Möbius Transformation is Bijection/Restriction to Reals

Let $a, b, c, d \in \R$ be real numbers. Let $f: \R^* \to \R^*$ be the Möbius transformation restricted to the real numbers: :$\map f x = \begin {cases} \dfrac {a x + b} {c x + d} & : x \ne -\dfrac d c \\ \infty & : x = -\dfrac d c \\ \dfrac a c & : x = \infty \\ \infty & : x = \infty \text { and } c = 0 \end {cases}$ Then: :$f: \R^* \to \R^*$ is a bijection {{iff}}: :$a c - b d \ne 0$


Definite Integral from 0 to 1 of Logarithm of x over One plus x

:$\displaystyle \int_0^1 \frac {\ln x} {1 + x} \rd x = -\frac {\pi^2} {12}$


Primitive of Reciprocal of x by Root of Power of x minus Power of a

:$\displaystyle \int \frac {\d x} {x \sqrt {x^n - a^n} } = \frac 2 {n \sqrt {a^n} } \arccos \sqrt {\frac {a^n} {x^n} }$


Set of Sets can be Defined as Family

Let $\Bbb S$ be a set of sets. Then $\Bbb S$ can be defined as an indexed family of sets.


Primitive of Sine of a x over x

{{begin-eqn}} {{eqn | l = \int \frac {\sin a x \d x} x       | r = \sum_{k \mathop \ge 0} \frac {\paren {-1}^k \paren {a x}^{2 k + 1} } {\paren {2 k + 1} \paren {2 k + 1}!} + C }} {{eqn | r = a x - \frac {\paren {a x}^3} {3 \times 3!} + \frac {\paren {a x}^5} {5 \times 5!} - \cdots }} {{end-eqn}}


Primitive of Reciprocal of Sine of a x/Logarithm of Cosecant minus Cotangent Form

:$\ds \int \frac {\d x} {\sin a x} = \frac 1 a \ln \size {\csc a x - \cot a x} + C$


Primitive of Reciprocal of Square of 1 minus Sine of a x

:$\displaystyle \int \frac {\mathrm d x} {\left({1 - \sin a x}\right)^2} = \frac 1 {2 a} \tan \left({\frac \pi 4 + \frac {a x} 2}\right) + \frac 1 {6 a} \tan^3 \left({\frac \pi 4 + \frac {a x} 2}\right) + C$


Converse Hinge Theorem

If two triangles have two pairs of sides which are the same length, the triangle in which the third side is longer also has the larger angle contained by the first two sides. {{:Euclid:Proposition/I/25}}


Definite Integral over Reals of Exponential of -(a x^2 plus b x plus c)

:$\displaystyle \int_{-\infty}^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x = \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} }$


Definite Integral to Infinity of Exponential of -(a x^2 plus b over x^2)

:$\displaystyle \int_0^\infty \map \exp {-\paren {a x^2 + \frac b {x^2} } } \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-2 \sqrt {a b} }$


Invariance of Extremal Length under Conformal Mappings

Let $X, Y$ be Riemann surfaces (usually, subsets of the complex plane). Let $\phi: X \to Y$ be a conformal isomorphism between $X$ and $Y$. Let $\Gamma$ be a family of rectifiable curves (or, more generally, of unions of rectifiable curves) in $X$. Let $\Gamma'$ be the family of their images under $\phi$. Then $\Gamma$ and $\Gamma'$ have the same extremal length: :$\map \lambda \Gamma = \map \lambda {\Gamma'}$


Set is Subset of Intersection of Supersets/Set of Sets

Let $T$ be a set. Let $\mathbb S$ be a set of sets. Suppose that for each $S \in \mathbb S$, $T \subseteq S$. Then: :$T \subseteq \displaystyle \bigcap \mathbb S$


Membership Relation is Not Symmetric

Let $\Bbb S$ be a set of sets in the context of pure set theory Let $\RR$ denote the membership relation on $\Bbb S$: :$\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \RR \iff a \in b$ $\RR$ is not in general a symmetric relation.


Complementary Error Function of Zero

:$\displaystyle \map \erfc 0 = 1$


Primitive of Cosine of a x over x

{{begin-eqn}} {{eqn | l = \int \frac {\cos a x \rd x} x       | r = \ln \size x + \sum_{k \mathop \ge 1} \frac {\paren {-1}^k \paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C }} {{eqn | r = \ln \size x - \frac {\paren {a x}^2} {2 \times 2!} + \frac {\paren {a x}^4} {4 \times 4!} - \frac {\paren {a x}^6} {6 \times 6!} - \cdots + C }} {{end-eqn}}


Primitive of Cosine of a x over x squared

:$\displaystyle \int \frac {\cos a x \ \mathrm d x} {x^2} = \frac {-\cos a x} x - a \int \frac {\sin a x \ \mathrm d x} x$


Definite Integral to Infinity of x over Hyperbolic Sine of a x

:$\displaystyle \int_0^\infty \frac x {\sinh a x} \rd x = \frac {\pi^2} {4 a^2}$


Empty Set from Principle of Non-Contradiction

The empty set can be characterised as: :$\O := \set {x: x \in E \text { and } x \notin E}$ where $E$ is an arbitrary set.


Primitive of Reciprocal of Square of 1 plus Cosine of a x

:$\displaystyle \int \frac {\mathrm d x} {\left({1 + \cos a x}\right)^2} = \frac 1 {2a} \tan \frac {a x} 2 + \frac 1 {6 a} \tan^3 \frac {a x} 2 + C$


Definite Integral from 0 to 2 Pi of Reciprocal of One minus 2 a Cosine x plus a Squared

:$\displaystyle \int_0^{2 \pi} \frac {\d x} {1 - 2 a \cos x + a^2} = \frac {2 \pi} {1 - a^2}$


Divisibility by 7

An integer $X$ with $n$ digits ($X_0$ in the ones place, $X_1$ in the tens place, and so on) is divisible by $7$ {{iff}}: :$\displaystyle \sum_{i \mathop = 0}^{n - 1} \paren {3^i X_i}$ is divisible by $7$.


Intersection is Empty and Union is Universe if Sets are Complementary

Let $A$ and $B$ be subsets of a universe $\Bbb U$. Then: :$A \cap B = \O$ and $A \cup B = \Bbb U$ {{iff}}: :$B = \relcomp {\Bbb U} A$ where $\relcomp {\Bbb U} A$ denotes the complement of $A$ with respect to $\Bbb U$.


Definite Integral to Infinity of Reciprocal of Exponential of x minus One minus Exponential of -x over x

:$\displaystyle \int_0^\infty \paren {\frac 1 {e^x - 1} - \frac {e^{-x} } x} \rd x = \gamma$


Set Intersection expressed as Intersection Complement

Let $A$ and $B$ be subsets of a universal set $\Bbb U$. Let $\uparrow$ denote the operation on $A$ and $B$ defined as: :$\paren {A \uparrow B} \iff \paren {\relcomp {\Bbb U} {A \cap B} }$ where $\relcomp {\Bbb U} A$ denotes the complement of $A$ in $\Bbb U$. Then: :$A \cap B = \paren {A \uparrow B} \uparrow \paren {A \uparrow B}$


Primitive of Power of Secant of a x by Tangent of a x

:$\displaystyle \int \sec^n a x \tan a x \ \mathrm d x = \frac {\sec^n a x} {n a} + C$


Primitive of Reciprocal of Square of Sine of a x by Square of Cosine of a x

:$\displaystyle \int \frac {\d x} {\sin^2 a x \cos^2 a x} = \frac {-2 \cot 2 a x} a + C$


Singleton of Subset is Element of Powerset of Powerset

Let $S \subseteq T$ where $S$ and $T$ are both sets. Then: :$\set S \in \powerset {\powerset T}$ where $\powerset T$ denotes the power set of $T$.


Power Function is Completely Multiplicative

Let $K$ be a field. Let $z \in K$. Let $f_z: K \to K$ be the mapping defined as: :$\forall x \in K: f_z \left({x}\right) = x^z$ Then $f_z$ is completely multiplicative.


Primitive of Cosine of a x over p plus q of Sine of a x

:$\displaystyle \int \frac {\cos a x \rd x} {p + q \sin a x} = \frac 1 {a q} \ln \left\vert{p + q \sin a x}\right\vert + C$


Primitive of Cosine of a x over Power of p plus q of Sine of a x

:$\displaystyle \int \frac {\cos a x \ \mathrm d x} {\left({p + q \sin a x}\right)^n} = \frac {-1} {a q \left({n - 1}\right) \left({p + q \sin a x}\right)^{n - 1} } + C$


Number of Quadratic Residues of Prime

Let $p$ be an odd prime. Then $p$ has $\dfrac {p-1} 2$ quadratic residues and $\dfrac {p-1} 2$ quadratic non-residues. The quadratic residues are congruent modulo $p$ to the integers $1^2, 2^2, \ldots, \left({\dfrac {p-1} 2}\right)$.


Limit to Infinity of Fresnel Sine Integral Function

:$\displaystyle \lim_{x \mathop \to \infty} \map {\operatorname S} x = \frac 1 2$


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 {{iff}}: :$\forall i, j: 1 \le i, j \le r: \gcd \set {n_i, n_j}$ 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 {{iff}} $x \not \equiv 2 \pmod 4$.


