hcju/proofwikips huggingface dataset:
Column Operation is Equivalent to Post-Multiplication by Product of Elementary Matrices
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\Gamma$ be a column operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$. Then there exists a unique invertible square matrix $\mathbf K$ of order $n$ such that: :$\mathbf A \mathbf K = \mathbf B$ where $\mathbf K$ is the product of a finite sequence of elementary column matrices.
Existence of Inverse Elementary Row Operation/Scalar Product of Row
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be the elementary row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. {{begin-axiom}} {{axiom | n = \text {ERO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply row $i$ by $\lambda$         | m = r_i \to \lambda r_i }} {{end-axiom}} Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary row operation: :$e' := r_i \to \dfrac 1 \lambda r_i$
Existence of Inverse Elementary Row Operation/Add Scalar Product of Row to Another
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be the elementary row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. {{begin-axiom}} {{axiom | n = \text {ERO} 2         | t = For some $\lambda \in K$, add $\lambda$ times row $k$ to row $l$         | m = r_k \to r_k + \lambda r_l }} {{end-axiom}} Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary row operation: :$e' := r'_k \to r'_k - \lambda r'_l$
Existence of Inverse Elementary Row Operation/Exchange Rows
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be the elementary row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. {{begin-axiom}} {{axiom | n = \text {ERO} 3         | t = Exchange rows $k$ and $l$         | m = r_k \leftrightarrow r_l }} {{end-axiom}} Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary row operation: :$e' := r_k \leftrightarrow r_l$ That is: :$e' = e$
Existence of Inverse Elementary Column Operation/Scalar Product of Column
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be the elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. {{begin-axiom}} {{axiom | n = \text {ECO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply column $k$ by $\lambda$         | m = \kappa_k \to \lambda \kappa_k }} {{end-axiom}} Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary column operation: :$e' := \kappa_k \to \dfrac 1 \lambda \kappa_k$
Existence of Inverse Elementary Column Operation/Add Scalar Product of Column to Another
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be the elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. {{begin-axiom}} {{axiom | n = \text {ECO} 2         | t = For some $\lambda \in K$, add $\lambda$ times column $l$ to column $k$         | m = \kappa_k \to \kappa_k + \lambda \kappa_l }} {{end-axiom}} Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary column operation: :$e' := \kappa_k \to \kappa_k - \lambda \kappa_l$
Existence of Inverse Elementary Column Operation/Exchange Columns
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be the elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. {{begin-axiom}} {{axiom | n = \text {ECO} 3         | t = Interchange columns $k$ and $l$         | m = \kappa_k \leftrightarrow \kappa_l }} {{end-axiom}} Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary column operation: :$e' := \kappa_k \leftrightarrow \kappa_l$ That is: :$e' = e$
Elementary Column Operations as Matrix Multiplications
Let $e$ be an elementary column operation. Let $\mathbf E$ be the elementary column matrix of order $n$ defined as: :$\mathbf E = e \paren {\mathbf I}$ where $\mathbf I$ is the unit matrix. Then for every $m \times n$ matrix $\mathbf A$: :$e \paren {\mathbf A} = \mathbf A \mathbf E$ where $\mathbf A \mathbf E$ denotes the conventional matrix product.
