nyuuzyou/wikis huggingface dataset:
Asymmetric Relation is Antisymmetric

== Theorem ==
Let $\RR$ be an asymmetric relation.
Then $\RR$ is also antisymmetric.
== Proof ==
Let $\RR$ be asymmetric.
Then from the definition of asymmetric:
:$\tuple {x, y} \in \RR \implies \tuple {y, x} \notin \RR$
Thus:
:$\neg \exists \tuple {x, y} \in \RR: \tuple {y, x} \in \RR$
Thus:
:$\set {\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR} = \O$
Thus:
:$\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR \implies x = y$
is vacuously true.
== Sources ==
* : Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations: $159$
Category:Asymmetric Relations
Category:Antisymmetric Relations
Asymptote/Examples/x^2 + 1 over x

== Examples of Asymptotes ==
:
Consider the curve whose equation in the Cartesian plane is given by:
:$y = x^2 + \dfrac 1 x$
This has $2$ asymptotes:
:the straight line $x = 0$
:the parabola $y = x^2$.
== Sources ==
* 
* 
Category:Examples of Asymptotes
Asymptote to Folium of Descartes

== Theorem ==
Consider the folium of Descartes $F$, given in parametric form as:
:$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$
The straight line whose equation is given by:
:$x + y + a = 0$
is an asymptote to $F$.
== Proof ==
First we note that from Behaviour of Parametric Equations for Folium of Descartes according to Parameter:
:when $t = 0$ we have that $x = y = 0$
:when $t \to \pm \infty$ we have that $x \to 0$ and $y \to 0$
:when $t \to -1^+$ we have that $1 + t^3 \to 0+$, and so:
::$x \to -\infty$
::$y \to +\infty$
:when $t \to -1^-$ we have that $1 + t^3 \to 0-$, and so:
::$x \to +\infty$
::$y \to -\infty$
We have that:
So setting $t = -1$:
The result follows.
== Sources ==
* : $\S 11$: Special Plane Curves: Folium of Descartes: $11.27$
* 
* 
* : $\S 9$: Special Plane Curves: Folium of Descartes: $9.27.$
* 
Category:Folium of Descartes
Asymptotes of Kappa Curve

== Theorem ==
Let $\KK$ be a kappa curve expressed in Cartesian coordinates as:
:$x^4 + x^2 y^2 = a^2 y^2$
$\KK$ has two asymptotes:
:the line $x = a$
:the line $x = -a$.
:
== Proof ==
== Sources ==
* 
* 
Category:Kappa Curve
Asymptotes to Hyperbola in Reduced Form

== Theorem ==
Let $\KK$ be a hyperbola embedded in a cartesian plane in reduced form with the equation:
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$
$\KK$ has two asymptotes which can be described by the equation:
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 0$
that is: 
:$y = \pm \dfrac b a x$
== Proof ==
From Equation of Hyperbola in Reduced Form: Polar Frame, $\KK$ can be described in polar coordinates as:
:$\dfrac {\cos^2 \theta} {a^2} - \dfrac {\sin^2 \theta} {b^2} = \dfrac 1 {r^2}$
When $\theta = 0$ we have that $r = a$.
As $\theta$ increases, $\cos^2 \theta$ decreases and $\sin^2 \theta$ increases.
Hence $\dfrac 1 {r^2}$ decreases and $r$ increases as a consequence.
This continues until:
:$\dfrac {\cos^2 \theta} {a^2} - \dfrac {\sin^2 \theta} {b^2} = 0$
that is:
This can be expressed in the form:
:$b^2 x^2 - a^2 y^2 = 0$
This is a homogeneous quadratic equation in $2$ variables of the form:
:$a' x^2 + 2 h' x y + b' y^2 0 $
where:
:$h' = 0$
:$a' = b^2$
:$b' = a^2$
From Homogeneous Quadratic Equation represents Two Straight Lines through Origin, this is the equation for $2$ straight lines through the origin:
:$y = \pm \dfrac b a x$
== Sources ==
* : Chapter $\text {V}$. The Hyperbola: $3$. Asymptotes
* 
* 
Category:Hyperbolas
Asymptotes to Rectangular Hyperbola