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 {{iff}}: :$\dfrac {BL} {LC} \times \dfrac {CM} {MA} \times \dfrac {AN} {NB} = 1$


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: :$\displaystyle \map {a_*} n = \frac 1 b \sum_{k \mathop = 0}^{b - 1} \map a k \xi^{-n k}$


Linear Second Order ODE/y'' - y = 3 exp -x

The second order ODE: :$(1): \quad y'' - y = 3 e^{-x}$ has the general solution: :$y = C_1 e^x + C_2 e^{-x} - \dfrac {3 x e^{-x} } 2$


Elements of Geometric Sequence from One which Divide Later Elements

Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric Sequence of integers. Let $a_0 = 1$. Let $m \in \Z_{> 0}$. Then: :$\forall r \in \set {0, 1, \ldots, m}: a_k \divides a_m$ where $\divides$ denotes divisibility. {{:Euclid:Proposition/IX/11}}


Elements of Geometric Sequence from One Divisible by Prime

Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers. Let $a_0 = 1$. Let $p$ be a prime number such that: :$p \divides a_n$ where $\divides$ denotes divisibility. Then $p \divides a_1$. {{:Euclid:Proposition/IX/12}}


Divisibility of Elements of Geometric Sequence from One where First Element is Prime

Let $Q_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of length $n$ consisting of integers only. Let $a_0 = 1$. Let $a_1$ be a prime number. Then the only divisors of $a_n$ are $a_j$ for $j \in \set {1, 2, \ldots, n}$. {{:Euclid:Proposition/IX/13}}


Linear Second Order ODE/y'' - 2 y' + y = exp x

The second order ODE: :$(1): \quad y'' - 2 y' + y = e^x$ has the general solution: :$y = C_1 e^x + C_2 x e^x + \dfrac {x^2 e^x} 2$


Linear First Order ODE/y' - y = x^2

The linear first order ODE: :$(1): \quad \dfrac {\d y} {\d x} - y = x^2$ has the general solution: :$y = C e^x - \paren {x^2 + 2 x + 2}$


Linear Second Order ODE/y'' + y = exp -x cos x

The second order ODE: :$(1): \quad y'' + y = e^{-x} \cos x$ has the general solution: :$y = \dfrac {e^{-x} } 5 \paren {\cos x - 2 \sin x} + C_1 \sin x + C_2 \cos x$


Restriction of Homeomorphism is Homeomorphism

Let $T_1 = \left({S_1, \tau_1}\right)$, $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces. Let $f: S_1 \to S_2$ be a homeomorphism between $T_1$ and $T_2$. Let $S$ be a subset of $S_1$. Let $f {\restriction_{S \times f\left[{S}\right]}} : S \to f \left[{S}\right]$ be the restriction of $f$ to $S \times f \left[{S}\right]$. Let $S$ and $f \left[{S}\right]$ bear their respective subspace topologies. Then $f {\restriction_{S \times f \left[{S}\right]}}$ is a homeomorphism.


Disjoint Independent Events means One is Void

Let $A$ and $B$ be events in a probability space. Suppose $A$ and $B$ are: :disjoint :independent. Then either $\map \Pr A = 0$ or $\map \Pr B = 0$. That is, if two events are disjoint and independent, at least one of them can't happen.


Product of Rational Cuts is Rational Cut

Let $p \in\ Q$ and $q \in \Q$ be rational numbers. Let $p^*$ and $q^*$ denote the rational cuts associated with $p$ and $q$. Then: :$p^* q^* = \paren {p q}^*$ Thus the operation of multiplication on the set of rational cuts is closed.


Set of Rational Cuts forms Ordered Field

Let $\RR$ denote the set of rational cuts. Let $\struct {\RR, +, \times, \le}$ denote the ordered structure formed from $\RR$ and: :the operation $+$ of addition of cuts :the operation $\times$ of multiplication of cuts :the ordering $\le$ of cuts. Then $\struct {\RR, + \times, \le}$ is an ordered field.


Infimum of Subset of Real Numbers May or May Not be in Subset

Let $S \subset \R$ be a proper subset of the set $\R$ of real numbers. Let $S$ admit an infimum $m$. Then $m$ may or may not be an element of $S$.


Apotome is Irrational

Every apotome is irrational, i.e.: : $\displaystyle \forall a, b \in \set {x \in \R_{>0} : x^2 \in \Q}: \paren {\frac a b \notin \Q \land \paren {\frac a b}^2 \in \Q} \implies \paren {\paren {a - b} \notin \Q \land \paren {a - b}^2 \notin \Q}$ {{:Euclid:Proposition/X/73}}


Basis Condition for Coarser Topology/Corollary 1

If $\BB_1$ and $\BB_2$ satisfy: :$\forall U \in \BB_1: \forall x \in U: \exists V \in \BB_2: x \in V \subseteq U$ then $\tau_1$ is coarser than $\tau_2$.


Derivatives of PGF of Discrete Uniform Distribution

Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$. Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are: :$\dfrac {\d^m} {\d s^m} \map {\Pi_X} s = \begin{cases} \displaystyle \dfrac 1 n \sum_{k \mathop = m}^n k^{\underline m} s^{k - m} & : m \le n \\ 0 & : k > n \end{cases}$ where $k^{\underline m}$ is the falling factorial.


Set of Doubletons of Natural Numbers is Countable

Let $S$ be the set defined as: :$S = \set {\set {n_1, n_2}: n_1, n_2 \in \N, n_1 \ne n_2}$ where $\N$ denotes the set of natural numbers. Then $S$ is countably infinite.


Frege Set Theory is Logically Inconsistent

The system of axiomatic set theory that is Frege set theory is inconsistent.


Empty Class is Unique

There is exactly one empty class.


Existence of Set is Equivalent to Existence of Empty Set

Let $V$ be a basic universe. Let $P$ be the axiom: :$V$ has at least one element. Then $P$ is equivalent to the axiom of the empty set: :The empty class $\O$ is a set.


Associativity on Four Elements

Let $\struct {S, \circ}$ be a semigroup. Let $a, b, c, d \in S$. Then: :$a \circ b \circ c \circ d$ gives a unique answer no matter how the elements are associated.


Intersection of Class is Subset of Intersection of Subclass

Let $V$ be a basic universe. Let $A$ and $B$ be classes of $V$: :$A \subseteq V, B \subseteq V$ such that it is not the case that $A = B = \O$. Let $\displaystyle \bigcap A$ and $\displaystyle \bigcap B$ denote the intersection of $A$ and intersection of $B$ respectively. Let $A$ be a subclass of $B$: :$A \subseteq B$ Then $\displaystyle \bigcap B$ is a subset of $\displaystyle \bigcap A$: :$\displaystyle \bigcap B \subseteq \displaystyle \bigcap A$


Frobenius's Theorem

An algebraic associative real division algebra $A$ is isomorphic to $\R, \C$ or $\Bbb H$.


Areas of Circles are as Squares on Diameters/Lemma


Dominance Relation is Ordering

Let $S$ and $T$ be cardinals. Let $S \preccurlyeq T$ denote that $S$ is dominated by $T$. Let $\mathbb S$ be any set of cardinals. Then the relational structure $\struct {\mathbb S, \preccurlyeq}$ is an ordered set. That is, $\preccurlyeq$ is an ordering (at least partial) on $\mathbb S$.


Basic Universe is Inductive

Let $V$ be a basic universe. Then $V$ is an inductive class.


Image of Subset is Image of Restriction

Let $f: S \to T$ be a mapping. Let $X \subseteq S$. Let $f {\restriction_X}$ be the restriction of $f$ to $X$. Then: :$f \sqbrk X = \Img {f {\restriction_X} }$ where $\Img f$ denotes the image of $f$, defined as: :$\Img f = \set {t \in T: \exists s \in S: t = \map f s}$


Binomial Distribution PMF

The probability mass function (pmf) of a binomially distributed random variable $X$ is equal to: :$\displaystyle \Pr \left({X = x}\right) = \binom n x p^x(1-p)^{n-x}$ where $n$ is the number of trials and $p$ is the probability of success.


Successor Mapping of Peano Structure has no Fixed Point

Let $\PP = \struct {P, s, 0}$ be a Peano structure. Then: :$\forall n \in P: \map s n \ne n$ That is, the successor mapping has no fixed points.


Closed Set is F-Sigma Set

Let $T = \struct {S, \tau}$ be a topological space. Let $V$ be a closed set of $T$. Then $V$ is an $F_\sigma$ set of $T$.


Union of Open Intervals of Positive Reals is Set of Strictly Positive Reals

Let $\R_{> 0}$ be the set of strictly positive real numbers. For all $x \in \R_{> 0}$, let $A_x$ be the open real interval $\openint 0 x$. Then: :$\displaystyle \bigcup_{x \mathop \in \R_{> 0} } A_x = \R_{> 0}$


Intersection of Closed Intervals of Positive Reals is Zero

Let $\R_{> 0}$ be the set of strictly positive real numbers. For all $x \in \R_{> 0}$, let $B_x$ be the closed real interval $\closedint 0 x$. Then: :$\displaystyle \bigcap_{x \mathop \in \R_{> 0} } B_x = \set 0$


Minimally Closed Class under Progressing Mapping is Well-Ordered

$N$ is well-ordered under the inclusion relation.


Sum of Independent Poisson Random Variables is Poisson

Let $X$ and $Y$ be independent discrete random variables with: :$X \sim \Poisson {\lambda_1}$ and :$Y \sim \Poisson {\lambda_2}$ for some $\lambda_1, \lambda_2 \in \R_{> 0}$. Then their sum $X + Y$ is distributed: :$X + Y \sim \Poisson {\lambda_1 + \lambda_2}$


Order of General Linear Group over Galois Field

Let $\GF$ be a Galois field with $p$ elements. Then the order of the general linear group $\GL {n, \GF}$ is: :$\displaystyle \prod_{j \mathop = 1}^n \paren {p^n - p^{j - 1} }$


Existence of Latin Squares

For each $n \in \Z_{>0}$ there exists at least one Latin square of order $n$.


Lipschitz Condition implies Uniform Continuity

Let $\left({M_1, d_1}\right)$ and $\left({M_2, d_2}\right)$ be metric spaces. Let $g: M_1 \to M_2$ satisfy the Lipschitz condition. Then $g$ is uniformly continuous on $M_1$.


Repeated Composition of Injection is Injection

Let $S$ be a set. Let $f: S \to S$ be an injection. Let the sequence of mappings: :$f^0, f^1, f^2, \ldots, f^n, \ldots$ be defined as: :$\forall n \in \N: \map {f^n} x = \begin {cases} x & : n = 0 \\ \map f x & : n = 1 \\ \map f {\map {f^{n - 1} } x} & : n > 1 \end{cases}$ Then for all $n \in \N$, $f^n$ is an injection.


Renaming Mapping from Set of Mappings on Single Element

Let $X$ and $Y$ be sets. Let $E$ be the set of all mappings from $X$ to $Y$. Let $b \in X$. Let $\mathcal R \subseteq E \times E$ be the relation on $E$ defined as: :$\mathcal R := \set {\tuple {f, g} \in \mathcal R: \map f b = \map g b}$ Let $e_b: E / \mathcal R \to Y$ be the renaming mapping induced by $\mathcal R$. Then $e_b$ is a bijection.


Power of Random Variable with Continuous Uniform Distribution has Beta Distribution

Let $X \sim \ContinuousUniform 0 1$ where $\ContinuousUniform 0 1$ is the continuous uniform distribution on $\closedint 0 1$. Let $n$ be a positive real number.  Then: :$X^n \sim \BetaDist {\dfrac 1 n} 1$ where $\operatorname {Beta}$ is the beta distribution.


