Module on Cartesian Product is Module

Let $\struct {R, +_R, \times_R}$ be a ring. Let $n \in \N_{>0}$. Let $\struct {R^n, +, \times}_R$ be the '''$R$-module $R^n$'''. Then $\struct {R^n, +, \times}_R$ is an $R$-module.


Module on Cartesian Product of Ring with Unity is Unitary Module

Let $\struct {R, +_R, \times_R}$ be a ring with unity. Let $n \in \N_{>0}$. Let $\struct {R^n, +, \times}_R$ be the '''$R$-module $R^n$'''. Then $\struct {R^n, +, \times}_R$ is a unitary $R$-module.


Ring of Polynomial Forms over Field is Vector Space

Let $\struct {F, +, \times}$ be a field whose unity is $1_F$. Let $F \sqbrk X$ be the ring of polynomials over $F$. Then $F \sqbrk X$ is an vector space over $F$.


Norm is Complete Iff Equivalent Norm is Complete

Let $R$ be a division ring. Let $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ be equivalent norms on $R$. Then: :$\struct {R,\norm {\,\cdot\,}_1}$ is complete {{iff}} $\struct {R,\norm {\,\cdot\,}_2}$ is complete.


Ring of Polynomial Forms over Field is Vector Space/Corollary

Let $S \subseteq F \sqbrk X$ denote the subset of $F \sqbrk X$ defined as: :$S = \set {\mathbf x \in F \sqbrk X: \map \deg {\mathbf x} < d}$ for some $d \in \Z_{>0}$. Then $S$ is an vector space over $F$.


P-adic Norm of p-adic Number is Power of p

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $x \in \Q_p: x \ne 0$. Then: :$\exists v \in \Z: \norm x_p = p^{-v}$


Borel-TIS inequality

Let $T$ be a topological space. Let $\sequence {f_t}_{t \mathop \in T}$ be a centred (i.e. mean zero) Gaussian process on $T$, such that: :$\norm f_T := \sup_{t \mathop \in T} \size {f_t}$ is almost surely finite. Let: :$\sigma_T^2 := \sup_{t \mathop \in T} \operatorname E \size {f_t}^2$ Then $\map {\operatorname E} {\norm f_T}$ and $\sigma_T$ are both finite, and, for each $u > 0$: :$\map {\operatorname P} {\norm f_T > \map {\operatorname E} {\norm f_T} + u} \le \map \exp {\dfrac {-u^2} {2 \sigma_T^2} }$


Negation of Propositional Function in Two Variables

Let $\map P {x, y}$ be a propositional function of two Variables. Then: :$\neg \forall x: \exists y: \map P {x, y} \iff \exists x: \forall y: \neg \map P {x, y}$ That is: :''It is not the case that for all $x$ a value of $y$ can be found to satisfy $\map P {x, y}$'' means the same thing as: :''There exists at least one value of $x$ such that for all $y$ it is not possible to satisfy $\map P {x, y}$''


Valuation Ideal of P-adic Numbers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. The valuation ideal induced by norm $\norm {\,\cdot\,}_p$ is the principal ideal: :$p \Z_p = \set {x \in \Q_p: \norm x_p < 1}$


Integers are Arbitrarily Close to P-adic Integers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Let $x \in \Z_p$. Then for $n \in \N$ there exists unique $\alpha \in \Z$: :$(1): \quad 0 \le \alpha \le p^n - 1$ :$(2): \quad \norm { x -\alpha}_p \le p^{-n}$


Integers are Dense in P-adic Integers

The integers $\Z$ are dense in the metric space $\struct{\Z_p, d_p}$.


P-adic Integer is Limit of Unique Coherent Sequence of Integers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Let $x \in \Z_p$. Then there exists a unique coherent sequence $\sequence {\alpha_n}$: :$\displaystyle \lim_{n \mathop \to \infty} \alpha_n = x$


Linearly Independent Set is Contained in some Basis/Proof 1

Let $E$ be a vector space of $n$ dimensions. Let $H$ be a linearly independent subset of $E$. {{:Linearly Independent Set is Contained in some Basis}}


Linearly Independent Set is Contained in some Basis/Proof 2


Vector Space over Division Subring is Vector Space

Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$. Let $S$ be a division subring of $R$, such that $1_R \in S$. The vector space $\struct {R, +, \circ_S}_S$ over $\circ_S$ is a $S$-vector space.


Vector Space on Field Extension is Vector Space

Let $\struct {K, +, \times}$ be a field. Let $L / K$ be a field extension over $K$. Let $\struct {L, +, \times}_K$ be the a vector space of $L$ over $K$. Then $\struct {L, +, \times}_K$ is a vector space.


Sheldon Conjecture

There is only $1$ Sheldon prime, and that is $73$.


Condition for Linear Divisor of Polynomial

Let $\map P x$ be a polynomial in $x$. Let $a$ be a constant. Then $x - a$ is a divisor of $\map P x$ {{iff}} $a$ is a root of $P$.


Polynomial with Algebraic Number as Root is Multiple of Minimal Polynomial

Let $F$ be a field. Let $\map P x$ be a polynomial in $F$. Let $z$ be a root of $\map P x$. Then $\map P x$ is a multiple of the minimal polynomial $\map m x$ in $z$ over $F$.


Simple Algebraic Field Extension consists of Polynomials in Algebraic Number

Let $F$ be a field. Let $\theta \in \C$ be algebraic over $F$. Let $\map F \theta$ be the simple field extension of $F$ by $\theta$. Then $\map F \theta$ consists of polynomials that can be written in the form $\map f \theta$, where $\map f x$ is a polynomial over $F$.


Element of Simple Algebraic Field Extension of Degree n is Polynomial in Algebraic Number of Degree Less than n

Let $F$ be a field. Let $\theta \in \C$ be algebraic over $F$ of degree $n$. Let $\map F \theta$ be the simple field extension of $F$ by $\theta$. Then any element of $\map F \theta$ can be written as $\map f \theta$, where $\map f x$ is a polynomial over $F$ of degree at most $n - 1$.


Degree of Simple Algebraic Field Extension equals Degree of Algebraic Number

Let $F$ be a field. Let $\theta \in \C$ be algebraic over $F$ of degree $n$. Let $\map F \theta$ be the simple field extension of $F$ by $\theta$. Then $\map F \theta$ is a finite extension of $F$ whose degree is: :$\index {\map F \theta} F = n$


Degree of Element of Finite Field Extension divides Degree of Extension

Let $F$ be a field whose zero is $0$ and whose unity is $1$. Let $K / F$ be a finite field extension of degree $n$. Let $\alpha \in K$ be algebraic over $F$. Then the degree of $\alpha$ is a divisor of $n$.


Algebraic Element of Degree 3 is not Element of Field Extension of Degree Power of 2

Let $K / F$ be a finite field extension of degree $2^m$. Let $\alpha \in K$ be algebraic over $F$ with degree $3$. Then $\alpha \notin K$.


Construction of Point in Cartesian Plane with Rational Coordinates

Let $\CC$ be a Cartesian plane. Let $P = \tuple {x, y}$ be a rational point in $\CC$. Then $P$ is constructible using a compass and straightedge construction.


Construction of Integer Multiple of Line Segment

Let $AB$ be a line segment in the plane. Let $AC$ be a line segment in the plane through a point $C$ Let $D$ be a point on $AC$ such that $AD = n AB$ for some $n \in \Z$. Then $AD$ is constructible using a compass and straightedge construction.


Point in Plane is Constructible iff Coordinates in Extension of Degree Power of 2

Let $\CC$ be a Cartesian plane. Let $S$ be a set of points in $\CC$. Let $F$ be the smallest field containing all the coordinates of the points in $S$. Let $P = \tuple {a, b}$ be a point in $\CC$. Then: :$P$ is constructible from $S$ using a compass and straightedge construction {{iff}}: :the coordinates of $P$ are contained in a finite extension $K$ of $F$ whose degree is given by: ::$\index K F = 2^m$ :for some $m \in \Z_{\ge 0}$.


Construction of Lattice Point in Cartesian Plane

Let $\CC$ be a Cartesian plane. Let $P = \tuple {a, b}$ be a lattice point in $\CC$. Then $P$ is constructible using a compass and straightedge construction.


Construction of Regular Heptagon by Compass and Straightedge Construction is Impossible

There is no compass and straightedge construction for a regular heptagon.


Rational Numbers with Denominator Power of Two form Integral Domain

Let $\Q$ denote the set of rational numbers. Let $S \subseteq \Q$ denote the set of set of rational numbers of the form $\dfrac p q$ where $q$ is a power of $2$: :$S = \set {\dfrac p q: p \in \Z, q \in \set {2^m: m \in \Z_{\ge 0} } }$ Then $\struct {S, +, \times}$ is an integral domain.


Normal Subgroup of Order 25 in Group of Order 100

Let $G$ be a group of order $100$. Then $G$ has a normal subgroup of order $25$.


Normal p-Subgroup contained in All Sylow p-Subgroups

Let $G$ be a finite group. Let $p$ be a prime number. Let $H$ be a normal subgroup of $G$ which is a $p$-group. Then $H$ is a subset of every Sylow $p$-subgroup of $G$.


Group of Order p^2 q is not Simple

Let $p$ and $q$ be prime numbers such that $p \ne q$. Let $G$ be a group of order $p^2 q$. Then $G$ is not simple.


Simple Group of Order Less than 60 is Prime

Let $G$ be a simple group. Let $\order G < 60$, where $\order G$ denotes the order of $G$. Then $G$ is a prime group.


Group of Order 42 has Normal Subgroup of Order 7

Let $G$ be of order $42$. Then $G$ has a normal subgroup of order $7$.


Group of Order 54 has Normal Subgroup of Order 27

Let $G$ be of order $54$. Then $G$ has a normal subgroup of order $27$.


Group of Order 40 has Normal Subgroup of Order 5

Let $G$ be of order $40$. Then $G$ has a normal subgroup of order $5$.


Finite Group with One Sylow p-Subgroup per Prime Divisor is Isomorphic to Direct Product

Let $G$ be a finite group whose order is $n$ and whose identity element is $e$. Let $G$ be such that it has exactly $1$ Sylow $p$-subgroup for each prime divisor of $n$. Then $G$ is isomorphic to the internal direct product of all its Sylow $p$-subgroups.


Number of Subgroups of Prime Power Order is Congruent to 1 modulo Prime

Let $G$ be a finite group whose order is $n$. Let $p$ be a prime number such that $p^k$ is a divisor of $n$. Then the number of subgroups of order $p^k$ is congruent to $1$ modulo $p$.


Count of Distinct Homomorphisms between Additive Groups of Integers Modulo m

Let $m, n \in \Z_{>0}$ be (strictly) positive integers. Let $\struct {\Z_m, +}$ denote the additive group of integers modulo $m$. The number of distinct homomorphisms $\phi: \struct {\Z_m, +} \to \struct {\Z_n, +}$ is $\gcd \set {m, n}$.


P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 1

:$\forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1}}$


P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 3

$\sequence {\alpha_n}$ is a unique sequence satisfying properties $(1)$, $(2)$ and $(3)$ above.


P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 2

:$\displaystyle \lim_{n \mathop \to \infty} \alpha_n = x$


Group Types of Order Prime Squared

Let $p$ be a prime number. Let $G$ be a group of order $p^2$. Then $G$ is isomorphic either to $\Z_{p^2}$ or to $\Z_p \times \Z_p$, where $\Z_p$ denotes the additive group of integers modulo $p$.


Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers

$\struct {G, \circ}$ is isomorphic to the additive group of real numbers $\struct {\R, +}$.


Open Set Characterization of Denseness/Analytic Basis

Let $\mathcal B \subseteq \tau$ be an analytic basis for $\tau$. Then $S$ is (everywhere) dense in $X$ {{iff}} every non-empty open set of $\mathcal B$ contains an element of $S$.


Open Set Characterization of Denseness/Open Ball

Let $\struct{X, d}$ be a metric space. Let $\tau_d$ be the topology induced by the metric $d$. Let $S \subseteq X$. Then $S$ is (everywhere) dense in $\struct{X, \tau_d}$ {{iff}} every open ball contains an element of $S$.


P-adic Integers is Metric Completion of Integers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $d$ be the subspace metric of the $p$-adic metric on the $p$-adic integers $\Z_p$. Then $\struct {\Z_p, d}$ is the metric completion of the integers $\Z$.


P-adic Metric on P-adic Numbers is Non-Archimedean Metric

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


Order of Automorphism Group of Cyclic Group

Let $C_n$ denote the cyclic group of order $n$. Let $\Aut {C_n}$ denote the automorphism group of $C_n$. Then: :$\order {\Aut {C_n} } = \map \phi n$ where: :$\order {\, \cdot \,}$ denotes the order of a group :$\map \phi n$ denotes the Euler $\phi$ function.


Order of Automorphism Group of Dihedral Group

Let $D_n$ denote the dihedral group of order $n$. Let $\Aut {D_n}$ denote the automorphism group of $D_n$. Then: :$\order {\Aut {D_n} } = 2 \map \phi n$ where: :$\order {\, \cdot \,}$ denotes the order of a group :$\map \phi n$ is the Euler $\phi$ function.


