hcju/proofwikips huggingface dataset:
Subgroup of Circle Group Generated by Distinct Roots of Unity
Let $K$ be the circle group. Let $m, n \in \Z_{>0}$ be (strictly) positive integers. Let $d = \lcm \set {m, n}$ be the least common multiple of $m$ and $n$. Let $\alpha$ be a primitive $n$th root of unity. Let $\beta$ be a primitive $m$th root of unity. Let $\gamma$ be a primitive $d$th root of unity. Let $H = \gen {\alpha, \beta}$ be the subgroup of $K$ generated by $\alpha, \beta$. Then $H = \gen \gamma$.
Multiplicative Group of Complex Roots of Unity is Subgroup of Circle Group
Let $n \in \Z$ be an integer such that $n > 0$. Let $\struct {U_n, \times}$ denote the multiplicative group of complex $n$th roots of unity. Let $\struct {K, \times}$ denote the circle group. Then $\struct {U_n, \times}$ is a subgroup of $\struct {K, \times}$.
Intersection of Multiplicative Groups of Complex Roots of Unity
Let $\struct {K, \times}$ denote the circle group. Let $m, n \in \Z_{>0}$ be (strictly) positive integers. Let $c = \lcm \set {m, n}$ be the lowest common multiple of $m$ and $n$. Let $\struct {U_n, \times}$ denote the multiplicative group of complex $n$th roots of unity. Let $\struct {U_m, \times}$ denote the multiplicative group of complex $m$th roots of unity. Let $H = U_m \cap U_n$. Then $H = U_c$.
Direct Product of Normal Subgroups is Normal
Let $G$ and $G'$ be groups. Let: :$H \lhd G$ :$H' \lhd G'$ where $\lhd$ denotes the relation of being a normal subgroup. Then: :$\paren {H \times H'} \lhd \paren {G \times G'}$ where $H \times H'$ denotes the group direct product of $H$ and $H'$
Normalizer of Reflection in Dihedral Group
Let $n \in \N$ be a natural number such that $n \ge 3$. Let $D_n$ be the dihedral group of order $2 n$, given by: :$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$ Let $\map {N_{D_n} } {\set \beta}$ denote the normalizer of the singleton containing the reflection element $\beta$. Then: :$\map {N_{D_n} } {\set \beta} = \begin{cases} \set {e, \beta} & : n \text { odd} \\ \set {e, \beta, \alpha^{n / 2}, \alpha^{n / 2} \beta} & : n \text { even} \end{cases}$
Center of Quaternion Group
Let $Q = \Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ be the quaternion group. Let $\map Z {\Dic 2}$ denote the center of $\Dic 2$. Then: :$\map Z {\Dic 2} = \set {e, a^2}$
Product of Generating Elements of Quaternion Group
Let $Q = \Dic 2$ be the quaternion group: :$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ Then for all $k \in \Z_{\ge 0}$: :$b a^k = a^{-k} b$
Conjugacy Classes of Quaternion Group
Let $Q = \Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ be the quaternion group. The conjugacy classes of $\Dic 2$ are: :$\set e, \set {a^2}, \set {a, a^3}, \set {b, a^2 b}, \set {a b, a^3 b}$
Conjugacy Action on Subsets is Group Action
Let $\powerset G$ be the set of all subgroups of $G$. For any $S \in \powerset G$ and for any $g \in G$, the conjugacy action: :$g * S := g \circ S \circ g^{-1}$ is a group action.
Normed Vector Space Requires Multiplicative Norm on Division Ring
Let $R$ be a normed division ring with a submultiplicative norm $\norm {\, \cdot \,}_R$. Let $V$ be a vector space that is not a trivial vector space. Let $\norm {\, \cdot \,}: V \to \R_{\ge 0}$ be a mapping from $V$ to the positive real numbers satisfying the vector space norm axioms. Then $\norm {\, \cdot \,}_R$ is a multiplicative norm. That is: :$\forall r, s \in R: \norm {r s}_R = \norm r_R \norm s_R$
Ring with Multiplicative Norm has No Proper Zero Divisors
Let $\struct {R, +, \circ}$ be a ring. Let its zero be denoted by $0_R$. Let $\norm {\,\cdot\,}$ be a multiplicative norm on $R$. Then $R$ has no proper zero divisors.  That is: :$\forall x, y \in R^*: x \circ y \ne 0_R$ where $R^*$ is defined as $R \setminus \set {0_R}$.
