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Combined display of all available logs of ProofWiki. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).
- 16:27, 3 February 2023 Anghel talk contribs created page Path Component of Locally Path-Connected Space is Closed (Created page with "== Theorem == Let $T = \left({S, \tau}\right)$ be a locally path-connected topological space. Let $G$ be a path component of $T$. Then $G$ is open in $T$. == Proof == Let $x \in \partial G$, where $\partial G$ denotes the boundary of $G$. By {{Open-set-axiom|3}}, it follows that...")
- 20:48, 29 January 2023 Anghel talk contribs created page Book:Christian Berg/Mål- og integralteori (Created page with "{{Book|Mål- og integralteori|2001|Universitetsbogladen|87-7834-424-7|Christian Berg|Tage Gutmann Madsen}} This book is based on a set of notes originally written in 1970 by Tage Gutmann Madsen to be used at the basic course about measure theory at the University of Copenhagen. === Subject Matter === * Measure Theory == Download == The book can be downloaded as a PDF...")
- 20:41, 29 January 2023 Anghel talk contribs created page Book:Christian Berg/Metriske rum (Created page with "{{Book|Metriske rum|1997|Universitetsbogladen|87-91180-01-5|Christian Berg}} This book is based on a set of notes written in 1990 by the author to be used at the basic course about metric spaces at the University of Copenhagen. === Subject Matter === * Metric Spaces == Download == The book can be downloaded as a PDF file from the University of Copenhagen for free. [https://web.math.ku.dk/noter/filer/matemati...")
- 14:09, 29 January 2023 Anghel talk contribs created page Definition:Inner Product/Also Defined as (Created page with "== Also defined as == Let $V$ be a vector space over a field $\GF$ that is a subfield of $\R$ or $\C$. Let $\innerprod \cdot \cdot: V \times V \to \GF$ be an inner product on $\GF$. <onlyinclude> Some texts define an '''inner product''' only for vector spaces over $\R$ or $\C$. This ensures that for all $v \in V$, the Definition:Inner Pr...")
- 22:08, 28 January 2023 Anghel talk contribs created page Definition:Nondegenerate Symmetric Bilinear Form (This clears 6 red links.) Tag: New redirect
- 22:05, 28 January 2023 Anghel talk contribs created page Definition:Symmetric Bilinear Form/Nondegenerate (Created page with "== Definition == <onlyinclude> Let $\Bbb K$ be a field. Let $V$ be a vector space over $\Bbb K$. Let $b: V \times V \to \Bbb K$ be a symmetric bilinear form. Let $b$ be a nondegenerate bilinear form. Then $b$ is a '''nondegenerate symmetric bilinear form'''. </onlyinclude> == Sources == * {{MathWorld|Symmetric Biline...")
- 21:20, 28 January 2023 Anghel talk contribs moved page Gram-Schmidt Orthogonalization/Inner Product Space to Gram-Schmidt Orthogonalization/Corollary 2 without leaving a redirect
- 21:17, 28 January 2023 Anghel talk contribs moved page Gram-Schmidt Orthogonalization/Corollary to Gram-Schmidt Orthogonalization/Corollary 1 without leaving a redirect (Room for more corollaries)
- 21:17, 28 January 2023 Anghel talk contribs moved page Talk:Gram-Schmidt Orthogonalization/Corollary to Talk:Gram-Schmidt Orthogonalization/Corollary 1 without leaving a redirect (Room for more corollaries)
- 18:04, 27 January 2023 Anghel talk contribs created page Union of Simply Connected Sets with Path-Connected Intersection is Simply Connected (Created page with "== Theorem == Let $\struct {T, \tau}$ be a topological space. Consider all subsets of $T$ as subspaces, equipped with the subspace topology induced by $T$. Let $U$ and $V$ be open subsets of $T$ that are simply connected. Let $U \cap V$ be Definition:Non-Empty Set|no...")
- 16:18, 27 January 2023 Anghel talk contribs created page Cartesian Product of Intervals is Simply Connected (Created page with "== Theorem == Let $n \in \N$. For all $k \in \set {1, \ldots, n}$, let $\Bbb I_k$ be a real interval of any of the real interval types. Let $\tau_0$ denote the subspace topology on the cartesian product $\Bbb I_1 \times \ldots \times \Bbb I_n$, induced by the Euclidean topology on $\R...")
- 16:12, 27 January 2023 Anghel talk contribs created page Cartesian Product of Intervals is Convex Set (Created page with "== Theorem == Let $n \in \N$. For all $k \in \set {1, \ldots, n}$, let $\Bbb I_k$ be a real interval of any of the real interval types. Then the cartesian product $\Bbb I_1 \times \ldots \times \Bbb I_n$ is a convex set. == Proof == Let $\mathbf x, \mathbf y \in \Bbb I_1 \times \ldots \times \Bbb I_n$ with: {{begin-eqn}} {{eqn...")
