hcju/proofwikips huggingface dataset:
Thirteen Catalan Polyhedra
There exist exactly $13$ distinct Catalan polyhedra: :Triakis tetrahedron :Triakis octahedron :Disdyakis dodecahedron :Tetrakis hexahedron :Triakis icosahedron :Disdyakis triacontahedron :Pentakis dodecahedron :Rhombic dodecahedron :Rhombic triacontahedron :Deltoidal icositetrahedron :Deltoidal hexecontahedron :Pentagonal icositetrahedron :Pentagonal hexecontahedron
Necessary and Sufficient Condition for Integral Parametric Functional to be Independent of Parametric Representation
Let $x: \R \to \R$ and $y: \R \to \R$ be real functions. Let $J \sqbrk {x, y}$ be a functional of the form $\displaystyle J \sqbrk {x, y} = \int_{t_0}^{t_1} \map \Phi {t, x, y, \dot x, \dot y} \rd t$ where $\dot y$ denotes the derivative of $y$ {{WRT|Differentiation}} $t$: :$\dot y = \dfrac {\d y} {\d t}$ Then $J \sqbrk {x, y}$ depends only on the curve in the $x y$-plane defined by the parametric equations: :$x = \map x t$, $y = \map y t$  and not on the choice of the parametric representation of the curve  the integrand $\Phi$ does not involve $t$ explicitly  and is a positive-homogeneous function of degree $1$ in $\dot x$ and $\dot y$.
Intersection of Upper Set with Directed Set is Directed Set
Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $A, B$ be subsets of $S$ such that :$A \cap B \ne \varnothing$ and :$A$ is an upper set, :$B$ is a directed set. Then $A \cap B$ is a directed set.
Convex Set is Contractible
Let $V$ be a topological vector space over $\R$ or $\C$. Let $A\subset V$ be a convex subset. Then $A$ is contractible.
Open Subgroup is Closed
Let $G$ be a topological group. Let $H\leq G$ be an open subgroup. Then $H$ is closed.
Topological Group is Hausdorff iff Identity is Closed
Let $G$ be a topological group. Let $e$ be its identity element. Then $G$ is Hausdorff  $\left\{ {e}\right\}$ is closed in $G$.
Jensen's Inequality (Complex Analysis)
Let $D \subset \C$ be an open set with $0 \in D$. Let $R > 0$ be such that $\map B {0, R} \subset D$. Let $f: D \to \C$ be analytic with $\map f 0 \ne 0$. Let $\cmod {\map f z} \le M$ for $\cmod z \le R$. Let $0 < r <R$. Then the number of zeroes of $f$, counted with multiplicity, for $\cmod z \le r$, is at most: :$\dfrac {\map \log {M / \cmod {\map f 0} } } {\map \log {R / r} }$
Complement of Subset with Property (S) is Closed under Directed Suprema
Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $X$ be a subset of $S$ with property (S). Then $\complement_S\left({X}\right)$ is closed under directed suprema.
Lifting The Exponent Lemma for Sums
Let $x, y \in \Z$ be integers with $x + y \ne 0$. Let $n \ge 1$ be an odd natural number. Let $p$ be an odd prime. Let: :$p \divides x + y$ and: :$p \nmid x y$ where $\divides$ and $\nmid$ denote divisibility and non-divisibility respectively. Then: :$\map {\nu_p} {x^n + y^n} = \map {\nu_p} {x + y} + \map {\nu_p} n$ where $\nu_p$ denotes $p$-adic valuation.
Lifting The Exponent Lemma for p=2
Let $x, y \in \Z$ be distinct odd integers. Let $n \ge 1$ be a natural number. Let: :$4 \divides x - y$ where $\divides$ denotes divisibility. Then :$\map {\nu_2} {x^n - y^n} = \map {\nu_2} {x - y} + \map {\nu_2} n$ where $\nu_2$ denotes $2$-adic valuation.
Classification of Convex Polyhedra whose Faces are Regular Polygons
The convex polyhedra whose faces are all regular polygons are as follows: :The $5$ Platonic solids :The regular prisms and regular antiprisms, countably infinite in number :The $13$ Archimedean polyhedra :The $92$ Johnson polyhedra.