Finite Ring with Multiplicative Norm is Field
Let $R$ be a finite ring with a multiplicative norm. Then $R$ is a field.
Composition of Isometries is Isometry
Let: :$\struct {X_1, d_1}$ :$\struct {X_2, d_2}$ :$\struct {X_3, d_3}$ be metric spaces. Let: :$\phi: \struct {X_1, d_1} \to \struct {X_2, d_2}$ :$\psi: \struct {X_2, d_2} \to \struct {X_3, d_3}$ be isometries. Then the composite of $\phi$ and $\psi$ is also an isometry.
Subgroup Action is Group Action
Let $\struct {G, \circ}$ be a group. Let $\struct {H, \circ}$ be a subgroup of $G$. Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as: :$\forall h \in H, g \in G: h * g := h \circ g$ Then $*$ is a group action.
Orbit of Subgroup Action is Coset
Let $\struct {G, \circ}$ be a group. Let $\struct {H, \circ}$ be a subgroup of $G$. Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as: :$\forall h \in H, g \in G: h * g := h \circ g$ The orbit of $x \in G$ is the right coset by $x$ of $H$: :$\Orb x = H x$
Removable Singularity at Infinity implies Constant Function
Let $f : \C \to \C$ be an entire function. Let $f$ have an removable singularity at $\infty$.  Then $f$ is constant.
Stabilizers of Elements in Same Orbit are Conjugate Subgroups
Let $G$ be a group acting on a set $X$. Let: :$y, z \in \Orb x$ where $\Orb x$ denotes the orbit of some $x \in X$. Then their stabilizers $\Stab y$ and $\Stab z$ are conjugate subgroups.
Stabilizer of Subgroup Action on Left Coset Space
Let $G$ be a group. Let $H$ and $K$ be subgroups of $G$. Let $K$ act on the left coset space $G / H^l$ by: :$\forall \tuple {k, g H} \in K \times G / H^l: k * g H := \paren {k g} H$ The stabilizer of $g H$ is $K \cap H^g$, where $H^g$ denotes the $G$-conjugate of $H$ by $g$.
Cardinality of Set Difference with Subset
Let $S$ and $T$ be sets such that $T$ is finite. Let $T \subseteq S$. Then: :$\card {S \setminus T} = \card S - \card T$ where $\card S$ denotes the cardinality of $S$.
Riemann Zeta Function at Non-Positive Integers
Let $n \ge 0$ be a integer.  Then:  :$\map \zeta {-n} = \paren {-1}^n \dfrac {B_{n + 1} } {n + 1}$  where: :$B_n$ is the $n$th Bernoulli number :$\zeta$ is the Riemann Zeta function
Remainder on Division is Least Positive Residue
Let $a, b \in \Z$ be integers such that $a \ge 0$ and $b \ne 0$. Let $r$ be the remainder resulting from the operation of integer division of $a$ by $b$: $a = q b + r, 0 \le r < \size b$ Then $r$ is equal to the least positive residue of $a \pmod b$.
Floor Function/Examples/Floor of 3
:$\floor 3 = 3$
Exponential on Real Numbers is Injection
Let $\exp: \R \to \R$ be the exponential function: :$\map \exp x = e^x$ Then $\exp$ is an injection.
Surjective Restriction of Real Exponential Function
Let $\exp: \R \to \R$ be the exponential function: :$\map \exp x = e^x$ Then the restriction of the codomain of $\exp$ to the strictly positive real numbers: :$\exp: \R \to \R_{>0}$ is a surjective restriction. Hence: :$\exp: \R \to \R_{>0}$ is a bijection.