Variance of Linear Transformation of Random Variable

Let $X$ be a random variable.  Let $a, b$ be real numbers.  Then we have:  :$\var {a X + b} = a^2 \var X$ where $\var X$ denotes the variance of $X$.


Versed Sine Function is Even

The versed sine is an even function: :$\forall \theta \in \R: \map \vers {-\theta} = \vers \theta$


Supremum Metric on Continuous Real Functions is Metric

Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval. Let $\mathscr C \left[{a \,.\,.\, b}\right]$ be the set of all continuous functions $f: \left[{a \,.\,.\, b}\right] \to \R$. Let $d$ be the supremum metric on  $\mathscr C \left[{a \,.\,.\, b}\right]$. Then $d$ is a metric.


P-Norm of Real Sequence is Strictly Decreasing Function of P

Let $p \ge 1$ be a real number. Let $\ell^p$ denote the $p$-sequence space. Let $\mathbf x = \sequence {x_n} \in \ell^p$. Suppose $\mathbf x$ is not a sequence of zero elements. Let $\norm {\mathbf x}_p$ denote the $p$-norm. Then the mapping $p \to \norm {\mathbf x}_p$ is strictly decreasing {{WRT}} $p$.


Extension of Half-Range Fourier Cosine Function to Symmetric Range

Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$. Let $\map f x$ be represented by the half-range Fourier cosine series $\map S x$: :$\map f x \sim \map S x = \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$ where for all $n \in \Z_{> 0}$: :$a_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \cos \frac {n \pi x} \lambda \rd x$ Then $\map S x$ also represents the extension to the even function $g: \openint {-\lambda} \lambda \to \R$ of $f$, defined as: :$\forall x \in \openint {-\lambda} \lambda: \map g x = \begin {cases} \map f x & : x > 0 \\ \map f {-x} & : x < 0 \\ \displaystyle \lim_{x \mathop \to 0} \map g x & : x = 0 \end {cases}$


P-adic Metric is Metric

Let $p \in \N$ be a prime. Let $\norm{\,\cdot\,}_p: \Q \to \R_{\ge 0}$ be the $p$-adic norm on $\Q$. Let $d_p$ be the $p$-adic metric on $\Q$: :$\forall x, y \in \Q: \map {d_p} {x, y} = \norm{x - y}_p$ Then $d_p$ is a metric.


Sum over Integers of Sine of n + alpha of theta over n + alpha

For $0 < \theta < 2 \pi$: :$\displaystyle \sum_{n \mathop \in \Z} \dfrac {\sin \paren {n + \alpha} \theta} {n + \alpha} = \pi$


Mittag-Leffler Expansion for Cotangent Function/Real Domain

:$\displaystyle \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \pi \cot \pi \alpha$


Sum of Cosines of Arithmetic Sequence of Angles/Formulation 1

{{begin-eqn}} {{eqn | l = \sum_{k \mathop = 0}^n \map \cos {\theta + k \alpha}       | r = \cos \theta + \map \cos {\theta + \alpha} + \map \cos {\theta + 2 \alpha} + \map \cos {\theta + 3 \alpha} + \dotsb }} {{eqn | r = \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} } \map \cos {\theta + \frac {n \alpha} 2} }} {{end-eqn}}


Count of Binary Operations Without Identity

Let $S$ be a set whose cardinality is $n$. The number $N$ of possible different binary operations which do not have an identity element that can be applied to $S$ is given by: :$N = n^{\left({\left({n-1}\right)^2 + 1}\right)} \left({n^{2 \left({n-1}\right)} - 1}\right)$


Addition of Coordinates on Cartesian Plane under Chebyshev Distance is Continuous Function

Let $\R^2$ be the real number plane. Let $d_\infty$ be the Chebyshev distance on $\R^2$. Let $f: \R^2 \to \R$ be the real-valued function defined as: :$\forall \left({x_1, x_2}\right) \in \R^2: f \left({x_1, x_2}\right) = x_1 + x_2$ Then $f$ is continuous.


Equivalence of Definitions of Order of Group Element

{{TFAE|def = Order of Group Element}} Let $G$ be a group whose identity is $e$. Let $x \in G$.


Maximum Rule for Continuous Functions

Let $\struct {S, \tau}$ be a topological space. Let $f, g: S \to \R$ be continuous real-valued functions. Let $\max \set {f, g}: S \to \R$ denote the pointwise maximum of $f$ and $g$. Then: :$\max \set {f, g}$ is continuous.


Continuity of Mapping to Cartesian Product under Chebyshev Distance

Let $M_1 = \left({A_1, d_1}\right), M_2 = \left({A_2, d_2}\right), \ldots, M_n = \left({A_n, d_n}\right)$ be metric spaces. Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$. Let $d_\infty: \mathcal A \times \mathcal A \to \R$ be the Chebyshev distance on $\mathcal A$: : $\displaystyle d_\infty \left({x, y}\right) = \max_{i \mathop = 1}^n \left\{ {d_i \left({x_i, y_i}\right)}\right\}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$. For all $i \in \left\{ {1, 2, \ldots, n}\right\}$, let $\operatorname{pr}_i: \mathcal A \to A_i$ be the $i$th projection on $\mathcal A$: :$\forall a \in \mathcal A: \operatorname{pr}_i \left({a}\right) = a_i$ where $a = \left({a_1, a_2, \ldots, a_n}\right) \in \mathcal A$. Let $M' = \left({X, d'}\right)$ be a metric space. Let $f: X \to \mathcal A$ be a mapping. Then $f$ is continuous on $X$ {{iff}} each of $\operatorname{pr}_i \circ f: X \to A_i$ is continuous on $X$.


Union of Mappings with Disjoint Domains is Mapping

Let $S_1, S_2, T_1, T_2$ be sets. Let $f: S_1 \to T_1$ and $g: S_2 \to T_2$ be mappings. Let $h = f \cup g$ be their union. If $S_1 \cap S_2 = \O$, then $h: S_1 \cup S_2 \to T_1 \cup T_2$ is a mapping whose domain is $S_1 \cup S_2$.


Natural Number m is Less than n iff m is an Element of n

Let $\omega$ be the set of natural numbers defined as the von Neumann construction. Let $m, n \in \omega$. Then: :$m < n \iff m \in n$ That is, every natural number is the set of all smaller natural numbers.


Limit of Image of Sequence/Real Number Line

Let $f$ be a real function which is continuous on the interval $\Bbb I$. Let $\left \langle {x_n} \right \rangle$ be a sequence of points in $\Bbb I$ such that: : $\displaystyle \lim_{n \mathop \to \infty} x_n = \xi$ where: : $(1): \quad \xi \in \Bbb I$ : $(2): \quad \displaystyle \lim_{n \mathop \to \infty} x_n$ denotes the limit of $x_n$. Then: : $\displaystyle \lim_{n \mathop \to \infty} f \left({x_n}\right) = f \left({\xi}\right)$ That is: : $\displaystyle \lim_{n \mathop \to \infty} f \left({x_n}\right) = f \left({\lim_{n \mathop \to \infty} x_n}\right)$


Like Electric Charges Repel

Let $a$ and $b$ be stationary particles, each carrying an electric charge of $q_a$ and $q_b$ respectively. Let $q_a$ and $q_b$ be of the same sign. That is, let $q_a$ and $q_b$ be like charges. Then the forces exerted by $a$ on $b$, and by $b$ on $a$, are such as to cause $a$ and $b$ to repel each other.


P-Sequence Space with P-Norm forms Banach Space

Let $\ell^p$ be a p-sequence space. Let $\norm {\, \cdot \,}_p$ be a p-norm. Then $\struct {\ell^p, \norm {\, \cdot \,}_p}$ is a Banach space.


Neighbourhood of Point Contains Point of Subset iff Distance is Zero

Let $M = \struct {X, d}$ be a metric space. Let $A \subseteq X$ be a non-empty subset of $X$. Let $x \in X$. Then every neighborhood of $x$ contains a point of $A$ {{iff}}: :$\map d {x, A} = 0$ where $\map d {x, A}$ denotes the distance from $x$ to $A$.


Set is Open iff Union of Open Balls

Let $M = \struct {A, d}$ be a metric space. Let $U \subseteq A$. Then $U$ is open in $M$ {{iff}} it is a union of open balls.


Empty Set is Open and Closed in Metric Space

Let $M = \left({A, d}\right)$ be a metric space. Then the empty set $\varnothing$ is both open and closed in $M$.


Equivalence of Definitions of Independent Subgroups

{{TFAE|def = Independent Subgroups}} Let $G$ be a group whose identity is $e$. Let $\left \langle {H_n} \right \rangle$ be a sequence of independent subgroups of $G$.


Condition for Factoring of Quotient Mapping between Modulo Addition Groups

Let $m, n \in \Z_{>0}$ be strictly positive integers. Let $\struct {\Z, +}$ denote the additive group of integers. Let $\struct {\Z_m, +_m}$ and $\struct {\Z_n, +_n}$ denote the additive groups of integers modulo $m$ and $n$ respectively. Let $f: \Z \to \Z_n$ be the quotient epimorphism from $\struct {\Z, +}$ to $\struct {\Z_n, +_n}$. Let $q: \Z \to \Z_m$ be the quotient epimorphism from $\struct {\Z, +}$ to $\struct {\Z_m, +_m}$. Then Then $N \subseteq K$ Then: :there exists a group homomorphism $\psi: \struct {\Z_m, +_m} \to \struct {\Z_n, +_n}$ {{iff}}  :$m \divides n$ where $\divides$ denotes divisibility.


Let the quotient structure defined by $\boxtimes$ be: : $\displaystyle \left({T', \oplus'}\right) := \left({\frac {S \times C} \boxtimes, \oplus_\boxtimes}\right)$ where $\oplus_\boxtimes$ is the operation induced on $\displaystyle \frac {S \times C} \boxtimes$ by $\oplus$. === Quotient Structure is Commutative Semigroup === {{:Construction of Inverse Completion/Quotient Structure is Commutative Semigroup}} === Quotient Mapping is Injective === {{:Construction of Inverse Completion/Quotient Mapping is Injective}} === Quotient Mapping is Monomorphism === {{:Construction of Inverse Completion/Quotient Mapping is Monomorphism}} === Image of Quotient Mapping is Subsemigroup === {{:Construction of Inverse Completion/Image of Quotient Mapping is Subsemigroup}} === Quotient Mapping to Image is Isomorphism === {{:Construction of Inverse Completion/Quotient Mapping to Image is Isomorphism}} === Image of Cancellable Elements in Quotient Mapping === {{:Construction of Inverse Completion/Quotient Mapping/Image of Cancellable Elements}}


Inclusion Mapping on Metric Space is Continuous

Let $M = \left({A, d}\right)$ be a metric space. Let $\left({H, d_H}\right)$ be a metric subspace of $M$. Then the inclusion mapping $i_H: H \to A$ is continuous.


