class_NN1 of_IO weakly_RR interacting_JJ fermion_NN1 systems_NN2 to_II this_DD1 list_NN1 ._. 
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<s>
Going_VVG back_RP to_II (_( 13_MC )_) ,_, the_AT above_JJ yields_NN2 @F_FO ._. 
</s>
<s>
Finally_RR ,_, since_CS @S_FO ,_, we_PPIS2 have_VH0 d2_FO >_FO p1_FO ._. 
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<s>
We_PPIS2 then_RT extend_VV0 @S._FO disjoint_JJ from_II previously_RR chosen_VVN @S_FO ._. 
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<s>
The_AT construction_NN1 (_( 6.5_MC )_) now_RT gives_VVZ a_AT1 path_NN1 from_II @S_FO to_II the_AT element_NN1 @S_FO ._. 
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<s>
We_PPIS2 can_VM then_RT choose_VVI an_AT1 extension_NN1 fi_NN2 of_IO fi_NN2 which_DDQ is_VBZ isotopic_JJ to_II ai_NNU o_ZZ1 fi+1_FO ._. 
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<s>
Therefore_RR ,_, one_PN1 obtains_VVZ @F_FO ,_, from_II which_DDQ ,_, integrating_VVG in_II t_ZZ1 and_CC using_VVG Propositions_NN2 3.1-3.2_MCMC ,_, we_PPIS2 have_VH0 @F_FO ,_, proving_VVG the_AT conclusion_NN1 ._. 
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<s>
Implementation_NN1 of_IO the_AT model_NN1 development_NN1 sequence_NN1 The_AT model_NN1 development_NN1 sequence_NN1 was_VBDZ implemented_VVN in_II two_MC iterations_NN2 over_RG two_MC teaching_NN1 semesters_NN2 as_II a_AT1 part_NN1 of_IO an_AT1 elective_JJ course_NN1 of_IO "_" Mathematical_JJ Modeling_NN1 for_IF Teachers_NN2 ._. "_" 
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<s>
In_II the_AT example_NN1 above_RL ,_, we_PPIS2 can_VM imagine_VVI that_CST E1_FO is_VBZ the_AT bottom_NN1 third_MD and_CC E2_FO is_VBZ the_AT top_JJ third_MD ._. 
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<s>
For_IF n_ZZ1 >_FO 1_MC1 we_PPIS2 have_VH0 @L_FO ._. 
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<s>
Proof_NN1 ._. 
</s>
<s>
In_II this_DD1 section_NN1 we_PPIS2 present_VV0 an_AT1 analysis_NN1 of_IO the_AT eigenvalues_NN2 and_CC eigenfunctions_NN2 of_IO limit_NN1 problem_NN1 (_( 3.7_MC )_) obtained_VVD in_II Section_NN1 3_MC ._. 
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<s>
In_II contrast_NN1 to_II Gelman_NP1 ,_, some_DD researchers_NN2 argued_VVD that_DD1 understanding_NN1 of_IO the_AT counting_NN1 principles_NN2 followed_VVD ,_, rather_CS21 than_CS22 preceded_VVN ,_, children_NN2 '_NULL s_ZZ1 use_NN1 of_IO accurate_JJ counting_NN1 procedures_NN2 (_( e.g._REX ,_, Briars_NP2 &;_NULL Siegler_NN1 ,_, 1984_MC ;_; Siegler_NP1 ,_, 1991_MC )_) ._. 
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<s>
In_II this_DD1 section_NN1 ,_, we_PPIS2 present_VV0 extensive_JJ computational_JJ experiments_NN2 to_TO evaluate_VVI the_AT SDDiP_NN1 Algorithm_NN1 2_MC on_II three_MC classes_NN2 of_IO extremely_RR challenging_JJ real-world_NN1 multistage_VV0 stochastic_JJ programs_NN2 ,_, namely_REX a_AT1 power_NN1 generation_NN1 expansion_NN1 planning_NN1 problem_NN1 ,_, a_AT1 financial_JJ portfolio_NN1 optimization_NN1 problem_NN1 ,_, and_CC an_AT1 airline_NN1 revenue_NN1 management_NN1 problem_NN1 ._. 
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<s>
It_PPH1 remains_VVZ now_RT to_TO discuss_VVI the_AT handling_NN1 of_IO the_AT resonant_JJ products_NN2 under_II the_AT paracontrolled_JJ assumption_NN1 ,_, namely_REX @S_FO and_CC @S_FO ._. 
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<s>
The_AT next_MD lemma_NN1 is_VBZ a_AT1 paralinearisation_NN1 result_NN1 adapted_VVN to_II our_APPGE nonlinear_JJ context_NN1 ._. 
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<s>
We_PPIS2 can_VM choose_VVI these_DD2 coordinates_NN2 y1_FO and_CC y2_FO so_CS21 that_CS22 dy1_FO and_CC dy2_FO have_VH0 the_AT same_DA sign_NN1 on_II each_DD1 edge_NN1 ._. 
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<s>
By_II continuity_NN1 ,_, Rj(g)_NP1 is_VBZ stable_JJ for_IF i_ZZ1 <_FO n_ZZ1 ,_, where_CS n_ZZ1 is_VBZ big_JJ if_CS g_ZZ1 is_VBZ sufficiently_RR close_JJ to_II f_ZZ1 ._. 
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<s>
Since_CS the_AT quasilinear_JJ diffusions_NN2 we_PPIS2 consider_VV0 have_VH0 singularities_NN2 at_II @S_FO ,_, and_CC for_IF the_AT readers_NN2 '_NULL convenience_NN1 ,_, we_PPIS2 recall_VV0 the_AT appropriate_JJ definitions_NN2 in_II Section_NN1 1.5_MC below_RL ._. 
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<s>
By_II virtue_NN1 of_IO (_( 4.11_MC )_) -(4.13)_JJ and_CC (_( 4.16_MC )_) ,_, we_PPIS2 find_VV0 that_CST @S_FO is_VBZ a_AT1 solution_NN1 of_IO the_AT nonlinear_JJ problem_NN1 (_( 2.13_MC )_) on_II &lsqb;_( 0_MC ,_, T_ZZ1 &rsqb;_) x_ZZ1 Q_ZZ1 ,_, provided_CS @S_FO solve_VV0 @F_FO ._. 
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<s>
Thanks_II21 to_II22 (_( 4.12_MC )_) ,_, we_PPIS2 find_VV0 that_CST (_( V_ZZ1 ,_, W_ZZ1 )_) =_FO 0_MC satisfies_VVZ (_( 4.18_MC )_) for_IF t_ZZ1 <_FO 0_MC ._. 
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<s>
Hence_RR ,_, based_VVN on_II (_( 4.2_MC )_) ,_, we_PPIS2 should_VM choose_VVI Ap_NP1 to_TO lie_VVI in_II the_AT direction_NN1 @S_FO ._. 
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<s>
Keeping_VVG in_II mind_NN1 that_CST (_( 4.2_MC )_) is_VBZ an_AT1 approximation_NN1 that_CST is_VBZ relevant_JJ only_RR for_IF small_JJ Ap_NP1 ,_, we_PPIS2 will_VM limit_VVI ourselves_PPX2 to_II a_AT1 small_JJ step_NN1 in_II that_DD1 direction_NN1 ._. 
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<s>
Let_VV0 S_ZZ1 and_CC θ_NULL be_VBI as_CSA above_RL ._. 
</s>
<s>
The_AT main_JJ technical_JJ point_NN1 of_IO departure_NN1 for_IF our_APPGE method_NN1 is_VBZ that_CST the_AT optional_JJ stopping_VVG argument_NN1 is_VBZ notmerely_RR a_AT1 technical_JJ device_NN1 to_TO prove_VVI FDR_NP1 control_NN1 for_IF a_AT1 fixed_JJ algorithm_NN1 like_II the_AT BH_NP1 ,_, Storey–_NP1 BH_NP1 or_CC '_NULL Knockoff+_FO '_NULL procedures_NN2 ._. 
</s>
<s>
It_PPH1 is_VBZ a_AT1 generalization_NN1 of_IO the_AT condition_NN1 (_( 1.4_MC )_) for_IF the_AT forced_JJ mean_JJ curvature_NN1 equation_NN1 ,_, which_DDQ ,_, as_CSA mentioned_VVN above_RL ,_, has_VHZ been_VBN shown_VVN to_TO be_VBI necessary_JJ for_IF homogenization_NN1 ,_, even_RR in_II the_AT periodic_JJ case_NN1 &lsqb;_( 12_MC &rsqb;_) ._. 
</s>
<s>
For_IF a_AT1 member_NN1 0_MC of_IO the_AT class_NN1 d_ZZ1 ,_, we_PPIS2 define_VV0 its_APPGE isotropic_JJ spectral_JJ density_NN1 as_CSA @F_FO ,_, and_CC throughout_II the_AT paper_NN1 ,_, we_PPIS2 use_VV0 the_AT notation_NN1 :_: @S_FO ,_, and_CC @S_FO for_IF the_AT radial_JJ parts_NN2 of_IO Fourier_NP1 transforms_VVZ of_IO 2_MC and_CC 2_MC ,_, respectively_RR ._. 
</s>
<s>
Suppose_VV0 @S_FO is_VBZ twice_RR differentiable_JJ on_II Q_ZZ1 and_CC satisfies_VVZ @S_FO where_RRQ qs(a)_NN1 is_VBZ an_AT1 s-degree_JJ polynomial_NN1 in_II a_AT1 ._. 
</s>
<s>
Let_VV0 L_ZZ1 be_VBI such_CS21 that_CS22 @S_FO ._. 
</s>
<s>
Then_RT @S_FO is_VBZ L-smooth_JJ relative_II21 to_II22 @S_FO ._. 
</s>
<s>
To_II this_DD1 end_NN1 ,_, we_PPIS2 need_VV0 a_AT1 well-known_JJ variant_NN1 of_IO Ramsey_NP1 '_NULL s_ZZ1 theorem_NN1 ,_, whose_DDQGE short_JJ proof_NN1 we_PPIS2 include_VV0 for_IF the_AT convenience_NN1 of_IO the_AT reader_NN1 ._. 
</s>
<s>
Denote_VV0 them_PPHO2 respectively_RR by_II @S_FO (_( we_PPIS2 keep_VV0 the_AT dependence_NN1 on_II the_AT coupling_NN1 parameter_NN1 t_ZZ1 implicit_JJ )_) ._. 
</s>
<s>
Thereafter_RT ,_, we_PPIS2 establish_VV0 appropriate_JJ Lyapunov-type_JJ estimates_NN2 for_IF exponentially_RR growing_VVG Lyapunov-type_JJ functions_NN2 in_II Lemma_NN1 2.2_MC and_CC Corollary_NN1 2.3_MC in_II Subsection_NN1 2.1_MC ._. 
</s>
<s>
She_PPHS1 continued_VVD thinking_VVG with_IW perceptual_JJ expressions_NN2 such_II21 as_II22 a.when_NNU trying_VVG to_TO visualize_VVI ,_, it_PPH1 seems_VVZ to_II me_PPIO1 that_DD1 Point_NN1 B_ZZ1 will_VM change_VVI less_RRR ._. "_" 
</s>
<s>
Further_RRR ,_, the_AT bounding_JJ exercise_NN1 results_NN2 in_II points_NN2 on_II the_AT plot_NN1 showing_VVG the_AT bounds_NN2 on_II the_AT partial_JJ @S_FO of_IO the_AT unobserved_JJ confounder_NN1 if_CS it_PPH1 were_VBDR k_ZZ1 times_NNT2 as_RG strong_JJ as_II the_AT observed_JJ covariate_NN1 Female_NN1 ._. 
</s>
<s>
If_CS @S_FO is_VBZ non-increasing_JJ in_II @S_FO ,_, we_PPIS2 can_VM give_VVI a_AT1 lower_JJR bound_NN1 for_IF @S_FO :_: @F_FO ,_, so_CS the_AT estimate_NN1 given_VVN by_II inequality_NN1 (_( 1.4_MC )_) is_VBZ asymptotically_RR optimal_JJ in_II this_DD1 case_NN1 ._. 
</s>
<s>
In_II the_AT context_NN1 of_IO Theorem_NN1 8_MC ,_, this_DD1 implies_VVZ that_CST the_AT left-hand_JJ side_NN1 of_IO (_( 1.3_MC )_) is_VBZ close_JJ to_II zero_NN1 for_IF a_AT1 certain_JJ range_NN1 of_IO primes_NN2 ._. 
</s>
<s>
Ellis_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
(_( 2016_MC )_) argued_VVD that_DD1 reasoning_NN1 with_IW covarying_VVG quantities_NN2 is_VBZ necessary_JJ for_IF students_NN2 hoping_VVG to_TO develop_VVI robust_JJ understandings_NN2 of_IO relationships_NN2 between_II two_MC quantities_NN2 ._. 
</s>
<s>
Free_JJ homotopies_NN2 of_IO loops_NN2 and_CC outer_JJ automorphisms_NN2 In_II this_DD1 section_NN1 ,_, h_ZZ1 denotes_VVZ a_AT1 homeomorphism_NN1 of_IO a_AT1 connected_JJ surface_NN1 M_NN1 without_IW boundary_NN1 ._. 
</s>
<s>
Let_VV0 p_ZZ1 be_VBI a_AT1 common_JJ period_NN1 of_IO x_ZZ1 ,_, y_ZZ1 ._. 
</s>
<s>
Set_VV0 K_ZZ1 :_: =_FO 4p_NNU ._. 
</s>
<s>
As_CSA in_II the_AT proof_NN1 of_IO the_AT previous_JJ theorem_NN1 ,_, the_AT H_ZZ1 '_NULL errors_NN2 for_IF both_DB2 discretizations_NN2 can_VM be_VBI bounded_VVN by_II an_AT1 interpolation_NN1 error_NN1 ._. 
</s>
<s>
Theorem_NN1 7.1_MC (_( Existence_NN1 for_IF the_AT regularized_JJ Prandtl_NP1 equation_NN1 )_) ._. 
</s>
<s>
The_AT explicit_JJ examples_NN2 of_IO A2_FO and_CC A0_FO cases_NN2 can_VM be_VBI found_VVN in_II longer_JJR arXiv_NN1 version_NN1 of_IO the_AT manuscript_NN1 &lsqb;_( 28_MC &rsqb;_) ._. 
</s>
<s>
Moreover_RR ,_, by_II the_AT ideal_JJ property_NN1 of_IO Hilbert-Schmidt_NP1 operators_NN2 ,_, setting_VVG ,_, for_IF any_DD e_ZZ1 >_FO 0_MC ,_, @F_FO ,_, we_PPIS2 have_VH0 @S_FO ._. 
</s>
<s>
Then_RT it_PPH1 follows_VVZ by_II Proposition_NN1 5.1_MC that_DD1 ,_, for_IF any_DD s_ZZ1 >_FO 0_MC ,_, there_RL exist_VV0 predictable_JJ processes_NN2 @F_FO ,_, with_IW @S_FO for_IF P-almost_RR all_DB @S_FO ,_, such_CS21 that_CS22 @F_FO in_II @S_FO for_IF all_DB @S_FO ._. 
</s>
<s>
Moreover_RR ,_, @S._FO in_II @S_FO and_CC @S-almost_FO surely_RR ._. 
</s>
<s>
Following_VVG &lsqb;_( Mt1_FO ,_, Mt2_FO &rsqb;_) ,_, one_PN1 can_VM associate_VVI to_II A_ZZ1 and_CC the_AT root_NN1 datum_NN1 an_AT1 integral_JJ Kac-Moody_JJ group_NN1 GZ_NP1 (_( a_AT1 group_NN1 ind-scheme_NN1 over_II Z_ZZ1 )_) ,_, together_RL with_IW a_AT1 Borel_NN1 subgroup_NN1 BZ_NP1 (_( see_VV0 &lsqb;_( AMRW_NP1 ,_, §10.2_FO &rsqb;_) for_IF further_JJR remarks_NN2 and_CC &lsqb;_( RW_NP1 ,_, §9.1_FO &rsqb;_) for_IF an_AT1 overview_NN1 of_IO the_AT construction_NN1 )_) ._. 
</s>
<s>
Assuming_VVG that_CST all_DB column_NN1 vectors_NN2 @S_FO and_CC @S_FO are_VBR gathered_VVN in_II @S_FO matrices_NN2 F_ZZ1 and_CC B_ZZ1 ,_, respectively_RR ,_, we_PPIS2 first_MD define_VV0 the_AT @S_FO auxiliary_JJ matrices_NN2 @F_FO to_TO form_VVI the_AT vector_NN1 of_IO objectives_NN2 @F_FO and_CC the_AT matrix_NN1 of_IO gradients_NN2 @F_FO ._. 
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<s>
For_IF any_DD @S_FO ,_, we_PPIS2 get_VV0 @F_FO ._. 
</s>
<s>
At_II time_NNT1 tj_NNU ,_, we_PPIS2 apply_VV0 a_AT1 source_NN1 term_NN1 which_DDQ is_VBZ zero_MC everywhere_RL except_CS in_II the_AT cell_NN1 @S_FO ,_, where_CS it_PPH1 takes_VVZ the_AT form_NN1 @F_FO ._. 
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<s>
The_AT other_JJ parameters_NN2 are_VBR @S_FO ,_, with_IW a_AT1 mesh_NN1 size_NN1 approximately_RR @S_FO ._. 
</s>
<s>
The_AT beam_NN1 trajectory_NN1 is_VBZ a_AT1 sequence_NN1 of_IO horizontal_JJ paths_NN2 from_II left_NN1 to_II right_NN1 ,_, as_CSA can_VM be_VBI seen_VVN on_II Figure_NN1 18_MC where_RRQ the_AT temperature_NN1 field_NN1 is_VBZ plotted_VVN during_II the_AT building_NN1 process_NN1 for_IF the_AT reference_NN1 shape_NN1 of_IO Figure_NN1 6_MC :_: from_II top_NN1 to_II bottom_NN1 ,_, snapshots_NN2 at_II times_NNT2 @S._FO @T_FO Figure_NN1 18_MC ._. 
</s>
<s>
Before_II turning_VVG to_II these_DD2 details_NN2 ,_, though_CS ,_, we_PPIS2 will_VM explain_VVI the_AT motivations_NN2 behind_II the_AT different_JJ types_NN2 of_IO convergence_NN1 analyzed_VVD in_II this_DD1 paper_NN1 ._. 
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<s>
This_DD1 is_VBZ straightforward_JJ once_CS we_PPIS2 have_VH0 a_AT1 bound_NN1 on_II @S_FO ._. 
</s>
<s>
Then_RT there_EX is_VBZ a_AT1 sequence_NN1 @S_FO such_CS21 that_CS22 @S_FO 1_MC1 ._. 
</s>
<s>
Theorem_NN1 2_MC ._. 
</s>
<s>
If_CS N_ZZ1 =_FO nb_NNU then_RT we_PPIS2 put_VV0 @S_FO ._. 
</s>
<s>
Now_RT ,_, we_PPIS2 note_VV0 that_CST the_AT sesquilinear_JJ forms_NN2 @S_FO and_CC @S_FO are_VBR bounded_VVN forms_NN2 (_( with_IW constants_NN2 which_DDQ depend_VV0 on_II the_AT index_NN1 of_IO refraction_NN1 )_) ._. 
</s>
<s>
Fix_VV0 @S_FO and_CC @S_FO ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI the_AT event_NN1 that_CST @F_FO ,_, then_RT there_EX is_VBZ a_AT1 function_NN1 fyn_NN1 satisfying_VVG @S_FO and_CC @S_FO such_CS21 that_CS22 for_IF @S_FO ,_, @F_FO ._. 
</s>
<s>
The_AT poles_NN2 for_IF w_NNU of_IO the_AT same_DA integrands_NN2 are_VBR at_II @S_FO ;_; of_IO these_DD2 ,_, the_AT poles_NN2 at_II @S_FO are_VBR contained_VVN inside_II the_AT contour_NN1 @S_FO and_CC the_AT others_NN2 are_VBR left_VVN outside_RL ._. 
</s>
<s>
I_PPIS1 hand_VV0 a_AT1 project-based_JJ thing_NN1 to_II a_AT1 student_NN1 ,_, then_RT I_PPIS1 sometimes_RT feel_VV0 I_PPIS1 open_VVI them_PPHO2 up_II21 to_II22 a_AT1 lot_NN1 of_IO stress_NN1 and_CC frustration_NN1 ,_, which_DDQ stops_VVZ that_DD1 learning_NN1 process_NN1 .._... 
</s>
<s>
In_RR21 particular_RR22 ,_, the_AT form_NN1 of_IO attention_NN1 that_CST was_VBDZ classified_VVN as_CSA perceiving_VVG properties_NN2 increased_VVN across_II school_NN1 cycles_NN2 ._. 
</s>
<s>
Furthermore_RR ,_, since_CS @S_FO in_II Br_JJ ,_, f?_FO is_VBZ real_JJ valued_JJ and_CC Lipschitz_NP1 on_II (_( -1_MC ,_, @S_FO &rsqb;_) ._. 
</s>
<s>
If_CS the_AT problem_NN1 (_( 1.1_MC )_) has_VHZ the_AT @S_FO (_( 2.1_MC )_) instead_II21 of_II22 (_( 2.6_MC )_) ,_, then_RT the_AT variational_JJ equation_NN1 (_( 2.8_MC )_) still_RR holds_VVZ true_JJ with_IW @S_FO and_CC @S_FO ,_, where_CS q_ZZ1 is_VBZ the_AT conjugate_NN1 of_IO @S_FO so_CS21 that_CS22 @S_FO ._. 
</s>
<s>
This_DD1 can_VM be_VBI proved_VVN following_II the_AT classical_JJ idea_NN1 of_IO Brouwer_NN1 of_IO cancelling_VVG point_NN1 inverses_NN2 with_IW opposite_JJ local_JJ degree_NN1 ,_, but_CCB in_II a_AT1 careful_JJ layered_JJ way_NN1 so_BCL21 as_BCL22 to_TO be_VBI able_JK to_TO control_VVI the_AT Lipschitz_NP1 constants_NN2 ._. 
</s>
<s>
Our_APPGE experiments_NN2 demonstrate_VV0 the_AT SDDiP_NN1 '_NULL s_ZZ1 effectiveness_NN1 with_IW state-variable_JJ binarization_NN1 for_IF large-scale_JJ problems_NN2 ._. 
</s>
<s>
Exploiting_VVG negative_JJ curvature_NN1 Our_APPGE first_MD sub-routine_NN1 either_RR declares_VVZ the_AT problem_NN1 locally_RR "_" almost_RR convex_JJ "_" or_CC finds_VVZ a_AT1 direction_NN1 of_IO f_ZZ1 that_CST has_VHZ negative_JJ curvature_NN1 ,_, meaning_VVG a_AT1 direction_NN1 v_ZZ1 such_CS21 that_CS22 @S_FO ._. 
</s>
<s>
The_AT idea_NN1 to_TO make_VVI progress_NN1 on_II f_ZZ1 by_II moving_VVG in_II directions_NN2 of_IO descent_NN1 on_II the_AT Hessian_NN1 is_VBZ of_RR21 course_RR22 well-known_JJ ,_, and_CC relies_VVZ on_II the_AT fact_NN1 that_CST if_CS at_II a_AT1 point_NN1 z_ZZ1 the_AT function_NN1 f_ZZ1 is_VBZ "_" very_RG "_" non-convex_JJ ,_, i.e._REX @S_FO for_IF some_DD @S_FO ,_, then_RT we_PPIS2 can_VM reduce_VVI the_AT objective_NN1 significantly_RR (_( by_II a_AT1 constant_JJ fraction_NN1 of_IO @S_FO at_RR21 least_RR22 )_) by_II taking_VVG a_AT1 step_NN1 in_II a_AT1 direction_NN1 of_IO negative_JJ curvature_NN1 ._. 
</s>
<s>
We_PPIS2 thus_RR refer_VV0 to_II that_DD1 last_MD article_NN1 for_IF the_AT discussion_NN1 about_II non-trivial_JJ representations_NN2 and_CC general_JJ bundles_NN2 ._. 
</s>
<s>
Compared_VVN with_IW the_AT classic_JJ worst-case_JJ example_NN1 given_VVN in_II &lsqb;_( 32_MC &rsqb;_) ,_, the_AT tridiagonal_JJ matrix_NN1 A_ZZ1 above_RL consists_VVZ of_IO a_AT1 different_JJ diagonal_JJ element_NN1 k_ZZ1 (_( instead_II21 of_II22 2_MC )_) ._. 
</s>
<s>
If_CS 1∈Pc(M)Pacc(M)_FO then_RT (_( 38_MC )_) holds_VVZ only_RR for_IF t∈_FO &lsqb;_( 0,1_MC )_) ._. 
</s>
<s>
Hence_RR we_PPIS2 get_VV0 @F_FO ,_, where_CS @S_FO are_VBR defined_VVN by_II @F_FO ._. 
</s>
<s>
As_CSA @S_FO does_VDZ not_XX ramify_VVI in_II L_ZZ1 ,_, this_DD1 also_RR implies_VVZ that_CST @S_FO does_VDZ not_XX ramify_VVI in_II E._NP1 As_CSA we_PPIS2 may_VM assume_VVI @S_FO and_CC @S_FO are_VBR isomorphic_JJ over_II Ffi_NP1 ,_, the_AT group_NN1 @S_FO also_RR splits_VVZ over_II @S_FO ,_, ._. 
</s>
<s>
Since_CS @S_FO ,_, it_PPH1 follows_VVZ that_CST @S_FO ._. 
</s>
<s>
Also_RR it_PPH1 follows_VVZ from_II Lemma_NN1 5.5_MC and_CC the_AT assumption_NN1 that_CST @S_FO is_VBZ connected_VVN that_CST @S_FO is_VBZ connected_VVN ._. 
</s>
<s>
For_IF this_DD1 strategy_NN1 to_TO succeed_VVI ,_, we_PPIS2 must_VM deal_VVI with_IW the_AT non-standard_JJ nature_NN1 of_IO the_AT cylinder-like_JJ sets_NN2 Qr1,2_FO (_( zo_NN1 )_) ._. 
</s>
<s>
Using_VVG this_DD1 conclusion_NN1 ,_, we_PPIS2 carry_VV0 out_RP an_AT1 induction_NN1 argument_NN1 over_II scales_NN2 ,_, which_DDQ will_VM imply_VVI that_CST the_AT Lp0_FO -bound_JJ from_II above_RL holds_VVZ even_CS21 if_CS22 we_PPIS2 do_VD0 not_XX impose_VVI the_AT a_JJ21 priori_JJ22 assumption_NN1 ._. 
</s>
<s>
A_AT1 structure_NN1 group_NN1 G_ZZ1 of_IO linear_JJ transformations_NN2 of_IO T_ZZ1 ,_, such_CS21 that_CS22 for_IF every_AT1 @S_FO every_AT1 a_AT1 e_ZZ1 A_ZZ1 and_CC every_AT1 @S_FO Ta_UH one_PN1 has_VHZ @S_FO ,_, with_IW @S_FO ._. 
</s>
<s>
To_TO study_VVI the_AT case_NN1 when_CS t?1_FO ,_, we_PPIS2 expand_VV0 out_RP MT1_FO :_: @F_FO ._. 
</s>
<s>
To_TO prove_VVI Proposition_NN1 3.5_MC ,_, we_PPIS2 set_VV0 @S_FO ._. 
</s>
<s>
Applying_VVG the_AT involution_NN1 from_II the_AT proof_NN1 of_IO Lemma_NN1 5.1_MC in_II @S_FO (_( flipping_VVG the_AT sign_NN1 of_IO the_AT last_MD label_NN1 that_CST attacks_VVZ a_AT1 label_NN1 with_IW the_AT same_DA absolute_JJ value_NN1 )_) ,_, we_PPIS2 see_VV0 that_CST the_AT terms_NN2 for_IF w_ZZ1 e_ZZ1 A_ZZ1 "_" ,_, such_CS21 that_CS22 @S_FO for_IF some_DD @S_FO ,_, cancel_VV0 out_RP ._. 
</s>
<s>
Hence_RR ,_, if_CS the_AT object_NN1 is_VBZ far_RR away_RL ,_, then_RT it_PPH1 is_VBZ almost_RR identical_JJ to_II using_VVG far-field_JJ reconstruction_NN1 ,_, and_CC we_PPIS2 do_VD0 not_XX need_VVI to_TO refine_VVI better_RRR than_CSN @S._FO (_( iv_MC )_) A_ZZ1 similar_JJ analysis_NN1 can_VM be_VBI performed_VVN in_II the_AT general_JJ case_NN1 when_CS Tr_JJ and_CC Ts_ZZ2 may_VM not_XX coincide_VVI ,_, but_CCB we_PPIS2 skip_VV0 the_AT analysis_NN1 for_IF the_AT sake_NN1 of_IO simplicity_NN1 ._. 
</s>
<s>
Within_II that_DD1 group_NN1 ,_, Pearson_NP1 correlation_NN1 coefficients_NN2 were_VBDR computed_VVN between_II Age_NN1 in_II months_NNT2 ,_, and_CC the_AT Arithmetic_NN1 and_CC Attitude_NN1 scores_NN2 ._. 
</s>
<s>
Assume_VV0 X_ZZ1 =_FO E(X/XT)_NN1 satisfies_VVZ @S_FO as_II31 well_II32 as_II33 @S_FO ,_, then_RT we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Confirmation_NN1 of_IO survey_NN1 and_CC assessment_NN1 constructs_NN2 The_AT interview_NN1 data_NN corroborated_VVN and_CC illuminated_VVN the_AT findings_NN2 of_IO the_AT survey_NN1 data_NN on_II the_AT MBI_NP1 (_( pedagogical_JJ beliefs_NN2 )_) and_CC MTEBI_NP1 (_( teaching_VVG efficacy_NN1 beliefs_NN2 )_) and_CC the_AT SCK_NP1 assessment_NN1 using_VVG Table_NN1 2_MC Pearson_NP1 correlations_NN2 between_II final_JJ scores_NN2 of_IO pedagogical_JJ beliefs_NN2 (_( MBI_NP1 )_) ,_, teaching_VVG efficacy_NN1 beliefs_NN2 (_( MTEBI_NP1 )_) ,_, SCK_NP1 (_( LMT_NP1 )_) ,_, and_CC observed_VVD teaching_VVG practices_NN2 (_( SBLEOP_NP1 )_) @T_FO ._. 
</s>
<s>
Table_NN1 3_MC Percentage_NN1 of_IO participants_NN2 '_NULL by_II score_NN1 for_IF five_MC SBLEOP_NP1 observed_VVD classroom_NN1 events_NN2 from_II observations_NN2 1_MC1 and_CC 2_MC @T._FO the_AT LMT_NP1 ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI a_AT1 level_JJ map_NN1 of_IO bounded_JJ pro-modules_NN2 ,_, and_CC assume_VV0 that_CST the_AT underlying_JJ map_NN1 of_IO pro-spectra_NN2 is_VBZ an_AT1 equivalence_NN1 ._. 
</s>
<s>
This_DD1 approach_NN1 allows_VVZ us_PPIO2 to_TO make_VVI several_DA2 aspects_NN2 of_IO &lsqb;_( BD18_FO &rsqb;_) and_CC &lsqb;_( BD17_FO &rsqb;_) explicit_JJ in_II a_AT1 conceptual_JJ way_NN1 ._. 
</s>
<s>
The_AT child_NN1 who_PNQS provides_VVZ a_AT1 correct_JJ cardinal_JJ response_NN1 after_II counting_VVG eight_MC bears_NN2 but_CCB who_PNQS recounts_VVZ when_RRQ asked_VVD how_RGQ many_DA2 after_II counting_VVG a_AT1 large_JJ collection_NN1 of_IO pennies_NNU2 may_VM realize_VVI that_DD1 for_IF the_AT large_JJ collection_NN1 they_PPHS2 have_VH0 not_XX quite_RR counted_VVN accurately_RR ,_, and_CC thus_RR understands_VVZ that_CST it_PPH1 does_VDZ not_XX make_VVI sense_NN1 to_TO provide_VVI a_AT1 similar_JJ cardinal_JJ response_NN1 in_II this_DD1 case_NN1 ._. 