Chinese Remainder Theorem/General Result 2
Let $n_1, n_2, \ldots, n_k$ be positive integers. Let $b_1, \ldots, b_k$ be integers such that: : $\forall i \ne j: \gcd \left({n_i, n_j}\right) \mathrel \backslash b_i - b_j$ where $\backslash$ denotes divisibility. Then the system of linear congruences: : $x \equiv b_1 \pmod {n_1}$ : $x \equiv b_2 \pmod {n_2}$ : $\ldots$ : $x \equiv b_k \pmod {n_k}$ has a simultaneous solution which is unique modulo $\lcm \left({n_1, \ldots, n_k}\right)$.
Necessary Condition for Integral Functional to have Extremum for given Function/Dependent on Nth Derivative of Function
Let $\map F {x, y, z_1, \ldots, z_n}$ be a function in differentiability class $C^2$  all its variables. Let $y = \map y x \in C^n\openint a b$ such that: :$\map y a = A_0, \map {y'} a = A_1, \ldots, \map {y^{\paren {n - 1} } } a = A_{n - 1}$ and: :$\map y b = B_0, \map {y'} b = B_1, \ldots, \map {y^{\paren {n - 1} } } b = B_{n - 1}$ Let $J \sqbrk y$ be a functional of the form: :$\displaystyle J \sqbrk y = \int_a^b \map F {x, y, y', \ldots, y^{\paren n} } \rd x$ Then a necessary condition for $J \sqbrk y$ to have an extremum (strong or weak) for a given function $\map y x$ is that $\map y x$ satisfy Euler's equation: :$F_y - \dfrac \d {\d x} F_{y'} + \dfrac {\d^2} {\d x^2} F_{y''}  -\cdots + \paren {-1}^n \dfrac {\d^n} {\d x^n} F_{y^{\paren n} } = 0$
Equivalence of Definitions of Absolute Convergence of Product of Complex Numbers
Let $\left\langle{a_n}\right\rangle$ be a sequence of complex numbers. Let $\log$ denote the complex logarithm. {{TFAE|def = Absolute Convergence of Product}}
Square of Quadratic Gauss Sum
Let $p$ be an odd prime. Let $a$ be an integer coprime to $p$. Let $g \left({a, p}\right)$ denote the quadratic Gauss sum of $a$ and $p$. Then: :${g \left({a, p}\right)}^2 = \left({\dfrac {-1} p}\right) \cdot p$
Absolutely Convergent Product is Convergent
Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field. Let $\mathbb K$ be complete. Let the infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + a_n}$ be absolutely convergent. Then it is convergent.
Simplest Variational Problem with Subsidiary Conditions
Let $J \sqbrk y$ and $K \sqbrk y$ be (real) functionals, such that :$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$ :$\ds K \sqbrk y = \int_a^b \map G {x, y, y'} \rd x = l$ where $l$ is a constant. Let $y = \map y x$ be an extremum of $F \sqbrk y$, and satisfy boundary conditions: :$\map y a = A$ :$\map y b = B$ Then, if $y = \map y x$ is not an extremal of $K \sqbrk y$, there exists a constant $\lambda$ such that $y = \map y x$ is an extremal of the functional: :$\ds \int_a^b \paren {F + \lambda G} \rd x$ or, in other words, $y = \map y x$ satisfies: :$F_y - \dfrac {\d} {\d x} F_{y'} + \lambda \paren {G_y - \dfrac {\d} {\d x} G_{y'} } = 0$
Complement of Closed under Directed Suprema Subset is Inaccessible by Directed Suprema
Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $X$ be a closed under directed suprema subset of $S$. Then $\complement_S\left({X}\right)$ is inaccessible by directed suprema.
Complement of Upper Set is Lower Set
Let $\left({S, \preceq}\right)$ be an ordered set. Let $L$ be an upper set. Then $S \setminus L$ is a lower set.
