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- 17:57, 21 September 2021 Pointwise Maximum of Simple Functions is Simple/Proof 1 (hist | edit) [1,211 bytes] Caliburn (talk | contribs) (Created page with "== Proof == <onlyinclude> From Pointwise Sum of Simple Functions is Simple Function: :$f + g$ is simple. From Scalar Multiple of Simple...")
- 17:56, 21 September 2021 Pointwise Maximum of Simple Functions is Simple/Proof 2 (hist | edit) [1,424 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == {{:Pointwise Maximum of Simple Functions is Simple}} == Proof == <onlyinclude> Let: :$h = \max \set {f, g}$ From Pointwise Minimum of Measurable Functions i...")
- 17:10, 21 September 2021 Pointwise Maximum of Simple Functions is Simple (hist | edit) [746 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\struct {X, \Sigma}$ be a measurable space. Let $f, g : X \to \R$ be Definition:Simple Function|simple func...")
- 17:05, 21 September 2021 Maximum Function in terms of Absolute Value (hist | edit) [1,243 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $x$ and $y$ be real numbers. Then: :$\ds \max \set {x, y} = \frac 1 2 \paren {\paren {x + y} + \size {x - y} }$ </...")
- 16:58, 21 September 2021 Pointwise Minimum of Simple Functions is Simple (hist | edit) [1,606 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\struct {X, \Sigma}$ be a measurable space. Let $f, g : X \to \R$ be Definition:Simple Function|simple func...")
- 18:51, 20 September 2021 Absolute Value of Measurable Function is Measurable (hist | edit) [1,671 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\struct {X, \Sigma}$ be a measurable space. Let $f : X \to \overline \R$ be a Definition:Measurable Functi...")
- 10:05, 20 September 2021 Leigh.Samphier/Sandbox/Induction Example (hist | edit) [1,446 bytes] Leigh.Samphier (talk | contribs) (Created page with "== Proof == Proof by induction: For all $n \in \N_{>0}$, let $\map P n$ be the proposition: :$\ds n^2 = \s...")
- 16:59, 19 September 2021 Pullback of Riemannian Metric by Smooth Mapping is Riemannian Metric iff Mapping is Immersion (hist | edit) [1,026 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $\struct {\tilde M, \tilde g}$ be a Riemannian manifold with or without boundary. Let...")
- 01:33, 19 September 2021 Leigh.Samphier/Sandbox/Equivalence of Definitions of Convergence in Normed Division Rings (hist | edit) [4,291 bytes] Leigh.Samphier (talk | contribs) (Created page with "== Theorem == Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a Definition:Cauchy Sequence...")
- 01:10, 19 September 2021 Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Proof 2 (hist | edit) [1,360 bytes] Leigh.Samphier (talk | contribs) (Created page with "{{Proofread}} == Theorem == {{:Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence}} == Proof == From P-adic Integers are Compact Subspace: :$\...")
- 00:27, 19 September 2021 Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Proof 1 (hist | edit) [1,484 bytes] Leigh.Samphier (talk | contribs) (Created page with "{{Proofread}} == Theorem == {{:Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence}} == Proof == {{qed}} == Sources == * {{BookReference|p-adic An...")
- 08:23, 18 September 2021 Equations defining Plane Reflection/Cartesian (hist | edit) [2,981 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\LL$ be a straight line through the origin $O$ of a Definition:Cartesian Plane|cartesian plan...") originally created as "Equations defining Plane Reflection"
- 12:43, 14 September 2021 Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence (hist | edit) [965 bytes] Leigh.Samphier (talk | contribs) (Created page with "{{Proofread}} == Theorem == Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$. Le...")
- 06:44, 14 September 2021 Similarity Mapping on Plane Commutes with Half Turn about Origin (hist | edit) [2,515 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\beta \in \R_{>0}$ be a (strictly) positive real number. Let $s_{-\beta}: \R^2 \to \R^2$ be the Definition:...")
- 06:43, 14 September 2021 Similarity Mapping on Plane with Negative Parameter (hist | edit) [2,352 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\beta \in \R_{<0}$ be a (strictly) negative real number. Let $s_\beta: \R^2 \to \R^2$ be the Definition:Sim...")
- 06:38, 14 September 2021 Similarity Mapping on Plane with Scale Factor Minus 1 (hist | edit) [1,348 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $s_{-1}: \R^2 \to \R^2$ be a similarity mapping on $\R^2$ whose scale factor is $-1$. Then $...")