</s>
<s>
We_PPIS2 point_VV0 out_RP that_CST this_DD1 problem_NN1 is_VBZ analogous_JJ to_II the_AT following_JJ question_NN1 :_: does_VDZ the_AT singular_NN1 set_VVI of_IO Oseen-Frank_JJ minimizers_NN2 consist_VV0 locally_RR of_IO isolated_JJ points_NN2 ?_? 
</s>
<s>
In_II the_AT second_MD and_CC third_MD of_IO the_AT series_NN ,_, we_PPIS2 will_VM study_VVI the_AT existence_NN1 ,_, uniqueness_NN1 ,_, and_CC stability_NN1 of_IO strictly_RR positive_JJ entire_JJ solutions_NN2 of_IO @S_FO and_CC the_AT existence_NN1 of_IO transition_NN1 fronts_NN2 of_IO @S_FO ,_, respectively_RR ._. 
</s>
<s>
Although_CS Mrs._NNB Purl_NP1 did_VDD not_XX explicitly_RR refer_VVI to_II LT-based_JJ lesson_NN1 goals_NN2 in_II31 terms_II32 of_II33 broader_JJR curriculum_NN1 goals_NN2 ,_, she_PPHS1 sometimes_RT related_JJ students_NN2 '_NULL strategies_NN2 to_II previous_JJ units_NN2 of_IO instruction_NN1 (_( e.g._REX ,_, skip_VV0 counting_VVG in_II a_AT1 unit_NN1 on_II number_NN1 )_) ._. 
</s>
<s>
In_RR21 particular_RR22 ,_, it_PPH1 is_VBZ shown_VVN in_II &lsqb;_( 10_MC ,_, Lemma_NN1 4.2_MC &rsqb;_) that_CST (_( 22_MC )_) in_II (_( P.2_FO )_) is_VBZ satisfied_VVN with_IW @S_FO ._. 
</s>
<s>
Computational_JJ evidence_NN1 in_II portfolio_NN1 management_NN1 and_CC queueing_VVG confirm_VV0 that_CST our_APPGE data-driven_NN1 sets_VVZ significantly_RR outperform_VV0 traditional_JJ robust_JJ optimization_NN1 techniques_NN2 whenever_RRQV data_NN are_VBR available_JJ ._. 
</s>
<s>
An_AT1 important_JJ feature_NN1 of_IO our_APPGE approach_NN1 is_VBZ that_CST the_AT pressure_NN1 and_CC vortex_VV0 sources_NN2 ,_, restricted_VVN to_II the_AT boundary_NN1 ,_, are_VBR the_AT only_JJ unknowns_NN2 in_II the_AT method_NN1 ._. 
</s>
<s>
The_AT purposes_NN2 (_( in_II Table_NN1 2_MC )_) were_VBDR echoed_VVN through_II reflective_JJ descriptions_NN2 of_IO specific_JJ K-8_FO school_NN1 ,_, curriculum_NN1 ,_, teaching_VVG ,_, and_CC student_NN1 learning_NN1 connections_NN2 the_AT MTEs_NP2 provided_VVD for_IF PTs_NN2 during_II content_JJ courses_NN2 ._. 
</s>
<s>
Let_VV0 us_PPIO2 write_VVI Ufi(z)_NP1 for_IF the_AT first_MD component_NN1 of_IO Un_NP1 (_( z_ZZ1 ,_, t_ZZ1 )_) ._. 
</s>
<s>
In_RR21 addition_RR22 ,_, we_PPIS2 apply_VV0 our_APPGE results_NN2 in_II Sect._NP1 3_MC in_II the_AT numerical_JJ analysis_NN1 of_IO the_AT contact_NN1 model_NN1 ._. 
</s>
<s>
Ei_NP1 and_CC E2_FO are_VBR Q-Cartier_NP1 prime_JJ divisors_NN2 ._. 
</s>
<s>
Now_RT ,_, the_AT number_NN1 of_IO possible_JJ choices_NN2 of_IO (_( j1_FO ,_, j2_FO ,_, ..._... ,_, j3N2_FO )_) is_VBZ bounded_VVN by_II @S_FO and_CC ,_, given_VVN (_( j1_FO ,_, j2_FO ,_, ..._... ,_, j3N2_FO )_) ,_, the_AT number_NN1 of_IO choices_NN2 of_IO @F_FO is_VBZ bounded_VVN by_II N6No_FO ._. 
</s>
<s>
By_II adopting_VVG this_DD1 scheme_NN1 one_PN1 can_VM immediately_RR have_VHI zh_NNU in_II (_( 3.21_MC )_) replaced_VVD by_II just_RR z_ZZ1 itself_PPX1 (_( with_IW scaling_NN1 )_) :_: @F_FO by_II choosing_VVG @S_FO ,_, where_CS @S_FO is_VBZ ,_, for_REX21 instance_REX22 as_II in_II Example_NN1 3.1_MC ,_, chosen_VVN to_TO be_VBI @S_FO for_IF some_DD @S_FO ._. 
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<s>
This_DD1 motivates_VVZ the_AT idea_NN1 that_CST (_( 3.22_MC )_) can_VM serve_VVI as_II a_AT1 remedy_NN1 to_II the_AT shortcoming_NN1 of_IO the_AT balanced_JJ scheme_NN1 (_( 3.18_MC )_) ._. 
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<s>
Lemma_NN1 5.2_MC (_( Solutions_NN2 with_IW decreasing_JJ total_JJ variation_NN1 )_) ._. 
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<s>
These_DD2 include_VV0 deciding_VVG when_RRQ to_TO have_VHI PTs_NN2 continue_VVI working_VVG on_II an_AT1 activity_NN1 and_CC when_RRQ to_TO move_VVI on_RP ,_, what_DDQ level_NN1 of_IO justification_NN1 is_VBZ sufficient_JJ ,_, when_RRQ should_VM mathematical_JJ ideas_NN2 be_VBI introduced_VVN by_II the_AT instructor_NN1 and_CC when_RRQ should_VM they_PPHS2 arise_VVI from_II the_AT PTs_NN2 '_NULL activity_NN1 ,_, as_II31 well_II32 as_II33 other_JJ instructional_JJ decisions_NN2 ._. 
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<s>
The_AT isotopy_NN1 class_NN1 of_IO the_AT realization_NN1 is_VBZ independent_JJ of_IO the_AT cyclic_JJ ordering_NN1 of_IO the_AT @S_FO arrayed_VVN about_II z0_FO E_ZZ1 Aq_NP1 ._. 
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Now_RT we_PPIS2 can_VM prove_VVI the_AT well-posedness_NN1 of_IO (_( 1.1_MC )_) ._. 
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But_CCB we_PPIS2 can_VM not_XX hope_VVI for_IF this_DD1 in_RR21 general_RR22 ._. 
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<s>
In_RR21 particular_RR22 we_PPIS2 have_VH0 @F_FO ,_, where_CS x_ZZ1 '_NULL is_VBZ any_DD lift_NN1 of_IO (_( x_ZZ1 ,_, a_ZZ1 )_) ._. 
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<s>
ROC_NP1 curves_NN2 for_IF estimating_VVG the_AT skeleton_NN1 of_IO the_AT CPDAG_NN1 and_CC the_AT directed_JJ part_NN1 of_IO the_AT CPDAG_NN1 are_VBR obtained_VVN by_II varying_VVG kn_NNU for_IF all_DB GES_NP2 based_VVD methods_NN2 ,_, and_CC by_II varying_VVG an_AT1 for_IF the_AT PC_NN1 algorithm_NN1 ._. 
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<s>
Thus_RR we_PPIS2 have_VH0 @F_FO ,_, which_DDQ indeed_RR depends_VVZ only_RR on_II k_ZZ1 ,_, A_ZZ1 and_CC &lsqb;_( p_ZZ1 &rsqb;_) ,_, but_CCB not_XX on_II the_AT choice_NN1 of_IO the_AT classification_NN1 data_NN ._. 
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<s>
My_APPGE hope_NN1 here_RL is_VBZ to_TO provoke_VVI such_DA research_NN1 ._. 
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<s>
It_PPH1 is_VBZ widely_RR known_VVN that_CST the_AT Earth_NN1 as_II31 well_II32 as_II33 most_DAT of_IO the_AT planets_NN2 in_II the_AT solar_JJ system_NN1 all_DB generate_VV0 magnetic_JJ fields_NN2 through_II the_AT motion_NN1 of_IO electrically_RR conducting_VVG fluids_NN2 &lsqb;_( 10,16_MC &rsqb;_) ._. 
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<s>
In_II fact_NN1 ,_, for_IF technical_JJ reasons_NN2 ,_, for_IF a_AT1 general_NN1 locally_RR finitely_RR presented_VVN derived_VVD Artin_NP1 K-stack_NN1 X_ZZ1 ,_, Pantev_NP1 et_RA21 al_RA22 ._. 
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<s>
&lsqb;_( 24_MC &rsqb;_) do_VD0 not_XX define_VVI p-forms_NN2 as_CSA elements_NN2 of_IO H-p(OlX)_NN1 ,_, and_RR31 so_RR32 on_RR33 ,_, as_CSA we_PPIS2 have_VH0 sketched_VVN above_RL ._. 
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<s>
Such_DA reduction_NN1 is_VBZ usually_RR achieved_VVN via_II varifold_JJ maximum_JJ principles_NN2 ,_, see_VV0 e.g._REX First_MD set_NN1 @S_FO for_IF any_DD @S_FO ._. 
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<s>
Note_VV0 that_CST the_AT mutations_NN2 in_II directions_NN2 @S_FO for_IF different_JJ k_ZZ1 commute_VV0 with_IW each_PPX221 other_PPX222 ._. 
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<s>
Then_RT p_ZZ1 extends_VVZ to_II a_AT1 global_JJ type_NN1 strongly_RR @S-generic_FO over_II M._NN1 We_PPIS2 will_VM only_RR highlight_VVI aspects_NN2 of_IO the_AT students_NN2 '_NULL process_VV0 that_CST were_VBDR shared_VVN across_II all_DB the_AT projects_NN2 ,_, unless_CS otherwise_RR noted_VVN ._. 
</s>
<s>
When_CS d_ZZ1 =_FO 2_MC ,_, the_AT distribution_NN1 of_IO this_DD1 maximum_JJ cardinality_NN1 is_VBZ the_AT same_DA as_CSA the_AT distribution_NN1 of_IO the_AT length_NN1 of_IO the_AT longest_JJT decreasing_JJ subsequence_NN1 of_IO a_AT1 uniform_JJ permutation_NN1 of_IO 1_MC1 ,_, ..._... ,_, n_ZZ1 ,_, a_AT1 famous_JJ object_NN1 of_IO study_NN1 in_II probability_NN1 and_CC combinatorics_NN2 ._. 
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<s>
On_II the_AT other_JJ hand_NN1 ,_, if_CS the_AT JK_NP1 particles_NN2 are_VBR initially_RR nematically_RR aligned_VVN ,_, that_REX21 is_REX22 ,_, @S_FO ,_, the_AT heading_NN1 angles_NN2 of_IO those_DD2 all_RR particles_NN2 are_VBR unchanged_JJ for_IF whole_JJ time_NNT1 t_ZZ1 ._. 
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<s>
In_II the_AT mean-field_JJ limit_NN1 for_IF the_AT IS_VBZ model_NN1 this_DD1 feature_NN1 can_VM be_VBI modeled_VVN by_II replacing_VVG @S_FO with_IW @F_FO ,_, where_RRQ again_RT T_ZZ1 is_VBZ a_AT1 non-increasing_JJ positive_JJ regular_JJ function_NN1 supported_VVN in_II &lsqb;_( 0,1_MC &rsqb;_) ,_, p_ZZ1 is_VBZ the_AT spatial_JJ density_NN1 ,_, Mx_MC ,_, r_ZZ1 is_VBZ the_AT mass_NN1 within_II distance_NN1 R_ZZ1 from_II x_ZZ1 ._. 
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<s>
Their_APPGE animation_NN1 indicates_VVZ that_CST the_AT introduction_NN1 of_IO key_JJ academic_JJ vocabulary_NN1 and_CC fractional_JJ notation_NN1 was_VBDZ independent_JJ of_IO the_AT part-whole_JJ relations_NN2 highlighted_VVN in_II the_AT One_MC1 Brownie_NN1 to_II Share_NN1 activity_NN1 (_( Figs._NN2 5_MC ,_, 6_MC )_) ._. 
</s>
<s>
Lemma_NN1 7.3_MC (_( Neumann_NP1 &lsqb;_( 30_MC &rsqb;_) )_) Let_VV0 G_ZZ1 be_VBI a_AT1 group_NN1 ,_, and_CC let_VV0 Hi_UH ,_, 1≤i≤n_FO ,_, be_VBI subgroups_NN2 of_IO G._NP1 Suppose_VV0 there_EX are_VBR group_NN1 elements_NN2 gi∈G_FO so_CS21 that_CS22 @F_FO ._. 
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<s>
Then_RT there_EX is_VBZ i_ZZ1 such_CS21 that_CS22 |_NULL G:Hi_FO |_NULL <∞_FO ._. 
</s>
<s>
The_AT first_MD subcase_NN1 is_VBZ where_RRQ P_ZZ1 is_VBZ annular_JJ ._. 
</s>
<s>
Algorithm_NN1 1_MC1 takes_VVZ in_II functions_NN2 @S_FO as_CSA arguments_NN2 ._. 
</s>
<s>
These_DD2 results_NN2 can_VM be_VBI used_VVN to_TO guide_VVI future_JJ computational_JJ studies_NN2 of_IO nonlocal_JJ problems_NN2 ._. 
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<s>
Afterwards_RT ,_, we_PPIS2 show_VV0 that_CST the_AT evolution_NN1 @S_FO ,_, v(t)_NNU )_) satisfies_VVZ an_AT1 energydissipation_NN1 balance_NN1 (_( condition_NN1 (_( EBY_NP1 )_) in_II Definition_NN1 5.1_MC )_) ._. 
</s>
<s>
The_AT reader_NN1 is_VBZ welcome_JJ to_TO check_VVI that_CST (_( 3.31_MC )_) and_CC (_( 3.32_MC )_) are_VBR numerically_RR consistent_JJ with_IW (_( 3.29_MC )_) at_II v_ZZ1 =_FO 0_MC and_CC with_IW (_( 3.30_MC )_) at_II v_ZZ1 =_FO 1_MC1 within_II the_AT relative_JJ error_NN1 0.05%_FO and_CC 0.8%_FO ,_, respectively_RR ._. 
</s>
<s>
Here_RL we_PPIS2 recall_VV0 that_CST @S_FO is_VBZ determined_VVN by_II @S_FO less_RRR than_CSN an_AT1 error_NN1 tolerance_NN1 ._. 
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<s>
The_AT estimates_NN2 on_II A(p)_NP1 presented_VVD above_RL show_VV0 that_CST in_II all_DB three_MC examples_NN2 ,_, when_CS f_ZZ1 *_FU is_VBZ well_RR approximated_VVN by_II a_AT1 function_NN1 whose_DDQGE "_" degree_NN1 of_IO sparsity_NN1 "_" is_VBZ @S_FO ,_, then_RT @S_FO and_CC Theorem_NN1 3.2_MC may_VM be_VBI used_VVN ._. 
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<s>
We_PPIS2 are_VBR finally_RR ready_JJ to_TO complete_VVI the_AT proof_NN1 that_CST rf>_FO (_( v_ZZ1 )_) =_FO v_ZZ1 ,_, and_CC hence_RR that_RG 5_MC is_VBZ injective_JJ ._. 
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<s>
In_II this_DD1 sense_NN1 ,_, (_( 1.7_MC )_) can_VM be_VBI thought_VVN of_IO as_II a_AT1 first_MD step_NN1 towards_II a_AT1 better_JJR mathematical_JJ understanding_NN1 of_IO the_AT excitation_NN1 spectrum_NN1 of_IO Bose_NP1 gases_NN2 in_II the_AT Gross-Pitaevskii_JJ regime_NN1 corresponding_VVG to_II (_( 1.1_MC )_) ._. 
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<s>
Our_APPGE approach_NN1 is_VBZ to_TO divide_VVI and_CC conquer_VVI the_AT slice_NN1 spectral_JJ sequence_NN1 for_IF the_AT motivic_JJ sphere_NN1 spectrum_NN1 ._. 
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<s>
In_II the_AT current_JJ investigation_NN1 ,_, the_AT main_JJ research_NN1 questions_NN2 were_VBDR as_CSA follows_VVZ :_: (_( 1_MC1 )_) What_DDQ were_VBDR students_NN2 '_NULL goals_NN2 for_IF learning_VVG maths_NN1 in_II year_NNT1 8_MC and_CC year_NNT1 9_MC ?_? 
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<s>
Progressive_JJ hedging_NN1 algorithm_NN1 in_II stochastic_JJ programming_NN1 :_: minimization_NN1 mode_NN1 ._. 
</s>
<s>
From_II the_AT results_NN2 in_II Section_NN1 7.5_MC with_IW @S_FO and_CC again_RT Proposition_VV0 22_MC it_PPH1 now_RT follows_VVZ that_CST the_AT information_NN1 lower_RRR bound_VVN for_IF estimating_VVG @S_FO at_II @S_FO from_II observations_NN2 Y_ZZ1 in_II (_( 5_MC )_) equals_VVZ @F_FO ._. 
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<s>
Step_VV0 B_ZZ1 (_( Inductive_JJ step_NN1 )_) :_: Since_CS this_DD1 step_NN1 uses_VVZ similar_JJ arguments_NN2 as_II21 to_II22 the_AT initial_JJ step_NN1 and_CC it_PPH1 is_VBZ also_RR very_RG lengthy_JJ ,_, we_PPIS2 leave_VV0 its_APPGE proof_NN1 to_II Appendix_NN1 B._NP1 We_PPIS2 consider_VV0 the_AT spatially_RR inhomogeneous_JJ Landau_NP1 equation_NN1 with_IW initial_JJ data_NN that_CST is_VBZ bounded_VVN by_II a_AT1 Gaussian_JJ in_II the_AT velocity_NN1 variable_NN1 ._. 
</s>
<s>
The_AT data_NN were_VBDR analyzed_VVN to_TO identify_VVI the_AT relevant_JJ representations_NN2 and_CC respective_JJ transformations_NN2 used_VVN by_II the_AT teacher_NN1 and_CC students_NN2 :_: (_( 1_MC1 )_) In_II autonomous_JJ work_NN1 by_II the_AT students_NN2 ._. 
</s>
<s>
Since_CS @S_FO ,_, it_PPH1 follows_VVZ from_II (_( 5.5_MC )_) that_CST @F_FO ,_, where_CS the_AT constant_JJ C_ZZ1 depends_VVZ on_II the_AT parameters_NN2 @S_FO and_CC @S_FO ,_, but_CCB is_VBZ independent_JJ of_IO @S_FO (_( since_CS both_DB2 @S_FO and_CC @S_FO are_VBR fixed_JJ closed_JJ subsets_NN2 of_IO @S_FO ,_, independent_JJ of_IO @S_FO )_) ._. 
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<s>
Production_NN1 of_IO the_AT reversal_NN1 error_NN1 is_VBZ due_II21 to_II22 the_AT inability_NN1 of_IO the_AT solver_NN1 to_TO detect_VVI situations_NN2 in_II which_DDQ a_AT1 direct_JJ translation_NN1 strategy_NN1 leads_VVZ to_II an_AT1 incorrect_JJ equation_NN1 ._. 
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<s>
Lemma_NN1 4.4_MC Let_VV0 X_ZZ1 and_CC Y_ZZ1 be_VBI normal_JJ complex_JJ projective_JJ varieties_NN2 with_IW canonical_JJ singularities_NN2 ,_, and_CC let_VV0 β_NULL :_: Y_ZZ1 →_NULL X_ZZ1 be_VBI a_AT1 birational_JJ morphism_NN1 ._. 
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<s>
Both_DB2 gradient_NN1 descent_NN1 and_CC Krylov_NP1 subspace_NN1 methods_NN2 may_VM fail_VVI to_TO converge_VVI to_II the_AT global_JJ solution_NN1 of_IO problems_NN2 (_( P.TR_NP1 )_) and_CC (_( P.cu_NP1 )_) in_II the_AT "_" hard_JJ case_NN1 "_" Notation_NN1 If_CS S_ZZ1 is_VBZ a_AT1 metric_JJ space_NN1 ,_, we_PPIS2 denote_VV0 by_II B(E)_VBI the_AT a-algebra_NN1 of_IO Borel_NP1 sets_VVZ on_II S._NP1 The_AT Lebesgue_NN1 measure_NN1 in_II Rn_NP1 is_VBZ denoted_VVN by_II Ln_NP1 ,_, while_CS Hn-1_MC1 is_VBZ the_AT (_( n1_FO )_) -dimensional_JJ Hausdorff_NN1 measure_NN1 ._. 
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<s>
We_PPIS2 will_VM not_XX use_VVI Michael_NP1 '_NULL s_ZZ1 theorem_NN1 ,_, but_CCB prove_VV0 a_AT1 selection_NN1 result_NN1 which_DDQ is_VBZ adapted_VVN to_II our_APPGE simpler_JJR setting_NN1 (_( Proposition_NN1 3.1_MC )_) ._. 
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<s>
Lastly_RR ,_, for_IF Chinese_JJ biology_NN1 and_CC chemistry_NN1 students_NN2 ,_, 11_MC and_CC 6.5%_FO gain_VV0 first_MD class_NN1 degrees_NN2 ,_, respectively_RR ._. 
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<s>
For_IF any_DD @S_FO ,_, @S_FO for_IF some_DD @S_FO ._. 
</s>
<s>
In_II this_DD1 case_NN1 ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Noting_VVG that_CST @S_FO ,_, we_PPIS2 get_VV0 @F_FO ._. 
</s>
<s>
By_II Assumption_NN1 4.2_MC ,_, we_PPIS2 then_RT have_VH0 @F_FO ._. 
</s>
<s>
In_II the_AT four-step_JJ assignment_NN1 further_RRR described_VVN below_RL ,_, we_PPIS2 treated_VVD each_DD1 step_NN1 curriculum_NN1 analysis_NN1 ,_, written_JJ lesson_NN1 plan_NN1 ,_, animation_NN1 ,_, and_CC overall_JJ reflectionas_NN2 marking_VVG a_AT1 stage_NN1 in_II a_AT1 documentational_JJ genesis_NN1 process_NN1 ,_, thereby_RR generating_VVG new_JJ documents_NN2 as_CSA they_PPHS2 moved_VVD toward_II visions_NN2 of_IO enactment_NN1 ._. 
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<s>
Actually_RR @S_FO remains_VVZ @F_FO positive_JJ on_II A_ZZ1 ,_, @F_FO and_CC @S_FO is_VBZ bounded_VVN on_II A_ZZ1 ,_, the_AT uniform_NN1 and_CC @S_FO absolute_JJ convergence_NN1 imply_VV0 that_CST for_IF any_DD probability_NN1 measure_NN1 v_ZZ1 on_II A_ZZ1 ,_, @F_FO ._. 
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<s>
Expression_NN1 (_( 10_MC )_) for_IF the_AT definition_NN1 of_IO ipd+y_FO gives_VVZ @F_FO ,_, and_CC identity_NN1 (_( 11_MC )_) becomes_VVZ for_IF v_ZZ1 =_FO p_ZZ1 ,_, @F_FO ._. 
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<s>
Identity_NN1 (_( 11_MC )_) becomes_VVZ for_IF @S_FO @F_FO ._. 
</s>
<s>
If_CS there_EX is_VBZ no_AT path_NN1 between_II Xi_NN1 and_CC Xj_NP1 in_II Co_NP1 ,_, then_RT the_AT output_NN1 of_IO a_AT1 8-optimal_JJ oracle_NN1 forward_RL phase_NN1 of_IO (_( AR_UH )_) GES_NP2 does_VDZ not_XX contain_VVI an_AT1 edge_NN1 between_II Xi_NN1 and_CC Xj_NP1 ._. 
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<s>
In_II this_DD1 part_NN1 we_PPIS2 formulate_VV0 a_AT1 version_NN1 of_IO Ax-Schanuel_NP1 in_II the_AT setting_NN1 of_IO a_AT1 differential_JJ field_NN1 ._. 
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<s>
Here_RL is_VBZ how_RRQ I_PPIS1 see_VV0 the_AT relevant_JJ issues_NN2 ._. 
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<s>
Furthermore_RR ,_, optimal_JJ solutions_NN2 to_II these_DD2 problems_NN2 enjoy_VV0 a_AT1 strong_JJ ,_, finite-sample_JJ probabilistic_JJ guarantee_NN1 whenever_RRQV the_AT constraints_NN2 and_CC objective_JJ function_NN1 are_VBR concave_JJ in_II the_AT uncertainty_NN1 ._. 
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<s>
The_AT foregoing_JJ model_NN1 introduces_VVZ individual_JJ heterogeneity_NN1 and_CC local_JJ interactions_NN2 while_CS considering_VVG the_AT rate_NN1 of_IO interaction_NN1 no_RR as_CSA constan_NN1 '_NULL and_CC equal_JJ across_II all_DB individuals_NN2 (_( e.g._REX ,_, space_NN1 homogeneity_NN1 )_) ._. 
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<s>
A_AT1 local_JJ version_NN1 ,_, for_IF semilinear_JJ heat_NN1 equations_NN2 ,_, has_VHZ been_VBN obtained_VVN in_II &lsqb;_( 28_MC &rsqb;_) ,_, under_II a_AT1 smallness_NN1 condition_NN1 on_II the_AT target_NN1 to_TO be_VBI tracked_VVN ._. 
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<s>
Despite_II acting_VVG during_II class_NN1 within_II the_AT parameters_NN2 of_IO the_AT discourse_NN1 order_NN1 which_DDQ prescribes_VVZ appropriate_JJ (_( invisible_JJ )_) behaviour_NN1 for_IF a_AT1 girl_NN1 (_( "_" I_PPIS1 say_VV0 a_AT1 lot_NN1 in_II other_JJ lessons_NN2 ,_, but_CCB in_II maths_NN1 if_CS I_ZZ1 '_NULL m_MC not_XX sure_JJ about_II it_PPH1 ,_, I_PPIS1 won_VVD '_NULL t_ZZ1 put_VV0 my_APPGE hand_NN1 up_RP "_" )_) ,_, it_PPH1 seems_VVZ that_CST Anna_NP1 '_NULL s_ZZ1 engagement_NN1 in_II mathematics_NN1 somehow_RR crosses_VVZ the_AT boundaries_NN2 of_IO possibility_NN1 ,_, challenging_JJ gender_NN1 binaries_NN2 in_II her_APPGE heteroglossic_JJ self-authoring_NN1 as_CSA competitive_JJ ,_, careless_JJ and_CC "_" not_XX having_VHG to_TO work_VVI that_CST hard_RR "_" ._. 
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<s>
Let_VV0 @S_FO be_VBI a_AT1 real_JJ valued_JJ sequence_NN1 that_CST admits_VVZ logcorrelations_NN2 on_II @S_FO ._. 
</s>
<s>
We_PPIS2 call_VV0 the_AT system_NN1 (_( or_CC the_AT measure_NN1 p_ZZ1 )_) defined_VVD in_II Proposition_NN1 3.2_MC the_AT Furstenberg_NP1 system_NN1 (_( or_CC measure_NN1 )_) associated_VVN with_IW a_AT1 and_CC N._NNU On_II the_AT other_JJ hand_NN1 ,_, it_PPH1 is_VBZ not_XX known_VVN how_RRQ to_TO obtain_VVI such_DA a_AT1 compact_JJ description_NN1 for_IF a_AT1 quantum_NN1 computer_NN1 :_: there_EX is_VBZ no_AT equivalent_JJ for_IF the_AT random_JJ bits_NN2 ,_, and_CC a_AT1 characterization_NN1 of_IO the_AT state_NN1 truly_RR requires_VVZ an_AT1 exponential_NN1 number_NN1 of_IO complex_JJ coefficients_NN2 ._. 
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<s>
Thereafter_RT ,_, we_PPIS2 shall_VM focus_VVI much_DA1 of_IO our_APPGE attention_NN1 deriving_VVG an_AT1 appropriate_JJ set_NN1 ofPDE_NN1 which_DDQ govern_VV0 it_PPH1 ._. 
</s>
<s>
Given_VVN @S_FO ,_, recall_VV0 that_DD1 BZp_NP1 ;_; g_ZZ1 denotes_VVZ the_AT complement_NN1 of_IO the_AT points_NN2 of_IO @S_FO around_II which_DDQ g_ZZ1 is_VBZ an_AT1 open_JJ immersion_NN1 (_( see_VV0 §3.1.2_FO )_) ._. 
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<s>
By_II the_AT generalized_JJ Poincare_NN1 inequality_NN1 &lsqb;_( 26_MC ,_, Chapter_NN1 2_MC ,_, Section_NN1 1.4_MC &rsqb;_) ,_, it_PPH1 holds_VVZ that_CST @F_FO ,_, where_CS @S_FO is_VBZ the_AT Poincare_NN1 constant_NN1 ._. 
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<s>
The_AT meaning_NN1 of_IO both_DB2 line_NN1 segment_NN1 and_CC its_APPGE length_NN1 is_VBZ required_VVN for_IF constructing_VVG the_AT perpendicular_JJ bisector_NN1 ._. 
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<s>
Lef_VV0 u_ZZ1 be_VBI fhe_NN1 uieak_NN1 sohdicm_NN1 of_IO (_( 3.4_MC )_) such_CS21 that_CS22 it_PPH1 holds_VVZ :_: @S_FO ,_, where_CS @S_FO (_( see_VV0 (_( 1.1_MC )_) )_) ,_, By_II @S_FO we_PPIS2 denofe_NN1 a_AT1 family_NN1 of_IO resolving_VVG meshes_NN2 with_IW @S_FO and_CC by_II @S_FO we_PPIS2 denote_VV0 a_AT1 family_NN1 of_IO graded_JJ meshes_NN2 ._. 
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<s>
Suppose_VV0 that_CST @S_FO ,_, and_CC pick_VV0 an_AT1 index_NN1 r_ZZ1 in_II @S_FO ._. 
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<s>
Observe_VV0 that_CST Tm_NP1 does_VDZ not_XX commute_VVI with_IW Tm_NP1 but_CCB commutes_VVZ with_IW the_AT other_JJ @S_FO ;_; hence_RR @S_FO for_IF every_AT1 @S_FO ._. 
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<s>
Assume_VV0 by_II contradiction_NN1 that_CST ,_, for_IF every_AT1 x_ZZ1 in_II V_ZZ1 ,_, @S_FO is_VBZ a_AT1 linear_JJ combination_NN1 of_IO the_AT @S_FO ,_, j2S_FO ,_, and_CC write_VV0 @S_FO ,_, where_CS the_AT Gj_NP1 '_NULL s_ZZ1 are_VBR analytic_JJ functions_NN2 on_II U._NP1 The_AT derived_JJ models_NN2 can_VM discriminate_VVI between_II different_JJ physical_JJ effects_NN2 in_II observations_NN2 and_CC simulations_NN2 ,_, dictated_VVN by_II different_JJ scales_NN2 ._. 
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<s>
We_PPIS2 have_VH0 two_MC contradictory_JJ pictures_NN2 of_IO reality_NN1 ;_; separately_RR neither_DD1 of_IO them_PPHO2 fully_RR explains_VVZ the_AT phenomena_NN2 of_IO light_NN1 ,_, but_CCB together_RL they_PPHS2 do_VD0 ._. 
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<s>
For_IF the_AT proof_NN1 of_IO the_AT claim_NN1 we_PPIS2 may_VM replace_VVI k_ZZ1 by_II a_AT1 finite_JJ extension_NN1 ._. 