Square Root of Number Plus Square Root/Proof 1
Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then: {{:Square Root of Number Plus Square Root}}
Square Root of Number Plus Square Root/Proof 2
Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then: {{:Square Root of Number Plus Square Root}}
Square Root of Number Minus Square Root/Proof 1
Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then: {{:Square Root of Number Plus Square Root}}
Exchange of Columns as Sequence of Other Elementary Column Operations
Let $\mathbf A$ be an $m \times n$ matrix. Let $i, j \in \closedint 1 m: i \ne j$ Let $\kappa_k$ denote the $k$th column of $\mathbf A$ for $1 \le k \le n$: :$\kappa_k = \begin {pmatrix} a_{1 k} \\ a_{2 k} \\ \vdots \\ a_{m k} \end {pmatrix}$ Let $e$ be the elementary column operation acting on $\mathbf A$ as: {{begin-axiom}} {{axiom | n = \text {ERO} 3         | t = Interchange columns $i$ and $j$         | m = \kappa_i \leftrightarrow \kappa_j }} {{end-axiom}} Then $e$ can be expressed as a finite sequence of exactly $4$ instances of the other two elementary column operations. {{begin-axiom}} {{axiom | n = \text {ERO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply column $i$ by $\lambda$         | m = \kappa_i \to \lambda \kappa_i }} {{axiom | n = \text {ERO} 2         | t = For some $\lambda \in K$, add $\lambda$ times column $j$ to column $i$         | m = \kappa_i \to \kappa_i + \lambda \kappa_j }} {{end-axiom}}
Determinant of Elementary Column Matrix
Let $\mathbf E$ be an elementary column matrix. The determinant of $\mathbf E$ is as follows:
Determinant of Elementary Column Matrix/Scale Column
Let $e_1$ be the elementary column operation $\text {ECO} 1$: {{begin-axiom}} {{axiom | n = \text {ECO} 1         | t = For some $\lambda \ne 0$, multiply column $k$ by $\lambda$         | m = \kappa_k \to \lambda \kappa_k }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\mathbf E_1$ be the elementary column matrix corresponding to $e_1$. The determinant of $\mathbf E_1$ is: :$\map \det {\mathbf E_1} = \lambda$
Determinant of Elementary Column Matrix/Scale Column and Add
Let $e_2$ be the elementary column operation $\text {ECO} 2$: {{begin-axiom}} {{axiom | n = \text {ECO} 2         | t = For some $\lambda$, add $\lambda$ times column $j$ to column $i$         | m = \kappa_i \to \kappa_i + \lambda \kappa_j }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\mathbf E_2$ be the elementary column matrix corresponding to $e_2$. The determinant of $\mathbf E_2$ is: :$\map \det {\mathbf E_2} = 1$
Determinant with Column Multiplied by Constant
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$. Let $\mathbf B$ be the matrix resulting from one column of $\mathbf A$ having been multiplied by a constant $c$. Then: :$\map \det {\mathbf B} = c \map \det {\mathbf A}$ That is, multiplying one column of a square matrix by a constant multiplies its determinant by that constant.
Determinant of Elementary Column Matrix/Exchange Columns
Let $e_3$ be the elementary column operation $\text {ECO} 3$: {{begin-axiom}} {{axiom | n = \text {ECO} 3         | t = Exchange columns $i$ and $j$         | m = \kappa_i \leftrightarrow \kappa_j }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\mathbf E_3$ be the elementary column matrix corresponding to $e_3$. The determinant of $\mathbf E_3$ is: :$\map \det {\mathbf E_3} = -1$
Determinant of Rescaling Matrix/Corollary
Let $\mathbf A$ be a square matrix of order $n$. Let $\lambda$ be a scalar. Let $\lambda \mathbf A$ denote the scalar product of $\mathbf A$ by $\lambda$. Then: :$\map \det {\lambda \mathbf A} = \lambda^n \map \det {\mathbf A}$ where $\det$ denotes determinant.
Sequence of Row Operations is Row Operation
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\Gamma_1$ be a row operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$. Let $\Gamma_2$ be a row operation which transforms $\mathbf B$ to another new matrix $\mathbf C \in \map \MM {m, n}$. Then there exists another row operation $\Gamma$ which transforms $\mathbf A$ back to $\mathbf C$ such that $\Gamma$ consists of $\Gamma_1$ followed by $\Gamma_2$.
Sequence of Column Operations is Column Operation
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\Gamma_1$ be a column operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$. Let $\Gamma_2$ be a column operation which transforms $\mathbf B$ to another new matrix $\mathbf C \in \map \MM {m, n}$. Then there exists another column operation $\Gamma$ which transforms $\mathbf A$ back to $\mathbf C$ such that $\Gamma$ consists of $\Gamma_1$ followed by $\Gamma_2$.