== Theorem ==
Let $\KK$ be a rectangular hyperbola.
The asymptotes of $\KK$ are perpendicular.
== Proof ==
Let $\KK$ be embedded in a cartesian plane in reduced form with the equation:
:$x^2 - y^2 = a^2$
From Asymptotes to Hyperbola in Reduced Form, the asymptotes of $\KK$ can be expressed in the form:
:$y = \pm \dfrac b a x$
By definition of a rectangular hyperbola, $\KK$ is such that:
:$a = b$
Hence the asymptotes of $\KK$ can be expressed in the form:
:$y = \pm x$
The slope of $y = x$ is $1$.
The slope of $y = -x$ is $-1$.
We have that:
:$1 = - \dfrac 1 {-1}$
Hence the result, from Condition for Straight Lines in Plane to be Perpendicular: Slope Form.
== Sources ==
* 
* 
Category:Rectangular Hyperbolas
Asymptotic Distribution/Examples/Arbitrary Example 1

== Example of Asymptotic Distribution ==
Let $X_1, X_2, \ldots, X_n$ be $n$ independent observations from a probability distribution with expectation $\mu$ and finite variance $\sigma_2$.
Then $Y_n = \ds \sum_{k \mathop = 1}^n X_k$ has expectation $n \mu$ and variance $n \sigma_2$.
Both of these tend to infinity, unless $\mu = 0$.
== Proof ==
== Sources ==
* 
* 
Category:Asymptotic Distribution
Asymptotic Expansion for Complementary Error Function

== Theorem ==
:$\ds \map \erfc x \sim \frac {e^{-x^2} } {\sqrt \pi x} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n - 1}!!} {\paren {2 x^2}^n}$
where:
:$\erfc$ denotes the complementary error function
:$n!!$ denotes the double factorial of $n$
:$\sim$ denotes asymptotic equivalence as $x \to \infty$.
== Proof ==
== Sources ==
* : $\S 35$: Complementary Error Function $\ds \map \erfc x = 1 - \map \erf x = \frac 2 {\sqrt \pi} \int_x^\infty e^{-u^2} \rd u$: $35.5$
Category:Asymptotic Expansions
Category:Complementary Error Function
Asymptotic Expansion for Cosine Integral Function

== Theorem ==
:$\ds \map \Ci x \sim \frac {\cos x} x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n + 1}!} {x^{2 n + 1} } - \frac {\sin x} x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n}!} {x^{2 n} }$
where: 
:$\Ci$ denotes the cosine integral function
:$\sim$ denotes asymptotic equivalence as $x \to \infty$.
== Proof ==
== Sources ==
* : $\S 35$: Cosine Integral $\ds \map \Ci x = \int_x^\infty \frac {\cos u} u \rd u$: $35.16$
Category:Cosine Integral Function
Category:Asymptotic Expansions
Asymptotic Expansion for Error Function

== Theorem ==
:$\ds \map \erf x \sim 1 - \frac {e^{-x^2} } {\sqrt \pi x} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n - 1}!!} {\paren {2 x^2}^n}$
where: 
:$\erf$ denotes the error function
:$n!!$ denotes the double factorial of $n$
:$\sim$ denotes asymptotic equivalence as $x \to \infty$.
== Proof ==
== Sources ==
* : $\S 35$: Error Function $\ds \map \erf x = \frac 2 {\sqrt \pi} \int_0^x e^{-u^2} \rd u$: $35.2$
Category:Error Function
Category:Asymptotic Expansions
Asymptotic Expansion for Exponential Integral Function/Formulation 1