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<s>
We_PPIS2 shall_VM encounter_VVI two_MC particular_JJ classes_NN2 of_IO functions_NN2 Z_ZZ1 in_II the_AT sequel_NN1 ._. 
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<s>
Our_APPGE proof_NN1 of_IO the_AT upper_JJ bound_NN1 follows_VVZ the_AT ideas_NN2 of_IO &lsqb;_( 6_MC &rsqb;_) and_CC &lsqb;_( 7_MC &rsqb;_) ._. 
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<s>
Thus_RR ,_, our_APPGE goal_NN1 in_II this_DD1 section_NN1 will_VM be_VBI to_TO construct_VVI a_AT1 biregular_JJ bipartite_JJ graph_NN1 G_ZZ1 on_II the_AT vertex_NN1 sets_NN2 G_ZZ1 and_CC Guv_NP1 ,_, where_CS G_ZZ1 is_VBZ the_AT set_NN1 of_IO adjacency_NN1 matrices_NN2 of_IO simple_JJ d-regular_JJ graphs_NN2 on_II n_ZZ1 vertices_VVZ ,_, and_CC Guv_NN1 is_VBZ the_AT subset_NN1 of_IO G_ZZ1 of_IO matrices_NN2 with_IW uv_JJ entry_NN1 equal_JJ to_II 1_MC1 ._. 
</s>
<s>
We_PPIS2 can_VM now_RT define_VVI a_AT1 filtration_NN1 on_II @S_FO where_RRQ the_AT rth_NNU filtered_JJ piece_NN1 is_VBZ spanned_VVN over_II @S_FO by_II the_AT M_NN1 with_IW @S_FO for_IF an_AT1 arbitrary_JJ fixed_JJ W-invariant_JJ quadratic_JJ form_NN1 @S_FO ._. 
</s>
<s>
By_II Proposition_NN1 2.5_MC multiplication_NN1 respects_VVZ this_DD1 filtration_NN1 ._. 
</s>
<s>
In_II fact_NN1 ,_, to_TO recover_VVI ergodicity_NN1 by_II adaptivity_NN1 has_VHZ already_RR been_VBN proposed_VVN in_II &lsqb;_( 34_MC &rsqb;_) ._. 
</s>
<s>
Next_MD we_PPIS2 introduce_VV0 the_AT Gaufi-Legendre_NP1 quadrature_NN1 rules_NN2 ._. 
</s>
<s>
Assume_VV0 that_CST for_IF some_DD constant_JJ @S_FO ,_, @L_FO ._. 
</s>
<s>
Then_RT @S_FO for_IF some_DD constant_JJ @S_FO depending_II21 on_II22 s_ZZ1 and_CC dimension_NN1 only_RR ._. 
</s>
<s>
We_PPIS2 now_RT consider_VV0 prediction_NN1 of_IO a_AT1 Gaussian_JJ field_NN1 at_II a_AT1 new_JJ location_NN1 s0_FO ,_, using_VVG the_AT GW_NP1 model_NN1 ,_, under_II fixed_JJ domain_NN1 asymptotics_NN2 ._. 
</s>
<s>
See_VV0 &lsqb;_( 9_MC ,_, 10_MC &rsqb;_) and_CC &lsqb;_( 46_MC ,_, Section_NN1 5_MC &rsqb;_) for_IF more_DAR discussion_NN1 of_IO this_DD1 '_NULL multi-scale_JJ '_NULL phenomenon_NN1 ._. 
</s>
<s>
A_AT1 general_JJ error_NN1 bound_VVN for_IF k-medoids_NN2 clustering_NN1 ._. 
</s>
<s>
Here_RL the_AT Lipschitz_NP1 constant_NN1 is_VBZ independent_JJ of_IO the_AT regularization_NN1 parameters_NN2 @S_FO and_CC @S_FO ._. 
</s>
<s>
Proof_NN1 ._. 
</s>
<s>
This_DD1 is_VBZ not_XX surprising_JJ as_CSA saccades_NN2 and_CC fixations_NN2 alternate_VV0 in_II regular_JJ reading_NN1 and_CC event_NN1 detection_NN1 algorithms_NN2 typically_RR infer_VV0 one_PN1 from_II the_AT other_JJ (_( Salvucci_NP1 &;_NULL Goldberg_NP1 ,_, 2000_MC )_) ._. 
</s>
<s>
Also_RR ,_, it_PPH1 lacks_VVZ a_AT1 weak_JJ version_NN1 ,_, and_CC its_APPGE coordinate_NN1 formulation_NN1 is_VBZ wrong_JJ (_( a_AT1 correct_JJ one_PN1 is_VBZ given_VVN in_II Section_NN1 2.5_MC )_) ._. 
</s>
<s>
Thus_RR ,_, property_NN1 (_( R5_FO )_) is_VBZ simply_RR a_AT1 mathematical_JJ articulation_NN1 of_IO this_DD1 convention_NN1 ._. 
</s>
<s>
We_PPIS2 invoke_VV0 Theorem_NN1 5.2_MC to_TO obtain_VVI a_AT1 lower_JJR bound_NN1 for_IF the_AT stray_JJ field_NN1 energy_NN1 ._. 
</s>
<s>
Then_RT let_VV0 E=E(1)∞H_FO ,_, where_CS H_ZZ1 is_VBZ the_AT closed_JJ half_NN1 plane_NN1 with_IW 0∈HH_FO and_CC q1_FO ,_, q2∈H_FO ._. 
</s>
<s>
Moreover_RR ,_, there_EX exists_VVZ a_AT1 set_NN1 A_ZZ1 c_ZZ1 D2_FO (_( P*_FO )_) of_IO initial_JJ data_NN ofpositive_JJ measure_NN1 with_IW thefollowing_VVG property:for_FO all_RR Z0_FO e_ZZ1 A_ZZ1 ,_, there_RL exist_VV0 uncountably-many_DA2 associated_JJ distinct_JJ global-in-time_JJ physical_JJ weak_JJ solutions_NN2 of_IO (_( 4_MC )_) ._. 
</s>
<s>
This_DD1 is_VBZ regarded_VVN as_CSA profound_JJ understanding_NN1 of_IO fundamental_JJ SMTPCK_NP1 ,_, which_DDQ is_VBZ to_II teacher_NN1 educators_NN2 and_CC PCK_NP1 what_DDQ Ma_NP1 '_NULL s_ZZ1 Profound_JJ Understanding_NN1 of_IO Fundamental_JJ Mathematics_NN1 (_( Ma_NP1 1999_MC )_) is_VBZ to_II school_NN1 teachers_NN2 and_CC mathematics_NN1 ._. 
</s>
<s>
Hence_RR ,_, we_PPIS2 obtain_VV0 for_IF all_DB @S_FO ,_, @S_FO that_DD1 @F_FO ._. 
</s>
<s>
Then_RT u_ZZ1 is_VBZ 1D_NNU profile_NN1 ,_, namely_REX ,_, u(x)=_FO (_( ex_NN1 )_) for_IF some_DD e∈S2_FO ,_, where_CS :_: R_ZZ1 →_NULL R_ZZ1 is_VBZ a_AT1 nondecreasing_NN1 bounded_VVN stable_JJ solution_NN1 to_II (_( 1.5_MC )_) in_II dimension_NN1 one_MC1 ._. 
</s>
<s>
For_IF the_AT purpose_NN1 of_IO analysis_NN1 ,_, the_AT least-squares_NN2 solution_NN1 to_II (_( 5.4_MC )_) is_VBZ often_RR written_VVN as_II the_AT solution_NN1 to_II the_AT corresponding_JJ set_NN1 of_IO normal_JJ equations_NN2 ._. 
</s>
<s>
Working_VVG with_IW the_AT original_JJ sources_NN2 &lsqb;_( of_IO Euler_NP1 and_CC Dirichlet_NP1 &rsqb;_) was_VBDZ really_RR difficult_JJ for_IF the_AT teacher_NN1 students_NN2 ._. 
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<s>
The_AT examples_NN2 here_RL have_VH0 four_MC objectives_NN2 ._. 
</s>
<s>
In_II §9_FO we_PPIS2 consider_VV0 Galois_NP1 lattices_NN2 ._. 
</s>
<s>
Likewise_RR ,_, we_PPIS2 assume_VV0 that_CST the_AT problem_NN1 (_( PS_NN1 )_) has_VHZ at_RR21 least_RR22 one_MC1 optimal_JJ solution_NN1 (_( sufficient_JJ conditions_NN2 ensuring_VVG existence_NN1 are_VBR standard_JJ ;_; see_VV0 ,_, e.g._REX ,_, &lsqb;_( 17_MC &rsqb;_) )_) ._. 
</s>
<s>
It_PPH1 is_VBZ convenient_JJ to_TO introduce_VVI the_AT quantity_NN1 @F_FO ,_, where_CS @S_FO is_VBZ a_AT1 d-dimensional_JJ linear_JJ subspace_NN1 ._. 
</s>
<s>
Moreover_RR ,_, while_CS fast_JJ algorithms_NN2 for_IF harmonic_JJ and_CC heat_NN1 potentials_NN2 are_VBR available_JJ in_II the_AT three-dimensional_JJ setting_NN1 ,_, suitable_JJ quadrature_NN1 rules_NN2 and_CC high-order_JJ surface_NN1 representations_NN2 are_VBR still_JJ areas_NN2 of_IO active_JJ research_NN1 ,_, especially_RR for_IF moving_VVG geometries_NN2 ._. 
</s>
<s>
These_DD2 convergence_NN1 results_NN2 do_VD0 not_XX hinge_NN1 on_II any_DD assumptions_NN2 about_II the_AT regularity_NN1 of_IO solutions_NN2 ._. 
</s>
<s>
Let_VV0 @S_FO ,_, assume_VVI that_CST @S_FO and_CC @S_FO hold_VV0 ,_, and_CC @S_FO is_VBZ the_AT Dirichlet_NN1 boundary_NN1 operator_NN1 ._. 
</s>
<s>
Smoothness_NN1 of_IO Sij_NP1 also_RR requires_VVZ s_ZZ1 +_FO s_ZZ1 to_TO approach_VVI K_ZZ1 at_II fixed_JJ points_NN2 fast_RR enough_RR ._. 
</s>
<s>
As_CSA in_II Lemma_NN1 2.9_MC we_PPIS2 consider_VV0 the_AT two_MC cases_NN2 ._. 
</s>
<s>
The_AT resulting_JJ distribution_NN1 of_IO 10%_NNU worst-case_JJ return_NN1 is_VBZ shown_VVN in_II the_AT top_JJ right_JJ panel_NN1 of_IO Fig._NN1 4_MC and_CC the_AT average_NN1 of_IO these_DD2 runs_NN2 is_VBZ shown_VVN Table_NN1 3_MC under_II column_NN1 ZAvg_NN1 ._. 
</s>
<s>
As_CSA might_VM have_VHI been_VBN guessed_VVN from_II the_AT crossvalidation_NN1 results_NN2 ,_, @S_FO delivers_VVZ more_RGR stable_JJ and_CC better_JJR performance_NN1 than_CSN either_RR @S_FO or_CC @S._FO @S_FO slightly_RR outperforms_VVZ CM_NNU ,_, and_CC its_APPGE distribution_NN1 is_VBZ shifted_VVN right_RR ._. 
</s>
<s>
Since_CS the_AT single-sample_JJ algorithm_NN1 illuminates_VVZ the_AT performance_NN1 advantage_NN1 gained_VVN from_II this_DD1 approach_NN1 ,_, we_PPIS2 examine_VV0 it_PPH1 first_MD ._. 
</s>
<s>
In_RR21 particular_RR22 an_AT1 equivalence_NN1 of_IO pro-simplicial_JJ rings_NN2 is_VBZ a_AT1 morphism_NN1 f:A_FO →_NULL B_ZZ1 inducing_VVG isomorphisms_NN2 of_IO pro-groups_NN2 i(A)_NNU →_NULL i(B)_NN1 for_IF all_DB i≥0_FO ._. 
</s>
<s>
In_II this_DD1 section_NN1 we_PPIS2 describe_VV0 a_AT1 quantum_NN1 algorithm_NN1 ,_, known_VVN as_II Simon_NP1 '_NULL s_ZZ1 algorithm_NN1 &lsqb;_( 36_MC &rsqb;_) ,_, that_DD1 gives_VVZ an_AT1 expected_JJ exponential_NN1 speedup_NN1 with_II31 respect_II32 to_II33 classical_JJ algorithms_NN2 ._. 
</s>
<s>
We_PPIS2 compute_VV0 @S_FO for_IF @S_FO and_CC @S_FO and_CC all_DB their_APPGE orientations_NN2 for_IF the_AT ramification_NN1 profile_NN1 @S_FO consisting_VVG of_IO a_AT1 three-cycle_NN1 ._. 
</s>
<s>
Given_VVN a_AT1 subspace_NN1 Vh_NP1 of_IO @S_FO ,_, a_AT1 conforming_VVG scheme_NN1 for_IF (_( 1.2_MC )_) is_VBZ written_VVN @F_FO ._. 
</s>
<s>
If_CS @S_FO and_CC @S_FO is_VBZ globally_RR Lipschitz_VV0 continuous_JJ ,_, we_PPIS2 have_VH0 @S_FO and_CC the_AT key_NN1 of_IO the_AT convergence_NN1 analysis_NN1 is_VBZ that_CST the_AT chain_NN1 rule_NN1 @S_FO enables_VVZ us_PPIO2 to_TO take_VVI @S_FO as_II a_AT1 test_NN1 function_NN1 in_II the_AT scheme_NN1 ._. 
</s>
<s>
However_RR ,_, in_II this_DD1 case_NN1 ,_, corollary_NN1 1_MC1 can_VM be_VBI used_VVN directly_RR toderive_VV0 the_AT following_JJ corollary_NN1 ._. 
</s>
<s>
Due_II21 to_II22 the_AT availability_NN1 of_IO heat_NN1 kernel_NN1 and_CC distance_NN1 distortion_NN1 estimates_NN2 ,_, it_PPH1 is_VBZ likely_JJ that_CST such_DA flows_NN2 offer_VV0 a_AT1 useful_JJ model_NN1 case_NN1 for_IF a_AT1 synthetic_JJ definition_NN1 of_IO Ricci_JJ flowsperhaps_NN2 via_II optimal_JJ transport_NN1 ,_, generalizing_VVG the_AT approach_NN1 of_IO Sturm_NP1 (_( see_VV0 &lsqb;_( Stu16_FO &rsqb;_) )_) ._. 
</s>
<s>
Thus_RR @F_FO This_DD1 integral_JJ is_VBZ positive_JJ ,_, since_CS @S_FO is_VBZ the_AT (_( positive_JJ )_) den_NN1 '_NULL ity_NN1 of_IO Y_ZZ1 ,_, except_CS when_CS @S_FO is_VBZ concentrated_VVN on_II the_AT diagonal_JJ @S_FO which_DDQ happens_VVZ only_RR if_CS @S_FO is_VBZ a_AT1 Dirac_JJ mass_NN1 ,_, that_REX21 is_REX22 ,_, if_CS @S_FO has_VHZ zero_MC variance_NN1 (_( in_II that_DD1 case_NN1 @S_FO for_IF any_DD @S_FO )_) ._. 
</s>
<s>
In_II this_DD1 simple_JJ example_NN1 ,_, the_AT origin_NN1 is_VBZ the_AT unique_JJ maximizer_NN1 (_( and_CC hence_RR unique_JJ Nash_NP1 equilibrium_NN1 )_) of_IO u.Moreover_NNU ,_, wetriviallyhave_VV0 @S_FO with_IW equality_NN1 if_CS and_CC only_RR if_CS x_ZZ1 =_FO 0_MC ,_, so_CS the_AT origin_NN1 satisfies_VVZ the_AT global_JJ version_NN1 of_IO (_( VS_II )_) ;_; however_RR ,_, u_ZZ1 is_VBZ not_XX even_RR pseudo-concave_JJ if_CS d_ZZ1 >_FO 2_MC ,_, so_CS the_AT game_NN1 can_VM not_XX be_VBI monotone_NN1 ._. 
</s>
<s>
Next_MD ,_, in_II Fig._NN1 7_MC ,_, we_PPIS2 compare_VV0 the_AT @S_FO error_NN1 of_IO VEM_NP1 defined_VVD in_II (_( 67_MC )_) with_IW the_AT standard_NN1 @S_FO error_NN1 of_IO hp_NNU FEM_NN1 employing_VVG the_AT same_DA meshes_NN2 and_CC discretization_NN1 parameters_NN2 discussed_VVN for_IF the_AT comparison_NN1 of_IO @S_FO errors_NN2 on_II the_AT skeleton_NN1 ._. 
</s>
<s>
Now_RT let_VV0 X_ZZ1 be_VBI a_AT1 very_RG general_JJ hypersurface_NN1 of_IO degree_NN1 @S_FO with_IW d_ZZ1 odd_JJ as_CSA in_II the_AT theorem_NN1 ._. 
</s>
<s>
The_AT potential_NN1 for_IF students_NN2 using_VVG Textbooks_NN2 4A_FO and_CC 8A_FO to_TO engage_VVI with_IW more_RGR varied_JJ types_NN2 of_IO understanding_NN1 (_( Table_NN1 5_MC )_) ,_, tasks_NN2 of_IO higher_JJR complexity_NN1 (_( Table_NN1 8_MC )_) and_CC tasks_NN2 requiring_VVG higher_JJR levels_NN2 of_IO understanding_NN1 (_( Table_NN1 7_MC )_) is_VBZ greater_JJR than_CSN that_DD1 of_IO their_APPGE grade-level_JJ counterparts_NN2 using_VVG Textbooks_NN2 4B_FO and_CC 8B_FO ._. 
</s>
<s>
The_AT first_MD trend_NN1 is_VBZ a_AT1 highly_RR visible_JJ one_PN1 ,_, and_CC one_PN1 that_CST has_VHZ already_RR been_VBN extremely_RR consequential_JJ ._. 
</s>
<s>
However_RR ,_, at_II e_ZZ1 =_FO 0.1_MC ,_, the_AT diffusion_NN1 approximation_NN1 starts_VVZ to_TO lose_VVI validity_NN1 and_CC deviates_VVZ from_II the_AT P5_FO solution_NN1 over_RG 10%_NNU in_II places_NN2 ._. 
</s>
<s>
All_DB these_DD2 generalizations_NN2 are_VBR in_II Euclidean_JJ settings_NN2 ._. 
</s>
<s>
Yet_RR ,_, after_CS the_AT students_NN2 asked_VVD even_RR more_DAR questions_NN2 ,_, she_PPHS1 tried_VVD again_RT :_: "_" So_RR think_VV0 -_- ifI_NN2 want_VV0 you_PPY to_TO set_VVI up_RP a_AT1 problem_NN1 so_CS21 that_CS22 the_AT solution_NN1 would_VM be_VBI a_AT1 to_II the_AT zero_NN1 power_NN1 "_" (_( 1057_MC )_) ._. 
</s>
<s>
In_II fact_NN1 ,_, @F_FO ,_, where_CS we_PPIS2 have_VH0 used_VVN that_CST @S_FO so_CS21 that_CS22 @S_FO ._. 
</s>
<s>
Thus_RR ,_, we_PPIS2 conclude_VV0 the_AT proof_NN1 ._. 
</s>
<s>
The_AT fact_NN1 that_CST the_AT law_NN1 of_IO CLEk_NP1 that_DD1 is_VBZ constructed_VVN in_II this_DD1 way_NN1 does_VDZ not_XX depend_VVI on_II the_AT root_NN1 is_VBZ non-trivial_JJ ,_, and_CC relies_VVZ on_II another_DD1 construction_NN1 of_IO these_DD2 CLE_NN1 '_NULL s_ZZ1 using_VVG the_AT Brownian_JJ loop-soups_NN2 (_( see_VV0 &lsqb;_( 25_MC &rsqb;_) )_) ._. 
</s>
<s>
The_AT understandinggap_NN1 between_II the_AT two_MC 8th-grade_JJ textbooks_NN2 increases_VVZ when_RRQ algorithmic_JJ tasks_NN2 are_VBR excluded_VVN from_II the_AT understanding_NN1 level_NN1 computation_NN1 ._. 
</s>
<s>
Therefore_RR ∈C_FO ,_, and_CC this_DD1 proves_VVZ part_NN1 (_( ii_MC )_) ._. 
</s>
<s>
To_II this_DD1 end_NN1 ,_, consider_VV0 the_AT measures_NN2 τ_NULL ,_, τ_NULL and_CC τ_NULL induced_VVN on_II the_AT countable_JJ probability_NN1 space_NN1 Qk_NP1 by_II @S_FO on_II R_ZZ1 :_: that_REX21 is_REX22 ,_, τ_NULL (_( I_ZZ1 )_) =_FO τ_NULL (_( I_ZZ1 )_) and_CC similarly_RR for_IF τ_NULL ,_, τ_NULL ._. 
</s>
<s>
Clearly_RR τ_NULL =_FO (_( 1δ_FO )_) τ_NULL +δ_FO τ_NULL ,_, so_CS @S_FO ._. 
</s>
<s>
Let_VV0 f=d_FO τ_NULL /d_FU τ_NULL ,_, so_CS f(I)=_FO τ_NULL (_( I_ZZ1 )_) /_FO τ_NULL (_( I_ZZ1 )_) ._. 
</s>
<s>
Children_NN2 '_NULL s_ZZ1 appropriation_NN1 of_IO the_AT informal_JJ multiplication_NN1 method_NN1 The_AT following_JJ stories_NN2 from_II child_NN1 participants_NN2 demonstrate_VV0 the_AT discontinuity_NN1 between_II home_NN1 and_CC school_NN1 learning_NN1 ,_, experienced_VVD especially_RR by_II immigrant_JJ families_NN2 ._. 
</s>
<s>
Students_NN2 are_VBR often_RR expected_VVN to_TO solve_VVI tasks_NN2 of_IO this_DD1 kind_NN1 by_II comparing_VVG means_NN of_IO the_AT treatment_NN1 and_CC control_NN1 group_NN1 in_BCL21 order_BCL22 to_TO test_VVI the_AT hypothesis_NN1 that_CST calcium_NN1 has_VHZ no_AT effect_NN1 on_II blood_NN1 pressure_NN1 ._. 
</s>
<s>
The_AT step_NN1 size_NN1 parameters_NN2 are_VBR chosen_VVN as_CSA derived_VVN in_II subsection_NN1 6.1_MC ;_; in_RR21 particular_RR22 ,_, we_PPIS2 choose_VV0 @L_FO ._. 
</s>
<s>
Note_VV0 that_CST the_AT contraction_NN1 rates_NN2 of_IO one_MC1 epoch_NN1 thetan_NN1 already_RR indicate_VV0 that_CST SPDHG_NP1 (_( n_ZZ1 50_MC )_) may_VM be_VBI faster_JJR than_CSN PDHG_NP1 and_CC SPDHG_NP1 (_( n_ZZ1 10_MC )_) ._. 
</s>
<s>
We_PPIS2 also_RR consider_VV0 Krylov_NP1 subspace_NN1 solutions_NN2 and_CC establish_VV0 sharp_JJ convergence_NN1 guarantees_VVZ to_II the_AT solutions_NN2 of_IO both_RR trust-region_JJ and_CC cubic-regularized_JJ problems_NN2 ._. 
</s>
<s>
It_PPH1 now_RT follows_VVZ from_II Proposition_NN1 6.5_MC that_CST applying_VVG first_MD @S_FO and_CC then_RT @S_FO is_VBZ different_JJ from_II applying_VVG @S_FO and_CC then_RT @S_FO ._. 
</s>
<s>
This_DD1 is_VBZ a_AT1 contradiction_NN1 ._. 
</s>
<s>
Hence_RR ,_, by_II the_AT continuity_NN1 of_IO the_AT determinant_NN1 and_CC eigenvalues_NN2 of_IO a_AT1 matrix_NN1 ,_, we_PPIS2 have_VH0 that_DD1 there_EX exists_VVZ kd_NNU >_FO 0_MC such_CS21 that_CS22 ,_, for_IF k_ZZ1 >_FO kd_NNU ,_, the_AT matrix_NN1 is_VBZ invertible_JJ and_CC @F_FO ,_, where_CS @S_FO denotes_VVZ the_AT eigenvalue_NN1 of_IO a_AT1 matrix_NN1 with_IW smallest_JJT absolute_JJ value_NN1 ._. 
</s>
<s>
This_DD1 approach_NN1 recasts_VVZ the_AT problem_NN1 as_II an_AT1 integral_JJ equation_NN1 in_II a_AT1 bounded_JJ domain_NN1 ,_, and_CC it_PPH1 proceeds_VVZ by_II computing_VVG certain_JJ singular_JJ exponents_NN2 a_AT1 that_RG make_VV0 @S_FO analytic_JJ near_II the_AT boundary_NN1 for_IF every_AT1 polynomial_NN1 @S_FO ._. 
</s>
<s>
As_CSA shown_VVN in_II Theorem_NN1 3.7_MC a_AT1 infinite_JJ sequence_NN1 of_IO such_DA values_NN2 of_IO a_AT1 is_VBZ given_VVN by_II @S_FO for_IF all_DB @S_FO ._. 
</s>
<s>
Morever_VV0 ,_, Section_NN1 3.2_MC shows_VVZ that_CST the_AT weighted_JJ operator_NN1 @S_FO maps_NN2 polynomials_NN2 of_IO degree_NN1 n_ZZ1 into_II polynomials_NN2 of_IO degree_NN1 nand_VV0 it_PPH1 provides_VVZ explicit_JJ closed-form_JJ expressions_NN2 for_IF the_AT images_NN2 of_IO each_DD1 polynomial_NN1 @S_FO ._. 
</s>
<s>
A_AT1 quick_JJ adaptation_NN1 of_IO the_AT last_MD proof_NN1 gives_VVZ a_AT1 better_JJR result_NN1 (_( see_VV0 ,_, e.g._REX ,_, &lsqb;_( 7_MC &rsqb;_) and_CC the_AT continuity_NN1 result_NN1 theorem_NN1 )_) ._. 
</s>
<s>
Before_II starting_VVG the_AT proof_NN1 of_IO Theorem_NN1 4_MC ,_, let_VV0 us_PPIO2 recall_VVI the_AT Fan_NN1 inequalities_NN2 in_II Lemma_NN1 52_MC (_( see_VV0 for_REX21 instance_REX22 &lsqb;_( 42_MC ,_, Theorem_NN1 1.6_MC &rsqb;_) or_CC &lsqb;_( 43_MC &rsqb;_) )_) ._. 
</s>
<s>
By_II (_( 5.38_MC )_) and_CC (_( 5.39_MC )_) ,_, we_PPIS2 conclude_VV0 @F_FO ._. 
</s>
<s>
This_DD1 is_VBZ a_AT1 generalisation_NN1 of_IO &lsqb;_( GWZ17_FO ,_, Section_NN1 4_MC &rsqb;_) ,_, which_DDQ is_VBZ motivated_VVN by_II &lsqb;_( Bat99_FO ,_, DL02_FO &rsqb;_) ,_, and_CC also_RR &lsqb;_( Yas06_FO &rsqb;_) ._. 
</s>
<s>
We_PPIS2 note_VV0 that_CST ,_, even_RR for_IF very_RG small_JJ @S_FO ,_, the_AT update_NN1 (_( 4.6_MC )_) is_VBZ not_XX guaranteed_VVN to_TO reduce_VVI the_AT overall_JJ cost_NN1 functionwe_NN1 have_VH0 traded_VVN the_AT mean_JJ for_IF a_AT1 single_JJ sample_NN1 ._. 
</s>
<s>
Here_RL are_VBR some_DD properties_NN2 of_IO these_DD2 :_: (_( a_ZZ1 )_) If_CS X_ZZ1 is_VBZ a_AT1 derived_JJ K-scheme_NN1 ,_, then_RT @S_FO for_IF k_ZZ1 >_FO 0_MC ._. 
</s>
<s>
In_II each_DD1 optimization_NN1 iteration_NN1 ,_, the_AT high-fidelity_JJ model_NN1 is_VBZ evaluated_VVN at_II the_AT minimizer_NN1 of_IO the_AT low-fidelity_JJ model_NN1 ._. 
</s>
<s>
As_CSA demonstrated_VVN in_II the_AT examples_NN2 below_RL ,_, the_AT teacher_NN1 opened_VVD the_AT activity_NN1 with_IW ritual-enabling_NN1 OTLs_VVZ that_CST aimed_VVD at_II setting_VVG the_AT ground_NN1 for_IF the_AT exploration_NN1 ._. 
</s>
<s>
In_II this_DD1 section_NN1 ,_, we_PPIS2 analyze_VV0 the_AT asymptotic_JJ behavior_NN1 ,_, as_CSA @S_FO ,_, of_IO the_AT solutions_NN2 of_IO the_AT differential_JJ equation_NN1 @F_FO ._. 
</s>
<s>
If_CS @S_FO for_IF m_ZZ1 p_ZZ1 ,_, then_RT on_II @S_FO one_MC1 employs_VVZ @S_FO ._. 
</s>
<s>
Hence_RR ,_, one_PN1 gets_VVZ a_AT1 version_NN1 of_IO (_( 2.6_MC )_) with_IW tm_NNU replaced_VVN by_II @S_FO ,_, which_DDQ (_( in_II31 view_II32 of_II33 @S_FO )_) leads_VVZ to_II the_AT desired_JJ version_NN1 of_IO (_( 2.7_MC )_) at_II @S_FO ._. 
</s>
<s>
However_RR ,_, the_AT situation_NN1 is_VBZ not_XX always_RR so_RG clearly_RR defined_VVN ._. 
</s>
<s>
Kazhdan_NN1 '_NULL s_ZZ1 property_NN1 (_( T_ZZ1 )_) is_VBZ a_AT1 rigidity_NN1 property_NN1 for_IF unitary_JJ representations_NN2 of_IO a_AT1 locally_RR compact_JJ group_NN1 ,_, which_DDQ has_VHZ found_VVN numerous_JJ applications_NN2 in_II various_JJ areas_NN2 of_IO pure_JJ and_CC applied_JJ mathematics_NN1 ,_, see_VV0 &lsqb;_( 3_MC &rsqb;_) ._. 
</s>
<s>
Note_VV0 that_CST both_DB2 the_AT mapping_NN1 class_NN1 @S_FO and_CC the_AT contact_NN1 surgery_NN1 along_II A_ZZ1 depend_VV0 on_II parametrizations_NN2 @S_FO ,_, which_DDQ are_VBR often_RR non-canonical_JJ ._. 
</s>
<s>
However_RR ,_, he_PPHS1 didn_VV0 '_NULL t_ZZ1 know_VV0 about_II how_RRQ to_TO add_VVI variables_NN2 ,_, that_REX21 is_REX22 ,_, adding_VVG x_ZZ1 and_CC 2x_FO ._. 
</s>
<s>
But_CCB immediately_CS she_PPHS1 brought_VVD to_II the_AT forefront_NN1 her_APPGE past_JJ position_NN1 as_II a_AT1 student_NN1 to_TO explain_VVI that_CST she_PPHS1 had_VHD had_VHN problems_NN2 with_IW some_DD of_IO the_AT oldest_JJT teachers_NN2 when_RRQ she_PPHS1 was_VBDZ a_AT1 student_NN1 ._. 
</s>
<s>
We_PPIS2 describe_VV0 the_AT family_NN1 of_IO leaves_NN2 of_IO G_ZZ1 (_( see_VV0 &lsqb;_( 1_MC1 ,_, Remark_VV0 3.12_MC &rsqb;_) )_) ._. 
</s>
<s>
It_PPH1 also_RR follows_VVZ easily_RR from_II Hopf_NP1 '_NULL s_ZZ1 lemma_NN1 that_CST v_ZZ1 changes_NN2 sign_VV0 in_II B(P)y2>0_FO for_IF any_DD >0_FO since_CS it_PPH1 vanishes_VVZ on_II y1-axis_NN1 ._. 