Equivalence of Definitions of Determinant
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. {{TFAE|def = Determinant of Matrix|view = the determinant of $\mathbf A$}}
Intersection With Singleton is Disjoint if Not Element
Let $S$ be a set. Let $\set x$ be the singleton of $x$. Then: :$x \notin S$  $\set x \cap S = \O$
Product of Matrix with Adjugate equals Determinant by Unit Matrix
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$. Let $\adj {\mathbf A}$ be the determinant of $\mathbf A$. Then: :$\paren {\adj {\mathbf A} } \mathbf A = \map \det {\mathbf A} \mathbf I = \mathbf A \paren {\adj {\mathbf A} }$ where $\mathbf I$ denotes the unit matrix of order $n$.
Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant
Let $\mathbf A = \sqbrk a_n$ be an invertible square matrix of order $n$. Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$. Let $\adj {\mathbf A}$ be the determinant of $\mathbf A$. Then: :$\mathbf A^{-1} = \dfrac 1 {\map \det {\mathbf A} } \cdot \adj {\mathbf A}$ where $\mathbf A^{-1}$ denotes the inverse of $\mathbf A$
Greedy Algorithm yields Maximal Set
Let $\struct{S,\mathscr F}$ be an independence system. Let $w : S \to \R_{\ge 0}$ be a weight function. Then the Greedy Algorithm selects a maximal set $A_0$ in $\mathscr F$.
Greedy Algorithm may not yield Maximum Weight
Let $\struct{S,\mathscr F}$ be an independence system. Let $w : S \to \R_{\ge 0}$ be a weight function. Then the maximal set $A_0 \in \mathscr F$ selected by the Greedy Algorithm may not have maximum weight.
Greedy Algorithm guarantees Maximum Weight iff Matroid
Let $S$ be a finite set. Let $\mathscr I$ be a non-empty set of subsets of $S$. Then $\mathscr I$ is the set of independent subsets of a matroid on $S$ : :$(1) \quad \struct{S, \mathscr I}$ is an independence system :$(2) \quad$ for all non-negative weight functions $w : S \to \R_{\ge 0}$, the Greedy Algorithm selects $A_w \in \mathscr I$: :::::$\forall B \in \mathscr I: \map {w^+} {A_w} \ge \map {w^+} B$ :where $w^+$ denotes the extended weight function of $w$.
Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication form Vector Space
Let $I := \closedint a b$ be a closed real interval. Let $\map \CC I$ be a space of continuous on closed interval real-valued functions. Let $\struct {\R, +_\R, \times_\R}$ be the field of real numbers. Let $\paren +$ be the pointwise addition of real-valued functions. Let $\paren {\, \cdot \,}$ be the pointwise scalar multiplication of real-valued functions. Then $\struct {\map \CC I, +, \, \cdot \,}_\R$ is a vector space.
Independent Subset is Contained in Base
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\mathscr B$ denote the set of all bases of $M$. Let $A \in \mathscr I$. Then: :$\exists B \in \mathscr B : A \subseteq B$
Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements/Corollary
Let $\struct{S, \preceq}$ be a finite ordered set. Let $x \in S$. Then there exists a maximal element $M \in S$ and a minimal element $m \in S$ such that: :$m \preceq x \preceq M$
Equivalent Conditions for Element is Loop
Let $M = \struct{S, \mathscr I}$ be a matroid. Let $\sigma$ denote the closure operator on $M$. Let $\rho$ denote the rank function of $M$. Let $\mathscr B$ denote the set of all bases of $M$. Let $x \in S$.  :$(1)\quad x$ is a loop :$(2)\quad x \in \map \sigma \O$  :$(3)\quad \map \rho {\set x} = 0$ :$(4)\quad \set x$ is a circuit :$(5)\quad x$ is not an element of any $B \in \mathscr B$
Power Set of Doubleton
Let $x, y$ be distinct objects. Then the power set of the doubleton $\set {x, y}$ is: :$\powerset {\set {x, y}} = \big \{ \O, \set x, \set y, \set {x,y} \big \}$
Doubleton of Elements is Subset
Let $S$ be a set. Let $\set {x,y}$ be the doubleton of distinct $x$ and $y$. Then: :$x, y \in S \iff \set {x,y} \subseteq S$
Sum of Unitary Divisors of Power of Prime
Let $n = p^k$ be the power of a prime number $p$. Then the sum of all positive unitary divisors of $n$ is $1 + n$.
