hcju/proofwikips huggingface dataset:
Solution to Simultaneous Linear Congruences
Let: {{begin-eqn}} {{eqn | l = a_1 x       | o = \equiv       | r = b_1       | rr= \pmod {n_1}       | c =  }} {{eqn | l = a_2 x       | o = \equiv       | r = b_2       | rr= \pmod {n_2}       | c =  }} {{eqn | o = \ldots       | c =  }} {{eqn | l = a_r x       | o = \equiv       | r = b_r       | rr= \pmod {n_r}       | c =  }} {{end-eqn}} be a system of simultaneous linear congruences. This system has a simultaneous solution : :$\forall i, j: 1 \le i, j \le r: \gcd \set {n_i, n_j}$ divides $b_j - b_i$. If a solution exists then it is unique modulo $\lcm \set {n_1, n_2, \ldots, n_r}$.
Primitive of Reciprocal of Tangent of a x
:$\displaystyle \int \frac {\d x} {\tan a x} = \frac 1 a \ln \size {\sin a x} + C$
Primitive of Power of Tangent of a x
:$\displaystyle \int \tan^n a x \ \mathrm d x = \frac {\tan^{n - 1} a x} {\left({n - 1}\right) a} - \int \tan^{n - 2} a x \ \mathrm d x + C$
Logarithm of One plus x in terms of Gaussian Hypergeometric Function
:$\displaystyle \map \ln {1 + x} = x \, {}_2 \map {F_1} {1, 1; 2; -x}$
Laplace Transform of Exponential times Cosine
:$\map {\laptrans {e^{b t} \cos a t} } s = \dfrac {s - b} {\paren {s - b}^2 + a^2}$
Complement of Direct Image Mapping of Injection equals Direct Image of Complement
Let $f: S \to T$ be an injection. Let $f^\to: \powerset S \to \powerset T$ denote the direct image mapping of $f$. Then: :$\forall A \in \powerset S: \map {\paren {\complement_{\Img f} \circ f^\to} } A = \map {\paren {f^\to \circ \complement_S} } A$ where $\circ$ denotes composition of mappings.
Direct Image of Inverse Image of Direct Image equals Direct Image Mapping
Let $f: S \to T$ be a mapping. Let: :$f^\to: \powerset S \to \powerset T$ denote the direct image mapping of $f$ :$f^\gets: \powerset T \to \powerset S$ denote the inverse image mapping of $f$ where $\powerset S$ denotes the power set of $S$. Then: :$f^\to \circ f^\gets \circ f^\to = f^\to$ where $\circ$ denotes composition of mappings.
Primitive of x by Cotangent of a x
:$\displaystyle \int x \cot a x \rd x = \frac 1 {a ^ 2} \paren {a x - \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} - \cdots + \frac {\paren {-1}^n 2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1} !} + \cdots} + C$ where $B_{2 n}$ denotes the $2 n$th Bernoulli number.
Primitive of Reciprocal of p plus q by Cotangent of a x
:$\displaystyle \int \frac {\d x} {p + q \cot a x} = \frac {p x} {p^2 + q^2} - \frac q {a \paren {p^2 + q^2} } \ln \size {p \sin a x + q \cos a x} + C$
Primitive of Reciprocal of Secant of a x
:$\displaystyle \int \frac {\d x} {\sec a x} = \frac {\sin a x} a + C$
Primitive of Reciprocal of Cosecant of a x
:$\displaystyle \int \frac {\d x} {\csc a x} = \frac {-\cos a x} a + C$
Primitive of x by Arcsine of x over a
:$\displaystyle \int x \arcsin \frac x a \ \mathrm d x = \left({\frac {x^2} 2 - \frac {a^2} 4}\right) \arcsin \frac x a + \frac {x \sqrt {a^2 - x^2} } 4 + C$
Elements of Primitive Pythagorean Triples Modulo 4
Let $x \in \Z: x > 2$. Then $x$ is an element of some primitive Pythagorean triple  $x \not \equiv 2 \pmod 4$.