== Theorem ==
Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
:$\ds \map \Ei x \sim \frac {e^{-x} } x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {n!} {x^n}$
where $\sim$ denotes asymptotic equivalence as $x \to \infty$.
== Proof ==
== Sources ==
* : $\S 35$: Exponential Integral $\ds \map \Ei x = \int_x^\infty \frac {e^{-u} } u \rd u$: $35.9$
Category:Asymptotic Expansion for Exponential Integral Function
Asymptotic Expansion for Fresnel Cosine Integral Function

== Theorem ==
where:
:$\operatorname C$ denotes the Fresnel cosine integral function
:$\sim$ denotes asymptotic equivalence as $x \to \infty$.
== Proof ==
== Sources ==
* : $\S 35$: Fresnel Cosine Integral $\ds \map {\operatorname C} x = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u$: $35.22$
Category:Fresnel Cosine Integral Function
Category:Asymptotic Expansions
Asymptotic Expansion for Fresnel Sine Integral Function

== Theorem ==
:$\map {\mathrm S} x \sim \dfrac 1 2 - \dfrac 1 {\sqrt {2 \pi} } \paren {\map \cos {x^2} \paren {\dfrac 1 x - \dfrac {1 \times 3} {2^2 x^5} + \dfrac {1 \times 3 \times 5 \times 7} {2^4 x^9} - \ldots} + \map \sin {x^2} \paren {\dfrac 1 {2 x^3} - \dfrac {1 \times 3 \times 5} {2^3 x^7} + \ldots} }$
where
:$\mathrm S$ denotes the Fresnel sine integral function
:$\sim$ denotes asymptotic equivalence as $x \to \infty$.
== Proof ==
== Sources ==
* : $\S 35$: Fresnel Sine Integral $\ds \map {\mathrm S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$: $35.19$
Category:Fresnel Sine Integral Function
Category:Asymptotic Expansions
Asymptotic Expansion for Sine Integral Function

== Theorem ==
:$\ds \map \Si x \sim \frac \pi 2 - \frac {\sin x} x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n + 1}!} {x^{2 n + 1} } - \frac {\cos x} x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {2 n}!} {x^{2 n} }$
where:
:$\Si$ denotes the sine integral function
:$\sim$ denotes asymptotic equivalence as $x \to \infty$.
== Proof ==
== Sources ==
* : $\S 35$: Sine Integral $\ds \map \Si x = \int_0^x \frac {\sin u} u \rd u$: $35.12$
Category:Asymptotic Expansions
Category:Sine Integral Function
Asymptotic Formula for Bernoulli Numbers

== Theorem == 
The Bernoulli numbers with even index can be approximated by the asymptotic formula:
:$B_{2 n} \sim \paren {-1}^{n + 1} 4 \sqrt {\pi n} \paren {\dfrac n {\pi e} }^{2 n}$
where:
:$B_n$ denotes the $n$th Bernoulli number
:$\sim$ denotes asymptotically equal.
== Proof ==
== Also presented as ==
== Sources ==
* : $\S 21$: Asymptotic Formula for Bernoulli Numbers: $21.12$
* : $\S 23$: Bernoulli and Euler Numbers: Asymptotic Formula for Bernoulli Numbers: $23.12.$
* 
Category:Asymptotic Formula for Bernoulli Numbers
Category:Bernoulli Numbers
Category:Asymptotics
Asymptotic Formula for Bernoulli Numbers/Also presented as

== Asymptotic Formula for Bernoulli Numbers: Also presented as == 
Asymptotic Formula for Bernoulli Numbers can also be presented in the form:
:$B_{2 n} \sim \paren {-1}^{n + 1} 4 n^{2 n} \paren {\pi e}^{-2 n} \sqrt {\pi n}$
The following can also be seen:
:${B_n}^* \sim 4 n^{2 n} \paren {\pi e}^{-2 n} \sqrt {\pi n}$
where ${B_n}^*$ denotes the archaic form of the Bernoulli numbers.
== Sources ==
* : $\S 21$: Asymptotic Formula for Bernoulli Numbers: $21.12$
* : $\S 23$: Bernoulli and Euler Numbers: Asymptotic Formula for Bernoulli Numbers: $23.12.$
Category:Asymptotic Formula for Bernoulli Numbers
Asymptotic Growth of Euler Phi Function

