hcju/proofwikips huggingface dataset:
Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Lemma 1
Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Lemma 1 0 77823 408895 2019-06-18T15:50:09Z Prime.mover 59 Created page with "== Example of Convergent Real Sequence ==  Let $h, k \in \R_{>0}$.  Let $\sequence {x_n}$ be the Definition:Real Sequence|real sequen..." wikitext text/x-wiki == Example of Convergent Real Sequence == Let $h, k \in \R_{>0}$. Let $\sequence {x_n}$ be the real sequence defined as: :$x_n = \begin {cases} h & : n = 1 \\ \dfrac k {1 + x_{n - 1} } & : n > 1 \end {cases}$ Then:  :$\forall n \in \N_{>1}: k > x_n > 0$  == Proof == The proof proceeds by induction. For all $n \in \Z_{>1}$, let $\map P n$ be the proposition: :$k > x_n > 0$ === Basis for the Induction === $\map P 2$ is the case: :$k > x_2 > 0$ We have: {{begin-eqn}} {{eqn | l = x_2       | r = \dfrac k {1 + x_1}       | c =  }} {{eqn | o = <       | r = \dfrac k 1       | c = as $x_1 > 0$ }} {{eqn | r = k       | c =  }} {{end-eqn}} Also, as $k > 0$ and $x_1 > 0$ we have that: :$\dfrac k {1 + x_1} > 0$ Thus $\map P 2$ is seen to hold. This is the basis for the induction. === Induction Hypothesis === Now it needs to be shown that if $\map P r$ is true, where $r \ge 2$, then it logically follows that $\map P {r + 1}$ is true. So this is the induction hypothesis: :$k > x_r > 0$ from which it is to be shown that: :$k > x_{r + 1} > 0$ === Induction Step === This is the induction step: {{begin-eqn}} {{eqn | l = x_{r + 1}       | r = \dfrac k {1 + x_r}       | c =  }} {{eqn | o = <       | r = \dfrac k 1       | c = Induction Hypothesis: $x_r > 0$ }} {{eqn | r = k       | c =  }} {{end-eqn}} Also, as $k > 0$ and $x_r > 0$ we have that: :$\dfrac k {1 + x_r} > 0$ So $\map P r \implies \map P {r + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\forall n \in \N_{>1}: k > x_n > 0$  == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n|next = Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Mistake}}: $\S 5$: Subsequences: Exercise $\S 5.7 \ (3)$
Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Lemma 2
== Example of Convergent Real Sequence == Let $h, k \in \R_{>0}$. Let $\sequence {x_n}$ be the real sequence defined as: :$x_n = \begin {cases} h & : n = 1 \\ \dfrac k {1 + x_{n - 1} } & : n > 1 \end {cases}$  Consider the subsequences $\sequence {x_{2 n} }$ and $\sequence {x_{2 n - 1} }$. One of them is strictly increasing and the other is strictly decreasing.  == Proof == We have that: {{begin-eqn}} {{eqn | l = x_{n + 1} - x_{n - 1}       | r = \dfrac k {1 + x_n} - \dfrac k {1 + x_{n - 2} }       | c =  }} {{eqn | r = \dfrac {k \paren {x_{n - 2} - x_n} } {\paren {1 + x_n} \paren {1 + x_{n - 2} } }       | c =  }} {{end-eqn}} and so $x_{n + 1} - x_{n - 1}$ has the opposite sign to $x_{n - 2} - x_n$. It can be proved by induction that one of the sequences $\sequence {x_{2 n} }$ and $\sequence {x_{2 n - 1} }$ increases and one decreases. {{finish|Provide the workings for the above}} In fact: :$\sequence {x_{2 n - 1} }$ is strictly increasing  $x_3 > x_1$ and is strictly decreasing  $x_3 < x_1$. {{finish|Prove the above as well.}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Mistake|next = Convergent Real Sequence/Examples/x n = root x n-1 y n-1, 1 over y n = half (1 over x n + 1 over y n-1)}}: $\S 5$: Subsequences: Exercise $\S 5.7 \ (3)$
Square of Chi Random Variable has Chi-Squared Distribution
