{"came_across": ["came_accross", "came_upon", "stumbled_across", "recently_came_across", "stumbled_upon", "ran_across"], "talking_about": ["talking"], "figured_out": [], "relatively_prime": ["coprime", "co_prime", "pairwise_relatively_prime"], "decimal_representation": ["terminating_decimal_expansion", "decimal_expansion", "repeating_decimal_expansion"], "decimal_expansion": ["terminating_decimal_expansion", "decimal_representation", "repeating_decimal_expansion", "decimal_expansions"], "bigger_than": ["greater_than", "less_than", "grater_than"], "currently_learning": ["recently_started_learning", "currently_studying", "recently_learned", "recently_learnt"], "greatly_appreciate": ["appreciate", "greatly_appreciated", "highly_appreciate", "sincerely_appreciate"], "one_to_one_correspondence_between": ["one_to_one_correspondence", "one_one_correspondence", "one_to_one_correspondences"], "inner_products": ["_inner_product", "semi_inner_product", "inner_product"], "linear_combination": ["r_combinations", "r_combination", "_linear_combination"], "quantum_physics": ["quantum_chemistry", "quantum_mechanics", "quantum_mechanical"], "rather_than": [], "pure_imaginary": ["pure_imaginary_quaternions", "purely_imaginary", "pure_imaginary_quaternion"], "math_skills": ["proof_writing_skills", "skills"], "fourier_series": ["fourier_bessel_series", "fourier"], "online_resources": ["online_resource", "online_sources", "resources_online", "websites", "tutorials"], "binary_operation": ["unary_operation", "associative_binary_operation", "nullary_operation"], "pure_math": ["mathematics", "pure_mathematics", "pure_maths", "pure_mathematician", "computer_science", "maths"], "john_baez": ["timothy_gowers", "william_thurston", "baez", "barry_mazur", "tim_gowers", "brian_conrad"], "terry_tao": ["terence_tao", "terrence_tao"], "gets_closer": [], "professor_told_us": ["teacher_told_us", "professor_told_me", "professor_told"], "\u2207f": [], "del_operator": ["koopman_operator", "prox_operator", "_operator"], "taller_than": ["taller"], "playing_around": ["fiddling_around", "experimenting", "toying_around", "mucking_around", "messing_around"], "algebraically_closed_field": ["non_algebraically_closed_fields", "algebraically_closed_fields", "non_algebraically_closed_field"], "algebraically_closed_fields": ["algebraically_closed_field", "non_algebraically_closed_fields", "non_algebraically_closed_field"], "both_sides": [], "figure_out": [], "pretty_straight_forward": ["fairly_straight_forward", "pretty_straightforward", "relatively_straight_forward"], "come_across": ["encountered", "stumbled_upon", "recently_come_across", "come_accross"], "indefinite_integral": ["anti_derivative", "definite_integral", "indefinite_integration", "antiderivative", "indefinite_integrals"], "pure_maths": ["math_competitions", "pure_mathematics", "software_engineering", "pure_mathematician", "pure_math", "university_level", "mechanical_engineering"], "best_bet": [], "michael_spivak": ["michael_atiyah", "james_stewart", "tom_apostol", "michael_artin", "michael_sipser", "spivak"], "makes_sense": [], "taylor_series": ["taylor_series_expansion", "taylor_series_expansions", "taylor_expansion", "maclaurin_series", "taylor_expansions", "taylor_serie", "maclaurin_expansion"], "she_knows": ["everyone_knows"], "little_rusty": ["rusty", "bit_rusty", "pretty_rusty"], "distance_traveled": ["traveled", "total_distance_traveled", "distance_travelled"], "fair_coin": ["tossing", "fair_coins", "tossed", "coin", "biased_coin", "fair_coin_flip", "fair_coin_toss"], "decide_whether": ["decide_wether"], "cross_sectional_area": ["cross_sectional_areas", "cross_section", "cross_sectional"], "convince_yourself": ["persuade_yourself"], "000_000": ["000_000_000", "000_000_000_000", "000", "300_000", "10000", "1000"], "straight_lines": ["straight_lines_passing_through", "lines", "straight_line"], "constructive_mathematics": ["metamathematics", "mainstream_mathematics", "meta_mathematics"], "allows_us": ["enables_us"], "000_000_000": ["600_000", "000_000", "000_000_000_000"], "distinction_between": [], "wikipedia_entry": ["wikipedia", "wikipedia_page", "wikipedia_articles", "wiki_article", "wiki_page", "wikipedia_article"], "functional_analysis": ["microlocal_analysis", "brezis_functional_analysis", "introductory_functional_analysis"], "nonprincipal_ultrafilter": ["non_principal_ultrafilter", "principal_ultrafilter", "nonprincipal_ultrafilters"], "contest_math": ["math_competitions", "maths_competition", "math_contests", "contest", "olympiads"], "arises_naturally": ["arise_naturally"], "advanced_calculus": ["honors_calculus", "kirby_calculus", "spivaks_calculus"], "worry_about": [], "relationship_between": [], "operatorname_arctanh": ["operatorname_arccosh", "operatorname_arcsinh", "operatorname_arcosh"], "sits_inside": [], "symbolic_manipulation": ["clever_manipulation", "symbolic_manipulations", "algebraic_manipulation"], "worrying_about": ["without_worrying_about", "caring_about", "worrying"], "math_education": ["undergraduate_education", "schooling", "mathematical_education", "education"], "everyone_else": ["everybody_else", "everyone", "everybody"], "sufficiently_large": ["sufficently_large"], "banach_tarski_paradox": ["banach_tarski", "burali_forti_paradox", "banach\u2013tarski_paradox"], "monty_hall": ["monty_hall_paradox"], "frac11": ["frac17_", "frac11_", "frac11_frac12"], "frac19": ["frac33", "frac17_", "frac17"], "dfrac1": ["dfrac2", "dfrac4", "dfrac3"], "definite_integral": ["indefinite_integration", "indefinite_integral", "definite_integrals"], "who_loves": ["everybody_loves", "who_likes", "lynn_loves"], "on_line_encyclopedia": ["online_encyclopedia", "encyclopedias", "encyclopedia"], "mathematical_community": ["mathematican", "mathematical_biology", "theoretical_computer_science"], "numerical_methods": ["iterative_methods"], "universal_quantifiers": ["quantifier", "quantifiers", "universal_quantifier", "existential_quantifier", "universal_quantification", "existential_quantifiers"], "preserved_under": ["preserved"], "inverse_symbolic_calculator": ["inverse_symbolic_calculators", "graphic_calculator", "scientific_calculator"], "nth_fibonacci": ["fibonacci", "fibonaccis", "fibbonacci"], "martin_gardner": ["scientific_american", "gardner", "martin_aigner", "peter_winkler"], "totally_ordered": ["cyclically_ordered", "partially_ordered", "well_ordered", "linearly_ordered"], "nonunital": ["non_unital_rings", "non_unital", "commutative_unital", "unital"], "concise_introduction": ["comprehensive_introduction", "6th_ed", "elliott_mendelson", "friendly_introduction", "\u201cintroduction", "gentle_introduction"], "farther_away": ["farther_apart", "farther", "far_away"], "equally_distributed": ["non_identically_distributed", "identically_distributed", "symmetrically_distributed"], "restrict_yourself": ["restrict_ourselves", "restrict_oneself", "restrict_attention"], "reflexive_antisymmetric": ["irreflexive", "reflexive_anti_symmetric", "reflexive_transitive", "quasi_order", "reflexive_irreflexive_symmetric"], "equivalence_relations": ["equivalance_relation", "equivalence_relation"], "people_who": ["people_who_voted", "people", "students_who"], "halting_problem": ["halting_oracle", "halting"], "bill_dubuque": ["hardmath", "gerry_myerson", "will_jagy", "alex_kruckman", "jyrkilahtonen", "hurkyl", "arturo_magidin"], "ian_stewart": ["stewarts", "stewart", "james_stewart"], "arbitrarily_small": ["made_arbitrarily_small"], "mac_lane": ["mac_lane_moerdijk", "cwm", "saunders_mac_lane", "maclane"], "kenneth_rosen": ["susanna", "kenneth_ross", "grimaldi", "kenneth", "applications_7th_edition", "susanna_epp", "kenneth_kunen"], "ask_yourself": [], "floating_around": ["mucking_around"], "frac12_frac13_frac14": ["1__frac12_frac13_", "frac13_frac13_frac13", "frac12_frac14_frac18", "frac13_frac14", "frac15_frac16", "frac11_frac12"], "frac9": ["frac95", "frac5", "frac97", "frac4", "frac98", "frac3", "frac7", "frac8"], "circular_motion": ["motion", "rigid_body_motion", "circular_membrane"], "pointing_towards": ["heading_towards", "pointing_upward", "pointing_upwards"], "reasoning_behind": ["lurking_behind", "logic_behind", "hiding_behind"], "endofunctors": ["pseudofunctors", "profunctors", "endofunctor"], "monoidal_categories": ["symmetric_monoidal_categories", "braided_monoidal_categories", "bicategories", "monoidal_category", "enriched_categories"], "monoidal_category": ["symmetric_monoidal_category", "monoidal_categories", "monoidal", "strict_monoidal_category", "symmetric_monoidal", "braided_monoidal_category"], "branch_cut": ["branch_cuts", "_branch_cut"], "unit_disk": ["punctured_unit_disk", "punctured_unit_disc", "unit_disc"], "blows_up": [], "imaginary_axis": [], "regular_polygons": ["irregular_polygons", "regular_polygon", "irregular_polygon"], "regular_polygon": ["regular_polygons", "regular_polygon_inscribed", "_sided_polygon", "regular_n_gon", "_gon", "n_gon", "irregular_polygon"], "few_days_ago": ["week_ago", "few_months_ago", "few_weeks_ago", "earlier_today", "few_years_ago"], "strikes_me": [], "chain_rule": ["multivariable_chain_rule"], "trouble_understanding": ["having_trouble_understanding", "trouble_grasping"], "figuring_out": ["figuring", "trouble_figuring_out"], "logically_valid": ["logically_sound", "intuitionistically_valid", "logically_entails"], "algebraic_closure": [], "leading_coefficient": [], "reflexivity_symmetry": ["reflexivity_transitivity", "reflexivity_antisymmetry", "reflexivity_symmetry_transitivity"], "3rd_ed": ["5th_edition", "4th_ed", "2nd_edition", "rd_edition", "3rd_edition", "third_edition"], "blue_balls": ["blue_marbles", "black_balls", "non_blue_balls", "white_balls", "red_balls", "red_marbles"], "red_balls": ["blue_balls", "blue_marbles", "black_balls", "white_balls", "red_marbles"], "fiddling_around": ["playing_around", "toying_around", "fooling_around"], "more_importantly": ["most_importantly"], "natural_deduction": ["natural_deduction_systems", "fitch_style_natural_deduction", "sequent_calculus", "hilbert_style", "propositional_logic", "natural_deduction_style"], "injects_into": [], "hairy_ball_theorem": ["borsuk_ulam_theorem"], "angle_bac": ["angle_acb", "angle_abd", "angle_baq", "angle_cbd", "angle_cab", "angle_bam", "angle_baf", "angle_dac"], "triangle_abc": ["triangle_pqr", "triangle_abd", "bigtriangleup_abc", "triangle_oab", "acute_triangle_abc", "triangle_abm", "triangle_abf"], "bd_dc": ["ce_ea", "bm_mc", "af_fb", "ae_ec", "ae_eb"], "angle_cad": ["angle_abd", "angle_cao", "angle_cab", "angle_cae", "angle_bae", "angle_dae", "angle_bda", "angle_acd"], "ravi_vakil": ["vakil", "robin_hartshorne", "gathmann"], "time_ago": ["sometime_ago", "awhile_ago", "decade_ago"], "diff_eq": [], "talks_about": ["discusses"], "without_resorting": ["without_recourse", "without_relying"], "equally_likely": ["equally_probable", "equally_likely_outcomes"], "dynamic_programming": ["imperative_programming", "nonlinear_programming"], "turns_out": [], "zero_divisors": ["zero_divisor", "non_zero_divisors", "non_zero_divisor"], "deeper_understanding": [], "feel_confident": ["confident"], "uncountably_many": ["countably_many", "denumerably_many", "countably_infinitely_many"], "countably_many": ["finitely_many", "uncountably_many", "denumerably_many", "countably_infinitely_many"], "uniquely_determined": ["uniquely_determines", "completely_determined", "determined"], "person_who": ["person"], "little_bit": ["little", "bit", "little_bit_trickier"], "2_9x": ["2_14x", "3_11x", "3_10x", "6_9x", "4_9x", "2_10x", "2_18x", "3_8x"], "higher_dimensions": ["higher_dimensional", "higher_dimensional_analogues", "higher_dimensional_analogue"], "random_walks": ["asymmetric_random_walk", "random_walk", "random_walker"], "binary_strings": ["bit_strings", "ternary_strings", "_bit_binary_strings", "_bit_binary_string", "strings", "bitstrings"], "without_loss": ["wlog", "generality"], "manhattan_distance": ["manhattan", "taxicab_distance", "levenshtein_distance"], "tells_us": [], "intuitive_explanation": ["explanation", "fuller_explanation", "intuitive_explanations"], "rain_tomorrow": ["predicts_rain", "rains_tomorrow", "rain_today", "tomorrow", "rains_today", "rained_today"], "minimal_polynomial": [], "pretty_straightforward": ["fairly_straight_forward", "pretty_straight_forward", "fairly_straightforward"], "frac00": ["dfrac00"], "trig_identities": ["trigonometric_identities"], "affine_schemes": ["affine_scheme", "affine_scheme_operatorname_spec", "non_affine_scheme", "locally_ringed_spaces", "affine_varieties", "ringed_spaces"], "graded_ring": ["graded_module", "_graded_ring", "graded_rings"], "affine_scheme": ["affine_scheme_operatorname_spec", "affine_schemes", "non_affine_scheme"], "graded_module": ["graded_ring", "d_module", "z_module", "graded_modules"], "weierstrass_factorization": ["casorati\u2013weierstrass", "weierstrass_wp"], "ll_leave": ["i\u2019ll_leave"], "into_account": [], "latin_squares": ["mutually_orthogonal_latin_squares", "latin_square"], "magic_squares": [], "distance_travelled": ["travelled", "distance_traveled", "total_distance_traveled"], "co_homology": ["homology", "co__homology", "homology_groups", "cohomology", "singular_homology", "k_homology", "simplicial_homology"], "differential_forms": ["0_forms"], "ratio_test": ["comparsion_test", "comparison_test", "comparison_tests"], "gauss_jordan_elimination": ["row_reduction", "gauss_jordan", "guassian_elimination", "gauss_elimination", "gaussian_elimination"], "principal_ideal_domain": ["pid", "ufd", "principal_ideal_domains", "dedekind_domain"], "begin_align": ["begin_alignat", "begin_aligned"], "heartsuit": ["heartsuit_spadesuit", "spadesuit_heartsuit", "spadesuit", "clubsuit"], "spadesuit": ["heartsuit_spadesuit", "heartsuit", "spadesuit_heartsuit", "clubsuit"], "similarity_transformations": ["transformations", "gauge_transformations", "lorentz_transformations"], "eludes_me": ["eludes"], "eluding_me": ["bothering_me", "stumping_me", "bugging_me"], "math_olympiad": ["olympiad", "math_contest", "maths_olympiad", "usamo", "mathematical_olympiad", "math_competition", "mathematics_olympiad", "math_olympiads"], "algebraic_variety": ["quasi_projective_variety", "variety", "algebraic_varieties", "affine_variety", "projective_variety"], "compound_interest": ["compounding_interest", "compounding", "compounded_interest"], "character_table": ["character_tables"], "langlands_program": ["langlands", "iwasawa_theory"], "young_diagrams": ["ferrers_diagrams", "young_tableaux", "young_diagram"], "young_diagram": ["ferrers_diagrams", "ferrers_diagram", "young_diagrams"], "rectangular_prism": ["rectangular_prisms", "cuboid", "rectangular_cuboid", "triangular_prism"], "move_forward": [], "keep_track": ["kept_track"], "optimality_conditions": ["wolfe_conditions", "kuhn_tucker_conditions", "kkt_conditions"], "passing_through": ["passing_thru", "passes_through", "straight_lines_passing_through", "straight_line_passing_through"], "upper_bounds": ["upper_bounding", "upperbounds", "lower_bounds"], "imaginary_parts": ["imaginary_part"], "its_own_consistency": ["own_consistency", "omega__consistency", "arithmetically_sound"], "general_relativity": ["celestial_mechanics", "special_relativity", "relativity"], "hyperbolic_trig_functions": ["hyperbolic_trig"], "fair_amount": ["lot", "considerable_amount", "tremendous_amount"], "gentle_introduction": ["brief_introduction", "comprehensive_introduction", "concise_introduction"], "particularly_interested": ["particularly_fond", "primarily_interested", "interested"], "what\u2019s_going": [], "birthday_paradox": ["monty_hall_paradox", "birthday_party"], "global_minimum": ["global_minimiser", "global_maximum", "global_minima"], "physical_interpretation": ["geometrical_interpretation", "graphical_interpretation", "physical_interpretations"], "winning_strategy": ["non_losing_strategy", "trading_strategy", "betting_strategy"], "numerical_evidence": ["empirical_evidence", "experimental_evidence", "numerical_evidence_suggests"], "lexicographic_order": ["lexicographical_order", "lexicographic_ordering", "reverse_lexicographic_order"], "directed_graph": ["directed_acyclic_graph", "digraph", "undirected", "undirected_graph", "directed_graphs"], "geometric_interpretation": ["geometrical_interpretation", "graphical_interpretation", "physical_interpretation"], "physical_phenomena": ["physical_phenomenon", "phenomena", "physical_world"], "math_overflow": ["mathoverflow_thread", "mathoverflow", "quora", "stack_overflow", "cstheory", "cross_validated", "physics_stack_exchange"], "rounded_down": ["rounded"], "most_importantly": ["more_importantly"], "triangle_acd": ["triangle_aed", "triangle_ahc", "triangle_adc", "triangle_adb", "triangle_abe", "triangle_abd"], "triangle_abd": ["triangle_aed", "triangle_adc", "triangle_adb", "triangle_abe", "triangle_abm", "triangle_abf", "triangle_acd"], "2_4ac_2a": ["2_4ac_4a", "_4ac_2a", "4ac_2a"], "alternating_harmonic_series": ["alternating_harmonic"], "conditionally_convergent_series": ["unconditionally_convergent", "conditionally_convergent", "absolutely_convergent_conditionally_convergent"], "efficient_algorithm": ["spigot_algorithm", "efficient", "efficient_algorithms"], "bin_packing": ["sphere_packings", "sphere_packing", "densest_packing"], "external_force": ["centrifugal_force", "frictional_force", "external_forces"], "smoothly_varying": ["continuously_varying"], "feasible_region": [], "increase_decrease": ["decrease", "increase"], "change_of_variables": ["change_of_variable"], "almost_everywhere": ["_almost_everywhere", "mu__almost_everywhere"], "traveling_along": ["traveling", "particle_moving_along", "travelling"], "without_invoking": ["without_appealing"], "initial_conditions": ["boundary_conditions", "robin_boundary_conditions"], "end_align": [], "riemann_stieltjes_integral": ["riemann\u2013stieltjes_integral", "riemann_stieltjes_integrals", "riemann_stieltjes_integrable"], "sometimes_referred": ["commonly_referred", "referred"], "begin_cases": [], "three_non_collinear_points": ["non_collinear", "noncollinear", "three_non_collinear"], "south_poles": ["equator", "north_pole", "south_pole"], "taylor_series_expansion": ["taylor_series_expansions", "taylor_expansion", "taylor_series", "maclaurin_series", "taylor_serie", "maclaurin_expansion"], "moment_generating": ["mgf", "cumulant_generating"], "passes_through": ["passing_through", "passes_trough", "passes_thru", "passes_thorough"], "feel_free": ["please_feel_free"], "significant_digit": ["significant_digits"], "abel_ruffini_theorem": ["abel\u2013ruffini", "abel_ruffini"], "invariant_under": ["invariant_under_conjugation", "invariant_under_rotations", "invariant_under_translations"], "disjoint_cycles": ["disjoint_cycles_commute", "edge_disjoint_cycles", "disjoint_2_cycles"], "_th_cyclotomic_polynomial": ["th_cyclotomic", "cyclotomic_polynomials", "cyclotomic_polynomial"], "mathematical_education": ["undergraduate_education", "chemical_education", "math_education", "mathematics\u201d"], "central_tendency": [], "absolute_deviation": ["squared_deviations", "absolute_deviations", "std_deviation"], "nullspaces": ["nullspace"], "opposite_sides": ["opposing_sides"], "4_8x": ["4_12x", "5_8x", "2_9x", "3_16x", "4_7x", "3_8x", "6_8x"], "keith_conrad": ["keith", "keith_conrad_jordan", "conrad"], "chinese_remainder_theorem": ["crt", "chinese_reminder_theorem", "chinese_remainder", "crt_chinese_remainder"], "coxeter_groups": ["17_wallpaper_groups", "wallpaper_groups", "braid_groups"], "optimal_strategy": ["greedy_strategy", "betting_strategy", "optimal_strategies"], "truth_assignments": ["truth_assignment", "truth_value_assignments", "atomic_propositions"], "hyperboloid_model": ["poincar\u00e9_disk_model", "hyperboloid", "hyperboloids", "poincare_disk_model"], "continued_fractions": ["continued_fraction_expansions", "continued_fraction", "continued_fraction_expansion"], "under_consideration": [], "currently_working": ["currently_studying", "currently_revising", "currently_reading"], "enough_reputation": ["reputation", "low_reputation"], "hodge_dual": ["hodge_star", "hodge_duality", "hodge_star_operator"], "my_head_around": ["head_around"], "baire_category_theorem": ["baire_category", "jordan_brouwer_separation_theorem", "heine_borel_theorem"], "anything_else": ["anything"], "qiaochu": ["lee_mosher", "mariano", "eric_wofsey", "hurkyl", "qiaochu_yuan", "qiaochuyuan"], "ve_never_heard": ["never_heard"], "highly_unlikely": ["highly_improbable", "extremely_unlikely", "unlikely"], "completely_determined": ["uniquely_determines", "uniquely_determined"], "previous_paragraph": ["preceding_paragraph"], "residue_classes": ["congruence_classes_modulo", "residues_modulo", "congruence_classes"], "inclusion_exclusion_principle": ["inclusion_exclusion_pie", "inclusion_exclusion", "inclusion\u2013exclusion_principle"], "deciding_whether": [], "almost_surely": ["almost_sure", "_almost_surely"], "formal_languages": ["decidable_languages", "natural_language", "languages"], "imaginary_part": ["imaginary_parts"], "compact_support": ["compactly_supported", "smooth_compactly_supported"], "everywhere_else": ["everywhere_except"], "restricting_ourselves": ["restrict_ourselves", "limiting_ourselves", "restricting_attention"], "computer_programs": ["computer_programmers", "computer_programming", "computer_program"], "available_online": ["freely_available", "publicly_available", "downloadable", "freely_available_online"], "special_relativity": ["general_relativity", "relativity"], "surjects_onto": ["retracts_onto", "projects_onto", "surject_onto"], "intuitively_speaking": ["morally_speaking", "intuitively", "loosely_speaking"], "mathworld_page": ["mathworld", "mathworks", "mathworld_article"], "strongly_connected_components": ["biconnected_component", "biconnected_components", "connected_components"], "math_majors": ["undergrads", "computer_science_majors", "majors"], "transition_probabilities": ["transition_rates"], "pure_mathematics": ["mathematics", "pure_maths", "pure_math"], "community_wiki": [], "undergraduate_courses": ["high_school_curriculum", "undergraduate", "undergraduate_curriculum", "introductory_courses", "curricula", "graduate_courses"], "conic_section": ["parabola", "conic", "conic_sections"], "posterior_distribution": ["weibull_distribution", "posterior", "posterior_predictive_distribution"], "geometrical_interpretation": ["graphical_interpretation", "geometric_interpretation", "physical_interpretation"], "gone_through": [], "detailed_description": ["brief_description", "detailed_discussion", "detailed_exposition"], "matlab_code": ["matlab_script", "mathematica_code", "python_code", "matlab_octave_code"], "coordinate_systems": ["coordinate_system", "curvilinear_coordinate_system"], "curry_howard_isomorphism": ["curry_howard", "curry_howard_correspondence"], "princeton_companion": ["princeton", "princeton_university", "princeton_review"], "continued_fraction_expansion": ["continued_fractions", "continued_fraction_expansions", "continued_fraction"], "fairly_straightforward": ["fairly_straight_forward", "relatively_straightforward", "pretty_straightforward", "straightforward"], "started_learning": ["recently_started_learning", "begun_studying", "begun_learning"], "representable_functors": ["hom_functors", "representable_functor", "representables", "hom_functor", "comma_categories", "representable_presheaves", "contravariant_functors"], "opposite_parity": ["opposite_parities"], "breaks_down": [], "mathematica_code": ["python_code", "mathematica_script", "matlab_code", "java_code"], "archimedian": ["archimedean_places", "archimedean", "archimidean"], "one_one_correspondence": ["one_to_one_correspondence", "one_to_one_correspondences", "1_to_1_correspondence"], "sufficiently_high": [], "perfect_square": ["non_perfect_square", "perfect_cube", "perfect_squares"], "disjoint_union": ["disjoint_unions", "union"], "slight_modification": ["modification", "slight_modifications", "minor_modification"], "qiaochu_yuan": ["eric_wofsey", "zhen_lin", "gerry_myerson", "qiaochu", "qiaochuyuan", "arturo_magidin"], "local_extrema": ["minima", "local_maxima_minima", "local_maxima", "relative_extrema", "local_minima", "extrema", "global_extrema", "local_extremum"], "strictly_increasing": ["strictly_decreasing", "monotonic", "non_strictly_increasing", "monotonically_increasing", "monotone_increasing", "non_decreasing", "stricly_increasing"], "strictly_decreasing": ["monotonically_decreasing", "increasing", "monotonely_decreasing", "decreasing", "monotonically_increasing", "strictly_increasing", "monotonously_decreasing"], "taylor_expansion": ["taylor_series_expansion", "taylor_series", "taylor_development", "taylor_expansions", "taylor_expand"], "intermediate_value": ["ivt"], "local_extremum": ["local_extrema", "global_extremum", "global_extrema"], "strictly_monotonic": ["strictly_monotonic_increasing", "strictly_monotonous", "strictly_monotone"], "purely_imaginary": ["immaginary", "purely_imaginary_quaternions", "pure_imaginary"], "converges_absolutely": ["diverges", "absolutely_convergent"], "informally_speaking": ["technically_speaking", "morally_speaking", "formally_speaking"], "de_rham_cohomology": ["de_rham", "cohomology", "_cohomology", "de_rahm_cohomology", "derham_cohomology"], "convince_myself": ["convinced_myself"], "walter_rudin": ["rudin", "pugh", "baby_rudin_3rd_edition", "shakarchi", "walter_rudin_3rd_edition", "analysis_3rd_ed", "papa_rudin", "fitzpatrick"], "hamel_basis": ["hamel_bases"], "classical_logic": ["classical_propositional_logic", "intuitionist_logic", "non_classical_logics", "non_classical_logic", "intuitionistic", "propositional_logic", "intuitionistic_logic"], "boil_down": [], "ordered_pairs": ["ordered_triples", "ordered_pair", "unordered_pairs"], "throwing_away": [], "personal_experience": ["limited_experience"], "player_plays": ["team_plays", "player2", "player_loses"], "induction_hypothesis": ["inductive_hypothesis"], "inductive_hypothesis": ["inductive_step", "induction_hypothesis"], "inductive_step": ["inductive_hypothesis"], "behaves_nicely": ["behaves_badly", "behave_nicely"], "van_loan": ["pitsianis", "watkins", "golub_van_loan", "golub", "golub_kahan"], "slower_than": ["faster", "slower", "grow_slower_than", "grows_slower_than"], "please_feel_free": ["feel_free"], "modular_forms": ["cusp_forms"], "continued_fraction": ["continued_fractions", "continued_fraction_expansions", "continued_fraction_expansion"], "crop_up": ["crops_up"], "recurrence_relations": ["recurrences", "reccurence_relation", "recurrence_relation", "recurence_relation"], "high_school_students": ["high_schoolers", "high_school_senior", "high_school_student", "middle_school_student"], "euler_lagrange_equations": ["euler_lagrange_el", "euler_lagrange", "euler_lagrange_equation"], "maxspec": ["mspec", "m_spec", "spec"], "ringed_space": ["locally_ringed", "locally_ringed_spaces", "ringed_spaces"], "please_explain": ["plz_explain", "kindly_explain", "explain"], "spits_out": ["spits"], "gets_mapped": [], "too_lazy": [], "smith_normal_form": ["smith_normal"], "am_gm_inequality": ["amgm_inequality", "am_gm", "am\u2013gm_inequality", "weighted_am_gm_inequality"], "green_marbles": ["blue_marbles", "white_marbles", "green_marble", "red_balls", "black_marbles", "red_marbles"], "blue_marbles": ["white_marbles", "blue_marble", "red_balls", "black_marbles", "green_marbles", "red_marbles"], "red_marbles": ["blue_marbles", "white_marbles", "red_marble", "red_balls", "black_marbles", "green_marbles"], "positively_oriented": ["negatively_oriented", "_oriented", "positively_oriented_contour"], "ap_bp": ["ap_pb", "ap_bq_cr", "ap_bp_cp"], "2_xz": ["3_yz", "yz_x", "3_xz", "2_xy_xz_yz", "2_xy_xz", "2_zx", "2_yz"], "ae_ec": ["ce_ea", "af_fb", "bd_dc", "bp_pc", "ae_eb"], "spherical_shell": ["cylindrical_shell", "cylindrical_shells", "spherical_shells"], "segment_joining": ["segment_connecting", "line_segment_joining", "straight_line_segment_connecting", "line_segment_connecting"], "cdot1_cdot1": ["cdot2_cdot1", "cdot5_cdot8", "cdot1_1", "cdot3_cdot4", "cdot2_cdot3_cdot4_cdot5", "cdot12_cdot11", "cdot2_cdot3_cdot4"], "valuation_ring": ["valuation_rings", "discrete_valuation_ring", "dvr", "discrete_valuation_rings"], "discrete_valuation_ring": ["discrete_valuation", "valuation_ring", "dvr", "discrete_valuation_rings"], "valuation_rings": ["dvrs", "valuation_ring", "discrete_valuation_ring", "discrete_valuation_rings"], "laurent_series": ["laurent_series_expansion", "laurent_expansion", "laurent_serie"], "flag_variety": ["schubert_variety", "fano_variety", "sub_variety"], "invariant_under_rotations": ["invariant_under_translations", "invariant_under"], "locally_ringed_spaces": ["locally_ringed", "ringed_space", "affine_schemes", "ringed_spaces"], "locally_ringed": ["ringed", "ringed_space", "locally_ringed_spaces"], "open_affines": ["_affine", "affine_opens", "affine_subschemes", "affines", "open_subschemes"], "volume_enclosed": ["area_enclosed", "solid_enclosed", "region_enclosed"], "spherical_polar_coordinates": ["cylindrical_polar_coordinates", "spherical_coordinate", "spherical_co_ordinates", "spherical_coordinates"], "fifth_postulate": ["5th_postulate", "proclus", "parallel_postulate", "ruler_postulate", "playfair"], "unit_disc": ["punctured_unit_disk", "punctured_unit_disc", "unit_disk"], "nested_intervals": ["intervals"], "leaving_aside": [], "universal_covering": ["universal_cover", "contractible_universal_cover", "2_fold_covering"], "galois_correspondence": ["galois_correspondance", "langlands_correspondence"], "finitely_generated_abelian_groups": ["finitely_generated_modules", "torsion_free_abelian_groups", "abelian_groups"], "jordan_normal": ["jordan_canonical_form", "jordan_canonical", "jnf", "jordan_canonical_forms"], "projective_resolution": ["projective_resolutions", "flasque_resolution", "injective_resolutions", "free_resolution"], "binary_trees": ["_ary_trees", "full_binary_trees", "rooted_binary_trees", "rooted_trees", "trees"], "periodic_orbits": ["heteroclinic_orbits", "periodic_orbit", "homoclinic_orbit"], "acute_triangles": ["scalene_triangles", "acute_angled_triangles", "obtuse_triangles"], "tensor_products": ["tensor_product"], "give_rise": ["gave_rise"], "universal_property": [], "completely_stumped": ["totally_lost", "completely_clueless", "completely_lost", "kinda_lost", "stumped"], "browsing_through": ["perusing", "scouring", "browsed_through", "skimming_through", "browsed", "sifting_through", "browsing"], "please_forgive": ["please_excuse", "pardon", "please_forgive_me", "forgive"], "intergers": ["integers", "intgers", "interger"], "projective_variety": ["quasi_projective_variety", "quasiprojective_variety", "smooth_projective_variety", "algebraic_variety", "affine_variety", "subvariety"], "missing_something": ["overlooking_something"], "asymptotic_growth": ["asymptotic_growth_rate", "asymptotic_behaviour", "asymptotic_behavior"], "linearly_ordered": ["partially_ordered", "totally_ordered", "densely_ordered"], "principal_ideal": ["principal_ideal_domain", "non_principal_ideal", "principal_ideals", "ideal"], "principal_ideals": ["non_principal_ideal", "ideals", "non_principal_ideals", "principal_ideal"], "pulls_back": [], "no_avail": [], "hoping_someone": ["hoping_someone_could", "was_hoping_someone"], "enlighten_me": ["enlightened", "please_enlighten_me", "enlighten"], "lambert_w_function": ["lambert", "lambert_w", "lambert_"], "professional_mathematicians": ["academics", "professionals", "experienced_mathematicians", "professional_mathematician", "non_mathematicians"], "forgive_me": ["apologize", "forgive", "apologise", "sorry", "please_forgive_me", "please_forgive", "apologies"], "major_axis": ["semi_major_axis", "semimajor_axis", "minor_axis"], "their_respective": ["their_respective_domains", "respective"], "\u2212t": ["t\u2212t"], "land_lnot": ["lnot", "lor", "lor_neg", "lor_lnot", "land_neg"], "been_staring": ["ve_been_staring", "been_wracking", "stared", "been_wrestling"], "xy_yz": ["xy_yz_zx", "xy_xz", "3_yz", "xy_yz_xz"], "principal_axes": ["semi_principal_axes"], "nowhere_near": [], "bugging_me": ["been_bugging_me", "bothering_me", "stumping_me"], "cumulative_hierarchy": ["borel_hierarchy", "von_neumann_hierarchy", "_hierarchy", "veblen_hierarchy"], "morse_theory": ["morse_bott", "hodge_theory"], "giving_rise": [], "cellular_homology": ["co__homology", "co_homology", "singular_homology", "simplicial_homology", "cellular_cohomology"], "spanning_trees": ["edge_disjoint_spanning_trees", "spanning_tree", "spanning_subgraphs"], "sitting_inside": [], "programming_languages": ["computer_languages", "programming_language", "computer_programming_languages"], "everybody_knows": ["nobody_knows", "everyone_knows"], "cartesian_products": ["co_products", "cartesian_product"], "filtered_colimits": ["_filtered_colimits", "filtered_colimit", "directed_colimits", "preserves_filtered_colimits", "colimits"], "abelian_groups": ["non_isomorphic_abelian_groups", "torsion_free_abelian_groups", "finitely_generated_abelian_groups"], "naturally_identified": ["identified", "canonically_identified", "uniquely_identifies"], "principal_ultrafilter": ["principal_ultrafilters", "nonprincipal_ultrafilter", "non_principal_ultrafilter"], "nonstandard_model": ["non_standard_models", "nonstandard_models", "non_standard_model"], "radon_nikodym_derivative": ["radon_nikod\u00fdm", "radon_nikodym", "radon\u2013nikodym_derivative"], "generatingfunctionology": ["wilf", "herbert_wilf"], "mathematical_maturity": ["mathematically_mature", "mathematical_education", "mathematical_rigor", "mathematical_sophistication"], "computer_scientists": ["computer_science_majors", "computer_science", "computer_scientist"], "baby_rudin": ["rudin", "pma_rudin", "papa_rudin"], "cech_cohomology": ["\u010dech_cohomology", "chech_cohomology", "sheaf_cohomology"], "bott_tu": ["bott", "allen_hatcher", "spanier"], "future_reference": ["future_readers"], "guy_who": [], "internal_nodes": [], "excluded_middle": ["non_contradiction", "excluded_middle_lem", "double_negation_elimination"], "detailed_explanation": ["detailed_explanations", "detailed", "brief_explanation", "thorough_explanation"], "hypothenuse": ["catheti", "cathetus", "hypoteneuse", "shorter_leg", "hypothenuses", "hypotenuse", "hypotenuses"], "intuitive_explanations": ["explanation", "explanations", "intuitive_explanation"], "without_needing": [], "purely_algebraically": ["algebraically", "algebraical", "algebraicaly"], "pmatrix": ["begin_pmatrix"], "sigma__finite_measure": ["sigma__finite", "sigma__finite_measures", "sigma_finite", "finite_measure"], "infinite_dimensional_normed": ["infinite_dimensional", "infinite_dimensional_banach_spaces", "infinite_dimensional_banach"], "nate_eldredge": ["kavi_rama_murthy", "phoemuex", "henno_brandsma", "user254665", "user21820", "alex_kruckman", "kevin_carlson"], "sigma__finite": ["sigma_finite", "sigma__finite_measure", "sigma__finite_measures", "countably_additive", "sigma__additive", "non__sigma__finite"], "homotopy_classes": ["homotopy_type", "homotopy_cofiber", "isotopy_classes"], "roughly_speaking": ["loosely_speaking", "broadly_speaking"], "_4ac": ["4ac", "4abx_4ac", "2_4ac"], "repeated_squaring": [], "tight_bound": ["tighter_bound", "tightest_bound", "tight_bounds"], "global_maximum": ["global_maximizer", "global_minimum", "global_maxima"], "hamilton_cycle": ["eulerian_circuit", "hamilton_path", "hamiltonian_cycle", "hamilton_circuit", "hamilton_cycles"], "asymptotic_behavior": ["asymptotics", "asymptotic_behaviour", "chaotic_behaviour"], "approaches_infinity": [], "nothing_else": [], "corrected_version": ["revised_version", "slightly_modified_version", "modified_version"], "affine_variety": ["quasi_projective_variety", "algebraic_variety", "quasi_affine_variety", "affine_varieties", "projective_variety"], "_variables": ["variables", "_variable", "two_variables"], "global_sections": ["global_section", "global_sections_functor", "global_section_functor"], "without_appealing": ["without_invoking"], "affine_varieties": ["quasi_affine_varieties", "projective_varieties", "algebraic_varieties", "affine_schemes", "quasi_projective_varieties"], "synthetic_geometry": ["hyperbolic_geometry", "projective_geometry", "toric_geometry"], "puiseux_series": ["puiseux"], "michael_hardy": ["jonas_meyer", "didier_piau", "michaelhardy", "andr\u00e9_nicolas", "david_mitra", "michael_lugo", "achille_hui", "michael_hoppe"], "chosen_arbitrarily": [], "concave_upward": ["concave_upwards", "concave_downwards", "concave_downward"], "yoneda_embedding": ["fully_faithful_functors", "fully_faithful_functor", "representable_functor"], "embeds_into": ["embeds"], "erlang_distribution": ["f_distribution", "skellam_distribution", "gumbel_distribution"], "poisson_processes": ["compound_poisson_process", "independent_poisson_processes", "poisson_process"], "poisson_process": ["inhomogeneous_poisson_process", "poisson_processes", "compound_poisson_process"], "please_excuse": ["pardon", "excuse", "please_forgive_me", "please_forgive", "apologies"], "time_elapsed": ["elapsed_time", "elapsed"], "angle_aob": ["angle_boc", "angle_cab", "angle_aoc", "angle_aod", "angle_poq", "angle_boa", "angle_aop"], "curved_surface": ["lateral_surface", "ruled_surface", "curved_surfaces"], "students_who": ["students_enrolled", "people_who", "students"], "slope_intercept_form": ["slope_intercept", "point_slope"], "intuition_tells": ["gut_feeling_tells", "my_gut_tells"], "consistency_strength": ["consistency"], "cardinal_arithmetic": ["cardinal_arithmetics", "mental_arithmetic"], "great_deal": ["lot"], "inaccessible_cardinals": ["woodin_cardinals", "inaccessibles", "weakly_inaccessible_cardinals", "strongly_inaccessible_cardinals", "inaccessible_cardinal"], "miles_reid": ["reid", "undergraduate_algebraic_geometry", "bosch", "paolo_aluffi", "david_eisenbud"], "approx2": ["approx7", "approx5", "approx8"], "few_hundred": ["few_dozen", "hundred", "one_hundred"], "been_bugging_me": ["been_bothering_me", "stumping_me", "bugging_me"], "strongly_recommend": ["warmly_recommend", "recommend", "highly_recommend", "heartily_recommend"], "high_school_student": ["high_schooler", "school_student", "high_school_students", "highschool_student", "computer_science_student"], "having_trouble": ["having_troubles", "struggling", "having_difficulties", "having_difficulty"], "anyone_shed": ["anyone_advise", "anyone_shed_light", "shed"], "gravitational_field": ["gravitational_potential", "gravitational_pull", "gravitational_force"], "reflection_principle": ["schwarz_reflection_principle", "superposition_principle"], "prior_knowledge": ["foreknowledge", "knowledge", "zero_knowledge"], "cauchy_condensation_test": ["p_series_test", "condensation_test", "leibniz_criterion", "comparsion_test", "cauchy_condensation"], "condensation_test": ["comparsion_test", "cauchy_condensation", "cauchy_condensation_test", "raabe"], "gravitational_force": ["gravitational", "net_force", "gravitational_acceleration", "gravitational_field", "gravity"], "n\u22651": ["n\u2265", "n\u22650"], "theoretical_computer_science": ["computer_scientists", "computer_science_major", "computer_science"], "rigid_motions": ["rigid_motion"], "nonstandard_analysis": ["non_standard_analysis"], "highly_recommend": ["recommend", "warmly_recommend", "highly_recommended", "heartily_recommend", "strongly_recommend"], "row_echelon_form": ["reduced_echelon_form", "echelon_form", "rref", "row_reduced_echelon", "row_echelon", "reduced_row_echelon"], "reduced_row_echelon": ["reduced_echelon_form", "row_echelon_form", "rref", "echelon_form", "row_reduced_echelon", "row_echelon", "reduced_echelon"], "personally_prefer": [], "laplace_expansion": ["cofactor_expansion_along", "cofactor_expansion", "co_factor_expansion"], "cofactor_expansion": ["co_factor_expansion", "laplace_expansion", "cofactor_expansion_along"], "echelon_form": ["reduced_echelon_form", "row_echelon_form", "rref", "echelon_forms", "echelon", "reduced_row_echelon"], "000_dollars": ["\u00a310", "\u20ac1", "billion_dollars", "\u00a31000", "million_dollars", "000_usd", "dollars", "\u00a35"], "fair_coins": ["fair_coin", "tossing", "fair_coin_flips", "unfair_coins", "biased_coins"], "crops_up": ["crop_up", "pops_up"], "project_euler": [], "2x_y": ["2x_2y", "x_2y", "_2x_y", "3x_y"], "absorbing_states": ["accepting_states", "absorbing_state", "transient_states", "absorbing_barriers"], "arise_naturally": ["arises_naturally"], "color_0a0": ["color_90f", "color_0af", "color_c0f", "color_c00"], "color_c00": ["color_90f", "color_0af", "color_c0f", "color_0a0"], "primitive_root": [], "upper_semi_continuous": ["upper_semicontinuous", "semi_continuous", "lower_semi_continuous"], "mo_thread": ["mathoverflow_thread", "linked_thread", "torsten", "mo_post"], "355_113": ["333_106"], "gt0": [], "eigen_vectors": ["eigevectors", "eigen_vector", "eigenvectors"], "eigen_values": ["eignevalues", "eigen_value", "eignvalues", "eigenvalues"], "topologically_transitive": ["topologically_mixing", "2_transitive", "doubly_transitive", "3_transitive"], "clever_trick": ["clever_tricks"], "descartes_rule": ["descartes", "descarte"], "local_minima": ["minima", "local_maxima", "local_minima_maxima", "global_minima", "local_minimums", "local_extrema", "global_maxima"], "restrict_attention": ["restrict_ourselves", "restrict_oneself", "confine_attention"], "non_homeomorphic": ["homeomorphic", "nonhomeomorphic"], "n\u22652": ["n\u22654", "n\u2265"], "minor_axes": ["semi_minor_axes", "semi_major", "semiaxes", "semi_axes", "semimajor", "semiminor_axes", "minor_axis", "major_minor_axes"], "times9": ["times99", "times3_times4", "times7", "times16", "times9_times9", "times8_times9"], "uniform_continuity": ["equicontinuity", "continuity", "uniform_equicontinuity", "uniform_convergence"], "clearing_denominators": ["clearing_fractions", "rationalizing_denominators"], "cartier_divisors": ["cartier_divisor", "effective_cartier_divisors", "weil_divisors"], "coset_representatives": ["representatives", "coset_representative"], "minus_signs": ["plus_minus_sign", "minus_sign"], "effective_cartier_divisor": ["cartier_divisor", "effective_cartier_divisors", "cartier_divisors"], "global_section": ["global_sections", "global_sections_functor", "global_section_functor"], "weil_divisors": ["weil_divisor", "effective_cartier_divisors", "cartier_divisors"], "short_exact_sequence": ["non_split_short_exact", "short_exact", "short_exact_sequences"], "brauer_group": [], "fully_grasp": ["fully_grasped", "fully_comprehend"], "hodge_decomposition": ["levi_decomposition", "bruhat_decomposition", "iwasawa_decomposition"], "arbitrarily_close": ["arbitrarily_closely", "gets_arbitrarily_close"], "contrapositively": ["contrapositive", "contrapositives", "contrapositve"], "common_divisor": ["greatest_common_divisor", "greatest_common_divisors"], "similar_fashion": [], "freely_available_online": ["stephen_simpson", "e_book", "publicly_available", "through_guided_discovery", "freely_available", "monumental", "available_online", "freely_downloadable"], "few_years_ago": ["few_weeks_ago", "fifty_years_ago", "few_days_ago", "few_months_ago"], "triangulated_categories": ["triangulated_category", "fibered_categories", "enriched_categories", "2_categories"], "graduate_students": ["undergraduate_student", "graduate_student", "year_graduate_student"], "connected_components": ["connected_component", "biconnected_components", "strongly_connected_components"], "university_library": ["university_level", "university_libraries", "public_library"], "professional_mathematician": ["trained_mathematician", "native_english_speaker", "mathematician", "professional_mathematicians", "computer_programmer", "pure_mathematician", "software_developer"], "his_her": ["her", "his_her_own"], "past_exam": ["past_exams", "old_exams", "old_exam", "old_qualifying_exam", "qualifying_exam", "past_exam_paper", "past_exam_papers"], "end_array": ["begin_array_clcr", "begin_array_ccl", "begin_array_cc", "begin_array"], "theroem": ["thoerem", "therorem", "theorem", "theoem"], "few_days": ["few_weeks", "past_few_days", "few_months"], "lend_itself": ["lends_itself"], "significant_digits": ["decimal_places", "significant_digit", "significant_figures"], "python_program": ["python_code", "pari_gp_program", "matlab_script", "computer_program", "python_script", "java_program"], "physical_quantities": [], "splitting_field": ["splitting_fields"], "hands_dirty": [], "sequentially_compact": ["feebly_compact", "countably_compact", "weakly_sequentially_complete", "weakly_sequentially_compact"], "conjugacy_class": ["_conjugacy_class", "_conjugacy_classes", "conjugacy_classes"], "brief_explanation": ["detailed_explanation", "fuller_explanation", "thorough_explanation"], "key_observation": [], "joint_pdf": ["marginal_density", "joint_distribution", "joint_cdf", "joint_pmf", "marginal_pdf", "joint_density"], "combinatorial_interpretation": ["combinatoric"], "homemorphism": ["homoemorphism", "diffemorphism", "homeorphism", "homeomorphism"], "depends_heavily": ["depends", "relies_heavily", "depends_crucially"], "rectangular_grid": ["rectangular_strips", "rectangular_garden", "rectangular_cuboid"], "52_cards": ["52__card_deck", "shuffled_deck", "playing_cards", "standard_52_card_deck", "52_card_deck", "deck", "cards"], "a\u2229c": ["b\u2229c", "a\u2229b_\u222a", "b\u222ac", "a\u2229b\u2229c", "\u2229c"], "b\u2229c": ["a\u222ab\u222ac", "a\u2229c", "b\u222ac", "a\u2229b\u2229c", "\u2229c"], "a\u2229b\u2229c": ["b\u2229c", "a\u2229b", "\u2229c", "a\u2229c"], "he_writes": ["author_writes"], "mo_post": ["mathoverflow_thread", "mo_thread", "blog_posting"], "silly_mistake": ["mistake_somewhere", "silly_mistakes", "silly_mistake_somewhere", "careless_mistake", "stupid_mistake"], "klein_four": ["klein_four_group", "klein_4_group", "klein_"], "semi_direct_products": ["semidirect_products", "semi_direct_product", "semidirect_product"], "dihedral_group": ["binary_icosahedral_group", "hyperoctahedral_group", "dihedral_groups"], "counted_twice": [], "choose3": ["choose1", "choose13", "choose10", "choose2", "choose4", "choose5"], "brown_eyes": ["blue_eyes", "brown_eyed", "brown_hair", "green_eyes"], "looks_promising": ["promising", "looked_promising"], "recently_learned": ["recently_started_learning", "currently_learning", "recently_learnt"], "maclaurin_series": ["taylor_series_expansion", "taylor_series", "maclauren_series", "mclaurin_series", "maclaurin_expansion"], "software_package": ["software_libraries", "softwares", "software", "software_packages"], "non_standard_analysis": ["nonstandard_analysis"], "everyone_knows": ["she_knows", "nobody_knows", "everybody_knows"], "begin_array": ["begin_array_llll", "end_array", "begin_array_lcr", "begin_array_lcc"], "months_ago": ["days_ago", "weeks_ago", "hours_ago", "few_months_ago", "years_ago"], "perfect_matching": ["bipartite_subgraph", "bipartite_graph", "hamiltonian_cycle", "perfect_matchings", "bipartite_matching"], "unilateral_shift": ["bilateral_shift"], "homology_groups": ["homology", "co__homology", "homologies", "cohomology", "co_homology", "singular_homology"], "global_minima": ["global_minimum", "local_minimums", "local_minima", "global_minimizers", "global_maxima"], "global_maxima": ["minima", "local_maxima", "global_minima", "global_maximizer", "local_minimums", "local_maximums", "global_maximum"], "broken_down_into": ["broken_down"], "nintegrate": [], "python_code": ["python_program", "python_script", "mathematica_code", "matlab_code", "java_code"], "persistent_homology": ["floer_homology", "co_homology", "khovanov_homology"], "grothendieck_universe": ["set_theoretic_universe", "grothendieck_universes", "tarski_grothendieck"], "dihedral_groups": ["dihedral_group", "hyperoctahedral_group", "dihedral"], "years_later": ["hundred_years_ago", "fifty_years_ago", "twenty_years_ago"], "sylow_p_subgroups": ["_sylows", "sylow_p_subgroup", "_sylow_subgroups", "sylow_3_subgroups", "p_subgroups", "p_sylow_subgroups", "sylow_subgroups"], "acts_faithfully": ["acts_freely", "acts_transitively", "acts_regularly", "act_faithfully"], "sylow_3_subgroups": ["sylow_5_subgroups", "sylow_7_subgroups", "_sylows", "sylow_3_subgroup", "sylow_2_subgroups", "3_sylow_subgroups"], "horizontal_asymptotes": ["vertical_asymptote", "vertical_asymptotes", "horizontal_asymptote"], "monic_irreducible_polynomial": ["monic_polynomial", "monic_irreducible_polynomials", "monic_irreducibles", "monic_irreducible"], "inaccessible_cardinal": ["inaccessible_cardinals", "strongly_inaccessible_cardinal", "weakly_inaccessible_cardinal", "strongly_inaccessible_cardinals"], "brushing_up": ["currently_revising", "brushing", "refreshing", "rushing", "self_teaching", "brush_up", "pretty_rusty"], "software_engineer": ["phd_student", "software_engineering", "computer_programmer", "programmer", "software_developer", "computer_science_student"], "advanced_topics": ["advanced", "topics"], "james_stewart": ["michael_spivak", "early_transcendentals", "calculus_early_transcendentals", "stewarts", "stewart", "ian_stewart"], "poincare_duality": ["poincar\u00e9_duality", "poincar\u00e9_dual", "poincare_dual", "lefschetz_duality"], "hodge_theory": ["voisin", "chern_weil_theory", "hodge_tate", "gauge_theories"], "smooth_projective_variety": ["quasi_projective_variety", "smooth_projective_varieties", "projective_variety"], "loosely_speaking": ["roughly_speaking", "broadly_speaking", "vaguely_speaking"], "mathematics_magazine": ["mathematical_gazette", "mathematics_magazine_vol", "american_mathematical_monthly"], "main_diagonal": ["diagonal", "sub_diagonal", "superdiagonal"], "end_smallmatrix_bigr": ["end_smallmatrix", "begin_smallmatrix", "bigl_begin_smallmatrix"], "mental_picture": [], "inner_automorphisms": ["outer_automorphisms", "inner_automorphism", "outer_automorphism"], "freely_generated": ["finitely_generated", "singly_generated", "principally_generated"], "deformation_retracts": ["deformation_retracting", "deformation_retracts_onto", "deformation_retract", "deformation_retracted"], "stanford_encyclopedia": ["encyclopedias", "stewart_shapiro", "mailing_list", "encyclopaedia", "encyclopedia", "sep_article"], "cumulative_distribution": ["cummulative_distribution", "empirical_cumulative_distribution", "cdf", "cumulative_density"], "busy_beaver": ["beaver"], "orbit_stabiliser_theorem": ["orbit_stabilizer_theorem", "orbit_stabilizer", "orbit_stabiliser"], "large_cardinals": ["woodin_cardinals", "mahlo_cardinals", "inaccessible_cardinals"], "currently_studying": ["studing", "currently_learning", "currently_self_studying", "studying", "currently_revising"], "neutral_element": [], "paying_attention": ["restricting_attention", "paid_attention"], "_sylow_subgroup": ["p_sylow_subgroup", "sylow_p_subgroup", "sylow", "sylow_", "5_sylow_subgroup", "_sylow", "3_sylow_subgroup", "_subgroup"], "_sylow_subgroups": ["_sylows", "_subgroups", "p_sylow_subgroups", "sylow_", "2_sylow_subgroups", "sylow_3_subgroups", "3_sylow_subgroups", "sylow_subgroups"], "schur_zassenhaus": ["cantor_zassenhaus", "zassenhaus"], "_sylow": ["_sylow_subgroup", "_sylows", "3_sylow_subgroup", "sylow", "7_sylow", "sylow_", "5_sylow", "_subgroup"], "decomposition_svd": ["decomposition"], "1_kx": ["nkx", "n_kx"], "2nd_ed": ["3rd_ed", "2nd_edition", "4th_ed"], "sylow_3_subgroup": ["sylow_7_subgroup", "2_sylow_subgroup", "sylow_2_subgroup", "sylow_3_subgroups", "sylow_5_subgroup", "3_sylow_subgroup"], "gauss_curvature": ["gaussian_curvature", "ricci_curvature", "signed_curvature"], "bxy_cy": ["2bxy_cy"], "perfect_squares": ["non_perfect_squares", "perfect_square", "perfect_cubes"], "\u221ax": [], "anybody_knows": ["anyone_knows", "nobody_knows"], "circumscribed_circle": ["circumcircle", "excircle", "circumscribed", "incircle", "circumscribes", "circum_circle", "nine_point_circle"], "angle_dcb": ["angle_eac", "angle_dca", "angle_dce", "angle_dba", "angle_dae", "angle_ead"], "minkowski_bound": [], "based_upon": [], "memory_serves": [], "principal_ideal_domains": ["ufds", "dedekind_domains", "pids", "principal_ideal_domain", "pr\u00fcfer_domains", "pid"], "monte_carlo_methods": ["monte_carlo", "monte_carlo_simulations", "monte_carlo_sampling"], "days_ago": ["months_ago", "weeks_ago", "hours_ago", "couple_days_ago"], "gelfand_transform": [], "financial_mathematics": ["mathematical_finance", "pure_mathematics", "risk_management", "recreational_mathematics"], "stochastic_process": ["_valued_stochastic_process", "discrete_time_stochastic_process", "markov_process", "continuous_time_stochastic_process"], "forces_acting": ["force_acting"], "force_acting": ["forces_acting", "force_exerted", "exerted"], "exceptional_divisor": ["exceptional_divisors", "weil_divisor"], "binomial_expansion": [], "gauss_hypergeometric": ["regularized_hypergeometric", "appell_hypergeometric", "generalized_hypergeometric"], "hardy_littlewood_maximal": ["hardy_littlewood_conjecture", "hardy_littlewood_sobolev_inequality", "hardy_littlewood"], "1_tx": ["1_ty"], "jointly_gaussian": ["uncorrelated", "jointly_gaussian_random_variables", "jointly_normally_distributed"], "lower_bounds": ["lower_bounded", "lower_bound", "upper_bounds"], "jacobson_radical": ["nilradical", "non_radical", "nil_radical"], "hamiltonian_path": ["hamilton_path", "hamiltonian_paths", "hamiltonian_cycle"], "primeq": ["primepi"], "congruence_classes": ["congruence_classes_modulo", "residue_classes", "equivalence_classes"], "ordered_pair": ["ordered_pairs", "unordered_pair"], "keeping_track": ["kept_track"], "matlab_script": ["python_program", "mathematica_script", "python_script", "java_program", "perl_script"], "original_poster": [], "undergraduate_student": ["grad_student", "undergrad", "undergraduate", "graduate_student", "year_undergraduate_student", "year_graduate_student", "enthusiast", "engineering_student"], "cocountable_topology": ["cofinite_topology", "co_finite_topology", "co_countable_topology"], "taking_into_account": [], "equally_likely_outcomes": ["equally_probable_outcomes", "equally_likely", "equiprobable_outcomes"], "lu_decomposition": ["qr_decomposition", "lu_factorization", "ldu_decomposition"], "orientable_surfaces": ["orientable_surface", "non_orientable_surfaces", "2_manifolds", "nonorientable_surface"], "shuffled_deck": ["52__card_deck", "standard_52_card_deck", "52_card_deck", "well_shuffled_deck", "poker_deck", "well_shuffled"], "diagram_chase": ["diagram_chasing"], "path_components": ["path_component"], "originally_posted": ["originally_posed", "crossposted", "reposted"], "rfloor__lfloor": ["rfloor_n", "rfloor_j", "rfloor_", "rfloor_2"], "terrence_tao": ["dudley", "poma", "terence_tao", "blogpost", "terry_tao"], "powerful_tool": ["powerfull", "powerful_tools", "powerful"], "ordinary_differential": ["ordinary_differential_equations", "differential", "integro_differential", "nonlinear_ordinary_differential", "ode"], "who_knows": ["someone_who_knows", "anyone_who_knows"], "iterative_methods": ["quasi_newton_methods", "iterative_solvers", "collocation_methods"], "stephen_abbott": ["arthur_mattuck", "poma", "abbot", "stephen_willard", "stephen_boyd", "henk_tijms", "abbott"], "binary_tree": ["full_binary_tree", "rooted_binary_tree", "tree", "binary_trees"], "fritz_john": ["fritz"], "mathematical_sciences": ["physical_sciences", "social_sciences", "mathematical_biology"], "rejection_sampling": ["subsampling", "re_sampling", "gibbs_sampling"], "empirical_cdf": ["ecdf", "empirical_cumulative_distribution", "empirical_risk", "empirical"], "test_statistic": ["z_statistic", "t_statistic", "likelihood_ratio_test_statistic"], "2_4ac": ["4ac", "2_4ac_4c", "_4ac"], "inference_rules": ["deductive_rules", "logical_deductions", "inference_rule", "tautologies"], "line_segment_connecting": ["segment_connecting", "line_segment_joining", "straight_line_segment_connecting", "segment_joining"], "2hx": ["2xh", "4xh", "3x_2h_3xh", "3hx", "3xh"], "fuchsian_groups": ["fuchsian_group"], "color_green": ["color_red", "color_green1", "color_orange", "color_blue", "color_grey"], "color_blue": ["color_red", "color_green1", "color_orange", "color_green"], "color_red": ["color_green1", "color_blue", "color_green"], "color_teal": ["color_lime", "color_orange", "color_olive"], "arbitrarily_high": [], "technical_details": ["gory_details", "details", "technicalities"], "gives_rise": [], "doubly_infinite": [], "analytically_continued": ["continued_analytically"], "uniform_integrability": ["_integrability", "integrability", "non_integrability"], "95_confidence": ["95_confidence_interval", "_confidence", "95_confident"], "iid_random_variables": ["identically_distributed_random_variables", "iid", "random_variables", "independent_identically_distributed", "_distributed_random_variables"], "_3z": ["2_6z", "_6z", "_5z", "x_3z", "4_3z", "2_3z"], "3_xy": ["2_xy", "4_4xy", "3y_xy", "2y_xy", "xy_2y"], "spectral_radius": [], "integrally_closed": ["noetherian_integrally_closed", "integrally_closed_domains", "integrally_closed_noetherian", "dedekind_domain", "ufd"], "coordinate_charts": ["coordinate_patches", "coordinate_chart", "charts", "coordinate_patch"], "lower_half_plane": ["lower_halfplane", "upper_half_plane", "left_half_plane"], "absolute_value_signs": ["absolute_value_bars"], "joint_distribution": ["marginal_distribution", "joint_pdf", "marginal_distributions", "joint_pmf", "joint_density"], "principal_bundle": ["_principal_bundle", "fibre_bundle", "principal_bundles", "principal_g_bundle", "fiber_bundle", "_bundle"], "frame_bundle": ["_bundle", "spinor_bundle", "hopf_bundle"], "wait_until": ["wait_till"], "maximum_modulus_principle": ["modulus_principle", "schwarz_lemma"], "trapezoidal_rule": ["trapezoidal", "composite_trapezoidal_rule"], "strongly_suggests": ["strongly_suspect"], "algebraically_closed": ["algebraically_closed_field", "algebrically_closed", "non_algebraically_closed_field"], "_coefficients": ["_coefficient", "coefficients"], "sectional_curvatures": ["non_positive_sectional_curvature", "sectional_curvature", "nonpositive_sectional_curvature"], "equiv1_bmod": ["equiv1_bmod4", "equiv1_pmod", "equiv1_mod3", "equiv1_mod", "equiv1_bmod3"], "digital_root": [], "bernoulli_trials": ["bernouli_trials", "trials", "independent_bernoulli_trials", "iid_bernoulli_trials", "bernoulli_trial"], "interplay_between": ["interplay"], "borel_measures": ["signed_measures", "borel_measure", "radon_measures"], "successor_ordinal": ["successor_cardinal", "successor_ordinals", "ordinal"], "power_set": ["powerset"], "uniform_convergence": ["pointwise_convergence", "uniform_continuity", "epi_convergence"], "regular_languages": ["non