== Theorem ==
Let $\phi$ be the Euler $\phi$ function.
For any $\epsilon > 0$ and sufficiently large $n$:
:$n^{1 - \epsilon} < \map \phi n < n$
== Proof ==
It is clear that $\map \phi n < n$ for all $n$, so it is sufficient to prove that:
:$\ds \lim_{n \mathop \to \infty} \frac {n^{1 - \epsilon} } {\map \phi n} = 0$
By Multiplicative Function that Converges to Zero on Prime Powers it is sufficient to prove that:
:$\ds \lim_{p^k \mathop \to \infty} \frac {p^{k \paren {1 - \epsilon} } } {\map \phi {p^k} } = 0$
as $p^k$ ranges through all prime powers.
By Euler Phi Function of Prime Power we have:
:$\map \phi {p^k} = p^k - p^{k - 1}$
for a prime power $p^k$.
Therefore:
Therefore:
:$\ds \lim_{p^k \mathop \to \infty} \frac {p^{k \paren {1 - \epsilon} } } {\map \phi {p^k} } \le \lim_{p^k \mathop \to \infty} \frac 2 {p^{k \epsilon} } = 0$
Category:Analytic Number Theory
Category:Euler Phi Function
Asymptotically Equal Real Functions/Examples/sin x and x

== Example of Asymptotically Equal Real Functions ==
Let $f: \R \to \R$ be the real function defined as:
:$\forall x \in \R: \map f x = \sin x$
Let $g: \R \to \R$ be the real function defined as:
:$\forall x \in \R: \map g x = x$
Then:
:$f \sim g$
as $x \to 0$.
== Proof ==
== Sources ==
* 
Category:Examples of Asymptotically Equal Real Functions
Asymptotically Equal Real Functions/Examples/x^2+1 and x^2

== Example of Asymptotically Equal Real Functions ==
Let $f: \R \to \R$ be the real function defined as:
:$\forall x \in \R: \map f x = x^2 + 1$
Let $g: \R \to \R$ be the real function defined as:
:$\forall x \in \R: \map g x = x^2$
Then:
:$f \sim g$
as $x \to +\infty$.
== Proof ==
== Sources ==
* 
Category:Examples of Asymptotically Equal Real Functions
Asymptotically Equal Real Functions/Examples/x and x+1

== Example of Asymptotically Equal Real Functions ==
Let $f: \R \to \R$ be the real function defined as:
:$\forall x \in \R: \map f x = x + 1$
Let $g: \R \to \R$ be the real function defined as:
:$\forall x \in \R: \map g x = x$
Then:
:$f \sim g$
as $x \to +\infty$.
== Proof ==
Since:
:$\dfrac {\map f x} {\map g x} = 1 + \dfrac 1 x$
we have:
:$\ds \lim _{x \mathop \to +\infty} \dfrac {\map f x} {\map g x} = 1$
== Sources ==
Category:Examples of Asymptotically Equal Real Functions
At Least One Third of Zeros of Riemann Zeta Function on Critical Line

== Theorem ==
At least $\dfrac 1 3$ of the nontrivial zeroes of the Riemann $\zeta$ function lie on the critical line.
== Proof ==
== Historical Note ==
== Sources ==
* 
* : $0,33333 33333 33 \ldots$
* : $0,5$
Category:Riemann Zeta Function
At Least One Third of Zeros of Riemann Zeta Function on Critical Line/Historical Note

== Historical Note on At Least One Third of Zeros of Riemann Zeta Function on Critical Line ==
This result, published in April $1974$, was one of the last things demonstrated.
In August $1974$ he published a further result: that the inequality is in fact strict.
In $1975$ he died of a brain tumour.
The result has since been improved, and it is now known that at least $\dfrac 2 5$ of the nontrivial zeroes of the Riemann zeta function lie on the critical line.
Category:Historical Notes
Atlas Belongs to Unique Differentiable Structure