Set of Transpositions is not Subgroup of Symmetric Group

Let $S$ be a finite set with $n$ elements such that $n > 2$. Let $G = \struct {\map \Gamma S, \circ}$ denote the symmetric group on $S$. Let $H \subseteq G$ denote the set of all transpositions of $S$ along with the identity mapping which moves no elements of $S$. Then $H$ does not form a subgroup of $G$.


Homomorphism from Reals to Circle Group/Corollary

Let $\struct {\R, +}$ be the additive group of real numbers. Let $\struct {C_{\ne 0}, \times}$ be the multiplicative group of complex numbers. Let $\phi: \struct {\R, +} \to \struct {C_{\ne 0}, \times}$ be the mapping defined as: :$\forall x \in \R: \map \phi x = \cos x + i \sin x$ Then $\phi$ is a (group) homomorphism.


Projection onto Ideal of External Direct Sum of Rings

Let $\left({R_1, +_1, \circ_1}\right), \left({R_2, +_2, \circ_2}\right), \ldots, \left({R_n, +_n, \circ_n}\right)$ be rings. Let $\displaystyle \left({R, +, \circ}\right) = \prod_{k \mathop = 1}^n \left({R_k, +_k, \circ_k}\right)$ be their direct product. For each $k \in \left[{1 \,.\,.\, n}\right]$, let: :$R'_k = \left\{{\left({x_1, \ldots, x_n}\right) \in R: \forall j \ne k: x_j = 0}\right\}$ Let $\operatorname{pr}_k: R \to R'_k$ be the projection on the $k$th coordinate of $\left({R, +, \circ}\right)$ onto $R'_k$. Then $\operatorname{pr}_k$ is an epimorphism.


Edgeless Graph is Bipartite

Let $N_n$ denote the edgeless graph with $n$ vertices. Then $N_n$ is a bipartite graph.


Set of Subfields forms Complete Lattice

Let $\struct {F, +, \circ}$ be a field. Let $\mathbb F$ be the set of all subfields of $F$. Then $\struct {\mathbb F, \subseteq}$ is a complete lattice.


Simple Graph of Maximum Size is Complete Graph

Let $G$ be a simple graph of order $n$ such that $n \ge 1$. Let $G$ have the largest size of all simple graphs of order $n$. Then: :$G$ is the complete graph $K_n$ :its size is $\dfrac {n \paren {n - 1} } 2$.


Sum of Integrals on Complementary Sets

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $A, E \in \Sigma$ with $A \subseteq E$. Let $f$ be a $\mu$-integrable function on $X$. Then  :$\displaystyle \int_E f \rd \mu = \int_A f \rd \mu + \int_{E \mathop \setminus A} f \rd \mu$


Characteristics of Cycle Graph

Let $G = \struct {V, E}$ be an (undirected) graph whose order is greater than $2$. Then $G$ is a cycle graph {{iff}}: :$G$ is connected :every vertex of $G$ is adjacent to $2$ other vertices :every edge of $G$ is adjacent to $2$ other edges.


Neighborhood of Point in Metrizable Space contains Closed Neighborhood

Let $T = \struct {S, \tau}$ be a metrizable topological space. Let $x \in S$ be an arbitrary point of $T$. Let $N$ be a neighborhood of $x$. Then $N$ has as a subset a neighborhood $V$ of $x$ such that $V$ is closed.


Induced Neighborhood Space is Neighborhood Space

Let $S$ be a set. Let $\tau$ be a topology on $S$, thus forming the topological space $\left({S, \tau}\right)$. Let $\left({S, \mathcal N}\right)$ be the neighborhood space induced by $\left({S, \tau}\right)$. Then $\left({S, \mathcal N}\right)$ is a neighborhood space.


Order 1 Simple Graph is Unique up to Isomorphism

Let $G_1 = \struct {\map V {G_1}, \map E {G_1} }$ and $G_2 = \struct {\map V {G_2}, \map E {G_2} }$ be simple graphs of order $1$. Then $G_1$ and $G_2$ are isomorphic.


Definition of Polynomial from Polynomial Ring over Sequence

Let $\struct {R, +, \circ}$ be a ring with unity. Let $\struct {P \sqbrk R, \oplus, \odot}$ be the polynomial ring over the set of all sequences in $R$: :$P \sqbrk R = \set {\sequence {r_0, r_1, r_2, \ldots} }$ where the operations $\oplus$ and $\odot$ on $P \sqbrk R$ be defined as: {{:Definition:Operations on Polynomial Ring of Sequences}} Let $\struct {R \sqbrk X, +, \circ}$ be the ring of polynomials over $R$ in $X$. {{explain|Strictly speaking the definition in terms of Definition:Polynomial Form is needed here, with $X$ specifically being an Definition:Indeterminate (Polynomial Theory) or Definition:Transcendental over Integral Domain.}} Then $\struct {R \sqbrk X, +, \circ}$ and $\struct {P \sqbrk R, \oplus, \odot}$ are isomorphic.


Bertrand's Theorem

Let $U: \R_{>0} \to \R$ be analytic for $r > 0$. Let $M > 0$ be a nonvanishing angular momentum such that a stable circular orbit exists. Suppose that every orbit sufficiently close to the circular orbit is closed. Then $U$ is either $k r^2$ or $-\dfrac k r$ (for $k > 0$) up to an additive constant.


Cube Root of 2 is Irrational

:$\sqrt [3] 2$ is irrational.


Integers under Multiplication form Countably Infinite Semigroup

The set of integers under multiplication $\struct {\Z, \times}$ is a countably infinite semigroup.


Third Derivative of Natural Logarithm Function

Let $\ln x$ be the natural logarithm function. Then: :$\map {D^3_x} {\ln x} = \dfrac 2 {x^3}$


Bounded Piecewise Continuous Function may not have One-Sided Limits

Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$, $a < b$.  Let $f$ be a bounded piecewise continuous function. {{:Definition:Bounded Piecewise Continuous Function}} Then it is not necessarily the case that $f$ is a piecewise continuous function with one-sided limits: {{:Definition:Piecewise Continuous Function with One-Sided Limits}}


Linear Second Order ODE/y'' = 1 over 1 - x^2

The second order ODE: :$(1): \quad y'' = \dfrac 1 {1 - x^2}$ has the general solution: :$y = x \tanh^{-1} x + \map \ln {1 - x^2} + C x + D$


Universal Property of Polynomial Ring/Free Monoid on Set

Let $R, S$ be commutative and unitary rings. Let $\left\langle{s_j}\right\rangle_{j \mathop \in J}$ be an indexed family of elements of $S$. Let $\psi: R \to S$ be a ring homomorphism. Let $R \left[{\left\{{X_j: j \in J}\right\}}\right]$ be a polynomial ring. Then there exists a unique evaluation homomorphism $\phi: R \left[{\left\{{X_j: j \in J}\right\}}\right] \to S$ at $\langle s_j\rangle_{j \in J}$ extending $\psi$.


Polynomial Addition is Associative

Addition of polynomials is an associative operation.


Null Polynomial is Additive Identity

The set of polynomial forms has an additive identity. {{explain|Context}}


De Morgan's Laws (Set Theory)/Proof by Induction/Difference with Union/Proof

Let $\mathbb T = \set {T_i: i \mathop \in I}$, where each $T_i$ is a set and $I$ is some finite indexing set. Then: : $\displaystyle S \setminus \bigcup_{i \mathop \in I} T_i = \bigcap_{i \mathop \in I} \paren {S \setminus T_i}$


Parity Addition is Commutative

Let $R := \struct {\set {\text{even}, \text{odd} }, +, \times}$ be the parity ring. The operation $+$ is commutative: :$\forall a, b \in R: a + b = b + a$


Parity Multiplication is Associative

Let $R := \struct {\set {\text{even}, \text{odd} }, +, \times}$ be the parity ring. The operation $\times$ is associative: :$\forall a, b, c \in R: \paren {a \times b} \times c = a \times \paren {b \times c}$


Events One of Which equals Union

Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$, so that $A \subseteq \Omega$ and $B \subseteq \Omega$. Let $A$ and $B$ be such that: :$A \cup B = A$ Then whenever $B$ occurs, it is always the case that $A$ occurs as well.


Direct Product iff Nontrivial Idempotent

Let $R$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Then $R$ is the direct product of two non-trivial rings {{iff}} $R$ contains an idempotent element not equal to $0_R$ or $1_R$.


Real Numbers under Multiplication do not form Group

The algebraic structure $\struct {\R, \times}$ consisting of the set of real numbers $\R$ under multiplication $\times$ is not a group.


Total Variation is Non-Negative

Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a function of bounded variation. Let $V_f$ be the total variation of $f$ on $\closedint a b$. Then:  :$V_f \ge 0$ with equality {{iff}} $f$ is constant.


Numbers of Type Integer a plus b root 2 Form Ordered Integral Domain

Let $\Z \sqbrk {\sqrt 2}$ denote the set: :$\Z \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Z}$ that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are integers. Then the algebraic structure: :$\struct {\Z \sqbrk {\sqrt 2}, +, \times}$ where $+$ and $\times$ are conventional addition and multiplication on real numbers, is an ordered integral domain.


Basis Test for Isolated Point

Let $T = \struct {S, \tau}$ be a topological space. Let $\BB$ be a synthetic basis of $T$. Let $H \subseteq S$. Then $x \in H$ is an isolated point of $H$ {{iff}}: :$\exists U \in \BB : U \cap H = \set x$


Boundary of Boundary is not necessarily Equal to Boundary

Let $T$ be a topological space. Let $H \subseteq T$. Let $\partial H$ denote the boundary of $H$. While it is true that: :$\map \partial {\partial H} \subseteq \partial H$ it is not necessarily the case that: :$\map \partial {\partial H} = \partial H$


Right Congruence Class Modulo Subgroup is Right Coset

Let $\mathcal R^r_H$ be the equivalence defined as right congruence modulo $H$. The equivalence class $\eqclass g {\mathcal R^r_H}$ of an element $g \in G$ is the right coset $H g$. This is known as the '''right congruence class of $g \bmod H$'''.