Valuation Ring of P-adic Norm on Rationals/Corollary 1

The set of integers $\Z$ is a subring of $\OO$.


Valuation Ring of P-adic Norm is Subring of P-adic Integers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Let $\Z_{\ideal p}$ be the induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$. Then: :$(1): \quad \Z_{\ideal p} = \Q \cap \Z_p$. :$(2): \quad \Z_{\ideal p}$ is a subring of $\Z_p$.


Valuation Ring of P-adic Norm is Subring of P-adic Integers/Corollary 1

The set of integers $\Z$ is a subring of $\Z_p$.


Laplace Transform of 1

Let $f: \R \to \R$ be the function defined as: :$\forall t \in \R: \map f t = 1$ Then the Laplace transform of $\map f t$ is given by: :$\laptrans {\map f t} = \dfrac 1 s$ for $\map \Re s > 0$.


Primitive of Exponential of a x by Hyperbolic Sine of b x

:$\displaystyle \int e^{a x} \sinh b x \rd x = \frac {e^{a x} \paren {a \sinh b x - b \cosh b x} } {a^2 - b^2} + C$


Primitive of Exponential of a x by Hyperbolic Cosine of b x

:$\displaystyle \int e^{a x} \cosh b x \rd x = \frac {e^{a x} \paren {a \cosh b x + b \sinh b x} } {a^2 - b^2} + C$


Laplace Transform of Derivative/Discontinuity at t = 0

Let $f$ fail to be continuous at $t = 0$, but let: :$\displaystyle \lim_{t \mathop \to 0} \map f t = \map f {0^+}$ exist. Then $\laptrans f$ exists for $\map \Re s > a$, and: :$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f {0^+}$


Laplace Transform of Derivative/Discontinuity at t = a

Let $f$ have a jump discontinuity at $t = a$. Then: :$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0 - e^{a s} \paren {\map f {a^+} - \map f {a^-} }$


Laplace Transform of Integral

:$\displaystyle \laptrans {\int_0^t \map f u \rd u} = \dfrac {\map F s} s$ wherever $\laptrans f$ exists.


Integral of Laplace Transform

:$\displaystyle \laptrans {\dfrac {\map f t} t} = \int_s^{\to \infty} \map F u \rd u$ wherever $\displaystyle \lim_{t \mathop \to 0} \dfrac {\map f t} t$ and $\laptrans f$ exist.


Inclusion Mapping on Subring is Homomorphism

Let $\struct {R, +, \circ}$ be a ring. Let $\struct {S, +{\restriction_S}, \circ {\restriction_S}}$ be a subring of $R$. Let $i_S: S \to R$ be the inclusion mapping from $S$ to $R$. Then ${i_S}$ is a ring homomorphism.


Inclusion Mapping on Subring is Monomorphism

Let $\struct {R, +, \circ}$ be a ring. Let $\struct {S, +{\restriction_S}, \circ {\restriction_S} }$ be a subring of $R$. Let $i_S: S \to R$ be the inclusion mapping from $S$ to $R$. Then $i_S$ is a ring monomorphism.


Negative of Subring is Negative of Ring

Let $\struct {R, +, \circ}$ be a ring. For each $x \in R$ let $-x$ denote the ring negative of $x$ in $R$. Let $\struct {S, + {\restriction_S}, \circ {\restriction_S}}$ be a subring of $R$. For each $x \in S$ let $\mathbin \sim x$ denote the ring negative of $x$ in $S$. Then: :$\forall x \in S: \mathbin \sim x = -x$


Limit to Infinity of Laplace Transform

Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$. Then: :$\displaystyle \lim_{s \mathop \to \infty} \map F s = 0$


Initial Value Theorem of Laplace Transform

Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$. Then: :$\displaystyle \lim_{t \mathop \to 0} \map f t = \lim_{s \mathop \to \infty} s \, \map F s$ if those limits exist.


Final Value Theorem of Laplace Transform

Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$. Then: :$\displaystyle \lim_{t \mathop \to \infty} \map f t = \lim_{s \mathop \to 0} s \, \map F s$ if those limits exist.


Evaluation of Integral using Laplace Transform

Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$. Then: :$\displaystyle \int_0^{\to \infty} \map f t \rd t = \map F 0$ assuming the integral is convergent.


Valuation Ring of Non-Archimedean Division Ring is Clopen

Let $\struct {R, \norm {\,\cdot\,} }$ be a non-Archimedean normed division ring with zero $0_R$. Let $\mathcal O$ be valuation ring induced by $\norm{\,\cdot\,}$. Then $\mathcal O$ is a both open and closed in the metric induced by $\norm{\,\cdot\,}$.


Valuation Ring of Non-Archimedean Division Ring is Clopen/Corollary 1

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Then the $p$-adic integers $\Z_p$ is both open and closed in the $p$-adic metric.


Series Expansion of Bessel Function of the First Kind

{{begin-eqn}} {{eqn | l = \map {J_n} x       | r = \dfrac {x^n} {2^n \, \map \Gamma {n + 1} } \paren {1 - \dfrac {x^2} {2 \paren {2 n + 2} } + \dfrac {x^4} {2 \times 4 \paren {2 n + 2} \paren {2 n + 4} } - \cdots}       | c =  }} {{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {n + k + 1} } \paren {\dfrac x 2}^{n + 2 k}       | c =  }} {{end-eqn}}


Recurrence Formula for Bessel Function of the First Kind

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Then: :$\map {J_{n + 1} } x = \dfrac {2 n} x \map {J_n} x - \map {J_{n - 1} } x$ And:  :$\map {J_{n + 1} } x = -2 \map {J_n'} x + \map {J_{n - 1} } x$ {{refactor|page per result}}


Derivative of x^n by Bessel Function of the First Kind of Order n of x

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Then: :$\map {\dfrac \d {\d x} } {x^n \map {J_n} x} = x^n \map {J_{n - 1} } x$


Generating Function for Bessel Function of the First Kind of Order n of x

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Then: :$\map \exp {\dfrac {x \paren {t - \frac 1 t} } 2} = \displaystyle \sum_{n \mathop = -\infty}^\infty \map {J_n} x t^n$


Series Expansion of Bessel Function of the First Kind/Negative Index

{{begin-eqn}} {{eqn | l = \map {J_{-n} } x       | r = \dfrac {x^{-n} } {2^{-n} \, \map \Gamma {1 - n} } \paren {1 - \dfrac {x^2} {2 \paren {2 - 2 n} } + \dfrac {x^4} {2 \times 4 \paren {2 - 2 n} \paren {4 - 2 n} } - \cdots}       | c =  }} {{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {k + 1 - n} } \paren {\dfrac x 2}^{2 k - n}       | c =  }} {{end-eqn}}


Bessel Function of the First Kind for Imaginary Argument

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Then: :$\map {J_n } {i x} = i^{-n} \, \map {I_n} x$ where: :$i$ denotes the imaginary unit :$\map {I_n} x$ denotes the modified Bessel function of the first kind of order $n$.


Integral to Infinity of Shifted Dirac Delta Function by Continuous Function

Let $\map \delta x$ denote the Dirac delta function. Let $g$ be a continuous real function. Let $a \in \R_{\ge 0}$ be a positive real number. Then: :$\displaystyle \int_0^{+ \infty} \map \delta {x - a} \, \map g x \rd x = \map g a$


Function which is Zero except on Countable Set of Points is Null

Let $S \subseteq \R$ be a subset of $\R$ such that $S$ is countable, either finite or countably infinite. Let $f: \R \to \R$ be a real function such that: :$\forall x \in \R \setminus S: \map f x = 0$ That is, except perhaps for the elements of $S$, the value of $f$ is zero. Then $f$ is a null function.


Laplace Transform of Bessel Function of the First Kind of Order Zero

Let $J_0$ denote the Bessel function of the first kind of order $0$. Then the Laplace transform of $J_0$ is given as: :$\laptrans {\map {J_0} t} = \dfrac 1 {\sqrt {s^2 + 1} }$


Laplace Transform of Bessel Function of the First Kind

Let $J_n$ denote the Bessel function of the first kind of order $n$. Then the Laplace transform of $J_n$ is given as: :$\laptrans {\map {J_n} {a t} } = \dfrac {\paren {\sqrt {s^2 + a^2} - s}^n} {a^n \sqrt {s^2 + a^2} }$


Laplace Transform of Sine of Root

:$\laptrans {\sin \sqrt t} = \dfrac {\sqrt \pi} {2 s^{3/2} } \map \exp {-\dfrac 1 {4 s} }$ where $\laptrans f$ denotes the Laplace transform of the function $f$.


Laplace Transform of Cosine of Root over Root

:$\laptrans {\dfrac {\cos \sqrt t} {\sqrt t} } = \sqrt {\dfrac \pi s} \, \map \exp {-\dfrac 1 {4 s} }$ where $\laptrans f$ denotes the Laplace transform of the function $f$.


Laplace Transform of Error Function of Root

:$\laptrans {\map \erf {\sqrt t} } = \dfrac 1 {s \sqrt {s + 1} }$ where: :$\laptrans f$ denotes the Laplace transform of the function $f$ :$\erf$ denotes the error function


Laplace Transform of Sine Integral Function

:$\laptrans {\map \Si t} = \dfrac 1 s \arctan \dfrac 1 s$ where: :$\laptrans f$ denotes the Laplace transform of the function $f$ :$\Si$ denotes the sine integral function


Laplace Transform of Cosine Integral Function

:$\laptrans {\map \Ci t} = \dfrac {\map \ln {s^2 + 1} } {2 s}$ where: :$\laptrans f$ denotes the Laplace transform of the function $f$ :$\Ci$ denotes the cosine integral function.


Laplace Transform of Heaviside Step Function

Let $\map {u_c} t$ denote the Heaviside step function: :$\map {u_c} t = \begin{cases} 1 & : t > c \\ 0 & : t < c \end{cases}$ The Laplace transform of $\map {u_c} t$ is given by: :$\laptrans {\map {u_c} t} = \dfrac {e^{-s c} } s$ for $\map \Re s > c$.


Laplace Transform of Dirac Delta Function

Let $\map \delta t$ denote the Dirac delta function. The Laplace transform of $\map \delta t$ is given by: :$\laptrans {\map \delta t} = 1$


Laplace Transform of Null Function

Let $\mathcal N: \R \to \R$ be a null function. The Laplace transform of $\map {\mathcal N} t$ is given by: :$\laptrans {\map {\mathcal N} t} = 0$


Laplace Transform of Function of t minus a/Proof 1

Let $f$ be a function such that $\laptrans f$ exists. Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$. Let $a \in \C$ or $\R$ be constant. {{:Laplace Transform of Function of t minus a}}


Laplace Transform of Function of t minus a/Proof 2


Laplace Transform of Sine of t over t

Let $\sin$ denote the real sine function. Let $\laptrans f$ denote the Laplace transform of a real function $f$. Then: :$\laptrans {\dfrac {\sin t} t} = \arctan \dfrac 1 s$


Integral to Infinity of Function over Argument

:$\displaystyle \int_0^\infty {\dfrac {\map f t} t} = \int_0^{\to \infty} \map F u \rd u$ provided the integrals converge.


Integral to Infinity of Exponential of -t^2

:$\displaystyle \int_0^\infty \map \exp {-t^2} \rd t = \dfrac {\sqrt \pi} 2$


Laplace Transform of Real Power

Let $n$ be a constant real number such that $n > -1$ Let $f: \R \to \R$ be the real function defined as: :$\map f t = t^n$ Then $f$ has a Laplace transform given by: {{begin-eqn}} {{eqn	| l = \laptrans {\map f t} 	| r = \int_0^\infty e^{-s t} t^n \rd t }} {{eqn	| r = \frac {\map \Gamma {n + 1} } {s^{n + 1} } }} {{end-eqn}} where $\Gamma$ denotes the gamma function.


Laplace Transform of Reciprocal of Square Root

:$\laptrans {\dfrac 1 {\sqrt t} } = \sqrt {\dfrac \pi s}$ where $\laptrans f$ denotes the Laplace transform of the real function $f$.


Gamma Function of 3 over 2

:$\map \Gamma {\dfrac 3 2} = \dfrac {\sqrt \pi} 2$


Gamma Function of Minus 3 over 2

:$\map \Gamma {-\dfrac 3 2} = \dfrac {4 \sqrt \pi} 3$


Gamma Function of Zero

:$\map \Gamma 0$ is not defined.


Gamma Function of Minus 1

:$\map \Gamma {-1}$ is not defined.


Gamma Function of Minus 2

:$\map \Gamma {-2}$ is not defined.


Integral to Infinity of Bessel Function of First Kind order Zero

:$\displaystyle \int_0^\infty \map {J_0} t \rd t = 1$ where $J_0$ denotes theBessel function of the first kind of order $0$.