== Theorem ==
Let $M$ be a locally Euclidean space of dimension $d$.
Let $\AA$ be an atlas on $M$.
Then there exists a unique differentiable structure $\FF$ on $M$ with $\AA \in \FF$.
== Proof ==
Let $\FF$ be the equivalence class of $\AA$ under the relation of compatibility.
By Compatibility of Atlases is Equivalence Relation, this is indeed an equivalence relation.
By definition we have $\AA \in \FF$.
By Relation Partitions Set iff Equivalence, $\FF$ is an element of the partition of equivalence classes.
By definition, the elements of a partition are pairwise disjoint.
Therefore if $\GG \ne \FF$ is an element of the partition, we must have:
:$\AA \notin \GG$
Therefore $\AA$ belongs to exactly one differentiable structure.
Category:Manifolds
Atom of Countably Generated Sigma-Algebra is Measurable

== Theorem ==
Let $X$ be a set.
Let $\Sigma$ be a countably generated $\sigma$-algebra of $X$.
Then:
:$\ds \forall x \in X : \quad \sqbrk x_\Sigma \in \Sigma$
where $\sqbrk x_\Sigma$ denotes the atom.
== Proof ==
By assumption, there is a countable collection $\GG \subseteq \Sigma$ such that:
:$\ds \Sigma = \map \sigma \GG$
Let $x \in X$.
It suffices to show:
:$\ds \sqbrk x_\Sigma = \bigcap_{ x \in A \in \GG } A \cap \bigcap_{ x \not \in A \in \GG} X \setminus A$
$\subseteq$ is clear.
We need to show $\supseteq$.
Category:Definitions/Sigma-Algebras
Atom of Lattice/Examples/Singleton in Lattice of Sets

== Examples of Atoms of a Lattice ==
Let $\powerset S$ denote the power set of a set $S$.
Let $\struct {\powerset S, \subseteq}$ be the relational structure defined on $\powerset S$ by the subset relation $\subseteq$.
From Power Set is Lattice, $\struct {\powerset S, \subseteq}$ is a lattice.
The atoms of $\struct {\powerset S, \subseteq}$ are the singleton subsets of $S$.
== Proof ==
From Empty Set is Bottom of Lattice of Power Set, the bottom of $\struct {\powerset S, \subseteq}$ is the empty set $\O$.
Let $A$ be an atom of $\struct {\powerset S, \subseteq}$.
Recalling the definition of atom.
:$\forall B \in S: B \subsetneqq A \implies B = \O$
We note that $\O$ itself is not an atom of $\struct {\powerset S, \subseteq}$ by definition.
Let $T \subseteq S$ be a subset of $S$ with strictly more than $1$ element.
Let $a, b \in T$.
Then
:$\set a \subsetneqq T$
but:
:$\set a \ne \O$
Hence for such a $T$, there exists a proper subset of $T$ which is not an atom of $\struct {\powerset S, \subseteq}$.
Hence a subset of $S$ with more than $1$ element is not an atom of $\struct {\powerset S, \subseteq}$.
Let $a \in S$ be arbitrary.
Then we have:
:$\set a \in \powerset S$
that is:
:$\set a \subseteq S$
Let $B \subseteq \set a$ such that $B \ne \set a$.
Let $b \in B$ such that $b \ne a$.
Then:
:$b \notin \set a$
and so:
:$B \not \subsetneqq A$
Hence if $B \subsetneqq A$ it follows that:
:$B = \O$
Hence, by definition, $\set a$ is an atom of $\struct {\powerset S, \subseteq}$.
== Sources ==
* 
* 
Category:Examples of Atoms of Lattices
Atomic Electric Field at Point within Body of Matter

