Cleanup

\( \texttt{counting_inversions} \) is Correct direct title match 100%

\( \texttt{counting_inversions} \) is Correct proposition computer_science/algorithms/divide_and_conquer/counting_inversions.html
\( \texttt{counting_inversions} \) is Correct proposition computer_science/algorithms/greedy/interval_partitioning.html

A Sequence of Sets Increase to Another direct title match 100%

A Sequence of Sets Increase to Another definition probability/scratch/limits_of_sets.html
A Sequence of Sets Increase to Another definition probability/scratch/probability_measures.html

addition direct title match 100%

addition definition algebra/complex/algebra_of_the_complex_plane.html
Addition definition algebra/complex/complex_numbers.html

Antiderivative direct title match 100%

Antiderivative definition analysis/multi_variable/the_fundamental_theorem_of_calculus.html
Antiderivative definition analysis/single_variable/indefinite_integrals.html

Automorphism direct title match 100%

Automorphism definition algebra/galois_theory/the_galois_group.html
Automorphism definition category_theory/category.html
Automorphism definition linear_operators/everything.html

Banach Space direct title match 100%

Banach Space definition linear_operators/banach_spaces.html
Banach Space definition linear_operators/metric_and_banach_spaces.html

Bounded direct title match 100%

Bounded definition analysis/multi_variable/closed_and_open_subsets.html
Bounded definition analysis/single_variable/boundedness.html

Cantor Function direct title match 100%

Cantor Function definition measure_theory/lebesgue_measure.html
Cantor Function theorem measure_theory/lebesgue_measure.html

Closed Graph Theorem direct title match 100%

Closed Graph Theorem theorem linear_operators/everything.html
Closed Graph Theorem theorem linear_operators/metric_and_banach_spaces.html

Closure direct title match 100%

Closure definition analysis/multi_variable/closed_and_open_subsets.html
Closure definition computer_science/programming_languages/evaluation.html
Closure definition topology/interior_and_closure.html

Compact direct title match 100%

Compact definition analysis/multi_variable/closed_and_open_subsets.html
Compact definition topology/connectedness_and_compactness.html

Complete direct title match 100%

Complete definition computer_science/algorithms/artificial_intelligence/searching.html
Complete definition graph_theory/spectral.html

Complete Metric Space direct title match 100%

Complete Metric Space definition linear_operators/banach_spaces.html
Complete Metric Space definition linear_operators/metric_and_banach_spaces.html

complex number times it's conjugate equals the modulus squared direct title match 100%

complex number times it's conjugate equals the modulus squared proposition algebra/complex/algebra_of_the_complex_plane.html
complex number times it's conjugate equals the modulus squared proposition algebra/complex/complex_numbers.html

conditional probability direct title match 100%

Conditional Probability definition probability/probability_models.html
conditional probability definition probability/scratch/probability_measures.html

conjugate direct title match 100%

conjugate definition algebra/complex/algebra_of_the_complex_plane.html
Conjugate definition algebra/complex/complex_numbers.html

conjugate cancels direct title match 100%

conjugate cancels proposition algebra/complex/algebra_of_the_complex_plane.html
conjugate Cancels proposition algebra/complex/complex_numbers.html

Connected direct title match 100%

Connected definition graph_theory/spectral.html
Connected definition topology/connectedness_and_compactness.html

continuity of a function at a point direct title match 100%

Continuity of a Function at a Point definition analysis/multi_variable/limits_and_continuity.html
continuity of a function at a point definition analysis/single_variable/continuity.html

continuous function direct title match 100%

Continuous Function definition analysis/multi_variable/limits_and_continuity.html
continuous function definition analysis/single_variable/continuity.html
Continuous Function definition topology/continuous_functions.html

Degree of a Vertex direct title match 100%

Degree of a Vertex definition graph_theory/graphs.html
Degree of a Vertex definition graph_theory/spectral.html

equality direct title match 100%

equality definition algebra/complex/algebra_of_the_complex_plane.html
Equality definition algebra/complex/complex_numbers.html

event direct title match 100%

Event definition probability/probability_measures.html
event definition probability/scratch/probability_measures.html

Expression direct title match 100%

Expression definition computer_science/programming_languages/functional_programming.html
Expression definition computer_science/programming_languages/lambda_calculus.html

extracting the real part of a complex number direct title match 100%

extracting the real part of a complex number proposition algebra/complex/algebra_of_the_complex_plane.html
extracting the real part of a complex number proposition algebra/complex/complex_numbers.html

Graph direct title match 100%

Graph definition graph_theory/graphs.html
Graph definition graph_theory/spectral.html

Hilbert Adjoint Operator direct title match 100%

Hilbert Adjoint Operator definition linear_operators/hilbert_spaces.html
Hilbert Adjoint Operator definition linear_operators/unbounded_linear_operators.html

Hilbert Space direct title match 100%

Hilbert Space definition linear_operators/everything.html
Hilbert Space definition linear_operators/hilbert_space_theory.html

imaginary part distributes direct title match 100%

imaginary part distributes proposition algebra/complex/algebra_of_the_complex_plane.html
imaginary part distributes proposition algebra/complex/complex_numbers.html

Increasing Sequence of Sets direct title match 100%

Increasing Sequence of Sets definition probability/scratch/limits_of_sets.html
Increasing Sequence of Sets definition probability/scratch/probability_measures.html