Integral to Infinity of e^-t by Error Function of Root t

:$\displaystyle \int_0^\infty e^{-t} \erf \sqrt t \rd t = \dfrac {\sqrt 2} 2$


Second Derivative of Laplace Transform

Let $f: \R \to \R$ or $\R \to \C$ be a continuous function, twice differentiable on any closed interval $\closedint 0 a$. Let $\laptrans f = F$ denote the Laplace transform of $f$. Then, everywhere that $\dfrac {\d^2} {\d s^2} \laptrans f$ exists: :$\dfrac {\d^2} {\d s^2} \laptrans {\map f t} = \laptrans {t^2 \, \map f t}$


Laplace Transform of Multiple Integral

:$\displaystyle \laptrans {\underbrace {\int_0^t \dotsm \int_0^t}_{\text {$n$ times} } \map f u \rd u^n} = \dfrac {\map F s} {s^n}$ wherever $\laptrans f$ exists.


Integral of Reciprocal is Divergent/Unbounded Above

:$\displaystyle \int_1^n \frac {\d x} x \to +\infty$ as $n \to + \infty$


Integral of Reciprocal is Divergent/To Zero

:$\displaystyle \int_\gamma^1 \frac {\d x} x \to -\infty$ as $\gamma \to 0^+$


Convergence of P-Series/Lemma

Let $p = x + i y$ be a complex number where $x, y \in \R$ such that: :$x > 0$ :$x \ne 1$ Then: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^x}$ converges {{iff}} $\displaystyle \lim_{P \mathop \to \infty} \dfrac {P^{1 - x} } {1 - x}$ converges.


Integral to Infinity of Reciprocal of Power of x

The improper integral :$\displaystyle \int_1^\infty \dfrac {\d t} {t^x}$ exists {{iff}} $x > 1$.


Convergence of P-Series/Real

Let $p \in \R$ be a real number. Then the $p$-series: :$\displaystyle \sum_{n \mathop = 1}^\infty n^{-p}$ is convergent {{iff}} $p > 1$.


Real Number Ordering is Transitive

Let $a, b, c \in \R$ such that $a > b$ and $b > c$. Then: :$a > c$


Real Number Ordering is Compatible with Multiplication/Positive Factor

:$\forall a, b, c \in \R: a < b \land c > 0 \implies a c < b c$


Real Number Ordering is Compatible with Multiplication/Negative Factor

:$\forall a, b, c \in \R: a < b \land c < 0 \implies a c > b c$


Rational Power of Product of Real Numbers

Let $r, s \in \R_{> 0}$ be (strictly) positive real numbers. <onlyinclude> Let $x \in \Q$ be a rational number. Let $r^x$ be defined as $r$ to the power of $x$. Then: :$\paren {r s}^x = r^x s^x$


Sign of Quadratic Function Between Roots

Let $a \in \R_{>0}$ be a (strictly) positive real number. Let $\alpha$ and $\beta$, where $\alpha < \beta$, be the roots of the quadratic function: :$\map Q x = a x^2 + b x + c$ whose discriminant $b^2 - 4 a c$ is (strictly) positive. Then: :$\begin {cases} \map Q x < 0 & : \text {when $\alpha < x < \beta$} \\ \map Q x > 0 & : \text {when $x < \alpha$ or $x > \beta$} \end {cases}$


Minimum Value of Real Quadratic Function

Let $a \in \R_{>0}$ be a (strictly) positive real number. Consider the quadratic function: :$\map Q x = a x^2 + b x + c$ $\map Q x$ achieves a minimum at $x = -\dfrac b {2 a}$, at which point $\map Q x = c - \dfrac {b^2} {4 a}$.


Number of Type Rational r plus s Root 2 is Irrational

Let $r, s \in \Q$ be rational numbers. Then $r + s \sqrt 2$ is irrational.


Roots of Quadratic with Rational Coefficients of form r plus s Root 2

Consider the quadratic equation: :$(1): \quad a^2 x + b x + c = 0$ where $a, b, c$ are rational. Let $\alpha = r + s \sqrt 2$ be one of the roots of $(1)$. Then $\beta = r - s \sqrt 2$ is the other root of $(1)$.


Supremum is not necessarily Greatest Element

Let $\struct {S, \preceq}$ be an ordered set. Let $T$ admit a supremum in $S$. Then the supremum of $T$ in $S$ is not necessarily the greatest element of $T$.


Closed Interval Defined by Absolute Value

:$\set {x \in \R: \size {\xi - x} \le \delta} = \closedint {\xi - \delta} {\xi + \delta}$ where $\closedint {\xi - \delta} {\xi + \delta}$ is the closed real interval between $\xi - \delta$ and $\xi + \delta$.


Open Interval Defined by Absolute Value

:$\set {x \in \R: \size {\xi - x} < \delta} = \openint {\xi - \delta} {\xi + \delta}$ where $\openint {\xi - \delta} {\xi + \delta}$ is the open real interval between $\xi - \delta$ and $\xi + \delta$.


Distance from Subset of Real Numbers to Element

:$x \in S \implies \map d {x, S} = 0$


Distance from Subset of Real Numbers to Supremum

Let $S$ be bounded above such that $\xi = \sup S$. Then: :$\map d {\xi, S} = 0$


Distance from Subset of Real Numbers to Infimum

Let $S$ be bounded below such that $\xi = \inf S$. Then: :$\map d {\xi, S} = 0$


Real Number at Distance Zero from Closed Real Interval is In Interval

Let $I \subseteq \R$ be a closed real interval. Then: :$\map d {x, I} = 0 \implies x \in I$


Existence of Real Number at Distance Zero from Open Real Interval not in Interval

Let $I \subseteq \R$ be an open real interval such that $I \ne \O$ and $I \ne \R$. Then: :$\exists x \notin I: \map d {x, I} = 0$


Distance from Subset of Real Numbers to Element/Proof 1

Let $S$ be a subset of the set of real numbers $\R$. Let $x \in \R$ be a real number. Let $\map d {x, S}$ be the distance between $x$ and $S$. Then: {{:Distance from Subset of Real Numbers to Element}}


Distance from Subset of Real Numbers to Element/Proof 2


Distance from Subset of Real Numbers to Supremum/Proof 1

Let $S$ be a subset of the set of real numbers $\R$. Let $x \in \R$ be a real number. Let $\map d {x, S}$ be the distance between $x$ and $S$. {{:Distance from Subset of Real Numbers to Supremum}}


Distance from Subset of Real Numbers to Supremum/Proof 2


Distance from Subset of Real Numbers to Infimum/Proof 1

Let $S$ be a subset of the set of real numbers $\R$. Let $x \in \R$ be a real number. Let $\map d {x, S}$ be the distance between $x$ and $S$. {{:Distance from Subset of Real Numbers to Infimum}}


Distance from Subset of Real Numbers to Infimum/Proof 2


Infimum of Set of Reciprocals of Positive Integers

Let $S$ be the subset of the set of real numbers defined as: :$S = \set {\dfrac 1 n: n \in \Z_{>0} }$ Then: :$\inf S = 0$ where $\inf S$ denotes the infimum of $S$.


Set of Numbers of form n - 1 over n is Bounded Above

Let $S$ be the subset of the set of real numbers $\R$ defined as: :$S = \set {\dfrac {n - 1} n: n \in \Z_{>0} }$ $S$ is bounded above with supremum $1$. $S$ has no greatest element.


Set of Rational Numbers Strictly between Zero and One has no Greatest or Least Element

Let $S \subseteq \Q$ be the subset of the set of rational numbers defined as: :$S = \set {r \in \Q: 0 < r < 1}$ Then $S$ has no greatest or smallest element. However, $S$ has a supremum $1$ and an infimum $0$.


Between two Real Numbers exists Irrational Number

Let $a, b \in \R$ be real numbers where $a < b$.  Then there exists an irrational number $\xi \in \R \setminus \Q$ such that: :$a < \xi < b$


Sequence of Powers of Reciprocals is Null Sequence/Real Index

Let $r \in \R_{>0}$ be a strictly positive real number. Let $\sequence {x_n}$ be the sequence in $\R$ defined as: : $x_n = \dfrac 1 {n^r}$ Then $\sequence {x_n}$ is a null sequence.


Odd Order Derivative of Even Function Vanishes at Zero

Let $X$ be a symmetric subset of $\R$ containing $0$.  Let $n$ be a positive integer.  Let $f:X \to \R$ be an even function.  Let $f$ be at least $\paren{2 n + 1}$-times differentiable. Then: :$\map {f^{\paren {2 n + 1} } } 0 = 0$


Reciprocal of Null Sequence/Corollary

:$x_n \to \infty$ as $n \to \infty$ {{iff}} $\size {\dfrac 1 {x_n} } \to 0$ as $n \to \infty$


Index of Subsequence not Less than its Index

Let $\sequence {x_n}_{n \mathop \ge 1}$ be a sequence in a set $S$. Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$. Then: :$\forall n \in \N_{>0}: n_r \ge r$


Even Order Derivative of Odd Function Vanishes at Zero

Let $X$ be a symmetric subset of $\R$ containing $0$.  Let $n$ be a positive integer.  Let $f:X \to \R$ be an odd function.  Let $f$ be at least $\paren {2 n}$-times differentiable. Then:  :$\map {f^{\paren {2 n} } } 0 = 0$


Geometric Mean of Reciprocals is Reciprocal of Geometric Mean

Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be strictly positive real numbers. Let $G_n$ denote the geometric mean of $x_1, x_2, \ldots, x_n$. Let ${G_n}'$ denote the geometric mean of their reciprocals $\dfrac 1 {x_1}, \dfrac 1 {x_2}, \ldots, \dfrac 1 {x_n}$. Then: :${G_n}' = \dfrac 1 {G_n}$


Geometric Mean is Never Less than Harmonic Mean

Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be strictly positive real numbers. Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$. Let $H_n$ be the harmonic mean of $x_1, x_2, \ldots, x_n$. Then $G_n \ge H_n$.


Harmonic Mean of two Real Numbers is Between them

Let $a, b \in \R_{\ne 0}$ be non-zero real numbers such that $a < b$. Let $\map H {a, b}$ denote the narmonic mean of $a$ and $b$. Then: :$a < \map H {a, b} < b$


Arithmetic Mean of two Real Numbers is Between them

Let $a, b \in \R_{\ne 0}$ be non-zero real numbers such that $a < b$. Let $\map A {a, b}$ denote the narmonic mean of $a$ and $b$. Then: :$a < \map A {a, b} < b$


Raw Moment of Bernoulli Distribution

Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$. Let $n$ be a strictly positive integer.  Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:  :$\expect {X^n} = p$


Equivalence of Definitions of Local Basis

Let $T = \struct {S, \tau}$ be a topological space. Let $x$ be an element of $S$. {{TFAE|def = Local Basis}}


Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets

Let $\BB$ be a set of open neighborhoods of $x$ such that: :$\forall U \in \tau: x \in U \implies \exists H \in \BB: H \subseteq U$


Equivalence of Definitions of Local Basis/Neighborhood Basis of Open Sets Implies Local Basis for Open Sets

Let $\BB$ be a set of open neighborhoods of $x$ such that: :every neighborhood of $x$ contains a set in $\BB$.


Arcsine of Zero is Zero

:$\arcsin 0 = 0$ where $\arcsin$ is the arcsine function.


Arcsine of One is Half Pi

:$\arcsin 1 = \dfrac \pi 2$ where $\arcsin$ is the arcsine function.


Real Sequence (1 + x over n)^n is Convergent

The sequence $\sequence {s_n}$ defined as: :$s_n = \paren {1 + \dfrac x n}^n$ is convergent.


Raw Moment of Poisson Distribution

Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$. Let $n$ be a strictly positive integer.  Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:  :$\displaystyle \expect {X^n} = \sum_{k \mathop = 0}^n \lambda^k {n \brace k}$ where $\displaystyle {n \brace k}$ is a Stirling number of the second kind.


Raw Moment of Exponential Distribution

Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$ Let $n$ be a strictly positive integer.  Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:  :$\expect {X^n} = n! \beta^n$


Central Moment of Exponential Distribution

Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$ Let $n$ be a strictly positive integer.  Then the $n$th central moment $\mu_n$ of $X$ is given by:  :$\displaystyle \mu_n = n! \beta^n \sum_{k \mathop = 0}^n \frac {\paren {-1}^k} {k!}$


Product of Limits of Real Sequences (1 + x over n)^n and (1 - x over n)^n equals 1

Let $\sequence {a_n}$ be the sequence defined as: :$a_n = \paren {1 + \dfrac x n}^n$ Let $\sequence {b_n}$ be the sequence defined as: :$b_n = \paren {1 - \dfrac x n}^n$ Then the product of the limits of $\sequence {a_n}$ and $\sequence {b_n}$ equals $1$


Farey Sequence is not Convergent

Consider the Farey sequence: :$F = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$ $F$ is not convergent.


Skewness of Exponential Distribution

Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$ Then the skewness $\gamma_1$ of $X$ is equal to $2$.