Closed Set iff Lower and Closed under Directed Suprema in Scott Topological Ordered Set
Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with Scott topology where $\left({S, \preceq}\right)$ is an up-complete ordered set. Let $A$ be a subset of $S$. Then $A$ is closed  $A$ is lower and closed under directed suprema.
Complement of Inaccessible by Directed Suprema Subset is Closed under Directed Suprema
Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $X$ be an inaccessible by directed suprema subset of $S$. Then $\complement_S\left({X}\right)$ is closed under directed suprema.
Convergence of Complex Sequence in Polar Form
Let $z \ne 0$ be a complex number with modulus $r$ and argument $\theta$. Let $\sequence {z_n}$ be a sequence of nonzero complex numbers. Let $r_n$ be the modulus of $z_n$ and $\theta_n$ be an argument of $z_n$. Then $z_n$ converges to $z$  the following hold: :$(1): \quad r_n$ converges to $r$ :$(2): \quad$ There exists a sequence $\sequence {k_n}$ of integers such that $\theta_n + 2 k_n \pi$ converges to $\theta$.
Complex Modulus of Sum of Complex Numbers
Let $z_1, z_2 \in \C$ be complex numbers. Let $\theta_1$ and $\theta_2$ be arguments of $z_1$ and $z_2$, respectively. Then: :$\cmod {z_1 + z_2}^2 = \cmod {z_1}^2 + \cmod {z_2}^2 + 2 \cmod {z_1} \cmod {z_2} \, \map \cos {\theta_1 - \theta_2}$
Complex Modulus of Difference of Complex Numbers
Let $z_1, z_2 \in \C$ be complex numbers. Let $\theta_1$ and $\theta_2$ be arguments of $z_1$ and $z_2$, respectively. Then: :$\cmod {z_1 - z_2}^2 = \cmod {z_1}^2 + \cmod {z_2}^2 - 2 \cmod {z_1} \cmod {z_2} \, \map \cos {\theta_1 - \theta_2}$
Uniform Absolute Convergence of Infinite Product of Complex Functions
Let $X$ be a compact metric space. Let $\left\langle{f_n}\right\rangle$ be a sequence of continuous functions $X \to \C$. Let $\displaystyle \sum_{n \mathop = 1}^\infty f_n$ converge uniformly absolutely on $X$. Then: :$f \left({x}\right) = \displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n \left({x}\right)}\right)$ converges uniformly absolutely on $X$ :$f$ is continuous :there exists $n_0 \in \N$ such that $\displaystyle \prod_{n \mathop = n_0}^\infty \left({1 + f_n \left({x}\right)}\right)$ has no zeroes.
Simplest Variational Problem with Subsidiary Conditions for Curve on Surface
Let $J \sqbrk {y, z}$ be a (real) functional of the form: :$\ds J \sqbrk y = \int_a^b \map F {x, y, z, y', z'} \rd x$ Let there exist admissible curves $y, z$ lying on the surface: :$\map g {x, y, z} = 0$ which satisfy boundary conditions: :$\map y a = A_1, \map y b = B_1$ :$\map z a = A_2, \map z b = B_2$ Let $J \sqbrk {y, z}$ have an extremum for the curve $y = \map y x, z = \map z x$. Let $g_y$ and $g_z$ not simultaneously vanish at any point of the surface $g = 0$. Then there exists a function $\map \lambda x$ such that the curve $y = \map y x, z = \map z x$ is an extremal of the functional: :$\ds \int_a^b \paren {F + \map \lambda x g} \rd x$ In other words, $y = \map y x$ satisfies the differential equations: {{begin-eqn}} {{eqn | l = F_y + \lambda g_y - \frac \d {\d x} F_{y'}       | r = 0 }} {{eqn | l = F_z + \lambda g_z - \frac \d {\d x} F_{z'}       | r = 0       | c =  }} {{end-eqn}}
Empty Intersection iff Subset of Relative Complement
Let $S$ be a set. Let $A, B$ be subset of $S$. Then $A \cap B = \varnothing \iff A \subseteq \complement_S\left({B}\right)$
Subset in Subsets
Let $S, B$ be sets. Let $A$ be subset of $S$. Then: :$A \subseteq B \iff \forall x \in S: x \in A \implies x \in B$
Empty Set as Subset
Let $S$ be a set. Let $A$ be a subset of $S$. Then: :$A = \O \iff \forall x \in S: x \notin A$
Pythagorean Triangles whose Areas are Repdigit Numbers
The following Pythagorean triangles have areas consisting of repdigit numbers:
Exponential of Series Equals Infinite Product
Let $\left\langle{z_n}\right\rangle$ be a sequence of complex numbers. Suppose $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ converges to $z \in \C$. Then $\displaystyle \prod_{n \mathop = 1}^\infty \exp \left({z_n}\right)$ converges to $\exp z$.
