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29 April 2024
- 00:2700:27, 29 April 2024 diff hist +2,898 N Inner Jordan Content is Well-Defined Created page with "== Theorem == Let $M \subseteq \R^n$ be a bounded subspace of Euclidean $n$-space. Let: :$\ds R = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$ :$\ds R' = \prod_{i \mathop = 1}^n \closedint {a'_i} {b'_i}$ be closed $n$-rectangles that contain $M$. Let $V_R, V_{R'} \in \R_{\ge 0}$ be defined as: :$\ds V_R = \prod_..."
28 April 2024
- 09:5909:59, 28 April 2024 diff hist +122 Definition:Inner Jordan Content Is this better? I definitely wrote it in a confusing way, but I hope this clears it up. I also added an explanation in prose.
- 06:4406:44, 28 April 2024 diff hist 0 m Outer Jordan Content of Dilation No edit summary
- 06:4306:43, 28 April 2024 diff hist +4,385 N Outer Jordan Content of Dilation Created page with "== Theorem == Let $M \subseteq \R^n$ be a bounded subspace of Euclidean $n$-space. Let $c_1, c_2, \dotsc, c_n \in \R_{\ge 0}$ be non-negative real numbers. Let $M' \subseteq \R^n$ be defined as: :$M' = \set {\tuple {c_1 x_1, c_2 x_2, \dotsc, c_n x_n} : \tuple {x_1, x_2, \dotsc, x_n} \in \R^n}$ Then: :$\map {m^*} {M'} = c_1 c_2 \dotsm c..."
- 04:5704:57, 28 April 2024 diff hist +96 Outer Jordan Content Never Smaller than Inner Jordan Content No edit summary current
- 04:3204:32, 28 April 2024 diff hist +670 N Set of Intersections with Superset is Cover Created page with "== Theorem == Let $S$ be a set. Let $\CC$ be a cover of $S$. Let $T \supseteq S$ be a superset of $S$. Then: :$\set {C \cap T : C \in \CC}$ is a cover of $S$. == Proof == Let $x \in S$ be arbitrary. By definition of cover, there is some $C \in \CC$ such that: :$x \in C$ By definition of superset: :$x \in T$ Ther..." current
- 03:4503:45, 28 April 2024 diff hist +727 N Jordan Content is Monotone Created page with "== Theorem == Let $A, B \subseteq \R^n$ be bounded subspaces of Euclidean $n$-space. Suppose that $A \subseteq B$. Further suppose that the Jordan content of both $A$ and $B$ exists. Then: :$\map m A \le \map m B$ where $m$ denotes the Jordan content. == Proof == By definition of the Definition:Jordan Content|J..." current
- 03:2503:25, 28 April 2024 diff hist +16 m Outer Jordan Content is Subadditive No edit summary current
- 03:2403:24, 28 April 2024 diff hist +1,903 N Outer Jordan Content is Subadditive Created page with "== Theorem == Let $A, B \subseteq \R^n$ be bounded subspaces of Euclidean $n$-space. Then: :$\map {m^*} {A \cup B} \le \map {m^*} A + \map {m^*} B$ where $m^*$ denotes the outer Jordan content. == Proof == Let $\epsilon > 0$ be arbitrary. By Characterizing Property of Infimum of Subset of Real Numbers, select: :$C$ to be a Definiti..."
27 April 2024
- 22:2122:21, 27 April 2024 diff hist +931 N Union of Covers is Cover of Union Created page with "== Theorem == Let $\sequence {S_i}_{i \in I}$ be an indexed family of sets. For each $i \in I$, let $\CC_i$ be a cover of $S_i$. Then, $\ds \bigcup_{i \mathop \in I} \CC_i$ is a cover of $\ds \bigcup_{i \mathop \in I} S_i$. == Proof == Let $\ds x \in \bigcup_{i \mathop \in I} S_i$ be arbitrary. By definition of union, there is some $i \in I..." current
- 20:2220:22, 27 April 2024 diff hist +1,021 N Inner Jordan Content is Monotone Created page with "== Theorem == Let $A, B \subseteq \R^n$ be bounded subspaces of Euclidean $n$-space. Suppose that $A \subseteq B$. Then: :$\map {m_*} A \le \map {m_*} B$ where $m_*$ denotes the inner Jordan content. == Proof == Let $R \subseteq \R^n$ be a closed $n$-rectangle that contains $..." current
- 20:0320:03, 27 April 2024 diff hist +1,300 N Outer Jordan Content is Monotone Created page with "== Theorem == Let $A, B \subseteq \R^n$ be bounded subspaces of Euclidean $n$-space. Suppose that $A \subseteq B$. Then: :$\map {m^*} A \le \map {m^*} B$ where $m^*$ denotes the outer Jordan content. == Proof == Let $C$ be a finite covering of $B$ by closed rectangles. Since $A..." current
- 19:5419:54, 27 April 2024 diff hist +515 N Cover is Cover of Subset Created page with "== Theorem == Let $S$ be a set. Let $\CC$ be a cover of $S$. Let $T \subseteq S$ be a subset of $S$. Then, $\CC$ is a cover of $T$. == Proof == By definition of a cover: :$\ds S \subseteq \bigcup C$ But then, by Subset Relation is Transitive: :$\ds T \subseteq \bigcup C$ Therefore, $C$ is a cover of $T$ by..." current
- 19:5419:54, 27 April 2024 diff hist +7 Subset of Cover is Cover of Subset Proofread, category added.