_regular_language", "non_regular_languages", "context_free_languages"], "statistical_tests": ["statistical_hypothesis_testing", "statistical", "t_tests"], "unordered_pairs": ["ordered_pairs", "unordered_triples", "unordered_pair"], "unordered_pair": ["ordered_pairs", "ordered_pair", "unordered_pairs"], "peter_weyl_theorem": ["atiyah_singer_index_theorem", "peter_weyl"], "180_circ": ["n180_circ", "180_\u00b0", "180_circ_", "180\u00b0"], "b_ax": [], "riemann_integrability": ["improper_riemann_integrable", "lebesgue_integrability", "non_riemann_integrable"], "improper_integral": ["improper_riemann_integrals", "improper_integrals", "improper_riemann_integral"], "countably_many_discontinuities": ["discontinuities", "countably_many_nowhere_dense", "jump_discontinuities"], "riemann_lebesgue_lemma": ["lebesgue_dominated_convergence", "riemann_lebesgue", "riemann\u2013lebesgue_lemma"], "ac_bc": ["2_ab_ac_bc", "ab_bc", "c_bc"], "bisects_angle": ["bisector", "bisects", "ad_bisects_angle"], "polar_co_ordinates": ["polar_coordinate", "polar_coordinates", "polar_cordinates"], "angle_cab": ["angle_bac", "angle_acb", "angle_cao", "angle_cae", "angle_cad", "angle_cba", "angle_dac", "angle_bca"], "100_x": ["400_x", "100_y", "200_x"], "punctured_plane": ["punctured_unit_disc", "punctured_disk", "punctured_torus"], "punctured_disk": ["punctured_disc", "punctured_unit_disc", "punctured_unit_disk"], "essential_singularity": ["non_isolated_singularity", "isolated_singularity", "removable_singularity", "essential_singularities"], "bit_rusty": ["rusty", "pretty_rusty", "little_rusty"], "frac1z": ["z__frac1z", "frac1z_"], "punctured_disc": ["punctured_unit_disk", "punctured_unit_disc", "punctured_disk"], "iz__e__iz": ["iy__e__iy", "_e__iz", "ix__e__ix", "_iz_2i", "iw__e__iw"], "ab_a": ["ba_b", "ab_ab", "2_a_b", "ab_b", "ab_ac", "2ab_a", "_3ab"], "quasicoherent_sheaf": ["coherent_sheaf", "quasi_coherent_sheaf", "quasicoherent_sheaves", "quasicoherent", "quasi_coherent"], "universal_cover": ["contractible_universal_cover", "2_fold_covering", "universal_covering"], "einstein_notation": ["einstein_summation", "einstein_summation_notation", "einstein_summation_convention"], "maurer_cartan_form": ["maurer_cartan"], "principal_bundles": ["g_bundles", "_principal_bundles", "principal_bundle", "fiber_bundles"], "leibniz_rule": ["leibnitz_rule"], "recently_come_across": ["recently_stumbled_upon", "recently_stumbled_across", "recently_came_across", "recently_encountered", "come_accross"], "pay_attention": [], "narrow_down": [], "subsequential_limits": ["sub_sequential_limits", "subsequential_limit"], "subsequential_limit": ["subsequential_limits", "sub_sequential_limits"], "rigid_motion": ["rigid_motions"], "\u00e0_la": [], "american_mathematical_monthly_vol": ["american_mathematical_society", "american_mathematical_monthly", "american_mathematical_society_vol"], "lower_darboux": ["upper_darboux_sums", "upper_darboux", "lower_darboux_sums"], "upper_darboux": ["lower_darboux_sums", "upper_darboux_sums", "lower_darboux"], "lower_darboux_sums": ["riemann_sums", "upper_darboux_sums", "darboux_sums", "lower_darboux"], "descending_chain": ["descending_chains", "ascending_chains", "ascending_chain"], "projection_onto": ["projector_onto", "orthogonal_projection_onto", "projection"], "convinced_myself": ["convincing_myself", "convince_myself"], "finite_dimensionality": ["\ufb01nite_dimensional", "finite_dimension", "finite_dimensional"], "alternating_signs": [], "begin_array_cc": ["begin_array_clcr", "end_array", "begin_array_ccl", "begin_array_ccc"], "ca_cb": ["a_cb"], "empirical_evidence": ["numerical_evidence", "experimental_evidence", "numerical_evidence_suggests"], "everybody_else": ["everyone", "everyone_else", "everybody"], "stone_weierstrass": ["stone_weierstrass_theorem", "stone_weierstra\u00df", "stone_weirstrass"], "critical_strip": ["vertical_strip"], "personal_favorite": [], "partitioned_into": [], "parallelogram_law": [], "unbiased_coin": ["biased_coins", "unfair_die", "unfair_coin", "biased_coin"], "perfectly_fine": ["perfectly_valid"], "2abi": ["2bi", "2ab_b", "3_3ab", "2iab", "3a_2b_b", "2abi_b"], "engineering_student": ["grad_student", "masters_student", "computer_science_student", "high_school_senior", "undergraduate_student", "amateur_mathematician"], "acute_angle": ["acute_angles", "angle", "obtuse_angle", "acute_angled"], "ix__e__ix": ["3ix__e__3ix", "iz__e__iz", "inx__e__inx", "iax__e__iax"], "angled_triangle": ["right_angled_triangle", "isosceles_triangle", "obtuse_angled_triangle", "acute_angled_triangle"], "strictly_convex": ["convex", "strictly_concave"], "author_writes": ["he_writes"], "dominated_convergence_theorem": ["mct", "lebesgue_dominated_convergence", "dominated_convergence", "monotone_convergence_theorem", "monotone_convergence"], "events_occurring": ["non_occurrence", "event_occurring", "accidents_occur"], "canonically_identified": ["identified", "naturally_identified", "uniquely_identifies"], "decomposed_into": ["decomposed"], "markov_property": ["strong_markov_property"], "treated_separately": ["handled_separately"], "schwarz_lemma": ["maximum_modulus_principle", "schwarz_pick_lemma", "schwarz_pick"], "jonas_meyer": ["phoemuex", "jonas", "rahul_narain", "david_speyer", "nate_eldredge", "5xum", "daw", "willie_wong"], "pairwise_distinct": [], "english_alphabet": ["latin_alphabet", "3_letter", "26_letter", "hebrew_alphabet", "three_letter", "_letter_alphabet"], "blaschke_product": ["blaschke", "blaschke_products"], "my_humble_opinion": ["personal_opinion", "opinion"], "brute_force_search": ["brute_force", "exhaustive_search", "brute_forced"], "expressive_power": ["expressive", "expressiveness"], "combinatorial_species": ["combinatorialist", "combinatorial", "combinatorial_designs"], "dynamical_system": ["autonomous_dynamical_system", "dynamical_systems", "nonlinear_dynamical_system"], "300_000": ["600_000", "000", "200_000", "500_000"], "mathematical_rigor": ["mathematical_education", "rigour", "rigor", "mathematical_maturity", "mathematical_sophistication", "informality"], "confidence_level": ["confidence_interval", "confidence_levels", "significance_level"], "population_standard_deviation": ["standard_deviations", "standard_deviations_away", "standard_deviation"], "stack_exchange": ["stackoverflow", "forum", "math_stack_exchange", "math_stackexchange", "stack_overflow", "physics_stack_exchange", "stackexchange"], "exponentially_distributed_random_variables": ["identically_distributed_random_variables", "normally_distributed_random_variables", "binomially_distributed_random_variable"], "exponentially_distributed": ["log_normally_distributed", "poisson_distributed", "distributed_exponentially", "exponentially_distributed_random_variables", "binomially_distributed"], "powerful_tools": ["powerfull", "powerful_tool", "powerful"], "d_finite": ["_finite", "\u03c3_finite", "tarski_finite", "dedekind_finite"], "greatly_simplified": ["greatly_simplifies"], "practical_purposes": ["computational_purposes", "educational_purposes", "pedagogical_purposes"], "randomly_selecting": ["randomly_select", "randomly_selects", "randomly_selected"], "classical_mechanics": ["celestial_mechanics", "analytical_mechanics", "fluid_mechanics"], "indefinite_integrals": ["indefinite_integral", "indefinite_integration", "definite_integrals"], "diagonal_entries": ["non_diagonal_entries", "entries", "subdiagonal_entries", "off_diagonal_entries"], "slightly_differently": ["phrased_differently", "differently"], "groebner_basis": ["grobner_basis", "gr\u00f6bner_basis", "groebnerbasis"], "groebner_bases": ["grobner_bases", "groebner_basis", "gr\u00f6bner_bases"], "counting_multiplicities": ["multiplicites", "multiplicities", "multiplicity"], "12xy": ["3_6xy", "2y_2x", "24xy", "16xy", "3_2xy", "36xy", "5xy_3y", "2y_2xy"], "2bxy_cy": ["2dx_2ey", "2bxy", "bxy_cy"], "gloss_over": ["glosses_over", "glossed_over"], "python_script": ["python_code", "python_program", "matlab_script", "java_program", "python_scipy"], "without_mentioning": [], "greater_generality": [], "trial_division": [], "2y_4": ["2y_6", "2y_4x", "2y_5", "3y_6", "2x_4y", "3y_2x"], "archimedean_property": ["archimidean_property", "archimedian_property", "archimedean_principle"], "xgx": ["hgx", "gg_xg", "gx_xg", "xhx", "hgh", "gyg"], "gram_schmidt_procedure": ["gram_schmidt_algorithm", "gram\u2013schmidt_process", "gram_schmidt", "gramm_schmidt_process", "gram_schmidt_orthogonalization", "gram_schmidt_process"], "allen_hatcher": ["hatcher", "tammo_tom_dieck", "hatchers", "tom_dieck", "switzer"], "gram_schmidt_process": ["gram_schmidt_procedure", "gram_schmidt_algorithm", "gram\u2013schmidt_process", "gram_schmidt", "gramm_schmidt_process", "gram_schmidt_orthogonalization"], "deformation_retraction": ["deformation_retracting", "deformation_retractions", "deformation_retract"], "acts_trivially": ["acts_non_trivially", "acts_freely", "acts_irreducibly", "acts_nontrivially"], "act_trivially": ["act_transitively", "act_nontrivially", "acts_nontrivially"], "2x2_matrices": ["_matrices", "matrices", "m_matrices"], "anyone_knows": ["anybody_knows"], "ai_bj_ck": ["ai_bj"], "geq5": ["geq7"], "principal_branch": ["_branch", "single_valued_branch", "_branch_cut"], "immediate_consequence": ["immediate_consequences", "immediate_corollary"], "equiv1_pmod": ["equiv1_pmod9", "equiv1_pmod2", "equiv0_pmod", "equiv_1", "equiv2_pmod", "equiv1_mod", "pmod", "equiv1_pmod8"], "listplot": ["listlineplot", "ezplot", "subplot"], "ex_falso_quodlibet": ["ex_falso_sequitur_quodlibet", "ex_falso", "double_negation_elimination", "falsum", "efq", "ponens"], "leading_principal_minors": ["principal_minor", "leading_principal_submatrices", "principal_minors"], "upper_triangular_matrices": ["unitriangular_matrices", "upper_unitriangular_matrices", "upper_triangular_unipotent"], "principal_minors": ["_minors", "principal_minor", "leading_principal_minors"], "numerical_stability": ["numerical_instability", "instability", "a_stability"], "multiply_both_sides": ["multiplying_both_sides"], "taken_into_account": [], "homomorphic_image": ["homomorphic_images"], "_int_dt": ["_ikt", "_ixt", "_int", "_ist_dt", "_ikt_dt"], "deleted_neighborhood": ["punctured_neighborhood", "punctured_neighbourhood", "deleted_neighbourhood"], "motivation_behind": ["motivations_behind", "intution_behind", "intuition_behind"], "orthogonality_relations": ["orthogonality"], "hamiltonian_circuit": ["hamiltonian_circuits", "hamilton_circuit", "hamiltonian_cycle"], "xsin": ["ysin", "xsinx"], "implicit_differentiation": ["implicit_differentation"], "greatest_lower_bound": ["greatest_lower", "greatest_lower_bounds"], "logarithmic_differentiation": ["automatic_differentiation", "anti_differentiation", "antidifferentiation"], "contour_integration": ["tabular_integration"], "removable_discontinuity": ["vertical_asymptote", "non_removable_singularity", "jump_discontinuity", "removable_discontinuities"], "full_subcategory": ["full_subcategories", "reflective_subcategory", "wide_subcategory", "subcategory"], "please_forgive_me": ["apologize", "forgive_me", "please_forgive", "apologise", "please_excuse", "apologies"], "cantor_bernstein": ["cantor_schr\u00f6der_bernstein", "cantor_schroder_bernstein", "cantor_bernstein_schr\u00f6der", "cantor_schroeder_bernstein", "schr\u00f6der_bernstein", "cantor_bernstein_theorem"], "rs_sr": ["srsr"], "video_lectures": ["video_tutorials", "lecture_videos", "videos", "video_lecture"], "personal_preference": ["personal_taste", "personal_preferences", "personal_opinion"], "heaviside_step": ["heavisidetheta", "heaviside"], "fibered_product": ["fibered_products"], "distributive_law": ["distributivity", "distributive_property", "distributive_laws"], "i_ba": ["i_ab"], "statistical_physics": ["theoretical_physicist", "theoretical_physics", "statistical_mechanics"], "2x_8": ["3x_4", "5x_3", "2_4x_4", "2x_7", "12x_8", "2_2x_8", "2_4x_5", "2x_4"], "row_reduced": ["reduced_echelon_form", "row_reduced_echelon", "row_reducing", "row_reduce", "reduced_row_echelon"], "row_reduction": ["row_reducing", "gauss_elimination", "gauss_jordan_elimination", "row_reduce", "row_reduced", "gaussian_elimination"], "left_inverse": ["right_inverse", "left_invertible", "left_inverses"], "dfrac4": ["dfrac3", "dfrac5", "dfrac7", "dfrac94", "dfrac8", "dfrac9"], "please_bear": ["please_excuse", "please_forgive"], "polar_coordinate": ["cylindrical_coordinate", "polar_coordiantes", "polar_coordinates", "polar_coords", "spherical_coordinate"], "homorphism": ["homomophism", "homomrphism", "homomorphism", "homomoprhism", "homormorphism", "comorphism", "homomorphsim"], "equiv0_pmod": ["equiv0_pmod2", "equiv1_pmod", "equiv0_pmod8", "equiv_1", "equiv2_pmod", "equiv1_mod", "pmod", "equiv0_pmod9"], "martin_brandenburg": ["henno", "henno_brandsma", "martin_argerami", "jeremy_rickard", "alex_kruckman", "alex_youcis", "najib_idrissi"], "ascending_chain_condition": ["descending_chain", "ascending_chains", "ascending_chain"], "countable_unions": ["under_countable_unions", "unions"], "little_trickier": ["trickier", "little_bit_trickier", "bit_trickier"], "homomorphic_images": ["homomorphic_image"], "who_likes": ["who_loves", "who_owns"], "cubic_bezier_curve": ["quadratic_bezier_curve", "cubic_bezier_curves", "cubic_bezier", "b\u00e9zier_curve", "bezier_curve", "cubic_b\u00e9zier_curve"], "infinitely_often": [], "mostowski_collapse": ["mostowski", "transitive_collapse", "levy_collapse"], "0_y": [], "pairwise_non_isomorphic": ["non_isomorphic", "nonisomorphic", "pairwise_nonisomorphic"], "powerseries": ["power_series"], "wreath_product": ["wreath_products", "semi_direct_product"], "numerical_experiments": ["numerical_experiments_suggest", "numerical_simulations", "numerical_experimentation"], "squaring_both_sides": ["raising_both_sides", "multiplying_both_sides", "cubing_both_sides"], "extremely_helpful": ["helpfull", "immensely_helpful", "gratefully_appreciated", "helpful", "hugely_appreciated", "tremendously_helpful"], "quadratic_bezier_curve": ["quadratic_b\u00e9zier_curve", "cubic_bezier_curve", "cubic_b\u00e9zier_curve", "b\u00e9zier_curve"], "zeta_zeta_z": ["zeta_a", "zeta_z"], "glue_together": ["stitch_together", "glue", "glues_together"], "pictured_below": [], "_1_a": ["4a_3", "_1_b", "_2_a"], "0_u": ["0_v"], "taylor_expansions": ["taylor_expand", "taylor_expansion", "taylor_series_expansions", "taylor_series"], "sufficient_statistic": ["z_statistic", "t_statistic", "minimal_sufficient_statistic"], "reverse_implication": ["forward_implication", "reverse_containment", "reverse_inclusion"], "robinson_arithmetic": ["presburger_arithmetic", "non_arithmetic", "peano_arithmetic", "first_order_peano_arithmetic"], "intuition_behind": ["intuitions_behind", "intution_behind", "motivation_behind"], "sufficiently_close": [], "slln": ["wlln", "lln", "strong_law"], "homotopy_invariance": ["homotopy_lifting", "homotopy_lifting_property", "homotopy_type"], "universal_quantification": ["universal_quantifier", "universal_quantifiers", "existential_quantification"], "stephen_boyd": ["stephenson", "vandenberghe", "boyd_vandenberghe", "lieven_vandenberghe", "boyd"], "25_x": ["25_2x", "25_y", "49_x", "16_x"], "truth_assignment": ["truth_assignments", "truth_value_assignments"], "change_of_basis": ["change_of_base_formula"], "hardest_part": [], "quadratic_non_residues": ["quadratic_nonresidues", "nonresidues", "quadratic_non_residue", "non_residues", "non_quadratic_residues", "quadratic_nonresidue"], "trouble_figuring_out": ["having_trouble_figuring_out", "having_trouble_figuring", "trouble_figuring"], "mathscr": ["mathcal"], "partially_ordered": ["cyclically_ordered", "totally_ordered", "poset", "preordered"], "confusion_arises": ["confusion_stems", "confusion"], "dirac_deltas": ["dirac_delta", "dirac_delta_function", "diracs"], "pulling_back": [], "successor_cardinal": ["successor_cardinals", "weakly_inaccessible", "uncountable_cofinality", "weakly_inaccessible_cardinal", "worldly_cardinal", "mahlo_cardinal", "strongly_inaccessible"], "pad\u00e9_approximants": ["pad\u00e9_approximant", "pad\u00e9_approximant_built", "pade_approximants"], "2nd_edition": ["2nd_ed", "4th_edition", "rd_edition", "3rd_edition", "third_edition", "fourth_edition"], "network_flow": ["max_flow", "networkx", "min_cost", "network_flows", "bipartite_matching"], "latin_square": ["latin_squares"], "4th_ed": ["6th_ed", "2nd_ed", "5th_ed", "4th_edition", "3rd_ed", "fourth_edition", "4ed", "8th_ed"], "graduate_texts": ["undergraduate_texts", "undergraduate_thesis", "graduate_studies", "springer_verlag", "revised_edition", "springer_gtm", "springer_lecture_notes"], "convex_bodies": ["convex_polytopes"], "perfectly_valid": ["perfectly_legitimate", "perfectly_fine", "perfectly_reasonable"], "stopping_time": ["stopped_martingale", "_stopping_time", "stopped_process"], "delve_into": ["dive_into", "delve"], "modding_out": ["quotienting", "modding"], "american_mathematical_monthly": ["american_mathematical_society", "american_mathematical_monthly_vol", "mathematics_magazine", "american_mathematical_society_vol"], "jump_discontinuities": ["countably_many_discontinuities", "jump_discontinuity", "removable_discontinuities", "discontinuities"], "strong_law": ["slln"], "gilbert_strang": ["friedberg_insel", "peter_lax", "strang", "gilbert", "jim_hefferon", "hoffman_kunze", "friedberg"], "implicitly_assumed": ["tacitly_assumes", "implicitly_assumes", "tacitly_assumed"], "implicitly_assumes": ["implicitly_assumed", "tacitly_assumes", "tacitly_assumed"], "relative_frequencies": ["cumulative_frequency", "frequencies", "low_frequencies"], "twin_primes": ["cuban_primes", "twin_primes_conjecture", "twin_prime"], "twin_prime": ["twin_primes", "twin_primes_conjecture", "sophie_germain_primes"], "othogonal": ["orhogonal", "orthogonal", "nonorthogonal"], "outgoing_edges": ["incoming_edges", "ingoing", "out_edges", "outgoing_edge"], "carl_mummert": ["mauro_allegranza", "noah_schweber", "danielv", "miha", "alex_kruckman", "arthur_fischer", "andres_caicedo"], "parallel_transport": ["parallel_transports", "parallel_transported", "parallel_transporting"], "parallel_transport_along": ["parallel_transports", "parallel_transported", "parallel_transport"], "begin_eqnarray": [], "end_eqnarray": [], "30_60_90_triangle": ["3_4_5_triangle", "right_angled_triangle", "45_45_90_triangle", "30_60_90_triangles"], "5n_1": ["5n_2", "5n_5", "3_4n", "2_12n", "6n_4", "4n_7", "5n_4"], "leq3": [], "rationale_behind": ["reasoning_behind", "logic_behind", "hiding_behind"], "cauchy_criterion": ["_criterion", "leibniz_criterion", "cauchy_condensation_test"], "cos2t": ["sin2t", "cos2m", "cos2c"], "duplication_formula": ["legendre_duplication_formula"], "scientific_computing": ["scientific_calculators", "operational_research", "financial_engineering", "riley_hobson", "scientific", "computer_architecture", "nocedal_wright"], "beppo_levi": ["ldct"], "land_neg": ["lor", "land_lnot", "lor_neg", "lor_lnot", "neg"], "jordan_blocks": ["jordan_block", "nilpotent_jordan_block", "largest_jordan_block"], "jordan_block": ["jordan_blocks", "nilpotent_jordan_block", "largest_jordan_block"], "khan_academy": ["khan_academy_video", "khanacademy", "khan_academy_videos", "kahn_academy"], "acting_freely": ["acting_transitively", "acts_freely", "acting_faithfully", "cocompactly", "acts_smoothly", "properly_discontinuously"], "serre_spectral_sequence": ["hochschild_serre_spectral", "adams_spectral_sequence", "leray_spectral_sequence"], "past_few_days": ["past_few_months", "past_few_weeks", "few_days"], "_linear_combination": ["linear_combination", "r_combination"], "leadsto": [], "vphantom": ["vphantom_tfrac12", "overbrace_vphantom", "vphantom_frac11"], "suitably_chosen": [], "maclaurin_expansion": ["taylor_series_expansion", "taylor_series", "mclaurin_expansion", "maclaurin_series", "maclaurin_expansions"], "greatest_common_divisor": ["greatest_common_divisors", "common_divisor", "gcd", "greatest_common"], "qualifying_exams": ["past_exams", "prelims", "old_exams", "past_qualifying_exam", "qualifying_exam", "comprehensive_exam", "analysis_qualifying_exam", "quals"], "galois_conjugates": ["_conjugates", "conjugates"], "kronecker_symbol": ["kronecker_deltas", "kronecker_delta", "kronecker"], "minimum_spanning_tree": ["mst", "spanning_tree", "spanning_trees"], "mistake_somewhere": ["messed_up_somewhere", "silly_mistake_somewhere", "mistake", "stupid_mistake", "silly_mistake"], "reminds_me": ["reminds_us", "reminds"], "perron_frobenius_theorem": ["perron_frobenius", "perron\u2013frobenius"], "david_speyer": ["theo_buehler", "yacp", "george_lowther", "georges_elencwajg", "david_ullrich", "david_mitra", "fedja", "marc_van_leeuwen"], "partial_fraction_decomposition": ["fraction_decomposition"], "associativity_commutativity": ["commutativity_associativity", "associativity_distributivity", "associativity"], "surreal_numbers": ["surreals", "surreal", "real_number"], "limit_inferior": ["limes_superior", "limes_inferior", "limit_superior"], "limit_superior": ["limes_superior", "limes_inferior", "limit_inferior"], "significant_figures": ["significant_digits", "decimal_places", "six_significant_figures", "twenty_significant_figures", "ten_significant_figures"], "do_carmo": ["carmo", "riemannian_geometry", "docarmo"], "jordan_canonical_form": ["jcf", "jordan_canonical_forms", "jnf", "jordan_normal", "jordan_canonical"], "sufficiently_small": [], "statistically_independent": ["stochastically_independent", "mutually_independent", "jointly_normally_distributed", "conditionally_independent"], "de_moivre": ["de_moivre\u2019s", "demoivre", "moivre", "de_moivres"], "global_optimum": ["global_optimality", "local_optima", "global_optima"], "came_accross": ["chanced_upon", "came_across", "bumped_into", "recently_came_across", "cam_across", "stumbled_upon"], "approx3": ["approx8", "approx4", "approx5", "approx_1", "approx_0"], "asymptotic_expansion": ["asymptotics", "asymptotic_expansions"], "countably_infinitely_many": ["countably_many", "countably_infinite", "uncountably_many"], "area_enclosed": ["region_enclosed", "solid_enclosed", "volume_enclosed"], "region_enclosed": ["area_enclosed", "solid_enclosed", "volume_enclosed"], "directed_graphs": ["undirected_graphs", "directed_graph", "multigraphs", "directed_acyclic_graphs", "digraphs"], "36_37_38": ["44_45_46", "36_38_39", "32_33_34", "41_42_43", "45_46_47"], "fields_medal": ["nobel_prize", "fields_medalist", "medalist", "fields_medal_winner", "nobel"], "_submanifold": ["embedded_submanifold", "_dimensional_submanifold", "immersed_submanifold", "submanifold"], "precalc": ["11th_grade", "10th_grade", "pre_calculus", "pre_calc", "highschool", "precalculus"], "_morphisms": ["1_morphisms", "morphisms", "2_morphisms"], "_functor": ["hom_functor", "_functors", "2_functor"], "finitely_supported": ["compactly_supported", "disjointly_supported"], "pigeon_hole_principle": ["pidgeonhole_principle", "pigeon_hole", "pigeonhole_principle"], "trig_substitution": ["u_substitution", "trig_sub", "trig_substitutions", "trigonometric_substitution"], "congruence_classes_modulo": ["residue_classes", "residues_modulo", "congruence_classes"], "eulerian_circuit": ["eulerian_cycle", "eulerian_path", "eulerian_circuits", "eulerian_trail", "euler_circuit", "eulerian_tour"], "eulerian_path": ["eulerian_cycle", "euler_trail", "eulerian_circuit", "eulerian_trail", "euler_circuit"], "pdx_qdy": ["qdy"], "arturo_magidin": ["mercio", "hagen_von_eitzen", "gerry_myerson", "mjd", "alex_kruckman", "arturo", "qiaochuyuan"], "picard_lindelof": ["phragmen_lindelof", "picard_lindel\u00f6f", "cauchy_lipschitz", "poincare_bendixson", "picard\u2013lindel\u00f6f_theorem", "picards"], "grater_than": ["greater_than", "less_than", "bigger_than"], "365_days": ["weekdays", "leap_days", "366_days"], "resources_online": ["online_resources", "resources", "tutorials_online"], "surface_patch": [], "multiplying_both_sides": ["raising_both_sides", "multiply_both_sides", "squaring_both_sides", "cubing_both_sides"], "tom_apostol": ["michael_spivak", "apostolos", "zorich", "apostol"], "newtons_method": ["newton\u2019s_method", "newton_raphson_method", "newton\u2013raphson_method"], "1_2y": ["2y_2xy", "1_2xy", "y_2xy"], "_3y": ["_4y", "_2x_2y", "_5y", "_6y", "_2y", "4y"], "augmented_matrix": ["rref"], "pivot_columns": ["columns", "non_pivot_column", "pivots", "non_pivot_columns"], "teach_myself": ["prepare_myself", "self_teaching_myself", "teaching_myself", "self_learn"], "genfrac_0pt": ["genfrac"], "grows_linearly": ["grow_linearly"], "sufficiently_strong": ["sufficiently_powerful", "effectively_axiomatized", "sufficiently_expressive", "computably_axiomatizable", "recursively_axiomatized", "sufficiently_rich"], "highly_doubt": [], "av_bw": ["av_aw", "av_bv"], "was_hoping_someone": ["hoping_someone_could", "hoping_someone", "someone"], "weaker_version": [], "symmetric_bilinear_forms": ["bilinear_forms", "non_degenerate_bilinear_forms", "nondegenerate_bilinear_forms"], "bilinear_forms": ["sesquilinear_forms", "symmetric_bilinear_forms", "nondegenerate_bilinear_forms"], "standard_deviations": ["population_standard_deviation", "standard_deviations_away", "standard_deviation"], "michael_artin": ["miles_reid", "charles_pinter", "michael_atiyah", "nathan_jacobson", "undergraduate_algebraic_geometry", "michael_albanese", "michael_rosen", "paolo_aluffi"], "faithful_functor": ["fully_faithful_functors", "fully_faithful_functor", "forgetful_functor"], "finset": ["chaus", "mathsf_finset", "inset", "cmon"], "affinely_independent": ["affinely_dependent", "lineary_independent", "lineraly_independent"], "smallgroup": ["mathtt_smallgroup", "allsmallgroups", "smallgroup_64"], "anyone_who_knows": ["someone_who_knows", "who_knows", "someone_who_understands"], "frac1n_": ["frac1n_1", "frac12_n", "frac1k_", "frac1nx", "frac1n__frac1", "frac3n", "frac1m_"], "randomly_distributed": ["equally_distributed", "uniformly_distributed", "evenly_distributed"], "importance_sampling": ["gibbs_sampling", "re_sampling", "monte_carlo_sampling"], "transcendence_degree": ["transcendence_bases", "transcendence_degrees", "transcendence_basis"], "having_difficulty": ["having_troubles", "facing_difficulty", "having_trouble", "having_difficulties", "struggling"], "epsilon__delta": ["epsilon_n", "epsilon_d", "epsilon_delta"], "instantaneous_rate": ["instantaneous_rates", "instantaneous_speed", "instantaneous_velocity"], "hamilton_path": ["hamilton_cycle", "hamiltonian_paths", "hamiltonian_path", "hamiltonian_cycle"], "distributive_laws": ["distributivity", "distributive_property", "distributive_law"], "logical_equivalence": ["logical_entailment", "logical_equivalences", "logical_equivalencies"], "\u00b7_\u00b7_\u00b7": ["\u00d7_\u00b7_\u00b7", "\u00b7_\u00b7_\u00b7_\u00b7", "\u00b7_\u00b7"], "ordered_tuples": ["ordered_n_tuple", "2_tuples", "6_tuples"], "mathematically_sound": ["mathematically_minded", "logically_sound", "mathematically_rigorous"], "pretty_cool": ["pretty_darn"], "enables_us": ["allows_us", "enables"], "con_zfc": ["zfc_con_zfc", "con_pa", "con_zf", "con_sf_zfc", "mathsf_zfc", "zfc"], "con_pa": ["neg_con_pa", "arithmetically_sound", "sf_pa", "pra", "con_zfc", "pa_con_pa"], "central_moments": ["raw_moments", "method_of_moments", "moments"], "root_finder": ["root_finders", "root_finding"], "gradient_descent": ["subgradient_descent", "proximal_gradient_descent", "projected_gradient_descent"], "wirtinger_derivatives": ["derivatives", "_derivatives"], "quadratic_residues_modulo": ["non_quadratic_residues", "quadratic_residues", "quadratic_non_residues"], "legendre_symbols": ["jacobi_symbol", "legendre_symbol"], "jacobi_symbol": ["legendre_symbols", "legendre_symbol"], "tight_bounds": ["sharper_bounds", "tighter_bounds", "tight_bound"], "lateral_surface": ["lateral_surface_area", "conical_surface", "curved_surface"], "introductory_texts": ["introductory_courses", "introductory", "introductory_textbooks", "texts"], "gone_wrong": ["went_wrong", "gone_wrong_somewhere"], "leq_1": ["leq_3", "leq_2", "geq_1"], "_1_i": ["_2_i", "1_i", "_1_2i"], "author_introduces": ["introduces"], "fmincon": ["cplex", "fminunc", "quadprog", "matlabs", "cvxpy", "fmin"], "gets_sent": [], "_differentiable": ["once_differentiable", "right_differentiable", "diferentiable"], "fubini_study_metric": ["fubini_study"], "extends_continuously": ["extended_continuously", "extends_holomorphically", "continuously_extendable"], "small_perturbations": ["perturbations", "perturbation", "slight_perturbation"], "tubular_neighbourhood": ["collar_neighborhood", "collar_neighbourhood", "tubular_neighborhood"], "cw_decomposition": ["cellular_decomposition", "ear_decomposition", "_decomposition"], "equally_probable": ["equally_probable_outcomes", "equi_probable", "equiprobable"], "very_ample": ["ample", "very_ample_divisor"], "effective_divisor": ["ample_divisor", "very_ample_divisor", "effective_cartier_divisor"], "written_additively": ["written_multiplicatively"], "3x_3": ["3x_6", "3x_8", "3x_9"], "taylor_development": ["taylor_series_expansion", "taylor_expansion", "taylor_expansions"], "ran_across": ["chanced_upon", "came_across", "came_upon", "stumbled_across", "cam_across", "stumbled_upon"], "zero_sum_game": ["bimatrix_game", "zero_sum_games", "nim_game", "zero_sum"], "euler_maclaurin_formula": ["euler_maclaurin_summation_formula", "euler\u2013maclaurin_formula", "euler_maclaurin"], "spit_out": [], "very_thankful": ["extremely_grateful", "thankful", "greatful", "grateful", "very_grateful"], "predicate_symbol": ["binary_predicate_symbol", "predicate_symbols", "unary_predicate", "unary_predicate_symbol"], "atomic_formulas": ["atomic_sentences", "atomic_formulae", "propositional_formulas"], "indefinite_integration": ["definite_integral", "indefinite_integrals", "indefinite_integral"], "ask_ourselves": [], "varies_continuously": ["vary_continuously"], "profinite_completion": ["profinite", "free_cocompletion", "_completion", "pontryagin_dual"], "equiv0_mod": ["equiv0_bmod5", "equiv0_pmod", "equiv0_bmod3", "pmod", "equiv0_pmod9"], "subextensions": ["subextension", "galois_subextensions", "subfields", "extensions", "galois_extensions"], "balls_numbered": ["tickets_numbered"], "bumped_into": ["chanced_upon", "came_accross", "stumbled_onto", "cam_across", "ran_across"], "per_request": ["upon_request"], "3x_x": ["6x_x", "4x_x", "5x_x"], "weakly_convergent_sequences": ["weakly_convergent", "weak__convergent_subsequence", "weakly_convergent_subsequence"], "old_exam": ["old_prelim", "old_exams", "old_qualifying_exam", "past_exam_paper", "old_qual", "past_exam"], "segment_connecting": ["segment_joining", "line_segment_joining", "straight_line_segment_connecting", "line_segment_connecting"], "educated_guess": ["educated_guesses", "educated_guessing"], "him_her": ["him_herself", "him"], "few_weeks_ago": ["weeks_ago", "couple_days_ago", "sometime_ago", "week_ago", "few_months_ago", "few_days_ago"], "co_countable": ["1st_countable", "cocountable", "co_finite", "co_countable_topology"], "equiv_1": ["equiv1_pmod", "equiv2_pmod", "equiv3_pmod", "equiv4_pmod", "equiv_9", "equiv_7", "equiv_10", "equiv_2"], "marginal_density": ["marginal_distribution", "marginal_pdfs", "marginal_pdf", "joint_pdf", "marginal_densities", "joint_density"], "confuses_me": ["bothers_me"], "card_deck": ["52__card_deck", "playing_cards", "standard_52_card_deck", "52_card_deck", "well_shuffled_deck", "poker_deck"], "times0": [], "studied_maths": ["studied_maths_upto"], "having_trouble_wrapping": ["having_trouble_grasping", "having_trouble_visualizing", "trouble_wrapping", "having_trouble_picturing"], "american_mathematical_society": ["mathematical_society", "bulletin", "american_mathematical_society_vol", "cambridge_philosophical_society", "doi_10", "proceedings", "london_mathematical_society"], "tripping_me_up": ["bothering_me", "stumping_me"], "orbit_stabilizer_theorem": ["orbit_stabiliser_theorem", "orbit_stabilizer", "orbit_stabiliser"], "perfect_cube": ["perfect_square", "perfect_cubes", "non_perfect_square"], "been_tasked": ["been_toying", "ve_been_tasked"], "experimental_data": ["experimental_evidence", "experimental_design", "experimental"], "k_form": ["0_form", "3_form", "1_form"], "landing_heads": ["lands_heads", "landed_heads", "coin_lands_heads"], "righthand_side": ["lefthand_side", "righthandside", "left_hand_side"], "2_2m": ["2_2mn", "4_2m", "2_3m", "3_2m", "3_3m"], "n_i_j": ["n_k_j", "m_j_1", "n_j_k", "k_i_j", "n_i_2", "n_i_1", "n_j_2"], "closed_under_complementation": ["under_countable_unions"], "mayer_vietoris_sequence": ["mayer_vietoris", "mayer\u2013vietoris", "meyer_vietoris"], "ascending_chain": ["descending_chains", "descending_chain", "ascending_chains"], "proof_assistants": ["isabelle_hol", "teaching_assistants", "mechanized", "assistants", "foundational_systems", "isabelle", "theorem_provers"], "alternates_between": [], "th_percentile": ["75th_percentile", "5th_percentile", "95th_percentile"], "non_commutative_ring": ["unital_commutative_ring", "commutative_ring", "non_commutative_rings"], "nontrivial_idempotents": ["central_idempotents", "orthogonal_idempotents", "idempotents"], "dover_books": ["undergraduate_texts", "revised_edition", "universitext", "dover", "dover_edition", "dover_publications"], "revised_version": ["unedited", "slightly_modified_version", "corrected_version", "modified_version"], "willie_wong": ["jonas_meyer", "didier_piau", "najib_idrissi", "raskolnikov", "hans_lundmark", "5xum", "ragib_zaman", "kenny_wong"], "totally_disconnected": ["locally_path_connected", "extremally_disconnected", "hereditarily_disconnected"], "associative_law": ["distributive_laws", "associativity", "distributive_law"], "keeps_track": [], "nonidentity_elements": ["non_identity", "nonidentity", "nonidentity_element"], "begin_array_rcl": ["begin_array_rll", "begin_array_rl", "begin_array_lcc"], "1_2i": ["2_2i", "5_2i", "_1_2i"], "nearest_neighbors": ["nearest_neighbours", "nearest_neighbor", "nearest_neighbour"], "having_trouble_understanding": ["having_trouble_grasping", "having_troubles_understanding", "having_difficulty_understanding"], "fair_die": ["fair_6_sided_die", "six_sided_die", "fair_dice", "six_sided_dice", "6_sided_die", "fair_six_sided_die", "_sided_die"], "de_morgan": ["demorgan", "de_morgans_law", "de_morgans"], "multi_set": ["multi_sets", "multiset"], "lyapunov_stable": ["liapunov_stable", "asymptotically_stable_equilibrium", "asymptotically_stable", "lyapunov_stability", "exponentially_stable"], "cdot2x": ["cdot4x", "1_12x", "cdot3x", "cdot2y", "6x_", "8x_"], "went_wrong": ["gone_wrong"], "divisibility_tests": ["primality_tests", "divisibility", "primality_testing"], "declaremathoperator": ["declaremathoperator_rk"], "hocolim": ["holim", "colim"], "asymptotic_behaviour": ["chaotic_behaviour", "asymptotics", "asymptotic_behavior"], "coupon_collector": ["coupon_collectors"], "continuously_extended": ["continuously_shrunk", "extended_continuously", "continuously_extendable"], "extended_continuously": ["continuously_extended", "continuously_extendable", "deformed_continuously"], "vary_continuously": ["varies_continuously"], "local_maxima": ["minima", "local_maxima_minima", "local_minima_maxima", "local_minima", "local_maximums", "local_extrema", "global_maxima", "extrema"], "teaching_myself": ["self_learning", "self_studying", "currently_self_studying", "doubting_myself", "self_teaching_myself", "teach_myself"], "fx_fy": [], "expected_payoff": ["expected_payout", "payoff", "expected_winnings"], "weeks_ago": ["week_ago", "months_ago", "days_ago", "hours_ago"], "simultaneously_diagonalizable": ["simultaneously_triangularizable", "simultaneously_diagonalisable", "simultaneously_diagonalized", "simultaneously_diagonalize"], "sorted_list": ["sorted_lists"], "optimal_stopping": ["optional_stopping"], "terminal_object": ["_object", "sub_object", "terminal_objects"], "mutually_independent": ["identically_distributed", "statistically_independent", "conditionally_independent", "independent", "lineraly_independent"], "bd_cd": ["abd_acd", "ad_bd_cd", "bd_ac"], "equiv1_mod": ["equiv1_pmod", "equiv1_mod3", "equiv0_pmod", "equiv1_bmod", "equiv3_pmod", "pmod"], "synthetic_division": ["long_division"], "lbrace1": ["lbrace2", "lbrace0", "lbrace"], "ah_bh": ["ah_bh_ch", "ahbh", "ha_hb", "ah_hb"], "limited_experience": ["inexperience", "experience", "limited_knowledge"], "ross_millikan": ["rossmillikan", "jmoravitz", "ross_milikan", "hagen_von_eitzen", "hardmath", "andr\u00e9_nicolas", "lulu"], "working_backwards": [], "2_2xy": ["2_3xy", "4_2xy", "2xy", "2_2xy_y", "3_2xy", "4xy", "2_3y", "2_4xy"], "ive_been": ["ve_been_scratching", "ve_been_mulling", "ve_been_toying"], "regular_pentagon": ["regular_heptagon", "regular_pentagons", "irregular_pentagon", "regular_hexagon", "pentagon"], "duality_gap": ["strong_duality_holds"], "strong_duality": ["fenchel_duality", "strong_duality_holds"], "teacher_gave": ["lecturer_gave", "professor_gave", "teacher_gave_us", "professor_gave_us"], "leonbloy": ["user17762", "didier_piau", "raymond_manzoni", "angryavian", "paramanand_singh", "user236182", "eric_naslund", "user1337"], "_vectorspace": ["_vectorspaces", "vectorspace", "subvectorspace"], "longest_path": ["shortest_paths", "shortest_path"], "hours_ago": ["months_ago", "weeks_ago", "days_ago", "minutes_ago"], "without_worrying_about": ["without_worrying", "worrying_about"], "vertical_strip": ["vertical_strips", "horizontal_strip", "horizontal_strips"], "moebius_band": ["mobius_band", "m\u00f6bius_band", "m\u00f6bius_strip", "mobius_bands"], "under_countable_unions": ["countable_unions", "closed_under_complementation", "unions"], "structural_induction": ["_induction", "non_induction", "coinduction"], "unions_intersections": ["unions", "pairwise_intersections", "intersections"], "unary_operation": ["binary_operation", "nullary_operation", "_ary_operation"], "lebesgue_differentiation": ["term_wise_differentiation", "lebesgue\u2019s", "lebesgue_dominated_convergence"], "spherical_coordinate": ["cylindrical_coordinate", "spherical_coordinates", "polar_coordinate", "cylindrical_coordinates", "hyperspherical_coordinates", "cartesian_coordinates"], "infinite_dimensional_banach": ["infinite_dimensional_banach_spaces", "infinite_dimensional_normed", "infinite_dimensional_hilbert"], "inquality": ["inequalty", "inequality"], "faithful_representation": ["_representation", "v_representation", "sub_representation"], "software_packages": ["libraries", "softwares", "software_package", "software_libraries"], "large_deviations": ["absolute_deviations", "deviations"], "having_trouble_figuring_out": ["trouble_figuring_out", "having_trouble_figuring", "having_difficulty_figuring_out"], "berry_esseen": ["de_moivre_laplace"], "associative_unital": ["unital_associative_algebra", "k__algebra", "commutative_unital", "unital_associative"], "license_plates": ["license_plate"], "remains_unchanged": ["remain_unchanged", "unchanged", "stays_unchanged"], "elementary_topos": ["pretopos", "grothendieck_topos"], "2yx": ["2_yx", "3yx", "4yx", "2_zx"], "1_\u03