Equivalence of Definitions of Filter Basis

{{TFAE|def = Filter Basis}} Let $S$ be a set. Let $\FF$ be a filter on $S$.


Lexicographic Order on Pair of Well-Ordered Sets is Well-Ordering

Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be ordered sets. Let $\preccurlyeq$ be the lexicographic order on $S_1 \times S_2$''': :$\tuple {x_1, x_2} \preccurlyeq \tuple {y_1, y_2} \iff \tuple {x_1 \prec_1 y_1} \lor \tuple {x_1 = y_1 \land x_2 \preceq_2 y_2}$ Then: :$\preccurlyeq$ is a well-ordering on $S_1 \times S_2$ {{iff}} :both $\preceq_1$ and $\preceq_2$ are well-orderings.


Product of Functions of Bounded Variation is of Bounded Variation

Let $a, b$ be real numbers with $a < b$. Let $f, g : \closedint a b \to \R$ be functions of bounded variation. Let $V_f$ and $V_g$ be the total variations of $f$ and $g$ respectively.  Then $f \times g$ is of bounded variation with: :$V_{f \times g} \le A V_f + B V_g$ where: :$V_{f \times g}$ denotes the total variation of $f \times g$ :$A, B$ are non-negative real numbers.


Difference of Functions of Bounded Variation is of Bounded Variation

Let $a, b$ be real numbers with $a < b$. Let $f, g : \closedint a b \to \R$ be functions of bounded variation. Let $V_f$ and $V_g$ be the total variations of $f$ and $g$ respectively.  Then $f - g$ is of bounded variation with: :$V_{f - g} \le V_f + V_g$ where $V_{f - g}$ denotes the total variation of $f - g$.


Approximation to Stirling's Formula for Gamma Function

Let: :$D_\epsilon = \left\{{z \in \C : \left\vert{\operatorname{Arg} \left({z}\right)}\right\vert < \pi - \epsilon,\ \left\vert{z}\right\vert > 1}\right\}$ where: :$\left\vert{\operatorname{Arg} \left({z}\right)}\right\vert$ denotes the absolute value of the principal argument of $z$ :$\left\vert{z}\right\vert$ denotes the modulus of $z$ :$\epsilon \in \R_{>0}$. Then for all $z \in D_\epsilon$, the gamma function of $z$ satisfies: :$\Gamma \left({z}\right) = \sqrt{\dfrac {2 \pi} z} \left({\dfrac z e}\right)^z\left({1 + \mathcal O \left({z^{-1} }\right)}\right)$ where $\mathcal O \left({z^{-1} }\right)$ denotes big-O of $z^{-1}$.


Cauchy Sequence in Metric Space is not necessarily Convergent

Let $M = \struct {A, d}$ be a metric space. Let $\sequence {x_n}$ be a Cauchy sequence in $M$. Then it is not necessarily the case that $M$ is a convergent sequence in $M$.


Integers are not Close Packed

The integers $\Z$ are not close packed. That is: :$\forall n \in \Z: \not \exists m \in \Z: n < m < n + 1$


Linear Combination of Integers is Ideal

Let $a, b$ be any integers. Let $\Bbb S = \set {a x + b y: x, y \in \Z}$. Then the algebraic structure: :$\struct {\Bbb S, +, \times}$ is an ideal of $\Z$.


Negative of Sum of Real Numbers/Corollary

:$\forall x, y \in \R: -\paren {x - y} = -x + y$


Real Division by One

:$\forall x, y \in \R: \dfrac x 1 = x$


Reciprocal of Real Number is Non-Zero

:$\forall x \in \R: x \ne 0 \implies \dfrac 1 x \ne 0$


Differentiable Function with Bounded Derivative is Absolutely Continuous

Let $a, b$ be real numbers with $a < b$. Let $f: \closedint a b \to \R$ be a continuous function. Let $f$ be differentiable on $\openint a b$, with bounded derivative. Then $f$ is absolutely continuous.


Limit Points in Open Extension Space/Subset

Let $U \subseteq S^*_p$. Then $p$ is a limit point of $U$.


Polynomial Forms is PID Implies Coefficient Ring is Field

Let $D$ be an integral domain. Let $D \sqbrk X$ be the ring of polynomial forms in $X$ over $D$. Let $D sqbrk X$ be a principal ideal domain; Then $D$ is a field.


Sum of Strictly Positive Real Numbers is Strictly Positive

:$x, y \in \R_{>0} \implies x + y \in \R_{>0}$


Lagrange's Theorem (Number Theory)

Let $f$ be a polynomial in one variable of degree $n$ over $\Z_p$ for some prime $p$. Then $f$ has at most $n$ roots in $\Z_p$.


Finite Intersection of Regular Open Sets is Regular Open

Let $T$ be a topological space. Let $n \in \N$. Suppose that: :$\forall i \in \set {1, 2, \dotsc, n}: H_i \subseteq T$ where all the $H_i$ are regular open in $T$. That is: :$\forall i \in \set {1, 2, \dotsc, n}: H_i = H_i^{- \circ}$ where $H_i^{- \circ}$ denotes the interior of the closure of $H_i$. Then $\displaystyle \bigcap_{i \mathop = 1}^n H_i$ is regular open in $T$. That is: :$\displaystyle \bigcap_{i \mathop = 1}^n H_i = \paren {\bigcap_{i \mathop = 1}^n H_i}^{- \circ}$


Closure of Subspace of Normed Vector Space is Subspace

Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $Y \subseteq X$ be a subspace of $X$. Let $Y^-$ be the closure of $Y$. Then $Y^- \subseteq X$ is also a subspace of $X$.


Not every Open Set is F-Sigma Set

Let $T = \struct {S, \tau}$ be a topological space. Let $V$ be an open set of $T$. Then it is not necessarily the case that $V$ is an $F_\sigma$ set of $T$.


Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space

Let $\struct {X, \norm {\, \cdot \,}}$ is a normed vector space. Let $D \subseteq X$ be a subset of $X$. Let $D^-$ be the closure of $D$. Then $D$ is dense iff $D^- = X$.


Tychonoff Space is Preserved under Homeomorphism

If $T_A$ is a Tychonoff (completely regular) space, then so is $T_B$.


Normal Space is Preserved under Homeomorphism

If $T_A$ is a normal space, then so is $T_B$.


Left and Right Inverses of Square Matrix over Field are Equal

Let $\Bbb F$ be a field, usually one of the standard number fields $\Q$, $\R$ or $\C$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\map \MM n$ denote the matrix space of order $n$ square matrices over $\Bbb F$. Let $\mathbf B$ be a left inverse matrix of $\mathbf A$. Then $\mathbf B$ is also a right inverse matrix of $\mathbf A$. Similarly, let $\mathbf B$ be a right inverse matrix of $\mathbf A$. Then $\mathbf B$ is also a right inverse matrix of $\mathbf A$.


Principle of Finite Induction/Peano Structure

Let $\struct {P, s, 0}$ be a Peano structure. Let $S \subseteq P$. Suppose that: :$(1): \quad 0 \in S$ :$(2): \quad \forall n: n \in S \implies \map s n \in S$ Then: :$S = P$


System of Simultaneous Equations may have Multiple Solutions

Let $S$ be a system of simultaneous equations. Then it is possible that $S$ may have a solution set which is a singleton.


Fibonacci Number as Sum of Binomial Coefficients

{{begin-eqn}} {{eqn | lo= \forall n \in \Z_{>0}:       | l = F_n       | r = \sum_{k \mathop = 0}^{\floor {\frac {n - 1} 2} } \dbinom {n - k - 1} k       | c =  }} {{eqn | r = \binom {n - 1} 0 + \binom {n - 2} 1 + \binom {n - 3} 2 + \dotsb + \binom {n - j} {j - 1} + \binom {n - j - 1} j       | c = where $j = \floor {\frac {n - 1} 2}$ }} {{end-eqn}}


Countably Metacompact Lindelöf Space is Metacompact

Let $T = \struct {S, \tau}$ be a Lindelöf space which is also countably metacompact. Then $T$ is metacompact.


Dirichlet's Approximation Theorem

Let $\alpha, x \in \R$. Then there exist integers $a, q$ such that: : $\gcd \left\{{a, q}\right\} = 1$, $1 \le q \le x$ and: :$\left|{\alpha - \dfrac a q}\right| \le \dfrac 1 {q x}$


Secant in terms of Tangent

{{begin-eqn}} {{eqn | l = \sec x       | r = +\sqrt {\tan ^2 x + 1}       | c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ }} {{eqn | l = \sec x       | r = -\sqrt {\tan ^2 x + 1}       | c = if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ }} {{end-eqn}}


Compact Hausdorff Topology is Minimal Hausdorff

Let $T = \struct {S, \tau}$ be a Hausdorff space which is compact. Then $\tau$ is the minimal subset of the power set $\powerset S$ such that $T$ is a Hausdorff space.


Compact Hausdorff Topology is Maximally Compact

Let $T = \struct {S, \tau}$ be a Hausdorff space which is compact. Then $\tau$ is maximally compact.


Cosine is Reciprocal of Secant

:$\cos \theta = \dfrac 1 {\sec \theta}$


Column Equivalence is Equivalence Relation

Column equivalence is an equivalence relation.


Cosine in terms of Sine

{{begin-eqn}} {{eqn | l = \cos x       | r = +\sqrt {1 - \sin^2 x}       | c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ }} {{eqn | l = \cos x       | r = -\sqrt {1 - \sin^2 x}       | c = if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ }} {{end-eqn}}


Sign of Cosecant

Let $x$ be a real number. Then: {{begin-eqn}} {{eqn | l = \csc x       | o = >       | r = 0       | c = if there exists an integer $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$ }} {{eqn | l = \csc x       | o = <       | r = 0       | c = if there exists an integer $n$ such that $\paren {2 n + 1} \pi < x < \paren {2 n + 2} \pi$ }} {{end-eqn}} where $\csc$ is the real cosecant function.