Independence of Events direct title match 100%

Independence of Events definition probability/probability_measures.html
Independence of Events definition probability/probability_models.html

induction direct title match 100%

induction theorem fundamentals/induction.html
Induction theorem number_theory/induction.html

Inner Product Space direct title match 100%

Inner Product Space definition algebra/linear/vector_spaces/linear_transformations.html
Inner Product Space definition linear_operators/hilbert_space_theory.html

Intermediate Value direct title match 100%

Intermediate Value theorem analysis/multi_variable/intermediate_value_theorem.html
Intermediate Value theorem topology/connectedness_and_compactness.html

inverse direct title match 100%

inverse definition algebra/complex/algebra_of_the_complex_plane.html
inverse definition algebra/complex/complex_numbers.html
Inverse definition algebra/groups/binary_operations_and_groups.html

Inverse Image of a Composition direct title match 100%

Inverse Image of a Composition proposition fundamentals/functions.html
Inverse Image of a Composition proposition measure_theory/measurable_spaces_and_functions.html

Inversion Lookahead in Sorted Concatenation direct title match 100%

Inversion Lookahead in Sorted Concatenation lemma computer_science/algorithms/divide_and_conquer/counting_inversions.html
Inversion Lookahead in Sorted Concatenation lemma computer_science/algorithms/greedy/interval_partitioning.html

Invertible iff Bijective direct title match 100%

Invertible iff Bijective proposition fundamentals/functions.html
Invertible Iff Bijective theorem analysis/single_variable/inverse_functions.html

Left Coset direct title match 100%

Left Coset definition algebra/groups/cosets.html
Left Coset definition algebra/groups/cosets.html

Left Ideal direct title match 100%

Left Ideal definition algebra/galois_theory/rings.html
Left Ideal definition linear_operators/banach_algebras.html

Limit Point direct title match 100%

Limit Point definition analysis/multi_variable/closed_and_open_subsets.html
Limit Point definition topology/limit_points.html

Maximal Ideal direct title match 100%

Maximal Ideal definition algebra/galois_theory/prime_ideals_and_maximal_ideals.html
Maximal Ideal definition linear_operators/banach_algebras.html

Measurable Function direct title match 100%

Measurable Function definition measure_theory/measurable_functions.html
Measurable Function definition measure_theory/measurable_spaces_and_functions.html

Measurable Space direct title match 100%

Measurable Space definition measure_theory/measurable_functions.html
Measurable Space definition measure_theory/measurable_spaces_and_functions.html

Measure direct title match 100%

Measure definition measure_theory/measurable_functions.html
Measure definition measure_theory/measures.html

modulus ignores conjugates direct title match 100%

modulus ignores conjugates corollary algebra/complex/algebra_of_the_complex_plane.html
Modulus Ignores Conjugates corollary algebra/complex/complex_numbers.html

modulus is greater than it's components direct title match 100%

modulus is greater than it's components proposition algebra/complex/algebra_of_the_complex_plane.html
modulus is greater than it's components proposition algebra/complex/complex_numbers.html

modulus plus one of it's components is positive direct title match 100%

modulus plus one of it's components is positive proposition algebra/complex/algebra_of_the_complex_plane.html
modulus plus one of it's components is positive proposition algebra/complex/complex_numbers.html

No Inversion Lookahead in Sorted Concatenation direct title match 100%

No Inversion Lookahead in Sorted Concatenation lemma computer_science/algorithms/divide_and_conquer/counting_inversions.html
No Inversion Lookahead in Sorted Concatenation lemma computer_science/algorithms/greedy/interval_partitioning.html

Normed Vector Space direct title match 100%

Normed Vector Space definition linear_operators/banach_spaces.html
Normed Vector Space definition linear_operators/metric_and_banach_spaces.html

Open Cover direct title match 100%

Open Cover definition measure_theory/outer_measure.html
Open Cover definition topology/connectedness_and_compactness.html

Open Mapping Theorem direct title match 100%

Open Mapping Theorem theorem linear_operators/everything.html
Open Mapping Theorem theorem linear_operators/metric_and_banach_spaces.html

Open set direct title match 100%

Open Set definition computer_science/algorithms/artificial_intelligence/searching.html
Open set definition topology/topological_spaces.html

Operator Norm direct title match 100%

Operator Norm definition linear_operators/banach_spaces.html
Operator Norm definition linear_operators/metric_and_banach_spaces.html

partial derivative direct title match 100%

Partial Derivative definition analysis/multi_variable/differentiation.html
partial derivative definition analysis/single_variable/differentiation.html

partition direct title match 100%

Partition definition analysis/multi_variable/riemann_integration.html
partition definition fundamentals/sets/union_and_intersection.html

Path direct title match 100%

Path definition analysis/multi_variable/intermediate_value_theorem.html
Path definition graph_theory/graphs.html
Path definition graph_theory/spectral.html
Path definition topology/algebraic_topology.html
Path definition topology/connectedness_and_compactness.html

product rule direct title match 100%

product rule proposition fundamentals/cardinality.html
Product Rule theorem probability/probability_models.html

Random Variable direct title match 100%

Random Variable definition probability/probability_measures.html
Random Variable definition probability/random_variables.html
Random Variable definition probability/scratch/probability_measures.html

Rubiks Cube direct title match 100%

Rubiks Cube corollary algebra/groups/rubiks_cube.html
Rubiks Cube definition algebra/groups/rubiks_cube.html

