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- 03:13, 30 March 2020 Null Sequence in Exponential Sequence/Proof 2 (hist) [1,319 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Null Sequence in Exponential Sequence}} == Proof == <onlyinclude> Let $\sequence {E_n}$ be the sequence of Definition:Complex Funct...")
- 03:12, 30 March 2020 Null Sequence in Exponential Sequence/Proof 1 (hist) [1,209 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Null Sequence in Exponential Sequence}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = \paren {1 + \frac {a_n} n}^n | r = \sum_{k \mathop = 0}^n {...")
- 06:26, 29 March 2020 Leigh.Samphier/Sandbox/Product Space is T3 iff Factor Spaces are T3/Factor Spaces are T3 implies Product Space is T3 (hist) [1,966 bytes] Leigh.Samphier (talk | contribs) (Created page with "{{Proofread}} == Theorem == Let $\mathbb S = \family{\struct{S_\alpha, \tau_\alpha}}_{\alpha \mathop \in I}$ be an indexed family of Definiti...")
- 06:01, 29 March 2020 Leigh.Samphier/Sandbox/Product Space is T3 iff Factor Spaces are T3/Product Space is T3 implies Factor Spaces are T3 (hist) [3,553 bytes] Leigh.Samphier (talk | contribs) (Created page with "{{Proofread}} == Theorem == Let $\mathbb S = \family{\struct{S_\alpha, \tau_\alpha}}_{\alpha \mathop \in I}$ be an indexed family of Definiti...")
- 05:35, 28 March 2020 Vector Space has Unique Additive Identity/Proof 1 (hist) [829 bytes] Prime.mover (talk | contribs) (Created page with "{{MissingLinks}} == Theorem == {{:Vector Space has Unique Additive Identity}} == Proof == <onlyinclude> Let $V$ be a vector space. Let $0$ and $...")
- 03:42, 28 March 2020 Primitive of x by Logarithm of x squared plus a squared/Proof 2 (hist) [998 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Primitive of x by Logarithm of x squared plus a squared}} == Proof == <onlyinclude> Let $u := x^2 + a^2$. Then: {{begin-eqn}} {{eqn | l = \dfrac {\d u} {\d...")
- 03:09, 28 March 2020 Primitive of x by Logarithm of x squared plus a squared (hist) [641 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> :$\displaystyle \frac {x^2 \map \ln {x^2 + a^2} } 2 - x + a \arctan {\frac x a} + C$ </onlyinclude> == Primitive of x by Logarithm of x squared...")
- 03:09, 28 March 2020 Primitive of x by Logarithm of x squared plus a squared/Proof 1 (hist) [816 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Primitive of x by Logarithm of x squared plus a squared}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = \int x \map \ln {x^2 + a^2} \rd x | r = \...")
- 18:36, 26 March 2020 Extension of Half-Range Fourier Cosine Function to Symmetric Range (hist) [2,370 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$. Let $\map f x$ be rep...")
- 18:32, 26 March 2020 Extension of Half-Range Fourier Sine Function to Symmetric Range (hist) [2,237 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$. Let $\map f x$ be rep...")
- 18:31, 24 March 2020 P-Norm of Real Sequence is Strictly Decreasing Function of P (hist) [2,309 bytes] Julius (talk | contribs) (Created page with "{{WIP}} == Theorem == Let $p \ge 1$ be a real number. Let $\ell^p$ denote the $p$-sequence space. Let $\mathbf x...")
- 17:35, 24 March 2020 Derivative of P-Norm wrt P (hist) [1,653 bytes] Julius (talk | contribs) (Created page with "== Theorem == Let $p \ge 1$ be a real number. Let $\ell^p$ denote the $p$-sequence space. Let $\mathbf x = \seque...")
- 03:30, 24 March 2020 Versed Sine Function is Even (hist) [525 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == The versed sine is an even function: :$\forall \theta \in \R: \map \vers {-\theta} = \vers \theta$ ==...")
- 11:22, 23 March 2020 Fourier Series/Absolute Value of x over Minus Pi to Pi/Proof 2 (hist) [850 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Fourier Series/Absolute Value of x over Minus Pi to Pi}} == Proof == <onlyinclude> By Fourier Series for Absolute Value Function over Symmetric Range, th...")
- 11:17, 23 March 2020 Fourier Series/Absolute Value of x over Minus Pi to Pi/Proof 1 (hist) [1,162 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Fourier Series/Absolute Value of x over Minus Pi to Pi}} == Proof == <onlyinclude> By definition, the absolute value functi...")