Square Root of Number Minus Square Root/Proof 2

Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then: {{:Square Root of Number Minus Square Root}}


Atlas Belongs to Unique Differentiable Structure

Let $M$ be a locally Euclidean space of dimension $d$. Let $\mathcal A$ be an atlas on $M$. Then there exists a unique differentiable structure $\mathcal F$ on $M$ with $\mathcal A \in \mathcal F$.


Topological Space with One Quasicomponent is Connected

Let $T = \struct {S, \tau}$ be a topological space which has one quasicomponent. Then $T$ is connected.


Multiple of Column Added to Column of Determinant

Let $\mathbf A = \begin {bmatrix} a_{1 1} & \cdots & a_{1 r} & \cdots & a_{1 s} & \cdots & a_{1 n} \\ a_{2 1} & \cdots & a_{2 r} & \cdots & a_{2 s} & \cdots & a_{2 n} \\  \vdots & \ddots &  \vdots & \ddots &  \vdots & \ddots &  \vdots \\ a_{n 1} & \cdots & a_{n r} & \cdots & a_{n s} & \cdots & a_{n n} \\ \end {bmatrix}$ be a square matrix of order $n$. Let $\map \det {\mathbf A}$ denote the determinant of $\mathbf A$. Let $\mathbf B = \begin{bmatrix} a_{1 1} & \cdots & a_{1 r} + \lambda a_{1 s} & \cdots & a_{1 s} & \cdots & a_{1 n} \\ a_{2 1} & \cdots & a_{2 r} + \lambda a_{2 s} & \cdots & a_{2 s} & \cdots & a_{2 n} \\  \vdots & \ddots &                    \vdots & \ddots &  \vdots & \ddots &  \vdots \\ a_{n 1} & \cdots & a_{n r} + \lambda a_{n s} & \cdots & a_{n s} & \cdots & a_{n n} \\ \end{bmatrix}$. Then $\map \det {\mathbf B} = \map \det {\mathbf A}$. That is, the value of a determinant remains unchanged if a constant multiple of any column is added to any other column.


Integral Closure is Integrally Closed

Let $A \subseteq B$ be an extension of commutative rings with unity. Let $C$ be the integral closure of $A$ in $B$. Then $C$ is integrally closed.


Determinant of Lower Triangular Matrix

Let $\mathbf T_n$ be a lower triangular matrix of order $n$. Let $\map \det {\mathbf T_n}$ be the determinant of $\mathbf T_n$. Then $\map \det {\mathbf T_n}$ is equal to the product of all the diagonal elements of $\mathbf T_n$. That is: :$\displaystyle \map \det {\mathbf T_n} = \prod_{k \mathop = 1}^n a_{k k}$


Relationship between Component Types

Let $T = \struct {S, \tau}$ be a topological space. Let $p \in S$. Let: : $A$ be the arc component of $p$ : $P$ be the path component of $p$ : $C$ be the component of $p$ : $Q$ be the quasicomponent of $p$. Then: :$A \subseteq P \subseteq C \subseteq Q$ In general, the inclusions do not hold in the other direction.


Ultraconnected Space is T4

Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected. Then $T$ is a $T_4$ space.


Squares Ending in n Occurrences of m-Digit Pattern

Suppose there exists some integer $x$ such that $x^2$ ends in some $m$-digit pattern ending in an odd number not equal to $5$ and is preceded by another odd number, i.e.: :$\exists x \in \Z: x^2 \equiv \sqbrk {1 a_1 a_2 \cdots a_m} \pmod {2 \times 10^m}$ where $a_m$ is odd, $a_m \ne 5$ and $m \ge 1$. Then for any $n \ge 1$, there exists some integer with not more than $m n$-digits such that its square ends in $n$ occurrences of the $m$-digit pattern.


Locally Arc-Connected Space is Locally Path-Connected

Let $T = \struct {S, \tau}$ be a topological space which is locally arc-connected. Then $T$ is also locally path-connected.


Unique Factorization Domain is Integrally Closed

Let $A$ be a unique factorization domain (UFD). Then $A$ is integrally closed.


Lindelöf's Lemma

Let $C$ be a set of open real sets. Let $S$ be a real set that is covered by $C$. Then there exists a countable subset of $C$ that covers $S$.


Mellin Transform of Heaviside Step Function

Let $c$ be a constant real number. Let $u_c \left({t}\right)$ be the Heaviside step function. Let $\mathcal M$ be the Mellin transform. Then: :$\mathcal M \left\{ {u_c \left({t}\right)}\right\} \left({s}\right) = - \dfrac {c^s} s$ for $c > 0, \Re \left({s}\right) < 0$.


Complex Numbers as External Direct Product

Let $\struct {\C_{\ne 0}, \times}$ be the group of non-zero complex numbers under multiplication. Let $\struct {\R_{> 0}, \times}$ be the group of positive real numbers under multiplication. Let $\struct {K, \times}$ be the circle group. Then: :$\struct {\C_{\ne 0}, \times} \cong \struct {\R_{> 0}, \times} \times \struct {K, \times}$ {{explain|It is apparent that the second $\times$ is Cartesian product, but this is not obvious.}}


Definite Integral from 0 to Half Pi of Square of Logarithm of Cosine x

:$\displaystyle \int_0^{\pi/2} \paren {\map \ln {\cos x} }^2 \rd x = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}$


Metric Space Completeness is Preserved by Isometry

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $\phi: M_1 \to M_2$ be an isometry. If $M_1$ is complete then so is $M_2$.


3 Proper Integer Heronian Triangles whose Area and Perimeter are Equal

There are exactly $3$ proper integer Heronian triangles whose area and perimeter are equal. These are the triangles whose sides are: :$\tuple {6, 25, 29}$ :$\tuple {7, 15, 20}$ :$\tuple {9, 10, 17}$


Multiplication is Superfunction

The function $f: \C \to \C$, defined as: :$\map f z = z \times c$ is a superfunction for any complex number $c$.


Integration by Substitution/Definite Integral

The definite integral of $f$ from $a$ to $b$ can be evaluated by: :$\displaystyle \int_{\map \phi a}^{\map \phi b} \map f t \rd t = \int_a^b \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$ where $x = \map \phi u$.


Discrete Space is Non-Meager

Let $T = \left({S, \tau}\right)$ be a discrete topological space. Then $T$ is non-meager.


Subset of Indiscrete Space is Dense-in-itself

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Let $H \subseteq S$ be a subset of $S$ containing more than one point. Then $H$ is dense-in-itself.


Indiscrete Space is Non-Meager

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is non-meager.


Equivalence of Definitions of Minimal Element

Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. {{TFAE|def = Minimal Element}}


Non-Zero Natural Numbers under Addition form Semigroup

Let $\N_{>0}$ be the set of natural numbers without zero, that is: :$\N_{>0} = \N \setminus \set 0$ Let $+$ denote natural number addition. The structure $\struct {\N_{>0}, +}$ forms a semigroup.


Indiscrete Space is Second-Countable

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then  $T$ is a second-countable space.


Doubling the Cube by Compass and Straightedge Construction is Impossible

There is no compass and straightedge construction to allow a cube to be constructed whose volume is double that of a given cube.


Derivative of Inverse Hyperbolic Cotangent Function

:$\map {\dfrac \d {\d x} } {\coth^{-1} u} = \dfrac {-1} {u^2 - 1} \dfrac {\d u} {\d x}$  where $\size u > 1$


Indiscrete Space is T4

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space. Then $T$ is a $T_4$ space.


Derivative of Cosine of Function

:$\map {\dfrac \d {\d x} } {\cos u} = -\sin u \dfrac {\d u} {\d x}$


Equation of Circle/Polar

: $r^2 - 2 r r_0 \map \cos {\theta - \varphi} + \paren {r_0}^2 = R^2$


Singleton Partition yields Indiscrete Topology

Let $S$ be a set which is not empty. Let $\PP$ be the (trivial) singleton partition $\set S$ on $S$. Then the partition topology on $\PP$ is the indiscrete topology.


Derivative of Even Function is Odd

Let $f$ be a differentiable real function such that $f$ is even. Then its derivative $f'$ is an odd function.


Graph of Quadratic describes Parabola/Corollary 2

The locus of the equation of the square root function on the non-negative reals: :$\forall x \in \R_{\ge 0}: \map f x = \sqrt x$ describes half of a parabola.


Westwood's Puzzle

:500px Take any rectangle $ABCD$ and draw the diagonal $AC$. Inscribe a circle $GFJ$ in one of the resulting triangles $\triangle ABC$. Drop perpendiculars $IEF$ and $HEJ$ from the center of this incircle $E$ to the sides of the rectangle. Then the area of the rectangle $DHEI$ equals half the area of the rectangle $ABCD$.


Number of Parameters of ARMA Model

Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be an '''ARMA model''' on $S$ of order $p$: :$\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$ Then $M$ has $p + q + 2$ parameters. </onlyinclude>


Summation Formula for Polygonal Numbers

Let $P \left({k, n}\right)$ be the $n$th $k$-gonal number. Then: : $\displaystyle P \left({k, n}\right) = \sum_{j \mathop = 1}^n \left({\left({k - 2}\right) \left({j - 1}\right) + 1}\right)$


ARIMA Model subsumes Moving Average Model

Let $S$ be a stochastic process based on an equispaced time series. Let $M$ be a moving average model for $S$. Then $M$ is also an implementation of an ARIMA model.


Infinite Particular Point Space is not Strongly Locally Compact

Let $T = \struct {S, \tau_p}$ be an infinite particular point space. Then $T$ is not strongly locally compact.


Kronecker’s Theorem

Let $K$ be a field. Let $f$ be a polynomial over $K$ of degree $n \ge 1$. Then there exists a finite extension $L / K$ of $K$ such that $f$ has at least one root in $L$. Moreover, we can choose $L$ such that the degree $\index L K$ of $L / K$ satisfies $\index L K \le n$. {{explain|Work out exactly which definitions of Polynomial and Degree are appropriate here.}}


Cancellation Law for Field Product

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $a, b, c \in F$. Then: :$a \times b = a \times c \implies a = 0_F \text { or } b = c$


Edge is Bridge iff in All Spanning Trees

Let $G$ be a simple graph. Let $e$ be an edge of $G$. Then $e$ is a bridge in $G$ {{iff}} $e$ belongs to every spanning tree for $G$.


Leigh.Samphier/Sandbox/Set Difference of Distinct Equal Cardinality Sets is Not Empty

Let $S$ and $T$ be distinct finite sets. Let $\card S = \card T$. Then: :$S \setminus T \ne \O$


Particular Point Space is Locally Path-Connected

Let $T = \struct {S, \tau_p}$ be a particular point space. Then $T$ is locally path-connected.


