# Category:Explanation Required

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This category is for pages which have a point which needs explaining.

See the Explain template.

## Subcategories

This category has the following 37 subcategories, out of 37 total.

### A

- Asymptotic Distribution (2 P)

### D

- Definitions/Monoid Rings (4 P)
- Definitions/Smooth Curves (6 P)
- Definitions/Sobolev Spaces (1 P)
- Differential Forms (empty)
- Divergent Products (empty)

### E

### F

- Fatou's Lemma for Integrals (3 P)
- Formal Laurent Series (empty)

### H

### M

- Minimum Variance Unbiased Estimators (empty)
- Monoid Rings (empty)

### N

- Natural Transformations (empty)

### O

- Outer Jordan Content (3 P)

### R

### S

- Semidirect Products (4 P)
- Smooth Curves (empty)
- Sobolev Spaces (2 P)
- Stone-Weierstrass Theorem (3 P)
- Strong Operator Topology (empty)

### T

## Pages in category "Explanation Required"

The following 200 pages are in this category, out of 1,092 total.

(previous page) (next page)### A

- Abel's Theorem
- Abel's Theorem/Historical Note
- Acceleration of Particle moving in Circle
- Accumulation Point of Infinite Sequence in First-Countable Space is Subsequential Limit
- Accumulation Point of Sequence of Reciprocals and Reciprocals + 1
- Addition of Natural Numbers is Provable
- Definition:Aleph Mapping
- Alexander's Compactness Theorem
- Definition:Algebraic Integer
- Alternating Sum and Difference of Factorials to Infinity
- Analytic Continuation of Dirichlet L-Function
- Analytic Continuation of Riemann Zeta Function using Jacobi Theta Function
- Angle Between Non-Zero Vectors Always Defined
- Angular Velocity is Vector Quantity
- Approximation to Reciprocal times Derivative of Gamma Function
- Approximation to Stirling's Formula for Gamma Function
- Arc Length of Sector
- Definition:Archimedean Property
- Definition:Archimedean Property/Norm
- Arens-Fort Space is not Locally Connected
- Definition:Argument of Complex Number
- Argument Principle
- Definition:Arithmetical Hierarchy
- Artin's Theorem on Alternative Algebras
- Definition:Artinian Ring
- Arzelà-Ascoli Theorem
- Definition:Ascending Chain Condition/Ideals
- Definition:Asymptotic Distribution
- Automorphism Group/Examples/Cyclic Group C8
- Auxiliary Approximating Relation has Interpolation Property
- Ax-Grothendieck Theorem
- Axiom:Axiom of Choice for Finite Sets
- Axiom:Axiom of Choice for Finite Sets/Proof from Hall's Marriage Theorem
- Axiom of Choice implies Zorn's Lemma
- Axiom of Choice implies Zorn's Lemma/Proof 2
- Axiom:Axiom of Continuity

### B

- Baire-Osgood Theorem
- Banach-Alaoglu Theorem
- Banach-Alaoglu Theorem/Proof 1
- Banach-Alaoglu Theorem/Proof 2
- Barbier's Theorem
- Definition:Barrier
- Basis for Open Ordinal Topology
- Basis of Free Module is No Greater than Generator
- Bendixson-Dulac Theorem
- Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y')/Lemma 3
- Bertrand's Theorem
- Bertrand's Theorem/Asymptotic Proof
- Bertrand's Theorem/Lemma
- Bertrand's Theorem/Non-Perturbative Proof
- Bertrand-Chebyshev Theorem/Also known as
- Definition:Between (Geometry)/N-dimensional Euclidean space Intuition
- Definition:Bilinear Form (Functional Analysis)
- Definition:Bilinear Functional
- Boolean Algebra is Equivalent to Boolean Lattice
- Boolean Prime Ideal Theorem
- Boolean Prime Ideal Theorem/Proof 1
- Definition:Boolean Satisfiability Problem
- Bounded Function is of Exponential Order Zero
- Bourbaki-Witt Fixed Point Theorem
- Branch of Finite Tree is Finite
- Burnout Height of Upward Rocket under Constant Gravity
- Bézout's Identity
- Bézout's Identity/Proof 4

