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 161 pages are in this category, out of 161 total.
A
B
C
- Cardinal Equal to Collection of All Dominated Ordinals
- Characteristic of Ordered Integral Domain is Zero
- Characteristics of Traversable Graph
- Characterization of Lipschitz Continuity on Shift of Finite Type by Variations
- Definition:Chess
- Circle of Apollonius is Circle
- Circle of Apollonius is Circle/Proof 2
- Definition:Class Interval
- Definition:Classical Propositional Logic
- Closure of Irreducible Subspace is Irreducible
- Compact First-Countable Space is Sequentially Compact
- Compact Subspace of Linearly Ordered Space/Reverse Implication
- Compact Subspace of Linearly Ordered Space/Reverse Implication/Proof 2
- Compactness Theorem/Proof using Henkin Construction
- Composition of Mappings is Associative
- Condition for Ideal to be Total Ring
- Condition on Congruence Relations for Cancellable Monoid to be Group/Counterexample
- Conditions for Strong Minimum of Functional
- Definition:Connection on Manifold
- Definition:Consistent (Logic)/Set of Formulas/Propositional Logic
- Definition:Contravariant Hom Functor
- Definition:Convex Polygon/Definition 3
- Cosine of Sum/Proof 4
- Countable Union of Finite Sets is Countable
- Definition:Covariant Hom Functor
- Cycle does not Contain Subcycles
D
- De Morgan's Laws (Predicate Logic)/Denial of Universality/Formulation 2/Forward Implication
- 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
- Determinant of Linear Operator is Well Defined
- Dimension of R-Module R is 1
- Dirichlet Integral/Proof 2
- Disjunction has no Inverse
E
- Eigenvalues of Normal Operator have Orthogonal Eigenspaces
- Empty Set is Well-Ordered
- Equation for Perpendicular Bisector of Two Points
- Equivalence of Definitions of Normal Subgroup
- 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 Theorem for Planar Graphs
- Definition:Exact Sequence of Modules
- Existential Instantiation/Informal Statement
- Expansion Theorem for Determinants
- Exponential of Sum/Real Numbers/Corollary
F
G
H
I
- Image of Small Class under Mapping is Small
- Template:Improve
- Increasing Sequence in Ordered Set Terminates iff Maximal Element
- Increasing Sequence of Sets forms Nest
- Indexed Summation of Multiple of Mapping
- Intersection of Ordinals is Smallest
- Definition:Inverse Sine/Real/Arcsine
- Definition:Irreducible (Representation Theory)
- Definition:Irreducible (Representation Theory)/G-Module
L
- Least Upper Bound Property/Proof 1
- Definition:Legendre Transform
- Lexicographic Product of Family of Ordered Sets is Ordered Set
- Limit Ordinal Equals its Union
- Linear Combination of Measures
- Liouville's Theorem (Complex Analysis)
- Liouville's Theorem (Complex Analysis)/Proof 1
- Logarithmic Integral as Non-Convergent Series
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
- Newton's Identities/Proof 1
- Non-Equivalence as Equivalence with Negation/Formulation 1/Forward Implication/Proof
- Norm on Vector Space is Quasinorm
- Definition:Number Base/Integers
- Number of Arrangements of n Objects of m Types
- Number which is Square and Cube Modulo 7
O
- Open Sets of Double Pointed Topology/Corollary
- Order Isomorphism iff Strictly Increasing Surjection
- 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
- Orthonormal Subset of Hilbert Space Extends to Basis
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
- 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 Divisor Sum and Euler Phi Functions
- Products of 2-Digit Pairs which Reversed reveal Same Product
- Definition:Projection
- Projection from Product Category
- Definition:Proper Relational Structure
- Pullback of Quotient Group Isomorphism/Examples/C6 with A4
R
- Rational Numbers with Denominators Coprime to Prime under Addition form Group
- Real Numbers of Type Rational a plus b root 2 form Field
- Reciprocal of Function of Bounded Variation Bounded away from Zero is of Bounded Variation
- Recursively Defined Sequence/Examples/Term is Term of Index less 1 plus 6 times Term of Index less 2
- Relation between Two Ordinals
- Relationship between Transitive Closure Definitions
S
- Scheffé's Lemma/Corollary
- Seifert-van Kampen Theorem
- Definition:Set of Literals
- Definition:Set Partition
- Shape of Cosecant Function
- Shape of Cosine Function
- Shape of Secant Function
- Sigma-Algebra Generated by Finite Partition is Finite Sub-Sigma-Algebra
- Sign of Composition of Permutations
- Sine and Cosine are Periodic on Reals/Corollary/Cosine
- Sine and Cosine are Periodic on Reals/Corollary/Sine
- Smallest Field is Field
- Solution to Legendre's Differential Equation
- Stopped Supermartingale is Supermartingale
- Strictly Increasing Sequence induces Partition
- Strictly Increasing Sequence of Natural Numbers
- Subband of Induced Operation is Set of Subbands
- Subgroup is Subset of Conjugate iff Normal
- Subgroup is Superset of Conjugate iff Normal
- 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