Absolute Value of Infinite Product
Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field. Let $\sequence{a_n}$ be a sequence in $\mathbb K$.
Convergence of Series of Complex Numbers by Real and Imaginary Part
Let $\sequence {z_n}$ be a sequence of complex numbers. Then: : the series $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ converges to $Z \in \C$ : : the series: ::$\displaystyle \sum_{n \mathop = 1}^\infty \Re \paren {z_n}$ :and: ::$\displaystyle \sum_{n \mathop = 1}^\infty \Im \paren {z_n}$ :converge to $\Re \paren Z$ and $\Im \paren Z$ respectively.
Lower Closure of Element is Closed under Directed Suprema
Let $L = \struct {S, \preceq}$ be an up-complete ordered set. Let $x \in S$. Then $x^\preceq$ is closed under directed suprema, where $x^\preceq$ denotes the lower closure of $x$.
Pythagorean Triangle whose Hypotenuse and Leg differ by 1
Let $P$ be a Pythagorean triangle whose sides correspond to the Pythagorean triple $T$. Then: : the hypotenuse of $P$ is $1$ greater than one of its legs : : the generator for $T$ is of the form $G = \tuple {n, n + 1}$ where $n \in \Z_{> 0}$ is a (strictly) positive integer.
General Variation of Integral Functional/Dependent on N Functions
Let $J$ be a (real) functional of the form $\ds J \sqbrk {\ldots y_i \ldots} = \int_{x_0}^{x_1} \map F {x, \ldots y_i \ldots, \ldots y_i' \ldots} \rd x, i = \openint 1 n$ Then: $\ds \delta J = \int_{x_0}^{x_1} \sum_{i \mathop = 1}^n \paren {F_{y_i} - \frac \d {\d x} F_{y_i'} } \map {h_i} x + \intlimits {\sum_{i \mathop = 1}^n F_{y_i'} \delta y_i} {x \mathop = x_0} {x \mathop = x_1} + \intlimits {\paren {F - \sum_{i \mathop = 1}^n y_i'F_{y_i'} } \delta x} {x \mathop = x_0} {x \mathop = x_1}$
Kusmin-Landau Inequality
Let $I$ be the half-open interval $\hointl a b$. Let $f: I \to R$ be continuously differentiable. Let $f'$ be monotonic. Let $\norm {f'} \ge \lambda$ on $I$ for some $\lambda \in \R_{>0}$, where $\norm {\, \cdot \,}$ denotes the distance to nearest integer. Then: :$\displaystyle \sum_{n \mathop \in I} e^{2 \pi i \map f n} = \map O {\frac 1 \lambda}$ where the big-O estimate does not depend on $f$.
Closure of Singleton is Lower Closure of Element in Scott Topological Lattice
Let $T = \struct {S, \preceq, \tau}$ be a up-complete topological lattice with Scott topology. Let $x \in S$. Then: :$\set x^- = x^\preceq$ where :$\set x^-$ denotes the topological closure of $\set x$ :$x^\preceq$ denotes the lower closure of $x$.
Pythagorean Triangle cannot be Isosceles
Let $P$ be a Pythagorean triangle. Then $P$ is not isosceles.
Brahmagupta-Fibonacci Identity/Corollary
:$\left({a^2 + b^2}\right) \left({c^2 + d^2}\right) = \left({a c - b d}\right)^2 + \left({a d + b c}\right)^2$
Scott Topological Lattice is T0 Space
Let $T = \left({S, \preceq, \tau}\right)$ be a complete topological lattice with Scott topology. Then $T$ is a $T_0$ space.
