User contributions for CircuitCraft

6 May 2024

• 06:1406:14, 6 May 2024 diff hist +2,456‎ N Outer Jordan Content of Right Triangle ‎ Created page with "== Theorem == Let $T \subseteq \R^2$ be defined as: :$T = \set {\tuple {x, y} \in \R^2 : x \ge 0 \land y \ge 0 \land x + y \le 1}$ Then: :$\map {m^*} T = \dfrac 1 2$ == Proof == Let $\epsilon > 0$ be arbitrary. By the Axiom of Archimedes, let $n \in \N$ such that: :$n > 2 \epsilon$ Define $C \subseteq \powerset {\R^2}$ as: :$C = \set {\closedint {\dfrac p n} {\dfrac {p + 1} n} \times \closedint {\dfrac q n} {\dfrac {q + 1} n} : p, q \in \set {0, 1, \dotsc, n..."

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