</s>
<s>
Our_APPGE third_MD category_NN1 ,_, human_JJ mobility_NN1 ,_, engages_VVZ with_IW place_NN1 in_II a_AT1 different_JJ way_NN1 :_: mobility_NN1 indexes_NN2 a_AT1 human_JJ right_NN1 to_TO physically_RR traverse_VVI the_AT space_NN1 around_II us_PPIO2 ,_, which_DDQ translates_VVZ to_II a_AT1 right_NN1 to_II placemaking_VVG ._. 
</s>
<s>
It_PPH1 is_VBZ also_RR clear_VV0 that_CST @F_FO ._. 
</s>
<s>
We_PPIS2 now_RT show_VV0 that_CST f3cc_FO is_VBZ locally_RR bounded_VVN in_II Q_ZZ1 ,_, by_II following_VVG a_AT1 similar_JJ argument_NN1 as_II21 for_II22 ac_NN1 ._. 
</s>
<s>
By_II construction_NN1 ,_, V_ZZ1 is_VBZ the_AT smaller_JJR hinge_NN1 of_IO the_AT two_MC hinges_NN2 determined_VVN by_II 71,72CN_FO ._. 
</s>
<s>
A_AT1 Borel_NN1 measurable_JJ property_NN1 of_IO ultrafilters_NN2 is_VBZ said_VVN to_TO hold_VVI T_ZZ1 almost_RR everywhere_RL if_CS the_AT set_NN1 of_IO ultrafilters_NN2 p_ZZ1 with_IW the_AT property_NN1 has_VHZ full_JJ measure_NN1 with_II31 respect_II32 to_II33 every_AT1 @S_FO ._. 
</s>
<s>
Hence_RR ,_, as_CSA analysts_NN2 ,_, to_TO decide_VVI about_II the_AT nature_NN1 of_IO OTL_NP1 ,_, one_PN1 needs_VVZ to_TO know_VVI something_PN1 about_II the_AT history_NN1 of_IO learning_VVG in_II the_AT given_JJ classroom_NN1 ._. 
</s>
<s>
The_AT images_NN2 of_IO the_AT embeddings_NN2 @S_FO from_II the_AT proof_NN1 @S_FO ._. 
</s>
<s>
Lemma_NN1 6.12_MC may_VM be_VBI enlarged_VVN and_CC then_RT joined_VVN by_II a_AT1 thickened_JJ path_NN1 to_TO obtain_VVI a_AT1 submanifold_JJ @S_FO diffeomorphic_JJ to_II W21_FO and_CC disjoint_JJ from_II the_AT images_NN2 of_IO @S_FO for_IF @S_FO ._. 
</s>
<s>
Applying_VVG Lemma_NN1 6.13_MC to_II the_AT embeddings_NN2 @S1_FO ._. 
</s>
<s>
At_II the_AT same_DA time_NNT1 ,_, for_IF the_AT iPad_NN1 cohorts_NN2 ,_, attitudes_NN2 from_II pre-_JJ to_II post-survey_NN1 in_II each_DD1 year_NNT1 did_VDD not_XX decrease_VVI significantly_RR ,_, which_DDQ suggests_VVZ that_DD1 membership_NN1 in_II an_AT1 iPad_NN1 class_NN1 helped_VVD to_TO maintain_VVI students_NN2 '_NULL positive_JJ attitudes_NN2 ._. 
</s>
<s>
In_RR21 particular_RR22 ,_, this_DD1 defines_VVZ a_AT1 representation_NN1 of_IO @S_FO under_II the_AT equivalence_NN1 of_IO Proposition_NN1 6.5_MC ._. 
</s>
<s>
A_AT1 numerical_JJ study_NN1 is_VBZ presented_VVN to_TO test_VVI the_AT proposed_JJ formulation_NN1 ._. 
</s>
<s>
It_PPH1 is_VBZ possible_JJ to_TO understand_VVI stochastic_JJ quasi-Newton_NP1 methods_NN2 as_II an_AT1 analogous_JJ approximation_NN1 of_IO individual_JJ sample_NN1 functions_NN2 ._. 
</s>
<s>
Stating_VVG that_CST the_AT rider_NN1 would_VM go_VVI from_II point_NN1 A_ZZ1 to_TO point_VVI B_ZZ1 and_CC then_RT "_" back_RP to_II A_ZZ1 which_DDQ is_VBZ close_JJ to_II the_AT ground_NN1 but_CCB far_RR from_II the_AT power_NN1 line_NN1 ,_, "_" Luis_NP1 drew_VVD a_AT1 curve_NN1 from_II A_ZZ1 to_II B_ZZ1 and_CC then_RT continued_VVD the_AT curve_NN1 up_RP and_CC to_II the_AT right_NN1 ,_, labeling_VVG the_AT upper_JJ right_JJ endpoint_NN1 A_ZZ1 as_RR21 well_RR22 ,_, and_CC commenting_VVG that_CST the_AT graph_NN1 should_VM not_XX be_VBI linear_JJ ._. 
</s>
<s>
Table_NN1 2_MC :_: Unit_NN1 square_NN1 ,_, relative_JJ eigenvalue_NN1 errors_NN2 ,_, q_ZZ1 =_FO 1_MC1 ._. 
</s>
<s>
S_ZZ1 :_: &lsqb;_( writes_VVZ "_" RB_NP1 "_" or_CC "_" Red-Blue_JJ "_" &rsqb;_) I_ZZ1 :_: Can_VM you_PPY represent_VVI all_DB of_IO the_AT outfits_NN2 that_DD1 way_NN1 ?_? )_) ._. 
</s>
<s>
Possible_JJ approaches_NN2 to_TO alleviate_VVI this_DD1 problem_NN1 include_VV0 iterative_JJ correction_NN1 &lsqb;_( 15_MC &rsqb;_) and_CC the_AT use_NN1 of_IO the_AT split_JJ method_NN1 as_II a_AT1 preconditioner_NN1 for_IF the_AT monolithic_JJ scheme_NN1 &lsqb;_( 10_MC &rsqb;_) ._. 
</s>
<s>
Thompson_NP1 ,_, 2011_MC discussion_NN1 of_IO one-dimensional_JJ area_NN1 )_) ,_, and_CC they_PPHS2 might_VM not_XX recognize_VVI important_JJ unit_NN1 concepts_NN2 ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI a_AT1 generic_JJ real-analytic_JJ submanifold_JJ in_II CN_NP1 ,_, @S_FO of_IO finite_JJ type_NN1 at_II @S_FO ,_, and_CC @S_FO be_VBI the_AT tube_NN1 ._. 
</s>
<s>
For_IF the_AT numerical_JJ simulations_NN2 we_PPIS2 choose_VV0 @S_FO ._. 
</s>
<s>
Thus_RR is_VBZ a_AT1 closed_JJ loop_NN1 in_II @S_FO which_DDQ by_II Hatcher_NP1 &lsqb;_( Ha_UH &rsqb;_) is_VBZ homotopically_RR trivial_JJ since_CS @S_FO ._. 
</s>
<s>
Here_RL we_PPIS2 are_VBR using_VVG formulation_NN1 (_( 8_MC )_) (_( see_VV0 the_AT appendix_NN1 of_IO &lsqb;_( Ha_UH &rsqb;_) )_) of_IO Hatcher_NP1 '_NULL s_ZZ1 theorem_NN1 which_DDQ asserts_VVZ that@S_FO is_VBZ homotopy_NN1 equivalent_NN1 to_II Q_ZZ1 (_( O_ZZ1 (_( 3_MC )_) )_) ._. 
</s>
<s>
We_PPIS2 define_VV0 BA_NN1 as_II the_AT space_NN1 of_IO simplicial_JJ maps_NN2 f:P_FO →_NULL RD_NN1 such_CS21 that_CS22 f(v)_NNU (_( v∈A_FO )_) are_VBR affinely_RR dependent_JJ ._. 
</s>
<s>
Fix_VV0 a_AT1 nonzero_NN1 element_NN1 @F_FO ,_, where_CS m_ZZ1 is_VBZ the_AT maximal_JJ ideal_NN1 of_IO o_ZZ1 ._. 
</s>
<s>
Founded_VVN in_II 1920_MC during_II an_AT1 international_JJ conference_NN1 in_II Strasbourg_NP1 ,_, the_AT Union_NN1 was_VBDZ discontinued_VVN during_II the_AT growing_JJ political_JJ unrest_NN1 and_CC economically_RR harsh_JJ times_NNT2 of_IO the_AT early_JJ 1930s_MC2 ._. 
</s>
<s>
So_RR the_AT proof_NN1 of_IO Theorem_NN1 1.1_MC does_VDZ not_XX proceed_VVI by_II proving_VVG that_DD1 strong_JJ property_NN1 (_( T_ZZ1 )_) passes_VVZ to_II lattices_NN2 ._. 
</s>
<s>
Indeed_RR ,_, @S_FO is_VBZ not_XX a_AT1 norm_NN1 :_: the_AT limit_NN1 defining_VVG the_AT inner_JJ product_NN1 @S_FO need_VM not_XX exist_VVI for_IF all_DB @S_FO ,_, and_CC the_AT space_NN1 @S_FO need_VM not_XX be_VBI complete_JJ with_II31 respect_II32 to_II33 @S_FO ._. 
</s>
<s>
To_TO address_VVI the_AT latter_DA issue_NN1 ,_, we_PPIS2 make_VV0 use_NN1 of_IO the_AT following_JJ proposition_NN1 ._. 
</s>
<s>
We_PPIS2 substitute_VV0 @S_FO ._. 
</s>
<s>
If_CS @S_FO we_PPIS2 obtain_VV0 @F_FO and_CC @F_FO ._. 
</s>
<s>
If_CS @S_FO we_PPIS2 obtain_VV0 @F_FO and_CC @F_FO ._. 
</s>
<s>
The_AT bounds_NN2 (_( 1.10_MC )_) and_CC (_( 1.11_MC )_) both_DB2 hold_VV0 if_CS @S_FO is_VBZ replaced_VVN with_IW and_CC if_CS C_ZZ1 >_FO 0_MC is_VBZ a_AT1 constant_JJ that_CST depends_VVZ on_II the_AT ratio_NN1 @F_FO ,_, and_CC this_DD1 constant_JJ C_ZZ1 becomes_VVZ arbitrarily_RR large_JJ as_CSA this_DD1 ratio_NN1 tends_VVZ to_II infinity_NN1 ._. 
</s>
<s>
Then_RT ,_, there_EX exists_VVZ a_AT1 positive_JJ time_NNT1 T_ZZ1 such_CS21 that_CS22 the_AT Cauchy_JJ problem_NN1 for_IF (_( 1.1_MC )_) with_IW initial_JJ data_NN f0_FO has_VHZ a_AT1 unique_JJ solution_NN1 f_ZZ1 satisfying_JJ @S_FO and_CC @F_FO ,_, where_CS HCT(R)_NP1 denotes_VVZ the_AT nonhomogeneous_JJ Sobolev_NP1 space_NN1 L2(R)_FO n_ZZ1 Ha(R)_NP1 ._. 
</s>
<s>
Subsection_NN1 3.3_MC defines_VVZ upper_JJ bPOE_NN1 and_CC compares_VVZ its_APPGE mathematical_JJ properties_NN2 with_IW lower_JJR bPOE_NN1 ._. 
</s>
<s>
Indeed_RR ,_, the_AT convergence_NN1 guarantee_NN1 of_IO such_DA a_AT1 stochastic_JJ method_NN1 in_II expectation_NN1 should_VM be_VBI better_JJR than_CSN the_AT one_PN1 for_IF the_AT (_( cyclic_JJ )_) IQN_VV0 method_NN1 ,_, as_CSA has_VHZ been_VBN observed_VVN for_IF first-oder_JJ cyclic_JJ and_CC stochastic_JJ methods_NN2 ._. 
</s>
<s>
The_AT main_JJ difficulty_NN1 in_II the_AT proof_NN1 of_IO this_DD1 result_NN1 is_VBZ to_TO handle_VVI the_AT lack_NN1 of_IO skew_NN1 symmetry_NN1 between_II the_AT terms_NN2 b1_FO and_CC b0_FO ._. 
</s>
<s>
We_PPIS2 describe_VV0 a_AT1 routine_NN1 as_CSA discursive_JJ if_CS a_AT1 person_NN1 interprets_VVZ the_AT task_NN1 situation_NN1 as_CSA requiring_VVG a_AT1 communicational_JJ action_NN1 ._. 
</s>
<s>
Based_VVN on_II these_DD2 results_NN2 and_CC our_APPGE review_NN1 of_IO related_JJ literature_NN1 ,_, we_PPIS2 developed_VVD a_AT1 set_NN1 of_IO tasks_NN2 that_CST could_VM be_VBI used_VVN in_II a_AT1 taskbased_JJ interview_NN1 setting_VVG to_TO further_RRR elicit_VVI individuals_NN2 '_NULL ways_NN2 of_IO understanding_NN1 ,_, along_II21 with_II22 interactions_NN2 Table_NN1 1_MC1 ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI a_AT1 discrete_JJ subgroup_NN1 such_CS21 that_CS22 @S_FO is_VBZ a_AT1 Shimura_NN1 Variety_NN1 ._. 
</s>
<s>
Both_DB2 episodes_NN2 presented_VVN illustrate_VV0 typical_JJ characteristics_NN2 of_IO the_AT three_MC students_NN2 '_NULL participation_NN1 in_II whole-class_JJ discussions_NN2 as_CSA identified_VVN recurrently_RR ._. 
</s>
<s>
Generate_VV0 three_MC independent_JJ random_JJ variables_NN2 @S_FO and_CC |_NULL @S_FO Poisson_NP1 @S_FO ._. 
</s>
<s>
For_IF ε>0_FO ,_, a_AT1 continuous_JJ map_NN1 f:X_FO →_NULL Y_ZZ1 is_VBZ called_VVN an_AT1 ε-embedding_FO with_II31 respect_II32 to_II33 if_CSW it_PPH1 satisfies_VVZ @F_FO ._. 
</s>
<s>
Note_VV0 that_CST this_DD1 is_VBZ an_AT1 open_JJ condition_NN1 for_IF f_ZZ1 in_II the_AT compact-open_JJ topology_NN1 ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI a_AT1 generic_JJ real-analytic_JJ submanifold_JJ and_CC @S_FO be_VBI a_AT1 real-analytic_JJ set_NN1 and_CC p2M_FO ,_, with_IW @S_FO ._. 
</s>
<s>
A_AT1 holomorphic_JJ deformation_NN1 for_IF @S_FO at_II p_ZZ1 is_VBZ a_AT1 (_( germ_NN1 of_IO a_ZZ1 )_) holomorphic_JJ map_NN1 B_ZZ1 :_: (_( @S_FO for_IF some_DD integer_NN1 r_ZZ1 >1_FO satisfying_VVG the_AT following_JJ conditions_NN2 :_: @L_FO ._. 
</s>
<s>
We_PPIS2 now_RT focus_VV0 on_II describing_VVG some_DD properties_NN2 of_IO the_AT basic_JJ BA_NN1 operator_NN1 @F_FO ._. 
</s>
<s>
First_MD ,_, consider_VV0 the_AT function_NN1 @F_FO which_DDQ is_VBZ holomorphic_JJ in_II z_ZZ1 and_CC h_ZZ1 ,_, and_CC satisfies_VVZ the_AT following_JJ properties_NN2 :_: @F_FO ._. 
</s>
<s>
They_PPHS2 immediately_RR imply_VV0 @F_FO ._. 
</s>
<s>
One_MC1 interesting_JJ difference_NN1 between_II this_DD1 study_NN1 and_CC Tillema_NP1 '_NULL s_ZZ1 (_( 2014_MC )_) earlier_JJR study_NN1 was_VBDZ that_CST in_II this_DD1 study_NN1 there_EX was_VBDZ a_AT1 more_RGR explicit_JJ design_NN1 to_TO promote_VVI the_AT construction_NN1 of_IO ordered_JJ pairs_NN2 ,_, which_DDQ allowed_VVD for_IF identifying_VVG three_MC criteria_NN2 for_IF defining_VVG when_RRQ MC1_FO and_CC MC2_FO students_NN2 had_VHD constructed_VVN ordered_JJ pairs_NN2 ._. 
</s>
<s>
Such_DA computations_NN2 are_VBR performed_VVN in_II Sect._NP1 6_MC ,_, see_VV0 Theorem_NN1 8_MC ,_, and_CC they_PPHS2 involve_VV0 a_AT1 careful_JJ treatment_NN1 of_IO several_DA2 Fourier_NP1 expansions_NN2 ,_, as_II31 well_II32 as_II33 the_AT computation_NN1 of_IO several_DA2 integrals_NN2 using_VVG the_AT method_NN1 of_IO steepest_JJT descent_NN1 along_II adequate_JJ complex_JJ paths_NN2 ,_, playing_VVG both_RR with_IW the_AT eccentric_JJ and_CC the_AT true_JJ anomaly_NN1 ._. 
</s>
<s>
The_AT trauma_NN1 was_VBDZ really_RR hard_JJ to_TO handle_VVI in_II cognitive_JJ ,_, metacognitive_JJ (_( what_DDQ do_VD0 I_PPIS1 have_VHI to_TO do_VDI ?_? )_) and_CC emotional_JJ terms_NN2 ._. 
</s>
<s>
Both_DB2 operators_NN2 in_II (_( 1.1_MC )_) and_CC (_( 5.6_MC )_) satisfy_VV0 the_AT requirements_NN2 posed_VVN in_II &lsqb;_( 28_MC &rsqb;_) ._. 
</s>
<s>
Therefore_RR ,_, the_AT number_NN1 of_IO deviations_NN2 can_VM not_XX be_VBI interpreted_VVN as_II an_AT1 indicator_NN1 of_IO student_NN1 abilities_NN2 ,_, but_CCB the_AT nature_NN1 and_CC reasons_NN2 for_IF such_DA deviations_NN2 do_VD0 yield_VVI insight_NN1 into_II students_NN2 '_NULL thought_VVD processes_NN2 ._. 
</s>
<s>
Many_DA2 algorithms_NN2 have_VH0 been_VBN proposed_VVN for_IF altering_VVG the_AT learning_NN1 rate_NN1 dynamically_RR ,_, that_REX21 is_REX22 ,_, based_VVN on_II information_NN1 from_II previous_JJ iterates_NN2 ._. 
</s>
<s>
Further_RRR independent_JJ regularity_NN1 properties_NN2 of_IO (_( 2.22_MC )_) ._. 
</s>
<s>
Also_RR ,_, since_CS @S_FO is_VBZ compact_JJ ,_, tensor_NN1 nuclear_JJ rank_NN1 is_VBZ upper_JJ semicontinuous_JJ ._. 
</s>
<s>
At_II the_AT same_DA time_NNT1 ,_, they_PPHS2 recognise_VV0 that_CST a_AT1 part_NN1 can_VM be_VBI divided_VVN into_II other_JJ parts_NN2 ._. 
</s>
<s>
We_PPIS2 then_RT show_VV0 @S_FO is_VBZ a_AT1 prime_JJ divisor_NN1 and_CC @S_FO ._. 
</s>
<s>
Using_VVG Theorem_NN1 4.1_MC ,_, we_PPIS2 see_VV0 Fe_NP1 is_VBZ finitely_RR generated_VVN and_CC the_AT corresponding_JJ degeneration_NN1 of_IO (_( Xo_NP1 ,_, Ao_NP1 )_) is_VBZ a_AT1 special_JJ test_NN1 configuration_NN1 with_IW generalized_JJ Futaki_JJ invariant_JJ zero_NN1 ._. 
</s>
<s>
Suppose_VV0 @S_FO ,_, @S_FO are_VBR self-similar_JJ sets_NN2 ,_, with_IW A_AT1 not_XX a_AT1 singleton_NN1 ,_, and_CC B_ZZ1 homogeneous_JJ ,_, satisfying_VVG the_AT open_JJ set_NN1 condition_NN1 ,_, and_CC of_IO dimension_NN1 strictly_RR smaller_JJR than_CSN 1_MC1 ._. 
</s>
<s>
Moreover_RR ,_, assume_VV0 that_CST u_ZZ1 e_ZZ1 Uad_NP1 is_VBZ a_AT1 solution_NN1 to_II the_AT control_NN1 problem_NN1 (_( CP_NP1 )_) with_IW associated_JJ state_NN1 @S_FO ._. 
</s>
<s>
Furthermore_RR ,_, with_IW the_AT notations_NN2 (_( 3.50_MC )_) (_( 3.51_MC )_) and_CC (_( 4.4_MC )_) ,_, let_VV0 (_( p_ZZ1 ,_, pr_NNU ,_, q_ZZ1 ,_, qr_NNU )_) be_VBI the_AT solution_NN1 to_II the_AT adjoint_JJ problem_NN1 (_( 4.7_MC )_) -(4.9)_NNU satisfying_VVG the_AT regularity_NN1 requirements_NN2 (_( 4.5_MC )_) (_( 4.6_MC )_) ._. 
</s>
<s>
In_II31 addition_II32 to_II33 these_DD2 cities_NN2 ,_, large_JJ cities_NN2 such_II21 as_II22 Tokyo_NP1 ,_, Nagoya_NP1 ,_, and_CC Osaka_NP1 have_VH0 relatively_RR high_JJ percentages_NN2 of_IO registered_JJ immigrants_NN2 ._. 
</s>
<s>
Our_APPGE development_NN1 reveals_VVZ that_CST the_AT basic_JJ trajectory_NN1 stratification_NN1 approach_NN1 can_VM be_VBI useful_JJ well_NN1 beyond_II the_AT estimation_NN1 of_IO stationary_JJ averages_NN2 for_IF time-homogeneous_JJ Markov_NP1 processes_NN2 ._. 
</s>
<s>
They_PPHS2 used_VVD expressions_NN2 such_II21 as_II22 "_" the_AT height_NN1 will_VM increase_VVI rapidly_RR "_" and_CC "_" the_AT height_NN1 increases_VVZ slowly_RR "_" indicating_VVG their_APPGE gross_JJ quantification_NN1 of_IO the_AT rate_NN1 of_IO change_NN1 ._. 
</s>
<s>
Let_VV0 the_AT requirements_NN2 of_IO Lemma_NN1 2.1_MC be_VBI fulfilled_VVN ._. 
</s>
<s>
On_II the_AT other_JJ hand_NN1 ,_, the_AT teacher_NN1 has_VHZ experiences_NN2 and_CC a_AT1 perspective_NN1 that_CST influence_VV0 how_RRQ those_DD2 materials_NN2 are_VBR read_VVN ,_, which_DDQ refers_VVZ to_II the_AT instrumentalization_NN1 of_IO those_DD2 resources_NN2 ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI the_AT approximant_JJ function_NN1 of_IO v_ZZ1 in_II the_AT space_NN1 @S_FO obtained_VVN by_II gluing_VVG the_AT local_JJ spaces_NN2 @S_FO and_CC Proposition_NN1 4.2_MC in_II &lsqb;_( 15_MC &rsqb;_) )_) ;_; then_RT it_PPH1 holds_VVZ that_CST @F_FO ._. 
</s>
<s>
Now_RT let_VV0 @S_FO be_VBI the_AT interpolant_NN1 of_IO @S_FO in_II the_AT sense_NN1 of_IO the_AT @S_FO ,_, so_CS21 that_CS22 @F_FO ._. 
</s>
<s>
Let_VV0 us_PPIO2 define_VVI @S_FO ;_; then_RT for_IF every_AT1 element_NN1 @S_FO the_AT following_JJ facts_NN2 hold_VV0 :_: Since_CS @S_FO and_CC @S_FO are_VBR polynomials_NN2 of_IO degree_NN1 @S_FO in_II @S_FO ,_, by_II definition_NN1 of_IO @S_FO and_CC @S_FO ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Since_CS @S_FO and_CC @S_FO are_VBR polynomials_NN2 of_IO degree_NN1 k_ZZ1 1_MC1 in_II E_ZZ1 ,_, by_II definition_NN1 of_IO @S_FO and_CC homogeneous_JJ boundary_NN1 data_NN (_( 50_MC )_) ,_, we_PPIS2 get_VV0 @F._FO @L_FO ._. 
</s>
<s>
Now_RT we_PPIS2 recall_VV0 that_CST ,_, for_IF any_DD @S_FO ,_, the_AT quantity_NN1 @S_FO depends_VVZ only_RR on_II the_AT values_NN2 of_IO @S_FO ;_; see_VV0 Proposition_NN1 3.2_MC ._. 
</s>
<s>
For_IF every_AT1 @S_FO ,_, there_RL exist_VV0 a_AT1 subsequence_NN1 T_ZZ1 of_IO T_ZZ1 and_CC a_AT1 non-principal_JJ ultrafilter_NN1 @S_FO such_CS21 that_CS22 @S_FO exists_VVZ for_IF all_DB @S_FO and_CC @F_FO holds_VVZ ._. 
</s>
<s>
It_PPH1 follows_VVZ that_CST the_AT entire_JJ composition_NN1 is_VBZ an_AT1 isomorphism_NN1 ._. 
</s>
<s>
For_IF @S_FO ,_, the_AT functions_NN2 cq_NNU and_CC sq_NNU have_VH0 an_AT1 holomorphic_JJ extension_NN1 to_II the_AT complex_JJ strip_NN1 @S_FO ._. 
</s>
<s>
This_DD1 extension_NN1 is_VBZ maximal_JJ in_II the_AT sense_NN1 that_CST these_DD2 functions_NN2 have_VH0 singularities_NN2 at_II 2_MC +_FO wn_NNU ±_FO ipkq_NN1 (_( which_DDQ are_VBR ramification_NN1 singularities_NN2 )_) ._. 
</s>
<s>
We_PPIS2 also_RR define_VV0 @S_FO ._. 
</s>
<s>
BY_II using_VVG Lemma_NN1 4.5_MC and_CC (_( 4.12_MC )_) ,_, we_PPIS2 obtain_VV0 @F_FO ._. 
</s>
<s>
Then_RT for_IF some_DD @S_FO depending_II21 on_II22 6_MC but_CCB not_XX f_ZZ1 ,_, there_RL exist_VV0 type_NN1 (_( n_ZZ1 1_MC1 ,_, n_ZZ1 )_) rational_JJ functions_NN2 @S_FO ,_, @S_FO ,_, such_CS21 that_CS22 @F_FO as_CSA @S_FO for_IF some_DD C_ZZ1 >_FO 0_MC ,_, where_CS @S_FO ._. 
</s>
<s>
Moreover_RR ,_, each_DD1 rn_NNU can_VM be_VBI taken_VVN to_TO have_VHI simple_JJ poles_NN2 only_RR at_II @F_FO ,_, where_CS a_AT1 >_FO 0_MC is_VBZ arbitrary_JJ ._. 
</s>
<s>
The_AT same_DA reasoning_NN1 implies_VVZ that_CST conversely_RR the_AT conditional_JJ law_NN1 of_IO L_ZZ1 given_JJ L_ZZ1 ,_, E_ZZ1 ,_, and_CC @S_FO is_VBZ that_DD1 of_IO an_AT1 SLEK_NN1 process_NN1 from_II @S_FO to_II @S_FO in_II the_AT component_NN1 of_IO @S_FO with_IW @S_FO on_II its_APPGE boundary_NN1 ._. 
</s>
<s>
By_II the_AT transmission_NN1 condition_NN1 (_( 6.1c_FO )_) and_CC the_AT normalisation_NN1 (_( 6.2_MC )_) ,_, we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Algorithm_NN1 1_MC1 in_II &lsqb;_( 6_MC &rsqb;_) for_IF pseudocode_NN1 ._. 
</s>
<s>
This_DD1 is_VBZ an_AT1 extension_NN1 of_IO Tsujii_NP1 '_NULL s_ZZ1 &lsqb;_( 40_MC ,_, Theorem_NN1 1.4_MC &rsqb;_) to_II the_AT present_NN1 higher-dimensional_JJR situation_NN1 ._. 
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<s>
Ax_NP1 |_NULL of_IO order_NN1 @S_FO holding_VVG with_IW probability_NN1 @S_FO for_IF some_DD C_ZZ1 >_FO 0_MC sufficiently_RR large_JJ depending_II21 on_II22 C._NP1 It_PPH1 turns_VVZ out_RP that_CST this_DD1 is_VBZ impossible_JJ ,_, at_RR21 least_RR22 when_CS d_ZZ1 is_VBZ fixed_VVN as_CSA "_" grows_VVZ ,_, since_CS a_AT1 O(sfd)_NN1 eigenvalue_NN1 bound_NN1 is_VBZ only_RR expected_VVN to_TO hold_VVI with_IW probability_NN1 approaching_VVG one_PN1 polynomially_RR in_II this_DD1 case_NN1 &lsqb;_( indeed_RR ,_, in_II the_AT permutation_NN1 model_NN1 it_PPH1 is_VBZ not_XX hard_JJ to_TO see_VVI that_CST the_AT graph_NN1 is_VBZ disconnected_VVN with_IW probability_NN1 Q_ZZ1 (_( "_" c_ZZ1 )_) for_IF some_DD c_ZZ1 >_FO 0_MC depending_II21 on_II22 d_ZZ1 &rsqb;_) ._. 
</s>
<s>
This_DD1 line_NN1 of_IO research_NN1 has_VHZ also_RR pointed_VVN out_RP some_DD of_IO the_AT conflict_NN1 and_CC struggles_VVZ that_CST immigrant_JJ parents_NN2 experience_VV0 (_( de_NP1 Abreu_NP1 &;_NULL Cline_NP1 ,_, 2005_MC ;_; Civil_JJ &;_NULL Bernier_NP1 ,_, 2006_MC ;_; Crafter_NP1 ,_, 2012_MC ;_; Gorgorio_NP1 &;_NULL Abreu_NP1 ,_, 2009_MC )_) ._. 
</s>
<s>
Since_CS for_IF wave-current_JJ interactions_NN2 in_II which_DDQ the_AT waves_NN2 are_VBR long_RR compared_VVN with_IW the_AT mean_JJ depth_NN1 of_IO the_AT effective_JJ flow_NN1 region_NN1 the_AT importance_NN1 of_IO a_AT1 non-zero_JJ mean_JJ vorticity_NN1 preponderates_VVZ that_DD1 of_IO its_APPGE specific_JJ distribution_NN1 (_( see_VV0 &lsqb;_( 13_MC &rsqb;_) )_) ,_, the_AT simplest_JJT realistic_JJ setting_NN1 is_VBZ that_DD1 of_IO flows_NN2 with_IW constant_JJ vorticity_NN1 above_RL and_CC below_II the_AT thermocline_NN1 :_: negative_JJ above_RL ,_, to_TO permit_VVI a_AT1 reversal_NN1 from_II the_AT surface_NN1 westward_RL wind-drift_JJ to_II the_AT eastward-flowing_JJ subsurface_NN1 EUC_NN1 ,_, and_CC positive_JJ below_RL to_TO model_VVI a_AT1 flow_NN1 that_CST withers_VVZ with_IW increasing_JJ depth_NN1 ._. 
</s>
<s>
Since_CS Hi_UH ,_, f2_FO ,_, and_CC g_ZZ1 are_VBR Lipschitz_NP1 ,_, it_PPH1 follows_VVZ from_II (_( 3.4_MC )_) that_CST u_ZZ1 is_VBZ bounded_VVN ._. 
</s>
<s>
Second_MD ,_, they_PPHS2 need_VV0 to_TO determine_VVI how_RRQ to_II equal_JJ partition_NN1 one_MC1 side_NN1 of_IO the_AT rectangle_NN1 (_( a_AT1 length_NN1 )_) to_TO construct_VVI a_AT1 row_NN1 of_IO area_NN1 units_NN2 and_CC then_RT carefully_RR iterate_VV0 that_CST row_VV0 in_II the_AT orthogonal_JJ direction_NN1 (_( Cullen_NP1 et_RA21 al._RA22 ,_, 2018_MC )_) ._. 
</s>
<s>
In_II fact_NN1 ,_, it_PPH1 is_VBZ sufficient_JJ to_TO prove_VVI this_DD1 for_IF every_AT1 t_ZZ1 large_JJ enough_RR ._. 