- 20:57, 13 September 2021 Equality of Integers to the Power of Each Other/Proof 2 (hist | edit) [1,242 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Equality of Integers to the Power of Each Other}} == Proof == <onlyinclude> {{WLOG}} suppose $m > n$. Write $m = n + x$, where $x$ is an Definition:Intege...")
- 20:53, 13 September 2021 Equality of Integers to the Power of Each Other/Proof 1 (hist | edit) [1,385 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Equality of Integers to the Power of Each Other}} == Proof == <onlyinclude> We have: {{begin-eqn}} {{eqn | l = n^m | r = m^n | c = }} {{eqn | l...")
- 19:04, 13 September 2021 Unit Tangent Bundle is Connected iff Manifold is Connected (hist | edit) [779 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $\struct {M,g}$ be a Riemannian manifold of dimension $n > 1$. Let $...")
- 18:54, 13 September 2021 Characterization of Unit Tangent Bundle (hist | edit) [1,006 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $\struct {M, g}$ be a Riemannian manifold with or without boundary. Let $TM$ be the Definition:Tangent Bundle|tangent...")
- 12:10, 12 September 2021 Definition:P-adic Integer/Notation (hist | edit) [505 bytes] Leigh.Samphier (talk | contribs) (Created page with "== Notation == Let $\Z_p$ denote the $p$-adic integers for some prime $p$. <onlyinclude> The...") originally created as "Leigh.Samphier/Sandbox/Definition:P-adic Integer/Notation"
- 10:00, 12 September 2021 Generating Function for Elementary Symmetric Function/Outline (hist | edit) [863 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Generating Function for Elementary Symmetric Function}} == Outline == <onlyinclude> Generating function discovery methods...")
- 19:12, 11 September 2021 Quantity of Positive Integers Divisible by Particular Integer (hist | edit) [1,065 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == Let $d$ be a positive integer. Let $x \ge 1$ be a real number. Then: :$\ds \sum_{n \le x, \, d...")
- 12:03, 11 September 2021 Value of Vandermonde Determinant/Formulation 2 (hist | edit) [730 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $V_n$ be the '''Vandermonde determinant of order $n$''' defined as the following formulati...")
- 11:55, 11 September 2021 Value of Vandermonde Determinant/Formulation 1 (hist | edit) [2,012 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $V_n$ be the '''Vandermonde determinant of order $n$''' defined as the following formulati...")
- 20:07, 9 September 2021 Existence of Orthonormal Frames (hist | edit) [1,559 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $\struct{M, g}$ be a $n$-dimensional Riemannian manifold with or with...")
- 06:41, 9 September 2021 Inverse of Similarity Mapping (hist | edit) [2,448 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $G$ be a vector space over a field $K$. Let $\beta \in K$ such that $\beta \ne 0$. Le...")
- 06:31, 9 September 2021 Similarity Mapping is Automorphism (hist | edit) [3,595 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $G$ be a vector space over a field $K$. Let $\beta \in K$. Let $s_\beta: G \to G$ be...")
- 15:55, 6 September 2021 Smooth Mapping between Equidimensional Riemannian Manifolds is Local Isometry iff it is Isometry (hist | edit) [746 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $\struct {M, g}$ and $\struct {\tilde M, \tilde g}$ be Riemannian manifolds of the same Definition...")
- 15:14, 6 September 2021 Equivalence of Almost Equal Locally Integrable Functions and their Distributions (hist | edit) [855 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $f, g \in \map {L^1_{loc}} \R$ be locally integrable functions. Let $T_f, T_g$ be the Definition:Distribution|...")
- 15:01, 6 September 2021 Heaviside Step Function is Locally Integrable (hist | edit) [647 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $u_c$ be a Heaviside step function. Then $u_c$ is a Definition:Locally Integrable Function|locally integrable fun...")
- 20:26, 5 September 2021 Locally Integrable Function defines Distribution (hist | edit) [2,943 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $f \in \map {L^1_{loc}} {\R^d}$ be a locally integrable function. Let $\phi \in \map \DD {\R^d}$ be a Definiti...")
- 19:55, 5 September 2021 Distribution Space with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space (hist | edit) [1,031 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $\map {\DD'} {\R^d}$ be the distribution space. Let $\struct {\C, +_\C, \times_\C}$ be the Definition:Field of Complex...")