Maximum Rule for Real Sequences
:$\displaystyle \lim_{n \mathop \to \infty} \max \set {x_n, y_n} = \max \set {l, m}$
Division Subring of Normed Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $S$ be a division subring of $R$. Then: :$\struct {S, \norm {\, \cdot \,}_S}$ is a normed division subring of $\struct {R, \norm {\, \cdot \,} }$ where $\norm {\, \cdot \,}_S$ is the norm $\norm{\,\cdot\,}$ restricted to $S$.
Normed Division Ring is Dense Subring of Completion
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {R', \norm {\, \cdot \,}' }$ be a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$ Then: :$\struct {R, \norm {\, \cdot \,} }$ is isometrically isomorphic to a dense normed division subring of $\struct {R', \norm {\, \cdot \,}' }$.
Inverse of Isometric Isomorphism
Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings. Let $\phi:R \to S$ be a mapping. Then $\phi:R \to S$ is an isometric isomorphism  $\phi^{-1}: S \to R$ is also an isometric isomorphism.
Isometric Isomorphism is Norm-Preserving
Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings. Let $\phi: R \to S$ be a ring isomorphism. Then $\phi: R \to S$ is an isometric isomorphism  $\phi$ satisfies: :$\forall x \in R: \norm {\map \phi x}_S = \norm x_R $
Taylor Series of Holomorphic Function
:$\displaystyle \map f z = \sum_{n \mathop = 0}^\infty \frac {\map {f^n} a} {n!} \paren {z - a}^n$
Equivalence of Definitions of Square Function
Let $\F$ denote one of the standard classes of numbers: $\N$, $\Z$, $\Q$, $\R$, $\C$. {{TFAE|def = Square Function}}
Subring of Non-Archimedean Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with non-archimedean norm $\norm {\, \cdot \,}$. Let $\struct {S, \norm {\, \cdot \,}_S }$ be a normed division subring of $R$. Then: :$\norm {\, \cdot \,}_S$ is a non-archimedean norm.
Isometrically Isomorphic Non-Archimedean Division Rings
Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings. Let $\phi:R \to S$ be an isometric isomorphism. Then: :$\norm {\,\cdot\,}_R$ is a non-archimedean norm  $\norm {\,\cdot\,}_S$ is a non-archimedean norm.
Non-Archimedean Division Ring Iff Non-Archimedean Completion
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {R', \norm {\, \cdot \,}' }$ be a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$ Then: :$\norm {\, \cdot \,}$ is non-archimedean  $\norm {\, \cdot \,}'$ is non-archimedean.
Domain of Integer Square Function
The domain of the integer square function is the entire set of integers $\Z$.
Image of Integer Square Function
The image of the integer square function is the set of square numbers.
Restriction of Real Square Mapping to Positive Reals is Bijection
Let $f: \R \to \R$ be the real square function: :$\forall x \in \R: \map f x = x^2$ Let $g: \R_{\ge 0} \to R_{\ge 0} := f {\restriction_{\R_{\ge 0} \times R_{\ge 0} } }$ be the restriction of $f$ to the positive real numbers $\R_{\ge 0}$. Then $g$ is a bijective restriction of $f$.
Inverse of Real Square Function on Positive Reals
Let $f: \R_{\ge 0} \to R_{\ge 0}$ be the restriction of the real square function to the positive real numbers $\R_{\ge 0}$. The inverse of $f$ is $f^{-1}: \R_{\ge 0} \times R_{\ge 0}$ defined as: :$\forall x \in \R_{\ge 0}: \map {f^{-1} } x = \sqrt x$ where $\sqrt x$ is the positive square root of $x$.
Real Square Function is not Bijective
Let $f: \R \to \R$ be the real square function: :$\forall x \in \R: \map f x = x^2$ Then $f$ is not a bijection.
Inverse of Linear Function on Real Numbers
Let $a, b \in \R$ be real numbers such that $a \ne 0$. Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = a x + b$ Then the inverse of $f$ is given by: :$\forall y \in \R: \map {f^{-1} } y = \dfrac {y - b} a$
Linear Function on Real Numbers is Bijection
Let $a, b \in \R$ be real numbers. Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = a x + b$ Then $f$ is a bijection  $a \ne 0$.