Ceva's Theorem
Let $\triangle ABC$ be a triangle. Let $L$, $M$ and $N$ be points on the sides $BC$, $AC$ and $AB$ respectively. Then the lines $AL$, $BM$ and $CN$ are concurrent : :$\dfrac {BL} {LC} \times \dfrac {CM} {MA} \times \dfrac {AN} {NB} = 1$
Primitive of Arccosine of x over a over x
:$\displaystyle \int \frac {\arccos \frac x a \rd x} x = \frac \pi 2 \ln \size x - \int \frac {\arcsin \frac x a \rd x} x + C$
Left Ideal is Left Module over Ring/Ring is Left Module over Ring
Let $\struct {R, +, \times}$ be a ring. Then $\struct {R, +, \times}$ is a left module over $\struct {R, +, \times}$.
Primitive of x squared by Arccotangent of x over a
:$\displaystyle \int x^2 \operatorname{arccot} \frac x a \ \mathrm d x = \frac {x^3} 3 \operatorname{arccot} \frac x a + \frac {a x^2} 6 - \frac {a^3} 6 \ln \left({x^2 + a^2}\right) + C$
Primitive of Arccotangent of x over a over x
:$\displaystyle \int \frac {\operatorname{arccot} \frac x a \ \mathrm d x} x = \frac \pi 2 \ln \left\vert{x}\right\vert - \int \frac {\arctan \frac x a \ \mathrm d x} x$
Primitive of Arcsecant of x over a over x squared
:$\displaystyle \int \frac {\operatorname{arcsec} \frac x a} {x^2} \ \mathrm d x = \begin{cases} \displaystyle \frac {-\operatorname{arcsec} \frac x a} x + \frac {\sqrt{x^2 - a^2} } {a x} + C & : 0 < \operatorname{arcsec} \dfrac x a < \dfrac \pi 2 \\ \displaystyle \frac {-\operatorname{arcsec} \frac x a} x - \frac {\sqrt{x^2 - a^2} } {a x} + C & : \dfrac \pi 2 < \operatorname{arcsec} \dfrac x a < \pi \\ \end{cases}$
Primitive of Power of x by Arcsecant of x over a
:$\displaystyle \int x^m \operatorname{arcsec} \frac x a \ \mathrm d x = \begin{cases} \displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{arcsec} \frac x a - \frac a {m + 1} \int \frac {x^m \ \mathrm d x} {\sqrt {x^2 - a^2} } + C & : 0 < \operatorname{arcsec} \dfrac x a < \dfrac \pi 2 \\ \displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{arcsec} \frac x a + \frac a {m + 1} \int \frac {x^m \ \mathrm d x} {\sqrt {x^2 - a^2} } + C & : \dfrac \pi 2 < \operatorname{arcsec} \dfrac x a < \pi \\ \end{cases}$
Linear Function is Primitive Recursive
The function $f: \N \to \N$, defined as: :$\map f n = a n + b$ where $a$ and $b$ are constants, is primitive recursive.
Substitution of Constant yields Primitive Recursive Function
Let $f: \N^{k+1} \to \N$ be a primitive recursive function. Then $g: \N^k \to \N$ given by: :$g \left({n_1, n_2, \ldots, n_k}\right) = f \left({n_1, n_2, \ldots, n_{i-1}, a, n_i \ldots, n_k}\right)$ is primitive recursive.
Primitive of Reciprocal of Square of p plus q by Exponential of a x
:$\displaystyle \int \frac {\d x} {\paren {p + q e^{a x} }^2} = \frac x {p^2} + \frac 1 {a p \paren {p + q e^{a x} } } - \frac 1 {a p^2} \ln \size {p + q e^{a x} } + C$
Minimum Function is Primitive Recursive
The minimum function $\min: \N^2 \to \N$, defined as: :$\min \left({n, m}\right) = \begin{cases} n: & n \le m \\ m: & m \le n \end{cases}$ is primitive recursive.
Additive Regular Representations of Topological Ring are Homeomorphisms
Let $\struct {R, + , \circ, \tau}$ be a topological ring. Let $x \in R$. Let $\lambda_x$ and $\rho_x$ be the left and right regular representations of $\struct {R, +}$ with respect to $x$. Then $\lambda_x, \,\rho_x: \struct {R, \tau} \to \struct {R, \tau}$ are homeomorphisms with inverses $\lambda_{-x}, \,\rho_{-x}: \struct {R, \tau} \to \struct {R, \tau}$ respectively.
Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms
Let $\struct{R, + , \circ, \tau}$ be a topological ring with unity $1_R$. For all $y \in R$, let $\lambda_y$ and $\rho_y$ denote the left and right regular representations of $\struct{R, \circ}$ with respect to $y$. Let $x \in R$ be a unit of $R$ with product inverse $x^{-1}$. Then $\lambda_x, \, \rho_x: \struct{R, \tau} \to \struct{R, \tau}$ are homeomorphisms with inverse mappings $\lambda_{x^{-1} }, \, \rho_{x^{-1} }: \struct{R, \tau} \to \struct{R, \tau}$ respectively.
Primitive of Logarithm of x over x squared
:$\displaystyle \int \frac {\ln x} {x^2} \ \mathrm d x = \frac {-\ln x} x - \frac 1 x + C$
Sylow Subgroup is Hall Subgroup
Let $G$ be a group. Let $H$ be a Sylow $p$-subgroup of $G$. Then $H$ is a Hall subgroup of $G$.
Primitive of Reciprocal of Logarithm of x
:$\displaystyle \int \frac {\d x} {\ln x} = \map \ln {\ln x} + \ln x + \sum_{k \mathop \ge 2}^n \frac {\paren {\ln x}^k} {k \times k!} + C$
Primitive of Power of Logarithm of x
:$\displaystyle \int \ln^n x \rd x = x \ln^n x - n \int \ln^{n - 1} x \rd x + C$
Primitive of Logarithm of x squared minus a squared
:$\displaystyle \int \ln \left({x^2 - a^2}\right) \rd x = x \ln \left({x^2 - a^2}\right) - 2 x + a \ln \left({\frac {x + a} {x - a} }\right) + C$ for $x^2 > a^2$.
Conservation of Energy
Let $P$ be a physical system. Let it have the action $S$: :$\displaystyle S = \int_{t_0}^{t_1} L \rd t$ where $L$ is the standard Lagrangian, and $t$ is time. Suppose $L$ does not depend on time explicitly: :$\dfrac {\partial L} {\partial t} = 0$ Then the total energy of $P$ is conserved.
Primitive of x squared by Hyperbolic Sine of a x
:$\displaystyle \int x^2 \sinh a x \ \mathrm d x = \left({\frac {x^2} a + \frac 2 {a^3} }\right) \cosh a x - \frac {2 x \sinh a x} {a^2} + C$
Poisson Brackets of Harmonic Oscillator
Let $P$ be a classical harmonic oscillator. Let the real-valued function $\map x t$ be the position of $P$, where $t$ is time. Then $P$ has the following Poisson brackets: :$\sqbrk {x, p} = 1$ :$\sqbrk {x, H} = \dfrac p m$ :$\sqbrk {p, H} = - k x$
Primitive Recursive Function is Total Recursive Function
Every primitive recursive function is a total recursive function.
Primitive of Reciprocal of Hyperbolic Secant of a x
:$\displaystyle \int \frac {\mathrm d x} {\operatorname{sech} a x} = \frac {\sinh a x} a + C$
Triangle Angle-Side-Angle and Side-Angle-Angle Equality
=== Triangle Angle-Side-Angle Equality === {{:Triangle Angle-Side-Angle Equality}} === Triangle Side-Angle-Angle Equality === {{:Triangle Side-Angle-Angle Equality}}
Primitive of x by Square of Hyperbolic Cosecant of a x
:$\displaystyle \int x \csch^2 a x \rd x = \frac {-x \coth a x} a + \frac 1 {a^2} \ln \size {\sinh a x} + C$
Primitive of Reciprocal of Square of Hyperbolic Sine of a x
:$\displaystyle \int \frac {\mathrm d x} {\sinh^2 a x} = - \frac {\coth a x} a + C$
Primitive of Reciprocal of Square of Hyperbolic Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {\cosh^2 a x} = \frac {\tanh a x} a + C$
Sum of Sequence of Odd Index Fibonacci Numbers
{{begin-eqn}} {{eqn | lo= \forall n \ge 1:       | l = \sum_{j \mathop = 1}^n F_{2 j - 1}       | r = F_1 + F_3 + F_5 + \cdots + F_{2 n - 1}       | c =  }} {{eqn | r = F_{2 n}       | c =  }} {{end-eqn}}
Primitive of Hyperbolic Sine of a x by Cosine of p x
:$\displaystyle \int \sinh a x \cos p x \ \mathrm d x = \frac {a \cosh a x \cos p x + p \sinh a x \sin p x} {a^2 + p^2} + C$
Primitive of Power of x by Hyperbolic Sine of a x
:$\displaystyle \int x^m \sinh a x \ \mathrm d x = \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a x \ \mathrm d x + C$
