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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).

• 23:28, 12 November 2023 Usagiop talk contribs created page Definition:Weakly Mixing Measure-Preserving Transformation/Definition 4 (Created page with "== Definition == Let $\struct {X, \BB, \mu}$ be a probability space. Let $T: X \to X$ be a measure-preserving transformation. <onlyinclude> $T$ is said to be '''weakly mixing''' {{iff}}: :for every ergodic measure-preserving system $\struct {Y, \BB_Y, \nu, S}$, the Definition:Mea...")
• 00:42, 11 November 2023 Usagiop talk contribs created page Definition:Weakly Mixing Measure-Preserving Transformation/Definition 3 (Created page with "== Definition == Let $\struct {X, \BB, \mu}$ be a probability space. Let $T: X \to X$ be a measure-preserving transformation. <onlyinclude> $T$ is said to be '''weakly mixing''' {{iff}}: :$T \times T$ is weakly mixing with respect to $\mu \times \mu$ </onlyinclude> where: :$\mu \times \mu$ denotes the Definition:Product Mea...")
• 00:37, 11 November 2023 Usagiop talk contribs created page Definition:Weakly Mixing Measure-Preserving Transformation/Definition 2 (Created page with "== Definition == Let $\struct {X, \BB, \mu}$ be a probability space. Let $T: X \to X$ be a measure-preserving transformation. <onlyinclude> $T$ is said to be '''weakly mixing''' {{iff}} :$T \times T$ is ergodic with respect to $\mu \times \mu$ </onlyinclude> where: :$\mu \times \mu$ denotes the Definition:Product Measure|product...")
• 00:18, 11 November 2023 Usagiop talk contribs created page Definition:Weakly Mixing Measure-Preserving Transformation/Definition 1 (Created page with "== Definition == Let $\struct {X, \BB, \mu}$ be a probability space. Let $T: X \to X$ be a measure-preserving transformation. <onlyinclude> $T$ is said to be '''weakly mixing''' {{iff}}: :$\ds \forall A, B \in \BB : \lim_{N \mathop\to \infty} \frac 1 N \sum _{n \mathop = n}^{N-1} \size {\map \mu {A \cap T^{-n} \sqbrk B} - \map \mu A \map \mu B} = 0$ </onlyinclude> == Sources == * {{Bo...")
• 23:59, 2 November 2023 Usagiop talk contribs created page Refinement of Open Cover has Greater Entropy (Created page with "== Theorem == <onlyinclude> Let $X$ be a topological Space. Let $\alpha, \beta$ be open covers of $X$. Let $\map H \alpha$ and $\map H \beta$ be their entropies. Suppose that $\beta$ is a refinement of $\alpha$. Then: :$\map H \alpha \le \map H \beta$ </onlyinclude> == Proof == {{ProofWanted}} == Sources == * {{BookReference|An Intro...")
• 23:52, 2 November 2023 Usagiop talk contribs created page Characterization of Zero Entropy of Open Cover (Created page with "== Theorem == <onlyinclude> Let $X$ be a topological Space. Let $\alpha$ be an open cover of $X$. Let $\map H \alpha$ be the entropy of $\alpha$. Then $\map H \alpha = 0$ {{iff}} $X \in \alpha$. </onlyinclude> == Proof == {{ProofWanted}} == Sources == * {{BookReference|An Introduction to Ergodic Theory|2013|Peter Walters|ed = 4th|edpage = Fourth Edition}}: Chapter $7$...")
• 23:40, 2 November 2023 Usagiop talk contribs created page Definition:Join of Open Covers (Created page with "== Definition == Let $\struct {S, \tau}$ be a topological space. Let $\alpha, \beta \subseteq \tau$ be open covers for $S$. The '''join''' of $\alpha$ and $\beta$ is: :$\alpha \vee \beta := \set { A \cap B : A \in \alpha, B \in \beta }$ Similarly, for open covers $\alpha_1, \ldots, \alpha_n$: :$\ds \bigvee_{k=1}^n \alpha_k := \set { A_1 \cap \cdots \cap A_n : A_i \in \alpha_i \; \t...")
