# Category:Improvements

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These are pages containing proofs that, although correct, are not the best they could be.

See the Improve template.

## Pages in category "Improvements"

The following 103 pages are in this category, out of 103 total.

### A

### C

- Cantor's Theorem (Strong Version)/Proof 2
- Characteristic of Ordered Integral Domain is Zero
- Definition:Chess
- Definition:Class (Descriptive Statistics)
- Compact First-Countable Space is Sequentially Compact
- 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 Theorem
- Condition for Ideal to be Total Ring
- Conditions for Strong Minimum of Functional
- Continuum Property/Proof 1
- Definition:Contravariant Hom Functor
- Cosine of Sum/Proof 4
- Countable Union of Finite Sets is Countable
- Definition:Covariant Hom Functor

### D

- De Morgan's Laws imply Uniquely Complemented Lattice is Boolean Lattice
- Definite Integral from 0 to Half Pi of Even Power of Sine x
- Definite Integral from 0 to Half Pi of Even Power of Sine x/Proof 1
- Definite Integral from 0 to Half Pi of Odd Power of Sine x
- Definite Integral from 0 to Half Pi of Odd Power of Sine x/Proof 1
- User:Dfeuer/thought
- Disjunction has no Inverse

### E

- Eigenvalues of Normal Operator have Orthogonal Eigenspaces
- Empty Group Word is Reduced
- Equation for Perpendicular Bisector of Two Points
- Equivalence of Definitions of Transitive Closure (Set Theory)
- Equivalence of Subobjects is Equivalence
- Euler's Equation for Vanishing Variation is Invariant under Coordinate Transformations
- Euler's Number: Limit of Sequence implies Base of Logarithm
- Euler's Theorem for Planar Graphs
- Exponential of Sum/Real Numbers/Corollary

### F

### G

### I

### M

### N

- Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite
- Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions
- Necessary Condition for Integral Functional to have Extremum for given Function/Dependent on Nth Derivative of Function
- Non-Equivalence as Equivalence with Negation/Formulation 1/Forward Implication/Proof
- Definition:Number Base/Integers

### O

- Open Sets of Double Pointed Topology/Corollary
- Order Topology equals Dual Order Topology
- Order Topology on Convex Subset is Subspace Topology
- Order-Extension Principle/Strict/Proof 2
- Ordinal Subset is Well-Ordered
- Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral
- Oscillation at Point (Infimum) equals Oscillation at Point (Limit)/Lemma

### P

- P-Product Metrics on Real Vector Space are Topologically Equivalent
- P-Product Metrics on Real Vector Space are Topologically Equivalent/Inequality for General Case
- Perpendicular Bisector of Triangle is Altitude of Medial Triangle
- Primary Decomposition Theorem
- Prime Magic Square/Examples/Order 3/Smallest
- Primitive of x over Root of a x + b by Root of p x + q
- Product of Sigma and Euler Phi Functions
- Products of 2-Digit Pairs which Reversed reveal Same Product
- Projection from Product Category
- Pullback of Quotient Group Isomorphism/Examples/C6 with A4

### R

### S

- Seifert-van Kampen Theorem
- Shape of Cosecant Function
- Shape of Secant Function
- Sign of Composition of Permutations
- Solution to Legendre's Differential Equation
- Strictly Increasing Sequence induces Partition
- Strictly Increasing Sequence of Natural Numbers
- Subband of Induced Operation is Set of Subbands
- Subset Relation is Compatible with Subset Product
- Subset Relation is Compatible with Subset Product/Corollary 2
- Sufficient Conditions for Uncountability
- Sum of Indexed Suprema
- Sums of Squares of Diagonals of Order 3 Magic Square
- Supremum of Absolute Value of Difference equals Supremum of Difference