Summary of Topology on P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$. Then $\struct{\Q_p, \tau_p}$ is: :$(1): \quad$ Hausdorff :$(2): \quad$ second-countable :$(3): \quad$ totally disconnected :$(4): \quad$ locally compact
Napier's Cosine Rule for Right Spherical Triangles
Let $\triangle ABC$ be a right spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let the angle $\sphericalangle C$ be a right angle. Let the remaining parts of $\triangle ABC$ be arranged according to the '''interior''' of this circle, where the symbol $\Box$ denotes a right angle. :410px Let one of the parts of this circle be called a '''middle part'''. Let the two parts which do not neighbor the '''middle part''' be called '''opposite parts'''. Then the sine of the '''middle part''' equals the product of the cosine of the '''opposite parts'''.
Napier's Tangent Rule for Quadrantal Triangles
Let $\triangle ABC$ be a quadrantal triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let the side $c$ be a right angle. Let the remaining parts of $\triangle ABC$ be arranged according to the '''exterior''' of this circle, where the symbol $\Box$ denotes a right angle. :410px Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Then the sine of the '''middle part''' equals the product of the tangents of the '''adjacent parts'''.
Reciprocal of 7
The decimal expansion of the reciprocal of $7$ has the maximum period, that is: $6$: :$\dfrac 1 {7} = 0 \cdotp \dot 14285 \, \dot 7$ {{OEIS|A020806}}
Primitive of Reciprocal of Hyperbolic Cosine of a x minus 1
:$\ds \int \frac {\d x} {\cosh a x - 1} = \frac {-1} a \coth \frac {a x} 2 + C$
Primitive of Reciprocal of Square of Hyperbolic Cosine of a x plus 1
:$\displaystyle \int \frac {\mathrm d x} {\left({\cosh a x + 1}\right)^2} = \frac 1 {2 a} \tanh \frac {a x} 2 - \frac 1 {6 a} \tanh^3 \frac {a x} 2 + C$
Equation of Witch of Agnesi/Cartesian
The equation of the Witch of Agnesi is given in cartesian coordinates as: :$y = \dfrac {8 a^3} {x^2 + 4 a^2}$
Primitive of Hyperbolic Sine of p x by Hyperbolic Cosine of q x
:$\displaystyle \int \sinh p x \cosh q x \ \mathrm d x = \frac {\cosh \left({p + q}\right) x} {2 \left({p + q}\right)} + \frac {\cosh \left({p - q}\right) x} {2 \left({p - q}\right)} + C$
Proof of Theorem by Truth Table
Let $\phi$ be a propositional formula whose atoms are $p_1, p_2, \ldots, p_n$. Let $l$ be the line number of any row in the truth table of $\phi$. For all $i: 1 \le i \ne n$, let $\hat {p_i}$ be defined as: : $\hat {p_i} = \begin{cases} p_i & : \text {the entry in line } l \text { of } p_i \text { is } T \\ \neg p_i & : \text {the entry in line } l \text { of } p_i \text { is } F \end{cases}$ Then: * $\hat {p_1}, \hat {p_2}, \ldots, \hat {p_n} \vdash \phi$ is provable if the entry for $\phi$ in line $l$ is $T$ * $\hat {p_1}, \hat {p_2}, \ldots, \hat {p_n} \vdash \neg \phi$ is provable if the entry for $\phi$ in line $l$ is $F$
Soundness Theorem for Propositional Tableaus and Boolean Interpretations
Tableau proofs (in terms of propositional tableaus) are a sound proof system for boolean interpretations. That is, for every WFF $\mathbf A$: :$\vdash_{\mathrm{PT}} \mathbf A$ implies $\models_{\mathrm{BI}} \mathbf A$
Primitive of Reciprocal of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {\sinh a x \cosh^2 a x} = \frac 1 a \ln \left\vert{\tanh \frac {a x} 2}\right\vert + \frac {\operatorname{sech} a x} a + C$
Celestial Equator is Parallel to Geographical Equator
Consider the celestial sphere with observer $O$. The plane of the celestial equator is parallel to the plane of the geographical equator.