- 14:51, 27 January 2023 Anghel talk contribs created page Talk:Completion of Nondegenerate Bases (Created page with "== Comment to User:Usagiop == There is already a condition on $\tuple {v_1, \ldots, v_k}$, as they are stated to be a nondegenerate $k$-tuple. So it is impossible that $v_1 = v_2$, as $\set {v_1 , v_2}$ is required to span a 2-dimensional subspace (or at least, that's how I interpret the definition of nondegenerate $k$-tuple )...")
- 14:44, 27 January 2023 Anghel talk contribs created page Definition talk:Nondegenerate Tuple of Elements of Scalar Product Space (Created page with "== Comment to User:Usagiop == The formulation could be clearer. I would write the requirement as: :Suppose for each $j \in \set {1, \ldots, k}$ and for all $i_1 , \ldots, i_j \in \set {1, \ldots, k}$ with $i_1 < i_2 < \ldots < i_j$ that: ::the vectors $\tuple {v_{i_1}, \ldots, v_{i_j} }$ span a nondegenerate Definition:Finite Dimensional...")
- 22:42, 26 January 2023 Anghel talk contribs created page Definition:Unbounded Real Interval without Endpoints (Redirected page to Definition:Real Interval/Unbounded without Endpoints) Tag: New redirect
- 22:41, 26 January 2023 Anghel talk contribs created page Definition:Real Interval/Unbounded without Endpoints (Created page with "== Definition == <onlyinclude> The '''unbounded (open) interval without endpoints''' is equal to the set of real numbers: :$\openint \gets \to = \R$ </onlyinclude> == Notation == {{:Definition:Real Interval/Notation}} == Also denoted as == The notation using $\infty$ is usual: {{begin-eqn}} {{eqn | l = \openint {-\infty} \infty | o = :=...")
- 12:35, 26 January 2023 Anghel talk contribs created page Template talk:SubjectCategory (Created page with "== Adding another parameter to the template == On a category page like ''Category:Convergent Sequences (Normed Vector Spaces)'' , the '''context''' parameter is given as "Normed Vector Spaces", which only exist as a definition in the singular form Definition:Normed Vector Space. As this occurs on multiple categories, it is a source of eight red links of Definition:Normed Vector Spaces. Could we add another optional parameter '''contextdisp''', which will displa...")
- 19:58, 25 January 2023 Anghel talk contribs created page Interior of Jordan Curve is Simply Connected (Created page with "== Theorem == Let $\gamma : \closedint 0 1 \to \R^2$ be a Jordan curve. Let $\Int \gamma$ denote the interior of $\gamma$. Let $\tau_0$ denote the subspace topology on $\Int \gamma$, induced by the Euclidean topology on $\R^2$. Then $\struct {\Int \gamma, \tau_0}$ is Definition:Simply Connected|simply connect...")
- 19:11, 25 January 2023 Anghel talk contribs created page Jordan Curve Bounding Loop in Euclidean Plane (Created page with "== Theorem == Let $f : \closedint 0 1 \to \R^2$ be a loop in the Euclidean plane $\R^2$. Let $\epsilon \in \R_{>0}$. Then there exists a Jordan curve $\gamma : \closedint 0 1 \to \R^2$ such that $\Img f \subseteq \Int \gamma$, and for all $t \in \closedint 0 1$: :$\map d {\map \gamma t, \Img f} < \epsilon$ where $\map d {\map \gamma t, \Img f}$ denotes the Definition:Euc...")
- 21:38, 22 January 2023 Anghel talk contribs created page Equivalence of Definitions of Generator of Unitary Module (The inclusion of the definitions are clunky in this theorem, as I tried to format the text as neatly as possible.)
- 22:29, 21 January 2023 Anghel talk contribs created page Closed Ball is Simply Connected (Created page with "== Theorem == Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space over $\R$ or $\C$. Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the norm $\norm {\,\cdot\,}$ on $V$. Let $\tau$ be the the topology on $V$ induced by the metric $d$. Let $v \in V$ and $\epsilon \in \R_{>0}$. Let $\map {B_\epsil...")
- 21:49, 21 January 2023 Anghel talk contribs created page Interior of Jordan Curve is Subset of Image of Null-Homotopy (Created page with "== Theorem == Let $f : \closedint 0 1 \to \R^2$ be a Jordan curve. Let $H : \closedint 0 1 \times \closedint 0 1 \to \R^2$ be a path homotopy between $f$ and a constant loop. Then: :$\Int f \subseteq \Img H$ where $\Int f$ denotes the interior of $f$, and $\Img H$ denotes the image of $H$. == Proof...")
