Power of One plus x in terms of Gaussian Hypergeometric Function

:$\displaystyle {}_2 \map {F_1} {-p, 1; 1; -x} = \paren {1 + x}^p$


Laplace Transform of Exponential times Sine

:$\map {\laptrans {e^{b t} \sin a t} } s = \dfrac a {\paren {s - b}^2 + a^2}$


Inverse Image of Direct Image of Inverse Image equals Inverse Image Mapping

Let $f: S \to T$ be a mapping. Let: :$f^\to: \powerset S \to \powerset T$ denote the direct image mapping of $f$ :$f^\gets: \powerset T \to \powerset S$ denote the inverse image mapping of $f$ where $\powerset S$ denotes the power set of $S$. Then: :$f^\gets \circ f^\to \circ f^\gets = f^\gets$ where $\circ$ denotes composition of mappings.


Graph of Real Function in Cartesian Plane intersects Vertical at One Point

Let $f: \R \to \R$ be a real function. Let its graph be embedded in the Cartesian plane $\mathcal C$: :520px Every vertical line through a point $a$ in the domain of $f$ intersects the graph of $f$ at exactly one point $P = \tuple {a, \map f a}$.


Equation of Vertical Line

Let $\mathcal L$ be a vertical line embedded in the Cartesian plane $\mathcal C$. Then the equation of $\mathcal L$ can be given by: :$x = a$ where $\tuple {a, 0}$ is the point at which $\mathcal L$ intersects the $x$-axis. :520px


Equation of Horizontal Line

Let $\mathcal L$ be a horizontal line embedded in the Cartesian plane $\mathcal C$. Then the equation of $\mathcal L$ can be given by: :$y = b$ where $\tuple {0, b}$ is the point at which $\mathcal L$ intersects the $y$-axis. :520px


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

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 $R^S$. Let $\Img f \subseteq U_R$ where $\Img f$ is the image of $f$. Then: :$f \in R^S$ is a unit of $R^S$ and the inverse of $f$ is the mapping $f^{-1} : S \to U_R$ defined by: :$\forall x \in S : \map {f^{-1}} {x} = \map f x^{-1}$


Unit of Ring of Mappings iff Image is Subset of Ring Units/Unit of Ring of Mappings implies 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 $R^S$. Let $f \in R^S$ be a unit of $R^S$. Then: :$\Img f \subseteq U_R$ where $\Img f$ is the image of $f$. In which case, the inverse of $f$ is the mapping $f^{-1} : S \to U_R$ defined by: :$\forall x \in S : \map {f^{-1} } x = \map f x^{-1}$


Graph of Real Surjection in Coordinate Plane intersects Every Horizontal Line

Let $f: \R \to \R$ be a real function which is surjective. Let its graph be embedded in the Cartesian plane $\mathcal C$: :520px Every horizontal line through a point $b$ in the codomain of $f$ intersects the graph of $f$ on at least one point $P = \tuple {a, b}$ where $b = \map f a$.


Graph of Real Injection in Coordinate Plane intersects Horizontal Line at most Once

Let $f: \R \to \R$ be a real function which is injective. Let its graph be embedded in the Cartesian plane $\mathcal C$: :520px Let $\mathcal L$ be a horizontal line through a point $b$ in the codomain of $f$. Then $\mathcal L$ intersects the graph of $f$ on at most one point $P = \tuple {a, b}$ where $b = \map f a$.


Graph of Real Bijection in Coordinate Plane intersects Horizontal Line at One Point

Let $f: \R \to \R$ be a real function which is bijective. Let its graph be embedded in the Cartesian plane $\mathcal C$: :520px Every horizontal line through a point $b$ in the codomain of $f$ intersects the graph of $f$ on exactly one point $P = \tuple {a, b}$ where $b = \map f a$.


Composition of 3 Mappings where Pairs of Mappings are Bijections

Let $A$, $B$, $C$ and $D$ be sets. Let: :$f: A \to B$ :$g: B \to C$ :$h: C \to D$ be mappings. Let $g \circ f$ and $h \circ g$ be bijections. Then $f$, $g$ and $h$ are all bijections.


Mapping Composed with Bijection forming Bijection is Bijection

Let $A$, $B$ and $C$ be sets. Let $f: A \to B$ and $g: B \to C$ be mappings. Let the composite mapping $g \circ f$ be a bijection. Let either $f$ or $g$ be a bijection. Then both $f$ and $g$ are bijections.