Composition of Linear Real Functions
Let $a, b, c, d \in \R$ be real numbers. Let $\theta_{a, b}: \R \to \R$  be the real function defined as: :$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$ Let $\theta_{c, d} \circ \theta_{a, b}$ denote the composition of $\theta_{c, d}$ with $\theta_{a, b}$. Then: :$\theta_{c, d} \circ \theta_{a, b} = \theta_{a c, b c + d}$
Condition for Composition of Linear Real Functions to be Commutative
Let $a, b, c, d \in \R$ be real numbers. Let $\theta_{a, b}: \R \to \R$  be the real function defined as: :$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$ Let $\theta_{c, d} \circ \theta_{a, b}$ denote the composition of $\theta_{c, d}$ with $\theta_{a, b}$. Then: :$\theta_{c, d} \circ \theta_{a, b} = \theta_{a, b} \circ \theta_{c, d}$ : :$b c + d = a d + b$
Composition of Right Inverse with Mapping is Idempotent
Let $f: S \to T$ be a mapping. Let $g: T \to S$ be a right inverse mapping of $f$. Then: :$\paren {g \circ f} \circ \paren {g \circ f} = g \circ f$
Sets of Permutations of Equivalent Sets are Equivalent
Let $A$ and $B$ be sets such that: :$A \sim B$ where $\sim$ denotes set equivalence. Let $\map \Gamma A$ denote the set of permutations on $A$. Then: :$\map \Gamma A \sim \map \Gamma B$
Three Points in Ultrametric Space have Two Equal Distances/Corollary 2
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$, Let $x, y \in R$ and $\norm x \ne \norm y$. Then: :$\norm {x + y} = \norm {x - y} = \norm {y - x} = \max \set {\norm x, \norm y}$
Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls
:$y \in \map {B_r} x \implies \map {B_r} y = \map {B_r} x$
Topological Properties of Non-Archimedean Division Rings/Centers of Closed Balls
:$y \in \map { {B_r}^-} x \implies \map { {B_r}^-} y = \map { {B_r}^-} x$
Topological Properties of Non-Archimedean Division Rings/Open Balls are Clopen
:The open $r$-ball of $x$, $\map {B_r} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$.
Topological Properties of Non-Archimedean Division Rings/Closed Balls are Clopen
:The closed $r$-ball of $x$, $\map { {B_r}^-} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$.
Mittag-Leffler Expansion for Hyperbolic Cotangent Function
:$\displaystyle \pi \, \map \coth {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 + n^2}$ where:  :$z \in \C$ is not an integer multiple of $i$ :$\coth$ is the hyperbolic cotangent function.
Mittag-Leffler Expansion for Secant Function
:$\displaystyle \pi \, \map \sec {\pi z} = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 - 4 z^2}$ where:  :$z \in \C$ is not a half-integer :$\sec$ is the secant function.
Pringsheim's Theorem
Let $f$ be a holomorphic function defined on a unit disc centered at the origin of the complex plane and is denoted by its Taylor series: :$\map f z = \displaystyle \sum_{n \mathop = 0}^{\infty} c_n z^n$ Let: :$(1): \quad \forall n \ge 0: c_n \ge 0$ :$(2): \quad$ the radius of convergence of the Taylor series of function $f$ is $1$. Then $z = 1$ is an isolated singularity of $f$.
Mittag-Leffler Expansion for Tangent Function
:$\displaystyle \pi \map \tan {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {\paren {2 n + 1}^2 - 4 z^2}$ where:  :$z \in \C$ is not a half-integer :$\tan$ is the tangent function.
Mittag-Leffler Expansion for Hyperbolic Tangent Function
:$\displaystyle \pi \map \tanh {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {4 z^2 + \paren {2 n + 1}^2}$ where: :$z \in \C$ is not a half-integer multiple of $i$ :$\tanh$ is the hyperbolic tangent function.