• 00:09, 30 October 2023 Usagiop talk contribs created page Definition:Inverse Limit of Inverse System of Groups (Created page with "== Definition == Let $\sequence {G_n}_{n \in \N}$ be a inverse system with homomorphisms $\sequence {\theta_n}_{n \in \N_{>0} }$. The '''inverse limit''' $\varprojlim G_n$ is the group consisting of all coherent sequences. == Sources == * {{BookReference|Introduction to Commutative Algebra|1969|M.F. Atiyah|autho...")
• 23:59, 29 October 2023 Usagiop talk contribs created page Definition:Coherent Sequence of Inverse System of Groups (Created page with "== Definition == Let $\sequence {G_n}_{n \in \N}$ be a inverse system with homomorphisms $\sequence {\theta_n}_{n \in \N_{>0} }$. Let $\prod_{n \in \N} G_n$ be the Cartesian product of $\sequence {G_n}_{n \in \N}$. Then $\sequence {x_n}_{n \in \N} \in \prod_{n \in \N} G_n$ is a '''coherent sequence''' {{iff}}: :$\forall n \in \N : \map {\theta_{n+1} }...")
• 23:48, 29 October 2023 Usagiop talk contribs created page Definition:Inverse System of Groups (Created page with "== Definition == Let $\sequence {G_n}_{n \in \N}$ be a sequence of groups. For each $n \in \N$, let: :$\theta_{n+1} : G_{n+1} \to G_n$ be a homomorphism. Then $\sequence {G_n}_{n \in \N}$ together with $\sequence {\theta_n}_{n \in \N_{>0} }$ is called an '''inverse system'''. == Sources == * {{BookReference|Introduction to Commutative Algebra|1969|M.F. Atiyah|author2 = I.G. MacDonald}}:...")
• 23:23, 23 October 2023 Usagiop talk contribs created page Maximal Radical implies Primary Ideal (Created page with "== Theorem == Let $R$ be a commutative ring with unity. Let $\mathfrak a$ be an ideal of $R$. Let $\map \Rad {\mathfrak a}$ be the radical of $\mathfrak a$. Suppose that $\map \Rad {\mathfrak a}$ is a maximal ideal. Then $\mathfrak a$ is a primary ideal. == Proof == {{ProofWanted}} Category:R...")
• 23:18, 23 October 2023 Usagiop talk contribs moved page Radical of Primary Ideal is Smallest Prime Ideal to Radical of Primary Ideal is Smallest Larger Prime Ideal without leaving a redirect
• 00:02, 20 October 2023 Usagiop talk contribs created page Maximal Ergodic Theorem (Created page with "== Theorem == <onlyinclude> Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system. Let $g : X \to \overline \R$ be a $\mu$-integrable function. Let $\alpha \in \R$. Let: :$\ds E_\alpha := \set { x \in X : \sup_{n \ge 1} \frac 1 n \sum_{i = 0}^{n-1} \map g {T^i x} > \alpha }$ Then: :$\ds \alpha \map \mu {E_\alpha} \le \int_{E_\alpha} g \rd \mu \le...")
• 19:09, 17 October 2023 Usagiop talk contribs moved page Maximal Inequality to Maximal Inequality for Positive Operators without leaving a redirect (more specific name)
• 00:18, 17 October 2023 Usagiop talk contribs created page Maximal Inequality (Created page with "== Theorem == Let $\struct {X, \BB, \mu}$ be a probability space. Let $\map {L^1} \mu$ be the real-valued $L^1$ space of $\mu$. Let $U : \map {L^1} \mu \to \map {L^1} \mu$ be a positive linear operator, that is: :$\forall f \in \map {L^1} \mu : f \ge 0 \implies U f \ge 0$ Suppose: :$\norm U \le 1$ where $\...") Tag: Visual edit: Switched
• 23:59, 16 October 2023 Usagiop talk contribs created page Definition:Effective Domaion/Convex Real-Valued Function/Real Vector Space (Created page with "== Definition == <onlyinclude> Let $\R^n$ be an $n$-dimensional real vector space. Let $S \subseteq \R^n$ be a subset. Let $f: S \to \overline \R$ be a convex function. The '''effective domain''' of $f$ is: :$\Dom f := \set {x \in S : \map f x < + \infty }$ </onlyinclude> == Sources == * {{BookReference|Convex Analysis|1970|R...")