### C

- Definition:C*-Algebra
- Canonical Injection into Cartesian Product of Modules
- Canonical Injection into Cartesian Product of Modules/Proof 2
- Canonical Order Well-Orders Ordered Pairs of Ordinals
- Definition:Cantor Normal Form
- Cantor Set is Uncountable
- Cantor Set is Uncountable/Proof 1
- Cantor Space is Nowhere Dense
- Cantor Space is Nowhere Dense/Proof 1
- Cantor-Bernstein-Schröder Theorem
- Cantor-Bernstein-Schröder Theorem/Proof 1
- Cantor-Dedekind Hypothesis
- Carathéodory's Theorem (Analysis)
- Cardinal Number is Ordinal
- Cardinal Number Less than Ordinal
- Cardinality of Set Union/General Case
- Carroll Paradox
- Cartan-Hadamard Theorem
- Definition:Cartesian Product/Family of Sets/Axiom of Choice
- Cauchy Distribution is Symmetric about Median
- Definition:Cauchy Functional Equation
- Cauchy Mean Value Theorem
- Cauchy Mean Value Theorem/Geometrical Interpretation
- Cauchy's Convergence Criterion for Series
- Cauchy's Integral Formula
- Cauchy's Integral Formula/General Result
- Cauchy-Hadamard Theorem/Complex Case
- Cauchy-Hadamard Theorem/Complex Case/Proof 2
- Cavalieri's Principle
- Cayley-Hamilton Theorem/Matrix
- Definition:Central Field
- Central Field is Field of Functional
- Cesàro Mean
- Change of Coordinate Vectors Under Linear Transformation
- Definition:Character (Number Theory)
- Definition:Character (Representation Theory)
- Character of Representations over C are Algebraic Integers
- Characterisation of Linearly Independent Set through Free Module Indexed by Set
- Characterisation of Local Rings
- Characterisation of Ordered Fields
- Characteristic Function of Random Variable is Well-Defined
- Characteristic of Increasing Mapping from Toset to Order Complete Toset
- Characterization of Continuity of Linear Functional in Weak Topology
- Characterization of Lipschitz Continuity on Shift of Finite Type by Variations
- Characterization of Measures
- Chebyshev's Inequality
- Chebyshev's Sum Inequality
- Chebyshev's Sum Inequality/Continuous
- Chi-Squared Test for Goodness of Fit/Examples
- Chi-Squared Test for Goodness of Fit/Examples/Cast of Dice
- Definition:Chi-Squared Test/Goodness of Fit
- Chinese Remainder Theorem (Commutative Algebra)
- Chinese Remainder Theorem (Commutative Algebra)/Proof 1
- Chinese Remainder Theorem (Commutative Algebra)/Proof 2
- Circle is Curve of Second Degree
- Circle of Apollonius is Circle
- Circle of Apollonius is Circle/Proof 1
- Class Equality is Reflexive
- Class Equality is Transitive
- Class of All Ordinals is Ordinal
- Classical Probability is Probability Measure
- Classification of Compact One-Manifolds/Lemma 1
- Classification of Compact Two-Manifolds
- Closed Interval in Complete Lattice is Complete Lattice
- Closed Linear Subspaces Closed under Setwise Addition
- Closed Real Interval is Neighborhood Except at Endpoints
- Closed Unit Interval is not Countably Infinite Union of Disjoint Closed Sets
- Closure of Irreducible Subspace is Irreducible
- Closure of Irreducible Subspace is Irreducible/Proof 1
- Closure of product equals product of closures
- Closure of Subset of Double Pointed Topological Space
- CNF Satisfiability Problem is NP-Complete
- Definition:Cofinal Relation on Ordinals
- Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube
- Commutativity of Incidence Matrix with its Transpose for Symmetric Design
- Compact Subspace of Linearly Ordered Space
- Compact Subspace of Linearly Ordered Space/Reverse Implication
- Compact Subspace of Linearly Ordered Space/Reverse Implication/Proof 2
- Compactness Properties Preserved under Continuous Surjection
- Compactness Theorem
- Complete and Totally Bounded Metric Space is Sequentially Compact
- Complete and Totally Bounded Metric Space is Sequentially Compact/Proof 1
- Definition:Complete Ritz Sequence
- Completeness Theorem for Semantic Tableaus
- Completion Theorem (Metric Space)/Lemma 4
- Complex Function is Entire iff it has Everywhere Convergent Power Series
- Complex Numbers as External Direct Product
- Complex Sequence is Null iff Modulus of Sequence is Null
- Complex Sequence is Null iff Modulus of Sequence is Null/Proof 1
- Component of Vector is Scalar Projection on Standard Ordered Basis Element
- Composite of Continuous Mappings is Continuous/Corollary
- Condition for Agreement of Family of Mappings
- Condition for Cofinal Nonlimit Ordinals
- Condition for Composition Series
- Condition for Differentiable Functional to have Extremum
- Condition for Injective Mapping on Ordinals
- Condition for Rational to be Convergent
- Condition for Set Union Equivalent to Associated Cardinal Number
- Condition on Congruence Relations for Cancellable Monoid to be Group/Counterexample
- Conditions for C^1 Smooth Solution of Euler's Equation to have Second Derivative
- Conditions for Extremal Embedding in Field of Functional
- Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation/Corollary 2
- Conditions for Strong Minimum of Functional
- Conformality is Equivalence Relation on Set of Riemannian Metrics
- Conjugacy Action on Group Elements is Group Action
- Conjugacy Action on Group Elements is Group Action/Proof 2
- Definition:Conjugate Point (Calculus of Variations)
- Definition:Conjugate Point (Calculus of Variations)/Definition 1
- Definition:Conjugate Point (Calculus of Variations)/Definition 3
- Definition:Conjugate Point (Calculus of Variations)/wrt original Functional
- Connected Subspace of Linearly Ordered Space
- Definition:Connection
- Consistency Principle for Binary Mess
- Construction of Conic Section
- Construction of Inverse Completion/Quotient Mapping/Image of Cancellable Elements
- Definition:Content of Polynomial/GCD Domain
- Definition:Continued Fraction
- Definition:Continued Fraction Expansion
- Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself
- Definition:Continued Fraction/Expansion of Real Number
- Continuous Function on Closed Real Interval is Uniformly Continuous
- Continuous Function on Closed Real Interval is Uniformly Continuous/Proof 2
- Continuous iff Way Below iff There Exists Element that Way Below and Way Below
- Continuous Image of Compact Space is Compact/Corollary 3
- Continuous Image of Compact Space is Compact/Corollary 3/Proof 2
- Continuous Image of Connected Space is Connected
- Continuous Image of Connected Space is Connected/Corollary 3
- Continuous Image of Connected Space is Connected/Proof 1
- Continuous Implies Locally Bounded
- Continuous Linear Transformation Algebra with Supremum Operator Norm is Normed Algebra
- Continuous Linear Transformation Space as Banach Algebra
- Continuous Midpoint-Convex Function is Convex
- Definition:Continuous Total Preordering
- Definition:Continuously Embedded
- Contour Integral of Gamma Function
- Definition:Contravariant Tensor