</s>
<s>
In_II uncertainty_NN1 propagation_NN1 ,_, the_AT model_NN1 input_NN1 is_VBZ described_VVN by_II a_AT1 random_JJ variable_NN1 and_CC we_PPIS2 are_VBR interested_JJ in_II statistics_NN of_IO the_AT model_NN1 output_NN1 ._. 
</s>
<s>
The_AT second_MD order_NN1 variational_JJ equation_NN1 (_( 2.19_MC )_) and_CC the_AT Bogomol_NN1 '_NULL nyi_NN2 equation_NN1 (_( 4.13_MC )_) deserve_VV0 further_JJR study_NN1 ._. 
</s>
<s>
The_AT rest_NN1 of_IO the_AT argument_NN1 ,_, which_DDQ is_VBZ easier_JJR ,_, uses_VVZ this_DD1 fact_NN1 to_TO show_VVI that_CST the_AT composition_NN1 in_II (_( 3.12_MC )_) is_VBZ an_AT1 isomorphism_NN1 (_( see_VV0 Section_NN1 3.7_MC )_) ._. 
</s>
<s>
How_RRQ do_VD0 you_PPY feel_VVI about_II that_DD1 ?_? 
</s>
<s>
To_TO see_VVI that_CST the_AT trace_NN1 of_IO y_ZZ1 is_VBZ a_AT1 Euclidean_JJ straight_JJ line_NN1 in_II An_AT1 ,_, consider_VV0 its_APPGE Euclidean_JJ representation_NN1 @S_FO ._. 
</s>
<s>
By_II (_( 5.4_MC )_) we_PPIS2 have_VH0 @F_FO ._. 
</s>
<s>
Solving_VVG for_IF h(t)_NNU gives_VVZ @F_FO ._. 
</s>
<s>
Expressing_VVG (_( 5.4_MC )_) in_II Euclidean_JJ coordinates_NN2 and_CC using_VVG (_( 5.6_MC )_) ,_, we_PPIS2 get_VV0 after_II some_DD algebra_NN1 that_CST @S_FO ._. 
</s>
<s>
The_AT paper_NN1 proposes_VVZ improvements_NN2 to_II PA_NN1 ._. 
</s>
<s>
Finally_RR ,_, recalling_VVG the_AT definition_NN1 of_IO m_ZZ1 ,_, we_PPIS2 get_VV0 that_DD1 @F_FO ._. 
</s>
<s>
Notice_VV0 now_CS21 that_CS22 @S_FO ,_, therefore_RR @S_FO ,_, Hence_RR ,_, (_( 2.7_MC )_) ,_, together_RL with_IW (_( 2.6_MC )_) ,_, gives_VVZ (_( 2.4_MC )_) ,_, thanks_NN2 to_II Lemma_NN1 2.1_MC ._. 
</s>
<s>
The_AT intervention_NN1 group_NN1 teachers_NN2 met_VVN with_IW the_AT researchers_NN2 in_II two_MC separate_JJ groups_NN2 ,_, with_IW four_MC preschool_JJ teachers_NN2 in_II one_MC1 group_NN1 and_CC eight_MC in_II the_AT other_JJ ._. 
</s>
<s>
If_CS this_DD1 is_VBZ true_JJ ,_, then_RT any_DD algorithm_NN1 that_CST finds_VVZ a_AT1 second-order_JJ critical_JJ point_NN1 also_RR solves_VVZ the_AT nonconvex_NN1 problem_NN1 (_( P_ZZ1 )_) ._. 
</s>
<s>
Notice_VV0 that_CST the_AT size_NN1 of_IO the_AT details_NN2 in_II the_AT draws_NN2 corresponds_VVZ well_RR to_II our_APPGE intuition_NN1 about_II the_AT role_NN1 of_IO @S_FO as_II a_AT1 measure_NN1 of_IO the_AT size_NN1 of_IO the_AT details_NN2 ._. 
</s>
<s>
Gibbs_NP1 entropy_NN1 @F_FO ._. 
</s>
<s>
The_AT entropic_JJ regularizer_NN1 (_( 3.8_MC )_) is_VBZ 1-strongly_RR convex_JJ with_II31 respect_II32 to_II33 the_AT L_ZZ1 1-norm_JJ on_II Rd_NN1 ._. 
</s>
<s>
Under_II i/y(dP)_FU ,_, (_( Pj_NP1 )_) jEV_NN1 is_VBZ 1-dependent_JJ :_: if_CS @S_FO are_VBR such_CS21 that_CS22 @S_FO ,_, then_RT (_( Pi_NN1 )_) iEU_NNU and_CC (_( Pj_NP1 )_) jEU_NN1 '_NULL are_VBR independent_JJ ._. 
</s>
<s>
Thus_RR we_PPIS2 have_VH0 a_AT1 sequence_NN1 of_IO morphisms_NN2 @F_FO ._. 
</s>
<s>
By_II Lemma_NN1 3.1.5_MC (_( i_ZZ1 )_) ,_, the_AT composition_NN1 @S_FO is_VBZ non-zero_JJ ._. 
</s>
<s>
Equivalently_RR ,_, we_PPIS2 want_VV0 the_AT based_VVN lift_NN1 of_IO yp_NN1 and_CC each_DD1 lift_NN1 of_IO p_ZZ1 starting_VVG at_II the_AT preimage_NN1 of_IO v_ZZ1 in_II the_AT based_VVN elevation_NN1 of_IO @S_FO to_TO end_VVI with_IW edges_NN2 dual_JJ to_II distinct_JJ elevations_NN2 of_IO B._NP1 Here_RL a_AT1 based_VVN lift_NN1 or_CC elevation_NN1 is_VBZ a_AT1 lift_NN1 or_CC elevation_NN1 where_CS v_ZZ1 lifts_VVZ to_II a_AT1 specified_JJ basepoint_NN1 of_IO X._NP1 If_CS @S_FO satisfies_VVZ @S_FO ;_; for_IF all_DB @S_FO ,_, then_RT ,_, for_IF every_AT1 @S_FO ,_, there_EX exists_VVZ a_AT1 constant_JJ @S_FO independent_NN1 of_IO h_ZZ1 with_IW @F_FO ._. 
</s>
<s>
We_PPIS2 have_VH0 already_RR encountered_VVN an_AT1 example_NN1 of_IO continuum_NN1 limit_NN1 in_II Sec._NNU 4_MC ,_, where_CS the_AT collective_JJ behavior_NN1 of_IO the_AT system_NN1 with_IW constant_JJ communication_NN1 matrix_NN1 entries_NN2 is_VBZ approximated_VVN for_IF large_JJ values_NN2 of_IO N_ZZ1 by_II the_AT nonlinear_JJ Fokker-Planck_NP1 equation_NN1 (_( 4.7_MC )_) ._. 
</s>
<s>
We_PPIS2 have_VH0 now_RT several_DA2 times_NNT2 defined_VVD symbols_NN2 of_IO the_AT form_NN1 ._. 
</s>
<s>
(_( R_ZZ1 ;_; Y_ZZ1 )_) as_II the_AT minimum_NN1 of_IO @S_FO over_II some_DD collection_NN1 of_IO paths_NN2 ._. 
</s>
<s>
Notice_VV0 that_CST the_AT @S_FO ,_, are_VBR not_XX disjoint_JJ in_RR21 general_RR22 ._. 
</s>
<s>
In_RR21 addition_RR22 ,_, the_AT inclusion_NN1 @S_FO holds_VVZ for_IF functions_NN2 with_IW a_AT1 compact_JJ support_NN1 in_II K._NP1 Nonetheless_RR ,_, as_CSA we_PPIS2 document_VV0 ,_, the_AT teachers_NN2 '_NULL interest_NN1 in_II improving_VVG how_RRQ they_PPHS2 launched_VVD instructional_JJ activities_NN2 provided_VVN leverage_NN1 for_IF their_APPGE subsequent_JJ learning_NN1 ._. 
</s>
<s>
In_II fact_NN1 ,_, in_II one_MC1 dimension_NN1 we_PPIS2 always_RR have_VH0 @S_FO since_CS we_PPIS2 assume_VV0 a_AT1 bound_NN1 with_IW 0_MC <_FO a_AT1 <_FO 2_MC ._. 
</s>
<s>
Students_NN2 were_VBDR asked_VVN to_TO first_MD predict_VVI and_CC then_RT measure_VV0 ._. 
</s>
<s>
Observe_VV0 that_CST ,_, for_IF all_DB @S_FO ,_, @F_FO where_RRQ @S_FO (_( and_CC similarly_RR for_IF @S_FO )_) ._. 
</s>
<s>
Let_VV0 S_ZZ1 be_VBI a_AT1 numerical_JJ semigroup_NN1 with_IW m_ZZ1 =_FO 1000_MC and_CC c_ZZ1 =_FO 4000_MC ._. 
</s>
<s>
Note_VV0 that_CST if_CS y_ZZ1 =_FO wi_NN2 ,_, then_RT b_ZZ1 =1_FO and_CC the_AT last_MD term_NN1 in_II (_( 9.7_MC )_) vanishes_VVZ ._. 
</s>
<s>
Applying_VVG Lemma_NN1 9_MC to_II (_( 73_MC )_) &lsqb;_( cf._VV0 We_PPIS2 omit_VV0 the_AT details_NN2 of_IO the_AT computations_NN2 ._. 
</s>
<s>
Theorem_NN1 3.3_MC Assume_VV0 that_CST g∈W0_FO has_VHZ stable_JJ integer_NN1 shifts_NN2 and_CC that_CST R_ZZ1 is_VBZ relatively_RR separated_VVN ._. 
</s>
<s>
Our_APPGE Redback_NN1 should_VM provide_VVI opportunities_NN2 for_IF PTs_NN2 by_II engaging_VVG them_PPHO2 through_II constraining_VVG some_DD ways_NN2 of_IO thinking_NN1 and_CC supporting_JJ others_NN2 ._. 
</s>
<s>
Theorem_NN1 3.1_MC is_VBZ proved_VVN at_II the_AT beginning_NN1 of_IO Section_NN1 5_MC via_II reduction_NN1 to_II another_DD1 embedding_NN1 theorem_NN1 ._. 
</s>
<s>
In_II that_DD1 setting_NN1 ,_, one_PN1 must_VM note_VVI that_CST @S_FO ._. 
</s>
<s>
The_AT following_JJ proposition_NN1 is_VBZ a_AT1 summary_NN1 of_IO the_AT discussion_NN1 above_RL (_( a_AT1 rigorous_JJ proof_NN1 of_IO it_PPH1 ,_, and_CC more_RGR precise_JJ estimates_NN2 can_VM be_VBI found_VVN in_II Section_NN1 3.2_MC )_) ._. 
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<s>
Yet_RR ,_, by_II direct_JJ calculations_NN2 ,_, one_PN1 can_VM check_VVI that_CST (_( 2.4_MC )_) admits_VVZ the_AT conservation_JJ form_NN1 (_( 1.5_MC )_) stated_VVD in_II the_AT introduction_NN1 ._. 
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<s>
Convergence_NN1 Guarantees_VVZ for_IF the_AT Trust-Region_NP1 Problem_NN1 ._. 
</s>
<s>
Therefore_RR ,_, the_AT proposed_JJ estimator_NN1 is_VBZ also_RR adaptive_JJ to_II the_AT shape_NN1 of_IO the_AT distribution_NN1 ._. 
</s>
<s>
Thus_RR ,_, in_II31 terms_II32 of_II33 the_AT Sobolev_NP1 embedding_NN1 theorem_NN1 ,_, VZm_NP1 is_VBZ invertible_JJ in_II R3_FO for_IF sufficiently_RR small_JJ 8_MC ._. 
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<s>
Our_APPGE analysis_NN1 shows_VVZ the_AT method_NN1 is_VBZ robust_JJ and_CC convergent_JJ allowing_VVG for_IF varying_VVG and_CC arbitrarily_RR small_JJ apertures_NN2 ._. 
</s>
<s>
From_II the_AT global-in-time_JJ boundedness_NN1 of_IO SN_NN1 ,_, there_EX exists_VVZ a_AT1 weak_JJ limit_NN1 θ_NULL ∞_FO independent_NN1 of_IO τ_NULL such_DA that_CST SN_NN1 (_( θ_NULL ∞_FO ,_, 0_MC )_) SN_NN1 (_( τ_NULL )_) ≤Cε_FO ._. 
</s>
<s>
We_PPIS2 claim_VV0 that_CST the_AT external_JJ rays_NN2 R?j_FO landing_VVG at_II x?_FO j_ZZ1 converge_VV0 to_II the_AT external_JJ ray_NN1 landing_VVG at_II x1_FO ._. 
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<s>
The_AT materials_NN2 were_VBDR similar_JJ to_II those_DD2 of_IO the_AT previous_JJ experiment_NN1 ,_, except_CS21 that_CS22 each_DD1 description_NN1 had_VHD the_AT figure_NN1 on_II the_AT left_JJ and_CC the_AT text_NN1 on_II the_AT right_NN1 ._. 
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<s>
This_DD1 class_NN1 contains_VVZ all_DB the_AT examples_NN2 mentioned_VVN in_II the_AT previous_JJ two_MC classes_NN2 ._. 
</s>
<s>
Hence_RR ,_, we_PPIS2 can_VM conclude_VVI that_CST @S_FO in_II @S_FO as_CSA @S_FO ._. 
</s>
<s>
For_IF the_AT error_NN1 estimate_NN1 ,_, we_PPIS2 prepare_VV0 the_AT following_JJ ._. 
</s>
<s>
The_AT algorithm_NN1 then_RT proceeds_VVZ as_II a_AT1 collection_NN1 of_IO interdependent_JJ tasks_NN2 that_CST operate_VV0 on_II the_AT tile_NN1 data_NN layout_NN1 and_CC are_VBR scheduled_VVN in_II an_AT1 out-of-order_JJ fashion_NN1 using_VVG either_RR the_AT OpenMP_NN1 runtime_NNT1 for_IF PLASMA_NP1 or_CC the_AT powerful_JJ PaRSEC_NP1 distributed_JJ runtime_NNT1 system_NN1 for_IF DPLASMA_NN1 ._. 
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<s>
In_RR21 addition_RR22 ,_, for_IF a_AT1 real_JJ number_NN1 @S_FO we_PPIS2 denote_VV0 by_II @S_FO the_AT mapping_NN1 with_IW the_AT property_NN1 that_CST for_IF all_DB @S_FO it_PPH1 holds_VVZ that_CST @F_FO ._. 
</s>
<s>
Note_VV0 for_IF every_AT1 @S_FO and_CC every_AT1 @S_FO that_DD1 @S_FO is_VBZ the_AT maximum_JJ step_NN1 size_NN1 of_IO the_AT partition_NN1 0_MC ._. 
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<s>
Recall_VV0 that_CST the_AT simple_JJ closed_JJ curves_NN2 T1_FO ,_, r2_FO and_CC r3_FO are_VBR the_AT unique_JJ closed_JJ geodesics_NN2 in_II the_AT intersection_NN1 of_IO @S_FO with_IW disjoint_JJ balls_NN2 B1_FO ,_, B2_FO and_CC B3_FO ,_, and_CC that_CST @SBj_FO can_VM be_VBI assumed_VVN to_TO be_VBI arbitrarily_RR close_RR to_II a_AT1 large_JJ region_NN1 of_IO a_AT1 catenoid_JJ Cj_NP1 centered_VVD at_II the_AT center_NN1 of_IO Bj_NP1 (_( and_CC suitably_RR rescaled_VVN )_) ,_, @S_FO ._. 
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<s>
Once_RR21 again_RR22 ,_, the_AT Gaussian_JJ hypercontractivity_NN1 gives_VVZ us_PPIO2 the_AT needed_VVN bounds_NN2 ._. 
</s>
<s>
To_TO obtain_VVI a_AT1 picture_NN1 of_IO the_AT entire_JJ conditional_JJ distribution_NN1 of_IO response_NN1 given_VVN predictors_NN2 ,_, estimation_NN1 of_IO the_AT conditional_JJ quantile_JJ function_NN1 at_II several_DA2 quantile_JJ levels_NN2 is_VBZ required_VVN ._. 
</s>
<s>
Let_VV0 q(t)_NNU and_CC qn(t)_NNU be_VBI @S-solutions_FO to_II (_( KdV_NP1 )_) ,_, in_II the_AT sense_NN1 of_IO Corollary_NN1 5.2_MC ,_, with_IW initial_JJ data_NN @S_FO in_II H1(R)_FO ._. 
</s>
<s>
The_AT periodic_JJ TASEP_NN1 can_VM be_VBI described_VVN if_CS we_PPIS2 keep_VV0 track_NN1 of_IO N_ZZ1 consecutive_JJ particles_NN2 ,_, say_VV0 @S_FO ._. 
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<s>
We_PPIS2 can_VM now_RT establish_VVI the_AT announced_JJ proposition_NN1 ._. 
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<s>
Using_VVG the_AT notation_NN1 ,_, we_PPIS2 get_VV0 the_AT following_JJ Stratonovich-Ito_NP1 transformation_NN1 ;_; see_VV0 Lemma_NN1 2.1_MC of_IO &lsqb;_( 10_MC &rsqb;_) ._. 
</s>
<s>
This_DD1 stochastic_JJ process_NN1 encompasses_VVZ random_JJ elements_NN2 such_II21 as_II22 Ak_NP1 ,_, Xk_FO ,_, Sk_NP1 ,_, which_DDQ are_VBR directly_RR computed_VVN by_II the_AT algorithm_NN1 ,_, but_CCB also_RR some_DD quantities_NN2 that_CST can_VM be_VBI derived_VVN as_CSA functions_NN2 of_IO Ak_NP1 ,_, Xk_FO ,_, Sk_NP1 ,_, such_II21 as_II22 @S_FO and_CC a_AT1 quantity_NN1 F_ZZ1 ,_, which_DDQ we_PPIS2 will_VM use_VVI to_TO denote_VVI some_DD measure_NN1 of_IO progress_NN1 towards_II optimality_NN1 ._. 
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<s>
There_EX exists_VVZ a_AT1 constant_JJ Cn_NP1 >_FO 0_MC ,_, such_CS21 that_CS22 @F_FO ._. 
</s>
<s>
Moreover_RR ,_, the_AT use_NN1 of_IO decision_NN1 heuristics_NN2 (_( i.e._REX ,_, presence_NN1 of_IO a_AT1 wall_NN1 )_) seems_VVZ to_TO increase_VVI the_AT efficient_JJ information_NN1 processing_NN1 especially_RR in_II small_JJ groups_NN2 formed_VVN in_II dense_JJ population_NN1 spaces_NN2 ._. 
</s>
<s>
We_PPIS2 also_RR mention_VV0 the_AT papers_NN2 &lsqb;_( 16_MC ,_, 17_MC ,_, 19_MC ,_, 20_MC &rsqb;_) ,_, which_DDQ deal_VV0 with_IW the_AT optimal_JJ control_NN1 of_IO three-dimensional_JJ systems_NN2 ,_, however_RR in_II the_AT time-discretized_JJ version_NN1 ._. 
</s>
<s>
This_DD1 is_VBZ again_RT similar_JJ to_II the_AT low-rank_NN1 Newton_NP1 ADI_NP1 solver_NN1 &lsqb;_( 7_MC &rsqb;_) ,_, and_CC not_XX possible_JJ in_II the_AT current_JJ version_NN1 of_IO the_AT Cayley_NP1 subspace_NN1 iteration_NN1 ._. 
</s>
<s>
Let_VV0 (_( X_ZZ1 ,_, A_ZZ1 )_) be_VBI a_AT1 log_NN1 Fano_NN1 pair_NN ,_, @S_FO a_AT1 proper_JJ birational_JJ morphism_NN1 with_IW Y_ZZ1 normal_JJ ,_, and_CC E_ZZ1 an_AT1 effective_JJ Q-Cartier_NP1 Q-divisor_NN1 on_II Y_ZZ1 such_CS21 that_CS22 E_ZZ1 is_VBZ p-ample_JJ ._. 
</s>
<s>
Now_RT we_PPIS2 consider_VV0 the_AT abstract_JJ Pryms_NN2 P_ZZ1 and_CC Pg_NP1 over_II A_ZZ1 and_CC Ag_FO associated_VVN to_II G_ZZ1 and_CC Hg_FO ._. 
</s>
<s>
This_DD1 completes_VVZ the_AT proof_NN1 of_IO (_( 2.57_MC )_) ._. 
</s>
<s>
Sample_NN1 items_NN2 are_VBR "_" I_MC1 don_NN1 '_NULL t_ZZ1 value_NN1 the_AT chance_NN1 of_IO learning_VVG mathematics_NN1 "_" and_CC "_" I_PPIS1 seldom_RR find_VV0 learning_VVG mathematics_NN1 interesting_JJ "_" ._. 
</s>
<s>
The_AT optimization_NN1 problems_NN2 (_( 7.5_MC )_) and_CC (_( 2.7_MC )_) have_VH0 the_AT same_DA objective_JJ function_NN1 ,_, and_CC the_AT feasible_JJ set_NN1 of_IO (_( 2.7_MC )_) is_VBZ contained_VVN in_II that_DD1 of_IO (_( 7.5_MC )_) ,_, so_CS we_PPIS2 have_VH0 m1_FO >_FO m2_FO ._. 
</s>
<s>
Remark_VV0 8_MC Recall_VV0 that_CST the_AT resolvent_NN1 of_IO the_AT subdifferential_JJ of_IO a_AT1 proper_JJ lower_JJR semi-continuous_JJ convex_JJ function_NN1 @S_FO is_VBZ Moreau_NP1 '_NULL s_ZZ1 proximity_NN1 operator_NN1 @S_FO ._. 
</s>
<s>
Now_RT consider_VV0 the_AT setting_NN1 of_IO Problem_NN1 2_MC and_CC execute_VV0 Algorithm_NN1 12_MC with_IW @S_FO and_CC @S_FO ._. 
</s>
<s>
Then_RT ,_, using_VVG the_AT same_DA arguments_NN2 as_CSA in_II &lsqb;_( 15_MC ,_, Proposition_NN1 5.4_MC &rsqb;_) ,_, it_PPH1 follows_VVZ from_II Theorem_NN1 13_MC that_CST @S_FO converges_VVZ weakly_RR to_II a_AT1 solution_NN1 to_II (_( 5_MC )_) and_CC that_CST @S_FO converges_VVZ weakly_RR to_II a_AT1 solution_NN1 to_II (_( 6_MC )_) ._. 
</s>
<s>
Then_RT the_AT first_MD terms_NN2 of_IO Hilbert_NP1 expansion_NN1 of_IO equal_JJ order_NN1 in_II ?_NNU and_CC ?j_FO for_IF @S_FO ,_, and_CC @s_FO are_VBR :_: @F_FO ,_, and_CC @F_FO ,_, while_CS for_IF the_AT second_MD population_NN1 the_AT calculation_NN1 loZhe_VV0 subsequent_JJ terms_NN2 expansion_NN1 yields_VVZ @F_FO ,_, where_CS @S_FO stands_VVZ for_IF the_AT Kronecker_NN1 delta_NN1 ._. 
</s>
<s>
The_AT bound_NN1 in_II Theorem_NN1 is_VBZ better_JJR than_CSN this_DD1 symmetry-based_NN1 bound_VVD ,_, but_CCB only_RR by_II a_AT1 multiplicative_JJ constant_JJ factor_NN1 (_( 1+a_FO )_) /2_MF when_RRQ @S_FO ;_; it_PPH1 is_VBZ of_RR21 course_RR22 far_RG better_RRR (_( linear_JJ convergence_NN1 rather_II21 than_II22 sublinear_JJ convergence_NN1 )_) when_RRQ @S_FO ._. 
</s>
<s>
All_DB of_IO this_DD1 implies_VVZ that_CST if_CS m_ZZ1 is_VBZ large_JJ enough_RR and_CC V∈E_FO ,_, then_RT for_IF V_ZZ1 close_VV0 enough_RR to_II V_ZZ1 we_PPIS2 have_VH0 that_DD1 V_ZZ1 is_VBZ (_( ,_, 2ε_FO ,_, m_ZZ1 )_) -entropy_JJ porous_JJ ._. 
</s>
<s>
Properties_NN2 of_IO @S_FO We_PPIS2 denote_VV0 by_II @S_FO the_AT mean-value_NN1 of_IO x_ZZ1 at_II time_NNT1 t_ZZ1 ,_, namely_REX ,_, @F_FO ._. 
</s>
<s>
Also_RR ,_, &lsqb;_( BD18_FO ,_, §5.3_FO &rsqb;_) discusses_VVZ an_AT1 approach_NN1 to_II computing_VVG a_AT1 finite_JJ set_NN1 containing_VVG X(Q)_NP1 when_RRQ @S_FO ,_, but_CCB @S_FO ,_, and_CC is_VBZ similar_JJ to_II the_AT one_PN1 used_VVD here_RL ._. 
</s>
<s>
Corollary_NN1 1.2_MC For_IF r_ZZ1 ,_, d_ZZ1 ,_, d_ZZ1 satisfying_JJ (_( r_ZZ1 ,_, d_ZZ1 )_) =_FO (_( r_ZZ1 ,_, d_ZZ1 )_) =1_FO ,_, the_AT E-polynomials_NN2 of_IO M_ZZ1 (_( g_ZZ1 ,_, r_ZZ1 ,_, d_ZZ1 )_) and_CC M_ZZ1 (_( g_ZZ1 ,_, r_ZZ1 ,_, d_ZZ1 )_) coincide_VV0 ._. 
</s>
<s>
Finally_RR ,_, it_PPH1 is_VBZ worthwhile_JJ to_TO note_VVI here_RL that_CST the_AT adiabatic_JJ theorem_NN1 can_VM be_VBI substantially_RR strengthened_VVN if_CS the_AT Ha_UH have_VH0 a_AT1 spectral_JJ gap_NN1 ,_, i.e._REX if_CS the_AT spectra_NN2 @S_FO are_VBR bounded_VVN away_II21 from_II22 0_MC ,_, uniformly_RR in_II a_AT1 ._. 
</s>
<s>
In_II that_DD1 case_NN1 ,_, at_II times_NNT2 in_II which_DDQ the_AT first_MD m_MC -derivatives_NN2 of_IO the_AT Hamiltonian_JJ vanish_VV0 ,_, the_AT result_NN1 (_( 1.4_MC )_) holds_VVZ with_IW an_AT1 improved_JJ error_NN1 bound_VVD Cm_NNU em_FU ,_, where_CS the_AT integer_NN1 m_ZZ1 depends_VVZ on_II the_AT smoothness_NN1 in_II a_AT1 and_CC it_PPH1 can_VM be_VBI made_VVN arbitrarily_RR large_JJ if_CS a_AT1 Ha_UH is_VBZ C8_FO ._. 
</s>
<s>
Therefore_RR ,_, the_AT subspaces_NN2 introduced_VVN in_II (_( 1.2_MC )_) are_VBR @S-invariant_FO ._. 
</s>
<s>
Second_MD ,_, Helen_NP1 should_VM have_VHI ensured_VVN Boris_NP1 had_VHD his_APPGE own_DA login_NN1 in_II order_NN1 to_TO further_RRR separate_VVI him_PPHO1 as_CSA student_NN1 from_II Helen_NP1 as_CSA lecturer_NN1 ,_, although_CS she_PPHS1 clearly_RR indicated_VVD who_PNQS was_VBDZ speaking_VVG and_CC in_II what_DDQ role_NN1 ._. 
</s>
<s>
Criterion_NN1 for_IF ISS_NN1 of_IO parabolic_JJ systems_NN2 ._. 
</s>
<s>
Consequently_RR ,_, we_PPIS2 investigate_VV0 two-dimensional_JJ inviscid_JJ flows_NN2 which_DDQ present_VV0 no_AT variation_NN1 in_II the_AT meridional_JJ direction_NN1 ,_, regarding_VVG them_PPHO2 as_CSA wavecurrent_JJ interactions_NN2 due_II21 to_II22 localised_JJ wave_NN1 perturbations_NN2 of_IO a_AT1 pure_JJ current_JJ background_NN1 state_NN1 ._. 
</s>
<s>
Furthermore_RR ,_, the_AT traditional_JJ OVB_NN1 ,_, be_VBI it_PPH1 standardized_VVN or_CC not_XX ,_, does_VDZ not_XX generalize_VVI easily_RR to_II multipleconfounders_NN2 :_: how_RRQ should_VM we_PPIS2 assess_VVI the_AT effect_NN1 of_IO confounders_NN2 Political_JJ Attitudes_NN2 and_CC Wealth_NN1 ,_, actingtogether_VV0 ,_, perhaps_RR with_IW complex_JJ non-linearities_NN2 ?_? 
</s>
<s>
The_AT sets_NN2 A_ZZ1 and_CC @S_FO This_DD1 leads_VVZ us_PPIO2 to_TO consider_VVI the_AT following_JJ symbolic_JJ sets_NN2 :_: @F_FO ,_, which_DDQ will_VM be_VBI useful_JJ to_TO define_VVI a_AT1 Young_JJ tower_NN1 on_II the_AT set_NN1 @F_FO ._. 
</s>
<s>
We_PPIS2 notice_VV0 that_CST the_AT map_NN1 @S_FO is_VBZ a_AT1 homeomorphism_NN1 onto_II its_APPGE image_NN1 (_( it_PPH1 is_VBZ injective_JJ because_CS it_PPH1 is_VBZ a_AT1 restriction_NN1 of_IO the_AT first_MD return_NN1 map_NN1 to_II Ye_PPY induced_VVD by_II f_ZZ1 )_) ._. 
</s>
<s>
We_PPIS2 call_VV0 the_AT union_NN1 of_IO connected_JJ components_NN2 of_IO @S_FO that_CST carry_VV0 the_AT same_DA label_VV0 the_AT local_JJ surfaces_NN2 of_IO @S_FO ._. 
</s>
<s>
We_PPIS2 label_VV0 these_DD2 local_JJ surfaces_NN2 also_RR by_II an_AT1 integer_NN1 in_II 1_MC1 ,_, ..._... ,_, n_ZZ1 according_II21 to_II22 the_AT ramification_NN1 point_NN1 they_PPHS2 carry_VV0 ._. 
</s>
<s>
Combining_VVG the_AT result_NN1 in_II &lsqb;_( 14_MC &rsqb;_) with_IW (_( 3.6_MC )_) yields_NN2 (_( 3.7_MC )_) ._. 
</s>
<s>
With_IW the_AT largest-k_JJT norm_NN1 ,_, we_PPIS2 can_VM obtain_VVI simple_JJ ,_, but_CCB key_JJ representations_NN2 of_IO the_AT cardinality_NN1 constraint_NN1 of_IO (_( 2_MC )_) ._. 
</s>
<s>
Indeed_RR ,_, unlike_JJ inert_JJ matter_NN1 ,_, the_AT behavioral_JJ ability_NN1 of_IO heterogeneous_JJ human_JJ beings_NN2 to_TO develop_VVI walking_VVG strategies_NN2 and_CC to_TO adapt_VVI themselves_PPX2 to_II the_AT context_NN1 generates_VVZ observable_JJ effects_NN2 arising_VVG from_II causes_NN2 that_CST often_RR do_VD0 not_XX appear_VVI evident_JJ ._. 
</s>
<s>
If_CS @S_FO there_EX exists_VVZ a_AT1 point_NN1 in_II the_AT relative_JJ interior_NN1 of_IO UK(x)_NP1 (_( Slater_NP1 point_NN1 )_) and_CC @S_FO ,_, then_RT for_IF all_DB x_ZZ1 ,_, y_ZZ1 ,_, @F_FO ._. 
</s>
<s>
The_AT following_JJ result_NN1 provides_VVZ the_AT basic_JJ building_NN1 blocks_NN2 ._. 
</s>
<s>
Solvers_NN2 based_VVN on_II Krylov_NP1 subspace_NN1 methods_NN2 &lsqb;_( 121_MC ,_, 9_MC ,_, 122_MC ,_, 184_MC &rsqb;_) and_CC on_II Anderson_NP1 relaxation_NN1 &lsqb;_( 5_MC ,_, 211_MC ,_, 202_MC &rsqb;_) perform_VV0 intermediate_JJ computations_NN2 in_II low-dimensional_JJ subspaces_NN2 that_CST are_VBR updated_VVN as_II the_AT computation_NN1 proceeds_NN2 ._. 