• 23:35, 16 October 2023 Usagiop talk contribs created page Definition:Convex Real-Valued Function (Created page with "== Definition == === Real Vector Space === {{:Definition:Convex Real-Valued Function/Real Vector Space}} Category:Convex Analysis")
• 23:33, 16 October 2023 Usagiop talk contribs created page Definition:Convex Real-Valued Function/Real Vector Space (Created page with "== Definition == <onlyinclude> Let $\R^n$ be an $n$-dimensional real vector space. Let $S \subseteq \R^n$ be a subset. Let $f: S \to \overline \R$ be an extended real-valued function. $f$ is '''convex''' {{iff}} its epigraph is a convex subset...")
• 00:20, 12 October 2023 Usagiop talk contribs created page Stone-Weierstrass Theorem/Lemma (Created page with "== Lemma == Let $T = \struct {X, \tau}$ be a compact topological space. Let $\map C {X, \R}$ be the set of real-valued continuous functions on $T$. Let $\times$ be the pointwise multiplication on $\map C {X, \R}$. Let $\struct {\map C {X, \R}, \times}$ be...")
• 15:42, 11 October 2023 Usagiop talk contribs deleted page Stone-Weierstrass Theorem/Lemma (Author request)
• 00:10, 11 October 2023 Usagiop talk contribs created page Stone-Weierstrass Theorem/Lemma (Created page with "== Lemma == Let $T = \struct {X, \tau}$ be a compact topological space. Let $\map C {X, \R}$ be the set of real-valued continuous functions on $T$. Let $\times$ be the pointwise multiplication on $\map C {X, \R}$. Let $\struct {\map C {X, \R}, \times}$ be...")
• 00:43, 7 October 2023 Usagiop talk contribs created page L1 Mean Ergodic Theorem (Created page with "== Theorem == <onlyinclude> Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system. Let $\map {L^1_\C} \mu$ be the complex-valued $L^1$ space of $\mu$. Then for each $f \in \map {L^1_\C} \mu$ there is a $T$-invariant function $f' \in \map {L^1_\C} \mu$ such that: :$\ds \lim_{N \mathop \to \i...") Tag: Visual edit: Switched
• 23:31, 25 September 2023 Usagiop talk contribs moved page Definition:Space of Borel Probability Measures to Definition:Space of Borel Probability Measures on Compact Metric Space without leaving a redirect
• 23:28, 25 September 2023 Usagiop talk contribs created page Definition:Space of Borel Probability Measures (Created page with "== Definition == Let $\struct {X, d}$ be a compact metric space. Let $\map C {X,\R}$ be the space of real continuous functions. Let $\map \MM X$ be the set of all Borel probability measures on $X$. Consider the weak-* topology on $\map \MM X$, i.e....")
• 22:50, 24 September 2023 Usagiop talk contribs created page Definition:Barycenter (Locally Convex Space) (Created page with "== Definition == <onlyinclude> Let $\GF \in \set {\R, \C}$. Let $X$ be a vector space over $\GF$. Let $K \subseteq X$ be a compact subset. Suppose $K$ is a metrizable subspace. Let $\mu$ be a Borel probability measure on $K$. Then $x \in K$ is...")
• 00:31, 23 September 2023 Usagiop talk contribs created page Choquet's Theorem (Created page with "== Theorem == Let $X$ be a locally convex vector space over $\R$. Let $K$ be a none-empty metrizable compact convex subset of $X$. Let $K_e$ be the set of extreme points of $K$. Let $u \in K$. Then th...")