Sigma Algebra direct title match 100%

Sigma Algebra definition measure_theory/measurable_functions.html
Sigma Algebra definition measure_theory/measurable_spaces_and_functions.html

Subgroup direct title match 100%

Subgroup definition algebra/groups/binary_operations_and_groups.html
Subgroup definition algebra/groups/subgroups.html

The Closure is Closed direct title match 100%

The Closure is Closed corollary analysis/multi_variable/closed_and_open_subsets.html
The Closure is Closed corollary topology/interior_and_closure.html

TODO direct title match 100%

TODO corollary algebra/groups/binary_operations_and_groups.html
TODO corollary analysis/single_variable/mean_value_theorem.html
TODO definition fundamentals/propositional_logic.html
TODO proposition analysis/single_variable/exponential_function.html
TODO theorem analysis/single_variable/exponential_function.html
TODO theorem topology/subspace_topology.html

Union direct title match 100%

Union definition fundamentals/sets/union_and_intersection.html
Union proposition probability/scratch/introduction.html

Unitary Operator direct title match 100%

Unitary Operator definition linear_operators/everything.html
Unitary Operator definition linear_operators/operators_on_hilbert_spaces.html

value of the inverse direct title match 100%

value of the inverse theorem algebra/complex/algebra_of_the_complex_plane.html
value of the inverse theorem algebra/complex/complex_numbers.html

Zero Divisor direct title match 100%

Zero Divisor definition algebra/galois_theory/domains_and_fields.html
Zero Divisor definition algebra/galois_theory/rings.html

Fundamental Theorem of Calculus I similar title 97%

Fundamental Theorem of Calculus I theorem analysis/multi_variable/the_fundamental_theorem_of_calculus.html
Fundamental Theorem of Calculus II theorem analysis/multi_variable/the_fundamental_theorem_of_calculus.html

GCD as a Linear Combination similar title 96%

GCD as a Linear Combination proposition algebra/galois_theory/polynomial_rings_over_fields.html
GCD is a Linear Combination proposition number_theory/gcd_and_lcm.html

A Sequence of Sets Decrease to Another similar title 95%

A Sequence of Sets Decrease to Another definition probability/scratch/limits_of_sets.html
A Sequence of Sets Increase to Another definition probability/scratch/limits_of_sets.html
A Sequence of Sets Increase to Another definition probability/scratch/probability_measures.html

Dirichlet Charater1 similar title 95%

Dirichlet Charater1 proposition number_theory/dirichlet_theorem.html
Dirichlet Charater2 proposition number_theory/dirichlet_theorem.html

Dirichlet Theorem similar title 94%

Dirichlet Theorem theorem number_theory/dirichlet_theorem.html
Dirichlet Theorem1 theorem number_theory/dirichlet_theorem.html

The Kernel of a Crone Homomorphism is a Proper Ideal similar title 94%

The Kernel of a Crone Homomorphism is a Proper Ideal proposition algebra/galois_theory/ideals.html
The Kernel of a Ring Homomorphism is a Proper Ideal proposition algebra/galois_theory/rings.html

Natural Numbers 0 similar title 94%

Natural Numbers 0 definition fundamentals/sets/basics.html
Natural Numbers 1 definition fundamentals/sets/basics.html

foldl Expression similar title 94%

foldl Expression proposition computer_science/programming_languages/recursion.html
foldr Expression proposition computer_science/programming_languages/recursion.html

Lower Sum of a Bounded Function over a Partition similar title 94%

Lower Sum of a Bounded Function over a Partition definition analysis/multi_variable/riemann_integration.html
Upper Sum of a Bounded Function over a Partition definition analysis/multi_variable/riemann_integration.html

Outer Measure is Additive when One Set is Borel similar title 94%

Outer Measure is Additive when One Set is Borel theorem measure_theory/lebesgue_measure.html
Outer Measure is Additive when One Set is Open theorem measure_theory/lebesgue_measure.html

Inversion Lookahead in Sorted Concatenation similar title 93%

Inversion Lookahead in Sorted Concatenation lemma computer_science/algorithms/divide_and_conquer/counting_inversions.html
No Inversion Lookahead in Sorted Concatenation lemma computer_science/algorithms/divide_and_conquer/counting_inversions.html
Inversion Lookahead in Sorted Concatenation lemma computer_science/algorithms/greedy/interval_partitioning.html
No Inversion Lookahead in Sorted Concatenation lemma computer_science/algorithms/greedy/interval_partitioning.html

Continuous iff Every Basis Element is Open similar title 93%

Continuous iff Every Basis Element is Open proposition topology/continuous_functions.html
Continuous iff Every Subbasis Element is Open proposition topology/continuous_functions.html

Monotone Decreasing Sequence similar title 93%

Monotone Decreasing Sequence definition analysis/single_variable/limits.html
Monotone Increasing Sequence definition analysis/single_variable/limits.html

For Every Dfa There Is an Equivalent Cfg similar title 93%

For Every Dfa There Is an Equivalent Gnfa lemma computer_science/theory_of_computation/index.html
For Every Dfa There Is an Equivalent Cfg proposition computer_science/theory_of_computation/index.html

Sorted in Ascending Order similar title 92%

Sorted in Ascending Order definition fundamentals/sets/tuples.html
Sorted in Descending Order definition fundamentals/sets/tuples.html