- 11:00, 23 March 2020 Fourier Series/Absolute Value Function over Symmetric Range (hist) [1,875 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $\map f x: \openint {-\lambda} \lambda \to \R$ b...")
- 10:57, 23 March 2020 Absolute Value Function is Even Function (hist) [1,545 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\size {\, \cdot \,} : \R \to \R$ denote the absolute value function on $\R$: Then $\size {\, \cdot \,}$ is an...")
- 07:57, 23 March 2020 Odd Function/Examples (hist) [230 bytes] Ascii (talk | contribs) (Created page with "== Examples of Identity Functions == <onlyinclude> === Example: Odd Function === {{:Identity Function is...")
- 07:39, 23 March 2020 Definite Integral of Odd Function/Corollary (hist) [917 bytes] Prime.mover (talk | contribs) (Created page with "== Corollary to Definite Integral of Odd Function == <onlyinclude> Let $f$ be an odd function with a Definition:Primitive (Calculus)|primitiv...")
- 07:11, 23 March 2020 Half-Range Fourier Series/Identity Function (hist) [998 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $\map f x: \openint 0 \lambda \to \R$ be the D...") originally created as "Fourier Series/Identity Function over Half Range"
- 05:35, 23 March 2020 Fourier Series/Identity Function over Minus Pi to Pi/Proof 2 (hist) [737 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Fourier Series/Identity Function over Minus Pi to Pi}} == Proof == <onlyinclude> By Fourier Series for Identity Function over Symmetric Range, the Defi...")
- 05:29, 23 March 2020 Fourier Series/Identity Function over Minus Pi to Pi/Proof 1 (hist) [1,238 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Fourier Series/Identity Function over Minus Pi to Pi}} == Proof == <onlyinclude> From Odd Power is Odd Function, $x$ is a Definition:Odd Function|Odd F...")
- 04:03, 23 March 2020 Identity Function is Odd Function (hist) [522 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $I_\R: \R \to \R$ denote the identity function on $\R$. Then $I_\R$ is an odd function. == P...")
- 18:53, 22 March 2020 P-Norm is Norm (hist) [4,982 bytes] Julius (talk | contribs) (Created page with "== Theorem == P-norm is a norm on the real and complex De...")
- 06:56, 22 March 2020 Standard Continuous Uniform Distribution in terms of Exponential Distribution (hist) [3,233 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $X$ and $Y$ be independent random variables. Let $\beta$ be a Definition:Strictly Positive Real...")
- 20:15, 21 March 2020 Fourier Series/Sawtooth Wave/Special Cases/Half Interval Pi (hist) [1,875 bytes] Prime.mover (talk | contribs) (Created page with "== Special Case of Fourier Series for Sawtooth Wave == <onlyinclude> Let $\map f x$ be the real function defined on the Definition:Open Real...")
- 20:10, 21 March 2020 Fourier Series/Sawtooth Wave/Special Cases/Unit Half Interval (hist) [1,577 bytes] Prime.mover (talk | contribs) (Created page with "== Special Case of Fourier Series for Sawtooth Wave == <onlyinclude> Let $\map f x$ be the real function defined on the Definition:Open Real...")
- 20:06, 21 March 2020 Fourier Series/Sawtooth Wave/Special Cases (hist) [452 bytes] Prime.mover (talk | contribs) (Created page with "== Special Cases of Fourier Series for Sawtooth Wave == <onlyinclude> === Unit Half Interval === {{:Fouri...")
- 19:24, 21 March 2020 Fourier Series/Sawtooth Wave (hist) [1,693 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == 600px|thumb|right|Sawtooth Wave and $9$th Approximation Let $\map f x$ be the real function defined on t...")
- 14:37, 21 March 2020 Fourier Series/Square Wave (hist) [2,656 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\map f x$ be the real function defined on the open interval $\openint {-l} l$ as: :$\map f...")
- 14:37, 21 March 2020 Fourier Series/Square Wave/Special Cases/Unit Half Interval (hist) [1,544 bytes] Prime.mover (talk | contribs) (Created page with "== Special Cases of Fourier Series for Square Wave == <onlyinclude> Let $\map f x$ be the real function defined on the Definition:Open Real...")
- 14:36, 21 March 2020 Fourier Series/Square Wave/Special Cases (hist) [440 bytes] Prime.mover (talk | contribs) (Created page with "== Special Cases of Fourier Series for Square Wave == === Unit Half Interval === {{:Fourier Series/Square...")