</s>
<s>
Consider_VV0 a_AT1 toy_NN1 example_NN1 where_CS the_AT cases_NN2 on_II the_AT unionsupport_NN1 can_VM be_VBI divided_VVN into_II two_MC types_NN2 ,_, characterized_VVN by_II low_JJ and_CC high_JJ baseline_NN1 activities_NN2 ;_; and_CC amongthe_VV0 more_RGR active_JJ types_NN2 a_AT1 larger_JJR proportion_NN1 would_VM exhibit_VVI differential_JJ levels_NN2 between_II the_AT twoconditions_NN2 ._. 
</s>
<s>
For_IF convenience_NN1 ,_, we_PPIS2 have_VH0 added_VVN these_DD2 rates_NN2 in_II parentheses_NN2 in_II the_AT legends_NN2 in_II Figure_NN1 3_MC ._. 
</s>
<s>
For_CS we_PPIS2 observe_VV0 the_AT faster_JJR convergence_NN1 rates_NN2 in_II the_AT right_JJ panel_NN1 of_IO Table_NN1 1_MC1 ,_, although_CS a_AT1 closer_JJR inspection_NN1 indicates_VVZ that_CST the_AT convergence_NN1 is_VBZ slowing_VVG down_RP as_CSA N_ZZ1 increases_VVZ ._. 
</s>
<s>
Ip_VV0 ,_, our_APPGE expression_NN1 simplifies_VVZ further_RRR and_CC we_PPIS2 get_VV0 the_AT finite-_JJ formula_NN1 @S_FO ,_, which_DDQ scales_NN2 like_II a_AT1 ._. 
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<s>
There_EX exists_VVZ a_AT1 conjugate_NN1 charge_NN1 p_ZZ1 ,_, localised_VVD in_II S&lsqb;_NP1 ,_, such_CS21 that_CS22 @F_FO ._. 
</s>
<s>
To_TO challenge_VVI the_AT binary_NN1 of_IO human_NN1 and_CC tools_NN2 ,_, de_NP1 Freitas_NP2 and_CC Sinclair_NP1 refer_VV0 back_RP to_II an_AT1 image_NN1 from_II Merleau-Ponty_NN1 (_( 1945_MC )_) ,_, that_DD1 of_IO a_AT1 man_NN1 walking_VVG in_II a_AT1 dark_JJ room_NN1 with_IW a_AT1 stick_NN1 :_: does_VDZ the_AT man_NN1 feel_VVI his_APPGE hand_NN1 touching_VVG the_AT stick_NN1 or_CC does_VDZ the_AT man_NN1 feel_VVI the_AT end_NN1 of_IO the_AT stick_NN1 touching_VVG the_AT contours_NN2 of_IO the_AT dark_JJ room_NN1 ?_? 
</s>
<s>
Set_VV0 @F_FO ._. 
</s>
<s>
Define_VV0 @F_FO ._. 
</s>
<s>
Also_RR ,_, set_VV0 @F_FO ._. 
</s>
<s>
Using_VVG obvious_JJ notations_NN2 we_PPIS2 split_VV0 the_AT bilinear_JJ form_NN1 @S_FO as_CSA @F_FO ._. 
</s>
<s>
We_PPIS2 begin_VV0 by_II observing_VVG that_CST ,_, from_II Ref._NN1 16_MC (_( in_RR21 particular_RR22 cf._VV0 Here_RL q_ZZ1 =_FO 4_MC ,_, @S_FO ,_, and_CC @F_FO ,_, as_CSA is_VBZ easily_RR seen_VVN ._. 
</s>
<s>
We_PPIS2 show_VV0 how_RRQ to_TO approximate_VVI the_AT resulting_JJ brittle_JJ fracture_NN1 energy_NN1 ,_, under_II some_DD geometric_JJ assumptions_NN2 on_II the_AT Dirichlet_NN1 part_NN1 ofthe_NN1 domain_NN1 in_II the_AT first_MD case_NN1 ,_, and_CC for_IF a_AT1 very_RG large_JJ class_NN1 of_IO compliance_NN1 functions_NN2 (_( possibly_RR such_CS21 that_CS22 the_AT displacement_NN1 is_VBZ not_XX a_RR21 priori_RR22 forced_VVN to_TO be_VBI even_RR integrable_JJ )_) in_II the_AT second_MD case_NN1 ._. 
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<s>
After_II having_VHG constructed_VVN such_DA a_AT1 W-invariant_JJ (_( ergodic_JJ )_) measure_VV0 v_ZZ1 ,_, we_PPIS2 apply_VV0 our_APPGE third_MD ingredient_NN1 ,_, which_DDQ is_VBZ a_AT1 general_JJ result_NN1 in_II ergodic_JJ theory_NN1 and_CC a_AT1 consequence_NN1 of_IO Sinai_NP1 '_NULL s_ZZ1 factor_NN1 theorem_NN1 ,_, to_TO show_VVI that_CST the_AT space_NN1 X_ZZ1 can_VM be_VBI partitioned_VVN (_( up_II21 to_II22 a_AT1 part_NN1 of_IO small_JJ v-measure_NN1 )_) into_II finitely_RR many_DA2 subsets_NN2 UjAj_VV0 such_CS21 that_CS22 for_IF @S_FO and_CC for_IF each_DD1 j_ZZ1 the_AT set_NN1 @S_FO satisfies_VVZ the_AT above_JJ properties_NN2 (_( i_ZZ1 )_) and_CC (_( ii_MC )_) ._. 
</s>
<s>
Since_CS A_ZZ1 is_VBZ a_AT1 standard_JJ form_NN1 cdga_NN1 ,_, these_DD2 variables_NN2 xj_NNU for_IF i_ZZ1 <_FO 0_MC and_CC y?_ZZ1 generate_VV0 A_ZZ1 freely_RR over_II A(0)_FO as_II a_AT1 commutative_JJ graded_JJ algebra_NN1 ._. 
</s>
<s>
So_RR we_PPIS2 shall_VM first_MD discuss_VVI the_AT solving_NN1 of_IO this_DD1 linear_JJ system_NN1 ._. 
</s>
<s>
According_II21 to_II22 Thompson_NP1 ,_, what_DDQ is_VBZ important_JJ in_II quantitative_JJ reasoning_NN1 is_VBZ not_XX assigning_VVG numeric_JJ measures_NN2 to_II quantities_NN2 ,_, but_CCB rather_RR reasoning_VVG about_II the_AT relationships_NN2 between_II two_MC or_CC more_DAR quantities_NN2 ._. 
</s>
<s>
Moreover_RR ,_, the_AT restriction_NN1 of_IO u_ZZ1 to_II any_DD subdomain_NN1 @S_FO satisfies_VVZ @F_FO ._. 
</s>
<s>
We_PPIS2 now_RT apply_VV0 Lemma_NN1 5.3_MC and_CC Corollary_NN1 5.5_MC to_TO establish_VVI Proposition_NN1 5.2_MC ._. 
</s>
<s>
Therefore_RR ,_, while_CS applying_VVG special_JJ methods_NN2 to_TO increase_VVI the_AT level_NN1 of_IO motivation_NN1 and_CC the_AT interest_NN1 of_IO the_AT "_" weak_JJ "_" students_NN2 is_VBZ seen_VVN as_II a_AT1 time-consuming_JJ by_II the_AT teachers_NN2 ,_, in_II31 case_II32 of_II33 "_" strong_JJ "_" students_NN2 teachers_NN2 at_RR21 least_RR22 are_VBR ready_JJ to_TO work_VVI with_IW them_PPHO2 by_II devoting_VVG time_NNT1 to_II the_AT extra_JJ classes_NN2 or_CC suggesting_VVG problems_NN2 with_IW different_JJ formats_NN2 and_CC levels_NN2 of_IO difficulty_NN1 ._. 
</s>
<s>
Besides_RR ,_, we_PPIS2 also_RR give_VV0 a_AT1 proof_NN1 of_IO Proposition_NN1 2.1_MC and_CC of_IO Proposition_NN1 2.2_MC regarding_II the_AT strong_JJ formulation_NN1 of_IO the_AT quasistatic_JJ evolution_NN1 ._. 
</s>
<s>
We_PPIS2 define_VV0 the_AT projectivity_NN1 group_NN1 of_IO X_ZZ1 at_II v_ZZ1 as_CSA @S_FO by_II viewing_VVG the_AT usual_JJ projectivity_NN1 group_NN1 @S_FO as_II a_AT1 subgroup_NN1 @S_FO ._. 
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<s>
Thus_RR the_AT projectivity_NN1 group_NN1 of_IO A_ZZ1 at_II v_ZZ1 is_VBZ @S_FO ,_, since_CS the_AT se@S_FO is_VBZ canonically_RR isomorphic_JJ to_II the_AT set_NN1 of_IO ends_NN2 of_IO the_AT panel_NN1 tree_NN1 Tv_NN1 by_II Proposition_NN1 2.15_MC ._. 
</s>
<s>
Let_VV0 Pn_NP1 be_VBI the_AT characteristic_JJ polynomial_NN1 of_IO Gn_NNU ,_, and_CC z_ZZ1 <_FO 0_MC ._. 
</s>
<s>
In_II the_AT cell_NN1 cytoplasm_NN1 ,_, the_AT SARS-CoV-2_MC RNA_NN1 are_VBR recognized_VVN and_CC destroyed_VVN by_II RNA_NN1 helicases_NN2 ._. 
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<s>
Let_VV0 us_PPIO2 say_VVI that_CST a_AT1 nonnegative_JJ kernel_NN1 @S_FO belongs_VVZ to_II the_AT class_NN1 K0_FO if_CS @F_FO ._. 
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<s>
Correspondingly_RR ,_, we_PPIS2 define_VV0 the_AT extremal_JJ operators_NN2 M+_FO and_CC @S_FO by_II @F_FO ._. 
</s>
<s>
The_AT conception_NN1 of_IO limit_NN1 as_II a_AT1 cluster_NN1 point_NN1 ,_, on_II the_AT other_JJ hand_NN1 ,_, emerged_VVD from_II the_AT interaction_NN1 of_IO such_DA ideas_NN2 as_CSA "_" getting_VVG close_RR to_II "_" and_CC "_" equal_JJ to_II "_" allowing_VVG students_NN2 to_TO determine_VVI "_" convergence_NN1 of_IO oscillating_VVG sequences_NN2 differently_RR from_II students_NN2 who_PNQS imagined_VVD limits_NN2 as_CSA asymptotes_NN2 "_" (_( Roh_NP1 ,_, 2008_MC ,_, p._NN1 227_MC )_) ._. 
</s>
<s>
A_ZZ1 is_VBZ a_AT1 valuation_NN1 if_CS @F_FO whenever_RRQV @S_FO is_VBZ convex_JJ ._. 
</s>
<s>
Proposition_NN1 3.5_MC Let_VVN be_VBI any_DD H-invariant_JJ probability_NN1 on_II SL_NP1 (_( m_ZZ1 ,_, R_ZZ1 )_) /SL_NN1 (_( m_ZZ1 ,_, Z_ZZ1 )_) ._. 
</s>
<s>
Next_MD ,_, using_VVG the_AT chunks_NN2 identified_VVN and_CC building_NN1 from_II the_AT work_NN1 of_IO Brown_NP1 (_( 2009_MC )_) and_CC other_JJ curriculum_NN1 researchers_NN2 (_( i.e._REX ,_, Remillard_NP1 2005_MC )_) ,_, we_PPIS2 considered_VVD it_PPH1 important_JJ to_TO research_VVI instances_NN2 when_RRQ pairs_NN2 included_VVD ,_, adapted_VVD ,_, or_CC omitted_VVN an_AT1 element_NN1 from_II the_AT original_JJ materials_NN2 ,_, as_CSA we_PPIS2 consider_VV0 this_DD1 to_TO provide_VVI insight_NN1 about_II their_APPGE use_NN1 and_CC directly_RR answer_VV0 the_AT first_MD research_NN1 question_NN1 about_II prospective_JJ teachers_NN2 '_NULL interactions_NN2 with_IW curricular_JJ elements_NN2 ._. 
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<s>
We_PPIS2 are_VBR also_RR grateful_JJ to_II the_AT anonymous_JJ referee_NN1 ,_, whose_DDQGE suggestions_NN2 significantly_RR improved_VVD the_AT quality_NN1 of_IO the_AT paper_NN1 and_CC the_AT references_NN2 ._. 
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<s>
In_II the_AT ring_NN1 Aq_NP1 ,_, there_EX is_VBZ an_AT1 equality_NN1 @F_FO ,_, where_CS the_AT sign_NN1 is_VBZ positive_JJ in_II the_AT additive_JJ case_NN1 and_CC negative_JJ in_II the_AT subtractive_JJ case_NN1 ._. 
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<s>
There_EX is_VBZ also_RR a_AT1 right_JJ action_NN1 xxR_NNU of_IO Ln_NP1 on_II Mg_NNU that_DD1 commutes_VVZ with_IW the_AT left_JJ action_NN1 of_IO Ln_NP1 &lsqb;_( 51_MC &rsqb;_) ._. 
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<s>
Some_DD selected_JJ proofs_NN2 for_IF Section_NN1 4.1.2_MC ._. 
</s>
<s>
For_IF each_DD1 j_ZZ1 ,_, we_PPIS2 define_VV0 @S_FO ._. 
</s>
<s>
Clearly_RR ,_, δj_FO →_NULL 0_MC ._. 
</s>
<s>
Relationships_NN2 between_II student_NN1 strategy_NN1 use_NN1 and_CC teachers_NN2 '_NULL ways_NN2 of_IO attending_VVG to_II students_NN2 '_NULL responses_NN2 A_ZZ1 cross-tabulation_NN1 of_IO teacher_NN1 attention_NN1 to_II student_NN1 written_VVN response_NN1 by_II student_NN1 strategy_NN1 use_NN1 (_( counting_NN1 ;_; functional_JJ ;_; recursive_JJ ;_; chunking_VVG ;_; and_CC whole-object_NN1 )_) was_VBDZ done_VDN (_( Table_NN1 9_MC )_) ._. 
</s>
<s>
Switching_VVG to_II polar_JJ coordinates_NN2 according_II21 to_II22 @S_FO ,_, we_PPIS2 find_VV0 ,_, after_II completing_VVG the_AT radial_JJ integration_NN1 ,_, @F_FO ._. 
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<s>
They_PPHS2 call_VV0 this_DD1 latter_DA algorithm_NN1 the_AT normalized_JJ augmented_VVD Newton_NP1 recurrence_NN1 ._. 
</s>
<s>
The_AT monotonicity_NN1 of_IO @S_FO on_II the_AT state_NN1 space_NN1 @S_FO is_VBZ inherited_VVN from_II its_APPGE monotonicity_NN1 on_II the_AT space_NN1 of_IO finite_JJ configurations_NN2 ,_, Sfin_NP1 ,_, defined_VVN in_II (_( 2.19_MC )_) ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI some_DD radial_JJ decreasing_JJ non-negative_JJ cut-off_NN1 function_NN1 with_IW =_FO 1_MC1 in_II Bi_NP1 ,_, and_CC set_VV0 @S_FO ._. 
</s>
<s>
Then_RT ,_, as_CSA in_II the_AT interior_JJ case_NN1 ,_, for_IF @S_FO and_CC @S_FO ,_, we_PPIS2 use_VV0 the_AT Lipschitz_NP1 function_NN1 @F_FO as_II a_AT1 test_NN1 function_NN1 in_II (_( 6.5_MC )_) ._. 
</s>
<s>
Within_II this_DD1 CoP_NN1 ,_, we_PPIS2 were_VBDR able_JK to_TO establish_VVI trust_NN1 where_CS any_DD of_IO us_PPIO2 could_VM make_VVI observations_NN2 about_II the_AT course_NN1 materials_NN2 ,_, the_AT questions_NN2 we_PPIS2 asked_VVD ,_, the_AT decisions_NN2 we_PPIS2 were_VBDR pondering_VVG ,_, etc._RA ,_, and_CC these_DD2 observations_NN2 were_VBDR received_VVN by_II the_AT other_JJ instructors_NN2 as_CSA valid_JJ and_CC were_VBDR considered_VVN within_II the_AT context_NN1 as_CSA we_PPIS2 made_VVD instructional_JJ decisions_NN2 ._. 
</s>
<s>
As_CSA in_II Section_NN1 5.2_MC there_EX are_VBR several_DA2 cases_NN2 to_TO consider_VVI when_RRQ computing_NN1 @S_FO ,_, where_CS @S_FO ,_, in_II fewer_DAR than_CSN @S_FO operations_NN2 :_: (_( 1_MC1 )_) @S_FO is_VBZ an_AT1 integer_NN1 ,_, (_( 2_MC )_) @S_FO and_CC @S_FO ,_, (_( 3_MC )_) @S_FO and_CC @S_FO ,_, and_CC (_( 4_MC )_) @S_FO ,_, but_CCB the_AT difference_NN1 is_VBZ a_AT1 noninteger_JJ ._. 
</s>
<s>
Definition_NN1 2.5_MC (_( Convergence_NN1 and_CC convergence_NN1 scheme_NN1 )_) ._. 
</s>
<s>
Mixed_VVN finite_JJ element_NN1 (_( MFE_NP1 )_) methods_NN2 for_IF elasticity_NN1 with_IW stressdisplacement_NN1 formulations_NN2 provide_VV0 accurate_JJ stress_NN1 ,_, local_JJ momentum_NN1 conservation_NN1 ,_, and_CC robust_JJ treatment_NN1 of_IO almost_RR incompressible_JJ materials_NN2 ._. 
</s>
<s>
This_DD1 is_VBZ not_XX essential_JJ but_CCB it_PPH1 is_VBZ convenient_JJ because_CS below_RL we_PPIS2 can_VM choose_VVI a_AT1 constant_JJ cone_NN1 field_NN1 which_DDQ is_VBZ invariant_JJ ._. 
</s>
<s>
Notice_VV0 that_CST each_DD1 rectangle_NN1 has_VHZ its_APPGE leftmost_JJ lowermost_JJ vertex_NN1 always_RR at_II @S_FO and_CC that_CST the_AT first_MD rectangle_NN1 @S_FO consists_VVZ of_IO only_RR two_MC vertices_NN2 ,_, @S_FO ._. 
</s>
<s>
These_DD2 are_VBR done_VDN in_II Subsections_NN2 6.1_MC and_CC 6.2_MC ._. 
</s>
<s>
The_AT 3D_NNU Euler_NN1 equations_NN2 (_( 1.1_MC )_) with_IW axial_JJ symmetry_NN1 can_VM be_VBI conveniently_RR described_VVN in_II the_AT so-called_JJ vorticity-stream_JJ form_NN1 (_( cf._VV0 Using_VVG translations_NN2 and_CC dilations_NN2 the_AT single_JJ interval_NN1 problem_NN1 in_II any_DD given_JJ interval_NN1 @S_FO can_VM be_VBI recast_VVN as_II a_AT1 corresponding_JJ problem_NN1 in_II any_DD desired_JJ open_JJ interval_NN1 (_( a_AT1 ,_, b_ZZ1 )_) ._. 
</s>
<s>
In_II summary_NN1 ,_, the_AT reasoning_NN1 of_IO the_AT 12_MC pairs_NN2 of_IO students_NN2 about_II the_AT context_NN1 of_IO the_AT optimization_NN1 problem_NN1 presented_VVN in_II the_AT profit_NN1 maximization_NN1 task_NN1 revealed_VVD that_CST a_AT1 focus_NN1 on_II the_AT context_NN1 made_VVD visible_JJ students_NN2 '_NULL reasoning_VVG about_II marginal_JJ cost_NN1 ,_, marginal_JJ revenue_NN1 ,_, and_CC sequences_NN2 of_IO quantitative_JJ differences_NN2 ._. 
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<s>
For_IF this_DD1 reason_NN1 ,_, the_AT mean-field_JJ limit_NN1 procedure_NN1 can_VM not_XX be_VBI applied_VVN to_II a_AT1 networked_JJ collective_JJ behavior_NN1 model_NN1 in_II which_DDQ the_AT interactions_NN2 are_VBR given_VVN as_CSA in_II (_( 2.1_MC )_) ._. 
</s>
<s>
Let_VV0 X_ZZ1 be_VBI an_AT1 algebraic_JJ stack_NN1 with_IW affine_JJ diagonal_JJ ,_, locally_RR of_IO finite_JJ type_NN1 over_II an_AT1 algebraically_RR closed_JJ field_NN1 k_ZZ1 ._. 
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<s>
Fn_NP1 (_( Corollary_NN1 2.9_MC )_) ,_, all_DB these_DD2 lifts_NN2 coincide_VV0 ,_, defining_VVG a_AT1 homeomorphism_NN1 @S_FO on_II X._NP1 Choose_VV0 a_AT1 covering_NN1 @S_FO of_IO M_NN1 by_II balls_NN2 of_IO radius_NN1 r_ZZ1 ,_, so_CS21 that_CS22 balls_NN2 of_IO half_DB the_AT radius_NN1 still_RR cover_VV0 M._NN1 In_II the_AT next_MD section_NN1 we_PPIS2 prove_VV0 Theorem_NN1 1.1_MC ._. 
</s>
<s>
Pseudocodes_NN2 of_IO the_AT forward_NN1 and_CC the_AT backward_JJ phases_NN2 are_VBR given_VVN in_II Section_NN1 3_MC of_IO the_AT Supplementary_JJ Material_NN1 and_CC we_PPIS2 refer_VV0 to_II Figure_NN1 3_MC of_IO the_AT Supplementary_JJ Material_NN1 for_IF an_AT1 illustration_NN1 of_IO the_AT search_NN1 path_NN1 of_IO GES_NP2 for_REX21 Example_REX22 1_MC1 (_( Section_NN1 3_MC )_) ._. 
</s>
<s>
As_II a_AT1 result_NN1 ,_, substantial_JJ efforts_NN2 have_VH0 been_VBN deployed_VVN in_II many_DA2 countries_NN2 to_TO engage_VVI mathematics_NN1 teachers_NN2 in_II lesson_NN1 study_NN1 ._. 
</s>
<s>
Suppose_VV0 the_AT assumptions_NN2 in_II Theorem_NN1 2.1_MC hold_NN1 ._. 
</s>
<s>
Similarly_RR to_II the_AT single-orbital_JJ entropy_NN1 s(i)_FW ,_, the_AT two-orbital_JJ entropy_NN1 @S_FO ,_, where_CS @S_FO is_VBZ the_AT two-orbital_JJ density_NN1 matrix_NN1 obtained_VVN from_II @S_FO ._. 
</s>
<s>
Given_VVN the_AT single-_NN1 and_CC two-orbital_JJ entropies_NN2 ,_, we_PPIS2 can_VM compute_VVI the_AT mutual_JJ information_NN1 ,_, @S_FO for_IF @S_FO ._. 
</s>
<s>
This_DD1 quantifies_VVZ the_AT electron_NN1 correlations_NN2 between_II orbital_JJ i_ZZ1 and_CC j_ZZ1 as_CSA they_PPHS2 are_VBR embedded_VVN in_II the_AT whole_JJ system_NN1 &lsqb;_( 35_MC &rsqb;_) ._. 
</s>
<s>
Clearly_RR @S_FO ._. 
</s>
<s>
Since_CS @S_FO and_CC a_AT1 u-admissible_JJ manifold_NN1 intersects_VVZ an_AT1 s-admissible_JJ manifold_NN1 at_RR21 most_RR22 once_RR &lsqb;_( KH95_FO ,_, Cor_UH ._. 
</s>
<s>
Remarkably_RR ,_, this_DD1 result_NN1 holds_VVZ true_JJ distribution-free_JJ ,_, and_CC it_PPH1 is_VBZ applicable_JJ without_IW any_DD knowledge_NN1 on_II the_AT probability_NN1 P._NN1 While_CS we_PPIS2 provide_VV0 some_DD guidance_NN1 on_II the_AT selection_NN1 of_IO sampling_VVG rates_NN2 in_II an_AT1 unbiased_JJ noise_NN1 setting_VVG in_II Sect._NP1 5_MC ,_, our_APPGE numerical_JJ experiments_NN2 show_VV0 that_CST the_AT bounds_NN2 on_II probabilities_NN2 suggested_VVN by_II our_APPGE theory_NN1 to_TO be_VBI necessary_JJ for_IF almost_RR sure_JJ convergence_NN1 are_VBR far_RG21 from_RG22 tight_JJ ._. 
</s>
<s>
However_RR ,_, they_PPHS2 suffer_VV0 from_II loose_JJ constants_NN2 and_CC are_VBR incapable_JJ of_IO providing_VVG quantitative_JJ recommendations_NN2 ._. 
</s>
<s>
In_RR21 short_RR22 ,_, given_VVN the_AT Helmholtz_NP1 decomposition_NN1 of_IO the_AT forcing_NN1 term_NN1 F_ZZ1 ,_, the_AT unsteady_JJ Stokes_NP1 equations_NN2 have_VH0 an_AT1 explicit_JJ solution_NN1 in_II free_JJ space_NN1 by_II quadrature_NN1 ._. 
</s>
<s>
Note_VV0 that_CST this_DD1 bound_VVD also_RR holds_VVZ if_CS @S_FO ._. 
</s>
<s>
There_EX is_VBZ a_AT1 direct_JJ connection_NN1 between_II these_DD2 eigenvalues_NN2 and_CC the_AT polynomial_NN1 conservation_NN1 laws_NN2 mentioned_VVD earlier_RRR ;_; see_VV0 ,_, for_REX21 example_REX22 ,_, &lsqb;_( 50_MC ,_, §3_FO &rsqb;_) ._. 
</s>
<s>
It_PPH1 is_VBZ worth_II mentioning_VVG here_RL that_CST although_CS the_AT theorem_NN1 presented_VVD below_RL as_II31 well_II32 as_II33 Proposition_NN1 3.2_MC are_VBR written_VVN for_IF the_AT selected_JJ representation_NN1 of_IO v_ZZ1 in_II (_( 2.15_MC )_) ,_, they_PPHS2 are_VBR invariant_JJ with_II31 respect_II32 to_II33 different_JJ choices_NN2 of_IO @S_FO ,_, ki_NN2 ,_, and_CC jli_NN2 in_II (_( 2.15_MC )_) ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI a_AT1 global_JJ strongly_RR @S-generic_FO type_NN1 which_DDQ is_VBZ M-invariant_JJ (_( exists_VVZ by_II Fact_NN1 3.3(1)_FO )_) ,_, and_CC let_VV0 h_ZZ1 realize_VVI p_ZZ1 over_RG Mg_NNU ._. 
</s>
<s>
Because_II21 of_II22 the_AT special_JJ symmetry_NN1 properties_NN2 of_IO K4_FO ,_, the_AT four_MC graphs_NN2 we_PPIS2 get_VV0 in_II this_DD1 way_NN1 are_VBR actually_RR four_MC copies_NN2 of_IO the_AT same_DA elementary_JJ generalized_JJ melon_NN1 ._. 
</s>
<s>
We_PPIS2 also_RR compare_VV0 computational_JJ times_NNT2 in_II Fig._NN1 3_MC ._. 
</s>
<s>
In_II the_AT particular_JJ case_NN1 @S_FO ,_, such_DA a_AT1 formula_NN1 was_VBDZ already_RR given_VVN in_II &lsqb;_( 18_MC &rsqb;_) ._. 
</s>
<s>
This_DD1 '_NULL propagation_NN1 of_IO chaos_NN1 '_NULL has_VHZ been_VBN proved_VVN for_IF the_AT models_NN2 under_II consideration_NN1 and_CC ,_, under_II certain_JJ assumptions_NN2 ,_, information_NN1 on_II the_AT convergencerateis_NN1 also_RR available_JJ (_( see_VV0 Section_NN1 4_MC below_RL for_IF bibliographical_JJ references_NN2 )_) ._. 
</s>
<s>
A6_FO measures_VVZ coercivity_NN1 with_II31 respect_II32 to_II33 -stationary_JJ ,_, possibly_RR highly_RR localized_JJ deformations_NN2 ._. 
</s>
<s>
This_DD1 implies_VVZ that_CST the_AT Euclidean_JJ distance_NN1 between_II hn_NNU and_CC hn+1_FO tends_VVZ to_II 0_MC ._. 
</s>
<s>
It_PPH1 is_VBZ convenient_JJ to_TO define_VVI the_AT rescaled_JJ stepsize_VV0 @F_FO ._. 
</s>
<s>
The_AT fundamental_JJ unknowns_NN2 in_II the_AT mixed_JJ potential_JJ representation_NN1 are_VBR densities_NN2 supported_VVN on_II the_AT boundary_NN1 of_IO the_AT domain_NN1 ._. 
</s>
<s>
One_MC1 member_NN1 of_IO the_AT project_NN1 team_NN1 initiated_VVD the_AT analysis_NN1 by_II transcribing_VVG and_CC reviewing_VVG all_DB equal_JJ sign_NN1 data_NN from_II the_AT pre-_JJ and_CC post-test_JJ clinical_JJ interviews_NN2 and_CC the_AT pre-_JJ ,_, mid-_VV0 ,_, and_CC post-teaching_JJ experiment_NN1 interviews_NN2 for_IF the_AT four_MC interview_NN1 participants_NN2 in_II the_AT participating_JJ classroom_NN1 from_II School_NN1 A_ZZ1 (_( 20_MC interviews_NN2 total_JJ )_) ._. 
</s>
<s>
If_CS @S_FO ,_, we_PPIS2 will_VM have_VHI @S_FO for_IF all_DB @ST_FO ,_, then_RT we_PPIS2 define_VV0 the_AT solution_NN1 as_CSA @F_FO ,_, where_CS @S_FO is_VBZ the_AT shock_NN1 curve_NN1 with_IW two_MC steady_JJ states_NN2 va_FW and_CC vy_NN1 given_VVN by_II (_( 4.4_MC )_) ._. 
</s>
<s>
Otherwise_RR ,_, choose_VV0 a_AT1 subsequence_NN1 T_ZZ1 of_IO @S_FO such_CS21 that_CS22 @S_FO exists_VVZ ._. 
</s>
<s>
Then_RT f_ZZ1 is_VBZ an_AT1 equivalence_NN1 if_CS and_CC only_RR if_CS for_IF all_DB i>0_FO the_AT map_NN1 @F_FO is_VBZ an_AT1 isomorphism_NN1 of_IO pro-groups_NN2 ,_, see_VV0 &lsqb;_( 18_MC ,_, Cor._NP1 7.5_MC &rsqb;_) ._. 
</s>
<s>
Let_VV0 T_ZZ1 '_NULL be_VBI a_AT1 tableau_NN1 of_IO shape_NN1 pfdn_NNU with_IW content_NN1 @S_FO such_CS21 that_CS22 no_AT letter_NN1 appears_VVZ more_DAR than_CSN once_RR in_II a_AT1 column_NN1 ._. 
</s>
<s>
Thus_RR ,_, we_PPIS2 find_VV0 that_CST the_AT integrals_NN2 are_VBR well-defined_JJ using_VVG @S_FO as_CSA @S_FO ._. 
</s>
<s>
At_RR21 least_RR22 one_MC1 of_IO the_AT vectors_NN2 @S_FO does_VDZ not_XX belong_VVI to_II K._NP1 Consequently_RR ,_, among_II the_AT N_ZZ1 terms_NN2 @F_FO ,_, at_RR21 least_RR22 one_PN1 is_VBZ not_XX vanishing_VVG ._. 
</s>
<s>
Since_CS @S_FO ,_, the_AT same_DA result_NN1 holds_VVZ for_IF the_AT nonsplit_JJ curve_NN1 of_IO level_NN1 13_MC ._. 