Lower Sum Decreases over Refinements similar title 92%

Lower Sum Decreases over Refinements proposition analysis/multi_variable/riemann_integration.html
Upper Sum Decreases over Refinements proposition analysis/multi_variable/riemann_integration.html

Outer Measure is Additive when One Set is Borel similar title 92%

Outer Measure is Additive when One Set is Borel theorem measure_theory/lebesgue_measure.html
Outer Measure is Additive when One Set is Closed theorem measure_theory/lebesgue_measure.html

Outer Measure is Additive when One Set is Closed similar title 92%

Outer Measure is Additive when One Set is Closed theorem measure_theory/lebesgue_measure.html
Outer Measure is Additive when One Set is Open theorem measure_theory/lebesgue_measure.html

A Language Is Context Free Iff There Is Some Pushdown Automaton That Models It similar title 91%

If a Language Is Context Free Then There Is Some Pushdown Automaton That Models It lemma computer_science/theory_of_computation/index.html
A Language Is Context Free Iff There Is Some Pushdown Automaton That Models It proposition computer_science/theory_of_computation/index.html

topology generated by a basis similar title 91%

topology generated by a basis definition topology/basis.html
topology generated by a subbasis definition topology/basis.html

Every Finite Dimensional Subspace of a Normed Space Is Closed similar title 90%

Every Finite Dimensional Subspace of a Normed Space Is Closed proposition linear_operators/banach_spaces.html
Every Finite Dimensional Subspace of a Normed Space Is Complete proposition linear_operators/banach_spaces.html

The Product of Two Multiplicative Functions is Multiplicative similar title 90%

The Product of Two Multiplicative Functions is Multiplicative theorem number_theory/arithmetic_functions.html
The Quotient of Two Multiplicative Functions is Multiplicative theorem number_theory/arithmetic_functions.html

Linearly Dependent similar title 90%

Linearly Dependent definition algebra/linear/vector_spaces/span_and_linear_independence.html
Linearly Independent definition algebra/linear/vector_spaces/span_and_linear_independence.html

Dirichlet Character similar title 89%

Dirichlet Character definition number_theory/dirichlet_theorem.html
Dirichlet Charater1 proposition number_theory/dirichlet_theorem.html

Dirichlet Character similar title 89%

Dirichlet Character definition number_theory/dirichlet_theorem.html
Dirichlet Charater2 proposition number_theory/dirichlet_theorem.html

Relation Preserving Function similar title 89%

Relation Preserving Function definition fundamentals/functions.html
Relation Reversing Function definition fundamentals/functions.html

Real Part of Product of Two Pure Quaternions similar title 89%

Real Part of Product of Two Pure Quaternions corollary algebra/quaternions.html
Vector Part of Product of Two Pure Quaternions corollary algebra/quaternions.html

\( L ^ 1 \)-Space similar title 89%

\( L ^ 1 \)-Space definition measure_theory/integration.html
\( L ^ p \)-Space definition measure_theory/lp_spaces.html

Dimension of a Vector Space similar title 89%

Dimension of a Vector Space definition algebra/linear/vector_spaces/basis.html
Direction of a Vector Space definition topology/quotient_topology.html

Riemann Integrable similar title 89%

Riemann Integrable definition analysis/multi_variable/riemann_integration.html
Riemann Integral definition analysis/single_variable/riemann_integration.html

A Finite Automaton Accepts an Input similar title 89%

A Finite Automaton Accepts an Input definition computer_science/theory_of_computation/index.html
A Finite Automaton Rejects an Input definition computer_science/theory_of_computation/index.html

Powers Pass Through a Homomorphism similar title 88%

Powers pass through an Isomorphism corollary algebra/groups/isomorphisms.html
Powers Pass Through a Homomorphism proposition algebra/groups/isomorphisms.html

Sequence Goes to Negative Infinity similar title 88%

Sequence Goes to Negative Infinity definition analysis/single_variable/limits.html
Sequence Goes to Positive Infinity definition analysis/single_variable/limits.html

Regular Languages Are Closed Under Complementation similar title 88%

Regular Languages Are Closed Under Complementation proposition computer_science/theory_of_computation/index.html
Regular Languages Are Closed Under Concatenation proposition computer_science/theory_of_computation/index.html

A power of an Even Number is Even similar title 88%

A power of an Even Number is Even proposition number_theory/division.html
A power of an Even Number is Odd proposition number_theory/division.html

A Turing Machine Accepts an Input similar title 88%

A Turing Machine Accepts an Input definition computer_science/theory_of_computation/index.html
A Turing Machine Rejects an Input definition computer_science/theory_of_computation/index.html

geometric series with incrementing factor similar title 88%

geometric series with incrementing factor proposition analysis/single_variable/summations_and_series.html
geometric sum with incrementing factor proposition analysis/single_variable/summations_and_series.html

\( L ^ \infty \)-Space similar title 88%

L2 Space definition linear_operators/everything.html
\( L ^ \infty \)-Space definition measure_theory/lp_spaces.html

\( L ^ 1 \)-Norm similar title 88%

\( L ^ 1 \)-Norm definition measure_theory/integration.html
\( L ^ p \)-Norm definition measure_theory/lp_spaces.html

Arbitrary And Indexed Notation similar title 88%

Arbitrary And Indexed Notation definition fundamentals/logic.html
Arbitrary Union Indexed Notation definition fundamentals/sets/union_and_intersection.html