Sum of Unitary Divisors is Multiplicative
Let $\map {\sigma^*} n$ denote the sum of unitary divisors of $n$. Then the function: :$\displaystyle \sigma^*: \Z_{>0} \to \Z_{>0}: \map {\sigma^*} n = \sum_{\substack d \mathop \divides n \\ d \mathop \perp \frac n d} d$ is multiplicative.
Sum of Unitary Divisors of Integer
Let $n$ be an integer such that $n \ge 2$. Let $\map {\sigma^*} n$ be the sum of all positive unitary divisors of $n$. Let the prime decomposition of $n$ be: :$\displaystyle n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} =  p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$ Then: :$\displaystyle \map {\sigma^*} n = \prod_{1 \mathop \le i \mathop \le r} \paren {1 + p_i^{k_i}}$
Range of Infinite Sequence may be Finite
Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence. Then it is possible for the range of $\sequence {x_n}$ to be finite.
Largest Number not Expressible as Sum of Multiples of Coprime Integers
Let $a, b$ be coprime integers, each greater than $1$. Then the largest number not expressible as a sum of multiples of $a$ and $b$ is the number: :$a b - a - b = \paren {a - 1} \paren {b - 1} - 1$
Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication form Vector Space
Let $I := \closedint a b$ be a closed real interval. Let $\map \CC I$ be a space of continuous on closed interval real-valued functions. Let $\map {\CC^1} I$ be a space of continuously differentiable functions on closed interval $I$. Let $\struct {\R, +_\R, \times_\R}$ be the field of real numbers. Let $\paren +$ be the pointwise addition of real-valued functions. Let $\paren {\, \cdot \,}$ be the pointwise scalar multiplication of real-valued functions. Then $\struct {\map {\CC^1} I, +, \, \cdot \,}_\R$ is a vector space.
Fermat Quotient of 2 wrt p is Square iff p is 3 or 7/Generalization
The Fermat quotient of $2$ with respect to $p$: :$\map {q_p} 2 = \dfrac {2^{p - 1} - 1} p$ is a perfect power  $p = 3$ or $p = 7$.
Taxicab Norm is Norm
The taxicab norm is a norm on the real and complex numbers.
Multiplication by 2 over 3 in Egyptian Fractions
Let $\dfrac 1 n$ be an Egyptian fraction not equal to $\dfrac 2 3$. In order to multiply $\dfrac 1 n$ by $\dfrac 2 3$ and have it that $\dfrac 1 n \times \dfrac 2 3$ is also expressed in Egyptian form, we have: :$\dfrac 1 n \times \dfrac 2 3 = \dfrac 1 {2 n} + \dfrac 1 {6 n}$
Proper Fraction can be Expressed as Finite Sum of Unit Fractions
Let $\dfrac p q$ denote a proper fraction expressed in canonical form. Then it is always possible to express $\dfrac p q$ as the sum of a finite number of distinct unit fractions: {{begin-eqn}} {{eqn | l = \dfrac p q       | r = \sum_{\substack {1 \mathop \le k \mathop \le m \\ n_j \mathop \le n_{j + 1} } } \dfrac 1 {n_k}       | c =  }} {{eqn | r = \dfrac 1 {n_1} + \dfrac 1 {n_2} + \dotsb + \dfrac 1 {n_m}       | c =  }} {{end-eqn}}
Upper Limit of Number of Unit Fractions to express Proper Fraction from Greedy Algorithm
Let $\dfrac p q$ denote a proper fraction expressed in canonical form. Let $\dfrac p q$ be expressed as the sum of a finite number of distinct unit fractions using Fibonacci's Greedy Algorithm. Then $\dfrac p q$ is expressed using no more than $p$ unit fractions.
Smallest n for which 3 over n produces 3 Egyptian Fractions using Greedy Algorithm when 2 Sufficient
Consider proper fractions of the form $\dfrac 3 n$ expressed in canonical form. Let Fibonacci's Greedy Algorithm be used to generate a sequence $S$ of Egyptian fractions for $\dfrac 3 n$. The smallest $n$ for which $S$ consists of $3$ terms, where $2$ would be sufficient, is $25$.