- 14:36, 21 March 2020 Fourier Series/Square Wave/Special Cases/Half Interval Pi (hist) [1,894 bytes] Prime.mover (talk | contribs) (Created page with "== Special Cases of Fourier Series for Square Wave == <onlyinclude> Let $\map f x$ be the real function defined on the Definition:Open Real...")
- 06:31, 21 March 2020 Variance of Linear Transformation of Random Variable (hist) [830 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $X$ be a random variable. Let $a, b$ be real numbers. Then we have: :$\var {a...")
- 05:38, 21 March 2020 Fourier Series/1 over -1 to 0, Cosine of Pi x over 0 to 1/Mistake (hist) [2,155 bytes] Prime.mover (talk | contribs) (Created page with "== Source Work == * {{BookReference|Fourier Series|1961|I.N. Sneddon}}: Chapter One: $\S 3$. Other Types of Whole-Range Series: Example $3$ == Mistake == <onlyinclude> :''B...")
- 15:32, 20 March 2020 Primenumber growth within exponential gaps (hist) [15,575 bytes] Proofname123 (talk | contribs) (a mathematical function explained)
- 05:12, 20 March 2020 Cauchy's Mean Theorem/Proof of Equality Condition (hist) [7,306 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive. Let $A_n$ be the De...")
- 04:44, 20 March 2020 Sum of Infinite Geometric Sequence/Corollary 2/Mistake (hist) [912 bytes] Prime.mover (talk | contribs) (Created page with "== Source Work == {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein}}: :'''geometric series''' == Mistake == <onlyinclude> :'...")
- 12:21, 18 March 2020 Expectation of Linear Transformation of Random Variable/Continuous (hist) [1,224 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == Let $X$ be a continuous random variable. Let $a, b$ be real numbers. Then we have: :...")
- 12:17, 18 March 2020 Expectation of Linear Transformation of Random Variable/Discrete (hist) [1,078 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == Let $X$ be a continuous random variable. Let $a, b$ be real numbers. Then we have: :...")
- 12:10, 18 March 2020 Expectation of Linear Transformation of Random Variable (hist) [757 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $X$ be a random variable. Let $a, b$ be real numbers. Then we have: :$\expect...")
- 10:25, 18 March 2020 Covariance of Multiples of Random Variables (hist) [744 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $X, Y$ be random variables. Let $a, b$ be real numbers. Then: :$\cov {a X, b Y}...")
- 10:14, 18 March 2020 Covariance of Sums of Random Variables/Lemma (hist) [2,031 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $n$ be a strictly positive integer. Let $\sequence {X_i}_{1 \le i \le n}$ be a Definition:Sequence|...")
- 08:24, 18 March 2020 Covariance of Sums of Random Variables (hist) [1,226 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $n$ be a strictly positive integer. Let $\sequence {X_i}_{1 \le i \le n}$, $\sequence {Y_j}_{1 \le j...")
- 08:17, 18 March 2020 Covariance of Linear Combination of Random Variables with Another (hist) [861 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $X, Y, Z$ be random variables. Let $a, b$ be real numbers. Then: :$\cov {a X +...")
- 08:02, 18 March 2020 Covariance is Symmetric (hist) [510 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $X$ and $Y$ be random variables. Then $\cov {X, Y} = \cov {Y, X}$. </onlyinclude> == Proof == {{begin-eqn}} {...")
- 07:58, 18 March 2020 Covariance of Random Variable with Itself (hist) [456 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $X$ be a random variable. Then $\cov {X, X} = \var X$. </onlyinclude> == Proof == We have: {{begin-eqn}} {{...")
- 07:52, 18 March 2020 Random Variable has Zero Variance iff Almost Surely Constant (hist) [1,673 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $X$ be a random variable such that $\expect {X^2}$ exists. Then $\var X = 0$ {{iff}} there exists $c \in \R$ wi...")
- 07:29, 18 March 2020 Condition for Expectation of Non-Negative Random Variable to be Zero (hist) [249 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $X$ be a random variable. Let: :$\map \Pr {X \ge 0} = 1$ Then $\expect X = 0$ {{iff}} $\map \Pr {X = 0} =...")
- 18:34, 17 March 2020 Expectation Preserves Inequality (hist) [631 bytes] Caliburn (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $X$, $Y$ be random variables. Let $\map \Pr {X \ge Y} = 1$. Then: :$\expect X \ge \expect Y$ </onlyinclude>...")