Binary Iteration similar title 88%

Binary Operation definition algebra/groups/binary_operations_and_groups.html
Binary Iteration definition algebra/groups/homomorphisms.html

T1 Space similar title 88%

T1 Space definition topology/hausdorff_spaces.html
T2 Space definition topology/hausdorff_spaces.html

Left Iterated Binary Operation similar title 87%

Left Iterated Binary Operation definition algebra/groups/homomorphisms.html
Right Iterated Binary Operation definition algebra/groups/homomorphisms.html

Non-Deterministic Polynomial Time similar title 87%

Non-Deterministic Polynomial Time Hard definition computer_science/complexity/complexity.html
Non-Deterministic Polynomial Time definition computer_science/complexity/complexity.html

Integer Addition and Multiplication are Well-Defined similar title 87%

Integer Addition and Multiplication are Well-Defined proposition analysis/single_variable/dedekind_cuts.html
Rational Addition and Multiplication are Well-Defined proposition analysis/single_variable/dedekind_cuts.html

Dot Product of Two Quaternions similar title 87%

Dot Product of Two Quaternions definition algebra/quaternions.html
Product of Two Quaternions definition algebra/quaternions.html

Symmetric Group similar title 87%

Symmetric Group definition algebra/groups/symmetric_group.html
Symmetry Group definition algebra/groups/symmetry_group.html

Regular Languages Are Closed Under Intersection similar title 86%

Regular Languages Are Recognized Under Intersection corollary computer_science/theory_of_computation/index.html
Regular Languages Are Closed Under Intersection proposition computer_science/theory_of_computation/index.html

A Subspace of a T2 Space Is T2 similar title 86%

A Subspace of a T2 Space Is T2 corollary topology/hausdorff_spaces.html
A Subspace of a T2 Space Remains T2 corollary topology/hausdorff_spaces.html

Multiplication similar title 86%

Multiplication definition algebra/complex/complex_numbers.html
Multiplicative definition number_theory/arithmetic_functions.html

One Layer Productions for All Rules similar title 86%

One Layer Productions for All Rules definition computer_science/theory_of_computation/index.html
One Layer Productions of a Rule definition computer_science/theory_of_computation/index.html

Probability Mass Function of the Binomial Distribution similar title 86%

Probability Mass Function of the Binomial Distribution proposition probability/discrete_distributions.html
Probability Mass Function of the Negative Binomial Distribution proposition probability/discrete_distributions.html

A Compact Subset of a Metric Space Is Closed and Bounded similar title 85%

A Compact Subset of a Metric Space Is Closed and Bounded lemma linear_operators/banach_spaces.html
Every Compact Subspace of a Metric Space Is Closed and Bounded proposition topology/connectedness_and_compactness.html

Greatest Lower Bound Property for Dedekind Reals similar title 85%

Greatest Lower Bound Property for Dedekind Reals theorem analysis/single_variable/dedekind_cuts.html
Least Upper Bound Property for Dedekind Reals theorem analysis/single_variable/dedekind_cuts.html

Product Topology by Universal Properties similar title 85%

Quotient Topology by Universal Properties proposition topology/quotient_topology.html
Product Topology by Universal Properties theorem topology/product_topology.html

A Compact Operator Is Bounded similar title 85%

Every Compact Operator Is Bounded proposition linear_operators/compact_operators.html
A Compact Operator Is Bounded theorem linear_operators/compact_operators.html

Real Part of Quaternion Product similar title 85%

Real Part of Quaternion Product corollary algebra/quaternions.html
Vector Part of Quaternion Product corollary algebra/quaternions.html

Dominated Convergence Theorem for Expectations similar title 85%

Dominated Convergence Theorem for Expectations theorem probability/probability_measures.html
Monotone Convergence Theorem for Expectations theorem probability/probability_measures.html

Finite Intersection as Conditional Probability similar title 85%

Intersection as Conditional Probability corollary probability/probability_models.html
Finite Intersection as Conditional Probability lemma probability/probability_models.html

Homeomorphism similar title 85%

Homeomorphism definition topology/continuous_functions.html
Homomorphisms proposition algebra/groups/homomorphisms.html

local maximum similar title 85%

local maximum definition analysis/single_variable/mean_value_theorem.html
local minimum definition analysis/single_variable/mean_value_theorem.html

The Image of a Linear Opertor Is a Vector Space similar title 85%

The Image of a Linear Opertor Is a Vector Space proposition linear_operators/banach_spaces.html
The Nullspace of a Linear Operator Is a Vector Space proposition linear_operators/banach_spaces.html

Independence of Random Variables similar title 84%

Independence of Random Variables definition probability/probability_measures.html
Independent Random Variables definition probability/random_variables.html

Closure Basis Intersection Equivalence similar title 84%

Closure Basis Intersection Equivalence corollary topology/interior_and_closure.html
Closure Intersection Equivalence theorem topology/interior_and_closure.html

Absolutely Dominated up to a Constant Factor similar title 84%

Absolutely Dominated up to a Constant Factor definition computer_science/runtime/runtime.html
Eventually Dominated up to a Constant Factor definition computer_science/runtime/runtime.html

Function Homotopy Is an Equivalence Relation similar title 84%

Function Homotopy Is an Equivalence Relation proposition topology/algebraic_topology.html
Path Homotopy Is an Equivalence Relation proposition topology/algebraic_topology.html