Derivative of Inverse Hyperbolic Sine of x over a/Corollary 2

:$\map {\dfrac \d {\d x} } {\ln \size {x - \sqrt {x^2 + a^2} } } = -\dfrac 1 {\sqrt {x^2 + a^2} }$


Tarski-Vaught Test

Let $\mathcal M, \mathcal N$ be $\mathcal L$-structures such that $\mathcal M$ is a substructure of $\mathcal N$. {{wtd|The page Definition:Structure is a disambiguation page, in which the form in which it is used here may not be included. The level of clarity in this page generally needs improving. Hence the invocation of the Disambiguate template.}} {{Disambiguate|Definition:Structure}} Then $\mathcal M$ is an elementary substructure of $\mathcal N$ {{iff}}: :for every $\mathcal L$-formula $\phi \left({x, \bar v}\right)$ and for every $\bar a$ in $\mathcal M$: ::if there exists an $n$ in $\mathcal N$ such that $\mathcal N \models \phi \left({n, \bar a}\right)$ ::then there exists an $m$ in $\mathcal M$ such that $\mathcal N \models \phi \left({m, \bar a}\right)$. {{wtd|Before sense can be made of this page, Definition:Substructure and  Definition:Elementary Substructure need to be written.}} {{Disambiguate|Definition:Logical Formula}} The condition on the right side of the {{iff}} statement above can be rephrased as: :Every existential statement with parameters from $\mathcal M$ which is satisfied in $\mathcal N$ can be witnessed by an element from the substructure $\mathcal M$.


Conditional and Inverse are not Equivalent

A conditional statement: :$p \implies q$ is not logically equivalent to its inverse: :$\lnot p \implies \lnot q$


Weight of Discrete Topology equals Cardinality of Space

Let $T = \left({S, \tau}\right)$ be a discrete topological space. Then: :$w \left({T}\right) = \left\vert{S}\right\vert$ where  :$w \left({T}\right)$ denotes the weight of $T$ :$\left\vert{S}\right\vert$ denotes the cardinality of $S$.


Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2

:$\displaystyle \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \ln \size {\dfrac {a + x} {a - x} } + C$


Open Extension Topology is Topology

Let $T = \struct {S, \tau}$ be a topological space. Let $\tau^*_{\bar p}$ be the open extension topology of $\tau$. Then $\tau^*_{\bar p}$ is a topology on $S^*_p = S \cup \set p$.


Symmetric Difference with Intersection forms Boolean Ring

Let $S$ be a set. Then $\struct {\powerset S, *, \cap}$ is a Boolean ring.


1-Seminorm on Continuous on Closed Interval Real-Valued Functions is Norm

Let $\CC \closedint a b$ be the space of continuous on closed interval real-valued functions. Let $x \in \CC \closedint a b$ be a continuous real valued function. Let $\displaystyle \norm x_1 := \int_a^b \size {\map x t} \rd t$ be the 1-seminorm. Then $\norm {\, \cdot \,}_1$ is a norm on $\CC \closedint a b$.


Perpendicularity is Symmetric Relation

Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is perpendicular to $l_2$. Then $\perp$ is a symmetric relation on $S$.


Perpendicularity is Antitransitive Relation

Let $S$ be the set of straight lines in the plane. For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is perpendicular to $l_2$. Then $\perp$ is an antitransitive relation on $S$.


Multiplication using Parabola

:500pxrightthumb Let the parabola $P$ defined as $y = x^2$ be plotted on the Cartesian plane. Let $A = \tuple {x_a, y_a}$ and $B = \tuple {x_b, y_b}$ be points on the curve $\map f x$ so that $x_a < x_b$. Then the line segment joining $A B$ will cross the $y$-axis at $-x_a x_b$. Thus $P$ can be used as a nomogram to calculate the product of two numbers $x_a$ and $x^b$, as follows: :$(1) \quad$ Find the points $-x_a$ and $x_b$ on the $x$-axis. :$(2) \quad$ Find the points $A$ and $B$ where the lines $x = -x_a$ and $x = x_b$ cut $P$. :$(3) \quad$ Lay a straightedge on the straight line joining $A$ and $B$ and locate its $y$-intercept $c$. Then $x_a x_b$ can be read off from the $y$-axis as the position of $c$.


Excluded Point Space is Locally Path-Connected

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is locally path-connected.


Occurrence in Polish Notation has Unique Scope

Let $\mathcal F$ be a formal language in Polish notation. Let $\mathbf A$ be a well-formed formula of $\mathcal F$. Let $a$ be an occurrence in $\mathbf A$. Then $a$ has a unique scope.


Exclusive Or as Conjunction of Disjunctions

: $p \oplus q \dashv \vdash \left({p \lor q}\right) \land \left({\neg p \lor \neg q}\right)$


Space of Almost-Zero Sequences is Everywhere Dense in 2-Sequence Space

Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the 2-sequence space equipped with Euclidean norm. Let $c_{00}$ be the space of almost-zero sequences. Then $c_{00}$ is everywhere dense in $\struct {\ell^2, \norm {\, \cdot \,}_2}$


Ordering of Integers is Reversed by Negation

Let $x, y \in \Z$ such that $x > y$. Then: :$-x < -y$


Excluded Point Space is Sequentially Compact

Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space. Then $T$ is a sequentially compact space.


Sufficient Conditions for Weak Stationarity of Order 2

Let $S$ be a stochastic process giving rise to a time series $T$. Let the mean of $S$ be fixed. Let the autocovariance matrix of $S$ be of the form: :$\boldsymbol \Gamma_n = \begin {pmatrix} \gamma_0 & \gamma_1 & \gamma_2 & \cdots & \gamma_{n - 1} \\ \gamma_1 & \gamma_0 & \gamma_1 & \cdots & \gamma_{n - 2} \\ \gamma_2 & \gamma_1 & \gamma_0 & \cdots & \gamma_{n - 3} \\ \vdots   & \vdots   & \vdots   & \ddots & \vdots \\ \gamma_{n - 1} & \gamma_{n - 2} & \gamma_{n - 3} & \cdots & \gamma_0 \end {pmatrix} = \sigma_z^2 \mathbf P_n = \begin {pmatrix} 1 & \rho_1 & \rho_2 & \cdots & \rho_{n - 1} \\ \rho_1 & 1 & \rho_1 & \cdots & \rho_{n - 2} \\ \rho_2 & \rho_1 & 1 & \cdots & \rho_{n - 3} \\ \vdots   & \vdots   & \vdots   & \ddots & \vdots \\ \rho_{n - 1} & \rho_{n - 2} & \rho_{n - 3} & \cdots & 1 \end {pmatrix}$ Then $S$ is weakly stationary of order $2$.


Strict Ordering on Integers is Trichotomy

Let $\eqclass {a, b} {}$ and $\eqclass {c, d} {}$ be integers, as defined by the formal definition of integers. Then exactly one of the following holds: {{begin-eqn}} {{eqn | l = \eqclass {a, b} {}       | o = <       | r = \eqclass {c, d} {}       | c =  }} {{eqn | l = \eqclass {a, b} {}       | o = =       | r = \eqclass {c, d} {}       | c =  }} {{eqn | l = \eqclass {a, b} {}       | o = >       | r = \eqclass {c, d} {}       | c =  }} {{end-eqn}} That is, strict ordering is a trichotomy.


Excluded Set Topology is not T0

Let $T = \struct {S, \tau_{\bar H} }$ be an excluded set space where $H$ has at least two distinct points. Then $T$ is not a $T_0$ (Kolmogorov) space.


Boubaker's Theorem

Let $\left({R, +, \circ}\right)$ be a commutative ring. Let $\left({D, +, \circ}\right)$ be an integral subdomain of $R$ whose zero is $0_D$ and whose unity is $1_D$. Let $X \in R$ be transcendental over $D$. Let $D \left[{X}\right]$ be the ring of polynomial forms in $X$ over $D$. Finally, consider the following properties: {{begin-eqn}} {{eqn | n = 1       | l = \sum_{k \mathop = 1}^N {p_n \left({0}\right)}       | r = -2N }} {{eqn | n = 2       | l = \sum_{k \mathop = 1}^N {p_n \left({\alpha_k}\right)}       | r = 0 }} {{eqn | n = 3       | l = \left.{\sum_{k \mathop = 1}^N \frac {\mathrm d p_x \left({x}\right)} {\mathrm d x} }\right\vert_{x \mathop = 0}       | r = 0 }} {{eqn | n = 4       | l = \left.{\sum_{k \mathop = 1}^N \frac {\mathrm d {p_n}^2 \left({x}\right)} {\mathrm d x^2} }\right\vert_{x \mathop = 0}       | r = \frac 8 3 N \left({N^2 - 1}\right) }} {{end-eqn}} where, for a given positive integer $n$, $p_n \in D \left[{X}\right]$ is a non-null polynomial such that $p_n$ has $N$ roots $\alpha_k$ in $F$. Then the subsequence $\left \langle {B_{4 n} \left({x}\right)}\right \rangle$ of the Boubaker polynomials is the unique polynomial sequence of $D \left[{X}\right]$ which verifies simultaneously the four properties $(1) - (4)$.


Either-Or Topology is T0

Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a $T_0$ (Kolmogorov) space.


Either-Or Topology is not T1

Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is not a $T_1$ (Fréchet) space.


Cardinality of Set of Self-Mappings on Finite Set

Let $S$ be a finite set. Let the cardinality of $S$ be $n$. The cardinality of the set of all mappings from $S$ to itself (that is, the total number of self-maps on $S$) is: :$\card {S^S} = n^n$


Cover of Interval By Closed Intervals is not Pairwise Disjoint

Let $\closedint a b$ be a closed interval in $\R$. {{explain|Title mentions only "interval"; this does not affect truth of statement so may "closed" above line be removed as superfluous?}} Let $\JJ$ be a set of two or more closed intervals contained in $\closedint a b$ such that $\displaystyle \bigcup \JJ = \closedint a b$. Then the intervals in $\JJ$ are not pairwise disjoint.


Factorization of Natural Numbers within 4 n + 1 not Unique

Let: :$S = \set {4 n + 1: n \in \N} = \set {1, 5, 9, 13, 17, \ldots}$ be the set of natural numbers of the form $4 n + 1$. Then not all elements of $S$ have a complete factorization by other elements of $S$ which is unique.