Mittag-Leffler Expansion for Hyperbolic Secant Function
:$\displaystyle \pi \, \map \sech {\pi z} = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 + 4 z^2}$ where:  :$z \in \C$ is not a half-integer multiple of $i$  :$\sech$ is the hyperbolic secant function.
Mittag-Leffler Expansion for Hyperbolic Cosecant Function
:$\ds \pi \map \csch {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac z {z^2 + n^2}$ where: :$z \in \C$ is not an integer multiple of $i$ :$\csch$ is the hyperbolic cosecant function.
Real Numbers under Subtraction do not form Semigroup
The set of real numbers under subtraction $\struct {\R, -}$ does not form a semigroup.
Count of Binary Operations on Set/Sequence
The sequence of $N$ for each $n$ begins: $\begin{array} {c|rr} n & n^2 & N = n^{\paren {n^2} } \\ \hline 1 & 1 & 1 \\ 2 & 4 & 16 \\ 3 & 9 & 19 \ 683 \\ 4 & 16 & 4 \ 294 \ 967 \ 296 \\ \end{array}$ There are still only $4$ elements in a set, and already there are over $4$ thousand million different possible algebraic structures.
Count of Commutative Binary Operations on Set/Sequence
The sequence of $N$ for each $n$ begins: $\begin{array} {c|cr} n & \dfrac {n \paren {n + 1} }2 & n^{\frac {n \paren {n + 1} } 2} \\ \hline 1 & 1 & 1 \\ 2 & 3 & 8 \\ 3 & 6 & 729 \\ 4 & 10 & 1 \ 048 \ 576 \\ \end{array}$ and so on.
Mittag-Leffler’s Expansion Theorem
Let $f$ be a meromorphic function with only simple poles continuous, or with a removable singularity, at $0$.  Let $X$ be the set of poles of $f$.  For $N \in \N$, let $C_N$ be a circle, centred at the origin, of radius $R_N$, where $R_N \to \infty$ as $N \to \infty$, such that $\partial C_N$ contains no poles of $f$ for any $N$.  Let $M > 0$ be a real number independent of $N$ such that for all $z \in \partial C_N$, $\cmod {\map f z} < M$ , for all $N \in \N$. Then: :$\displaystyle \map f z = \map f 0 + \sum_{n \mathop \in X} \Res f n \paren {\frac 1 {z - n} + \frac 1 n}$  where: :$\Res f n$ is the residue of $f$ at $n$ :$z$ is not a pole of $f$ :$\displaystyle \map f 0 = \lim_{z \mathop \to 0} \map f z$ if $f$ has a removable singularity at $0$.
Normed Division Ring Operations are Continuous
Let $\struct {R, +, *, \norm {\,\cdot\,} }$ be a normed division ring. Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$. Let $p \in \R_{\ge 1} \cup \set \infty$. Let $d_p$ be the $p$-product metric on $R \times R$. Let $R^* = R \setminus \set 0$ Let $d^*$ be the restriction of $d$ to $R^*$. Then the following results hold:
Normed Division Ring Operations are Continuous/Addition
:$+ : \struct {R \times R, d_p} \to \struct{R,d}$  is continuous.
Normed Division Ring Operations are Continuous/Negation
:$\eta: \struct {R, d} \to \struct {R, d}: \map \eta x = -x$  is continuous.
Normed Division Ring Operations are Continuous/Multiplication
:$* : \struct {R \times R, d_p} \to \struct {R, d}$  is continuous.
Normed Division Ring Operations are Continuous/Corollary
Let $\tau$ be the topology induced by the metric $d$. Then: :$\struct {R, \tau}$ is a topological division ring.