Every Finite Set is Closed in a Hausdorff Space similar title 84%

Every Finite Set is Closed in a Hausdorff Space corollary topology/hausdorff_spaces.html
Every one point set is Closed in a Hausdorff Space proposition topology/hausdorff_spaces.html

Referential Transparency similar title 84%

Referentially Transparent definition computer_science/programming_languages/functional_programming.html
Referential Transparency definition computer_science/programming_languages/lambda_calculus.html

Product of Two Pure Quaternions similar title 84%

Product of Two Quaternions definition algebra/quaternions.html
Product of Two Pure Quaternions proposition algebra/quaternions.html

Equation for the Divisor Counting Function similar title 83%

Equation for the Divisor Counting Function proposition number_theory/arithmetic_functions.html
Equation for the Divisor Sum Function proposition number_theory/arithmetic_functions.html

Group Homomorphism similar title 83%

Group Homomorphism definition algebra/groups/homomorphisms.html
Group Isomorphism definition algebra/groups/isomorphisms.html

Regular Languages Are Closed Under Concatenation similar title 83%

Regular Languages Are Closed Under Concatenation proposition computer_science/theory_of_computation/index.html
Regular Languages Are Closed Under Intersection proposition computer_science/theory_of_computation/index.html

p Minus one Factorial is Congruent to Minus one Mod p similar title 83%

p Minus two Factorial is Congruent to One Mod p corollary number_theory/primes.html
p Minus one Factorial is Congruent to Minus one Mod p proposition number_theory/primes.html

Regular Languages Are Closed Under Intersection similar title 83%

Regular Languages Are Closed Under Intersection proposition computer_science/theory_of_computation/index.html
Regular Languages Are Closed Under Union proposition computer_science/theory_of_computation/index.html

Product Topology by Universal Properties similar title 83%

Product Topology by Universal Properties theorem topology/product_topology.html
Subspace Topology by Universal Properties theorem topology/subspace_topology.html

Quotient Topology by Universal Properties similar title 83%

Quotient Topology by Universal Properties proposition topology/quotient_topology.html
Subspace Topology by Universal Properties theorem topology/subspace_topology.html

A Positive Number is Greater than the Negative Version of Itself similar title 83%

A Positive Number is Greater than the Negative Version of Itself corollary analysis/single_variable/absolute_value.html
absolute value is greater than the negative version of itself corollary analysis/single_variable/absolute_value.html

Probability of an Interval in the Exponential Distribution similar title 83%

Probability of an Interval in the Exponential Distribution corollary probability/continuous_distributions.html
Probability of an Interval in the Uniform Distribution corollary probability/continuous_distributions.html

Pure Unit Quaternion Times a Parallel Pure Quaternion similar title 83%

Pure Unit Quaternion Times a Parallel Pure Quaternion lemma algebra/quaternions.html
Pure Unit Quaternion Times a Perpendicular Pure Quaternion lemma algebra/quaternions.html

Addition and Multiplication of Integers similar title 83%

Addition and Multiplication of Integers definition analysis/single_variable/dedekind_cuts.html
Addition and Multiplication of Rationals definition analysis/single_variable/dedekind_cuts.html

Approximation by Simple Functions similar title 83%

Approximation by Simple Functions theorem measure_theory/convergence_of_measurable_functions.html
Approximation in \( L ^ 1 \) by Simple Functions theorem measure_theory/integration.html

Countable Subadditivity of Measures similar title 83%

Countable Subadditivity of Measures proposition measure_theory/measures.html
Countable Subadditivity of Outer Measure proposition measure_theory/outer_measure.html

Specific Value of the Divisor Function similar title 83%

Specific Value of the Divisor Function corollary number_theory/quotient_and_remainder.html
Specific Value of the Remainder Function corollary number_theory/quotient_and_remainder.html

Ring Homomorphism similar title 82%

Ring Homomorphism definition algebra/galois_theory/homomorphisms.html
Ring Isomorphism definition algebra/galois_theory/homomorphisms.html

Ideal Generated By a Set shared words 100%

Ideal Generated by a Set is an Ideal corollary algebra/galois_theory/ideals.html
Ideal Generated By a Set definition algebra/galois_theory/ideals.html

Image of Inverse Image Equality shared words 100%

Image of Inverse Image Equality proposition fundamentals/functions.html
Inverse Image of Image Equality proposition fundamentals/functions.html

Set Difference with Itself is Empty shared words 100%

Set Difference with Itself is Empty proposition fundamentals/sets/set_difference.html
Set Difference with the Empty set is Itself proposition fundamentals/sets/set_difference.html

Acceptance Set for Context Free Grammars shared words 83%

Acceptance Set for Context Free Grammars definition computer_science/theory_of_computation/index.html
The Acceptance Set for Context Free Grammars Is Decidable proposition computer_science/theory_of_computation/index.html

Every Closed Subspace of a Compact Space Is Compact shared words 83%

Every Closed Subspace of a Compact Space Is Compact proposition topology/connectedness_and_compactness.html
Every Compact Subspace of a Hausdorff Space Is Closed proposition topology/connectedness_and_compactness.html

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Least Upper Bound Property definition analysis/single_variable/boundedness.html
\( \mathbb{ R } \) has the Least Upper Bound Property theorem analysis/single_variable/boundedness.html