Excess Kurtosis of Geometric Distribution

Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the excess kurtosis $\gamma_2$ of $X$ is given by:  :$\gamma_2 = 6 + \dfrac {p^2} {1 - p}$


Excess Kurtosis of Poisson Distribution

Let $X$ be a discrete random variable with a Poisson distribution with parameter $\lambda$. Then the excess kurtosis $\gamma_2$ of $X$ is given by:  :$\gamma_2 = \dfrac 1 \lambda$


Expectation of Chi-Squared Distribution

Let $n$ be a strictly positive integer.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the expectation of $X$ is given by:  :$\expect X = n$


Variance of Chi-Squared Distribution

Let $n$ be a strictly positive integer.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the variance of $X$ is given by:  :$\var X = 2 n$


Skewness of Chi-Squared Distribution

Let $n$ be a strictly positive integer.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the skewness $\gamma_1$ of $X$ is given by:  :$\gamma_1 = \sqrt{\dfrac 8 n}$


Raw Moment of Chi-Squared Distribution

Let $n$ and $m$ be strictly positive integers.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the $m$th raw moment $\expect {X^m}$ of $X$ is given by:  :$\displaystyle \expect {X^m} = \prod_{k \mathop = 0}^{m - 1} \paren {n + 2 k}$


Sequence of Square Roots of Natural Numbers is not Cauchy

Let $\sequence {x_n}_{n \mathop \in \N_{>0} }$ be the sequence in $\R$ defined as: :$x_n = \sqrt n$ Then, despite the fact that from Difference Between Adjacent Square Roots Converges: :$\size {\sqrt {n + 1} - \sqrt n} \to 0$ as $n \to \infty$ it is not the case that $\sequence {x_n}$ is a Cauchy sequence.


Square of Standard Gaussian Random Variable has Chi-Squared Distribution

Let $X \sim \Gaussian 0 1$ where $\Gaussian 0 1$ is the standard Gaussian distribution.  Then $X^2 \sim \chi^2_1$ where $\chi^2_1$ is the chi-square distribution with $1$ degree of freedom.


Sum of Chi-Squared Random Variables

Let $n_1, n_2, \ldots, n_k$ be strictly positive integers which sum to $N$. Let $X_i \sim \chi^2_{n_i}$ for $1 \le i \le k$, where $\chi^2_{n_i}$ is the chi-squared distribution with $n_i$ degrees of freedom.  Then:  :$\displaystyle X = \sum_{i \mathop = 1}^k X_i \sim \chi^2_N$


Moment Generating Function of Chi-Squared Distribution

Let $n$ be a strictly positive integer.  Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.  Then the moment generating function of $X$, $M_X$, is given by:  :$\displaystyle \map {M_X} t = \begin{cases} \paren {1 - 2 t}^{-n / 2} & t < \frac 1 2 \\ \text{does not exist} & t \ge \frac 1 2\end{cases}$


Moment Generating Function of Linear Combination of Independent Random Variables

Let $X_1, X_2, \ldots, X_n$ be independent random variables.  Let $k_1, k_2, \ldots, k_n$ be real numbers. Let:  :$\displaystyle X = \sum_{i \mathop = 1}^n k_i X_i$ Let $M_{X_i}$ be the moment generating function of $X_i$ for $1 \le i \le n$. Then:  :$\displaystyle \map {M_X} t = \prod_{i \mathop = 1}^n \map {M_{X_i}} {k_i t}$ for all $t$ such that $M_{X_i}$ exists for all $1 \le i \le n$.


Variance of Random Sample from Gaussian Distribution has Chi-Squared Distribution

Let $X_1, X_2, \ldots, X_n$ form a random sample of size $n$ from the Gaussian distribution $\Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$.  Let: :$\displaystyle \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$ and: :$\displaystyle s^2 = \frac 1 {n - 1} \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2$ Then:  :$\dfrac {\paren {n - 1} s^2} {\sigma^2} \sim \chi^2_{n - 1}$ where $\chi^2_{n - 1}$ is the chi-squared distribution with $n - 1$ degrees of freedom.


Expectation of F-Distribution

Let $n, m$ be strictly positive integers.  Let $X \sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\tuple {n, m}$ degrees of freedom. Then the expectation of $X$ is given by: :$\expect X = \dfrac m {m - 2}$ for $m > 2$, and does not exist otherwise.


Variance of F-Distribution

Let $n, m$ be strictly positive integers.  Let $X \sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\tuple {n, m}$ degrees of freedom. Then the variance of $X$ is given by:  :$\var X = \dfrac {2 m^2 \paren {m + n - 2} } {n \paren {m - 4} \paren {m - 2}^2}$ for $m > 4$, and does not exist otherwise.


Differential Equations for Shortest Path on 3d Sphere/Cartesian Coordinates

Let $M$ be a $3$-dimensional Euclidean space. Let $S$ be a sphere embedded in $M$. Let $\gamma$ be a curve on $S$. Let the chosen coordinate system be Cartesian. Let $\gamma$ begin at $\paren {x_0, y_0, z_0}$ and terminate at $\paren {x_1, y_1, z_1}$. Let $\map y x$, $\map z x$ be real functions. Let $\gamma$ connecting both endpoints be of minimum length. Then $\gamma$ satisfies the following equations of motion: :$2 y \map \lambda x - \dfrac \d {\d x} \dfrac {y'} {\sqrt {1 + y'^2 + z'^2} } = 0$ :$2 z \map \lambda x - \dfrac \d {\d x} \dfrac {z'} {\sqrt {1 + y'^2 + z'^2} } = 0$


Skewness of F-Distribution

Let $n, m$ be strictly positive integers.  Let $X \sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\tuple {n, m}$ degrees of freedom. Then the skewness $\gamma_1$ of $X$ is given by:  :$\gamma_1 = \dfrac {2 \paren {m + 2 n - 2} } {m - 6} \sqrt {\dfrac {2 \paren {m - 4} } {n \paren {m + n - 2} } }$ for $m > 6$, and does not exist otherwise.


Expectation of Chi Distribution

Let $n$ be a strictly positive integer.  Let $X \sim \chi_n$ where $\chi_n$ is the chi distribution with $n$ degrees of freedom.  Then the expectation of $X$ is given by:  :$\expect X = \sqrt 2 \dfrac {\map \Gamma {\paren {n + 1} / 2} } {\map \Gamma {n / 2} }$ where $\Gamma$ is the gamma function.


Mean of Random Sample from Chi-Squared Distribution has Gamma Distribution

Let $n$ be a strictly positive integer.  Let $X_1, X_2, \ldots, X_k$ form a random sample of size $k$ from the chi-squared distribution with $n$ degrees of freedom.  Then:  :$\displaystyle \overline X = \frac 1 k \sum_{i \mathop = 1}^k X_i \sim \map \Gamma {\frac {n k} 2, \frac k 2}$ where $\map \Gamma {\dfrac {n k} 2, \dfrac k 2}$ is the gamma distribution with parameters $\dfrac {n k} 2$ and $\dfrac k 2$.


Equivalence of Definitions of Separated Sets

Let $T = \struct {S, \tau}$ be a topological space. Let $A, B \subseteq S$. {{TFAE|def = Separated Sets}}


Equivalence of Definitions of Separated Sets/Definition 1 implies Definition 2

Let $A, B \subseteq S$ satisfy: :$A^- \cap B = A \cap B^- = \O$ where $A^-$ denotes the closure of $A$ in $T$, and $\O$ denotes the empty set.


Equivalence of Definitions of Separated Sets/Definition 2 implies Definition 1

Let $A, B \subseteq S$. Let $U, V \in \tau$ satisfy: :$A \subset U$ and $U \cap B = \O$ :$B \subset V$ and $V \cap A = \O$


Excess Kurtosis of Student's t-Distribution

Let $k$ be a strictly positive integer.  Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom. Then the excess kurtosis $\gamma_2$ of $X$ is given by:  :$\gamma_2 = \dfrac 6 {k - 4}$ for $k > 4$, and does not exist otherwise.


Connected Set in Subspace

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq B \subseteq S$. Let $T_B = \struct {B, \tau_B}$ be the topological space where $\tau_B$ is the subspace topology on $B$. Then :$A$ is connected in $T_B$ {{iff}} $A$ is connected in $T$.


Expectation of Erlang Distribution

Let $k$ be a strictly positive integer.  Let $\lambda$ be a strictly positive real number.  Let $X$ be a continuous random variable with an Erlang distribution with parameters $k$ and $\lambda$. Then the expectation of $X$ is given by: :$\expect X = \dfrac k \lambda$


Variance of Erlang Distribution

Let $k$ be a strictly positive integer.  Let $\lambda$ be a strictly positive real number.  Let $X$ be a continuous random variable with an Erlang distribution with parameters $k$ and $\lambda$. Then the variance of $X$ is given by:  :$\var X = \dfrac k {\lambda^2}$


Covariance as Expectation of Product minus Product of Expectations

Let $X$ and $Y$ be random variables.  Let the expectations of $X$ and $Y$ exist and be finite.  Then the covariance of $X$ and $Y$ is given by:  :$\cov {X, Y} = \expect {X Y} - \expect X \expect Y$


Covariance of Independent Random Variables is Zero

Let $X$ and $Y$ be independent random variables. Let the expectations of $X$ and $Y$ exist and be finite.  Then the covariance of $X$ and $Y$ is $0$.


Variance of Linear Combination of Random Variables

Let $X$ and $Y$ be random variables.  Let the variances of $X$ and $Y$ be finite.  Let $a$ and $b$ be real numbers.  Then the variance of $a X + b Y$ is given by:  :$\var {a X + b Y} = a^2 \, \var X + b^2 \, \var Y + 2 a b \, \cov {X, Y}$ where $\cov {X, Y}$ is the covariance of $X$ and $Y$.


Square of Expectation of Product is Less Than or Equal to Product of Expectation of Squares

Let $X$ and $Y$ be random variables.  Let the expectation of $X Y$, $\expect {X Y}$, exist and be finite.  Then:  :$\paren {\expect {X Y} }^2 \le \expect {X^2} \expect {Y^2}$


Square of Covariance is Less Than or Equal to Product of Variances

Let $X$ and $Y$ be random variables.  Let the variances of $X$ and $Y$ exist and be finite.  Then:  :$\paren {\cov {X, Y} }^2 \le \var X \, \var Y$ where $\cov {X, Y}$ denotes the covariance of $X$ and $Y$.


Absolute Value of Pearson Correlation Coefficient is Less Than or Equal to 1

Let $X$ and $Y$ be random variables.  Let the variances of $X$ and $Y$ exist and be finite.  Then:  :$\size {\map \rho {X, Y} } \le 1$ where $\map \rho {X, Y}$ denotes the Pearson correlation coefficient of $X$ and $Y$.


Equivalence of Definitions of Locally Connected Space/Definition 1 implies Definition 2

Let each point of $T$ have a local basis consisting entirely of connected sets in $T$.


Equivalence of Definitions of Locally Connected Space/Definition 2 implies Definition 1

Let $T$ be weakly locally connected at each point of $T$. That is, each point of $T$ has a neighborhood basis consisting of connected sets of $T$.


Equivalence of Definitions of Locally Connected Space/Definition 1 implies Definition 3

Let each point $x$ of $T$ have a local basis $\mathcal D_x$ consisting entirely of connected sets in $T$.


Equivalence of Definitions of Locally Connected Space/Definition 3 implies Definition 1

Let $T$ have a basis $\mathcal B$ consisting of connected sets in $T$.


Equivalence of Definitions of Locally Connected Space/Definition 4 implies Definition 3

Let the components of the open sets of $T$ are also open in $T$.


Connected Component is Closed

Let $T = \struct{S, \tau}$ be a topological space. Then every connected component of $T$ is closed.


Constant Mapping is Non-Commutative

Let $S$ be a set whose cardinality is greater than one. Let $f: S \to S$ and $g: S \to S$ be constant mappings on $S$. Then: :$f \circ g \ne g \circ f$ where $\circ$ denotes composition of mappings.


Quotient Mapping is Injection iff Equality

Let $\mathcal R$ be an equivalence relation on $S$. Then the quotient mapping $q_{\mathcal R}: S \to S / \mathcal R$ is an injection {{iff}} $\mathcal R$ is the equality relation.


Closure of Subset in Subspace

Let $T = \struct{S, \tau}$ be a topological space. Let $H$ be a subset of $S$. Let $T_H = \struct {H, \tau_H}$ be the topological subspace on $H$. Let $A$ be a subset of $H$. Then: :$\map {\cl_H} A = H \cap \map \cl A$ where  :$\map {\cl_H} A$ denotes the closure of $A$ in $T_H$ :$\map \cl A$ denotes the closure of $A$ in $T$


Equivalence of Definitions of Locally Path-Connected Space/Definition 1 implies Definition 2

Let each point of $T$ have a local basis consisting entirely of path-connected sets in $T$.


Equivalence of Definitions of Locally Path-Connected Space/Definition 2 implies Definition 1

Let each point of $T$ have a neighborhood basis consisting of path-connected sets in $T$.


Equivalence of Definitions of Locally Path-Connected Space/Definition 1 implies Definition 3

Let each point $x$ of $T$ have a local basis $\mathcal D_x$ consisting entirely of path-connected sets in $T$.


Equivalence of Definitions of Locally Path-Connected Space/Definition 3 implies Definition 1

Let $T$ have a basis $\mathcal B$ consisting of path-connected sets in $T$.


Equivalence of Definitions of Locally Path-Connected Space/Definition 3 implies Definition 4

Let $T$ have a basis consisting of path-connected sets in $T$.


Equivalence of Definitions of Locally Path-Connected Space/Definition 4 implies Definition 3

Let the path components of open sets of $T$ be also open in $T$.