- 20:41, 31 August 2021 Kernel of Linear Transformation is Submodule (hist | edit) [1,678 bytes] Prime.mover (talk | contribs) (Created page with "{{Refactor|In progress: one of $4$ results from Linear Transformation of Submodule}} == Theorem == Let $\struct {R, +_R, \times_R}$ be a Definition:Ring (Abstract Alge...")
- 20:39, 31 August 2021 Image of Linear Transformation is Submodule (hist | edit) [1,033 bytes] Prime.mover (talk | contribs) (Created page with "{{Refactor|In progress: one of $4$ results from Linear Transformation of Submodule}} == Theorem == Let $\struct {R, +_R, \times_R}$ be a Definition:Ring (Abstract Alge...")
- 20:05, 31 August 2021 Smooth Manifold admits Riemannian Metric (hist | edit) [464 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $M$ be a smooth manifold. Then $M$ admits a Riemannian metric. == Proof == {{ProofWante...")
- 19:31, 31 August 2021 Test Function Space with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space (hist | edit) [5,613 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $\map \DD {\R^d}$ be the test function space. Let $\struct {\C, +_\C, \times_\C}$ be the Definition:Field of Complex N...") originally created as "Test Function Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space"
- 06:13, 31 August 2021 Preimage of Submodule under Linear Transformation is Submodule (hist | edit) [3,374 bytes] Prime.mover (talk | contribs) (Created page with "{{Refactor|In progress: one of $4$ results from Linear Transformation of Submodule}} == Theorem == Let $\struct {R, +_R, \times_R}$ be a Definition:Ring (Abstract Alge...")
- 17:18, 30 August 2021 Image of Submodule under Linear Transformation is Submodule (hist | edit) [2,479 bytes] Prime.mover (talk | contribs) (Created page with "{{Refactor|In progress: one of $4$ results from Linear Transformation of Submodule}} == Theorem == Let $\struct {R, +_R, \times_R}$ be a Definition:Ring (Abstract Alge...")
- 16:39, 30 August 2021 Elements of Submodule form Subgroup (hist | edit) [678 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\struct {R, +, \circ}$ be a ring. Let $\struct {G, +_G}$ be an abelian group. Let $\s...") originally created as "Vectors of Submodule form Subgroup"
- 20:11, 29 August 2021 Fundamental Solutions to Distributional Homogeneous ODE with Constant Coefficients differ by Classical Solution (hist | edit) [2,281 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $E_*, E, T \in \map {\DD'} \R$ be distributions. Let $D$ be an ordinary Definition:Differe...")
- 20:33, 28 August 2021 Sum of Finite Set of Submodules is Supremum of Lattice of Submodules (hist | edit) [903 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $R$ be a ring. Let $M = \struct {G, +, \circ}_R$ be an $R$-module. Let $H_1, H_...")
- 20:33, 28 August 2021 Set of Submodules of Module is Complete Lattice (hist | edit) [759 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $R$ be a ring. Let $M = \struct {G, +, \circ}_R$ be an $R$-module. Definition:Ordering|Ord...")
- 20:18, 28 August 2021 Intersection of Set of Submodules containing Subset is Smallest Submodule (hist | edit) [931 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $R$ be a ring. Let $M = \struct {G, +, \circ}_R$ be an $R$-module. Let $S \subse...")
- 19:56, 28 August 2021 Sum of Submodules is Submodule (hist | edit) [677 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $R$ be a ring. Let $M = \struct {G, +, \circ}_R$ be an $R$-module. Let $H$ and $K$ be Defin...")
- 07:37, 27 August 2021 Module of All Mappings/Examples (hist | edit) [558 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Module of All Mappings == <onlyinclude> The most important case of this example is when $M = R$. === Definition:Module...")
- 06:01, 27 August 2021 Vector Space over Division Subring is Vector Space/Special Case (hist | edit) [1,447 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$. Let $S$ be a ...")
- 19:44, 24 August 2021 Solution to Distributional Ordinary Differential Equation with Constant Coefficients (hist | edit) [3,665 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $D$ be an ordinary differential operator with constant Def...")
- 19:19, 23 August 2021 Gram-Schmidt Orthogonalization/Inner Product Space (hist | edit) [2,334 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $\struct {V, \innerprod \cdot \cdot}$ be an $n$-dimensional inner product space....")
- 20:00, 22 August 2021 Polarization Identity (hist | edit) [614 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space. Then for all $v, w \in V$ it holds that: :$\ds \inner...")