Supremum Norm is Norm/Continuous on Closed Interval Real-Valued Function
Let $I = \closedint a b$ be a closed interval. Let $\struct {\map \CC I, +, \, \cdot \,}_\R$ be the vector space of real-valued functions, continuous on $I$. Let $\map x t \in \map \CC I$ be a continuous real function. Let $\size {\, \cdot \,}$ be the absolute value. Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on real-valued functions, continuous on $I$. Then $\norm {\, \cdot \,}_\infty$ is a norm over $\struct {\map \CC I, +, \, \cdot \,}_\R$.
Union with Disjoint Singleton is Dependent if Element Depends on Subset
Let $M = \struct{S, \mathscr I}$ be a matroid. Let $A \subseteq S$. Let $x \in S : x \notin A$. If $x$ depends on $A$ then $A \cup \set x$ is dependent
Element Depends on Independent Set iff Union with Singleton is Dependent
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $X \in \mathscr I$. Let $x \in S : x \notin X$. Then: :$x \in \map \sigma X$  $X \cup \set x$ is dependent.
Element Depends on Independent Set iff Union with Singleton is Dependent/Lemma
Let $A \in \mathscr I$ such that $A \subseteq X \cup \set x$. Then: :$\size A \le \size X$
Rank of Independent Subset Equals Cardinality
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\rho : \powerset S \to \Z$ be the rank function of $M$. Let $X \in \mathscr I$ Then: :$\map \rho X = \size X$
Generating Function for Lucas Numbers
Let $\map G z$ be the function defined as: :$\map G z = \dfrac {2 - z} {1 - z - z^2}$ Then $\map G z$ is a generating function for the Lucas numbers.
492 Cubed is Sum of 3 Positive Cubes in 13 Ways
The cube of $492$ can be expressed as the sum of $3$ positive cubes in $13$ different ways: {{begin-eqn}} {{eqn | l = 492^3       | r = 24^3 + 204^3 + 480^3 }} {{eqn | r = 48^3 + 85^3 + 491^3 }} {{eqn | r = 72^3 + 384^3 + 396^3 }} {{eqn | r = 113^3 + 264^3 + 463^3 }} {{eqn | r = 114^3 + 360^3 + 414^3 }} {{eqn | r = 149^3 + 336^3 + 427^3 }} {{eqn | r = 176^3 + 204^3 + 472^3 }} {{eqn | r = 190^3 + 279^3 + 449^3 }} {{eqn | r = 207^3 + 297^3 + 438^3 }} {{eqn | r = 226^3 + 332^3 + 414^3 }} {{eqn | r = 243^3 + 358^3 + 389^3 }} {{eqn | r = 246^3 + 328^3 + 410^3 }} {{eqn | r = 281^3 + 322^3 + 399^3 }} {{end-eqn}}
Maximum Area of Isosceles Triangle
Consider two line segments $A$ and $B$ of equal length $a$ which are required to be the legs of an isosceles triangle $T$. Then the area of $T$ is greatest when the apex of $T$ is a right angle. The area of $T$ in this situation is equal to $\dfrac {a^2} 2$.
Inscribing Equilateral Triangle inside Square with a Coincident Vertex
Let $\Box ABCD$ be a square. It is required that $\triangle DGH$ be an equilateral triangle inscribed within $\Box ABCD$ such that vertex $D$ of $\triangle DGH$ coincides with vertex $D$ of $\Box ABCD$.
Construction of Perpendicular using Rusty Compass
Let $AB$ be a line segment. Using a straightedge and rusty compass, it is possible to construct a straight line at right angles to $AB$ from the endpoint $A$, without extending $AB$ past $A$.
Division of Straight Line into Equal Parts using Rusty Compass
Let $AB$ be a line segment. Using a straightedge and rusty compass, it is possible to divide $AB$ into as many equal parts as required.
Construction of Regular Pentagon using Rusty Compass
Using a straightedge and rusty compass, it is possible to inscribe a regular pentagon inside a circle.