• 22:52, 21 September 2023 Usagiop talk contribs created page Definition talk:Riemannian Metric (Why $C^\infty$: new section)
• 22:06, 19 September 2023 Usagiop talk contribs created page Definition talk:Almost Sure Convergence (Created page with "What is the point of the following complicated and unreasonable expression? :$\forall \epsilon \in \R_{>0}: \ds \map \Pr {\lim_{n \mathop \to \infty} \size {X_n - X} < \epsilon} = 1$ Is this a mistake? The definition should be: :$\forall \epsilon \in \R_{>0}: \ds \map \Pr {\limsup_{n \mathop \to \infty} \size {X_n - X} < \epsilon} = 1$ Or, more simply: :$\ds \map \Pr {\lim_{n \mathop \to \infty} \size {X_n - X} = 0 } = 1$ Can someone please check the source? --~~~~")
• 20:44, 16 September 2023 Usagiop talk contribs created page Mediant is Between/Corollary 2/Proof 4 (Created page with "== Corollary to Mediant is Between == {{:Mediant is Between/Corollary 2}} == Proof == <onlyinclude> Let $\epsilon \in \R_{>0}$. Then: :$\dfrac a b < \dfrac {c + \epsilon} d$ By Mediant is Between: :$\dfrac a b < \dfrac {a + c + \epsilon} {b + d} < \dfrac {c + \epsilon} d$ Letting $\epsilon \to 0$: :$\dfrac a b \le \dfrac {a + c} {b + d} \le \dfrac c d$ Since {{hypothesis}}: :$\dfrac a b = \dfrac c d$ we have: :$\dfrac a b = \dfrac {a + c} {b + d} = \dfrac c...")
• 20:40, 16 September 2023 Usagiop talk contribs created page Mediant is Between/Corollary 2/Proof 3 (Created page with "== Corollary to Mediant is Between == {{:Mediant is Between/Corollary 2}} == Proof == <onlyinclude> Let: :$t := \dfrac a b = \dfrac c d$ Then: :$a = t c$ and: :$c = t d$ Hence: :$\dfrac {a + b}{c + d} = \dfrac {t \paren {c + d} }{c + d} = t$ {{qed}} </onlyinclude> Category:Mediant is Between")
• 00:46, 15 September 2023 Usagiop talk contribs created page Fermat's Two Squares Theorem/Uniqueness Lemma/Proof 3 (Created page with "== Lemma for Fermat's Two Squares Theorem == {{:Fermat's Two Squares Theorem/Uniqueness Lemma}} == Proof == <onlyinclude> Let $p = c^2 + d^2$. Then we have: :$(ac+bd)(ad+bc)=(a^2+b^2)cd+(c^2+d^2)ab=p(ab+cd)$ Thus $p \divides ac+bd$ or $p \divides ad+bc$. Swapping $c$ and $d$, if necessary, we may assume: :$p \divides ac+bd$ On the other hand: :$p^2=(ac+bd)^2+(ad-bc)^2\geq p^2+(ad-bc)^2$ Thus: :$ad-bc = 0$ Let: :$k := \dfrac a c = \dfrac b d$ Then: :$p = a^2 +...")
• 00:16, 13 September 2023 Usagiop talk contribs created page Entropy of Measure-Preserving Transformation with respect to Finite Sub-Sigma Algebra is Well-Defined (Created page with "== Theorem == Let $\struct {X, \BB, \mu}$ be a probability space. Let $T: X \to X$ be a $\mu$-preserving transformation. Let $\AA \subseteq \BB$ be a finite sub-$\sigma$-algebra. Then the entropy of $T$ with respect to $\AA$: :$\ds \map h {T, \AA}...")