Lower Closures are Equal implies Elements are Equal
Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $x, y \in S$ such that :$x^\preceq = y^\preceq$ where $x^\preceq$ denotes the lower closure of $x$. Then $x = y$
General Variation of Integral Functional/Dependent on N Functions/Canonical Variables
Let $\delta J$ be a general variation of integral functional dependent on n functions. Suppose a following coordinate transformation is done: :$\set {x, \ldots y_i, \ldots, \ldots, y_i', \ldots, F} \to \set {x, \ldots, y_i, \ldots, \ldots p', \ldots, H}, i = \tuple {1, \ldots, n}$ Then, in canonical variables: :$\displaystyle \delta J = \int_{x_0}^{x_1} \sum_{i \mathop = 1}^n \paren {F_{y_i} - \dfrac {\d {p_i} } {\d x} } \map {h_i} x \rd x + \intlimits {\sum_{i \mathop = 1}^n p_i \delta y_i - H \delta x} {x \mathop = x_0} {x \mathop = x_1}$ where: {{begin-eqn}} {{eqn | l = \bigvalueat {\delta x} {x \mathop = x_j}       | r = \delta x_j }} {{eqn | l = \bigvalueat {\delta y_i} {x \mathop = x_j}       | r = \delta_i^j }} {{eqn | l = j       | r = \tuple {0, 1} }} {{end-eqn}} {{explain|the meaning of $j {{=}} \tuple {0, 1}$ in this context, and hence the meaning of $x_j$ and $\delta x_j$, and also $\delta_i^j$, all of which are obscure}}
Square of Pythagorean Prime is Hypotenuse of Pythagorean Triangle
Let $p$ be a Pythagorean prime. Then $p^2$ is the hypotenuse of a Pythagorean triangle.
Square of Hypotenuse of Pythagorean Triangle is Difference of two Cubes/Refutation
Let $h$ be the hypotenuse of a Pythagorean triangle. Then it is not necessarily the case that: :$h^2 = a^3 - b^3$ for some $a, b \in \Z_{>0}$.
Pythagorean Triangle from Sum of Reciprocals of Consecutive Same Parity Integers
Let $a, b \in \Z_{>0}$ be (strictly) positive integers such that they are consecutively of the same parity. Let $\dfrac p q = \dfrac 1 a + \dfrac 1 b$. Then $p$ and $q$ are the legs of a Pythagorean triangle.
Rational Number plus Irrational Number is Irrational
Rational number plus irrational number is irrational. That is, let $x \in \Q$, $y \in \R \setminus \Q$ and $x + y = z$. Then $z \in \R \setminus \Q$.
Lower Closure of Element is Topologically Closed in Scott Topological Ordered Set
Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with Scott topology where $\left({S, \preceq}\right)$ is an up-complete ordered set. Let $x \in S$. Then $x^\preceq$ is topologically closed, where $x^\preceq$ denotes the lower closure of $x$.
Complement of Lower Closure of Element is Open in Scott Topological Ordered Set
Let $T = \struct {S, \preceq, \tau}$ be a relational structure with Scott topology where $\struct {S, \preceq}$ is an up-complete ordered set. Let $x \in S$. Then $\relcomp S {x^\preceq}$ is topologically open, where :$x^\preceq$ denotes the lower closure of $x$, :$\relcomp S {x^\preceq}$ denotes the relative complement of $x^\preceq$.
Open iff Upper and with Property (S) in Scott Topological Lattice
Let $T = \left({S, \preceq, \tau}\right)$ be an up-complete topological lattice. Let $A$ be a subset of $S$. Then $A$ is open  $A$ is upper and with property (S).