- 16:19, 20 January 2023 Anghel talk contribs created page Definition:Generator of Unitary Module (Redirected page to Redigerer Definition:Generator of Module/Unitary) Tag: New redirect
- 15:09, 19 January 2023 Anghel talk contribs moved page Definition:Proper Submodule to Definition:Submodule/Proper (Transclusion.)
- 13:00, 19 January 2023 Anghel talk contribs created page Equivalence of Definitions of Generator of Module (Created page with "== Theorem == Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \subseteq M$ be a subset of $M$. {{TFAE|def = Generator of Module}} === Definition 1 === {{:Definition:Generator of Module/Definition 1}} === Definition 2 === {{:Definition:Generator of Module/Definition 2}} == Proof == ==...")
- 22:34, 18 January 2023 Anghel talk contribs created page Linear Transformation of Generated Module/Proof 2 (Created page with "== Theorem == {{:Linear Transformation of Generated Module}} == Proof == <onlyinclude> This proof assumes that $R$ is a ring with unity, so $G$ and $H$ become unitary modules. Let $y \in G$ be arbitrary. Then by definition of generator, $y$ is the linear combination of elements of $S$: :$\ds y = \sum_{k \mat...")
- 22:30, 18 January 2023 Anghel talk contribs created page Linear Transformation of Generated Module/Proof 1 (Created page with "== Theorem == {{:Linear Transformation of Generated Module}} == Proof == <onlyinclude> By definitions of generator of module and generated submodule, it follows that: :$\ds G := \bigcap \set { M' \subseteq G : S \subseteq M', \textrm {$M'$ is a submodule of $G$} }$ Set $S' := \set {x \in G: \map \phi x = \map \psi x}$. From Elements of Module with Equal Images under Linear Transform...")
- 22:16, 18 January 2023 Anghel talk contribs created page Category:Linear Transformation of Generated Module (Created page with "{{SubjectCategory|result = Linear Transformation of Generated Module}} Category:Linear Transformations")
- 14:28, 18 January 2023 Anghel talk contribs created page Generated Submodule may not equal Set of Linear Combinations (Created page with "== Theorem == Let $R = \set { 2 k : k \in \Z}$ be the set of all even integers. Let $S = \set {2}$ be a subset of $R$. Then $\struct {R, +, \times}$ is a ring, where the submodule generated by $S$ is equal to: :$R = \set { 2 k : k \in \Z}$ and the set of all Definition:Linear Combination|linear combinatio...")
- 22:22, 17 January 2023 Anghel talk contribs moved page Definition:Generator of Module/Definition 3 to Definition:Generator of Module/Definition 1 without leaving a redirect
- 22:17, 17 January 2023 Anghel talk contribs moved page Definition:Generator of Module/Definition 1 to Definition:Generator of Module/Unitary without leaving a redirect
- 20:58, 17 January 2023 Anghel talk contribs created page Definition talk:Generator of Module (Created page with "This page also needs some corrections, which I will try to do soon. --~~~~")
- 20:55, 17 January 2023 Anghel talk contribs moved page Equivalence of Definitions of Generated Submodule to Equivalence of Definitions of Generated Submodule over Ring with Unity without leaving a redirect (Better name)
- 20:43, 17 January 2023 Anghel talk contribs moved page Definition:Generated Submodule/Definition 2 to Definition:Generated Submodule/Unitary without leaving a redirect (Unique name)
- 12:53, 17 January 2023 Anghel talk contribs created page Distance between Points in Regular Hexagon (Created page with "== Theorem == Let $H$ be a regular hexagon embedded in the Euclidean plane $\R^2$. Let $s \in \R_{>0}$ be the side length of $H$. Let $\mathbf x, \mathbf y \in \R^2$ such that $\mathbf x$ and $\mathbf y$ lie in the interior of $H$, or on the Definition:Circumference of Geometr...")
- 22:07, 15 January 2023 Anghel talk contribs created page Definition:Bounded Metric Space/Definition 4 (Created page with "== Definition == Let $M = \struct {A, d}$ be a metric space. Let $M' = \struct {B, d_B}$ be a subspace of $M$. <onlyinclude> Let $x_0 \in A$. '''$M'$ is bounded (in $M$)''' {{iff}}: :$\exists \epsilon \in \R_{>0}: B \subseteq \map {B_\epsilon} {x_0}$ where $\map {B_\epsilon} {x_0}$ is the open $\epsilon$-ball of $x_0$. </onlyinclude> == Also see == * Equivalence of Definitions of...")