bb": ["2_\u03bb"], "pairwise_independence": ["non_independence", "independence", "uncorrelatedness", "mutual_independence"], "regular_hexagon": ["hexagon", "regular_pentagon", "regular_octagon", "regular_tetrahedron", "dodecagon", "regular_heptagon", "irregular_hexagon", "regular_hexagons"], "math_contest": ["math_competitions", "programming_competition", "competitive_exam", "math_olympiad", "math_competition", "programming_contest", "math_contests"], "monotone_convergence": ["mct", "lebesgue_dominated_convergence", "dominated_convergence", "monotone_convergence_theorem", "dominated_convergence_theorem"], "dominated_convergence": ["dominated_convergence_theorem", "monotone_convergence_theorem", "monotone_convergence", "lebesgue_dominated_convergence"], "integrating_factor": [], "fat_cantor": ["middle_thirds_cantor", "ternary_cantor", "cantor_type"], "minimizing_geodesic": ["geodesic", "_geodesic", "minimizing_geodesics"], "hopf_rinow_theorem": ["hopf_rinow"], "da_db": ["da_db_dc"], "input_signal": ["audio_signal", "noisy_signal", "signal"], "player_wins": ["gambler_wins", "player_loses", "wins"], "lecture_slides": ["lecture_slide", "lecture_videos", "lecture_notes"], "dummy_variable": ["dummy_variables"], "annual_rate": ["compounded_monthly", "effective_interest_rate", "effective_annual_rate", "nominal_rate", "annual_interest_rate", "annual_effective_rate"], "branched_cover": ["branched_covering", "branched_coverings", "branched_covers"], "nonlinear_ode": ["nonlinear_ordinary_differential", "quasilinear_pde", "nonlinear_pdes", "nonlinear_odes"], "electric_potential": ["electric_flux", "electrostatic_potential", "electric_charge"], "whitney_embedding_theorem": ["nash_embedding", "kodaira_embedding"], "negligible_compared": ["negligible"], "3cos": ["_3cos", "4cos", "6cos", "_2cos", "5sin", "4sin"], "ux_vy": ["ux_vx", "ux_vy_wz"], "local_diffeomorphism": ["submersion", "diffeomorphism", "local_diffeomorphisms", "self_diffeomorphism"], "3x_5": ["3x_6", "2x_5", "3x_7"], "branching_process": ["galton_watson_process", "branching_processes", "galton_watson_branching_process"], "highly_appreciated": ["appreciated_thanks", "appreciated", "deeply_appreciated", "much_appreciated", "welcomed", "hugely_appreciated", "greatly_appreciated"], "without_affecting": ["without_altering"], "few_months": ["few_weeks", "past_few_months", "few_months_ago"], "bothering_me": ["been_bothering_me", "stumping_me", "bugging_me"], "co_finite": ["\u03c3_finite", "co_countable", "cocountable", "cofinite", "pro_finite"], "taxicab_metric": ["manhattan_metric", "taxi_cab_metric", "taxicab_norm"], "x3_y3": ["x1_y1", "x1_y1_x2_y2", "x1_y1_z1", "x3_x3_y3_y3", "x3_y3_z3", "x2_y2", "x3_y1", "x4_y4"], "pow_x1": ["pow_x2", "x3_pow", "pow_x3", "x1_pow_x3", "d1_pow"], "pow_x3": ["pow_x2", "x3_pow", "pow_x1", "x1_pow_x3", "d1_pow"], "pow_x2": ["x3_pow", "pow_x3", "pow_x1", "x1_pow_x3", "d1_pow"], "egyptian_fraction": ["egyptian_fractions"], "hardmath": ["rossmillikan", "erick_wong", "user21820", "raskolnikov", "peter_ko\u0161in\u00e1r", "eric_naslund", "hans_lundmark", "mixedmath"], "injectiveness": ["bijectiveness", "surjectiveness", "injectivity_surjectivity"], "currently_reading": ["currently_self_studying", "currently_revising", "currently_studying", "reading"], "boundary_conditions": ["robin_boundary_condition", "initial_conditions", "robin_boundary_conditions"], "preserves_colimits": ["preserve_colimits", "preserves_coproducts", "preserves_filtered_colimits", "cocontinuous"], "nested_radical": ["infinitely_nested_radical", "nested_radicals"], "nicely_behaved": ["poorly_behaved", "well_behaved", "badly_behaved"], "reference_frame": ["inertial_reference_frame", "body_frame", "inertial_frame"], "reduced_homology": ["homology", "zeroth_homology", "unreduced_homology", "singular_homology", "k_homology"], "deformation_retracts_onto": ["retracts_onto", "deformation_retract", "deformation_retracts"], "stone_cech_compactification": ["stone\u2013\u010dech_compactification", "\u010dech_stone_compactification", "stone_\u010dech_compactification", "cech_stone_compactification"], "maximal_atlas": ["smooth_atlas", "_atlas"], "tend_towards": ["tend_toward"], "radian_measure": [], "trignometric": ["goniometric", "trignometry", "trigonometric", "trigonmetric", "trigonometrical"], "pythagorean_triplets": ["pythagorean_triplet", "pythagorean_triple", "pythagorean_triples"], "outer_measure": ["lesbesgue_measure", "premeasure", "lebesgue_outer_measure"], "heine_borel_theorem": ["bolzano_weierstrass_theorem", "arzel\u00e0\u2013ascoli_theorem", "heine_borel"], "oksendal": ["bernt_oksendal", "\u00f8ksendal", "protter"], "expected_payout": ["expected_payoff", "expected_earnings", "expected_winnings"], "non_finitely_generated": ["finitely_generated", "non_finitely_generated_ideals", "finitely_generated_torsion_free", "non_noetherian_ring", "non_artinian"], "separating_hyperplane": ["_hyperplane", "supporting_hyperplanes", "supporting_hyperplane"], "edges_incident": ["incident", "edges"], "without_replacement": ["draws_without_replacement", "urn_without_replacement", "drawn_without_replacement"], "ternary_expansion": ["ternary_expansions", "binary_expansions", "decimal_expansion"], "lebesgue_stieltjes": ["lebesgue_stieltjes_measure", "lebesgue\u2013stieltjes", "riemann\u2013stieltjes"], "rising_factorial": ["rising_factorials", "pochhammer_symbol", "falling_factorial", "falling_factorials"], "truth_tables": ["truth_table", "logical_equivalencies"], "messing_around": ["mucking_around", "experimenting", "toying_around", "playing_around"], "author_claims": [], "introductory_functional_analysis": ["erwin_kreyszig", "brezis_functional_analysis", "kreyszig", "microlocal_analysis", "erwine_kreyszig", "functional_analysis"], "munkres_topology": ["munkres_topology_2nd_edition", "munkre", "munkres", "munkres_topology_2nd_ed"], "undergrad_level": ["undergrad", "beginning_graduate_level", "advanced_undergraduate", "beginning_graduate", "undergraduate_level", "advanced_undergraduate_level", "upper_division"], "someone_who_knows": ["anyone_who_knows", "who_knows", "someone_who_understands"], "richardson_extrapolation": ["richardson", "extrapolation"], "color_90f": ["color_0af", "color_c0f", "color_c00", "color_0a0"], "parabolic_cylinder": ["hyperbolic_paraboloid", "half_cylinder", "elliptical_cylinder"], "online_calculator": ["online_graphing_calculator", "ti_83_calculator", "online_calculators", "wolfram"], "error_tolerance": ["tolerance", "tolerant", "tolerances"], "half_open_intervals": ["half_open_interval", "half_open", "intervals"], "rough_sketch": ["brief_sketch", "rough_outline", "sketch"], "3_4i": ["2_4i", "_3_4i", "4_4i"], "1_x_y": ["2_x_y", "1_x_z", "1_x_y_z"], "gotten_anywhere": ["gotten_nowhere"], "dvol": [], "_9a": ["_12a", "2_9a", "34a", "_8a", "_9b", "_7a"], "bi_cj_dk": ["a_bi_cj_dk", "bi_cj"], "logical_consequence": ["logical_consequences", "semantic_consequence", "tautological_consequence"], "angle_cbd": ["angle_cbf", "angle_dca", "angle_dbc", "angle_bdc", "angle_cbm", "angle_dcb", "angle_aeb", "angle_cbp"], "angle_bdc": ["angle_dca", "angle_dbc", "angle_cbd", "angle_bmc", "angle_eab", "angle_bda", "angle_bfe", "angle_adb"], "wrapping_around": ["poking_around", "mucking_around", "toying_around"], "lebesgue_dominated_convergence": ["lebesgue_differentiation", "ldct", "dominated_convergence", "monotone_convergence_theorem", "dominated_convergence_theorem", "lebesgue_dominated"], "joint_density": ["joint_pdf", "joint_distribution", "marginal_density"], "biggm": [], "begin_aligned": ["begin_alignat", "begin_align"], "color_purple": ["color_pink", "color_orange", "color_olive"], "cdot3_cdot5": ["cdot3_cdot5_cdot7", "cdot3_cdot3", "cdot4_cdot6", "cdot3_cdot7", "cdot5_cdot7", "cdot2_cdot3", "cdot3_cdot4"], "cdot2_cdot3": ["cdot2_cdot1", "cdot3_cdot4", "cdot2_cdot2_cdot2", "cdot2_cdot3_cdot4_cdot5", "cdot4_cdot6", "cdot3_cdot5", "cdot2_cdot4"], "cdot4_cdot6": ["cdot4_cdot3", "cdot4_cdot6_cdots2n", "cdot4_cdot6_cdot8", "cdot2n", "cdot4_cdot5", "cdot6_cdot9", "cdot2_cdot3_cdot4_cdot5", "cdot4_cdot4"], "parametricplot": ["parametricplot3d"], "workingprecision": [], "line_segment_joining": ["segment_joining", "segment_connecting", "line_segment_connecting", "line_segment"], "geometrically_speaking": ["physically_speaking", "geometrically", "technically_speaking"], "bothers_me": ["annoys_me", "confuses_me"], "short_exact_sequences": ["non_split_short_exact", "short_exact", "short_exact_sequence"], "cayley_table": ["cayley_tables"], "_sided_dice": ["six_sided_dice", "sided_die", "sided_dice", "dices", "20_sided_dice", "10_sided_dice", "6_sided_dice", "_sided_die"], "1_nx": ["2_nx", "n_nx", "x_nx"], "upper_semicircle": ["semi_circle", "semicircle", "upper_semi_circle"], "absolutely_summable": ["_summable", "absolutely_convergent", "summable", "square_summable"], "aluffi": ["paolo_aluffi", "hungerford", "herstein"], "u_sub": ["u_substitution", "trig_sub"], "wiener_process": ["brownian_motion", "weiner_process", "wiener_processes"], "most_probable": [], "derivating": ["derivatve", "derivaties", "differentiating"], "numerical_recipes": ["seminumerical_algorithms"], "von_neumann_hierarchy": ["constructible_hierarchy", "cumulative_hierarchy", "_hierarchy", "von_neumann_universe"], "very_grateful": ["thankful", "greatful", "truly_grateful", "grateful", "glad", "very_thankful"], "grobner_basis": ["grobner_bases", "groebner_basis", "gr\u00f6bner_basis"], "toplogy": ["topology", "topolgy"], "disjoint_unions": ["unions", "union", "disjoint_union"], "native_english_speaker": ["native_speaker", "native_english", "english_speaker", "professional_mathematician"], "genetic_algorithm": ["cordic_algorithm", "simulated_annealing", "viterbi_algorithm", "genetic_algorithms"], "math_stack_exchange": ["mathstackexchange", "forums", "math_stackexchange", "stack_exchange", "physics_stack_exchange", "stackexchange", "mathematics_stack_exchange"], "few_months_ago": ["months_ago", "couple_days_ago", "week_ago", "few_weeks_ago", "few_days_ago", "few_years_ago"], "pi_2x": ["cos2x_1", "pi_2y", "cos2x_", "cos2x_sin2x", "pi_3x", "cos2x_cos3x", "pi_2"], "nilpotents": ["nilpotent"], "multiplicatively_closed": [], "fuchsian_group": ["fuchsian_groups", "fuchsian"], "having_trouble_visualizing": ["having_trouble_grasping", "having_trouble_formalizing", "having_trouble_picturing"], "begin_array_ccc": ["begin_array_ccc_ccc", "begin_array_ccl", "begin_array_cc", "begin_array_cccc"], "riemannian_curvature": ["gaussian_curvature", "sectional_curvature", "ricci_tensor", "curvature_tensor", "ricci_curvature", "ricci_scalar"], "caracteristic": ["charcteristic", "chracteristic", "characteristic"], "hh_ht_th": ["ttt_tth", "hh_tt", "hh_ht", "hhh_hht_hth_htt", "ht_th"], "math_competitions": ["competitions", "maths_competition", "math_contests", "software_engineering", "math_competition", "math_olympiads", "olympiads"], "projecting_onto": ["projecting"], "pa_qb": ["qa_qb", "a_qb"], "reminds_us": ["reminds", "reminds_me"], "professor_gave_us": ["teacher_gave", "teacher_gave_us", "lecturer_gave_us", "professor_gave"], "riemann_stieltjes_integrable": ["riemann_stieltjes", "riemann_stieltjes_integral", "riemann_stieltjes_integrals"], "\u2227_\u00acp": ["\u2227_\u00acq", "\u2228_\u00acr", "\u00ac_\u00acp", "\u2227_\u00acr", "\u2227_\u00acp_\u2228", "\u2192_\u00acp"], "\u00acr": ["\u00acp_\u00acq", "\u2227_\u00acq", "\u2228_\u00acp", "\u00acr_\u2228", "\u00acq", "\u2227_\u00acp", "\u2192_\u00acq"], "sa_tb": ["a_tb"], "randomreal": ["randominteger"], "winds_around": ["wraps_around", "encircles", "winds"], "binary_expansions": ["ternary_expansions", "ternary_expansion", "decimal_expansions"], "undetermined_coefficients": ["undetermined_coefficient"], "basel_problem": [], "geometric_multiplicities": ["geometric_multiplicity", "algebraic_multiplicity"], "2_ac": ["2_ab_bc_ac", "2_bc", "2_ab_ac_bc", "2_c"], "marginal_pdf": ["marginal_pmfs", "marginal_density", "marginal_distribution", "marginal_pdfs", "marginal_pmf", "joint_pdf", "marginal_densities"], "marginal_pdfs": ["marginal_pmfs", "marginal_pmf", "marginal_pdf", "joint_pdf", "marginal_densities"], "paper_titled": ["titled"], "_metric": ["sup_metric", "semi_metric", "pseudo_metric"], "tower_law": ["power_law"], "global_extrema": ["relative_extrema", "global_extremum", "absolute_extrema", "local_extrema", "global_maxima", "extremas"], "splitting_fields": ["galois_subextensions", "splitting_field"], "purely_inseparable": ["recursively_inseparable", "purely_inseparable_extension", "inseparable"], "oriented_counterclockwise": ["oriented_anticlockwise", "oriented_counter_clockwise", "oriented_clockwise", "traversed_counterclockwise"], "completely_lost": ["totally_lost", "completely_clueless", "completely_stumped", "kinda_lost", "kinda_stuck"], "polinomial": ["polynomial", "poylnomial", "poynomial", "polnomial"], "coordinate_chart": ["coordinate_patches", "coordinate_charts", "chart", "coordinate_patch"], "monotone_increasing": ["monotonic_increasing", "monotonely_increasing", "monotonic", "monotonically_decreasing", "monotonically_increasing", "strictly_monotone_increasing", "monotone_decreasing"], "_adic_completion": ["_adic_completions"], "independent_increments": ["stationary_increments"], "phase_portrait": ["phase_portraits", "portrait"], "stable_equilibrium": ["asymptotically_stable_equilibrium", "unstable_equilibria", "unstable_equilibrium"], "phase_portraits": ["phase_portrait"], "nullclines": ["nullcline", "clines", "null_clines"], "rigorous_treatment": ["thorough_treatment", "modern_treatment", "modern_treatments"], "fair_dice": ["fair_6_sided_dice", "fair_six_sided_dice", "six_sided_die", "six_sided_dice", "dice", "fair_die", "6_sided_dice"], "sided_die": ["six_sided_die", "20_sided_die", "sided_dice", "8_sided_die", "6_sided_die", "4_sided_die", "_sided_die"], "_sided_die": ["six_sided_die", "10_sided_die", "sided_die", "sided_dice", "8_sided_die", "_sided_dice", "fair_die", "4_sided_die"], "preceding_paragraph": ["previous_paragraph", "preceding_paragraphs"], "qualifying_exam": ["qual", "midterm", "past_qualifying_exam", "prelim", "old_qualifying_exam", "quals", "qualifying_exams"], "faculty_member": ["faculty", "faculty_members"], "ia_ja": ["j_a_jb", "ia_jb", "ja_jb"], "relatively_straightforward": ["relatively_straight_forward", "fairly_straightforward", "fairly_straight_forward"], "rooted_trees": ["binary_trees", "rooted_labeled_trees", "labeled_trees", "unrooted_trees"], "references_therein": ["references"], "real_analysis": ["functional_analysis", "complex_analysis", "microlocal_analysis"], "archimedean_principle": ["archimedean_property", "archimedian_principle", "archimedian_property"], "positive_orthant": ["non_negative_orthant", "nonnegative_orthant"], "extremely_unlikely": ["extremely_rare", "unlikely", "highly_unlikely"], "grow_faster_than": ["grow_faster", "grow_slower_than", "grows_faster_than", "growing_faster_than"], "directional_derivatives": ["directional_derivate", "distributional_derivatives", "directional_derivative"], "hodge_star": ["hodge_duality", "hodge_star_operator", "hodge_dual"], "cellular_approximation": [], "associative_binary_operation": ["binary_operation", "unary_operation", "nullary_operation"], "few_weeks": ["few_weeks_ago", "few_months", "past_few_weeks", "few_days"], "black_balls": ["blue_balls", "white_marbles", "blue_marbles", "white_balls", "red_balls"], "white_balls": ["blue_balls", "white_marbles", "blue_marbles", "black_balls", "red_balls"], "ldct": ["dct", "dominated_convergence", "beppo_levi", "lebesgue_dominated_convergence"], "triangle_pqr": ["triangle_pbc", "triangle_oab", "triangle_pab", "triangle_abc", "triangle_opq"], "online_tool": ["online_calculators", "online_tutorials", "online_calculator"], "polarization_identity": ["polarization", "polarisation_identity"], "lowerbound": ["lower_bound", "higherbound", "upperbound"], "recently_came_across": ["came_across", "recently_stumbled_upon", "came_upon", "recently_stumbled_across", "recently_come_across", "stumbled_across", "recently_encountered"], "comparison_tests": ["comparsion_test", "comparison_test", "comparision_test"], "primary_decomposition": ["primary_decompositions", "minimal_primary_decomposition"], "mild_conditions": ["wolfe_conditions", "conditions", "pre_condition", "mild_assumptions"], "quick_google_search": ["google_search", "quick_search", "quick_google", "google_searches"], "quasi_projective_varieties": ["projective_varieties", "affine_varieties", "quasiprojective_varieties", "quasi_projective_variety"], "ac_ad_bc": ["di_ac_bd_ad_bc", "ac_2bd_ad_bc", "ac_bd_ad_bc"], "dive_into": ["delve_into"], "remain_unchanged": ["unchanged", "remains_unchanged", "stays_unchanged"], "2_ax": ["2_ax_1", "2_ax_b", "3_ax"], "mu__almost_everywhere": ["_almost_everywhere", "almost_everywhere"], "2b_a": ["2b_b", "2b_ab", "2b_2a", "2c_2abc"], "lagrangian_multipliers": ["lagrange_multipliers", "lagrangian_multiplier", "langrange_multipliers"], "_varieties": ["varieties", "algebraic_varieties", "prevarieties"], "chernoff_bounds": ["chernoff", "chernoff_bound", "chernoff_hoeffding"], "markov_processes": ["markov_chains", "diffusions_markov_processes", "markov_process", "stochastic_processes"], "markov_process": ["markov_chain", "discrete_time_stochastic_process", "time_homogeneous_markov_process", "stochastic_process", "markov_processes", "continuous_time_markov_chain"], "randomly_select": ["randomly_selects", "randomly_selected", "randomly_selecting"], "memoryless_property": ["memorylessness", "memorylessness_property", "memoryless"], "kunneth": ["kunneth_formula", "k\u00fcnneth_formula", "k\u00fcnneth"], "meters_per_second": ["metres_per_second", "feet_per_second", "meters_per"], "angle_bca": ["angle_dbc", "angle_bpd", "angle_dba", "angle_bmc", "angle_dab", "angle_cba", "angle_bfe", "angle_pab"], "angle_cba": ["angle_eac", "angle_cbf", "angle_cbe", "angle_dba", "angle_cbm", "angle_obc", "angle_abo", "angle_dae"], "angle_abd": ["angle_dbc", "angle_cbd", "angle_bdc", "angle_afb", "angle_acd", "angle_abo", "angle_cad", "angle_abp"], "angle_adb": ["angle_adc", "angle_abd", "angle_bdc", "angle_afb", "angle_dab", "angle_aeb", "angle_dac"], "angle_cdb": ["angle_dce", "angle_dbc", "angle_dba", "angle_cbm", "angle_ebc", "angle_cbp", "angle_cda", "angle_dea"], "similar_vein": [], "_valued_random_variable": ["_distributed_random_variable", "iid_random_variables", "random_varible"], "finer_than": ["coarser_than", "strictly_finer_than"], "riemannian_connection": ["chern_connection", "riemannian", "riemannian_metric", "levi_civita_connection"], "frac78": ["frac74", "frac79", "frac28", "frac75", "frac76", "frac58"], "legal_move": ["legal_moves"], "legal_moves": ["pawn_moves", "legal_move"], "strongly_suspect": ["strongly_suggests"], "indexed_family": ["indexed_families", "_indexed_family", "_indexed_families"], "strong_induction": ["induction", "_induction", "non_induction"], "rdrd_theta": ["rdrd", "rdr", "rdzdrd", "da_rdrd_theta"], "grows_slower_than": ["grows_slower", "grow_slower_than", "grows_faster_than"], "had_hoped": ["hoped"], "calc_ii": ["calc_iii"], "ay_bz": ["cy_bz", "ay_bx"], "gh_hg": ["ghg", "hg_gh", "gn_ng", "hg_hg", "gn_hn", "ga_ag"], "_alg": ["_vect"], "pm1_pm2": ["pm1_pm3", "pm1_pm2_pm4", "pm1_pm2_pm3"], "stokes_theorem": ["generalized_stokes_theorem", "stokes", "stokes\u2019", "stoke"], "depressed_cubic": ["depressed_quartic"], "andrew_ng": ["andrew_granville", "andrew_pressley", "andrew"], "opposite_signs": [], "distributional_derivative": ["directional_derivate