- 19:3819:38, 27 April 2024 diff hist 0 m Definition:Jordan Content Probably more useful if transcluded.
- 19:3819:38, 27 April 2024 diff hist +14 m Definition:Jordan Content No edit summary
- 18:0118:01, 27 April 2024 diff hist +1 m Definition:Jordan Content No edit summary
- 18:0118:01, 27 April 2024 diff hist −388 Definition:Jordan Content I don't remember why I didn't separate out the inner Jordan content. Both sources use that terminology.
- 17:5617:56, 27 April 2024 diff hist +1,273 N Definition:Inner Jordan Content Created page with "== Definition == Let $M \subseteq \R^n$ be a bounded subspace of Euclidean $n$-space. Let $R \subseteq \R^n$ be closed $n$-rectangle that contains $M$. Then, the '''inner Jordan content''' of $M$ is defined and denoted as: :$\map {m_*} M = \map V R - \map {m^*} {R \setminus M}$ where: :$\map V R$ denotes: ::$\ds \m..."
- 17:4217:42, 27 April 2024 diff hist 0 m Definition:Outer Jordan Content No edit summary
- 17:4217:42, 27 April 2024 diff hist −380 Definition:Outer Jordan Content The Evolution of Integration - "The outer Jordan measure of M is the glb of the areas of the coverings of M consisting of finite unions of rectangles."
- 17:3117:31, 27 April 2024 diff hist 0 m Limit to Infinity of Binomial Coefficient over Power Reverted edits by CircuitCraft (talk) to last revision by Prime.mover current Tag: Rollback
- 17:3117:31, 27 April 2024 diff hist +358 Talk:Limit to Infinity of Binomial Coefficient over Power No edit summary
- 08:0508:05, 27 April 2024 diff hist −2 Definition:Jordan Content No edit summary
- 08:0108:01, 27 April 2024 diff hist +183 Definition:Outer Jordan Content No edit summary
- 07:4407:44, 27 April 2024 diff hist +409 Axiom:Content Axioms No edit summary
26 April 2024
- 15:2115:21, 26 April 2024 diff hist 0 m Limit to Infinity of Binomial Coefficient over Power First proof seems to support this. Tag: Reverted
- 15:1915:19, 26 April 2024 diff hist +337 User talk:CircuitCraft No edit summary current
- 03:0003:00, 26 April 2024 diff hist +1,573 N Limit to Infinity of Binomial Coefficient over Power/Proof 2 Created page with "== Theorem == {{:Limit to Infinity of Binomial Coefficient over Power}} == Proof == <onlyinclude> This proof applies to the special case where $k \in \Z$. Then, {{hypothesis}}, we need only consider: :$k \in \set {0, 1, 2, \dotsc}$ By Gamma Function Extends Factorial, it suffices to show: :$\ds \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k} = \frac 1 {k !}$ We have: {{begin-eqn}} {{eqn | q = \forall r \in \R | l = \frac {\dbinom r k} {r^k} |..."
- 02:4102:41, 26 April 2024 diff hist 0 m Definition:Power (Algebra)/Integer No edit summary current
- 02:3402:34, 26 April 2024 diff hist +59 Limit to Infinity of Binomial Coefficient over Power No edit summary
- 02:1602:16, 26 April 2024 diff hist 0 m Definition:Binomial Coefficient/Real Numbers No edit summary current
- 02:1402:14, 26 April 2024 diff hist +1,855 N Limit to Infinity of Binomial Coefficient over Power/Proof 1 Created page with "== Theorem == {{:Limit to Infinity of Binomial Coefficient over Power}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k} | r = \lim_{r \mathop \to \infty} \frac {\map \Gamma {r + 1} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} r^k} | c = Gamma Function Extends Factorial }} {{eqn | r = \lim_{r \mathop \to \infty} \frac 1 {\map \Gamma {k + 1} } \frac {\sqrt {2 \pi r} \paren {r / e}^r} {\sqrt {2 \p..."