Square of Chi Random Variable has Chi-Squared Distribution 0 77952 409556 2019-06-22T10:16:19Z Caliburn 3218 Created page with "== Theorem ==   Let $n$ be a strictly positive integer.  Let $X \sim \chi_n$ where $\chi_n$ is the Definition:Chi Distr..." wikitext text/x-wiki == Theorem ==   Let $n$ be a strictly positive integer. Let $X \sim \chi_n$ where $\chi_n$ is the chi distribution with $n$ degrees of freedom.  Then $X^2 \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  == Proof ==  Let $Y \sim \chi^2_n$.  We aim to show that:  :$\map \Pr {Y < x^2} = \map \Pr {X < x}$ for all $x \in \hointr 0 \infty$. We have: {{begin-eqn}} {{eqn	| l = \map \Pr {Y < x^2} 	| r = \int_0^{x^2} \frac 1 {2^{n / 2} \map \Gamma {n / 2} } t^{\paren {n / 2} - 1} e^{- t / 2} \rd t 	| c = {{Defof|Chi-Squared Distribution}} }} {{eqn	| r = \frac 2 {2^{n / 2} \map \Gamma {n / 2} } \int_0^x u \paren {u^2}^{\paren {n / 2} - 1} e^{- u^2 / 2} \rd u 	| c = substituting $t = u^2$ }} {{eqn	| r = \frac 1 {2^{\paren {n / 2} - 1} \map \Gamma {n / 2} } \int_0^x u u^{2 \paren {\paren {n / 2} - 1} } e^{- u^2 / 2} \rd u }} {{eqn	| r = \frac 1 {2^{\paren {n / 2} - 1} \map \Gamma {n / 2} } \int_0^x u^{n - 1} e^{- u^2 / 2} \rd u }} {{eqn	| r = \map \Pr {X < x} 	| c = {{Defof|Chi Distribution}} }} {{end-eqn}}
Multiple of Chi-Squared Random Variable has Gamma Distribution
Multiple of Chi-Squared Random Variable has Gamma Distribution 0 77989 409843 2019-06-23T13:22:53Z Caliburn 3218 Created page with "== Theorem ==   Let $n$ be a strictly positive integer.   Let $k > 0$ be a real number.   Let..." wikitext text/x-wiki == Theorem ==   Let $n$ be a strictly positive integer.  Let $k > 0$ be a real number.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then:  :$k X \sim \map \Gamma {\dfrac n 2, \dfrac 1 {2 k}}$ where $\map \Gamma {\dfrac n 2, \dfrac 1 {2 k}}$ is the gamma distribution with parameters $\dfrac n 2$ and $\dfrac 1 {2 k}$.  == Proof == Let: :$Y \sim \map \Gamma {\dfrac n 2, \dfrac 1 {2 k}}$ We aim to show that:  :$\map \Pr {Y < k x} = \map \Pr {X < x}$ for all real $x \ge 0$. We have:  {{begin-eqn}} {{eqn	| l = \map \Pr {Y < k x} 	| r = \frac 1 {\map \Gamma {n / 2} } \paren {\frac 1 {2 k} }^{n / 2} \int_0^{k x} t^{\paren {n / 2} - 1} e^{-\paren {1 / k} t / 2} \rd t 	| c = {{Defof|Gamma Distribution}} }} {{eqn	| r = \frac 1 {\map \Gamma {n / 2} } \paren {\frac 1 {2 k} }^{n / 2} \int_0^x k \paren {k u}^{\paren {n / 2} - 1} e^{-u / 2} \rd u 	| c = substituting $t = k u$ }} {{eqn	| r = \frac {k^{n / 2} } {\map \Gamma {n / 2} \paren {2 k}^{n / 2} } \int_0^x u^{\paren {n / 2} - 1} e^{-u / 2} \rd u }} {{eqn	| r = \frac 1 {2^{n / 2} \map \Gamma {n / 2} } \int_0^x u^{\paren {n / 2} - 1} e^{-u / 2} \rd u  }} {{eqn	| r = \map \Pr {X < x} 	| c = {{Defof|Chi-Squared Distribution}} }} {{end-eqn}}
Expectation of Student's t-Distribution
Expectation of Student's t-Distribution 0 78121 410154 2019-06-25T13:45:04Z Caliburn 3218 Created page with "== Theorem ==  Let $k$ be a strictly positive integer.   Let $X \sim t_k$ where $t_k$ is the Definition:Student's t-Dis..." wikitext text/x-wiki == Theorem ==  Let $k$ be a strictly positive integer.  Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom. Then the expectation of $X$ is equal to $0$ for $k > 1$, and does not exist otherwise.  == Proof ==
Raw Moment of Erlang Distribution