Path-Connected Set in Subspace

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq B \subseteq S$. Let $T_B = \struct {B, \tau_B}$ be the topological space where $\tau_B$ is the subspace topology on $B$. Then :$A$ is path-connected in $T_B$ {{iff}} $A$ is path-connected in $T$.


Precisely One Function in terms of And, Or and Not

Let $\map P {A, B, C}$ denote the precisely one function on the statements $A$, $B$ and $C$. Then: :$\map P {A, B, C} \dashv \vdash \paren {A \land \neg B \land \neg C} \lor \paren {\neg A \land B \land \neg C} \lor \paren {\neg A \land \neg B \land C}$ where: :$\land$ denotes conjunction :$\lor$ denotes disjunction :$\neg$ denotes negation


Lévy's Continuity Theorem

Let $\sequence {X_n}_{n \mathop \ge 1}$ be a sequence of discrete random variables with characteristic functions $\map {\phi_n} t := E \sqbrk {e^{i t X_n} }$. Let the sequence $\sequence {\phi_n}$ converge to some real function $\phi$: :$\forall t \in \R: \map {\phi_n} t \to \map \phi t$. Then the following statements are equivalent: :$(1): \quad$ The $\sequence {X_n}$ converges in distribution to some random variable $X$: ::$X_n \stackrel {dist} {\to} X$ with characteristic function $\map {\phi_X} t := \map \phi t$ :$(2): \quad$ The sequence $\sequence {X_n}$ is tight, that is: ::$\displaystyle \lim_{M \mathop \to \infty} \sup_{n \mathop \ge 1} P \sqbrk {\size {X_n} \ge M} = 0$


Composition of Three Mappings which form Identity Mapping

Let $A$, $B$ and $C$ be non-empty sets. Let $f: A \to B$, $g: B \to C$ and $h: C \to A$ be mappings. Let the following hold: {{begin-eqn}} {{eqn | l = h \circ g \circ f       | r = I_A }} {{eqn | l = f \circ h \circ g       | r = I_B }} {{eqn | l = g \circ f \circ h       | r = I_C }} {{end-eqn}} where: :$g \circ f$ (and so on) denote composition of mappings :$I_A$ (and so on) denote the identity mappings. Then each of $f$, $g$ and $h$ are bijections, and: {{begin-eqn}} {{eqn | l = f^{-1}       | r = h \circ g }} {{eqn | l = g^{-1}       | r = f \circ h }} {{eqn | l = h^{-1}       | r = g \circ f }} {{end-eqn}} where $f^{-1}$ (and so on) denote the inverse mappings.


Composition of Product Mappings on Natural Numbers

Let $a \in \N$ be a natural number. Let $\mu_a: \N \to \N$ be the mapping defined as: :$\forall x \in \N: \map {\mu_a} x = x a$ Then: :$\mu_{a b} = \mu_b \circ \mu_a$


Moment Generating Function of Linear Transformation of Random Variable

Let $X$ be a random variable.  Let $\alpha$ and $\beta$ be real numbers. Let $Z = \alpha X + \beta$. Let $M_X$ be the moment generating function of $X$.  Then the moment generating function of $Z$, $M_Z$, is given by:  :$\map {M_Z} t = e^{\beta t} \map {M_X} {\alpha t}$


Standard Gaussian Random Variable as Transformation of Gaussian Random Variable

Let $\mu$ be a real number.  Let $\sigma$ be a positive real number.  Let $X \sim \Gaussian \mu {\sigma^2}$ where $\Gaussian \mu {\sigma^2}$ is the Gaussian distribution with parameters $\mu$ and $\sigma^2$.  Then:  :$\dfrac {X - \mu} \sigma \sim \Gaussian 0 1$ where $\Gaussian 0 1$ is the standard Gaussian distribution.


Composition of Addition Mappings on Natural Numbers

Let $a \in \N$ be a natural number. Let $\alpha_a: \N \to \N$ be the mapping defined as: :$\forall x \in \N: \map {\alpha_a} x = x + a$ Then: :$\alpha_{a + b} = \alpha_b \circ \alpha_a$


Reciprocal of Random Variable with F-Distribution has F-Distribution

Let $n, m$ be strictly positive integers.  Let $X \sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\tuple {n, m}$ degrees of freedom. Then:  :$\dfrac 1 X \sim F_{m, n}$


Square of Random Variable with t-Distribution has F-Distribution

Let $k$ be a strictly positive integer.  Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom. Then:  :$X^2 \sim F_{1, k}$ where $F_{1, k}$ is the $F$-distribution with $\tuple {1, k}$ degrees of freedom.


Union of Connected Sets with Common Point is Connected

Let $T = \struct {S, \tau}$ be a topological space. Let $\family {B_\alpha}_{\alpha \mathop \in A}$ be a family of connected sets of $T$. Let $\exists x \in \displaystyle \bigcap \family {B_\alpha}_{\alpha \mathop \in A}$. Then :$\displaystyle \bigcup \family {B_\alpha}_{\alpha \mathop \in A}$ is a connected set of $T$.


Connected Subset of Union of Disjoint Open Sets

Let $T = \struct{S, \tau}$ be a topological space. Let $A$ be a connected set of $T$. Let $U, V$ be disjoint open sets. Let $A \subseteq U \cup V$. Then :either $A \subseteq U$ or $A \subseteq V$.


Rearrangement of Variables in Total Differential Equation

In a total differential equation, any one of the variables can be regarded as an independent variable, while the remainder can be treated as dependent variables.


Archimedean Principle/Variant

Let $x$ and $y$ be a natural numbers. Then there exists a natural number $n$ such that: :$n x \ge y$


Set Intersection Preserves Subsets/Families of Sets/Intersection is Empty Implies Intersection of Subsets is Empty

Let $I$ be an indexing set. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$. Let: :$\forall \beta \in I: A_\beta \subseteq B_\beta$ Then: :$\displaystyle \bigcap_{\alpha \mathop \in I} B_\alpha = \O \implies \bigcap_{\alpha \mathop \in I} A_\alpha = \O$


Union of Path-Connected Sets with Common Point is Path-Connected

Let $T = \struct {S, \tau}$ be a topological space. Let $\family {B_\alpha}_{\alpha \mathop \in A}$ be a family of path-connected sets of $T$. Let $\exists x \in \displaystyle \bigcap \family {B_\alpha}_{\alpha \mathop \in A}$. Then :$\displaystyle \bigcup \family {B_\alpha}_{\alpha \mathop \in A}$ is a path-connected set of $T$.


Upper Bound for Lucas Number

Let $L_n$ denote the $n$th Lucas number. Then: :$L_n < \paren {\dfrac 7 4}^n$


Condition for Increasing Binomial Coefficients

Let $n \in \Z_{> 0}$ be a (strictly) positive integer. Let $\dbinom n k$ denote a binomial coefficient for $k \in \N$. Then: :$\dbinom n k < \dbinom n {k + 1} \iff 0 \le k < \dfrac {n - 1} 2$


Condition for Equality of Adjacent Binomial Coefficients

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\dbinom n k$ denote a binomial coefficient for $k \in \Z$. Then: :$\dbinom n k = \dbinom n {k + 1}$ {{iff}}: :$n$ is an odd integer :$k = \dfrac {n - 1} 2$


Equivalence of Definitions of Component/Equivalence Class equals Union of Connected Sets

Let $T = \struct {S, \tau}$ be a topological space. Let $x \in T$. Let $\CC_x = \set {A \subseteq S: x \in A \land A \text{ is connected in } T}$ Let $C = \bigcup \CC_x$ Let $\sim$ be the equivalence relation defined by: :$y \sim z$ {{iff}} $y$ and $z$ are connected in $T$. Let $C’$ be the equivalence class of $\sim$ containing $x$. Then $C = C'$.


Equivalence of Definitions of Component/Union of Connected Sets is Maximal Connected Set

Let $T = \struct {S, \tau}$ be a topological space. Let $x \in T$. Let $\mathcal C_x = \set {A \subseteq S : x \in A \land A \text{ is connected in } T}$ Let $C = \bigcup \mathcal C_x$ Then $C$ is a maximal connected set of $T$.


Equivalence of Definitions of Component/Maximal Connected Set is Union of Connected Sets

Let $\tilde C$ be a maximal connected set of $T$ that contains $x$.


Binomial Coefficient n Choose j in terms of n-2 Choose r

Let $n \in \Z$ such that $n \ge 4$. Let $\dbinom n k$ denote a binomial coefficient for $k \in \Z$. Then: :$\dbinom n k = \dbinom {n - 2} {k - 2} + 2 \dbinom {n - 2} {k - 1} + \dbinom {n - 2} k$ for $2 \le k \le n - 2$.


Sum of Sequence of Binomial Coefficients by Powers of 2

{{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^n 2^j \binom n j       | r = \dbinom n 0 + 2 \dbinom n 1 + 2^2 \dbinom n 2 + \dotsb + 2^n \dbinom n n       | c =  }} {{eqn | r = 3^n       | c =  }} {{end-eqn}}


Sum of Sequence of n Choose 2

Let $n \in \Z$ be an integer such that $n > 2$. {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 2}^n \dbinom j 2       | r = \dbinom 2 2 + \dbinom 3 2 + \dbinom 4 2 + \dotsb + \dbinom n 2       | c =  }} {{eqn | r = \dbinom {n + 1} 3       | c =  }} {{end-eqn}} where $\dbinom n j$ denotes a binomial coefficient.


If n is Triangular then so is 25n + 3

Let $n$ be a triangular number. Then $25 n + 3$ is also triangular.


If n is Triangular then so is 49n + 6

Let $n$ be a triangular number. Then $49 n + 6$ is also triangular.


Sum of Sequence of Triangular Numbers

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $T_n$ denote the $n$th triangular number. Then: {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 1}^n T_j       | r = T_1 + T_2 + T_3 + \dotsb + T_n       | c =  }} {{eqn | r = \dfrac {n \paren {n + 1} \paren {n + 2} } 6       | c =  }} {{end-eqn}}


Square of Odd Multiple of 3 is Difference between Triangular Numbers

Let $n \in \Z_{\ge 0}$ be a positive integer. Let $T_n$ denote the $n$th triangular number. Let $m = 2 n + 1$ be an odd integer Then: :$\paren {3 m}^2 = T_{9 n + 4} - T_{3 n + 1}$


Square Sum of Three Consecutive Triangular Numbers

Let $T_n$ denote the $n$th triangular number for $n \in \Z_{>0}$ a (strictly) positive integer. Let $T_n + T_{n + 1} + T_{n + 2}$ be a square number. Then at least one value of $n$ fulfils this condition: :$n = 5$


Sufficient Condition for Square of Product to be Triangular

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $2 n^2 \pm 1 = m^2$ be a square number. Then $\paren {m n}^2$ is a triangular number.


Topology on Singleton is Indiscrete Topology

Let $S$ be a singleton. The only possible topology on $S$ is the indiscrete topology.


Existence of Divisor with Remainder between 2b and 3b

For every pair of integers $a, b$ where $b > 0$, there exist unique integers $q$ and $r$ where $2 b \le r < 3 b$ such that: :$a = q b + r$


Integer of form 6k + 5 is of form 3k + 2 but not Conversely

Let $n \in \Z$ be an integer of the form: :$n = 6 k + 5$ where $k \in \Z$. Then $n$ can also be expressed in the form: :$n = 3 k + 2$ for some other $k \in \Z$. However it is not necessarily the case that if $n$ can be expressed in the form: :$n = 3 k + 2$ then it can also be expressed in the form: :$n = 6 k + 5$


Odd Integer Modulo 4

Let $n$ be an odd integer. Then $n$ can be expressed either as: :$n = 4 k + 1$ or as: :$n = 4 k + 3$


N (n + 1) (2n + 1) over 6 is Integer

Let $n \in \Z$ be an integer. Then $\dfrac {n \paren {n + 1} \paren {2 n + 1} } 6$ is also an integer.


Number which is Square and Cube Modulo 7

Let $n \in \Z$ be an integer. Let $n$ be both a square and a cube at the same time. Then either: :$n \equiv 0 \pmod 7$ or: :$n \equiv 1 \pmod 7$


Weak Law of Large Numbers

Let $P$ be a population. Let $P$ have mean $\mu$ and finite variance.  Let $\sequence {X_n}_{n \mathop \ge 1}$ be a sequence of random variables forming a random sample from $P$. Let:  :$\ds {\overline X}_n = \frac 1 n \sum_{i \mathop = 1}^n X_i$ Then: :${\overline X}_n \xrightarrow p \mu$ where $\xrightarrow p$ denotes convergence in probability.


Variance of Linear Combination of Random Variables/Corollary

Let $X$ and $Y$ be independent random variables.


Singleton is Connected in Topological Space

Let $T = \struct {S, \tau}$ be a topological space. Let $x \in S$. Then the singleton $\set{x}$ is connected.


Equivalence of Definitions of Component/Lemma 1

:$C$ is connected in $T$ and $C \in \mathcal C_x$.