</s>
<s>
Examining_VVG children_NN2 '_NULL s_ZZ1 relational_JJ use_NN1 of_IO the_AT counting_NN1 principles_NN2 provides_VVZ a_AT1 window_NN1 into_II their_APPGE concurrent_JJ development_NN1 ,_, and_CC how_RRQ the_AT use_NN1 of_IO a_AT1 given_JJ principle_NN1 did_VDD not_XX emerge_VVI in_II the_AT same_DA ways_NN2 for_IF each_DD1 child_NN1 or_CC in_II a_AT1 specific_JJ sequence_NN1 across_II the_AT data_NN set_VV0 ._. 
</s>
<s>
The_AT following_JJ theorem_NN1 follows_VVZ from_II &lsqb;_( McM2_FO ,_, Theorem_NN1 8.1_MC &rsqb;_) ._. 
</s>
<s>
For_IF the_AT sake_NN1 of_IO clarity_NN1 ,_, we_PPIS2 shall_VM assume_VVI that_CST @S_FO and_CC set_VVD ,_, for_IF all_DB @S_FO ._. 
</s>
<s>
From_II Eq_NN1 ._. 
</s>
<s>
(_( 3.27_MC )_) we_PPIS2 get_VV0 for_IF @S_FO @F_FO F1_FO is_VBZ C1_FO in_II C_ZZ1 Z1_FO ,_, Zn_NP1 and_CC using_VVG (_( 3.27_MC )_) its_APPGE derivative_NN1 is_VBZ given_VVN by_II @F_FO ._. 
</s>
<s>
Let_VV0 @F_FO ._. 
</s>
<s>
In_II the_AT second_MD stage_NN1 of_IO our_APPGE analysis_NN1 ,_, we_PPIS2 looked_VVD for_IF patterns_NN2 (_( commonalities_NN2 and_CC difficulties_NN2 )_) in_II students_NN2 '_NULL verbal_JJ responses_NN2 and_CC written_JJ work_NN1 within_II each_DD1 of_IO the_AT a_JJ21 priori_JJ22 codes_NN2 identified_VVN in_II the_AT first_MD stage_NN1 ._. 
</s>
<s>
If_CS @S_FO ,_, the_AT subscript_NN1 A_ZZ1 will_VM be_VBI dropped_VVN from_II the_AT notation_NN1 ._. 
</s>
<s>
In_II this_DD1 section_NN1 we_PPIS2 recall_VV0 some_DD basic_JJ facts_NN2 on_II A2-buildings_NN2 and_CC their_APPGE boundaries_NN2 ._. 
</s>
<s>
The_AT solution_NN1 uniqueness_NN1 may_VM break_VVI down_RP when_CS @S_FO for_IF coefficients_NN2 a(x)_VV0 that_CST are_VBR not_XX continuous_JJ ._. 
</s>
<s>
Example_NN1 3.1_MC Let_VV0 us_PPIO2 calculate_VVI f_ZZ1 in_II a_AT1 few_DA2 cases_NN2 ._. 
</s>
<s>
Thus_RR ,_, to_TO analyze_VVI (_( 6.23_MC )_) further_RRR ,_, we_PPIS2 must_VM obtain_VVI a_AT1 more_RGR refined_JJ estimate_NN1 on_II the_AT error_NN1 in_II approximating_VVG these_DD2 quantities_NN2 by_II @S_FO ._. 
</s>
<s>
The_AT stabilizer_NN1 of_IO ve@S_FO in_II @S_FO equals_VVZ @S_FO ,_, i.e._REX ,_, the_AT image_NN1 of_IO the_AT stabilizer_NN1 of_IO ve@S_FO in_II @S_FO under_II the_AT map_NN1 @S_FO ._. 
</s>
<s>
Moreover_RR ,_, the_AT homomorphism_NN1 @S_FO extends_VVZ to_II a_AT1 closed_JJ immersion_NN1 from_II @S_FO to_II @S_FO which_DDQ we_PPIS2 continue_VV0 to_TO denote_VVI by_II @S_FO ._. 
</s>
<s>
Then_RT there_EX holds_VVZ @S_FO ,_, @S_FO ,_, and_CC for_IF @S_FO ,_, we_PPIS2 have_VH0 @L_FO ._. 
</s>
<s>
In_II fact_NN1 ,_, we_PPIS2 prove_VV0 a_AT1 stronger_JJR result_NN1 ,_, as_CSA we_PPIS2 relax_VV0 the_AT sectional_JJ curvature_NN1 to_II the_AT holomorphic_JJ sectional_JJ curvature_NN1 ,_, and_CC remove_VV0 the_AT assumption_NN1 of_IO simply-connectedness_NN1 ._. 
</s>
<s>
Consider_VV0 a_AT1 discrete_JJ Lagrange_NN1 multiplier_NN1 @S_FO and_CC aim_VV0 to_TO find_VVI the_AT saddle_NN1 points_NN2 of_IO the_AT functional_JJ @F_FO ,_, where_CS we_PPIS2 recall_VV0 that_CST @F_FO and_CC @S_FO is_VBZ a_AT1 solution_NN1 to_II (_( 1_MC1 )_) ._. 
</s>
<s>
We_PPIS2 use_VV0 the_AT fact_NN1 that_CST for_CS @S_FO and_CC @S_FO ,_, @F_FO ._. 
</s>
<s>
MC1_FO and_CC MC2_FO students7_FO solutions_NN2 of_IO CPPs_NP1 Steffe_NP1 (_( 2007_MC )_) has_VHZ identified_VVN these_DD2 multiplicative_JJ concepts_NN2 as_CSA distinct_JJ stages_NN2 of_IO reasoning_VVG that_CST can_VM last_VVI for_IF two_MC or_CC more_DAR years_NNT2 ._. 
</s>
<s>
For_IF any_DD fixed_JJ j_ZZ1 =_FO 1_MC1 ,_, ..._... ,_, d_ZZ1 ,_, denote_VV0 by_II Bj_NP1 (_( 1_MC1 )_) the_AT unit_NN1 ball_NN1 of_IO Hj_NP1 ,_, the_AT hypothesis_NN1 space_NN1 we_PPIS2 consider_VV0 in_II this_DD1 paper_NN1 is_VBZ defined_VVN by_II @F_FO ,_, which_DDQ corresponds_VVZ to_II the_AT class_NN1 of_IO functions_NN2 @S_FO that_CST decompose_VV0 as_CSA sums_NN2 of_IO univariate_NN1 functions_NN2 on_II each_DD1 coordinates_NN2 ._. 
</s>
<s>
As_CSA documented_VVN in_II 78_MC the_AT lognormal_JJ fit_NN1 was_VBDZ fairly_RR good_JJ for_IF women_NN2 in_II the_AT United_NP1 Kingdom_NP1 in_II 1973_MC (_( as_CSA it_PPH1 is_VBZ for_IF many_DA2 other_JJ years_NNT2 and_CC western_JJ countries_NN2 )_) ._. 
</s>
<s>
Hitting_VVG time_NNT1 For_IF a_AT1 given_JJ discrete_JJ time_NNT1 stochastic_JJ process_NN1 ,_, Zt_NP1 ,_, recall_VV0 the_AT concept_NN1 of_IO a_AT1 hitting_NN1 time_NNT1 for_IF an_AT1 event_NN1 Zt_NP1 e_ZZ1 S._NP1 In_II the_AT eleventh_MD grade_NN1 class_NN1 it_PPH1 &lsqb;_( motivation_NN1 &rsqb;_) is_VBZ already_RR inside_II them_PPHO2 ,_, they_PPHS2 are_VBR ready_JJ for_IF it_PPH1 ,_, they_PPHS2 know_VV0 it_PPH1 ._. 
</s>
<s>
A_AT1 useful_JJ generalization_NN1 of_IO convexity_NN1 used_VVN in_II OT_NP1 theory_NN1 comes_VVZ from_II so-called_JJ c-transforms_VVZ &lsqb;_( 51_MC ,_, section_NN1 1.2_MC &rsqb;_) ._. 
</s>
<s>
The_AT aim_NN1 of_IO this_DD1 study_NN1 was_VBDZ to_TO investigate_VVI whether_CSW a_AT1 different_JJ etude_NN1 (_( "_" Devising_VVG equations_NN2 "_" ,_, see_VV0 Section_NN1 2.2.2_MC )_) is_VBZ as_RG effective_JJ as_CSA traditional_JJ exercises_NN2 at_II developing_JJ students_NN2 '_NULL procedural_JJ fluency_NN1 in_II solving_VVG linear_JJ equations_NN2 ,_, relative_II21 to_II22 the_AT alternative_JJ hypothesis_NN1 that_CST the_AT etude_NN1 and_CC the_AT exercises_NN2 are_VBR not_XX equally_RR effective_JJ ._. 
</s>
<s>
The_AT definition_NN1 p0_FO is_VBZ slightly_RR more_RGR complicated_JJ and_CC as_CSA such_DA its_APPGE definition_NN1 will_VM be_VBI delayed_VVN to_II Section_NN1 4.4_MC below_RL ;_; see_VV0 (_( 4.25_MC )_) and_CC (_( 4.26_MC )_) ._. 
</s>
<s>
While_CS this_DD1 suggest_VV0 that_CST a_AT1 robust_JJ IFN_NN1 response_NN1 is_VBZ ongoing_JJ ,_, the_AT IFN_NN1 gene_NN1 is_VBZ not_XX upregulated_VVN ._. 
</s>
<s>
In_RR21 particular_RR22 ,_, if_CS the_AT increment_NN1 of_IO the_AT Euler-Maruyama_NP1 step_NN1 is_VBZ tamed_VVN by_II an_AT1 appropriate_JJ exponential_NN1 term_NN1 ,_, then_RT we_PPIS2 might_VM obtain_VVI a_AT1 scheme_NN1 that_CST admits_VVZ exponential_NN1 integrability_NN1 properties_NN2 ._. 
</s>
<s>
Relative_II21 to_II22 the_AT water_NN1 crisis_NN1 of_IO Flint_NP1 ,_, Michigan_NP1 but_CCB oriented_VVD more_RGR broadly_RR around_II the_AT question_NN1 of_IO responsibility_NN1 ,_, Aguirre_NP1 ,_, Anhalt_NP1 ,_, Cortez_NP1 ,_, Turner_NP1 and_CC Simic-Muller_NP1 (_( 2019_MC )_) present_VV0 a_AT1 mathematical_JJ modeling_NN1 task_NN1 (_( "_" Flint_NP1 Water_NN1 Task_NN1 "_" )_) that_CST they_PPHS2 have_VH0 used_VVN in_II teacher_NN1 education_NN1 contexts_NN2 ._. 
</s>
<s>
This_DD1 unified_JJ modelis_NN1 then_RT applied_VVN in_II one_MC1 or_CC more_DAR model-adaptation_JJ activities_NN2 ,_, which_DDQ are_VBR structurally_RR similar_JJ but_CCB contain_VV0 some_DD added_JJ complication_NN1 ._. 
</s>
<s>
We_PPIS2 next_MD give_VV0 some_DD assumptions_NN2 on_II @S_FO ._. 
</s>
<s>
Let_VV0 @S_FO ._. 
</s>
<s>
Then_RT for_IF all_DB @S_FO one_MC1 can_VM find_VVI 6_MC >_FO 0_MC such_CS21 that_CS22 for_IF any_DD choice_NN1 of_IO @S_FO ,_, @F_FO ._. 
</s>
<s>
Proof_NN1 ._. 
</s>
<s>
They_PPHS2 used_VVD a_AT1 tight_JJ relaxation_NN1 for_IF PEP_NN1 and_CC studied_VVD the_AT tight_JJ (_( exact_JJ )_) numerical_JJ worst-case_JJ bounds_NN2 of_IO FPGM_NP1 ,_, a_AT1 proximal_JJ gradient_NN1 version_NN1 of_IO OGM_NP1 ,_, and_CC some_DD variants_NN2 versus_II number_NN1 of_IO iterations_NN2 N._NP1 Their_APPGE numerical_JJ results_NN2 suggest_VV0 that_CST there_EX exists_VVZ an_AT1 OGM-type_JJ acceleration_NN1 of_IO PGM_NP1 that_CST has_VHZ a_AT1 worst-case_NN1 cost_VVN function_NN1 bound_VVD that_DD1 is_VBZ about_RG twice_RR smaller_JJR than_CSN that_DD1 of_IO FPGM_NP1 ,_, showing_VVG room_NN1 for_IF improvement_NN1 in_II accelerating_VVG PGM_NP1 ._. 
</s>
<s>
Since_CS a_AT1 is_VBZ a_AT1 regular_JJ sequence_NN1 ,_, we_PPIS2 can_VM compute_VVI these_DD2 using_VVG the_AT Koszul_NN1 complex_NN1 of_IO a_AT1 ._. 
</s>
<s>
Tensoring_VVG with_IW A_ZZ1 then_RT gives_VVZ the_AT Koszul_NN1 complex_NN1 of_IO the_AT sequence_NN1 a_AT1 ._. 
</s>
<s>
The_AT following_JJ two_MC theorems_NN2 establish_VV0 exactly_RR that_DD1 ._. 
</s>
<s>
The_AT result_NN1 concerning_II the_AT (_( weighted_JJ )_) total_JJ variation_NN1 follows_VVZ from_II the_AT similar_JJ analysis_NN1 in_II Lemma_NN1 6.7_MC ._. 
</s>
<s>
One_MC1 may_VM investigate_VVI the_AT limiting_JJ case_NN1 @S_FO further_RRR ,_, but_CCB this_DD1 requires_VVZ the_AT introduction_NN1 of_IO a_AT1 '_NULL logarithmic_JJ scale_NN1 '_NULL to_TO measure_VVI smoothness_NN1 of_IO functions_NN2 ,_, and_CC we_PPIS2 abstain_VV0 from_II doing_VDG so_RR for_IF ease_NN1 of_IO exposition_NN1 ._. 
</s>
<s>
Using_VVG Multiplicative_JJ comparison_NN1 in_II scenario_NN1 1_MC1 potentially_RR helped_VVD a_AT1 teacher_NN1 correctly_RR identify_VV0 the_AT situation_NN1 as_CSA not_XX proportional_JJ as_CSA was_VBDZ apparent_JJ by_II those_DD2 teachers_NN2 who_PNQS changed_VVD identification_NN1 in_II scenario_NN1 1_MC1 (_( see_VV0 Table_NN1 2_MC )_) ._. 
</s>
<s>
Other_JJ authors_NN2 advocate_VV0 smart_JJ particle_NN1 models_NN2 that_CST follow_VV0 decision-based_JJ dynamics_NN ._. 
</s>
<s>
Finally_RR ,_, as_CSA shown_VVN in_II Table_NN1 8_MC ,_, when_CS controls_NN2 for_IF the_AT classroom_NN1 climate_NN1 ,_, teacher_NN1 effort_NN1 ,_, and_CC teacher_NN1 experience_NN1 were_VBDR added_VVN in_II the_AT model_NN1 ,_, results_NN2 largely_RR remained_VVD the_AT same_DA for_IF both_RR years_NNT2 under_II consideration_NN1 both_RR with_IW and_CC without_IW school_NN1 fixed_JJ effects_NN2 ._. 
</s>
<s>
The_AT logical_JJ order_NN1 of_IO this_DD1 article_NN1 is_VBZ as_CSA follows_VVZ ._. 
</s>
<s>
By_II showing_VVG that_CST ,_, the_AT decay_NN1 mechanism_NN1 is_VBZ "_" stable_JJ "_" with_II31 respect_II32 to_II33 the_AT sort_NN1 of_IO perturbations_NN2 which_DDQ this_DD1 second_NNT1 linear_JJ operator_NN1 introduces_VVZ ,_, we_PPIS2 are_VBR able_JK to_TO keep_VVI the_AT decay_NN1 mechanism_NN1 and_CC close_VVI a_AT1 decay_NN1 estimate_NN1 for_IF p_ZZ1 and_CC show_VV0 that_CST p_ZZ1 ,_, while_CS not_XX decaying_VVG ,_, converges_VVZ as_CSA @S_FO ._. 
</s>
<s>
By_II &lsqb;_( 17_MC ,_, 18_MC &rsqb;_) ,_, the_AT SL_NP1 (_( m_ZZ1 ,_, R_ZZ1 )_) distance_NN1 is_VBZ quasi-isometric_JJ to_II the_AT word-length_NN1 in_II SL_NP1 (_( m_ZZ1 ,_, Z_ZZ1 )_) ._. 
</s>
<s>
By_II case_NN1 (_( ii_MC )_) of_IO Theorem_NN1 3.5_MC ,_, y_ZZ1 is_VBZ @S_FO ,_, @S_FO regular_JJ with_IW @S_FO ._. 
</s>
<s>
Notice_VV0 that_CST @F_FO ._. 
</s>
<s>
To_TO start_VVI with_IW ,_, we_PPIS2 will_VM show_VVI that_DD1 equation_NN1 (_( 3.5_MC )_) allows_VVZ to_TO obtain_VVI many_DA2 equivalent_JJ formulations_NN2 ,_, which_DDQ contain_VV0 the_AT quotient_NN1 @S_FO ,_, each_DD1 of_IO them_PPHO2 useful_JJ for_IF various_JJ purposes_NN2 ._. 
</s>
<s>
Main_JJ results_NN2 and_CC outline_NN1 of_IO the_AT paper_NN1 ._. 
</s>
<s>
Lastly_RR choose_VV0 @S_FO large_JJ enough_RR such_CS21 that_CS22 @F_FO ._. 
</s>
<s>
This_DD1 and_CC the_AT boundary_NN1 condition_NN1 @S_FO imply_VV0 @S_FO and_CC @S_FO ._. 
</s>
<s>
Hence_RR ,_, it_PPH1 is_VBZ easy_JJ to_TO see_VVI that_CST the_AT method_NN1 (_( 3.2_MC )_) is_VBZ consistent_JJ for_IF u_ZZ1 and_CC p_ZZ1 sufficiently_RR smooth_JJ as_CSA stated_VVN above_RL ,_, i.e._REX (_( 3.2_MC )_) is_VBZ satisfied_VVN with_IW uh_UH replaced_VVN by_II u_ZZ1 ._. 
</s>
<s>
However_RR ,_, we_PPIS2 note_VV0 that_CST without_IW further_JJR strong_JJ and_CC likely_JJ to_TO be_VBI unrealistic_JJ conditions_NN2 on_II the_AT shape_NN1 of_IO the_AT estimating_VVG functions_NN2 ,_, M0_FO can_VM not_XX be_VBI controlled_VVN as_II a_AT1 fixed_JJ set_NN1 even_RR at_II the_AT limiting_JJ case_NN1 when_CS @S_FO ,_, so_CS21 that_CS22 it_PPH1 will_VM depend_VVI on_II the_AT value_NN1 of_IO the_AT parameter_NN1 0_MC ._. 
</s>
<s>
Theorem_NN1 4.1_MC Let_VV0 @S_FO be_VBI a_AT1 maximally_RR monotone_JJ operator_NN1 such_CS21 that_CS22 @S_FO ._. 
</s>
<s>
Let_VV0 @S_FO be_VBI a_AT1 solution_NN1 of_IO the_AT continuous_JJ dynamic_NN1 (_( 64_MC )_) ,_, where_CS a_AT1 >_FO 2_MC and_CC @S_FO with_IW @S_FO ._. 
</s>
<s>
Assume_VV0 also_RR that_CST @S_FO ._. 
</s>
<s>
Then_RT ,_, x(t)_NNU converges_VVZ weakly_RR ,_, as_CSA @S_FO ,_, to_II an_AT1 element_NN1 of_IO S._NP1 Moreover_RR @S_FO ._. 
</s>
<s>
By_II shifting_VVG the_AT operator_NN1 by_II k0_FO units_NN2 ,_, we_PPIS2 can_VM assume_VVI @S_FO ._. 
</s>
<s>
Theorem_NN1 7.1_MC still_JJ holds_NN2 if_CS 0_MC is_VBZ a_AT1 local_JJ (_( n_ZZ1 +_FO 1_MC1 )_) -maximum_NN1 by_II Remark_NN1 6.5_MC ._. 
</s>
<s>
Using_VVG the_AT Kalton-Randrianarivony_NP1 concentration_NN1 inequality_NN1 ,_, it_PPH1 was_VBDZ shown_VVN in_II &lsqb;_( 6_MC &rsqb;_) that_CST if_CS X_ZZ1 coarse_JJ Lipschitz_NP1 embeds_VVZ into_II a_AT1 reflexive_NN1 Banach_NN1 space_NN1 that_CST is_VBZ asymptotically_RR uniformly_RR smooth_JJ ,_, then_RT X_ZZ1 must_VM be_VBI reflexive_NN1 ._. 
</s>
<s>
Why_RRQ did_VDD you_PPY do_VDI &lsqb;_( X_ZZ1 &rsqb;_) or_CC respond_VV0 the_AT way_NN1 you_PPY did_VDD ?_? 
</s>
<s>
For_IF every_AT1 @S_FO that_DD1 satisfies_VVZ @S_FO for_IF some_DD F0lner_FO sequence_NN1 T_ZZ1 ,_, one_PN1 can_VM find_VVI infinite_JJ sets_NN2 @S_FO with_IW @S_FO ._. 
</s>
<s>
They_PPHS2 showed_VVD that_CST pointwise_JJ IQCs_NN2 alone_RR exhibit_VV0 crude_JJ bounds_NN2 ,_, and_CC the_AT use_NN1 of_IO off-by-one_MC IQCs_NP2 improves_VVZ the_AT numerical_JJ solutions_NN2 greatly_RR ._. 
</s>
<s>
Before_II getting_VVG to_II the_AT proof_NN1 of_IO Theorem_NN1 1.1_MC ,_, we_PPIS2 must_VM recall_VVI some_DD ideas_NN2 related_VVN to_II the_AT Nielsen-Thurston_NP1 classification_NN1 for_IF elements_NN2 of_IO @S_FO ._. 
</s>
<s>
For_IF basics_NN2 on_II this_DD1 theory_NN1 ,_, including_II the_AT definition_NN1 of_IO a_AT1 pseudo-Anosov_NP1 mapping_NN1 class_NN1 ,_, see_VV0 the_AT book_NN1 by_II Farb_NP1 and_CC the_AT second_MD author_NN1 of_IO this_DD1 paper_NN1 &lsqb;_( 20_MC ,_, Chapter_NN1 13_MC &rsqb;_) ._. 
</s>
<s>
This_DD1 bound_VVD on_II the_AT finite_JJ number_NN1 of_IO ends_NN2 when_RRQ MM_NNU has_VHZ at_RR21 least_RR22 two_MC ends_NN2 implies_VVZ that_CST MM_NNU has_VHZ finite_JJ stability_NN1 index_NN1 which_DDQ is_VBZ bounded_VVN by_II a_AT1 constant_JJ that_CST only_RR depends_VVZ on_II its_APPGE genus_NN1 ._. 
</s>
<s>
Finally_RR ,_, the_AT presence_NN1 or_CC lack_NN1 of_IO multidimensionality_NN1 may_VM relate_VVI to_II the_AT items_NN2 that_CST comprise_VV0 these_DD2 assessments_NN2 ._. 
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<s>
The_AT results_NN2 of_IO model_NN1 lb_NNU suggest_VV0 that_CST none_PN of_IO the_AT conjectured_JJ ._. 
</s>
<s>
Because_II21 of_II22 these_DD2 constraints_NN2 ,_, as_II31 well_II32 as_II33 (_( 28_MC )_) and_CC (_( 29_MC )_) ,_, it_PPH1 turns_VVZ out_RP that_CST the_AT simplest_JJT approach_NN1 is_VBZ to_TO start_VVI by_II choosing_VVG b_ZZ1 as_II a_AT1 function_NN1 of_IO r_ZZ1 along_II the_AT event_NN1 horizon_NN1 ._. 
</s>
<s>
In_II the_AT rest_NN1 of_IO this_DD1 section_NN1 we_PPIS2 keep_VV0 the_AT notation_NN1 and_CC assumptions_NN2 of_IO Theorem_NN1 2.1_MC and_CC we_PPIS2 prove_VV0 it_PPH1 ._. 
</s>
<s>
In_RR21 particular_RR22 ,_, we_PPIS2 shall_VM see_VVI that_CST one_PN1 can_VM decouple_VVI the_AT nonlinear_JJ and_CC nonlocal_JJ aspects_NN2 ._. 
</s>
<s>
For_IF each_DD1 e_ZZ1 ,_, let_VV0 @S_FO be_VBI a_AT1 solution_NN1 to_II (_( 2.6_MC )_) with_IW @S_FO satisfying_VVG the_AT following_JJ property_NN1 :_: @F_FO and_CC @F_FO satisfying_JJ (_( 3.4_MC )_) ._. 
</s>
<s>
The_AT details_NN2 of_IO the_AT choice_NN1 of_IO the_AT sign_NN1 will_VM be_VBI clarified_VVN elsewhere_RL (_( it_PPH1 uses_VVZ the_AT residue_NN1 definition_NN1 of_IO the_AT localization_NN1 contribution_NN1 ,_, which_DDQ was_VBDZ worked_VVN out_RP in_II &lsqb;_( MNS_NP1 &rsqb;_) ,_, it_PPH1 is_VBZ similar_JJ to_II what_DDQ sometimes_RT is_VBZ referred_VVN to_II as_II the_AT Jeffrey-Kirwan_NP1 residue_NN1 &lsqb;_( JK_NP1 &rsqb;_) in_II the_AT mathematical_JJ literature_NN1 ,_, see_VV0 also_RR &lsqb;_( W_ZZ1 &rsqb;_) )_) ._. 
</s>
<s>
For_REX21 example_REX22 ,_, in_II surgery_NN1 allocation_NN1 ,_, yi_NNU represents_VVZ yes-no_JJ decisions_NN2 of_IO allocating_VVG J_ZZ1 surgeries_NN2 in_II operating_VVG room_NN1 (_( OR_CC )_) i_ZZ1 for_IF all_DB @S_FO ._. 
</s>
<s>
The_AT operational_JJ time_NNT1 limit_NN1 of_IO each_DD1 OR_CC (_( i.e._REX ,_, Tj_NP1 )_) is_VBZ usually_RR deterministic_JJ ,_, but_CCB the_AT processing_NN1 time_NNT1 of_IO each_DD1 surgery_NN1 (_( i.e._REX ,_, @S_FO )_) is_VBZ usually_RR random_JJ due_II21 to_II22 the_AT variety_NN1 of_IO patients_NN2 ,_, surgical_JJ teams_NN2 ,_, and_CC surgery_NN1 characteristics_NN2 ._. 
</s>
<s>
Examples_NN2 to_II which_DDQ our_APPGE theory_NN1 applies_VVZ include_VV0 stochastic_JJ block_NN1 models_NN2 ,_, power_NN1 law_NN1 graphs_NN2 and_CC sparse_JJ versions_NN2 of_IO W-random_JJ graphs_NN2 ._. 
</s>
<s>
These_DD2 references_NN2 also_RR contain_VV0 the_AT correct_JJ history_NN1 of_IO and_CC detailed_JJ attribution_NN1 for_IF the_AT results_NN2 we_PPIS2 simply_RR quote_VV0 here_RL ._. 
</s>
<s>
It_PPH1 is_VBZ should_VM be_VBI noted_VVN that_CST if_CS a_AT1 function_NN1 is_VBZ defined_VVN on_II O+_FO ,_, then_RT the_AT function_NN1 is_VBZ horizontally_RR periodic_JJ ,_, that_REX21 is_REX22 ,_, @S_FO for_IF any_DD integer_NN1 m_ZZ1 and_CC n_ZZ1 ._. 
</s>
<s>
The_AT homeomorphism_NN1 @S_FO fixes_VVZ each_DD1 end_NN1 of_IO M._NN1 Proof_NN1 ._. 
</s>
<s>
Finally_RR ,_, the_AT all-zero_JJ binary_JJ string_NN1 of_IO dimension_NN1 q_ZZ1 is_VBZ normally_RR denoted_VVN @S_FO ,_, rather_CS21 than_CS22 @S_FO ._. 
</s>
<s>
The_AT limit_NN1 @F_FO defines_VVZ @S_FO and_CC satisfies_VVZ @S_FO on_II @S_FO ._. 
</s>
<s>
It_PPH1 also_RR satisfies_VVZ the_AT desired_JJ equality_NN1 @F_FO ,_, since_CS @S_FO and_CC if_CS the_AT last_MD inequality_NN1 were_VBDR strict_JJ ,_, then_RT Step_NN1 2_MC could_VM be_VBI applied_VVN to_TO improve_VVI @S_FO ._. 
</s>
<s>
This_DD1 shows_VVZ existence_NN1 of_IO a_AT1 discrete_JJ solution_NN1 @S_FO of_IO (_( 2.2_MC )_) as_II31 well_II32 as_II33 the_AT uniform_JJ bound_NN1 @S_FO ._. 
</s>
<s>
Now_RT we_PPIS2 consider_VV0 the_AT case_NN1 of_IO multiple_NN1 (_( possibly_RR ,_, nonlinear_JJ )_) moment_NN1 constraints_NN2 ._. 
</s>
<s>
This_DD1 phenomenon_NN1 is_VBZ related_VVN to_II microscopic_JJ buckling_NN1 of_IO composite_JJ materials_NN2 ._. 
</s>
<s>
Across_II the_AT sample_NN1 ,_, children_NN2 presented_VVD 22_MC of24_FO possible_JJ combinations_NN2 of_IO principle_NN1 use_NN1 ._. 
</s>
<s>
In_II large-scale_JJ testing_NN1 problems_NN2 ,_, the_AT false_JJ discovery_NN1 rate_NN1 FDR_NP1 (_( Benjamini_NP1 and_CC Hochberg_NP1 ,_, 1995_MC )_) hasbeen_NN1 widely_RR used_VMK to_TO control_VVI the_AT inflation_NN1 of_IO type_NN1 I_ZZ1 errors_NN2 ._. 
</s>
<s>
Skipping_VVG a_AT1 tedious_JJ computation_NN1 ,_, one_MC1 can_VM undeniably_RR obtain_VVI 704_MC state_NN1 feedback_NN1 stabilizers_NN2 based_VVN on_II Theorem_NN1 4.10_MC ._. 
</s>
<s>
The_AT various_JJ subfigures_NN2 of_IO Figure_NN1 7.2_MC show_VV0 a_AT1 3-dimensional_JJ subset_NN1 H_ZZ1 of_IO the_AT form_NN1 @S_FO and_CC the_AT projection_NN1 of_IO @S_FO to_II H_ZZ1 ,_, where_CS the_AT A_ZZ1 changes_NN2 by_II isotopy_NN1 as_CSA we_PPIS2 progress_VV0 from_II a_ZZ1 )_) to_II f_ZZ1 )_) ._. 
</s>
<s>
Support_VV0 for_IF GPUs_NP2 also_RR makes_VVZ MatConvNet_NP1 efficient_JJ for_IF large_JJ scale_NN1 computations_NN2 ,_, and_CC pretrained_VVD networks_NN2 may_VM be_VBI downloaded_VVN for_IF immediate_JJ use_NN1 ._. 
</s>
<s>
The_AT study_NN1 adopted_VVD the_AT analytical_JJ lenses_NN2 of_IO Harel_NP1 and_CC Sowder_VV0 '_NULL s_ZZ1 (_( 2007_MC )_) proof_NN1 schemes_NN2 taxonomy_NN1 and_CC investigated_VVD the_AT question_NN1 :_: can_VV0 the_AT Harel_NN1 and_CC Sowder_NP1 proof_NN1 schemes_NN2 taxonomy_NN1 be_VBI deployed_VVN in_II the_AT characterization_NN1 of_IO secondary_JJ students_NN2 '_NULL first_MD encounter_NN1 with_IW proof_NN1 and_CC proving_VVG ?_? 
</s>
<s>
Then_RT ,_, at_II each_DD1 @S_FO ,_, one_PN1 chooses_VVZ an_AT1 optimal_JJ strategy_NN1 for_IF the_AT period_NN1 @S_FO ,_, in_II31 response_II32 to_II33 his_APPGE future_JJ selves_NN2 '_NULL given_VVN strategy_NN1 on_II &lsqb;_( t_ZZ1 +_FO 1_MC1 ,_, N_ZZ1 &rsqb;_) ._. 