Composite of Three Mappings in Cycle forming Injections and Surjection

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: :$h \circ g \circ f$ is an injection :$f \circ h \circ g$ is an injection :$g \circ f \circ h$ is a surjection. where: :$g \circ f$ (and so on) denote composition of mappings. Then each of $f$, $g$ and $h$ are bijections.


Component Mappings of Set Coproduct are Injective

Let $S_1$ and $S_2$ be sets. Let $\struct {C, i_1, i_2}$ be a coproduct of $S_1$ and $S_2$. Then $i_1$ and $i_2$ are injections.


Existence of Bijection between Coproducts of two Sets

Let $S_1$ and $S_2$ be sets. Let $\struct {C, i_1, i_2}$ and $\struct {D, j_1, j_2}$ be two coproducts on $S_1$ and $S_2$. Then there exists a unique bijection $\theta: D \to C$ such that: :$\theta \circ j_i = i_1$ :$\theta \circ j_2 = i_2$


Number of Friday 13ths in a Year

In any given year, there are between $1$ and $3$ (inclusive) months in which the $13$th falls on a Friday.


Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Equivalence Class Contains 1 Odd Number

Let $\eqclass n \alpha$ be the $\alpha$-equivalence class of a natural number $n$. Then $\eqclass n \alpha$ contains exactly $1$ odd number.


Equivalence Relation on Natural Numbers such that Quotient is Power of Two/One of Pair of Equivalent Elements is Divisor of the Other

Let $c, d \in \N$ such that $c \mathrel \alpha d$. Then either: :$c \divides d$ or: :$d \divides c$ where $\divides$ denotes divisibility.


Exists Divisor in Set of n+1 Natural Numbers no greater than 2n

Let $S$ be a set of $n + 1$ non-non-zero natural numbers all less than or equal to $2 n$. Then there exists $a, b \in S$ such that :$a \divides b$ where $\divides$ denotes divisibility.


Equality of Squares Modulo Integer is Equivalence Relation

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\RR_n$ be the relation on the set of integers $\Z$ defined as: :$\forall x, y \in \Z: x \mathrel {\RR_n} y \iff x^2 \equiv y^2 \pmod n$ Then $\RR_n$ is an equivalence relation.


Left Ideal is Left Module over Ring

Let $\struct {R, +, \times}$ be a ring. Let $J \subseteq R$ be a left ideal of $R$. Let $\circ : R \times J \to J$ be the restriction of $\times$ to $R \times J$. Then $\struct {J, +, \circ}$ is a left module over $\struct {R, +, \times}$.


Right Ideal is Right Module over Ring

Let $\struct {R, +, \times}$ be a ring. Let $J \subseteq R$ be a right ideal of $R$. Let $\circ : J \times R \to J$ be the restriction of $\times$ to $J \times R$. Then $\struct {J, +, \circ}$ is a right module over $\struct {R, +, \times}$.


Opposite Ring is Ring

Let $\struct {R, +, \times}$ be a ring. Let $\struct {R, +, *}$ be the opposite ring of $\struct {R, +, \times}$. Then $\struct {R, +, *}$ is a ring.


Opposite Ring of Opposite Ring

Let $\struct {R, +, \times}$ be a ring. Let $\struct {R, +, *}$ be the opposite ring of $\struct {R, +, \times}$. Let $\struct {R, +, \circ}$ be the opposite ring of $\struct {R, +, *}$. Then $\struct {R, +, \circ} = \struct {R, +, \times}$.


Left Module over Ring Induces Right Module over Opposite Ring

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {R, +_R, *_R}$ be the opposite ring of $\struct {R, +_R, \times_R}$. Let $\struct{G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$. Let $\circ’ : G \times R \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: x \circ’ \lambda = \lambda \circ x$ Then $\struct{G, +_G, \circ’}$ is a right module over $\struct {R, +_R, *_R}$.


Right Module over Ring Induces Left Module over Opposite Ring

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {R, +_R, *_R}$ be the opposite ring of $\struct {R, +_R, \times_R}$. Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$. Let $\circ’ : R \times G \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: \lambda \circ’ x = x \circ \lambda $ Then $\struct{G, +_G, \circ’}$ is a left module over $\struct {R, +_R, *_R}$.