Characteristic Version of Symmetric Difference shared words 80%

Characteristic Version of Symmetric Difference proposition fundamentals/sets/symmetric_difference.html
Characteristic Version of Symmetric Difference Generalized proposition fundamentals/sets/symmetric_difference.html

Complement Non-Deterministic Polynomial Time shared words 80%

Complement Non-Deterministic Polynomial Time definition computer_science/complexity/complexity.html
Non-Deterministic Polynomial Time definition computer_science/complexity/complexity.html

Every Lipschitz Function is Continuous shared words 80%

Every Lipschitz Function is Continuous proposition analysis/multi_variable/limits_and_continuity.html
Every Lipschitz function is Uniformly Continuous proposition analysis/multi_variable/uniform_continuity.html

Lebesgue Measure on Lebesgue Measurable Sets shared words 80%

Lebesgue Measure on Lebesgue Measurable Sets definition measure_theory/lebesgue_measure.html
Outer Measure is a Measure on Lebesgue Measurable Sets theorem measure_theory/lebesgue_measure.html

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Orthogonal Complement of an Inner Product Space definition linear_operators/hilbert_spaces.html
Orthogonal in an Inner Product Space definition linear_operators/hilbert_spaces.html

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Pure Quaternion from a 3D Vector definition algebra/quaternions.html
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Rational Number Pair Relation shared words 80%

Rational Number Pair Relation definition analysis/single_variable/dedekind_cuts.html
Rational Number Pair Relation is an Equivalence Relation proposition analysis/single_variable/dedekind_cuts.html

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Sorted in Strictly Ascending Order definition fundamentals/sets/tuples.html
Sorted in Strictly Ascending order Predicate definition fundamentals/sets/tuples.html

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Union Factors through Image proposition fundamentals/functions.html
Union Factors through Inverse Image proposition fundamentals/functions.html

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We can make the Absolute value of a Polynomial as small as we Want Inverse Edition corollary analysis/multi_variable/intermediate_value_theorem.html
We can make the Absolute value of a Polynomial as Small as we Want lemma analysis/multi_variable/intermediate_value_theorem.html

\( G_{\mathcal{B}} \) is a topology shared words 75%

collection generated by a basis definition topology/basis.html
\( G_{\mathcal{B}} \) is a topology theorem topology/basis.html

\( G_{\mathcal{B}} \) is a topology shared words 75%

topology generated by a basis definition topology/basis.html
\( G_{\mathcal{B}} \) is a topology theorem topology/basis.html

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A Compact Operator Is Bounded theorem linear_operators/compact_operators.html
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A Compact Operator on a Infinite Dimensional Space Is Not Invertible corollary linear_operators/compact_operators.html
The Identity Operator on an Infinite Dimensional Space Is Not Compact theorem linear_operators/compact_operators.html

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A Finite Automaton Models a Language definition computer_science/theory_of_computation/index.html
Language of a Finite Automaton definition computer_science/theory_of_computation/index.html

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A Language Is Context Free definition computer_science/theory_of_computation/index.html
Language of a Context Free Grammar definition computer_science/theory_of_computation/index.html

A Regular Expression Is a Regular Language shared words 75%

If a Language Is Regular Then It Is a Regular Expression corollary computer_science/theory_of_computation/index.html
A Regular Expression Is a Regular Language lemma computer_science/theory_of_computation/index.html

Absolutely Continuous Random Variable shared words 75%

Absolutely Continuous Random Variable definition probability/continuous_distributions.html
Continuous Random Variable definition probability/continuous_distributions.html

Adjoint of a Linear Operator shared words 75%

Adjoint of a Linear Operator definition linear_operators/banach_spaces.html
Self Adjoint Linear Operator definition linear_operators/everything.html

Adjoint of a Linear Operator shared words 75%

The Adjoint Is a Unique Linear Operator corollary linear_operators/banach_spaces.html
Adjoint of a Linear Operator definition linear_operators/banach_spaces.html

Adjoint of a Sesquilinear Form shared words 75%

Adjoint of a Sesquilinear Form definition linear_operators/everything.html
Self Adjoint Sesquilinear Form definition linear_operators/everything.html

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Arbitrary Intersection Indexed Notation definition fundamentals/sets/union_and_intersection.html

Bounded Linear Functional shared words 75%

Bounded Linear Functional definition linear_operators/linear_functionals.html
Bounded Linear Functional Integral lemma linear_operators/everything.html

Cauchy Subtraction Relation shared words 75%

Cauchy Subtraction Relation definition analysis/single_variable/real_numbers.html
The Cauchy Subtraction Relation is an Equivalence Relation proposition analysis/single_variable/real_numbers.html

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Pure Unit Quaternion definition algebra/quaternions.html
Conjugation of a Pure Quaternion by a Unit Quaternion is Pure lemma algebra/quaternions.html

Context Free Grammar shared words 75%

Context Free Grammar definition computer_science/theory_of_computation/index.html
Language of a Context Free Grammar definition computer_science/theory_of_computation/index.html

Divisor Counting Function shared words 75%

Divisor Counting Function definition number_theory/arithmetic_functions.html
Equation for the Divisor Counting Function proposition number_theory/arithmetic_functions.html

Divisor Counting Function shared words 75%

Divisor Counting Function definition number_theory/arithmetic_functions.html
The Divisor Counting Function Is Multiplicative proposition number_theory/quotient_and_remainder.html