Raw Moment of Erlang Distribution 0 78142 410309 2019-06-26T11:05:45Z Caliburn 3218 Created page with "== Theorem ==  Let $n, k$ be strictly positive integers.   Let $\lambda$ be a Definition:Strictly Positive Real Number|..." wikitext text/x-wiki == Theorem ==  Let $n, k$ be strictly positive integers.  Let $\lambda$ be a strictly positive real number.  Let $X$ have a continuous random variable with an Erlang distribution with parameters $k$ and $\lambda$. Then the $n$th raw moment of $X$ is given by: :$\displaystyle \expect {X^n} = \frac 1 {\lambda^n} \prod_{m \mathop = 0}^{n - 1} \paren {k + m}$   == Proof == From the definition of the Erlang distribution, $X$ has probability density function: :$\map {f_X} x = \dfrac {\lambda^k x^{k - 1} e^{- \lambda x} } {\map \Gamma k}$ From the definition of the expected value of a continuous random variable:  :$\displaystyle \expect {X^n} = \int_0^\infty x^n \map {f_X} x \rd x$ So: {{begin-eqn}} {{eqn	| l = \expect {X^n} 	| r = \frac {\lambda^k} {\map \Gamma k} \int_0^\infty x^{n + k - 1} e^{- \lambda x} \rd x }} {{eqn	| r = \frac {\lambda^k} {\lambda \map \Gamma k} \int_0^\infty \paren {\frac u \lambda}^{n + k - 1} e^{- u} \rd u 	| c = substituting $u = \lambda x$ }} {{eqn	| r = \frac {\lambda^k} {\lambda^{n + k} \map \Gamma k} \int_0^\infty u^{n + k - 1} e^{- u} \rd u }} {{eqn	| r = \frac 1 {\lambda^n \map \Gamma k} \map \Gamma {n + k} 	| c = {{Defof|Gamma Function}} }} {{eqn	| r = \frac 1 {\lambda^n} \frac {\map \Gamma k} {\map \Gamma k} \prod_{m \mathop = 0}^{n - 1} \paren {k + m} 	| c = Gamma Difference Equation }} {{eqn	| r = \frac 1 {\lambda^n} \prod_{m \mathop = 0}^{n - 1} \paren {k + m} }} {{end-eqn}}
Linear Transformation of Gaussian Random Variable
Linear Transformation of Gaussian Random Variable 0 78298 411601 2019-07-04T13:50:30Z Caliburn 3218 Created page with "== Theorem ==   Let $\mu$, $\alpha$ and $\beta$ be real numbers.   Let $\sigma$ be a Definition:Positive Real Number|positive real nu..." wikitext text/x-wiki == Theorem ==   Let $\mu$, $\alpha$ and $\beta$ be real numbers.  Let $\sigma$ be a positive real number.  Let $X \sim \Gaussian \mu {\sigma^2}$ where $\Gaussian \mu {\sigma^2}$ is the Gaussian distribution with parameters $\mu$ and $\sigma^2$.  Then:  :$\alpha X + \beta \sim \Gaussian {\alpha \mu + \beta} {\alpha^2 \sigma^2}$   == Proof == Let $Z = \alpha X + \beta$.  Let $M_Z$ be the moment generating function of $Z$.  We aim to show that:  :$Z \sim \Gaussian {\alpha \mu + \beta} {\alpha^2 \sigma^2}$ By Moment Generating Function of Gaussian Distribution and Moment Generating Function is Unique, it is sufficient to show that: :$\displaystyle \map {M_Z} t = \map \exp {\paren {\alpha \mu + \beta} t + \frac 1 2 \alpha^2 \sigma^2 t^2}$ We also have, by Moment Generating Function of Gaussian Distribution, that the moment generating function of $X$, $M_X$, is given by:  :$\displaystyle \map {M_X} t = \map \exp {\mu t + \frac 1 2 \sigma^2 t^2}$ We have: {{begin-eqn}} {{eqn	| l = \map {M_Z} t 	| r = e^{\beta t} \map {M_X} {\alpha t} 	| c = Moment Generating Function of Linear Combination of Independent Random Variables }} {{eqn	| r = e^{\beta t} \map \exp {\alpha \mu t + \frac 1 2 \sigma^2 \paren {\alpha t}^2} }} {{eqn	| r = \map \exp {\paren {\alpha \mu + \beta} t + \frac 1 2 \sigma^2 \alpha^2 t^2} 	| c = Exponential of Sum }} {{end-eqn}}
Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Freedom has F-Distribution
Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Freedom has F-Distribution 0 78327 411751 2019-07-05T11:40:51Z Caliburn 3218 Created page with "== Theorem ==  Let $n$ and $m$ be strictly positive integers.  Let $X$ and $Y$ be Definition:Independent Random Variabl..." wikitext text/x-wiki == Theorem ==  Let $n$ and $m$ be strictly positive integers. Let $X$ and $Y$ be independent random variables.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Let $Y \sim \chi^2_m$ where $\chi^2_m$ is the chi-squared distribution with $m$ degrees of freedom. Then:  :$\dfrac {X / n} {Y / m} \sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\tuple {n, m}$ degrees of freedom.   == Proof ==
Derivative of Function plus Constant
Derivative of Function plus Constant 0 78816 414752 2019-07-27T07:52:21Z Prime.mover 59 Created page with "== Theorem ==  Let $f$ be a real function which is differentiable on $\R$.  Let $c \in \R$..." wikitext text/x-wiki == Theorem ==  Let $f$ be a real function which is differentiable on $\R$. Let $c \in \R$ be a constant. Then: :$\map {D_x} {\map f x + c} = \map {D_x} {\map f x}$  == Proof == {{begin-eqn}} {{eqn | l = \map {D_x} {\map f x + c}       | r = \map {D_x} {\map f x} + \map f x \, c       | c = Sum Rule for Derivatives }} {{eqn | r = \map {D_x} {\map f x} + 0       | c = Derivative of Constant }} {{eqn | r = \map {D_x} {\map f x}       | c = }} {{end-eqn}}
Definite Integral from 0 to Half Pi of Logarithm of Sine x/Lemma
Definite Integral from 0 to Half Pi of Logarithm of Sine x/Lemma 0 78911 415395 2019-07-30T15:31:34Z Caliburn 3218 need this for another page or two wikitext text/x-wiki == Lemma for Definite Integral from 0 to Half Pi of Logarithm of Sine x ==  :$\displaystyle \int_0^\pi \map \ln {\sin x} \rd x = 2 \int_0^{\pi/2} \map \ln {\sin x} \rd x$  == Proof == We have:  {{begin-eqn}} {{eqn	| l = \int_{\pi/2}^\pi \map \ln {\sin x} \rd x 	| r = -\int_{\pi/2}^0 \map \ln {\map \sin {\pi - x} } \rd x 	| c = substituting $x \mapsto \pi - x$ }} {{eqn	| r = \int_0^{\pi/2} \map \ln {\sin x} \rd x 	| c = Reversal of Limits of Definite Integral, Sine of Supplementary Angle }} {{end-eqn}} We can therefore write:  {{begin-eqn}} {{eqn	| l = \int_0^\pi \map \ln {\sin x} \rd x 	| r = \int_0^{\pi/2} \map \ln {\sin x} \rd x + \int_{\pi/2}^\pi \map \ln {\sin x} \rd x 	| c = Sum of Integrals on Adjacent Intervals for Integrable Functions }} {{eqn	| r = 2 \int_0^{\pi/2} \map \ln {\sin x} \rd x }} {{end-eqn}} {{qed|lemma}}
Lowest Common Multiple/Examples/6 and 15
Lowest Common Multiple/Examples/6 and 15 0 79059 416693 2019-08-05T07:01:34Z Prime.mover 59 Created page with "== Example of Lowest Common Multiple of Integers ==  The Definition:Lowest Common Multiple|lowest common multi..." wikitext text/x-wiki == Example of Lowest Common Multiple of Integers ==  The lowest common multiple of $6$ and $15$ is: :$\lcm \set {6, 15} = 30$  == Proof == We find the greatest common divisor of $6$ and $15$ using the Euclidean Algorithm: {{begin-eqn}} {{eqn | n = 1       | l = 15       | r = 2 \times 6 + 3 }} {{eqn | n = 2       | l = 6       | r = 2 \times 3 }} {{end-eqn}} Thus $\gcd \set {6, 15} = 3$. Then: {{begin-eqn}} {{eqn | l = \lcm \set {6, 15}       | r = \dfrac {6 \times 15} {\gcd \set {6, 15} }       | c = Product of GCD and LCM }} {{eqn | r = \dfrac {\paren {2 \times 3} \times \paren {3 \times 5} } 3       | c =  }} {{eqn | r = 2 \times 3 \times 5       | c =  }} {{eqn | r = 30       | c =  }} {{end-eqn}}