• 23:26, 10 September 2023 Usagiop talk contribs created page Definition:Hyperbolic Set of Riemannian Manifold (Created page with "== Definition == Let $M$ be a $C^1$ Riemannian manifold. Let $U \subseteq M$ be a non-empty open subset. Let $f : U \to f \sqbrk U$ be a $C^1$ diffeomorphism. Let $\Lambda \subseteq U$ be a compact subset. Suppose that $\Lambda$ is Definition:In...")
• 00:33, 6 September 2023 Usagiop talk contribs created page Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum/Proof 2 (Created page with "== Theorem == {{:Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum}} == Proof == Let: :$\MM_\infty := \set { A \in \MM : \map \Card {\MM_A} = \infty}$ where: :$\MM_A$ denotes the trace $\sigma$-algebra of $A$ in $\MM$ Let $A \in \MM_\infty$. Then: :$\forall B \in \MM_A : B \in \MM_\infty \lor A \setminus B \in \MM_\infty$ since: :$\map \Card {\MM_B} + \map \Card {\MM_{A \setminus B} } \ge \map \Card {\MM_A...")
• 20:26, 4 September 2023 Usagiop talk contribs moved page Image of Fredholm Operator between Banach Spaces is Closed to Image of Fredholm Operator of Banach Spaces is Closed without leaving a redirect
• 23:13, 3 September 2023 Usagiop talk contribs created page Definition:Trivial Factor of Measure-Preserving Dynamical System (Created page with "== Definition == Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical systems. Let $\struct {Y, \CC, \nu, S}$ is called a factor of $\struct {X, \BB, \mu, T}$. $\struct {Y, \CC, \nu, S}$ is '''trivial''' {{iff}} $\struct {Y, \CC, \nu}$ is a singleton measure space, i.e.: :$\exists \set y \in...")
• 22:32, 3 September 2023 Usagiop talk contribs created page Definition:Pseudoinverse of Bounded Linear Transformation (Created page with "== Definition == <onlyinclude> Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be normed vector spaces. Let $S: X \to Y$ be a bounded linear transformation. Let $T: Y \to X$ be a bounded linear transformation. $S$ and $T$ are '''pseudoinverse''' to each other {{iff}}: :$T \circ S -...")
• 19:39, 1 September 2023 Usagiop talk contribs created page Image of Fredholm Operator between Banach Spaces is Closed (Created page with "== Theorem == <onlyinclude> Let $\Bbb F \in \set {\R, \C}$. Let $X, Y$ be Banach spaces over $\Bbb F$. Let $T: U \to V$ be a Fredholm operator. Let $\Img T$ be the image of $T$ . Then $\Img T$ is closed. </onlyinclude> == Proof == {{ProofWanted}} Category:Fredholm Operators Category:Functional Analysis")
• 22:35, 31 August 2023 Usagiop talk contribs moved page Definition:Index of Linear Transformation to Definition:Index of Fredholm Operator without leaving a redirect
• 22:33, 31 August 2023 Usagiop talk contribs moved page Definition:Index of Fredholm Operator to Definition:Index of Linear Transformation without leaving a redirect (revert)
• 22:27, 31 August 2023 Usagiop talk contribs moved page Definition:Index of Linear Transformation to Definition:Index of Fredholm Operator without leaving a redirect
• 21:53, 31 August 2023 Usagiop talk contribs created page Definition:Fredholm Operator (Created page with "#Redirect Definition:Linear Transformation of Finite Index")
• 22:04, 28 August 2023 Usagiop talk contribs moved page Definition:Multiplicatively Closed Subset is Saturated iff Complement is Union of Prime Ideals to Multiplicatively Closed Subset is Saturated iff Complement is Union of Prime Ideals without leaving a redirect
• 22:02, 28 August 2023 Usagiop talk contribs created page Definition:Multiplicatively Closed Subset is Saturated iff Complement is Union of Prime Ideals (Created page with "== Definition == Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a multiplicatively closed subset. Let $\Spec A$ be the prime spectrum of $A$. Then $S$ is saturated {{iff}}: :$\ds \exists \TT \subseteq \Spec A : A \setminus S = \bigcup_{\mathfrak p \math...")