Variance of Sample Mean

Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$. Let:  :$\displaystyle \overline X = \frac 1 n \sum_{i \mathop = 1}^n X_i$ Then:  :$\var {\overline X} = \dfrac {\sigma^2} n$


Common Divisor Divides Integer Combination/General Result

Let $c$ be a common divisor of a set of integers $A := \set {a_1, a_2, \dotsc, a_n}$. That is: :$\forall x \in A: c \divides x$ Then $c$ divides any integer combination of elements of $A$: :$\forall x_1, x_2, \dotsc, x_n \in \Z: c \divides \paren {a_1 x_2 + a_2 x_2 + \dotsb + a_n x_n}$


One is Common Divisor of Integers

Let $a, b \in \Z$ be integers. Then $1$ is a common divisor of $a$ and $b$.


Equivalence of Definitions of Path Component

{{TFAE|def = Path Component|view = Path Component|context = Topology (Mathematical Branch)|contextview = Topology}} Let $T = \struct {S, \tau}$ be a topological space. Let $x \in T$.


GCD of Integer and its Negative

Let $a \in \Z$ be an integer. Then: :$\gcd \set {a, -a} = \size a$ where: :$\gcd$ denotes greatest common divisor :$\size a$ denotes the absolute value of $a$.


Equivalence of Definitions of Path Component/Lemma 1

:$C$ is path-connected in $T$ and  $C \in \CC_x$.


Square Divides Product of Multiples

Let $a, b, c, \in \Z$ be integers. Let: :$a \divides b, a \divides c$ where $\divides$ denotes divisibility. Then: :$a^2 \divides b c$


Abel's Limit Theorem

Let $\sequence {a_i}$ be a convergent series. The limit which is assigned by the Abel summation method exists and equals the sum of the series.


Equivalence of Definitions of Path Component/Equivalence Class equals Union of Path-Connected Sets

Let $T = \struct {S, \tau}$ be a topological space. Let $x \in T$. Let $\CC_x = \left\{ {A \subseteq S : x \in A \land A }\right.$ is path-connected in $\left. {T}\right\}$. Let $C = \bigcup \CC_x$ Let $\sim$ be the equivalence relation defined by: :$y \sim z$ {{iff}} $y$ and $z$ are path-connected in $T$. Let $C'$ be the equivalence class of $\sim$ containing $x$. Then $C = C'$.


Image of Path is Path-Connected Set

Let $T = \struct {S, \tau}$ be a topological space. Let $I \subset \R$ be the closed real interval $\closedint a b$. Let $\gamma: I \to S$ be a path. Then: :$\map \gamma I$ is a path-connected set of $T$.


Closure of Hadamard Product

Let $\struct {S, \cdot}$ be an algebraic structure. Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$. For $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$, let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$. The operation $\circ$ is closed on $\map {\MM_S} {m, n}$ {{iff}} $\cdot$ is closed on $\struct {S, \cdot}$.


Associativity of Hadamard Product

Let $\struct {S, \cdot}$ be an algebraic structure. Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$. For $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$, let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$. The operation $\circ$ is associative on $\map {\MM_S} {m, n}$ {{iff}} $\cdot$ is associative on $\struct {S, \cdot}$.


Commutativity of Hadamard Product

Let $\struct {S, \cdot}$ be an algebraic structure. Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$. For $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$, let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$. The operation $\circ$ is commutative on $\map {\MM_S} {m, n}$ {{iff}} $\cdot$ is commutative on $\struct {S, \cdot}$.


Matrix Entrywise Addition over Ring is Closed

Let $\struct {R, +, \circ}$ be a ring. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$. For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. The operation $+$ is closed on $\map {\MM_R} {m, n}$.


Points are Path-Connected iff Contained in Path-Connected Set

Let $T = \struct {S, \tau}$ be a topological space. Let $x, y \in S$ Then: :$x, y$ are path-connected points in $T$ {{iff}} there exists a path-connected set of $T$ containing $x$ and $y$.


Compound Angle Formulas

=== Sine of Sum === {{:Sine of Sum}} === Cosine of Sum === {{:Cosine of Sum}} === Tangent of Sum === {{:Tangent of Sum}}


Equivalence of Definitions of Path Component/Union of Path-Connected Sets is Maximal Path-Connected Set

Let $T = \struct {S, \tau}$ be a topological space. Let $x \in T$. Let $\CC_x = \left\{ {A \subseteq S : x \in A \land A } \right.$ is path-connected in $\left. {T} \right\}$ Let $C = \bigcup \CC_x$ Then $C$ is a maximal path-connected set of $T$.


Solutions to Diophantine Equation 16x^2+32x+20 = y^2+y

The indeterminate Diophantine equation: :$16x^2 + 32x + 20 = y^2 + y$ has exactly $4$ solutions: :$\tuple {0, 4}, \tuple {-2, 4}, \tuple {0, -5}, \tuple {-2, -5}$


Equivalence of Definitions of Path Component/Maximal Path-Connected Set is Union of Path-Connected Sets

Let $\tilde C$ be a maximal path-connected set of $T$ that contains $x$.


Power Set is Closed under Set Complement

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A \in \powerset S: \relcomp S A \in \powerset S$


Normal Subgroup of Symmetric Group Order Greater than 4 is Alternating Group

Let $n \in \N$ be a natural number such that $n > 4$. Let $S_n$ denote the symmetric group on $n$ letters. Let $A_n$ denote the alternating group on $n$ letters. $A_n$ is the only proper non-trivial normal subgroup of $S_n$.


Path Components are Open iff Union of Open Path-Connected Sets/Path Components are Open implies Space is Union of Open Path-Connected Sets

Let the path components of $T$ be open sets.


Path Components are Open iff Union of Open Path-Connected Sets/Space is Union of Open Path-Connected Sets implies Path Components are Open

Let $T = \left({S, \tau}\right)$ be a topological space. Let $S$ be the union of open path-connected sets of $T$. Then: :The path components of $T$ are open sets.


Path Components are Open iff Union of Open Path-Connected Sets/Lemma 1

::$U \cap C \ne \O$ {{iff}} $U \ne \O$ and $U \subseteq C$


Integral Points of Elliptic Curve y^2 = x^3+3x

The elliptic curve: :$y^2 = x^3 + 3x$ has exactly $7$ lattice points: :$\tuple {0, 0}, \tuple {1, \pm 2}, \tuple {3, \pm 6}, \tuple {12, \pm 42}$


Components are Open iff Union of Open Connected Sets/Components are Open implies Space is Union of Open Connected Sets

Let $T = \struct {S, \tau}$ be a topological space. Let the components of $T$ be open sets. Then: :$S$ is a union of open connected sets of $T$.


Components are Open iff Union of Open Connected Sets/Space is Union of Open Connected Sets implies Components are Open

Let $T = \struct {S, \tau}$ be a topological space. Let $S$ be a union of open connected sets of $T$. Then: :The components of $T$ are open sets.


Components are Open iff Union of Open Connected Sets/Lemma 1

::$U \cap C \neq \O$ {{iff}} $U \neq \O$ and $U \subseteq C$


Integral to Infinity of Square of Sine p x over x Squared

:$\displaystyle \int_0^\infty \paren {\frac {\sin p x} x}^2 \rd x = \frac {\pi \size p} 2$


Integral to Infinity of One minus Cosine p x over x Squared

:$\displaystyle \int_0^\infty \frac {1 - \cos p x} {x^2} \rd x = \frac {\pi \size p} 2$


1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways

The number $1$ can be expressed as the sum of $4$ distinct unit fractions in $6$ different ways: {{begin-eqn}} {{eqn | l = 1       | r = \frac 1 2 + \frac 1 3 + \frac 1 7 + \frac 1 {42} }} {{eqn | r = \frac 1 2 + \frac 1 3 + \frac 1 8 + \frac 1 {24} }} {{eqn | r = \frac 1 2 + \frac 1 3 + \frac 1 9 + \frac 1 {18} }} {{eqn | r = \frac 1 2 + \frac 1 3 + \frac 1 {10} + \frac 1 {15} }} {{eqn | r = \frac 1 2 + \frac 1 4 + \frac 1 5 + \frac 1 {20} }} {{eqn | r = \frac 1 2 + \frac 1 4 + \frac 1 6 + \frac 1 {12} }} {{end-eqn}}


Empty Set is Subset of Power Set

The empty set is a subset of all power sets: :$\forall S: \O \subseteq \powerset S$


Equivalence of Definitions of Weakly Locally Connected at Point/Definition 1 implies Definition 2

Let $x$ have a neighborhood basis consisting of connected sets.


Equivalence of Definitions of Weakly Locally Connected at Point/Definition 2 implies Definition 1

Let every open neighborhood $U$ of $x$ contain an open neighborhood $V$ such that every two points of $V$ lie in some connected subset of $U$.


Cycle of Subsets implies Set Equality

Let $A_1, A_2, \dotsc, A_n$ be sets. Let: :$\forall k \in \set {2, 3, \dotsc, n}: A_{k - 1} \subseteq A_k$ and: :$A_n \subseteq A_1$ Then: :$\forall j, k \in \set {1, 2, \dotsc, n}: A_j = A_k$


Correspondence between Set and Ordinate of Cartesian Product is Mapping

Let $S$ and $T$ be sets such that $T \ne \O$. Let $S \times T$ denote their cartesian product. Let $t \in T$ be given. Let $j_t \subseteq S \times \paren {S \times T}$ be the relation on $S \times {S \times T}$ defined as: :$\forall s \in \S: \map {j_t} s = \tuple {s, t}$ Then $j_t$ is a mapping.


Mapping from Set to Ordinate of Cartesian Product is Injection

Let $S$ and $T$ be sets such that $T \ne \O$. Let $S \times T$ denote their cartesian product. Let $t \in T$ be given. Let $j_t \subseteq S \times \paren {S \times T}$ be the mapping from $S$ to $S \times T$ defined as: :$\forall s \in \S: \map {j_t} s = \tuple {s, t}$ Then $j_t$ is an injection.


Primitive of Sine Integral Function

:$\displaystyle \int \map \Si x \rd x = x \map \Si x + \cos x + C$


Preimage of Subset of Cartesian Product under Injection from Factor

Let $S$ and $T$ be sets such that $T \ne \O$. Let $S \times T$ denote their cartesian product. Let $t \in T$ be given. Let $j_t \subseteq S \times \paren {S \times T}$ be the injection from $S$ to $S \times T$ defined as: :$\forall s \in \S: \map {j_t} s = \tuple {s, t}$ Let $W \subseteq S \times T$. Let $V = {j_t}^{-1} \sqbrk W$ denote the preimage of $W$ under $j_t$. Then: :$V = \set {s: \tuple {s, t} \in W}$


Derivative of Error Function

:$\displaystyle \frac \d {\d x} \paren {\map \erf x} = \frac 2 {\sqrt \pi} e^{-x^2}$


Limit to Infinity of Error Function

:$\displaystyle \lim_{x \mathop \to \infty} \map \erf x = 1$


Dirichlet Beta Function in terms of Hurwitz Zeta Function

:$\displaystyle \map \beta s = \frac 1 {4^s} \paren {\map \zeta {s, \frac 1 4} - \map \zeta {s, \frac 3 4} }$


Inverse of Right-Total Relation is Left-Total

:$\RR$ is right-total {{iff}} $\RR^{-1}$ is left-total.


Inverse of Left-Total Relation is Right-Total

:$\mathcal R$ is left-total {{iff}} $\mathcal R^{-1}$ is right-total.


Inverse of Mapping is Right-Total Relation

Let $f$ be a mapping. Then its inverse $f^{-1}$ is a right-total relation.


Inverse of One-to-One Relation is One-to-One

The inverse of a one-to-one relation is a one-to-one relation.


Inverse of Injection is One-to-One Relation

Let $f$ be an injective mapping. Then its inverse $f^{-1}$ is a one-to-one relation.


Inverse of Surjection is Relation both Left-Total and Right-Total

Let $f$ be an surjective mapping. Then its inverse $f^{-1}$ is a relation which is both left-total and right-total.


Inverse is Mapping implies Mapping is Injection

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping. Then $f$ is an injection.


Inverse is Mapping implies Mapping is Surjection

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping. Then $f$ is a surjection.


Derivative Function is not Invertible

Let $\Bbb I = \closedint a b$ be a closed interval on the set of real numbers $\R$ such that $a < b$. Let $A$ denote the set of all continuous real functions $f: \Bbb I \to \R$. Let $B \subseteq A$ denote the set of all functions differentiable on $\Bbb I$ whose derivative is continuous on $\Bbb I$. Let $d: B \to A$ denote the mapping defined as: :$\forall \map f x \in B: \map d f = \map {D_x} f$ where $D_x$ denotes the derivative of $f$ {{WRT|Differentiation}} $x$. Then $d$ is not an invertible mapping.


Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 2 implies Definition 4

Let $\sequence {x_k}$ satisfy: :$\forall \epsilon > 0: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$ where $\map {B_\epsilon} l$ is the open $\epsilon$-ball of $l$.


Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 4 implies Definition 2

Let $\sequence {x_k}$ satisfy: :for every $\epsilon \in \R{>0}$, the open $\epsilon$-ball about $l$ contains all but finitely many of the $p_n$.


Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x

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


Map from Set of Continuous Functions on Interval to Set of their Integrals

Let $\Bbb I = \closedint a b$ be a closed interval on the set of real numbers $\R$ such that $a < b$. Let $A$ denote the set of all continuous real functions $f: \Bbb I \to \R$. Let $B \subseteq A$ denote the set of all functions differentiable on $\Bbb I$ whose derivative is continuous on $\Bbb I$. Let $C \subseteq B$ denote the subset of $B$ which consists of all elements of $B$ such that $\map f a = 0$. For each $f \in A$, let $h$ denote the mapping defined as: :$\map {\paren {\map h f} } x = \displaystyle \int_a^x \map f t \rd t$ Then: :$h: A \to C$


Definite Integral from 0 to Half Pi of Reciprocal of One plus Power of Tan x

:$\displaystyle \int_0^{\pi/2} \frac {\d x} {1 + \tan^m x} = \frac \pi 4$


Möbius Transformation is Bijection

Let $a, b, c, d \in \C$ be complex numbers. Let $f: \overline \C \to \overline \C$ be the Möbius transformation: :$\map f z = \begin {cases} \dfrac {a z + b} {c z + d} & : z \ne -\dfrac d c \\ \infty & : z = -\dfrac d c \\ \dfrac a c & : z = \infty \\ \infty & : z = \infty \text { and } c = 0 \end {cases}$ Then: :$f: \overline \C \to \overline \C$ is a bijection {{iff}}: :$a c - b d \ne 0$


Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x

:$\displaystyle \int_0^1 x^m \paren {\ln x}^n \rd x = \frac {\paren {-1}^n \map \Gamma {n + 1} } {\paren {m + 1}^{n + 1} }$


Definite Integral from 0 to 1 of x to the x

{{begin-eqn}} {{eqn	| l = \int_0^1 x^x \rd x 	| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } {n^n} }} {{eqn	| r = -\sum_{n \mathop = 1}^\infty \paren {-n}^{-n} }} {{eqn	| r = 0.78343 \ 05107 \ 12\ldots }} {{end-eqn}}


Definite Integral from 0 to 1 of x to the minus x

{{begin-eqn}} {{eqn	| l = \int_0^1 x^{-x} \rd x 	| r = \sum_{n \mathop = 1}^\infty n^{-n} }} {{eqn	| r = 1.29128 \ 5997 \ldots }} {{end-eqn}}


Definite Integral to Infinity of Exponential of -a x by Sine of b x

:$\displaystyle \int_0^\infty e^{-a x} \sin b x \rd x = \frac b {a^2 + b^2}$


Sum of Exponential of i k x

:$\displaystyle \sum_{k \mathop = 0}^n \map \exp {i k x} = \paren {i \sin \frac {n x} 2 + \cos \frac {n x} 2} \frac {\map \sin {\frac {\paren {n + 1} x} 2} } {\sin \frac x 2}$


Extension of Extension of Mapping is Extension

Let $A, B, C, S$ be sets such that $A \subseteq B \subseteq C$. Let $f: A \to S$, $g: B \to S$ and $h: C \to S$ be mappings such that: :$g$ is an extension of $f$ to $B$ :$h$ is an extension of $g$ to $C$. Then $h$ is an extension of $f$ to $C$.


Indexed Cartesian Space is Set of all Mappings

Let $I$ be an indexing set. Let $\displaystyle \prod_{i \mathop \in I} S$ denote the cartesian space of $S$ indexed by $I$. Then $\displaystyle \prod_{i \mathop \in I} S$ is the set of all mappings from $I$ to $S$, and hence the notation: :$S^I := \displaystyle \prod_{i \mathop \in I} S$


Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 2

Let $M = \struct {A, d}$ be a metric space or a pseudometric space. Let $\sequence {x_k}$ be a sequence in $A$. {{TFAE|def = Convergent Sequence|context = Metric Space|contextview = Metric Spaces}}


Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 3


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

:$\displaystyle \int_0^1 \frac {\ln x} {1 - x} \rd x = -\frac {\pi^2} 6$


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

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


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

:$\displaystyle \int_0^1 \frac {\map \ln {1 - x} } x \rd x = -\frac {\pi^2} 6$


Definite Integral to Infinity of Exponential of -a x by Cosine of b x

:$\displaystyle \int_0^\infty e^{-a x} \cos b x \rd x = \frac a {a^2 + b^2}$


Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x

:$\displaystyle \int_0^\infty \frac {e^{-a x} - e^{-b x} } x \rd x = \ln \frac b a$


Definite Integral to Infinity of Exponential of -a x^2

:$\displaystyle \int_0^\infty e^{-a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi a}$


Definite Integral from 0 to 1 of Difference of Powers of x over Logarithm of x

:$\displaystyle \int_0^1 \frac {x^m - x^n} {\ln x} \rd x = \map \ln {\frac {m + 1} {n + 1} }$


Union is Commutative/Family of Sets

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets. Let $\displaystyle I = \bigcup_{i \mathop \in I} S_i$ denote the union of  $\family {S_i}_{i \mathop \in I}$. Let $J \subseteq I$ be a subset of $I$. Then: :$\displaystyle \bigcup_{i \mathop \in I} S_i = \bigcup_{j \mathop \in J} S_j \cup \bigcup_{k \mathop \in \relcomp I J} S_k = \bigcup_{k \mathop \in \relcomp I J} S_k \cup \bigcup_{j \mathop \in J} S_j$ where $\relcomp I J$ denotes the complement of $J$ relative to $I$.


Intersection is Commutative/Family of Sets

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets. Let $\displaystyle I = \bigcap_{i \mathop \in I} S_i$ denote the intersection of $\family {S_i}_{i \mathop \in I}$. Let $J \subseteq I$ be a subset of $I$. Then: :$\displaystyle \bigcap_{i \mathop \in I} S_i = \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k = \bigcap_{k \mathop \in \relcomp I J} S_k \cap \bigcap_{j \mathop \in J} S_j$ where $\relcomp I J$ denotes the complement of $J$ relative to $I$.


Definite Integral to Infinity of Power of x by Logarithm of x over One plus x

:$\displaystyle \int_0^\infty \frac {x^{p - 1} \ln x} {1 + x} \rd x = -\pi^2 \csc p \pi \cot p \pi$


Definite Integral from 0 to Half Pi of Logarithm of Sine x

:$\displaystyle \int_0^{\pi/2} \map \ln {\sin x} \rd x = -\frac \pi 2 \ln 2$


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

:$\displaystyle \int_0^{\pi/2} \map \ln {\cos x} \rd x = -\frac \pi 2 \ln 2$


Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary/P-adic Norm

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\sequence {x_n}$ be a Cauchy sequence such that $\sequence {x_n}$ does not converge to $0$. Then: :$\exists N \in \N: \forall n, m \ge N: \norm {x_n}_p = \norm {x_m}_p$


Definite Integral to Infinity of Logarithm of Exponential of x plus One over Exponential of x minus One

:$\displaystyle \int_0^\infty \map \ln {\frac {e^x + 1} {e^x - 1} } \rd x = \frac {\pi^2} 4$


Definite Integral from 0 to Quarter Pi of Logarithm of One plus Tan x

:$\displaystyle \int_0^{\pi/4} \map \ln {1 + \tan x} \rd x = \frac \pi 8 \ln 2$


Primitive of Sine x by Logarithm of Sine x

:$\ds \int \sin x \map \ln {\sin x} \rd x = \cos x \paren {1 - \map \ln {\sin x} } + \ln \size {\tan \frac x 2} + C$


Euler-Mascheroni Constant as Difference of Integrals involving Cosine

:$\displaystyle \int_0^1 \frac {1 - \cos x} x \rd x - \int_1^\infty \frac {\cos x} x \rd x = \gamma$


Definite Integral from 0 to 1 of Arcsine of x over x

:$\displaystyle \int_0^1 \frac {\arcsin x} x = \frac \pi 2 \ln 2$


Definite Integral from 0 to 1 of Arctangent of x over x

:$\displaystyle \int_0^1 \frac {\arctan x} x \rd x = G$


Definite Integral from 0 to Half Pi of x over Sine x

:$\displaystyle \int_0^{\pi/2} \frac x {\sin x} \rd x = 2 G$


Definite Integral to Infinity of Arctangent of p x minus Arctangent of q x over x

:$\displaystyle \int_0^\infty \frac {\arctan p x - \arctan q x} x \rd x = \frac \pi 2 \ln \frac p q$


Definite Integral from 0 to Half Pi of Sine x by Logarithm of Sine x

:$\displaystyle \int_0^{\pi/2} \sin x \map \ln {\sin x} \rd x = \ln 2 - 1$


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

:$\displaystyle \int_0^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} } \map \erfc {\frac b {2 \sqrt a} }$


Set is Subset of Intersection of Supersets

Let $S$, $T_1$ and $T_2$ be sets. Let $S$ be a subset of both $T_1$ and $T_2$. Then: :$S \subseteq T_1 \cap T_2$ That is: :$\paren {S \subseteq T_1} \land \paren {S \subseteq T_2} \implies S \subseteq \paren {T_1 \cap T_2}$


Definite Integral to Infinity of Power of x by Exponential of -a x^2

:$\displaystyle \int_0^\infty x^m e^{-a x^2} \rd x = \frac {\map \Gamma {\paren {m + 1}/2} } {2 a^{\paren {m + 1}/2} }$


Integral Representation of Dirichlet Eta Function in terms of Gamma Function

:$\displaystyle \map \eta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {x^{s - 1} } {e^x + 1} \rd x$


Definite Integral to Infinity of x over Exponential of x plus One

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


Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x

:$\displaystyle \int_0^1 \ln x \map \ln {1 - x} \rd x = 2 - \frac {\pi^2} 6$


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

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


Power Series Expansion for Error Function

:$\displaystyle \map \erf x = \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }$


Set is Subset of Intersection of Supersets/General Result

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. Let $X$ be a set such that: :$\forall i \in I: X \subseteq S_i$ Then: :$X \subseteq \bigcup_{i \mathop \in I} S_i$ where $\displaystyle \bigcup_{i \mathop \in I} S_i$ is the intersection of $\family {S_i}$.


Error Function is Odd

:$\map \erf {-x} = -\map \erf x$


Error Function of Zero

:$\map \erf 0 = 0$


P-adic Expansion is a Cauchy Sequence in P-adic Norm

Let $p$ be a prime number. Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals numbers $\Q$. Let $\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$ be a $p$-adic expansion. Then the sequence of partial sums of the series: :$\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$ is a Cauchy sequence in the valued field  $\struct{\Q, \norm{\,\cdot\,}_p}$.


P-adic Expansion is a Cauchy Sequence in P-adic Norm/Converges to P-adic Number

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.


Characterization of Exponential Integral Function

:$\displaystyle \map \Ei x = -\gamma - \ln x + \int_0^x \frac {1 - e^{-u} } u \rd u$


Membership Relation is Not Reflexive

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 reflexive relation.


Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\sequence{x_n}$ be a sequence in the rational numbers $\Q$. Then $\sequence {x_n}$ is a Cauchy sequence in $\struct{\Q, \norm{\,\cdot\,}_p}$ {{iff}} $\sequence {x_n}$ is a Cauchy sequence in $\struct{\Q_p, \norm{\,\cdot\,}_p}$


Definite Integral to Infinity of Sine x over Root x

:$\displaystyle \int_0^\infty \frac {\sin x} {\sqrt x} \rd x = \sqrt {\frac \pi 2}$


Digamma Function of One Half

:$\displaystyle \map \psi {\frac 1 2} = -\gamma - 2 \ln 2$


Definite Integral to Infinity of Exponential of -x^2 by Logarithm of x

:$\displaystyle \int_0^\infty e^{-x^2} \ln x \rd x = -\frac {\sqrt \pi} 4 \paren {\gamma + 2 \ln 2}$


Definite Integral to Infinity of Cosine x over Root x

:$\displaystyle \int_0^\infty \frac {\cos x} {\sqrt x} \rd x = \sqrt {\frac \pi 2}$


Definite Integral to Infinity of Cube of Sine x over x Cubed

:$\displaystyle \int_0^\infty \frac {\sin^3 x} {x^3} \rd x = \frac {3 \pi} 8$


Equivalence of Definitions of Composition of Mappings

{{TFAE|def = Composition of Mappings}} Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$.


P-adic Numbers are Generated Ring Extension of P-adic Integers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Then: :$Q_p = \Z_p \sqbrk {1 / p}$ where $\Z_p \sqbrk {1 / p}$ denotes the ring extension generated by $1 / p$.