Definite Integral from 0 to Half Pi of Logarithm of Sine x/Proof 1
Definite Integral from 0 to Half Pi of Logarithm of Sine x/Proof 1 0 79137 417073 2019-08-07T15:17:48Z Caliburn 3218 Created page with "== Theorem == {{:Definite Integral from 0 to Half Pi of Logarithm of Sine x}}  == Proof ==  By Definite Integral from 0 to Half Pi of Logarithm of Sine x/Lemma|..." wikitext text/x-wiki == Theorem == {{:Definite Integral from 0 to Half Pi of Logarithm of Sine x}} == Proof ==  By Definite Integral from $0$ to $\dfrac \pi 2$ of $\map \ln {\sin x}$: Lemma, we have:  :$\displaystyle \int_0^\pi \map \ln {\sin x} \rd x = 2 \int_0^{\pi/2} \map \ln {\sin x} \rd x$ We also have:  {{begin-eqn}} {{eqn	| l = \int_0^{\pi/2} \map \ln {\sin x} \rd x 	| r = \int_0^{\pi/2} \map \ln {\map \sin {\frac \pi 2 - x} } \rd x 	| c = Integral between Limits is Independent of Direction }} {{eqn	| r = \int_0^{\pi/2} \map \ln {\cos x} \rd x 	| c = Sine of Complement equals Cosine }} {{end-eqn}} giving:  {{begin-eqn}} {{eqn	| l = 2 \int_0^{\pi/2} \map \ln {\sin x} \rd x 	| r = \int_0^{\pi/2} \map \ln {\sin x \cos x} \rd x 	| c = Sum of Logarithms }} {{eqn	| r = \int_0^{\pi/2} \map \ln {\frac 1 2 \sin 2 x} \rd x 	| c = Double Angle Formula for Sine }} {{eqn	| r = \frac \pi 2 \map \ln {\frac 1 2} + \int_0^{\pi/2} \map \ln {\sin 2 x} \rd x 	| c = Primitive of Constant, Sum of Logarithms }} {{eqn	| r = \frac \pi 2 \map \ln {\frac 1 2} + \frac 1 2 \int_0^\pi \map \ln {\sin u} \rd u 	| c = substituting $u = 2 x$ }} {{eqn	| r = -\frac \pi 2 \ln 2 + \int_0^{\pi/2} \map \ln {\sin u} \rd u 	| c = Logarithm of Reciprocal }} {{end-eqn}} Therefore:  :$\displaystyle \int_0^{\pi/2} \map \ln {\sin x} \rd x = -\frac \pi 2 \ln 2$
Equivalence Class/Examples/Months that Start on the Same Day of the Week
== Examples of Equivalence Class ==  Let $M$ be the set of months of the (calendar) year according to the (usual) Gregorian calendar. Let $\sim$ be the relation on $M$ defined as: :$\forall x, y \in M: x \sim y \iff \text {$x$ and $y$ both start on the same day of the week}$ The set of equivalence classes under $\sim$ depends on whether the year is a leap year. For a non-leap year, the set of equivalence classes is: :$\set {\set {\text {January}, \text {October} }, \set {\text {February}, \text {March}, \text {November} }, \set {\text {April}, \text {July} }, \set {\text {May} }, \set {\text {June} }, \set {\text {August} }, \set {\text {September}, \text {December} } }$ For a leap year, the set of equivalence classes is: :$\set {\set {\text {January}, \text {April}, \text {July} }, \set {\text {February}, \text {August} }, \set {\text {March}, \text {November} }, \set {\text {May} }, \set {\text {June} }, \set {\text {September}, \text {December} }, \set {\text {October} } }$  == Proof == We have that: :The months with $30$ days are: ::$\text {April}, \text {June}, \text {September}, \text {November}$ :The months with $31$ days are: ::$\text {January}, \text {March}, \text {May}, \text {July}, \text {August}, \text {October}, \text {December}$ :In a non-leap year, $\text {February}$ has $28$ days :In a leap year, $\text {February}$ has $29$ days. Let month $m$ have $m_d$ days in it. Let month $m$ start on day $d$, where $d$ is in the range $0$ to $6$ (which day of the week corresponds to which number is irrelevant at this stage). Then month $m + 1$ starts on day $\paren {d + m_d} \pmod 7$. For reference: {{begin-eqn}} {{eqn | l = 28       | o = \equiv       | r = 0 \pmod 7       | c =  }} {{eqn | l = 29       | o = \equiv       | r = 0 \pmod 7       | c =  }} {{eqn | l = 30       | o = \equiv       | r = 0 \pmod 7       | c =  }} {{eqn | l = 31       | o = \equiv       | r = 0 \pmod 7       | c =  }} {{end-eqn}} , let $\text {January}$ start on day $0$. Then the sequence of the days which are the $1$st of the month are as follows: For a non-leap year: :$\tuple {0, 3, 3, 6, 1, 4, 6, 2, 5, 0, 3, 5}$ {{OEIS|A189915}} For a leap year: :$\tuple {0, 3, 4, 0, 2, 5, 0, 3, 6, 1, 4, 6}$ {{OEIS|A189916}} The result follows.  == Sources == * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Equivalence Relation/Examples/Months that Start on the Same Day of the Week|next = Number of Friday 13ths in a Year}}: $\S 6$. Indexed families; partitions; equivalence relations: Exercise $7$
Symbols:Abbreviations/W/WFF
== Abbreviation: WFF ==  :Well-formed formula.  == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Mathematician:Hermann Klaus Hugo Weyl|next = Mathematician:Alfred North Whitehead|entry = wff}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Mathematician:Hermann Klaus Hugo Weyl|next = Mathematician:Alfred North Whitehead|entry = wff}}
Symbols:Number Theory/Does Not Divide
== Is Not a Divisor ==  :$x \nmid y$ This means '''$x$ is not a divisor of $y$'''. {{LatexFor|for = x \nmid y}}  == Also denoted as == This symbol is preferable to $x \mathrel {\not \backslash} y$ due to the somewhat confusing appearance of this symbol: {{LatexFor|for = x \mathrel {\not \backslash} y}}
Principle of Stationary Action
== Physical Law == The '''principle of stationary actions''' states that the equations of motion of a physical system can be acquired by finding a stationary point of the action. In other words, the first variation of the action has to vanish. == Also known as == The '''principle of stationary action''' is also known as the '''principle of least action'''. == Notes == {{refactor|Extract the appropriate bit of this into a historical note, and the other bits into separate pages of their own as statement and proof. Hence lose the "notes" section.}} The '''principle of least action''' attained its name due to classical problems of minimization. However, if broken trajectories are allowed, the action can sometimes acquire lower values than for any allowed smooth trajectory. Since smooth trajectories are more realistic, leastness has been weakened to stationarity.
Powers of 3 Modulo 8
Powers of 3 Modulo 8 0 80449 424107 2019-09-08T12:06:35Z Prime.mover 59 Created page with "== Theorem ==  Let $n \in \Z_{\ge 0}$ be a strictly positive integer.  Then: :$3^n \equiv \begin {cases} 1 \pmod 8 & : \t..." wikitext text/x-wiki == Theorem ==  Let $n \in \Z_{\ge 0}$ be a strictly positive integer. Then: :$3^n \equiv \begin {cases} 1 \pmod 8 & : \text {$n$ even} \\ 3 \pmod 8 & : \text {$n$ odd} \end {cases}$  == Proof 1 == {{:Powers of 3 Modulo 8/Proof 1}} == Proof 2 == {{:Powers of 3 Modulo 8/Proof 2}}
Common Logarithm/Examples/2.36
== Example of Common Logarithm ==  The common logarithm of $2 \cdotp 36$ is: :$\log_{10} 2 \cdotp 36 = 0 \cdotp 3729$  == Sources == * {{BookReference|Theory and Problems of Statistics|1972|Murray R. Spiegel|author2 = R.W. Boxer|ed = SI|edpage = SI Edition|prev = Range of Common Logarithm of Number between 1 and 10|next = Common Logarithm/Examples/23.6}}: Chapter $1$: Logarithms: '''Example 1.'''