</s>
<s>
By_II multiplying_VVG (_( 5.1_MC )_) by_II fn_NNU and_CC summing_VVG up_RP the_AT results_NN2 for_IF @S_FO ,_, we_PPIS2 obtain_VV0 @F_FO ,_, with_IW @F_FO ._. 
</s>
<s>
The_AT cutoff_NN1 procedure_NN1 :_: we_PPIS2 only_RR keep_VV0 the_AT cycles_NN2 that_CST are_VBR not_XX separated_VVN from_II the_AT external_JJ boundary_NN1 of_IO T_ZZ1 (_( p_ZZ1 by_II a_AT1 cycle_NN1 of_IO length_NN1 less_RRR than_CSN ep_NN1 ._. 
</s>
<s>
The_AT uniform_JJ convergence_NN1 of_IO A(z)_NP1 (_( defined_VVN in_II (_( 4.17_MC )_) )_) on_II compact_NN1 sets_VVZ @S_FO towards_II A(z)_NP1 follows_VVZ now_RT easily_RR from_II these_DD2 bounds_NN2 and_CC the_AT corresponding_JJ convergence_NN1 of_IO F_ZZ1 (_( x_ZZ1 ,_, z_ZZ1 )_) ._. 
</s>
<s>
Let_VV0 (_( M_ZZ1 ,_, g_ZZ1 )_) be_VBI a_AT1 two-dimensional_JJ Riemannian_JJ manifold_NN1 ._. 
</s>
<s>
However_RR ,_, it_PPH1 still_RR remains_VVZ open_JJ whether_CSW these_DD2 estimators_NN2 can_VM achieve_VVI the_AT minimax_NN1 rates_NN2 of_IO the_AT s-contamination_JJ models_NN2 ._. 
</s>
<s>
The_AT choice_NN1 of_IO the_AT plus_NN1 sign_NN1 in_II (_( 4.88_MC )_) yields_VVZ the_AT speed_NN1 of_IO linear_JJ waves_NN2 outrunning_VVG the_AT current_JJ ,_, while_CS (_( 3.82_MC )_) shows_VVZ that_CST the_AT minus_NN1 sign_NN1 corresponds_VVZ to_II the_AT linear_JJ waves_NN2 propagating_VVG westwards_RL ._. 
</s>
<s>
For_IF piecewise_JJ linears_NN2 ,_, the_AT method_NN1 has_VHZ been_VBN studied_VVN extensively_RR ._. 
</s>
<s>
In_RR21 addition_RR22 ,_, the_AT DC_NP1 components_NN2 f1_FO and_CC f2_FO are_VBR locally_RR Lipschitz_VV0 continuous_JJ ,_, and_CC let_VV0 L1_FO >_FO 0_MC and_CC L2_FO >_FO 0_MC be_VBI the_AT Lipschitz_NP1 constants_NN2 of_IO f1_FO and_CC f2_FO on_II F?1_FO ,_, respectively_RR ._. 
</s>
<s>
For_IF @S_FO ,_, @S_FO ,_, write_VV0 @S_FO so_CS21 that_CS22 @S_FO ._. 
</s>
<s>
We_PPIS2 are_VBR grateful_JJ to_II Avi_NP1 Wigderson_NP1 for_IF pointing_VVG that_CST our_APPGE proof_NN1 reveals_VVZ that_CST ,_, using_VVG occurrence_NN1 obstructions_NN2 ,_, one_PN1 can_VM not_XX prove_VVI significant_JJ lower_JJR bounds_NN2 in_II the_AT significantly_RR weaker_JJR model_NN1 of_IO power_NN1 sums_NN2 ._. 
</s>
<s>
This_DD1 is_VBZ the_AT sign_NN1 of_IO a_AT1 deep_JJ fact_NN1 :_: a-posteriori_NN2 observing_VVG k_ZZ1 in_II dimension_NN1 d_ZZ1 is_VBZ not_XX the_AT same_DA as_CSA working_VVG in_II dimension_NN1 k_ZZ1 ,_, or_CC ,_, said_VVD differently_RR ,_, simple_JJ solutions_NN2 (_( supported_VVN by_II k_ZZ1 constraints_NN2 )_) to_II complex_JJ problems_NN2 (_( in_II dimension_NN1 d_ZZ1 >_FO k_ZZ1 )_) are_VBR not_XX as_RG guaranteed_JJ as_CSA solutions_NN2 to_II simple_JJ problems_NN2 (_( in_II dimension_NN1 k_ZZ1 )_) ._. 
</s>
<s>
Some_DD students_NN2 believed_VVD that_CST the_AT longer_JJR decimal_JJ number_NN1 ,_, with_IW more_DAR digits_NN2 ,_, was_VBDZ larger_JJR (_( similar_JJ to_II Resnick_NP1 et_RA21 al_RA22 ._. 
</s>
<s>
1989_MC )_) ,_, while_CS other_JJ students_NN2 believed_VVD that_CST the_AT shorter_JJR decimal_NN1 was_VBDZ larger_JJR ._. 
</s>
<s>
The_AT proof_NN1 schemes_NN2 evidenced_VVN in_II the_AT proving_JJ activity_NN1 of_IO the_AT students_NN2 in_II their_APPGE studies_NN2 were_VBDR grouped_VVN in_II three_MC classes_NN2 :_: external_JJ conviction_NN1 proof_NN1 schemes_NN2 ,_, empirical_JJ proof_NN1 schemes_NN2 ,_, and_CC deductive_JJ proof_NN1 schemes_NN2 ,_, with_IW each_DD1 divided_VVD further_RRR into_II subclasses_NN2 ._. 
</s>
<s>
Thus_RR ,_, @S_FO ._. 
</s>
<s>
The_AT quantity_NN1 of_IO interest_NN1 that_CST we_PPIS2 approximate_VV0 is_VBZ defined_VVN as_CSA @F_FO ._. 
</s>
<s>
We_PPIS2 interpret_VV0 the_AT parameters_NN2 @S_FO as_CSA i.i.d_NNU ._. 
</s>
<s>
Under_II Assumption_NN1 1_MC1 and_CC a_AT1 <_FO L_ZZ1 ,_, the_AT stable_JJ set_NN1 of_IO the_AT strict_JJ saddle_NN1 points_NN2 has_VHZ measure_NN1 zero_NN1 ,_, meaning_VVG p(Wg)_NNU =_FO 0_MC ._. 
</s>
<s>
Every_AT1 curve_NN1 s∈S_FO also_RR determines_VVZ a_AT1 length_NN1 function_NN1 ?s:S_FO →_NULL &lsqb;_( 0_MC ,_, ∞_FO )_) via_II ?s(s)=i_NN2 (_( s_ZZ1 ,_, s_ZZ1 )_) ._. 
</s>
<s>
First_MD note_NN1 that_CST the_AT triangle_NN1 inequality_NN1 ,_, (_( 1_MC1 )_) ,_, and_CC (_( 40_MC )_) ensure_VV0 that_CST @F_FO ._. 
</s>
<s>
Ito_NN1 '_NULL s_ZZ1 formula_NN1 and_CC the_AT PDE(3)_FO imply_VV0 that_CST for_IF all_DB @S_FO ,_, @S_FO ,_, @S_FO it_PPH1 holds_VVZ @S_FO that_DD1 @F_FO ._. 
</s>
<s>
This_DD1 ,_, (_( 40_MC )_) ,_, and_CC (_( 43_MC )_) show_VV0 that_CST for_IF all_DB @S_FO ,_, @S_FO it_PPH1 holds_VVZ that_CST @F_FO ._. 
</s>
<s>
This_DD1 ensures_VVZ that_CST @S_FO ._. 
</s>
<s>
This_DD1 and_CC (_( 44_MC )_) prove_VV0 for_IF all_DB @S_FO ,_, @S_FO that_DD1 @F_FO ._. 
</s>
<s>
This_DD1 proves_VVZ (_( i_ZZ1 )_) ._. 
</s>
<s>
We_PPIS2 expect_VV0 that_CST in_II fact_NN1 it_PPH1 holds_VVZ for_IF a_AT1 typical_JJ partition_NN1 of_IO large_JJ enough_DD size_NN1 ._. 
</s>
<s>
In_II the_AT equation_NN1 ,_, the_AT multiplier_NN1 ,_, M_ZZ1 ,_, is_VBZ interpreted_VVN as_II the_AT number_NN1 ofgroups_NN2 ,_, the_AT multiplicand_NN1 ,_, N_ZZ1 ,_, is_VBZ interpreted_VVN as_II the_AT number_NN1 of_IO units_NN2 in_II each_DD1 group_NN1 ,_, and_CC the_AT product_NN1 ,_, P_ZZ1 ,_, is_VBZ interpreted_VVN as_II the_AT number_NN1 of_IO total_JJ units_NN2 in_II M_MC groups_NN2 ._. 
</s>
<s>
Comparing_VVG with_IW Lemma_NN1 3.5_MC and_CC using_VVG Proposition_NN1 3.7_MC and_CC its_APPGE proof_NN1 ,_, we_PPIS2 deduce_VV0 isomorphisms_NN2 of_IO Rv-modules_NN2 @F_FO ._. 
</s>
<s>
The_AT teachers_NN2 participate_VV0 in_II these_DD2 activities_NN2 for_IF their_APPGE continuous_JJ professional_JJ development_NN1 on_II a_AT1 voluntary_JJ basis_NN1 ,_, unlike_II the_AT mandatory_JJ konaikenshu_NN1 ._. 
</s>
<s>
Both_DB2 CWENOZ3_FO schemes_NN2 provide_VV0 similar_JJ results_NN2 ._. 
</s>
<s>
In_RR21 particular_RR22 ,_, we_PPIS2 will_VM establish_VVI Theorem_NN1 1.3_MC ._. 
</s>
<s>
Instead_II21 of_II22 seeking_VVG numerical_JJ schemes_NN2 for_IF the_AT continuous_JJ DO_VD0 equations_NN2 (_( 14_MC )_) --(16)_NN1 ,_, it_PPH1 is_VBZ numerically_RR useful_JJ to_TO apply_VVI the_AT DO_VD0 methodology_NN1 directly_RR on_II the_AT spatial_JJ discretization_NN1 chosen_VVN for_IF the_AT original_JJ SPDE_NN1 (_( 6_MC )_) ._. 
</s>
<s>
The_AT word_NN1 acbc1_FO has_VHZ a_AT1 "_" b_ZZ1 "_" in_II the_AT position_NN1 just_RR after_II the_AT middle_NN1 ,_, thus_RR the_AT second_MD equality_NN1 is_VBZ impossible_JJ ._. 
</s>
<s>
For_IF conservative_JJ problems_NN2 ,_, no_AT explicit_JJ RK_NP1 method_NN1 of_IO any_DD order_NN1 can_VM enforce_VVI discrete_JJ conservation_NN1 ,_, even_RR for_IF linear_JJ autonomous_JJ problems_NN2 ._. 
</s>
<s>
Applying_VVG Corollary_NN1 3.48_MC to_II @S_FO shows_VVZ the_AT left_JJ vertical_JJ map_NN1 in_II (_( 3.5_MC )_) is_VBZ @S-connected_FO ._. 
</s>
<s>
Definition_NN1 of_IO the_AT branching_JJ peeling_NN1 by_II layers_NN2 algorithm_NN1 ._. 
</s>
<s>
The_AT modeling_NN1 of_IO the_AT transition_NN1 probability_NN1 density_NN1 is_VBZ developed_VVN as_CSA in_II Subsection_NN1 3.2_MC accounting_NN1 for_IF @S_FO and_CC consequently_RR for_IF the_AT post-interaction_JJ velocities_NN2 ._. 
</s>
<s>
We_PPIS2 can_VM still_RR attain_VVI small_JJ gaps_NN2 ,_, i.e._REX ,_, good_JJ estimations_NN2 on_II the_AT optimal_JJ value_NN1 ._. 
</s>
<s>
Spatial_JJ perspectives_NN2 invite_VV0 transformations_NN2 of_IO place_NN1 as_II a_AT1 way_NN1 to_TO rectify_VVI injustices_NN2 ;_; for_REX21 example_REX22 ,_, the_AT identification_NN1 of_IO specific_JJ city_NN1 blocks_NN2 in_II the_AT previously_RR described_VVN Million_NNO Dollar_NNU1 Blocks_NN2 maps_NN2 suggests_VVZ the_AT need_NN1 for_IF increased_JJ community_NN1 investment_NN1 ,_, changes_NN2 to_II policing_VVG practices_NN2 ,_, interventions_NN2 through_II education_NN1 ,_, or_CC collective_JJ transformation_NN1 ._. 
</s>
<s>
One_MC1 of_IO them_PPHO2 is_VBZ stabilization_NN1 for_IF uncertain_JJ fractional_JJ PDE_NN1 systems_NN2 ,_, which_DDQ has_VHZ been_VBN discussed_VVN for_IF classic_JJ PDE_NN1 systems_NN2 in_II &lsqb;_( 8_MC ,_, 9_MC ,_, 10_MC ,_, 11_MC ,_, 12_MC ,_, 13_MC ,_, 14_MC ,_, 34_MC ,_, 35_MC &rsqb;_) and_CC the_AT abundant_JJ references_NN2 therein_RR ._. 
</s>
<s>
Their_APPGE study_NN1 showed_VVD that_CST students_NN2 taught_VVN by_II elementary_JJ school_NN1 teachers_NN2 who_PNQS scored_VVD 1_MC1 SD_NP1 above_II the_AT mean_JJ on_II their_APPGE MKT_NP1 assessment_NN1 experienced_JJ gains_NN2 in_II their_APPGE test_NN1 scores_VVZ equivalent_JJ to_II one-half_MF to_II two-thirds_MF of_IO a_AT1 month_NNT1 of_IO additional_JJ growth_NN1 compared_VVN to_II their_APPGE counterparts_NN2 taught_VVN by_II average-MKT_JJ teachers_NN2 ._. 
</s>
<s>
The_AT same_DA proof_NN1 shows_VVZ that_CST the_AT set_NN1 of_IO local_JJ minimum_JJ values_NN2 of_IO f_ZZ1 is_VBZ at_RR21 most_RR22 countable_JJ ._. 
</s>
<s>
We_PPIS2 introduce_VV0 a_AT1 target_NN1 system_NN1 ,_, @F_FO ,_, where_CS the_AT parameter_NN1 c_ZZ1 is_VBZ used_VVN to_TO regulate_VVI the_AT convergence_NN1 speed_NN1 ,_, which_DDQ is_VBZ seen_VVN from_II Lemma_NN1 2.1_MC ._. 
</s>
<s>
We_PPIS2 conclude_VV0 that_CST @F_FO ._. 
</s>
<s>
Finally_RR ,_, since_CS (_( X_ZZ1 ,_, )_) is_VBZ non-atomic_JJ ,_, @S_FO ,_, and_CC q_ZZ1 ?_NULL is_VBZ a_AT1 coarsening_NN1 of_IO p_ZZ1 ?_NULL ,_, there_EX is_VBZ a_AT1 refinement_NN1 of_IO with_IW @S_FO for_IF all_DB 0≤t<_FO |_NULL p_ZZ1 ?_NULL |_NULL ._. 
</s>
<s>
Thus_RR ,_, we_PPIS2 have_VH0 @F_FO for_IF all_DB @S_FO ._. 
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The_AT last_MD inequality_NN1 is_VBZ exactly_RR the_AT Cordes_NP1 condition_NN1 (_( 2.3_MC )_) with_IW @S_FO ._. 
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Note_VV0 that_CST the_AT uniform_JJ ellipticity_NN1 (_( 1.2_MC )_) implies_VVZ @S_FO ._. 
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Hence_RR ,_, the_AT Cordes_NP1 condition_NN1 (_( 2.3_MC )_) is_VBZ satisfied_VVN with_IW @S_FO under_II the_AT condition_NN1 of_IO (_( 1.2_MC )_) for_IF two_MC dimensional_JJ problems_NN2 ._. 
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Let_VV0 U_ZZ1 be_VBI an_AT1 iid_JJ sequence_NN1 of_IO uniform_NN1 &lsqb;_( 0,1_MC &rsqb;_) random_JJ variables_NN2 ,_, and_CC e_ZZ1 a_AT1 random_JJ variable_NN1 taking_VVG values_NN2 in_II E._NP1 Assume_VV0 that_CST for_IF every_AT1 @S_FO ,_, Ut_NP1 is_VBZ independent_JJ of_IO @S_FO ,_, then_RT @S_FO has_VHZ law_NN1 @S_FO ._. 
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Assuming_VVG this_DD1 proposition_NN1 ,_, let_VV0 us_PPIO2 define_VVI the_AT contours_NN2 @S_FO and_CC @S_FO ._. 
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As_CSA in_II Section_NN1 6.1_MC ,_, they_PPHS2 will_VM each_DD1 be_VBI the_AT union_NN1 of_IO two_MC contours_NN2 ,_, a_AT1 small_JJ piecewise_JJ linear_JJ part_NN1 near_RL and_CC a_AT1 large_JJ curved_JJ part_NN1 that_CST closely_RR follows_VVZ the_AT level_JJ line_NN1 @S_FO ._. 
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For_IF @S_FO with_IW @S_FO ,_, it_PPH1 holds_VVZ that_CST @F_FO ._. 
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In_II practice_NN1 ,_, we_PPIS2 use_VV0 the_AT pivoted_JJ Cholesky_JJ algorithm_NN1 (_( see_VV0 Section_NN1 2.1_MC )_) to_TO construct_VVI a_AT1 low_JJ rank_NN1 approximant_NN1 for_IF H_ZZ1 in_II @S_FO operations_NN2 ._. 
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Each_DD1 annulus_NN1 is_VBZ of_IO the_AT form_NN1 H/Z_ZZ1 ,_, where_CS H_ZZ1 is_VBZ the_AT hyperbolic_JJ half-plane_NN1 and_CC Z_ZZ1 acts_VVZ by_II translation_NN1 along_II the_AT boundary_NN1 ._. 
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As_II a_AT1 classroom_NN1 activity_NN1 ,_, how_RRQ teachers_NN2 can_VM shape_VVI classroom_NN1 cultures_NN2 in_II which_DDQ students_NN2 are_VBR involved_JJ in_II PP_NN1 and_CC in_II which_DDQ way_NN1 it_PPH1 becomes_VVZ an_AT1 accepted_JJ practice_NN1 in_II the_AT classroom_NN1 are_VBR important_JJ questions_NN2 (_( Cai_NP1 et_RA21 al._RA22 ,_, 2015_MC )_) ._. 
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In_II this_DD1 case_NN1 @F_FO ._. 
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As_II a_AT1 consequence_NN1 ,_, if_CS inequality_NN1 (_( 5.25_MC )_) holds_VVZ and_CC since_RR g(t)_NNU is_VBZ absolutely_RR continuous_JJ with_II31 respect_II32 to_II33 @S_FO (_( see_VV0 Remark_NN1 (_( 6_MC )_) )_) ,_, one_MC1 easily_RR concludes_VVZ with_IW the_AT exponential_NN1 convergence_NN1 of_IO the_AT relative_JJ Shannon_NP1 entropy_NN1 to_TO zero_VVI at_II the_AT explicit_JJ rate_NN1 2p_NNU ._. 
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The_AT formal_JJ definition_NN1 of_IO the_AT subdifferential_JJ appears_VVZ in_II section_NN1 2_MC and_CC is_VBZ standard_JJ in_II the_AT optimization_NN1 literature_NN1 (_( e.g._REX ,_, &lsqb;_( 57_MC ,_, Definition_NN1 8.3_MC &rsqb;_) )_) ._. 
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We_PPIS2 claim_VV0 that_CST the_AT projectivit@S_FO fixes_VVZ the_AT ball_NN1 of_IO radius_NN1 n_ZZ1 around_II xn_FO pointwise_RR so_CS21 that_CS22 u_ZZ1 is_VBZ indeed_RR horocyclic_JJ ._. 
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Suppose_VV0 that_CST the_AT initial_NN1 and_CC boundary_NN1 data_NN satisfy_VV0 @F_FO ,_, and_CC the_AT compatibility_NN1 conditions_NN2 @S_FO ._. 
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Let_VV0 @S_FO be_VBI the_AT solution_NN1 obtained_VVN in_II Proposition_NN1 2.1_MC and_CC let_VV0 @S_FO ._. 
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The_AT definition_NN1 of_IO strong_JJ regularity_NN1 is_VBZ recalled_VVN in_II Section_NN1 1_MC1 ._. 
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One_PN1 might_VM wonder_VVI ,_, why_RRQ not_XX take_VVI L_ZZ1 =_FO 0_MC ?_? 
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This_DD1 is_VBZ because_CS RO_NP1 optimizes_VVZ the_AT worst-case_NN1 instead_II21 of_II22 the_AT average_JJ performance_NN1 as_CSA in_II SO_RR ._. 
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Mrs._NNB Kadiddlehopper_NP1 observed_VVD Fernando_NP1 to_TO determine_VVI whether_CSW his_APPGE pieces_NN2 were_VBDR equal_JJ as_II31 well_II32 as_II33 how_RRQ he_PPHS1 did_VDD so_RR ._. 
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On_II the_AT assumptions_NN2 (_( A2_FO )_) and_CC (_( A3_FO )_) ._. 
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Since_CS both_DB2 terms_NN2 in_II (_( 2.27_MC )_) are_VBR normally_RR ordered_VVN ,_, (_( vi_MC )_) remains_VVZ valid_JJ as_RR21 well_RR22 ,_, by_II the_AT induction_NN1 assumption_NN1 ._. 
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By_II Proposition_NN1 20_MC (_( the_AT conditions_NN2 of_IO the_AT proposition_NN1 are_VBR met_VVN by_II our_APPGE choice_NN1 of_IO the_AT value_NN1 of_IO K_ZZ1 )_) ,_, we_PPIS2 have_VH0 @F_FO for_IF each_DD1 @S_FO ._. 
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We_PPIS2 combine_VV0 this_DD1 with_IW the_AT trivial_JJ estimate_NN1 (_( coming_VVG from_II Lemma_NN1 6_MC )_) @F_FO and_CC use_VV0 superadditivity_NN1 of_IO entropy_NN1 between_II scales_NN2 of_IO integral_JJ ratio_NN1 (_( Lemma_NN1 10_MC )_) to_TO obtain_VVI @F_FO ,_, for_IF some_DD number_NN1 c_ZZ1 that_CST depends_VVZ only_RR on_II a_AT1 ._. 
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Since_CS it_PPH1 is_VBZ not_XX easy_JJ to_TO call_VVI FPC.AS_NP1 when_RRQ A_ZZ1 is_VBZ not_XX a_AT1 scalar_JJ ,_, based_VVN on_II the_AT recommend_VV0 at_II ion_NN1 of_IO the_AT referee_NN1 ,_, we_PPIS2 use_VV0 another_DD1 popular_JJ active-set_NN1 based_VVN solver_NN1 PSSas_NP2 &lsqb;_( 44_MC &rsqb;_) to_TO replace_VVI FPC.AS_NP1 for_IF the_AT testing_NN1 ._. 
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Reshetnyak_VV0 '_NULL s_ZZ1 gluing_JJ theorem_NN1 Let_VV0 X±_FO be_VBI @S_FO spaces_NN2 with_IW closed_JJ convex_JJ subsets_NN2 @S_FO ._. 
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If_CS @S_FO is_VBZ an_AT1 isometry_NN1 ,_, then_RT the_AT space_NN1 X_ZZ1 arising_VVG from_II gluing_VVG X+_FO and_CC @S_FO along_II the_AT isometry_NN1 G_ZZ1 is_VBZ @S_FO ;_; cf._VV0 However_RR ,_, I_PPIS1 do_VD0 not_XX interpret_VVI his_APPGE counting_NN1 seven_MC flags_NN2 to_TO mean_VVI that_CST he_PPHS1 maintained_VVD the_AT seven_MC colors_NN2 while_CS he_PPHS1 was_VBDZ actually_RR making_VVG flags_NN2 because_CS he_PPHS1 expressed_VVD uncertainty_NN1 about_II the_AT number_NN1 of_IO flags_NN2 in_II each_DD1 column_NN1 ._. 
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In_II the_AT rest_NN1 of_IO the_AT tests_NN2 (_( Problems_NN2 1_MC1 ,_, 7-10_MCMC ,_, 13_MC ,_, and_CC 15_MC )_) ,_, MPBNGC_NP1 uses_VVZ the_AT least_DAT evaluations_NN2 ,_, while_CS the_AT difference_NN1 between_II DBDC_NP1 and_CC PBDC_NP1 is_VBZ rather_RG small_JJ except_II21 for_II22 in_II Problems_NN2 13_MC and_CC 15_MC ._. 
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In_II an_AT1 important_JJ work_NN1 &lsqb;_( 3_MC &rsqb;_) ,_, Christodoulou_NP1 gave_VVD a_AT1 precise_JJ description_NN1 of_IO shock_NN1 formation_NN1 for_IF irrotational_JJ relativistic_JJ fluids_NN2 starting_VVG with_IW small_JJ smooth_JJ initial_JJ data_NN ._. 
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These_DD2 conditions_NN2 on_II the_AT local_JJ quadrature_NN1 rules_NN2 are_VBR similar_JJ to_II those_DD2 highlighted_VVN for_IF finite_JJ elements_NN2 in_II &lsqb;_( 9_MC ,_, 10_MC &rsqb;_) but_CCB ,_, interestingly_RR ,_, they_PPHS2 appear_VV0 here_RL from_II the_AT need_NN1 of_IO estimating_VVG quite_RG different_JJ error_NN1 terms_NN2 than_CSN in_II the_AT case_NN1 of_IO linear_JJ models_NN2 as_CSA in_II these_DD2 references_NN2 ._. 
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Finally_RR we_PPIS2 compute_VV0 @S_FO ._. 
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Let_VV0 @S_FO denote_VVI again_RT a_AT1 lift_NN1 of_IO x_ZZ1 ._. 
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By_II (_( 10_MC and_CC the_AT orbit_NN1 formula_NN1 for_IF finite_JJ groups_NN2 we_PPIS2 have_VH0 @F_FO ._. 
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We_PPIS2 then_RT compute_VV0 the_AT single-orbital_JJ entropy_NN1 for_IF the_AT i-mode_JJ matricization_NN1 denoted_VVD s(i)_FW ,_, i.e._REX ,_, @S_FO ,_, where_CS @S_FO is_VBZ the_AT singleorbital_JJ density_NN1 matrix_NN1 ._. 
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Moreover_RR ,_, because_II21 of_II22 the_AT negative_JJ effects_NN2 of_IO prior_JJ self-efficacy_NN1 on_II the_AT effects_NN2 of_IO prompting_VVG multiple_JJ solutions_NN2 on_II perceived_JJ competence_NN1 ,_, we_PPIS2 expected_VVD prior_JJ self-efficacy_NN1 to_II also_RR negatively_RR influence_VV0 the_AT indirect_JJ effects_NN2 of_IO teaching_NN1 method_NN1 on_II self-efficacy_NN1 ._. 
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There_EX is_VBZ l1_FO >_FO l0_FO such_CS21 that_CS22 @F_FO ._. 
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Then_RT @F_FO ._. 
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The_AT push-forward_JJ distribution_NN1 @S_FO is_VBZ computed_VVN ._. 
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<s>
The_AT tridiagonal_JJ structure_NN1 allows_VVZ fast_RR linear_JJ system_NN1 solution_NN1 ,_, making_VVG the_AT reduced_JJ instance_NN1 solvable_JJ in_II time_NNT1 roughly_RR linear_JJ in_II t_ZZ1 ;_; see_VV0 Appendix_NN1 A_ZZ1 for_IF details_NN2 ._. 
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In_II this_DD1 paper_NN1 ,_, we_PPIS2 employ_VV0 one_MC1 of_IO the_AT standard_JJ model_NN1 selection_NN1 techniques_NN2 ,_, namely_REX Stein_NP1 unbiased_JJ risk_NN1 estimate_NN1 (_( SURE_JJ )_) ,_, in_II our_APPGE asymptotic_JJ framework_NN1 and_CC show_VV0 how_RRQ the_AT properties_NN2 of_IO the_AT solution_NN1 path_NN1 of_IO AMP_NN1 and_CC LASSO_NP1 enable_VV0 us_PPIO2 to_II not_XX only_RR obtain_VV0 an_AT1 efficient_JJ parameter_NN1 tuning_VVG scheme_NN1 ,_, but_CCB also_RR prove_VV0 the_AT consistency_NN1 of_IO these_DD2 schemes_NN2 under_II the_AT asymptotic_JJ setting_NN1 ._. 
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<s>
Equation_NN1 (_( 4.26_MC )_) in_II turn_NN1 implies_VVZ (_( a_ZZ1 )_) ._. 
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<s>
Students_NN2 also_RR need_VV0 additional_JJ opportunities_NN2 to_TO improve_VVI their_APPGE ability_NN1 to_TO construct_VVI ,_, design_NN1 ,_, and_CC program_NN1 robots_NN2 ,_, in_II order_NN1 to_TO flexibly_RR use_VVI this_DD1 tool_NN1 in_II mathematical_JJ learning_NN1 ._. 
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In_II consequence_NN1 ,_, we_PPIS2 fix_VV0 our_APPGE attention_NN1 in_II the_AT last_MD term_NN1 ._. 
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<s>
There_EX are_VBR ,_, of_RR21 course_RR22 ,_, a_AT1 wide-variety_NN1 of_IO common_JJ model_NN1 selection_NN1 procedures_NN2 (_( see_VV0 &lsqb;_( 2,31_MC &rsqb;_) )_) ,_, some_DD of_IO which_DDQ may_VM be_VBI more_RGR appropriate_JJ to_II the_AT specific_JJ application_NN1 than_CSN others_NN2 ._. 
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<s>
These_DD2 M_NNO intervals_NN2 may_VM each_DD1 cover_VVI more_DAR than_CSN one_MC1 of_IO the_AT original_JJ endpoints_NN2 ,_, say_VV0 m_ZZ1 of_IO them_PPHO2 ,_, and_CC there_EX are_VBR at_RR21 most_RR22 m_ZZ1 C_ZZ1 1_MC1 distinct_JJ ways_NN2 to_TO continue_VVI the_AT orbit_NN1 of_IO a_AT1 word_NN1 of_IO length_NN1 n_ZZ1 ._. 
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<s>
Finite_JJ asymptotic_JJ dimension_NN1 For_IF any_DD N_ZZ1 the_AT subspace_NN1 Prob(G)_NN1 (_( N_ZZ1 )_) of_IO probability_NN1 measures_NN2 supported_VVN on_II sets_NN2 of_IO cardinality_NN1 at_RR21 most_RR22 N+1_FO is_VBZ naturally_RR a_AT1 simplicial_JJ complex_NN1 of_IO dimension_NN1 N._NP1 Sampling_NN1 ,_, uniqueness_NN1 ,_, and_CC interpolating_VVG sets_NN2 for_IF shift-invariant_JJ spaces_NN2 Since_CS for_IF g∈W0_FO all_RR spaces_VVZ Vp(g)_NP1 are_VBR subspaces_NN2 of_IO C(R)_NP1 ,_, sampling_NN1 of_IO functions_NN2 f∈Vp(g)_FO is_VBZ well-defined_JJ ._. 
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<s>
Theorems_NN2 13_MC and_CC 14_MC assert_VV0 convergence_NN1 to_II within_II @S_FO of_IO f_ZZ1 only_RR ._. 
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The_AT above_JJ construction_NN1 allows_VVZ for_IF asymptotically_RR valid_JJ inference_NN1 for_IF 6(x)_FO ,_, but_CCB the_AT performance_NN1 of_IO the_AT forests_NN2 can_VM in_II practice_NN1 be_VBI improved_VVN by_II first_MD regressing_VVG out_RP the_AT effect_NN1 of_IO the_AT features_NN2 Xi_NN1 on_II all_DB the_AT outcomes_NN2 separately_RR ._. 
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