Limit to Infinity of Exponential Integral Function

:$\displaystyle \lim_{x \mathop \to \infty} \map \Ei x = 0$


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

:$\displaystyle \int_0^\infty \frac {x^n} {\sinh a x} \rd x = \frac {2^{n + 1} - 1} {2^n a^{n + 1} } \map \Gamma {n + 1} \map \zeta {n + 1}$


Fourier Series/Identity Function over Symmetric Range

Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $\map f x: \openint {-\lambda} \lambda \to \R$ be the identity function on the open real interval $\openint {-\lambda} \lambda$: :$\forall x \in \openint {-\lambda} \lambda: \map f x = x$ The Fourier series of $f$ over $\openint {-\lambda} \lambda$ can be given as: :$\map f x \sim \dfrac {2 \lambda} \pi \displaystyle \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin \frac {n \pi x} \lambda$


Restriction of Mapping is its Intersection with Cartesian Product of Subset with Image

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


Primitive of Reciprocal of One plus Fourth Power of x

:$\displaystyle \int \frac 1 {1 + x^4} \rd x = \frac 1 {2 \sqrt 2} \paren {\map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + \frac 1 2 \ln \size {\frac {x^2 + \sqrt 2 x + 1} {x^2 - \sqrt 2 x + 1} } } + C$


Sine Integral Function is Odd

:$\displaystyle \map \Si {-x} = -\map \Si x$


Sine Integral Function of Zero

:$\map \Si 0 = 0$


Arctangent of Root 3 over 3

:$\map \arctan {\dfrac {\sqrt 3} 3} = \dfrac \pi 6$


Cardinality of Set Union/2 Sets

Let $S_1$ and $S_2$ be finite sets. Then: :$\card {S_1 \cup S_2} = \card {S_1} + \card {S_2} - \card {S_1 \cap S_2}$


Cardinality of Set Union/3 Sets

Let $S_1$, $S_2$ and $S_3$ be finite sets. Then: {{begin-eqn}}      {{eqn | l = \card {S_1 \cup S_2 \cup S_3}       | r = \card {S_1} + \card {S_2} + \card {S_3}       | c = }} {{eqn | o =        | ro= -       | r = \card {S_1 \cap S_2} - \card {S_1 \cap S_3} - \card {S_2 \cap S_3}       | c = }} {{eqn | o =        | ro= +       | r = \card {S_1 \cap S_2 \cap S_3}       | c = }} {{end-eqn}}


Definite Integral to Infinity of Exponential of -i x^2

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


P-adic Number times Integer Power of p is P-adic Integer

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Then: :$\forall a \in \Q_p: \exists n \in \N: p^n a \in \Z_p$


Definite Integral to Infinity of Sine of a x^2

:$\ds \int_0^\infty \map \sin {a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi {2 a} }$


Definite Integral to Infinity of Cosine of a x^2

:$\displaystyle \int_0^\infty \map \cos {a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi {2 a} }$


Definite Integral to Infinity of Hyperbolic Sine of a x over Exponential of b x minus One

:$\displaystyle \int_0^\infty \frac {\sinh a x} {e^{b x} - 1} \rd x = \frac 1 {2 a} - \frac \pi {2 b} \cot \frac {a \pi} b$


Union with Intersection equals Intersection with Union iff Subset

Let $A$, $B$ and $C$ be sets. Then: :$\paren {A \cap B} \cup C = A \cap \paren {B \cup C} \iff C \subseteq A$


(A cap C) cup (B cap Complement C) = Empty iff B subset C subset Complement A

Let $A$, $B$ and $C$ be subsets of a universe $\Bbb U$. Then: :$\paren {A \cap C} \cup \paren {B \cap \map \complement C} = \O \iff B \subseteq C \subseteq \map \complement A$ where $\map \complement C$ denotes the complement of $C$ in $\Bbb U$.


Union of Intersections of 2 from 3 equals Intersection of Unions of 2 from 3

Let $A$, $B$ and $C$ be sets. Then: :$\paren {A \cap B} \cup \paren {B \cap C} \cup \paren {C \cap A} = \paren {A \cup B} \cap \paren {B \cup C} \cap \paren {C \cup A}$


Definite Integral to Infinity of Cosine of a x over Hyperbolic Cosine of b x

:$\displaystyle \int_0^\infty \frac {\cos a x} {\cosh b x} \rd x = \frac \pi {2 b} \sech \frac {a \pi} {2 b}$


Definite Integral to Infinity of Hyperbolic Sine of a x over Exponential of b x plus One

:$\displaystyle \int_0^\infty \frac {\sinh a x} {e^{b x} + 1} \rd x = \frac \pi {2 b} \csc \frac {a \pi} b - \frac 1 {2 a}$


Intersection Complement of Set with Itself is 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: :$A \uparrow A = \relcomp {\Bbb U} A$


Set Union 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: :$A \cup B = \paren {A \uparrow A} \uparrow \paren {B \uparrow B}$


Element in Set iff Singleton in Powerset

Let $S$ be a set. Then: :$x \in S \iff \set x \in \powerset S$ where $\powerset S$ denotes the power set of $S$.


Set Consisting of Empty Set is not Empty

Let $S$ be the set defined as: :$S = \set \O$ Then $S$ is not the empty set. That is: :$\O \ne \set \O$


Elements of Ordered Pair do not Commute

Let $\set {a, b}$ be a doubleton, so that $a$ and $b$ are distinct objects. Let $\tuple {a, b}$ denote the ordered pair such that the first coordinate is $a$ and the second coordinate is $b$. Then: :$\tuple {a, b} \ne \tuple {b, a}$


Definite Integral to Infinity of Cosine m x over x Squared plus a Squared

:$\displaystyle \int_0^\infty \frac {\cos m x} {x^2 + a^2} \rd x = \frac \pi {2 a} e^{-m a}$


Let $\struct{S, \tau_{_S}}$ be a topological space. Let $\struct{R, +, *, \norm{\,\cdot\,}}$ be a normed division ring. Let $\tau_{_R}$ be the topology induced by the norm $\norm{\,\cdot\,}$. Let $\lambda \in R$. Let $f,g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ be continuous mappings. Let $U = S \setminus \set{x : \map g x = 0}$ Let $g^{-1} : U \to R$ denote the mapping defined by: :$\forall x \in U : \map {g^{-1}} x = \map g x^{-1}$ Let $\tau_{_U}$ be the subspace topology on $U$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Sum Rule}} === Translation Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Translation Rule}} === Negation Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Negation Rule}} === Product Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Multiple Rule}} === Inverse Rule === {{:Combination Theorem for Continuous Mappings/Normed Division Ring/Inverse Rule}}


Combination Theorem for Continuous Mappings/Normed Division Ring/Sum Rule

:$f + g: \struct {S, \tau_{_S} } \to \struct{R, \tau_{_R} }$ is continuous.


Combination Theorem for Continuous Mappings/Normed Division Ring/Multiple Rule

:$\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous :$f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.


Combination Theorem for Continuous Mappings/Normed Division Ring/Product Rule

:$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.


Combination Theorem for Continuous Mappings/Normed Division Ring/Inverse Rule

:$g^{-1}: \struct {U, \tau_{_U} } \to \struct {R, \tau_{_R} }$ is continuous.


Derivative of Cosine Integral Function

:$\displaystyle \frac \d {\d x} \paren {\map \Ci x} = -\frac {\cos x} x$


Derivative of Exponential Integral Function

:$\displaystyle \frac \d {\d x} \paren {\map \Ei x} = -\frac {e^{-x} } x$


Derivative of Sine Integral Function

:$\displaystyle \frac \d {\d x} \paren {\map \Si x} = \frac {\sin x} x$


Primitive of Exponential Integral Function

:$\displaystyle \int \map \Ei x \rd x = x \map \Ei x - e^{-x} + C$


Direct Image Mapping of Domain is Image Set of Mapping

Let $S$ and $T$ be sets. Let $\powerset S$ and $\powerset T$ be their power sets. Let $f \subseteq S \times T$ be a mapping from $S$ to $T$. Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$: :$\forall X \in \powerset S: \map {f^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \map f s = t} & : X \ne \O \\ \O & : X = \O \end {cases}$ Then: :$\map {f^\to} S = \Img f$ where $\Img f$ is the image set of $f$.


Direct Image Mapping of Domain is Image Set of Relation

Let $S$ and $T$ be sets. Let $\powerset S$ and $\powerset T$ be their power sets. Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$. Let $\mathcal R^\to: \powerset S \to \powerset T$ be the direct image mapping of $\mathcal R$: :$\forall X \in \powerset S: \map {\mathcal R^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \tuple {x, t} \in \mathcal R} & : X \ne \O \\ \O & : X = \O \end {cases}$ Then: :$\map {\mathcal R^\to} {\Dom {\mathcal R} } = \Img {\mathcal R}$ where: :$\Dom {\mathcal R}$ is the domain of $\mathcal R$ :$\Img {\mathcal R}$ is the image set of $\mathcal R$.


Inverse Image Mapping of Codomain is Preimage Set of Mapping

Let $S$ and $T$ be sets. Let $\powerset S$ and $\powerset T$ be their power sets. Let $f \subseteq S \times T$ be a mapping from $S$ to $T$. Let $f^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $f$: :$\forall Y \in \powerset T: \map {f^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \map f s = t} & : \Img f \cap Y \ne \O \\ \O & : \Img f \cap Y = \O \end {cases}$ Then: :$\map {f^\gets} T = \Preimg f$ where $\Preimg f$ is the preimage set of $f$.


Inverse Image Mapping of Codomain is Preimage Set of Relation

Let $S$ and $T$ be sets. Let $\powerset S$ and $\powerset T$ be their power sets. Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$. Let $\mathcal R^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $\mathcal R$: :$\forall X \in \powerset S: \map {\mathcal R^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \tuple {x, t} \in \mathcal R} & : X \ne \O \\ \O & : X = \O \end {cases}$ Then: :$\map {\mathcal R^\gets} T = \Preimg {\mathcal R}$ where $\Preimg {\mathcal R}$ is the preimage of $\mathcal R$.


Direct Image Mapping of Mapping is Mapping

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping on $S \times T$. Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$: :$\forall X \in \powerset S: \map {f^\to} X = \set {t \in T: \exists s \in X: \map f s = t}$ Then $f^\to$ is indeed a mapping.


Inverse Image Mapping of Mapping is Mapping

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $f^\gets$ be the inverse image mapping of $f$: :$f^\gets: \powerset T \to \powerset S: \map {f^\gets} Y = f^{-1} \sqbrk Y$ Then $f^\gets$ is indeed a mapping.


Direct Image Mapping of Mapping is Empty iff Argument is Empty

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $f^\to$ be the direct image mapping of $f$: :$f^\to: \powerset S \to \powerset T: \map {f^\to} X = \set {t \in T: \exists s \in X: \map f s = t}$ Then: :$\map {f^\to} X = \O \iff X = \O$


Derivative of Fresnel Sine Integral Function

:$\displaystyle \frac {\d \operatorname S} {\d x} = \sqrt {\frac 2 \pi} \sin x^2$


Derivative of Fresnel Cosine Integral Function

:$\displaystyle \frac {\d \operatorname C} {\d x} = \sqrt {\frac 2 \pi} \cos x^2$


Power Series Expansion for Fresnel Cosine Integral Function

:$\displaystyle \map {\operatorname C} x = \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{4 n + 1} } {\paren {4 n + 1} \paren {2 n}!}$


Fresnel Sine Integral Function is Odd

:$\map {\operatorname S} {-x} = -\map {\operatorname S} x$


Fresnel Sine Integral Function of Zero

:$\map {\operatorname S} 0 = 0$


Fresnel Cosine Integral Function is Odd

:$\map {\operatorname C} {-x} = -\map {\operatorname C} x$


Fresnel Cosine Integral Function of Zero

:$\map {\operatorname C} 0 = 0$


Direct Image Mapping of Left-Total Relation is Empty iff Argument is Empty

Let $S$ and $T$ be sets. Let $\mathcal R: S \to T$ be a left-total relation on $S \times T$. Let $\mathcal R^\to$ be the direct image mapping of $\mathcal R$: :$\mathcal R^\to: \powerset S \to \powerset T: \map {\mathcal R^\to} X = \set {t \in T: \exists s \in X: \tuple {s, t} \in \mathcal R}$ Then: :$\map {\mathcal R^\to} X = \O \iff X = \O$


Limit to Infinity of Fresnel Cosine Integral Function

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


Composition of Direct Image Mappings of Mappings

Let $A, B, C$ be non-empty sets. Let $f: A \to B$ and $g: B \to C$ be mappings. Let: :$f^\to: \powerset A \to \powerset B$ and :$g^\to: \powerset B \to \powerset C$ be the direct image mappings of $f$ and $g$. Then: :$\paren {g \circ f}^\to = g^\to \circ f^\to$


Composition of Inverse Image Mappings of Mappings

Let $A, B, C$ be non-empty sets. Let $f: A \to B, g: B \to C$ be mappings. Let: :$f^\gets: \powerset B \to \powerset A$ and :$g^\gets: \powerset C \to \powerset B$ be the inverse image mappings of $f$ and $g$. Then: :$\paren {g \circ f}^\gets = f^\gets \circ g^\gets$


Unit of Ring of Mappings iff Image is Subset of Ring Units

Let $\struct {R, +, \circ}$ be a ring with unity $1_R$. Let $U_R$ be the set of units in $R$. Let $S$ be a set. Let $\struct {R^S, +', \circ'}$ be the ring of mappings on the set of mappings $R^S$. Then: :$f \in R^S$ is a unit of $R^S$ {{iff}} $\Img f \subseteq U_R$ where $\Img f$ is the image of $f$. In this case, the inverse of $f$ is the mapping $f^{-1} : S \to U_R$ defined by: :$\forall x \in S : \map {f^{-1} } x = \map f x^{-1}$


Relative Complement Mapping on Powerset is Bijection

Let $S$ be a set. Let $\complement_S: \powerset S \to \powerset S$ denote the relative complement mapping on the power set of $S$. Then $\complement_S$ is a bijection. Thus each $T \subseteq S$ is in one-to-one correspondence with its relative complement.