Jensen's Inequality (Real Analysis)/Corollary
== Corollary to Jensen's Inequality: Real Analysis ==  Let $I$ be a real interval. Let $\phi: I \to \R$ be a concave function. Let $x_1, x_2, \ldots, x_n \in I$. Let $\lambda_1, \lambda_2, \ldots, \lambda_n \ge 0$ be real numbers, at least one of which is non-zero. Then: :$\displaystyle \map \phi {\frac {\sum_{k \mathop = 1}^n \lambda_k x_k} {\sum_{k \mathop = 1}^n \lambda_k} } \ge \frac {\sum_{k \mathop = 1}^n \lambda_k \map \phi {x_k} } {\sum_{k \mathop = 1}^n \lambda_k}$ For $\phi$ strictly concave, equality holds  $x_1 = x_2 = \cdots = x_n$.  == Proof == By Real Function is Concave iff its Negative is Convex, $-\phi: I \to \R$ is a convex function. Therefore, we can apply Jensen's Inequality: Real Analysis with $-\phi$ to obtain: :$\displaystyle -\map \phi {\frac {\sum_{k \mathop = 1}^n \lambda_k x_k} {\sum_{k \mathop = 1}^n \lambda_k} } \le -\frac {\sum_{k \mathop = 1}^n \lambda_k \map \phi {x_k} } {\sum_{k \mathop = 1}^n \lambda_k}$ with equality for $-\phi$ strictly convex  $x_1 = x_2 = \cdots = x_n$. That is, for $\phi$ strictly concave, equality holds  $x_1 = x_2 = \cdots = x_n$. With that, we have established the equality case. Multiplying through $-1$ in our inequality gives:  :$\displaystyle \map \phi {\frac {\sum_{k \mathop = 1}^n \lambda_k x_k} {\sum_{k \mathop = 1}^n \lambda_k} } \ge \frac {\sum_{k \mathop = 1}^n \lambda_k \map \phi {x_k} } {\sum_{k \mathop = 1}^n \lambda_k}$ as required.
Axiom:Axioms of Uncertainty/Axiom 5
Axiom:Axioms of Uncertainty/Axiom 5 100 81786 431708 2019-10-18T00:07:10Z Prime.mover 59 Created page with "== Axiom == {{:Axiom:Axioms of Uncertainty}}  ${H_n}$ fulfils the following axiom:  :$\map {H_n} {\dfrac 1 n, \dfrac 1 n, \dotsc, \dfrac 1 n}..." wikitext text/x-wiki == Axiom == {{:Axiom:Axioms of Uncertainty}} ${H_n}$ fulfils the following axiom:  :$\map {H_n} {\dfrac 1 n, \dfrac 1 n, \dotsc, \dfrac 1 n} \le \map {H_{n + 1} } {\dfrac 1 {n + 1}, \dfrac 1 {n + 1}, \dotsc, \dfrac 1 {n + 1} }$  Thus, for example, a $2$-horse race is less uncertain than a $3$-horse race. == Sources == * {{BookReference|Codes and Cryptography|1988|Dominic Welsh|prev = Axiom:Axioms of Uncertainty/Axiom 4|next = Axiom:Axioms of Uncertainty/Axiom 6}}: $\S 1$: Entropy = uncertainty = information: $1.1$ Uncertainty
Axiom:Axioms of Uncertainty/Axiom 6
Axiom:Axioms of Uncertainty/Axiom 6 100 81787 431709 2019-10-18T00:08:51Z Prime.mover 59 Created page with "== Axiom == {{:Axiom:Axioms of Uncertainty}}  ${H_n}$ fulfils the following axiom:  :$H_n$ is a Definition:Continuous Real-Valued Function|..." wikitext text/x-wiki == Axiom == {{:Axiom:Axioms of Uncertainty}} ${H_n}$ fulfils the following axiom:  :$H_n$ is a continuous function of its arguments.  == Sources == * {{BookReference|Codes and Cryptography|1988|Dominic Welsh|prev = Axiom:Axioms of Uncertainty/Axiom 5|next = Axiom:Axioms of Uncertainty/Axiom 7}}: $\S 1$: Entropy = uncertainty = information: $1.1$ Uncertainty