Either-Or Topology is not T3

Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is not a $T_3$ space.


Definite Integral to Infinity of Reciprocal of x Squared plus a Squared/Corollary

:$\ds \int_0^\infty \dfrac {\d x} {1 + x^2} = \frac \pi 2$ for $a \ne 0$.


Either-Or Topology is First-Countable

Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a first-countable space.


Construction of Regular Heptadecagon

It is possible to construct a regular hepadecagon (that is, a regular polygon with $17$ sides) using a compass and straightedge construction.


Countable Stability implies Stability for All Infinite Cardinalities

Let $T$ be a complete $\mathcal L$-theory whose language $\mathcal L$ is countable. If $T$ is $\omega$-stable, then $T$ is $\kappa$-stable for all infinite $\kappa$.


Finite Complement Topology is Topology

Let $T = \struct {S, \tau}$ be a finite complement space. Then $\tau$ is a topology on $T$.


Event Space of Experiment with Final Sample Space has Even Cardinality

Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Omega$ be a finite set. Then the event space $\Sigma$ consists of an even number of subsets of $\Omega$.


Set of Elementary Events belonging to k Events is Event

Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $A_1, A_2, \ldots, A_m$ be events in the event space $\Sigma$ of $\EE$. Let $S$ denote the set of all elementary events of $\EE$ which are elements of exactly $k$ of the events $A_1, A_2, \ldots, A_m$. Then $S$ is an event of $\Sigma$.


Probability of Set Difference of Events

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$. Let $\map \Pr A$ denote the probability of event $A$ occurring. Then: :$\map \pr {A \setminus B} = \map \Pr A - \map \Pr {A \cap B}$


Distance Moved by Body from Rest under Constant Acceleration

Let a body $B$ be stationary. Let $B$ be subject to a constant acceleration. Then the distance travelled by $B$ is proportional to the square of the length of time $B$ is under the acceleration.


Set of Endomorphisms of Non-Abelian Group is not Ring

Let $\struct {G, \oplus}$ be a group which is non-abelian. Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$. Let $*: \mathbb G \times \mathbb G \to \mathbb G$ be the operation defined as: :$\forall u, v \in \mathbb G: u * v = u \circ v$ where $u \circ v$ is defined as composition of mappings. Then the algebraic structure $\struct {\mathbb G, \oplus, *}$ is not a ring.


Density not greater than Weight

Let $T = \left({S, \tau}\right)$ be a topological space. Then :$d \left({T}\right) \leq w \left({T}\right)$ where :$d \left({T}\right)$ denotes the density of $T$, :$w \left({T}\right)$ denotes the weight of $T$.


Countable Complement Space is not Countably Compact

Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not a countably compact space.


Closed Form for Number of Derangements on Finite Set

The number of derangements $D_n$ on a finite set $S$ of cardinality $n$ is: :$D_n = n! \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \cdots + \dfrac {\paren {-1}^n} {n!} }$


Characteristics of Birkhoff-James Orthogonality

Let $\struct {V, \norm {\,\cdot\,} }$ be a normed linear space. Let $x, y \in V$. Then $x$ and $y$ are '''Birkhoff-James orthogonal''' {{iff}} either: :$(1): \quad x = 0$ or: :$(2): \quad$ there exists a continuous functional $ f$ on $\struct {V, \norm {\,\cdot\,} }$ such that: ::::$\norm f = 1$ ::::$\map f x = \norm x$ ::::$\map f y = 0$


Countable Complement Space is Pseudocompact

Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is pseudocompact.


Countable Complement Space is not Countably Metacompact

Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$. Then $T$ is not countably metacompact.


Direct Product of Banach Spaces is Banach Space

Let $\struct {X, \norm {\, \cdot \,}}$ and $\struct {Y, \norm {\, \cdot \,}}$ be normed vector spaces. Let $V = X \times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations. Let $\norm {\tuple {x, y} }$ be the direct product norm. Suppose $X$ and $Y$ are Banach spaces. Then $V$ is a Banach space.


Rational Numbers are F-Sigma Set in Real Line

Let $\left({\R, \tau}\right)$ be the real number line considered asa  topological space with the usual (Euclidean) topology. Then: :$\Q$ is an $F_\sigma$ set in $\left({\R, \tau}\right)$.


Confocal Conics are Self-Orthogonal

The confocal conics defined by: :$\quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$ forms a family of orthogonal trajectories which is self-orthogonal. :500px


Compact Complement Topology is Connected

Let $T = \struct {\R, \tau}$ be the compact complement topology. Then $T$ is a connected space.


Compact Complement Topology is not Ultraconnected

Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $T$ is not an ultraconnected space.


Partial Derivatives of x tan^-1 (x^2 + y)

Let: :$\map f {x, y} = x \map \arctan {x^2 + y}$ Then: {{begin-eqn}} {{eqn | l = \map {f_1} {1, 0}       | r = \dfrac \pi 4 + 1 }} {{eqn | l = \map {f_2} {x, y}       | r = \dfrac x {1 + \paren {x^2 + y}^2} }} {{end-eqn}}


Partial Derivatives of x ln y^2 + y e^z

Let: :$\map f {x, y, z} = x \ln y^2 + y e^z$ Then: {{begin-eqn}} {{eqn | l = \map {f_1} {1, -1, 0}       | r = 0 }} {{eqn | l = \map {f_2} {x, x y, y + z}       | r = \dfrac 2 y + e^{y + z} }} {{end-eqn}}


Partial Derivative/Examples/u - v + 2 w, 2 u + v + 2 w, u - v + w

Let: {{begin-eqn}} {{eqn | l = u - v + 2 w       | r = x + 2 z }} {{eqn | l = 2 u + v - 2 w       | r = 2 x - 2 z }} {{eqn | l = u - v + w       | r = z - y }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \dfrac {\partial u} {\partial y}       | r = 0 }} {{eqn | l = \dfrac {\partial v} {\partial y}       | r = 2 }} {{eqn | l = \dfrac {\partial w} {\partial y}       | r = 1 }} {{end-eqn}}


Compact Complement Topology is Sequentially Compact

Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $T$ is a sequentially compact space.


Theories with Infinite Models have Models with Order Indiscernibles

Let $T$ be an $\LL$-theory with infinite models. Let $\struct {I, <$ be an infinite strict linearly ordered set. There is a model $\MM \models T$ containing an order indiscernible set $\set {x_i : i \in I}$.


Fort Space is Sequentially Compact

Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a sequentially compact space.


First Order ODE/x y' = y + 2 x exp (- y over x)

is a homogeneous differential equation with solution: :$e^{y / x} = \ln x^2 + C$


Fort Space is not Extremally Disconnected

Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is not an extremally disconnected space.


Non-Forking Types have Non-Forking Completions

Let $T$ be a complete $\mathcal L$-theory. Let $\mathfrak C$ be a monster model for $T$. Let $A\subseteq B$ be subsets of the universe of $\mathfrak C$. Let $\pi(\bar x)$ be an $n$-type over $B$. If $\pi$ does not fork over $A$, then there is a complete $n$-type $p$ over $B$ such that $\pi \subseteq p$ and $p$ does not fork over $A$.


Fortissimo Space is Completely Normal

Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is a completely normal space. Consequently, $T$ satisfies all weaker separation axioms.


First Order ODE/-1 over y sine x over y dx + x over y^2 sine x over y dy

is an exact differential equation with solution: :$\dfrac x y = C$


First Order ODE/dx = (y over (1 - x^2 y^2)) dx + (x over (1 - x^2 y^2)) dy

is an exact differential equation with solution: :$\map \ln {\dfrac {1 + x y} {1 - x y} } - 2 x = C$


First-Countability is not Continuous Invariant

Let $T_A = \struct {A, \tau_A}$ and $T_B = \struct {B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous mapping. If $T_A$ is a first-countable space, then it does not necessarily follow that $T_B$ is also first-countable.


First Order ODE/(x y - 1) dx + (x^2 - x y) dy = 0

The first order ODE: :$(1): \quad \paren {x y - 1} \rd x + \paren {x^2 - x y} \rd y = 0$ has the general solution: :$x y - \ln x - \dfrac {y^2} 2 + C$


First Order ODE/x dy = (x^5 + x^3 y^2 + y) dx

The first order ODE: :$(1): \quad x \rd y = \paren {x^5 + x^3 y^2 + y} \rd x$ has the general solution: :$\arctan \dfrac x y = -\dfrac {x^4} 4 + C$


Linear First Order ODE/y' + y = 2 x exp -x + x^2

The linear first order ODE: :$(1): \quad y' + y = 2 x e^{-x} + x^2$ has the general solution: :$y = x^2 e^{-x} + x^2 - 2 x + 2 + C e^{-x}$


First Order ODE/(exp y - 2 x y) y' = y^2

The first order ODE: :$(1): \quad \paren {e^y - 2 x y} y' = y^2$ has the general solution: :$x y^2 = e^y + C$


Differential Equation of Family of Linear Combination of Functions is Linear

Consider the one-parameter family of curves: :$(1): \quad y = C \map f x + \map g x$ The differential equation that describes $(1)$ is linear and of first order.


Trivial Topological Space is Non-Meager

Let $T = \struct {S, \tau}$ be a trivial topological space. Then $T$ is non-meager.


Hermitian Operators have Orthogonal Eigenvectors

The eigenvectors of a Hermitian operation are orthogonal.


First Order ODE/(exp x - 3 x^2 y^2) y' + y exp x = 2 x y^3

The first order ordinary differential equation: :$(1): \quad \paren {e^x - 3 x^2 y^2} y' + y e^x = 2 x y^3$ is an exact differential equation with solution: :$y e^x - x^2 y^3 = C$


First Order ODE/(y^2 exp x y + cosine x) dx + (exp x y + x y exp x y) dy = 0

The first order ordinary differential equation: :$(1): \quad \paren {y^2 e^{x y} + \cos x} \rd x + \paren {e^{x y} + x y e^{x y} } \rd y = 0$ is an exact differential equation with solution: :$y e^{x y} + \sin x = C$


First Order ODE/(y^2 - 3 x y - 2 x^2) dx = (x^2 - x y) dy

The first order ordinary differential equation: :$(1): \quad \paren {y^2 - 3 x y - 2 x^2} \rd x = \paren {x^2 - x y} \rd y$ is a homogeneous differential equation with solution: :$y^2 x^2 - 2 y x^3 + x^4 = C$