Disjunction of Conjunctions
: $\left({p \land q}\right) \lor \left({r \land s}\right) \vdash p \lor r$
Borel-Cantelli Lemma
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $E_n \subseteq \Sigma$ be a countable collection of measurable sets. If: :$\displaystyle \sum_{n \mathop = 1}^\infty \map \mu {E_n} < \infty$ then: :$\displaystyle \map \mu {\limsup_{n \mathop \to \infty} } {E_n} = 0$ where $\limsup$ denotes limit superior of sets.
Primitive of Inverse Hyperbolic Cosine of x over a
:$\displaystyle \int \cosh^{-1} \frac x a \ \mathrm d x = \begin{cases} x \cosh^{-1} \dfrac x a - \sqrt {x^2 - a^2} + C & : \cosh^{-1} \dfrac x a > 0 \\ x \cosh^{-1} \dfrac x a + \sqrt {x^2 - a^2} + C & : \cosh^{-1} \dfrac x a < 0 \end{cases}$
Vandermonde Matrix Identity for Hilbert Matrix
Define polynomial root sets $\set {1,2,\ldots, n}$ and $\set { 0,-1,\ldots,-n+1}$ for Definition:Cauchy Matrix. Let: {{begin-eqn}} {{eqn   | l = H   | r = \paren {\begin{smallmatrix}\displaystyle         1 & \dfrac {1} {2} & \cdots & \dfrac {1} {n} \\         \dfrac {1} {2} & \dfrac 1 {3}   & \cdots & \dfrac {1} {n+1} \\         \vdots                 & \vdots                 & \cdots & \vdots \\         \dfrac {1} {n} & \dfrac {1} {n+1} & \cdots & \dfrac {1} {2n-1} \\ \end{smallmatrix} }   | c = Hilbert matrix of order $n$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn   | l = H   | r = -P V_x^{-1} V_y Q^{-1}   | c = Vandermonde Matrix Identity for Cauchy Matrix and Hilbert Matrix is Cauchy Matrix }} {{end-eqn}} Definitions of Vandermonde matrices $V_x$, $V_y$ and diagonal matrices $P$, $Q$: :$\displaystyle V_x=\paren {\begin{smallmatrix} 1         & 1         & \cdots & 1 \\ 1       & 2       & \cdots & n \\ \vdots    & \vdots    & \ddots & \vdots \\ 1 & 2^{n-1} & \cdots & n^{n-1} \\ \end{smallmatrix} },\quad  V_y=\paren {\begin{smallmatrix} \displaystyle 1         & 1         & \cdots & 1 \\ 0       & -1       & \cdots & -n+1 \\ \vdots    & \vdots    & \ddots & \vdots \\ 0 & \paren {-1}^{n-1} & \cdots & \paren {-n+1}^{n-1} \\ \end{smallmatrix} }$ Vandermonde matrices :$\displaystyle P= \paren {\begin{smallmatrix} p_1(1) &  \cdots & 0 \\ \vdots   & \ddots  & \vdots \\ 0        & \cdots  & p_n(n) \\ \end{smallmatrix} }, \quad Q= \paren {\begin{smallmatrix} p(0)  & \cdots  & 0 \\ \vdots  & \ddots  & \vdots \\ 0       & \cdots  & p(-n+1) \\ \end{smallmatrix} }$  Diagonal matrices Definitions of polynomials $p$, $p_1$, $\ldots$, $p_n$: :$\displaystyle p(x) = \prod_{i \mathop = 1}^n \paren {x - i}$ :$\displaystyle p_k(x) = \dfrac{ \map p x}{x-k} = \prod_{i \mathop = 1,i \mathop \ne k}^n \, \paren {x - i}$, $1 \mathop \le k \mathop \le n$
Egorov's Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $D \in \Sigma$ be such that $\map \mu D < +\infty$. Let $\sequence {f_n}_{n \mathop \in \N}, f_n: D \to \R$ be a sequence of $\Sigma$-measurable functions. Suppose that $f_n$ converges a.e. to $f$, for some $\Sigma$-measurable function $f: D \to \R$. Then $f_n$ converges a.u. to $f$.