Count of Binary Operations with Identity/Sequence
The sequence of $N$ for each $n$ begins: $\begin{array} {c|cr} n & \paren {n - 1}^2 + 1 & n^{\paren {n - 1}^2 + 1}\\ \hline 1 & 1 & 1 \\ 2 & 2 & 4 \\ 3 & 5 & 243 \\ 4 & 10 & 1 \ 048 \ 576 \\ \end{array}$
Structure with Element both Identity and Zero has One Element
Let $\struct {S, \circ}$ be an algebraic structure. Let $z \in S$ such that $z$ is both an identity element and a zero element. Then: :$S = \set z$
Group/Examples/Linear Functions
Let $G$ be the set of all real functions $\theta_{a, b}: \R \to \R$ defined as: :$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$ where $a, b \in \R$ such that $a \ne 0$. The algebraic structure $\struct {G, \circ}$, where $\circ$ denotes composition of mappings, is a group. $\struct {G, \circ}$ is specifically non-abelian.
Summation Formula for Alternating Series
Let $C_N$ be the square with vertices $\paren {N + \dfrac 1 2} \paren {\pm 1 \pm i}$ for some real $N \in \N$.  Let $f$ be a function meromorphic on $C_N$. Let $\cmod {\map f z} < \dfrac M {\cmod z^k}$, for constants $k > 1$ and $M$ independent of $N$, for all $z \in \partial C_N$.  Let $X$ be the set of poles of $f$. Then:  :$\displaystyle \sum_{n \mathop \in \Z \mathop \setminus X} \paren {-1}^n \map f n = -\sum_{z_0 \mathop \in X} \Res {\pi \map \csc {\pi z} \map f z} {z_0}$ If $X \cap \Z = \O$, this becomes:  :$\displaystyle \sum_{n \mathop = -\infty}^\infty \paren {-1}^n \map f n = -\sum_{z_0 \mathop \in X} \Res {\pi \map \csc {\pi z} \map f z} {z_0}$
Integer Multiples under Addition form Subgroup of Integers
Let $\struct {\Z, +}$ denote the additive group of integers. Let $n \Z$ be the set of integer multiples of $n$. Then $\struct {n \Z, +}$ is a subgroup of $\struct {\Z, +}$. Hence $\struct {n \Z, +}$ can be justifiably referred to as the additive group of integer multiples.
Group is Generated by Itself
Let $G$ be a group. Then: :$G = \gen G$ where $\gen G$ denotes the group generated by $G$.
Summation Formula for Alternating Series over Half-Integers
Let $C_N$ be the square with vertices $N \paren {\pm 1 \pm i}$ for real $N \in \N$.  Let $f$ be a function meromorphic on $C_N$. Let $\cmod {\map f z} < \dfrac M {\cmod z^k}$, for constants $k > 1$ and $M$ independent of $N$, for all $z \in \partial C_N$.  Let $X$ be the set of poles of $f$. Let $Y$ be the set of poles of $\map f {\dfrac {2 z + 1} 2}$.  Then:  :$\displaystyle \sum_{n \in \Z \setminus Y} \paren {-1}^n \map f {\frac {2 n + 1} 2} = \sum_{z_0 \in X} \Res {\pi \sec \paren {\pi z} \map f z} {z_0}$ If $Y \cap \Z = \O$, this becomes:  :$\displaystyle \sum_{n \mathop = -\infty}^\infty \paren {-1}^n \map f {\frac {2 n + 1} 2} = \sum_{z_0 \in X} \Res {\pi \sec \paren {\pi z} \map f z} {z_0}$
Summation Formula over Half-Integers
Let $C_N$ be the square with vertices $N \paren {\pm 1 \pm i}$ for real $N \in \N$.  Let $f$ be a function meromorphic on $C_N$. Let $\cmod {\map f z} < \dfrac M {\cmod z^k}$, for constants $k > 1$ and $M$ independent of $N$, for all $z \in \partial C_N$.  Let $X$ be the set of poles of $f$. Let $Y$ be the set of poles of $\map f {\dfrac {2 z + 1} 2}$. Then:  :$\displaystyle \sum_{n \in \Z \setminus Y} \map f {\frac {2 n + 1} 2} = \sum_{z_0 \in X} \Res {\pi \tan \paren {\pi z} \map f z} {z_0}$ If $Y \cap \Z = \O$, this becomes:  :$\displaystyle \sum_{n \mathop = -\infty}^\infty \map f {\frac {2 n + 1} 2} = \sum_{z_0 \in X} \Res {\pi \tan \paren {\pi z} \map f z} {z_0}$