Necessary Condition for Integral Functional to have Extremum for given Function/Non-differentiable at Intermediate Point
Let $y, F$ be real functions. Let $y$ be continuously differentiable for $x \in \hointr a c \cap  \hointl c b$ and satisfy: :$\map y a = A$ :$\map y b = B$ Let $J\sqbrk y$ be a functional of the form :$\ds J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$ Then the functional $J$ has a weak extremum if $y$ satisfies the following system of equations: {{begin-eqn}} {{eqn | l = F_y - \dfrac \d {\d x} F_{y'}       | r = 0 }} {{eqn | l = \bigvalueat {F_{y'} } {x \mathop = c \mathop - 0}       | r = \bigvalueat {F_{y'} } {x \mathop = c \mathop + 0} }} {{eqn | l = \bigvalueat {\paren {F - y' F_{y'} } } {x \mathop = c \mathop - 0}       | r = \bigvalueat {\paren {F - y' F_{y'} } } {x \mathop = c \mathop + 0} }} {{end-eqn}} where, by the use of limit from the left and limit from the right, the following abbreviations are denoted as follows: :$\ds \bigvalueat {\map y x} {x \mathop = c \mathop + 0} = \lim_{x \mathop \to c^+} \map y x$ :$\ds \bigvalueat {\map y x} {x \to x \mathop = c \mathop - 0} = \lim_{x \mathop \to c^-} \map y x$ The last two equations are known as the '''Weierstrass-Erdmann corner conditions'''. {{refactor|move this definition to a separate page once a general principle can be distilled}}
Euler's Equation for Vanishing Variation in Canonical Variables
{{refactor|There are a number of pages linking here with the presentation of the link set as "momenta". This is going to need a definition of its own, but it is not clear what that is from looking at this page.}} Consider the following system of differential equations: :$(1): \quad \begin {cases} F_{y_i} - \dfrac \d {\d x} F_{y_i'} = 0 \\ \dfrac {\d {y_i} } {\d x} = y_i'\end{cases}$ where $i \in \set {1, \ldots, n}$. Let the coordinates $\tuple {x, \family {y_i}_{1 \mathop \le i \mathop \le n}, \family {y_i'}_{1 \mathop \le i \mathop \le n}, F}$ be transformed to canonical variables: :$\tuple {x, \family {y_i}_{1 \mathop \le i \mathop \le n}, \family {p_i}_{1 \mathop \le i \mathop \le n}, H}$ Then the system $(1)$ is transformed into: :$\begin {cases} \dfrac {\d y_i} {\d x} = \dfrac {\partial H} {\partial p_i} \\ \dfrac {\d p_i} {\d x} = -\dfrac {\partial H} {\partial y_i} \end {cases}$
Relational Structure with Topology of Subsets with Property (S) is Topological Space
Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with topology where :$\left({S, \preceq}\right)$ is an up-complete ordered set, :$\tau$ is the set of all subsets of $S$ with property (S). Then $\left({S, \tau}\right)$ is topological space.
Subgroup of Real Numbers is Discrete or Dense
Let $G$ be a subgroup of the additive group of real numbers. Then one of the following holds: :$G$ is dense in $\R$. :$G$ is discrete and there exists $a \in \R$ such that $G = a \Z$, that is, $G$ is cyclic.
Discrete Subgroup of Real Numbers is Closed
Let $G$ be a subgroup of the additive group of real numbers. Let $G$ be discrete. Then $G$ is closed.
Consecutive Integers with Same Sigma
Let $\sigma: \Z_{>0} \to \Z_{>0}$ be the $\sigma$ function, defined on the strictly positive integers. The equation: :$\map \sigma n = \map \sigma {n + 1}$ is satisfied by integers in the sequence: :$14, 206, 957, 1334, 1364, 1634, 2685, 2974, 4364, 14841, 18873, \ldots$ {{OEIS|A002961}}
Coarser Between Generator Set and Filter is Generator Set of Filter
Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $F$ be a filter on $L$. Let $G$ be a generator set of $F$. Let $A$ be a subset of $S$ such that :$G$ is coarser than $A$ and $A$ is coarser than $F$. Then $A$ is generator set of $F$.
Borsuk-Ulam Theorem
Let $n$ be a positive integer. Let $f: \mathbb S^n \to \R^n$ be a continuous mapping from an $n$-sphere to $\R^n$. Then there exists $x \in \mathbb S^n$ such that $\map f x = \map f {-x}$.