== Theorem ==
Let $B$ be a body of matter.
Let $P$ be a point inside $B$ whose position vector is $\mathbf r$.
The atomic electric field at $P$ is given by:
:$\ds \map {\mathbf E_{\text {atomic} } } {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all space} } \dfrac {\paren {\mathbf r - \mathbf r'} \map {\rho_{\text {atomic} } } {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd V'$
where:
:$\d V'$ is an infinitesimal volume element
:$\mathbf r'$ is the position vector of $\d V'$
:$\map {\rho_{\mathrm {atomic} } } {\mathbf r'}$ is the atomic charge density caused by the electric charges within the atoms that make up $B$
:$\varepsilon_0$ denotes the vacuum permittivity.
== Proof ==
From Electric Field Strength from Assemblage of Point Charges, the electric field strength caused by an assemblage of point charges $q_1, q_2, \ldots, q_n$ is given by:
:$\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \epsilon_0} \sum_i \dfrac {\paren {\mathbf r - \mathbf r_i} q_i} {\size {\mathbf r - \mathbf r_i}^3}$
where $\mathbf r_1, \mathbf r_2, \ldots, \mathbf r_n$ are the position vectors of $q_1, q_2, \ldots, q_n$ respectively.
We apply the same principle to the atomic charge density and convert the summation into a definite integral, as follows:
Consider an infinitesimal volume element $\d V'$:
:
The electric field strength caused by $\d V'$ is:
:$\map {\mathbf E_{\text {atomic} } } {\mathbf r'} = \dfrac 1 {4 \pi \epsilon_0} \dfrac {\paren {\mathbf r - \mathbf r'} \rd q} {\size {\mathbf r - \mathbf r'}^3}$
where $\d q$ is the electric charge on $\d V'$.
By definition of electric charge density:
:$\map {\rho_{\mathrm {atomic} } } {\mathbf r'} = \dfrac {\d q} {\d V'}$
and so:
:$\map {\mathbf E_{\text {atomic} } } {\mathbf r'} = \dfrac 1 {4 \pi \epsilon_0} \dfrac {\paren {\mathbf r - \mathbf r'} \map {\rho_{\mathrm {atomic} } } {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd V'$
Integrating over all space:
:$\ds \map {\mathbf E_{\text {atomic} } } {\mathbf r} = \dfrac 1 {4 \pi \varepsilon_0} \int_{\text {all space} } \dfrac {\paren {\mathbf r - \mathbf r'} \map {\rho_{\text {atomic} } } {\mathbf r'} } {\size {\mathbf r - \mathbf r'}^3} \rd V'$
Hence the result.
== Sources ==
* : Chapter $1$: Force and energy in electrostatics: $1.3$ Electric Fields in Matter: $1.3.2$ The atomic electric field: $(1.7)$
Category:Atomic Electric Fields
Attractor/Examples/Origin under Complex Square Function

== Example of Attractor ==
Consider the complex function $f: \C \to \C$ defined as:
:$\forall z \in \C: \map f z = z^2$
Then the origin $\tuple {0, 0}$ of the Argand plane is an attractor.
== Proof ==
Consider the set of points in the open (complex) disk with the center at the origin $\tuple {0, 0}$ and radius $1$ which is the set:
:$\map B {0, 1} = \set {z \in \C: \cmod z < 1}$
All of these points can be described in exponential form:
Since $\cmod z < 1$ and from Modulus of Exponential is Exponential of Real Part, then $\cmod r < 1$.
Applying our squaring complex function to any of the points in this set, we obtain:
Applying our squaring complex function to any of the points in this set $n$ times, we obtain:
Since $\cmod r < 1$ and from Natural Logarithm of 1 is 0 and Logarithm is Strictly Increasing, we know $\ln r < 0$.
Therefore, we have:
:$\ds \lim_{n \mathop \to \infty} 2^n \ln r = -\infty$
From Exponential Tends to Zero and Infinity, we have:
:$\ds \lim_{n \mathop \to \infty} \paren {e^{2^n \ln r + 2^n i \theta} } = 0$
== Sources ==
* 
* 
Category:Examples of Attractors
Attractors of Mandelbrot Set