Primitive of Root of x squared plus a squared/Logarithm Form
:$\displaystyle \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C$
Existence of Solution of 2nd Order Linear ODE
Let $P \left({x}\right)$, $Q \left({x}\right)$ and $R \left({x}\right)$ be continuous real functions on a closed interval $I = \left[{a . . b}\right]$. Let $x_0 \in I$, and let $y_0 \in \R$ and $y_0' \in \R$ be arbitrary. Then the initial value problem: :$\displaystyle \frac {d^2y}{dx^2} + P \left({x}\right) \frac {dy}{dx} + Q \left({x}\right) y = R \left({x}\right), y \left({x_0}\right) = y_0, y' \left({x_0}\right) = y_0'$ has one and only one solution $y = y \left({x}\right)$ on the interval $a \le x \le b$.
Cartesian Metric is Rotation Invariant
The cartesian metric does not change under rotation.
Primitive of x squared by Inverse Hyperbolic Cosine of x over a
:$\displaystyle \int x^2 \cosh^{-1} \frac x a \ \mathrm d x = \begin{cases} \displaystyle \frac {x^3} 3 \cosh^{-1} \frac x a - \frac {\left({x^2 + 2 a^2}\right) \sqrt {x^2 - a^2} } 9 + C & : \cosh^{-1} \frac x a > 0 \\ \displaystyle \frac {x^3} 3 \cosh^{-1} \frac x a - \frac {\left({x^2 + 2 a^2}\right) \sqrt {x^2 - a^2} } 9 + C & : \cosh^{-1} \frac x a < 0 \end{cases}$
Odd Vertices Determines Edge-Disjoint Trails
Let $G$ be a loop-multigraph with $2 n$ odd vertices, $n > 0$. Then $G$ has $n$ edge-disjoint trails such that every edge of $G$ is contained in one of these trails. Each of these trails starts and ends on an odd vertex.
Yff's Conjecture
Let $\triangle ABC$ be a triangle. Let $\omega$ be the Brocard angle of $\triangle ABC$. Then: :$8 \omega^3 < ABC$ where $A, B, C$ are measured in radians.
Reciprocal of One minus x in terms of Gaussian Hypergeometric Function
:$\dfrac 1 {1 - x} = {}_2 \map {F_1} {1, p; p; x}$
Rational Number Expressible as Sum of Reciprocals of Distinct Squares
Let $x$ be a rational number such that $0 < x < \dfrac {\pi^2} 6 - 1$. Then $x$ can be expressed as the sum of a finite number of reciprocals of distinct squares.
Adding Edge to Tree Creates One Cycle
Adding a new edge to a tree can create no more than one cycle.
Sophie Germain's Identity
:$x^4 + 4 y^4 = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}$
Coreflexive Relation Subset of Diagonal Relation
A coreflexive relation is a subset of the diagonal relation.
Strict Weak Ordering Induces Partition
Let $\struct {S, \prec}$ be a relational structure such that $\prec$ is a strict weak ordering on $S$. Then $S$ can be partitioned into equivalence classes whose equivalence relation is "is non-comparable". That is, each of the partitions $A$ of $S$ is a relational structure $\struct {\mathbb S, <}$ such that: :$\mathbb S$ is the set of these partitions of $S$; :$<$ is the strict total ordering on $\mathbb S$ '''induced by''' $\prec$.
Coherent Sequence Converges to P-adic Integer
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\sequence {\alpha_n}$ be a coherent sequence.  Let $\Z_p$ be the $p$-adic integers. Then the sequence $\sequence {\alpha_n}$ converges to some $x \in \Z_p$. That is, there exists $x \in \Z_p$ such that: :$\displaystyle \lim_{n \mathop \to \infty} \alpha_n = x$
Space in which All Convergent Sequences have Unique Limit not necessarily Hausdorff
Let $T = \left({S, \tau}\right)$ be a topological space. Let $T$ be such that all convergent sequences have a unique limit point. Then it is not necessarily the case that $T$ is a Hausdorff space.