- 21:57, 15 January 2023 Anghel talk contribs created page Definition:Bounded Metric Space/Definition 3 (Created page with "== Definition == Let $M = \struct {A, d}$ be a metric space. Let $M' = \struct {B, d_B}$ be a subspace of $M$. <onlyinclude> '''$M'$''' is bounded in $M$ {{iff}}: :$\exists x \in A, \epsilon \in \R_{>0}: B \subseteq \map {B_\epsilon} x$ where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$. That is, $M'$ can be fitted inside an Definiti...")
- 23:38, 13 January 2023 Anghel talk contribs created page Definition:Null-Homotopy (Redirected page to Definition:Homotopy/Null-Homotopic/Null-Homotopy) Tag: New redirect
- 20:38, 13 January 2023 Anghel talk contribs created page Constant Loop is Loop (Created page with "== Theorem == Let $\struct {T, \tau}$ be a topological space. Let $p \in T$. Let $c_p : \closedint 0 1 \to T$ be the constant mapping defined by: :$\forall t \in \closedint 0 1 : \map {c_p} t = p$ Then $p$ is a loop in $T$. == Proof == From Constant Mapping is Continuous, it follows that $c_p$ is continuous. By definit...")
- 20:29, 13 January 2023 Anghel talk contribs created page Definition:Constant Loop (Topology) (Redirected page to Definition:Loop (Topology)/Constant Loop) Tag: New redirect
- 20:28, 13 January 2023 Anghel talk contribs created page Definition:Set of All Loops (Topology) (Redirected page to Definition:Loop (Topology)/Set of All Loops) Tag: New redirect
- 22:56, 11 January 2023 Anghel talk contribs created page Borsuk Null-Homotopy Lemma/Corollary (Created page with "== Corollary == <onlyinclude> Let $a,b \in \R^2$. </onlyinclude> Let $\struct {A, \tau_A}$ be a compact topological space. <onlyinclude> Let $f : A \to \R^2 \setminus \set {a,b}$ be a continuous injective mapping. Let $f$ be null-homotopic. Then $a$ and $b$ lie in the same component of $\R^2 \s...")
- 22:53, 11 January 2023 Anghel talk contribs created page Borsuk Null-Homotopy Lemma (Created page with "== Theorem == Let $\Bbb S^2$ denote the unit sphere in $\R^3$. Let $a,b \in \Bbb S^2$. Let $\struct {A, \tau_A}$ be a compact topological space. Let $f : A \to \Bbb S^2 \setminus \set {a,b}$ be a continuous injective mapping. Let $f$ be null-homotopic. Then $a$ and $b$ lie in the same D...")
- 21:47, 9 January 2023 Anghel talk contribs created page Area of Regular Hexagon/Proof 2 (Created page with "== Theorem == {{:Area of Regular Hexagon}} == Proof == <onlyinclude> A regular hexagon is a regular 6-sided polygon. Therefore: {{begin-eqn}} {{eqn | l = \AA | r = \dfrac 1 4 \times 6 \times s^2 \times \cot \dfrac \pi 6 | c = Area of Regular Polygon }} {{eqn | r = \dfrac 3 2 \times s^2 \times \sqrt 3 | c = Cotangent of 30 Degrees|Cotangent of $30...")
- 21:40, 9 January 2023 Anghel talk contribs created page Area of Regular Hexagon/Proof 1 (Created page with "== Theorem == {{:Area of Regular Hexagon}} == Proof == <onlyinclude> From Regular Hexagon is composed of Equilateral Triangles, it follows that a regular hexagon can be dissected into six identical equilateral triangles: :300px Let $\AA_T$ be the area of the bottom triangle. Then by A...")
- 21:30, 9 January 2023 Anghel talk contribs created page Category:Area of Regular Hexagon (Created page with "{{SubjectCategory|result = Area of Regular Hexagon}} Category:Hexagons")
- 20:07, 9 January 2023 Anghel talk contribs created page Regular Hexagon is composed of Equilateral Triangles (Created page with "== Theorem == Let $P$ be a regular hexagon. Let $O$ be the center of $P$. Then there exists a triangulation of $P$ into six identical equilateral triangles. The three vertices of each triangle are $O$ and two Definition:Adjacent Vertices of Polygon|adj...")
- 16:21, 9 January 2023 Anghel talk contribs created page Regular Polygon composed of Isosceles Triangles (Created page with "== Theorem == Let $P$ be a regular $n$-gon. Let $O$ be the center of $P$. Then there exists a triangulation of $P$ into $n$ isosceles triangles. The three vertices of each triangle are $O$ and two Definition:Adjacent...")
- 23:44, 8 January 2023 Anghel talk contribs created page Components of Vector between two Points (I could not find this simple result. Since I need it, I have put it up.)
- 22:18, 5 January 2023 Anghel talk contribs deleted page File:Contours.png (Replaced by a better image)