Heaviside Expansion Formula
Let $P, Q$ be polynomials with coefficients in $\C$.  Let $\deg Q \ge \deg P + 1$. Let $\map Q z$ have a simple zero for $z \in X$.  Let $\map {\laptrans f} z = \dfrac {\map P z} {\map Q z}$.  Then: :$\displaystyle \map f t = \sum_{z \mathop \in X} e^{z t} \frac {\map P z} {\map {Q'} z}$
Action of Inverse of Group Element
Let $\struct {G, \circ}$ be a group. Let $S$ be a sets. Let $*: G \times S \to S$ be a group action. Then: :$g * a = b \iff g^{-1} * b = a$
Union Operation on Supersets of Subset is Closed
Let $S$ be a set. Let $T \subseteq S$ be a given subset of $S$. Let $\powerset S$ denote the power set of $S$ Let $\mathscr S$ be the subset of $\powerset S$ defined as: :$\mathscr S = \set {Y \in \powerset S: T \subseteq Y}$ Then the algebraic structure $\struct {\mathscr S, \cup}$ is closed.
Existence of Magma with no Proper Submagma
Let $n \in \Z_{>0}$ be a strictly positive integer. Let $S$ be a set of cardinality $n$: :$\card S = n$ Then there exists an operation $\circ$ on $S$ such that: :$\struct {S, \circ}$ is a magma :$\struct {S, \circ}$ has no submagma $\struct {T, \circ}$ such that $T$ is a non-empty proper subset of $S$.
Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup
Let $\struct {G, \circ}$ be an abelian group. Let $S \subset G$ be a non-empty subset of $G$ such that $\struct {S, \circ}$ is closed. Let $H$ be the set defined as: :$H := \set {x \circ y^{-1}: x, y \in S}$ Then $\struct {H, \circ}$ is a subgroup of $\struct {G, \circ}$.
Local Basis Test
Let $\struct {S, \tau}$ be a topological space. Let $x \in S$. Let $\BB$ be a local basis for $x$ in $\struct {S, \tau}$. Let $\CC$ be a set of open neighborhoods of $x$. Then: :$\CC$ is a local basis : ::$\forall B \in \BB \implies \exists C \in \CC: C \subseteq B$
Union of Subgroups/Corollary 1
Let $H \cup K$ be a subgroup of $G$. Then either $H \subseteq K$ or $K \subseteq H$.
Stabilizer of Element after Group Action
Let $\struct {G, \circ}$ be a group. Let $S$ be a set. Let $*_S: G \times S \to S$ be a group actions. Let $x \in S, a \in G$. Then: :$\Stab {a * x} = a^{-1} \circ \Stab x \circ a$
Group Action of Symmetric Group/Subset
Let $r \in \N: 0 < r \le n$. Let $B_r$ denote the set of all subsets of $\N_n$ of cardinality $r$: :$B_r  := \set {S \subseteq \N_n: \card S = r}$ Let $*$ be the mapping $*: S_n \times B_r \to B_r$ defined as: :$\forall \pi \in S_n, \forall S \in B_r: \pi * B_r = \pi \sqbrk S$ where $\pi \sqbrk S$ denotes the image of $S$ under $\pi$. Then $*$ is a group action.
Group Action of Symmetric Group on Subset is Transitive
Let $r \in \N: 0 < r \le n$. Let $B_r$ denote the set of all subsets of $\N_n$ of cardinality $r$: :$B_r  := \set {S \subseteq \N_n: \card S = r}$ Let $*$ be the mapping $*: S_n \times B_r \to B_r$ defined as: :$\forall \pi \in S_n, \forall S \in B_r: \pi * B_r = \pi \sqbrk S$ where $\pi \sqbrk S$ denotes the image of $S$ under $\pi$. Then $*$ is a transitive group action.