== Theorem ==
Let $M$ be the Mandelbrot set.
Let $T_c: \C \to \C$ be the complex function defined as:
:$\forall z \in \C: \map {T_c} z = z^2 + c$
Let $c \in M$.
Then $T_c$ has an attractor which is either a fixed point or a set of $n$ periodic points of period $n$.
== Proof ==
== Sources ==
* 
* 
Category:Mandelbrot Set
Category:Attractors
Aurifeuillian Factorization/Examples/2^4n+2 + 1

== Example of Aurifeuillian Factorization ==
:$2^{4 n + 2} + 1 = \paren {2^{2 n + 1} - 2^{n + 1} + 1} \paren {2^{2 n + 1} + 2^{n + 1} + 1}$
== Proof ==
From Sum of Squares as Product of Factors with Square Roots:
:$x^2 + y^2 = \paren {x + \sqrt {2 x y} + y} \paren {x - \sqrt {2 x y} + y}$
Let $x = 2^{2 n + 1}$ and $y = 1$.
Then:
== Historical Note ==
== Sources ==
* : $2^{58} + 1$
* : $2^{58} + 1$
Category:Aurifeuillian Factorizations
Aurifeuillian Factorization/Examples/2^4n+2 + 1/Historical Note

== Historical Note on Aurifeuillian Factorization of $2^{4 n + 2} + 1$ ==
According to in his of $1997$, this identity was established by , who generalised the result:
:$2^{58} + 1 = \paren {2^{29} - 2^{15} + 1} \paren {2^{29} + 2^{15} + 1}$
which had been discovered by in $1871$.
== Sources ==
* : $2^{58} + 1$
* : $2^{58} + 1$
Category:Historical Notes
Aurifeuillian Factorization/Examples/3^6n+3 + 1

== Example of Aurifeuillian Factorization ==
:$3^{6 n + 3} + 1 = 3^{2 n + 1} \paren {3^{2 n + 1} - 3^{n + 1} + 1} \paren {3^{2 n + 1} + 3^{n + 1} + 1}$
Category:Aurifeuillian Factorizations
Autocorrelation Matrix is Positive Definite

== Theorem ==
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$.
Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$:
:$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$
Let $\mathbf P_n$ denote the autocorrelation matrix associated with $S$ for $\sequence {s_n}$.
Then $\mathbf P_n$ is a positive definite matrix.
== Proof ==
== Sources ==
* :
::Part $\text {I}$: Stochastic Models and their Forecasting:
:::$2$: Autocorrelation Function and Spectrum of Stationary Processes:
::::$2.1$ Autocorrelation Properties of Stationary Models:
:::::$2.1.3$ Positive Definiteness and the Autocovariance Matrix
Category:Autocorrelation Matrices
Autocorrelation at Zero Lag for Strictly Stationary Stochastic Process is 1

== Example of Strictly Stationary Stochastic Process ==
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$.
Then the autocorrelation at zero lag is given by:
:$\rho_0 = 1$
== Proof ==
From Autocorrelation of Strictly Stationary Stochastic Process:
:$\rho_k = \dfrac {\gamma_k} {\gamma_0}$
where $\gamma_k$ denotes the autocovariance of $S$.
For zero lag, $k = 0$.
Hence:
:$\rho_0 = \dfrac {\gamma_0} {\gamma_0} = 1$
== Sources ==
* :
::Part $\text {I}$: Stochastic Models and their Forecasting:
:::$2$: Autocorrelation Function and Spectrum of Stationary Processes:
::::$2.1$ Autocorrelation Properties of Stationary Models:
:::::$2.1.2$ Stationary Stochastic Processes: Autocovariance and autocorrelation coefficients
Category:Stationary Stochastic Processes
Category:Autocorrelation