", "distributional_derivatives", "directional_derivative"], "tempered_distribution": ["tempered_distributions"], "calc_iii": ["multivariable_calc", "calc_ii"], "sheldon_ross": ["kai_lai_chung", "allan_gut", "sheldon"], "noetherian_schemes": ["_schemes", "locally_noetherian_scheme", "locally_noetherian_schemes", "noetherian_scheme"], "n_r_1": ["n_r_s", "m_r_1", "k_r_1"], "circular_table": [], "ultimate_goal": ["ultimate_aim"], "john_conway": ["conway", "david_bressoud", "barry_mazur", "paul_erdos", "john_horton_conway", "john_kelley", "schramm", "lars_ahlfors"], "heat_equation": ["1d_wave_equation", "burgers_equation", "heat_diffusion"], "2z_x": ["2z_2xyz", "2z_y", "2_xyz", "3_yz", "3_xz", "2_4xz", "2_zx"], "quadratic_reciprocity_law": ["reciprocity_law", "quadratic_reciprocity", "cubic_reciprocity"], "equiv3": ["equiv4", "pmod5", "equiv32", "equiv36", "equiv7", "equiv5", "equiv2", "equiv39"], "qquad_": ["20pt", "qquad1", "cdots_1", "hspace_2em", "large1", "tfrac94", "qquad2", "quad_"], "hospitals_rule": ["hopitals", "h\u00f4pitals_rule", "hopitals_rule", "hospital_rule", "hopital"], "made_arbitrarily_small": ["arbitrarily_small"], "squeeze_theorem": ["stolz\u2013ces\u00e0ro_theorem", "stolz_ces\u00e0ro_theorem", "squeeze_sandwich"], "4k_2": ["4k_1", "4k_4", "4k_3"], "key_ingredient": ["crucial_ingredient", "ingredient", "main_ingredient"], "head_around": ["my_head_around"], "hspace_10mm": ["hspace_10px", "hspace_12pt", "hspace_15mm"], "xa_ya": ["xa_xb", "xb_ya", "xa_yb", "xb_yb"], "worth_pointing_out": ["worth_remarking", "worth_emphasizing", "worth_mentioning", "worth_noting"], "strong_evidence": ["insufficient_evidence", "evidence", "empirical_evidence"], "gone_wrong_somewhere": ["gone_wrong", "mistake_somewhere", "messed_up_somewhere"], "grows_faster_than": ["grows_faster", "grow_faster_than", "grows_slower_than"], "ha_hb": ["ah_bh", "ha_ha", "ah_hb"], "quasi_isomorphisms": ["quasi_isomorphism", "weak_equivalences", "quasi_isomorphic", "_isomorphisms"], "weak_equivalences": ["quillen_equivalence", "quasi_isomorphisms", "autoequivalences"], "went_ahead": ["went_awry"], "distributive_property": ["distributivity", "distributive_laws", "distributive_law"], "completely_unrelated": ["unrelated", "seemingly_unrelated", "totally_unrelated"], "raise_both_sides": ["raising_both_sides", "exponentiate_both_sides", "cubing_both_sides"], "1_dim": [], "pure_mathematician": ["mathematician", "practitioner", "theoretical_physicist", "professional_mathematician", "hobbyist", "non_mathematician"], "logic_behind": ["reasoning_behind", "hiding_behind", "rationale_behind"], "gets_arbitrarily_close": ["arbitrarily_closely", "arbitrarily_close"], "subsequence_converging": ["subsequences_converging", "convergent_sub_sequence", "convergent_subsequence"], "stein_shakarchi": ["folland", "royden_fitzpatrick", "royden", "stein", "shakarchi", "carother", "rami_shakarchi"], "local_martingale": ["_local_martingale", "local_martingales", "martingale", "ui_martingale", "semimartingale"], "strictly_concave": ["strictly_quasi_concave", "strictly_convex", "concave"], "optional_stopping_theorem": ["optional_sampling_theorem", "optional_stopping"], "_4p": ["3_4p", "27q", "2_4p", "4_2p", "3p_4p"], "wallis_product": [], "generalized_continuum_hypothesis": ["continuum_hypothesis_ch", "sf_gch", "sf_ch", "mathsf_gch", "continuum_hypothesis"], "lt0": [], "everywhere_except": ["everywhere_else"], "logical_equivalences": ["logical_deductions", "logical_equivalence", "logical_equivalencies"], "skyscraper_sheaf": ["skyscraper_sheaves", "_sheaf", "skyscraper"], "removable_discontinuities": ["discontinuities", "removable_discontinuity", "jump_discontinuities"], "removable_singularities": ["non_removable_singularity", "removable_singularity", "essential_singularities"], "pythagorean_triplet": ["pythagorean_triples", "pythagorean_triple", "pythagorean_triplets"], "jump_discontinuity": ["removable_discontinuity", "discontinuity", "jump_discontinuities"], "gibbs_phenomenon": ["phenomenon", "runge_phenomenon", "phenomenom"], "greatest_common": ["greatest_common_divisor", "greatest_common_divisors"], "distributes_over": ["multiplication_distributes_over", "distributes_over_addition"], "operatorname_coker": ["operatorname_coim", "coker"], "rooted_tree": ["rooted_binary_tree", "rooted_trees", "unrooted_trees"], "brief_introduction": ["\u201cintroduction", "gentle_introduction", "concise_introduction"], "logically_equivalent": ["birationally_equivalent", "logically_entails", "topologically_equivalent"], "alexander_duality": ["poincare_duality", "poincar\u00e9_duality", "serre_duality", "lefschetz_duality"], "12_k": ["20_k", "13_k", "16_k", "11_k", "8_2k"], "infinite_dimensional_banach_spaces": ["infinite_dimensional_normed", "infinite_dimensional", "infinite_dimensional_banach"], "frac2n": ["frac1n_frac2n", "frac1n_", "frac3n"], "hamiltonian_paths": ["hamiltonian_cycles", "perfect_matchings", "hamiltonian_cycle", "hamiltonian_path"], "atiyah_macdonald": ["macdonald", "hungerford", "atiyah", "atiyah_mcdonald", "matsumura", "eisenbud"], "finitely_generated_torsion_free": ["finitely_generated", "left_module", "non_finitely_generated", "finitely_generated_module"], "canonical_projection": ["_projection"], "strictly_weaker": [], "assures_us": ["assures"], "harmonic_motion": ["damped_harmonic_oscillator", "shm", "harmonic_oscillators", "harmonic_oscillator"], "allow_ourselves": ["confine_ourselves", "concern_ourselves", "remind_ourselves"], "indeterminate_forms": ["indeterminate", "indeterminations"], "free_resolution": ["free_resolutions", "projective_resolution", "_resolution"], "scientific_calculator": ["scientific_calculators", "ti_83_calculator", "graphic_calculator", "calculator"], "xz__plane": ["xz_plane", "xy_plane", "yz__plane", "xy__plane", "yz_plane"], "yz__plane": ["xz_plane", "xy_plane", "xz__plane", "xy__plane", "yz_plane"], "biject": ["bijection", "bijected", "bijects"], "kx_ky": ["x_ky", "hx_ky", "kx_ky_kz"], "quick_search": ["quick_web_search", "google_search", "quick_google_search", "web_search", "quick_google"], "strictly_finer_than": ["finer_than", "strictly_finer", "strictly_coarser", "coarser_than"], "serre_fibration": ["_fibration", "serre_fibrations", "hurewicz_fibration"], "homotopy_lifting_property": ["homotopy_lifting", "lifting_criterion", "homotopy_invariance", "path_lifting"], "mapsto1": ["mapsto1_x", "mapsto7", "mapsto0"], "sufficiently_fast": ["sufficiently_rapidly"], "rough_estimate": ["crude_estimate", "over_estimate", "crude_estimates"], "cos2a": ["tan2a", "cos2b", "cos3a", "sin2a", "sin3a", "cos2c"], "m_test": ["_test", "weierstrass_m_test", "p_test", "weirstrass_m_test"], "hoping_someone_could": ["hoping_someone", "was_hoping_someone"], "infinitely_thin": ["infinitesimally_thin"], "borel_sigma__field": ["borel_sigma__algebra", "borel__sigma__algebra", "borel_sigma_algebra"], "countably_generated": ["singly_generated", "separably_generated", "non_finitely_generated"], "sigma__field_generated": ["sigma__fields", "sigma__field", "sub__sigma__field"], "sigma__field": ["sigma__algebra", "sigma__fields", "tail_sigma__field", "sigma_field", "sigma_algebra", "sub__sigma__field"], "laurent_polynomials": [], "3_6x": ["3_3x", "4_6x", "3_5x", "4_4x", "3_9x", "3_8x", "6_6x"], "absolute_value_bars": ["absolute_value_signs"], "acute_angles": ["acute_angled", "obtuse_angles", "acute_angle"], "upcoming_exam": ["exam_tomorrow", "prepping", "qualifying_exam", "upcoming_midterm", "upcoming_test", "exam_preparation", "qualifying_exams"], "irrationality_measure": [], "maximum_likelihood_estimate": ["mle", "maximum_likelihood_estimation", "mle_estimator", "maximum_likelihood_estimator", "maximum_likelihood_estimates"], "laurent_series_expansion": ["laurent_series", "laurent_expansion", "laurent_serie"], "spherical_caps": ["spheric", "hemispherical", "spherical"], "bit_sloppy": ["sloppy"], "comprehensive_introduction": ["friendly_introduction", "gentle_introduction", "concise_introduction"], "sigma__compact": ["sigma_compact", "\u03c3_compact", "countably_compact"], "physical_sciences": ["sciences", "social_sciences", "riley_hobson", "mathematical_sciences"], "m\u00f6bius_band": ["m\u00f6bius_bands", "mobius_strip", "moebius_band", "mobius_band", "m\u00f6bius_strip", "mobius_bands", "moebius_strip"], "statistical_learning": ["statistical_inference", "hastie", "deep_learning", "machine_learning"], "growth_rates": ["asymptotic_growth_rate", "growth_rate", "growths"], "research_project": ["research", "research_level", "researchgate"], "abstract_nonsense": [], "modified_version": ["revised_version", "slightly_modified_version", "corrected_version"], "_linearly_independent": ["linearly_independant", "linearly_independet", "linearly_independent"], "_coefficient": ["integer_coefficient", "_coefficients"], "xyxy": ["xyx", "yxy", "xyxyx", "yxyx", "xyxyxy", "xyx_yxy"], "cyclotomic_extension": ["cyclotomic_extensions", "cyclotomic", "cyclotomic_fields"], "biggl_biggl": ["biggr_biggl", "biggl", "biggr_biggr"], "regular_tetrahedron": ["regular_octahedron", "irregular_tetrahedron", "trirectangular_tetrahedron", "tetrahedron", "regular_hexagon"], "dashed_lines": ["dashed", "dotted_lines", "dashed_line"], "ex_ey": [], "cardinal_exponentiation": ["ordinal_exponentiation"], "highly_appreciate": ["apreciate", "very_appreciative", "appriciate", "extremely_grateful", "greatful", "greatly_appreciate", "highly_appreciated", "sincerely_appreciate"], "hospital_rule": ["hospitals_rule", "h\u00f4pitals_rule", "hopitals_rule", "hospital", "hopital", "h\u00f4pital"], "induces_isomorphisms": ["reflects_isomorphisms", "induce_isomorphisms", "_isomorphisms"], "weak_homotopy_equivalence": ["homotopy_equivalences", "weak_homotopy_equivalences", "homotopy_equivalence"], "professor_told_me": ["teacher_told_me", "professor_told_us", "professor_told"], "attaching_maps": ["attaching_cells", "attaching"], "pa_pb": ["ap_pb", "pa_pb_pc", "pa_pb_pc_pd"], "nonunits": ["nonunit", "nonzero_nonunits", "non_units"], "revolving_around": ["revolved_around", "revolve_around"], "minute_hand": ["minute_hands"], "pairwise_relatively_prime": ["relatively_prime", "pairwise_coprime", "pairwise_co_prime"], "embedded_submanifold": ["_submanifold", "_dimensional_submanifold", "embedded_submanifolds", "immersed_submanifold", "submanifold"], "rounding_error": ["round_off_error", "rounding_errors", "roundoff_error"], "2_7x": ["2_6x", "5_7x", "3_7x"], "cw_structure": ["delta__complex_structure"], "approximately_normally_distributed": ["log_normally_distributed", "lognormally_distributed", "normally_distributed"], "finitely_generated_projective_modules": ["finitely_generated_modules", "graded_modules", "finitely_generated_module", "projective_modules"], "maximal_tori": ["maximal_toral"], "probabilistic_interpretation": ["bhk_interpretation", "frequentist_interpretation"], "prior_posts": [], "trapping_region": [], "technical_reasons": ["pedagogical_reasons", "historical_reasons"], "reflection_across": ["mirror_reflection", "reflecting_across", "reflection"], "character_tables": ["laver_tables", "character_table"], "2x_a": ["2x_b", "b_2x", "2x_a_b", "2x_2a"], "london_mathematical_society": ["mathematical_society", "american_journal", "pacific_journal", "american_mathematical_society", "doi_10", "american_mathematical_society_vol", "cambridge_university_press"], "cambridge_university_press": ["birkh\u00e4user", "cambridge_university_press_cambridge", "wiley", "springer_verlag", "revised_edition", "universitext", "cambridge_university", "oxford_university_press"], "disjunctive_normal_form": ["conjunctive_normal", "dnf", "conjunctive_normal_form", "cnf", "prenex_normal_form"], "unfair_coin": ["unfair_coins", "unbiased_coin", "biased_coin", "fair_coin"], "glossing_over": ["glossed_over"], "falling_factorials": ["rising_factorials", "falling_factorial", "rising_factorial"], "2_5x": ["4_5x", "7_5x", "2_6x", "6_5x", "x_3", "4_3x", "2_4x", "2_x_2"], "bx_cy": ["axy_bx_cy", "bx_cy_dxy", "x_cy"], "n_1_j": ["n_k_j", "k_1_j", "m_j_1", "2j_n", "n_1_l", "n_j_2"], "xhx": ["xh_hx", "ghg", "gng", "gkg", "hgh", "xgx"], "historical_reasons": ["pedagogical_reasons", "technical_reasons", "historical_remarks"], "dirichlet_characters": ["dirichlet_character", "dirichlet_character_chi", "dirichlet_characters_modulo"], "2_x_y": ["a_x_y", "4_x_y", "3_x_y"], "densely_defined": ["densely_defined_selfadjoint", "densely_defined_operator", "closable"], "full_binary_tree": ["full_binary_trees", "rooted_binary_tree", "binary_tree"], "nobody_knows": ["anybody_knows", "everybody_knows", "everyone_knows"], "fubini_tonelli": ["fubini", "tonelli", "tonelli_fubini"], "kindly_explain": ["plz_explain", "please_explain", "explain"], "pun_intended": [], "2xp": ["_xp", "_2xp"], "invariant_under_conjugation": ["invariant_under_translations", "invariant_under"], "unlhd": ["trianglelefteq", "triangleleft", "lhd"], "coarser_than": ["finer_than", "coarser", "strictly_finer_than"], "commutative_unital_ring": ["unital_commutative_ring", "commutative_ring", "commutative_unital", "noetherian_ring", "commutative_unital_rings"], "right_hand_sides": ["rhs", "left_hand_side", "right_hand_side"], "independently_distributed": ["jointly_distributed", "uniformely_distributed", "identically_distributed"], "reverse_inclusion": ["reverse_containment", "set_inclusion", "reverse_implication"], "leibniz_criterion": ["p_series_test", "cauchy_condensation_test", "cauchy_criterion", "_criterion", "comparision_test", "cauchy_condensation", "alembert_criterion"], "r_mod": ["_mod", "a_mod"], "diagram_chasing": ["diagram_chase"], "angular_speed": ["angular_velocities", "angular_velocity", "angular_acceleration"], "universal_instantiation": ["instantiation", "existential_instantiation", "existential_elimination", "existential_generalization"], "existential_instantiation": ["instantiation", "existential_elimination", "universal_instantiation"], "arithmetic_geometric_mean_inequality": ["qm_am_inequality", "arithmetic_mean_geometric_mean", "arithmetic_geometric_mean"], "joint_pmf": ["marginal_pmf", "joint_pdf", "joint_distribution", "joint_cdf"], "3abc": ["6abc", "8abc", "_3abc", "2abc", "3bc", "ab_bc_ca", "5abc"], "af_bg": [], "lands_heads": ["coin_lands", "coin_lands_tails", "coin_lands_heads", "lands_tails", "landed_heads"], "transcendence_basis": ["transcendence_bases", "transcendence_degree", "transcendence_degrees"], "cocomplete_category": ["1_category", "grothendieck_topos", "bgg_category", "locally_presentable_category", "locally_finitely_presentable", "cocomplete", "well_powered", "n_category"], "chain_complexes": ["cochain_complex", "complexes", "cochain_complexes"], "efficient_algorithms": ["algorithms", "spigot_algorithms", "efficient_algorithm"], "l\u00e9vy_process": ["l\u00e9vy_processes", "levy_process", "cadlag_process"], "l\u00e9vy_processes": ["feller_processes", "ito_processes", "levy_processes"], "came_upon": ["came_across", "came_accross", "stumbled_across", "recently_came_across", "stumbled_upon", "ran_across"], "operatorname_inn": ["inn"], "reverse_containment": ["containment", "reverse_inclusion", "reverse_implication"], "sylow_5_subgroup": ["sylow_5_subgroups", "sylow_7_subgroup", "sylow_3_subgroup", "2_sylow_subgroup", "3_sylow_subgroup", "5_sylow_subgroup"], "begin_array_ccccc": ["begin_array_cccc", "begin_array_cccccc", "begin_array_ccccccc"], "particle_moves": ["particle_moving_along", "particle_moves_along", "particle_moving"], "particle_moving": ["particle_moves", "particle_moving_along", "particle_moves_along"], "tends_towards": ["tends_toward"], "borel_cantelli_lemma": ["borel\u2013cantelli_lemma", "borel_cantelli", "borel_cantelli_lemmas"], "professor_gave": ["teacher_gave", "lecturer_gave", "teacher_gave_us", "professor_gave_us"], "rank_nullity_theorem": ["rank_nullity", "rank\u2013nullity_theorem"], "messing_up": ["messing"], "cos0": ["sin0"], "tx_ty": ["tx_t", "x_ty"], "interarrival_time": ["inter_arrival", "interarrival", "inter_arrival_time"], "finite_subcollection": ["finite_subcovering", "finite_subcovers", "finite_subcover", "subfamily"], "reference_request": [], "cantor_pairing": [], "infintely_many": ["infinitely_many", "in\ufb01nitely_many", "inifinitely_many"], "integration_by_parts": ["integration", "integrations", "romberg_integration"], "geometric_realization": ["geometric_realisation"], "shed_light": ["anyone_shed_light", "shed", "sheds_light"], "feel_comfortable": ["feel_uncomfortable", "uncomfortable", "comfortable"], "steepest_ascent": ["steepest", "steepest_decent", "steepest_slope"], "kunneth_formula": ["kunneth", "k\u00fcnneth_formula", "uct", "k\u00fcnneth"], "nonstandard_models": ["non_standard_models", "unprovable_statements", "non_standard_model", "nonstandard_model"], "brouwer_fixed_point": ["lefschetz_fixed_point", "borsuk_ulam", "brouwer"], "having_troubles": ["struggling", "having_difficulties", "having_trouble", "having_difficulty"], "sigma__finite_measures": ["sigma__finite_measure", "non__sigma__finite", "sigma__finite", "mutually_singular"], "mutually_singular": ["sigma__finite_measures", "sigma_finite"], "angle_aoc": ["angle_boc", "angle_cob", "angle_aob", "angle_abo", "angle_aod", "angle_aop", "angle_oac", "angle_oba"], "nearest_singularity": [], "ordinal_exponentiation": ["cardinal_exponentiation"], "while_maintaining": ["maintaining"], "de_bruijn": ["de_bruijn_sequences"], "brownian_motions": ["_brownian_motion", "wiener_processes", "independent_brownian_motions", "brownian_motion", "pre_brownian_motion"], "independent_identically_distributed": ["identically_distributed_random_variables", "identically_distributed", "iid_random_variables", "iid", "non_identically_distributed", "random_variables"], "convince_ourselves": ["confine_ourselves"], "heat_kernel": ["heat_kernels"], "rapidly_decreasing": ["monotonously_decreasing", "monotonically_decreasing", "monotonely_decreasing"], "cauchy_hadamard": [], "john_stillwell": ["pierre_samuel", "procesi", "nathan_carter", "john_roe", "claudio", "john_kelley", "stillwell"], "fully_faithful": ["fully_faithful_functors", "fully_faithful_functor"], "cumulant_generating": ["moment_generating"], "improper_riemann_integral": ["improper_riemann_integrable", "improper_riemann_integrals", "riemann_integral", "improper_integral"], "practical_applications": ["practical_importance", "theoretical_aspects", "applications"], "parallelogram_abcd": ["trapezoid_abcd", "trapezium_abcd", "parallelogramm", "convex_quadrilateral_abcd", "parallelogram_bisect", "isosceles_triangle_abc", "parallelogram"], "2x_2y": ["_2x_2y", "2_3xy", "3y_z", "_x_2y", "x_2y"], "die_rolls": ["dice_roll", "dice_rolls", "dice_rolled", "die_roll"], "pretty_confident": ["reasonably_confident", "confident", "fairly_confident"], "set_theoretically": ["non_set_theoretic", "set_theoretic", "set_theoretical"], "_4t": ["5t", "_3t", "_6t", "_2t"], "2_6x": ["3_4x", "9_6x", "2_4x", "4_4x", "2_5x", "2_8x", "2_6x_7"], "hensel_lifting": ["hensel"], "mental_arithmetic": ["cardinal_arithmetic"], "x_y_1": ["_y_1", "x_y_2", "2x_y_1"], "_bilinear_map": ["_bilinear"], "back_substitute": ["back_substituting", "back_substitution", "resubstitute"], "hamel_bases": ["hamel_basis"], "sylow_5_subgroups": ["sylow_7_subgroups", "_sylows", "2_sylow_subgroups", "sylow_3_subgroups", "sylow_5_subgroup", "3_sylow_subgroups"], "xy_yz_zx": ["2_xy_yz_zx", "yz_zx", "3xyz", "xy_yz_xz"], "stiefel_whitney_classes": ["stiefel\u2013whitney", "stiefel_whitney", "stiefel_whitney_class"], "principal_minor": ["leading_principal_submatrices", "leading_principal_minors", "principal_minors"], "shortest_distance_between": ["shortest_distance"], "wraps_around": ["wrap_around", "wraps", "winds_around"], "acts_properly_discontinuously": ["acting_freely", "acts_freely", "properly_discontinuous_action", "properly_discontinuously", "acting_properly_discontinuously", "acts_smoothly"], "nsubseteq": ["subseteqq", "nsupseteq", "subseteq", "notin"], "2k_n": ["2k_i", "2k_n_1", "2k_k"], "2x_x": ["_2x_x", "3_2x_x", "3x_x"], "continuously_deformed": ["deformed_continuously", "continuously_shrunk", "homotoped", "continuously_deform"], "derek_holt": ["derek_elkins", "thomas_andrews", "dietrich_burde", "jyrki_lahtonen", "derek", "jack_schmidt", "jyrkilahtonen", "andreas_caranti"], "unbiased_estimate": ["unbiased_estimators", "biased_estimator", "unbiased_estimator"], "complicate_things": ["complicate", "complicate_matters", "complicates_things"], "stackrel_frown": ["arc_stackrel_frown", "overset_frown"], "enspace": ["enspace_enspace_enspace", "enspace_enspace"], "ricci_flow": ["ricci_flat", "ricci_soliton", "normalized_ricci_flow"], "ruling_out": [], "inference_rule": ["deductive_rules", "inference_rules", "inference"], "conjunction_elimination": ["disjunctive_syllogism", "disjunction_elimination", "land__elimination", "double_negation_elimination", "_elimination", "negation_elimination", "disjunction_introduction"], "immediate_corollary": ["immediate_consequences", "immediate_consequence"], "optional_sampling_theorem": ["optional_sampling", "optional_stopping_theorem", "optional_stopping"], "sufficiently_well_behaved": ["reasonably_well_behaved", "suitably_well_behaved", "well_behaved"], "snake_lemma": ["horseshoe_lemma", "five_lemma", "5_lemma"], "thereom": ["theroem"], "binomially_distributed": ["bernoulli_distributed", "exponentially_distributed", "binomially_distributed_random_variable"], "statistical_significance": ["physical_significance", "statistical", "geometrical_significance"], "ge5": [], "frac22": ["frac33", "frac28", "frac31", "frac17_", "frac14_frac15", "frac12__frac14", "frac21"], "jordan_canonical_forms": ["jordan_canonical_form", "jordan_canonical", "jordan_normal"], "picard_lindel\u00f6f": ["phragm\u00e9n_lindel\u00f6f", "cauchy_lipschitz", "picard\u2013lindel\u00f6f_theorem", "picard_lindelof"], "formally_speaking": ["morally_speaking", "technically_speaking", "informally_speaking"], "_ball_around": [], "rlap": [], "color_brown": ["color_b0f"], "g\u00f6del_sentence": ["godel_sentence", "arithmetical_sentence", "l_sentence"], "separable_hilbert": ["separable_hilbert_space", "non_separable_hilbert", "non_separable_hilbert_spaces"], "internal_direct": ["internal_direct_product", "external_direct", "external_direct_product"], "creative_telescoping": ["telescoping"], "english_sentences": ["english_language", "sentences"], "skew_lines": ["non_parallel_lines"], "stars_and_bars": ["stars_bars"], "axiomatic_systems": ["deductive_systems", "foundational_theories", "axiomatisations", "arithmetical_truths", "foundational_systems", "axiomatizations", "axiomatic_framework"], "x1_y1_x2_y2": ["x1_x2_y2", "p2_x2_y2", "x1_y1_z1", "x3_y3"], "grassmanian": ["grassmanians", "grassmannians", "grassmannian"], "order_preserving_bijection": ["order_preserving", "order_isomorphism", "order_preserving_injection"], "topology_inherited": ["subspace_topology_inherited"], "properly_discontinuous": ["properly_discontinuous_action", "properly_discontinuously", "acting_properly_discontinuously"], "ndsolve": ["dsolve", "fsolve", "sol_dsolve"], "acts_freely": ["acts_properly_discontinuously", "acts_faithfully", "acting_freely", "cocompactly", "acts_smoothly", "properly_discontinuously", "acts_trivially"], "nonidentity_element": ["non_identity", "nonidentity", "nonidentity_elements"], "computer_science_student": ["phd_student", "high_school_senior", "software_developer", "computer_science_majors", "computer_scientist", "hobbyist", "software_engineer", "computer_science_major"], "competitive_exam": ["prelim_exam", "math_contest", "programming_competition", "competitive_exams", "competitive", "math_competition", "programming_contest"], "lagrange_interpolating_polynomial": ["lagrange_interpolation", "interpolating_polynomial", "lagrange_interpolating"], "unnecessarily_complicated": ["over_complicated", "overly_complicated", "needlessly_complicated", "overcomplicated"], "neumann_boundary": ["neumann", "robin_boundary_conditions", "robin_boundary_condition"], "squared_distances": ["eucd_distance"], "non_standard_models": ["nonstandard_models", "peano_arithmetics", "non_standard_model", "nonstandard_model"], "frac13_frac13": ["frac13_frac23", "frac13_frac13_frac13", "frac13_frac14", "frac25_frac15", "frac16_frac16", "frac14_frac34", "frac14_frac15"], "substack": [], "ax_by": ["_ax_by", "ax_bx", "x_by"], "one_parameter_family": ["one_parameter_families", "1_parameter_family", "_parameter_family"], "littlewood_paley": ["littlewood", "hardy_littlewood_maximal", "littlewood_richardson"], "independent_brownian_motions": ["brownian_motions", "wiener_processes", "drifted_brownian_motion", "_brownian_motion"], "displaystyle_": ["displaystyle1_", "displaystyle1", "displaystyle0"], "infinite_descending_chain": ["descending_chains", "descending_chain", "infinite_descending_chains"], "commonly_denoted": [], "2xz": ["2xy_2xz_2yz", "6yz", "4yz", "2yz", "_2xz", "4xz", "2xz_2yz", "3xz"], "gauss_divergence": ["bregman_divergence", "_divergence", "divergence"], "algorithm_terminates": ["terminates"], "hom_sets": ["homsets", "homset", "hom_set"], "over7": ["over50", "over5", "over32"], "_extensions": ["_extension", "end_extensions", "extensions"], "newton_polygon": ["n_polygon"], "bivariate_normal_distribution": ["lognormal_distribution", "bivariate_normal", "log_normal_distribution"], "_bx": ["_a_bx", "_ax", "x_bx", "_cx"], "overdetermined_system": ["overdetermined_systems", "over_determined", "underdetermined_system", "overdetermined", "under_determined"], "continues_until": ["game_continues_until", "process_continues_until"], "newton_iteration": ["newton_raphson", "newton_iterates", "newton_raphson_iteration"], "error_propagation": ["back_propagation", "backpropagation", "belief_propagation"], "white_noise_process": ["white_noise", "cir_process", "gaussian_white_noise"], "\ufb01nite": ["in\ufb01nite"], "uv_vu": ["uv_uw", "uv_vw", "uvu"], "nullhomotopic": ["nulhomotopic", "null_homotopic", "non_nullhomotopic"], "ra_rb": ["ra_rb_rc"], "xh_yh": ["xyh", "xh_xh", "xh_hx"], "non_normal_subgroup": ["subnormal_subgroups", "non_normal_subgroups", "_hall_subgroup"], "expected_winnings": ["net_winnings", "expected_payout", "expected_earnings"], "accepting_state": ["accepting_states", "non_accepting_state", "_transition", "acceptor_state"], "central_binomial_coefficient": ["_binomial_coefficient", "binomial_coefficient", "trinomial_coefficient"], "horizontal_strip": ["vertical_strip", "vertical_strips", "horizontal_strips"], "begin_array_lcl": ["begin_array_lr", "begin_array_lcr", "begin_array_lcc"], "strictly_monotonically_increasing": ["monotonically_increasing", "strictly_increasing", "strictly_monotonic_increasing", "strictly_monotone_increasing"], "cellular_chain_complex": ["cellular_chain", "cochain_complex", "cellular_cohomology"], "triangular_faces": ["hexagonal_faces", "pentagonal_faces", "faces", "triangular_prisms"], "highly_composite": ["superior_highly_composite"], "moore_penrose_pseudoinverse": ["moore\u2013penrose_pseudoinverse", "moore_penrose_pseudo_inverse", "pseudo_inverse", "moore_penrose", "moore_penrose_inverse", "pseudoinverse"], "context_free_language": ["non_context_free", "non_regular_language", "context_free", "context_free_grammars", "context_free_languages"], "x_ex": [], "countably_many_copies": ["countably_many", "_many_copies"], "hausdorffness": ["hausdorff_ness", "hausdorf", "hausdorfness"], "bernoulli_random_variables": ["iid_random_variables", "binomially_distributed_random_variable", "iid_bernoulli_random_variables"], "slight_variation": ["slight_variations", "slight_variant"], "poker_hand": ["straight_flush", "poker_hands", "5_card_hand", "card_poker_hand", "five_card_poker_hand"], "straight_flush": ["five_card", "four_of_a_kind", "full_house", "flush", "royal_flush", "straight_flushes"], "mclaurin_series": ["maclaurin_series", "taylor_series", "maclauren_series"], "robert_israel": ["daniel_fischer", "omnomnomnom", "yves_daoust", "robertisrael", "robjohn", "mhenni", "anon"], "normally_distributed_random_variables": ["identically_distributed_random_variables", "exponentially_distributed_random_variables", "_distributed_random_variables", "random_variables"], "ruled_surface": ["2_surface", "_surface", "ruled_surfaces"], "chern_class": ["chern_classes", "chern_character"], "haar_measures": ["haar_measure", "left_invariant_haar_measure"], "eilenberg_maclane_spaces": ["eilenberg_mac_lane", "eilenberg_maclane_spectrum", "eilenberg_maclane"], "chern_character": ["brauer_character", "todd_class", "cocharacter", "brauer_characters"], "5th_edition": ["9th_edition", "fifth_edition", "wackerly", "sixth_edition", "eighth_edition", "6th_edition", "10th_edition"], "practically_speaking": ["technically_speaking", "physically_speaking", "mathematically_speaking"], "2mx": ["2my", "_2mx", "4mx"], "caring_about": ["worrying_about"], "2xf": ["4xf", "_2xf", "_xf"], "_invariant_subspace": ["t_invariant_subspace", "t_invariant_subspaces", "_invariant_subspaces"], "sqrt7": ["sqrt7_", "sqrt6_", "sqrt3", "sqrt5_sqrt7"], "week_ago": ["weeks_ago", "couple_days_ago", "few_months_ago", "few_weeks_ago", "few_days_ago", "month_ago"], "monic_irreducible_polynomials": ["monic_irreducibles", "monic_irreducible_polynomial", "monic_irreducible"], "terminating_decimal": ["terminating_decimal_expansion", "non_terminating_decimal", "repeating_decimal", "terminating_decimals"], "non_degenerate_bilinear_form": ["nondegenerate_bilinear_form", "non_degenerate_bilinear_forms", "nondegenerate_bilinear_forms"], "nilpotency_class": ["nilpotency_index", "nilpotency"], "2_dim": [], "short_exact": ["non_split_short_exact", "short_exact_sequences", "short_exact_sequence"], "detailed_discussion": ["thorough_discussion", "detailed_description", "brief_discussion"], "ac_bd_ad_bc": ["ac_bd_ac_bd", "di_ac_bd_ad_bc", "ac_2bd_ad_bc"], "ad_bc_bd": ["ac_bd", "ad_bc", "ac_bd_ac_bd"], "dddot": ["ddot"], "pattern_continues": [], "frac14_": ["frac14__frac14", "frac13_", "frac12__frac13", "frac14u", "frac18_", "frac14f", "frac12__frac14", "frac15_"], "begin_array_cccc": ["begin_array_ccc_ccc", "begin_array_ccccc", "begin_array_ccc", "begin_array_rrr", "begin_array_cccccc"], "buildrel": [], "\u230ax\u230b": [], "chief_factor": ["p_subgroup", "chief_factors", "sylow_p__subgroup", "p__group"], "cumulative_density": ["cummulative_distribution", "empirical_cumulative_distribution", "cdf", "cumulative_distribution"], "ecdf": ["empirical_cdf"], "black_marble": ["blue_marble", "red_marble", "black_marbles", "defective_bulb", "white_marble", "black_sock"], "reparametrize": ["reparametrizing", "reparameterize", "reparametrise"], "6n_1": ["6n_4", "6n_2", "6n_1_6n"], "uniquely_determines": ["completely_determined", "uniquely_determined"], "non_borel": ["non_lebesgue_measurable", "nonmeasurable", "non_measurable"], "leaf_nodes": ["_node", "leaf_node", "non_leaf_nodes"], "totally_ramified": ["tamely_ramified", "wildly_ramified", "totally_ramified_extension", "unramified", "ramifies"], "frac25": ["frac35", "frac23", "frac55", "frac33", "frac31", "frac15", "frac13", "frac58"], "continuum_many": ["continuum"], "renewal_process": ["pure_birth_process", "renewal_processes", "compound_poisson_process"], "finest_topology": ["coarsest_topology", "weakest_topology", "k_topology"], "numerically_stable": ["asymptotically_stable", "computationally_efficient", "marginally_stable", "numerically_unstable"], "mutual_information": [], "squared_errors": ["squared_residuals"], "vacuous_truth": ["vacuous_truths"], "sorgenfrey_line": ["sorgenfrey", "sorgenfrey_topology", "sorgenfrey_plane"], "hspace_1cm": ["hspace_12pt", "hspace_1in", "hspace_15mm"], "de_la": ["sur_la", "que_la", "nombre", "des_fonctions", "hypoth\u00e8se", "sur_les", "dans_la", "probl\u00e8me"], "regular_octagon": ["regular_hexagon", "regular_octahedron", "regular_heptagon", "dodecagon"], "past_exam_paper": ["past_exams", "old_exams", "old_qualifying_exam", "past_exam", "comprehensive_exam", "past_exam_papers"], "dirichlet_character": ["dirichlet_character_modulo", "dirichlet_characters", "primitive_dirichlet_character", "dirichlet_character_chi"], "radon_measure": ["signed_radon_measure", "signed_radon_measures", "radon_measures", "signed_measure"], "limiting_behavior": ["chaotic_behaviour", "limiting_behaviour", "asymptotic_behavior"], "dynkin_system": ["_system", "dynkin_systems"], "isolated_singularity": ["non_isolated_singularity", "isolated_singularities", "essential_singularity"], "subbasic": ["subbase", "subbasis", "subbases"], "stacks_project": ["stacksproject"], "uv__plane": ["xy__plane", "w_plane", "xz__plane"], "fourth_quadrants": ["4th_quadrant", "4th_quadrants", "fourth_quadrant", "3rd_quadrant"], "fourth_quadrant": ["4th_quadrant", "3rd_quadrant", "fourth_quadrants", "quadrant", "4th_quadrants"], "1st_quadrant": ["4th_quadrant", "first_quadrant", "3rd_quadrant"], "4th_quadrant": ["fourth_quadrants", "4th_quadrants", "fourth_quadrant", "3rd_quadrant"], "comprehension_axiom": ["restricted_comprehension", "replacement_schema", "unrestricted_comprehension", "comprehension_scheme", "comprehension", "comprehension_schema"], "ordered_triples": ["ordered_triplets", "unordered_triples", "ordered_tuples"], "having_difficulties": ["having_troubles", "struggling", "facing_difficulty", "having_trouble", "facing_difficulties", "having_difficulty"], "rnorm": ["dnorm", "qnorm", "runif", "pnorm"], "kan_complex": ["kan_complexes"], "3uv": ["3u_2v", "3_3uv", "2_3u", "3u_3v"], "sandwiched_between": [], "cusp_forms": ["p_forms", "maass_forms", "cuspforms"], "completely_metrizable": ["metrizable", "separable_completely_metrizable", "first_countable", "completely_metrisable", "metrisable", "hereditarily_separable"], "_3k": ["2_3k", "1_3k", "_4k", "2_8k", "2_5k"], "dummit_foote": ["hungerford", "foote", "herstein", "dummit", "gallian", "dummit_foote_abstract_algebra"], "cylindrical_coordinate": ["polar_coordinate", "spherical_coordinate", "cylindrical_coordinates", "cylindrical_co_ordinates"], "lawvere_theory": ["lawvere_theories", "eilenberg_moore_category"], "absorption_law": ["absorption", "absorption_laws"], "p\u2192": ["p\u2227r", "q\u2192p", "p\u2192q", "r\u2192q", "q\u2192", "p\u2192q_\u2192"], "_i2": ["_j2"], "likelihood_ratio": ["likelihood_ratio_statistic", "monotone_likelihood_ratio", "likelihood_ratio_test"], "n_dim": ["n_dimension", "n_dimensional"], "confusion_stems": ["confusion_arises"], "augmentation_ideal": [], "linearized_system": ["linearised_system"], "generalized_eigenspaces": ["generalised_eigenspace", "generalised_eigenspaces", "generalized_eigenspace", "eigenspaces"], "bbox_5px_border_1px": ["solid_c00", "bbox_10px_border_1px", "bbox_8px_border_1px", "bbox_15px_border_1px"], "eigenpairs": ["eigenvalue_eigenvector", "eigenvectors", "eigenpair"], "runge_kutta_methods": ["runge_kutta", "runge_kutta_method", "runge_kutta_4th"], "qing_liu": ["wedhorn", "liu"], "biased_coins": ["unfair_coins", "unbiased_coin", "fair_coins", "biased_coin"], "2zx": ["3zx", "_2yz", "2zy", "4zx", "2_2yz", "2xy_2yz"], "little_o_notation": ["little__o_notation", "big__o_notation", "big_o_notation", "little_o"], "selling_price": ["sales_price", "price", "sale_price", "wholesale_price"], "subobject_classifier": ["subobject", "subobjects"], "dummy_variables": ["dummy_variable", "metavariables"], "arzela_ascoli": ["arzela", "arzel\u00e1_ascoli", "ascoli_arzel\u00e0", "arzel\u00e0_ascoli"], "finite_rank_operators": ["finite_rank", "hilbert_schmidt_operators", "trace_class_operators"], "tube_lemma": ["horseshoe_lemma", "five_lemma", "5_lemma"], "filtered_colimit": ["directed_colimit", "_filtered_colimits", "filtered_colimits", "preserves_filtered_colimits"], "poor_english": ["native_english", "non_english", "bad_formatting", "poor_formatting", "incompetence", "bad_english", "naivety"], "mathsf_zf": ["sf_zfa", "zf_dc", "sf_zf", "sf_zfc", "mathsf_zfc", "zfc"], "mathematics_stack_exchange": ["math_stack_exchange", "math_stackexchange", "physics_stackexchange", "physics_stack_exchange"], "columnspace": ["rowspace", "column_space", "row_space", "rowsp"], "rowspace": ["rowsp", "columnspace", "colspace"], "begin_array_rr": ["begin_array_rll", "begin_array_rrr", "begin_array_rrrr"], "2m_3": ["2m_4", "2m_2", "2m_5"], "significance_level": ["confidence_levels", "confidence_level"], "falls_apart": ["fall_apart"], "projective_resolutions": ["free_resolutions", "projective_resolution", "injective_resolutions"], "sigma__additivity": ["countable_subadditivity", "sigma_additivity", "countable_additivity", "sigma__additive"], "young_tableaux": ["young_tableau", "tableaux", "standard_young_tableaux"], "christian_blatter": ["sangchul_lee", "rossmillikan", "lucian", "joriki", "robjohn", "christianblatter", "jimmyk4542", "robert_israel"], "deductive_system": ["deductive_systems", "meta_system", "henkin_semantics", "classical_first_order_logic", "deductive_rules"], "peter_smith": ["peter_scott", "karen_smith", "peter_scholze"], "title_suggests": [], "zhen_lin": ["lee_mosher", "zhen", "martin_brandenburg", "eric_wofsey", "qiaochuyuan", "qiaochu_yuan", "kevin_carlson"], "orthonormal_eigenbasis": ["orthonomal_basis", "eigenbasis", "orthonormal_basis"], "oeis_entry": ["oeis"], "polya_enumeration_theorem": ["polya_enumeration", "ham_sandwich_theorem"], "_convergent": ["convergente", "convergent", "_converges"], "affine_opens": ["open_affines", "affine_subschemes", "affines", "open_subschemes", "affine_patches"], "ka_kb": ["ka_kb_kc", "a_kb", "ja_kb"], "cauchy_euler": ["ricatti"], "_cocycle": ["2_cocycle", "cocycle", "1_cocycle"], "4yz": ["2xy_2xz", "6yz", "2_4xz", "6xz", "5yz"], "_yz": ["2_xyz", "1_yz", "xz_x", "_xz", "_2xz", "2_zx", "2_yz"], "2_8y": ["4_6y", "2_24y", "5_2y", "4_4xy", "2_12y", "3_4y", "2_8xy"], "8xy": ["6xy", "_8xy", "12xy", "24xy"], "jordan_decomposition": ["jordan_chevalley_decomposition", "hahn_decomposition", "cartan_decomposition"], "_8y": ["36y", "_6y", "2y_4x", "2_12y", "2_8y", "3_4y", "_12y"], "bernstein_polynomials": ["gegenbauer_polynomials", "zernike_polynomials"], "x_2y": ["2x_2y", "3x_2y", "4x_2y", "2x_y"], "cylic": [], "vectorspaces": ["_vectorspaces", "vector_spaces", "vectorspace"], "polynoms": ["polynomes", "polynomals", "polynome"], "seating_arrangements": ["arrangements", "seatings", "seating_arrangement"], "act_transitively": ["_transitively", "acting_transitively", "acts_transitively"], "co_primes": ["coprimes", "co_prime", "sexy_primes"], "1xf": ["bxf", "2dx_", "1xdx", "dx_2"], "hspace_3pt": ["hspace_3ex", "hspace_30pt", "hspace_3em"], "thom_class": [], "event_occurring": ["events_occurring"], "fsolve": ["lpsolve", "ndsolve", "dsolve", "fzero"], "exhaustive_search": ["brute_force_search"], "outward_normal": ["unit_outward_normal", "outward_pointing_normal", "outward_unit"], "primarily_interested": ["particularly_interested", "interested"], "recursive_calls": [], "mathord_sim": ["mathord_"], "mark_scheme": ["markscheme"], "ito_isometry": ["it\u00f4_isometry"], "_cf": [], "spanning_subgraph": ["loopless", "bipartite_subgraph", "spanning_subgraphs", "k_connected", "subgraph", "spanning_forest", "sub_graph"]}