Sum to Infinity of 2x^2n over n by 2n Choose n
For $\cmod x < 1$: :$\displaystyle \frac {2 x \arcsin x} {\sqrt {1 - x^2} } = \sum_{n \mathop = 1}^\infty \frac {\paren {2 x}^{2 n} } {n \dbinom {2 n} n}$
Definite Integral from 0 to Pi of Logarithm of a plus b Cosine x
:$\displaystyle \int_0^\pi \map \ln {a + b \cos x} \rd x = \pi \map \ln {\frac {a + \sqrt {a^2 - b^2} } 2}$
Arccosine in terms of Arctangent
:$\displaystyle \arccos x = 2 \map \arctan {\sqrt {\frac {1 - x} {1 + x} } }$
Definite Integral from 0 to Half Pi of Reciprocal of a plus b Cosine x
:$\displaystyle \int_0^{\pi/2} \frac 1 {a + b \cos x} \rd x = \frac 1 {\sqrt {a^2 - b^2} } \map \arccos {\frac b a}$
Definite Integral to Infinity of Exponential of -a x by Sine of b x over x
:$\displaystyle \int_0^\infty \frac {e^{-a x} \sin b x} x \rd x = \map \arctan {\frac b a}$
Definite Integral to Infinity of Sine of m x over Exponential of 2 Pi x minus One
:$\displaystyle \int_0^\infty \frac {\sin m x} {e^{2 \pi x} - 1} \rd x = \frac 1 4 \coth \frac m 2 - \frac 1 {2 m}$
Definite Integral to Infinity of Exponential of -a x^2 by Cosine of b x
:$\displaystyle \int_0^\infty e^{-a x^2} \cos b x \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-\frac {b^2} {4 a} }$
Fourier Series/Logarithm of Sine of x over 0 to Pi
:$\displaystyle \map \ln {\sin x} = -\ln 2 - \sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n$
Subset of Set Difference iff Disjoint Set
Let $S, T$ be sets. Let $A \subseteq S$ Then: :$A \cap T = \varnothing \iff A \subseteq S \setminus T$ where: :$A \cap T$ denotes set intersection :$\varnothing$ denotes the empty set :$S \setminus T$ denotes set difference.
Set Difference of Doubleton and Singleton is Singleton
Let $x, y$ be distinct objects. Then: :$\set{x, y} \setminus \set x = \set y$
Egyptian Formula for Area of Quadrilateral
Let $\Box ABCD$ be a quadrilateral. Let the sides of $\Box ABCD$ be $a$, $b$, $c$ and $d$ such that $a$ is opposite $c$ and $b$ is opposite $d$. Then the area of $\Box ABCD$ can be approximated by: :$\map \Area {\Box ABCD} \approx \dfrac {a + c} 2 \times \dfrac {b + d} 2$ The closer $\Box ABCD$ is to a rectangle, the better the approximation.
1 plus Perfect Power is not Power of 2
The equation: :$1 + a^n = 2^m$ has no solutions in the integers for $n, m > 1$. This is an elementary special case of Catalan's Conjecture.
Definite Integral from 0 to Half Pi of Logarithm of Sine x by Cosine of 2nx
For $n \in \N_{>0}$: :$\displaystyle \int_0^{\pi/2} \map \ln {\sin x} \cos 2 n x \ \d x = -\frac \pi {4 n}$
Distinct Matroid Elements are Parallel iff Each is in Closure of Other/Lemma
Let $a, b \in S$. Let $\set a$ and $\set b$ be independent. Then $\set {a, b}$ is dependent : :$a \in \map \sigma {\set b}$ and :$b \in \map \sigma {\set a}$
Definite Integral to Infinity of Sine m x over x by x Squared plus a Squared
:$\displaystyle \int_0^\infty \frac {\sin m x} {x \paren {x^2 + a^2} } \rd x = \frac \pi {2 a^2} \paren {1 - e^{-m a} }$
Definite Integral from 0 to Half Pi of Square of Logarithm of Sine x
:$\displaystyle \int_0^{\pi/2} \paren {\map \ln {\sin x} }^2 \rd x = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}$