• 00:04, 27 August 2023 Usagiop talk contribs created page Definition:Projective Algebraic Set (Created page with "== Definition == Let $K$ be a field. Let $n \in \N_{>0}$. Let $\map {\mathbb P^n} K$ be the $n$-projective space. Let $S = K \sqbrk {X_0, \ldots, X_n}$ be the ring of polynomial functions in $n + 1$ variables over $K$. Then a subset $X \subseteq \map {\mathbb P^n} K$ is a '''projective algebraic set''' {{iff}} there...")
• 23:43, 25 August 2023 Usagiop talk contribs created page Kernel of Character on Unital Commutative Banach Algebra is Maximal Ideal/Proof 1 (Created page with "== Theorem == {{:Kernel of Character on Unital Commutative Banach Algebra is Maximal Ideal}} == Proof == <onlyinclude> From Kernel of Ring Homomorphism is Ideal, $\ker \phi$ is an ideal of $A$. From First Isomorphism Theorem: Rings, we have: :$\phi \sqbrk A$ and $\dfrac A {\ker \phi}$ are isomorphic as rings. From Character on Banach Algebra is Surjective, we have...")
• 23:27, 25 August 2023 Usagiop talk contribs created page Kernel of Character on Unital Commutative Banach Algebra is Maximal Ideal/Proof 2 (Created page with "== Theorem == {{:Kernel of Character on Unital Commutative Banach Algebra is Maximal Ideal}} == Proof == Let $I$ be an ideal of $A$ such that: :$\ker \phi \subsetneq I$ We shall show $I = A$. Let: :$x \in I \setminus \ker \phi$ Then: :$\map \phi x \ne 0$ Let: :$\ds \tilde x := \frac 1 {\map \phi x} x$ Then: {{begin-eqn}} {{eqn | l = \map \phi { {\mathbf 1}_A - \tilde x} | r = \map \phi { {\mathbf 1}_A } - \map \phi {\tilde x} }}...")
• 22:59, 19 August 2023 Usagiop talk contribs created page Character on Banach Algebra is Surjective/Proof 2 (Created page with "== Theorem == {{:Character on Banach Algebra is Surjective}} == Proof == <onlyinclude> As $\phi$ is non-zero, there exists an $x_0 \in A$ such that: :$\map \phi {x_0} \in \C \setminus \set 0$ Thus, for each $a \in \C$: {{begin-eqn}} {{eqn | l = \frac a {\map \phi {x_0} } x_0 | o = \in | r = A | c = as $A$ is a $\C$-algebra }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = \map \phi {\frac a {\map \phi {x_0} } x_0} |...")
• 22:59, 19 August 2023 Usagiop talk contribs created page Character on Banach Algebra is Surjective/Proof 1 (Created page with "== Theorem == {{:Character on Banach Algebra is Surjective}} == Proof == <onlyinclude> From Image of Submodule under Linear Transformation is Submodule, $\phi \sqbrk A$ is a vector subspace of $\C$. From Dimension of Proper Subspace is Less Than its Superspace, we have: :$\dim \phi \sqbrk A \le \dim \C = 1$ and so we either have $\phi \sqbrk A = \set 0$ or $\phi \sqbrk A = \C$. Since $\phi \ne 0$ by the definition of a Definit...")
• 22:20, 19 August 2023 Usagiop talk contribs created page Product of Commuting Elements in Monoid is Unit iff Each Element is Unit/Proof 1 (Created page with "== Theorem == {{:Product of Commuting Elements in Monoid is Unit iff Each Element is Unit}} == Proof == <onlyinclude> === Necessary Condition === We proceed by induction on $n$. For all $n \ge 2$, let $\map P n$ be the proposition: :for every set of commuting elements $x_1, \ldots, x_n$ in $A$ ::if $\ds \prod_{i \mathop = 1}^n x_i \in \map G A$, the...")