Divisor Sum Function shared words 75%

Divisor Sum Function definition number_theory/arithmetic_functions.html
Equation for the Divisor Sum Function proposition number_theory/arithmetic_functions.html

Dominated Convergence Theorem shared words 75%

Dominated Convergence Theorem theorem measure_theory/integration.html
Dominated Convergence Theorem for Expectations theorem probability/probability_measures.html

Extended Ray Test for Measurability shared words 75%

Extended Ray Test for Measurability proposition measure_theory/measurable_spaces_and_functions.html
Ray Test for Measurability proposition measure_theory/measurable_spaces_and_functions.html

Finite Rank Linear Operator shared words 75%

Finite Rank Operator definition linear_operators/compact_operators.html
Finite Rank Linear Operator definition linear_operators/everything.html

Fractional Part of Floor shared words 75%

Fractional Part of Floor Bound corollary number_theory/floor_and_ceiling.html
Fractional Part of Floor definition number_theory/floor_and_ceiling.html

Function goes to Infinity at a Point From the Right shared words 75%

Function goes to Infinity at a Point From the Right definition analysis/single_variable/mean_value_theorem.html
Function goes to Infinity at a Point From the Right Alternate Characterization proposition analysis/single_variable/mean_value_theorem.html

Generlized Nondeterministic Finite Automaton shared words 75%

Generlized Nondeterministic Finite Automaton definition computer_science/theory_of_computation/index.html
Nondeterministic Finite Automaton definition computer_science/theory_of_computation/index.html

Geometric Power Series shared words 75%

Geometric Power Series with Offset corollary analysis/single_variable/summations_and_series.html
Geometric Power Series proposition analysis/single_variable/summations_and_series.html

If a Square Free Number Divides a Square then It divides the Square Root shared words 75%

If a Square Free Number Divides a Square then It divides the Square Root proposition number_theory/square_free_integers.html
If a Square Free Number Divides the Nth Power then it Divides the Nth Square Root proposition number_theory/square_free_integers.html

Image Maintains Subsets shared words 75%

Image Maintains Subsets proposition fundamentals/functions.html
Inverse Image Maintains Subsets proposition fundamentals/functions.html

Increasing Sequence of Sets shared words 75%

Increasing Sequence of Sets definition probability/scratch/limits_of_sets.html
Increasing Sequence of Sets definition probability/scratch/probability_measures.html
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Inner Product Space shared words 75%

Inner Product Space definition algebra/linear/vector_spaces/linear_transformations.html
Orthogonal in an Inner Product Space definition linear_operators/hilbert_spaces.html

Integer Pair Relation shared words 75%

Integer Pair Relation definition analysis/single_variable/dedekind_cuts.html
Integer Pair Relation is an Equivalence Relation proposition analysis/single_variable/dedekind_cuts.html

intersection factors from intersection shared words 75%

intersection factors from intersection proposition fundamentals/sets/union_and_intersection.html
intersection factors from union proposition fundamentals/sets/union_and_intersection.html

Least Common Multiple shared words 75%

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Least Common Multiple definition number_theory/gcd_and_lcm.html

Length of a Zero Indexed Tuple shared words 75%

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Length of a Zero Indexed Tuple proposition fundamentals/sets/tuples.html

Monotone Convergence Theorem shared words 75%

Monotone Convergence Theorem theorem measure_theory/integration.html
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Monte Carlo Estimator shared words 75%

Monte Carlo Estimator definition probability/monte_carlo.html
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Normed Vector Space shared words 75%

The Dual Space Is a Normed Vector Space corollary linear_operators/linear_functionals.html
Normed Vector Space definition linear_operators/banach_spaces.html

Product with a Real Quaternion shared words 75%

Real Part of Quaternion Product corollary algebra/quaternions.html
Product with a Real Quaternion proposition algebra/quaternions.html

Russian Roulette Estimator shared words 75%

Russian Roulette Estimator definition probability/monte_carlo.html
Russian Roulette Estimator is Unbiased proposition probability/monte_carlo.html

Self Adjoint Linear Operator shared words 75%

Self Adjoint Linear Operator definition linear_operators/everything.html
Self-Adjoint Operator definition linear_operators/operators_on_hilbert_spaces.html

Sequence Goes to Infinity shared words 75%

Sequence Goes to Infinity definition analysis/single_variable/limits.html
Sequence Goes to Negative Infinity definition analysis/single_variable/limits.html

Sequence Goes to Infinity shared words 75%

Sequence Goes to Infinity definition analysis/single_variable/limits.html
Sequence Goes to Positive Infinity definition analysis/single_variable/limits.html

Sorted in Ascending Order shared words 75%

Sorted in Ascending Order definition fundamentals/sets/tuples.html
Sorted in Strictly Ascending Order definition fundamentals/sets/tuples.html

Squared Sum Partition shared words 75%

Squared Sum Partition definition computer_science/complexity/complexity.html
Squared Sum Partition is NP-Complete proposition computer_science/complexity/complexity.html

Sum of Consecutive Integers shared words 75%

Sum of the Squares of Consecutive Integers proposition number_theory/induction.html
Sum of Consecutive Integers theorem fundamentals/summations.html

Vector Part of Quaternion Product shared words 75%

Vector Part of Quaternion Product corollary algebra/quaternions.html
Vector Part of the Quaternion definition algebra/quaternions.html

$Θ ρ ϵ η Π α τ π$

Cleanup

Empty Statements 23

Duplicates 214