- 02:1402:14, 26 April 2024 diff hist −1,437 Limit to Infinity of Binomial Coefficient over Power No edit summary
24 April 2024
- 02:5302:53, 24 April 2024 diff hist +2,608 N Principle of Open Induction for Real Numbers Created page with "== Theorem == Let $a < b$ be real numbers. Let $S$ be an open set of real numbers. Suppose that, for every $x \in \closedint a b$ such that: :$\hointr a x \subseteq S$ it also holds that: :$x \in S$ Then, $\closedint a b \subseteq S$. == Proof == {{AimForCont}} there exists some $x \in \closedint a b$ such that: :$x \notin S$ Let: :$T := \closedint a b \setminus S$ be the set o..."
22 April 2024
- 05:3705:37, 22 April 2024 diff hist 0 m Differentiation of Power Series No edit summary current
- 05:3705:37, 22 April 2024 diff hist 0 m Nth Derivative of Mth Power No edit summary current
25 March 2024
- 23:3723:37, 25 March 2024 diff hist +1,183 N Filter Containing Complements is Not Proper Created page with "== Theorem == Let $L = \struct {S, \lor, \land, \preceq}$ be a bounded lattice. Let $F \subseteq S$ be a filter on $L$. Suppose there exist $a, b \in F$ such that: :$b$ is a complement of $a$ Then, $F = S$. == Proof == By filter axiom $\paren 2$: :$\exists c \in F: c \preceq a \land c \preceq b$ By definition of Definition:Complement (Lattice..."
- 18:5118:51, 25 March 2024 diff hist +744 Definition talk:Limit Point/Filter No edit summary current
21 March 2024
- 21:0421:04, 21 March 2024 diff hist −1 m Riemann-Stieltjes Integral with Step Integrator No edit summary current
- 18:5118:51, 21 March 2024 diff hist +20 m Riemann-Stieltjes Integral with Step Integrator No edit summary
- 18:5018:50, 21 March 2024 diff hist +55 m Reduction of Riemann-Stieltjes Integral to Identity Integrator No edit summary current
- 18:5018:50, 21 March 2024 diff hist +3,987 Riemann-Stieltjes Integral with Step Integrator No edit summary
- 16:5716:57, 21 March 2024 diff hist +6,915 N Riemann-Stieltjes Integral with Step Integrator Created page with "== Theorem == Let $a < c < b$ be real numbers. Let $f$ be a real function that is bounded on $\closedint a b$. Let $\alpha$ be a real function on $\closedint a b$ such that: :$\forall x \in \hointr a c: \map \alpha x = \map \alpha a$ :$\forall x \in \hointl c b: \map \alpha x = \map \alpha b$ Suppose that: :Either $f$ is Definition:Left-C..."
- 05:1905:19, 21 March 2024 diff hist −104 Left-Hand and Right-Hand Differentiable Function is Continuous I extracted this theorem for something else, but it simplifies the logic here accidentally. current
- 05:1605:16, 21 March 2024 diff hist +1,566 N Continuous at Point iff Left-Continuous and Right-Continuous Created page with "== Theorem == Let $A \subseteq \R$ be an open set of real numbers. Let $f : A \to \R$ be a real function. Let $x_0 \in A$. Then: :$f$ is continuous at $x_0$ {{iff}}: :$f$ is both left-continuous and right-continuous at $x_0$ == Proof == === Necessary Condition === Su..." current
- 03:1303:13, 21 March 2024 diff hist 0 Book:Tom M. Apostol/Mathematical Analysis/Second Edition I checked both on my physical copy, as well as https://openlibrary.org/books/OL5291289M/Mathematical_Analysis and https://search.worldcat.org/title/827630. current
- 02:4302:43, 21 March 2024 diff hist +70 m Integration by Substitution/Riemann-Stieltjes Integral No edit summary
- 02:4202:42, 21 March 2024 diff hist +1 m Integration by Substitution/Riemann-Stieltjes Integral No edit summary
- 02:3802:38, 21 March 2024 diff hist +2,859 Integration by Substitution/Riemann-Stieltjes Integral Removed redirect to Integration by Substitution/Riemann-Stieltjes Integral/Increasing Tag: Removed redirect