Gershgorin Circle Theorem
Let $n$ be a positive integer. Let $A = \sqbrk {a_{i j} }$ be a complex square matrix of order $n$. Let $\lambda$ be an eigenvalue of $A$. Then there exists $i \in \set {1, 2, \ldots, n}$ such that: :$\lambda \in \map {\mathbb D} {a_{i i}, R_i}$ where: :$\displaystyle R_i = \sum_{j \mathop \ne i} \cmod {a_{ i j} }$ :$\map {\mathbb D} {a, R}$ denotes the complex disk of center $a$ and radius $R$.
Finite Infima Set of Coarser Subset is Coarser than Finite Infima Set
Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $A, B$ be subsets of $S$ such that :$A$ is coarser than $B$. Then $\operatorname{fininfs} \left({A}\right)$ is coarser than $\operatorname{fininfs} \left({B}\right)$ where $\operatorname{fininfs} \left({B}\right)$ denotes the finite infima set of $B$.
Approximations to Equilateral Triangles by Heronian Triangles
The sequence of best approximations to an equilateral triangle by a Heronian triangle begins: :The $\paren {3, 4, 5}$ triangle, with area $6$ :The $\paren {13, 14, 15}$ triangle, with area $84$, where $14 = 4^2 - 2$ :The $\paren {193, 194, 195}$ triangle, where $194 = 14^2 - 2$ :The $\paren {37 \, 633, 37 \, 634, 37 \, 635}$ triangle, where $37 \, 634 = 194^2 - 2$ and so on. {{OEIS|A003010}}
Legendre Transform is Involution
The Legendre transform is an Involution.
Upper Closure of Coarser Subset is Subset of Upper Closure
Let $L = \left({S, \preceq}\right)$ be a preordered set. Let $A, B$ be subsets of $S$ such that :$A$ is coarser than $B$. Then $A^\succeq \subseteq B^\succeq$
Set Coarser than Upper Set is Subset
Let $\left({S, \preceq}\right)$ be a preordered set. Let $A, B$ be subsets of $S$ such that :$A$ is coarser than $B$ and :$B$ is an upper set. Then $A \subseteq B$
Sum of 4 Unit Fractions that equals 1
There are $14$ ways to represent $1$ as the sum of exactly $4$ unit fractions.
Sum of 5 Unit Fractions that equals 1
There are $147$ ways to represent $1$ as the sum of exactly $5$ unit fractions.
Sum of 6 Unit Fractions that equals 1
There are $3462$ ways to represent $1$ as the sum of exactly $6$ unit fractions.
Conditions for Function to be Maximum of its Legendre Transform Two-variable Equivalent
Let $x, p \in \R$. Let $\map f x$ be a strictly convex real function. Let $f^*$ be a Legendre transformed $f$. Let $\map g {x, p} = - \map {f^*} p + x p$ Then: :$\displaystyle \map f x = \max_p \paren {-\map {f^*} p + x p}$ where $\displaystyle \max_p$ maximises the function with respect to a variable $p$.
Set is Coarser than Image of Mapping of Infima
Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $f, g:\N \to S$ be mappings such that :$\forall n \in \N: g\left({n}\right) = \inf \left\{ {f\left({m}\right): m \in \N \land m \le n}\right\}$ Then $f\left[{\N}\right]$ is coarser than $g\left[{\N}\right]$ where $f\left[{\N}\right]$ denotes the image of mapping $f$.
Solutions of Ramanujan-Nagell Equation
Solutions to the Ramanujan-Nagell equation: :$x^2 + 7 = 2^n$ exist for only $5$ values of $n$: :$3, 4, 5, 7, 15$ {{OEIS|A060728}} The corresponding values of $x$ are: :$1, 3, 5, 11, 181$ {{OEIS|A038198}}
Five Ramanujan-Nagell Numbers
There exist exactly $5$ Ramanujan-Nagell numbers: positive integers of the form $2^m - 1$ which are triangular: :$0, 1, 3, 15, 4095$ {{OEIS|A076046}}