Equal Sized Triangles on Equal Base have Same Height
Triangles of equal area which are on equal bases, and on the same side of it, are also in the same parallels. {{:Euclid:Proposition/I/40}}
Characteristic Function of Union/Variant 3
:$\chi_{A \mathop \cup B} = \max \set {\chi_A, \chi_B}$
Characteristic Function of Intersection/Variant 2
:$\chi_{A \cap B} = \min \left\{{\chi_A, \chi_B}\right\}$
Tangent Space is Vector Space
Let $M$ be a smooth manifold of dimension $n \in \N$.  Let $m \in M$ be a point.  Let $\left({U, \kappa}\right)$ be a chart with $m \in U$.  Let $T_m M$ be the tangent space at $m$.  Then $T_m M$ is a real vector space of dimension $n$, spanned by the basis:  :$\left\{ {\left.{\dfrac \partial {\partial \kappa^i} }\right\vert_m : i \in \left\{{1, \dotsc, n}\right\} }\right\}$ that is, the set of partial derivatives with respect to the $i$th coordinate function $\kappa^i$ evaluated at $m$.
Parametric Equation of Involute of Circle
Let $C$ be a circle of radius $a$ whose center is at the origin of a cartesian plane. The involute $V$ of $C$ can be described by the parametric equation: :$\begin {cases} x = a \paren {\cos \theta + \theta \sin \theta} \\ y = a \paren {\sin \theta - \theta \cos \theta} \end {cases}$
Basis Expansion of Irrational Number
A basis expansion of an irrational number never terminates and does not recur.
Composite Number has Prime Factor
Let $a$ be a composite number. Then there exists a prime number $p$ such that: :$p \divides a$ where $\divides$ means '''is a divisor of'''. {{:Euclid:Proposition/VII/31}}
General Associativity Theorem/Formulation 3
Let $\struct {S, \circ}$ be a semigroup. Let $a_i$ denote elements of $S$. Let $\circ$ be associative. Let $n \in \Z$ be a positive integer such that $n \ge 3$. Then all possible parenthesizations of the expression: :$a_1 \circ a_2 \circ \cdots \circ a_n$ are equivalent.
Partial Differential Equation of Planes in 3-Space
The set of planes in real Cartesian $3$-dimensional space can be described by the system of partial differential equations: {{begin-eqn}} {{eqn | l = \dfrac {\partial^2 z} {\partial x^2}       | r = 0 }} {{eqn | l = \dfrac {\partial^2 z} {\partial x \partial y}       | r = 0 }} {{eqn | l = \dfrac {\partial^2 z} {\partial y^2}       | r = 0 }} {{end-eqn}}
Equivalence of Definitions of Congruence
{{TFAE|def = Congruence (Number Theory)|view = Congruence|context = Number Theory}} Let $z \in \R$.
Congruence Modulo Zero is Diagonal Relation
Congruence modulo zero is the diagonal relation. That is: :$x \equiv y \pmod 0 \iff x = y$
Numbers between which exist two Mean Proportionals are Similar Solid
Let $a, b \in \Z$ be the extremes of a geometric sequence of integers whose length is $4$: :$\tuple {a, m_1, m_2, b}$ That is, such that $a$ and $b$ have $2$ mean proportionals. Then $a$ and $b$ are similar solid numbers. {{:Euclid:Proposition/VIII/21}}
If First of Three Numbers in Geometric Sequence is Square then Third is Square
Let $P = \tuple {a, b, c}$ be a geometric sequence of integers. Let $a$ be a square number. Then $c$ is also a square number. {{:Euclid:Proposition/VIII/22}}
Finite Fourier Series
Let $\map a n$ be any finite periodic function on $\Z$ with period $b$. Let $\xi = e^{2 \pi i/ b}$ be the first $b$th root of unity. Then: :$\displaystyle \map a n = \sum_{k \mathop = 0}^{b - 1} \map {a_*} k \xi^{n k}$ where: :$\displaystyle \map {a_*} n = \frac 1 b \sum_{k \mathop = 0}^{b - 1} \map a k \xi^{-n k}$