Ring is Commutative iff Opposite Ring is Itself

Let $\struct {R, +, \times}$ be a ring. Let $\struct {R, +, *}$ be the opposite ring of $\struct {R, +, \times}$. Then $\struct {R, +, \times}$ is a commutative ring {{iff}}: :$\struct {R, +, \times} = \struct {R, +, *}$


Left Module induces Right Module over same Ring iff Actions are Commutative

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$. Let $\circ': G \times R \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: x \circ' \lambda = \lambda \circ x$ Then $\struct {G, +_G, \circ'}$ is a right module over $\struct {R, +_R, \times_R}$ {{iff}}: :$\forall \lambda, \mu \in R: \forall x \in G: \paren {\lambda \times_R \mu} \circ x = \paren {\mu \times_R \lambda} \circ x$


Right Module induces Left Module over same Ring iff Actions are Commutative

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$. Let $\circ' : R \times G \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: \lambda \circ' x = x \circ \lambda $ Then $\struct {G, +_G, \circ'}$ is a left module over $\struct {R, +_R, \times_R}$ {{iff}}: :$\forall \lambda, \mu \in R: \forall x \in G: x \circ \paren{ \lambda \times_R \mu} = x \circ \paren {\mu \times_R \lambda}$


Euler's Theorem for Planar Graphs

Let $G = \struct {V, E}$ be a connected planar graph with $V$ vertices and $E$ edges. Let $F$ be the number of faces of $G$. Then: :$V - E + F = 2$


Fermat Problem

Let $\triangle ABC$ be a triangle Let the vertices of $\triangle ABC$ all have angles less than $120 \degrees$. Let $\triangle ABG$, $\triangle BCE$ and $\triangle ACF$ be equilateral triangles constructed on the sides of $ABC$. Let $AE$, $BF$ and $CG$ be constructed. Let $P$ be the point at which $AE$, $BF$ and $CG$ meet. :500px Then $P$ is the Fermat-Torricelli point of $\triangle ABC$. If one of vertices of $\triangle ABC$ be of $120 \degrees$ or more, then that vertex is itself the Fermat-Torricelli point of $\triangle ABC$.


Left Module over Commutative Ring induces Right Module

Let $\struct {R, +_R, \times_R}$ be a commutative ring. Let $\struct{G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$. Let $\circ’ : G \times R \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: x \circ’ \lambda = \lambda \circ x$ Then $\struct{G, +_G, \circ’}$ is a right module over $\struct {R, +_R, \times_R}$.


Right Module over Commutative Ring induces Left Module

Let $\struct {R, +_R, \times_R}$ be a commutative ring. Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$. Let $\circ' : R \times G \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: \lambda \circ’ x = x \circ \lambda$ Then $\struct{G, +_G, \circ'}$ is a left module over $\struct {R, +_R, \times_R}$.


Right Ideal is Right Module over Ring/Ring is Right Module over Ring

Let $\struct {R, +, \times}$ be a ring. Then $\struct {R, +, \times}$ is a right module over $\struct {R, +, \times}$.


Right Module Does Not Necessarily Induce Left Module over Ring

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$. Let $\circ' : R \times G \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: \lambda \circ' x = x \circ \lambda$ Then $\struct{G, +_G, \circ'}$ is not necessarily a left module over $\struct {R, +_R, \times_R}$


Left Module Does Not Necessarily Induce Right Module over Ring/Lemma

:$G$ is a left ideal of $\struct {\map {\MM_S} 2, +, \times}$.


Right Module Does Not Necessarily Induce Left Module over Ring/Lemma

:$G$ is a right ideal of $\struct {\map {\MM_S} 2, +, \times}$.


Ideal is Bimodule over Ring

Let $\struct {R, +, \times}$ be a ring. Let $J \subseteq R$ be an ideal of $R$. Let $\circ_l : R \times J \to J$ be the restriction of $\times$ to $R \times J$. Let $\circ_r : J \times R \to J$ be the restriction of $\times$ to $J \times R$. Then $\struct {J, +, \circ_l, \circ_r}$ is a bimodule over $\struct {R, +, \times}$.


Ideal is Bimodule over Ring/Ring is Bimodule over Ring

Let $\struct {R, +, \times}$ be a ring. Then $\struct {R, +, \times, \times}$ is a bimodule over $\struct {R, +, \times}$.


Left Module over Commutative Ring induces Bimodule

Let $\struct {R, +_R, \times_R}$ be a commutative ring. Let $\struct{G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$. Let $\circ’ : G \times R \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: x \circ’ \lambda = \lambda \circ x$ Then $\struct{G, +_G, \circ, \circ’}$ is a bimodule over $\struct {R, +_R, \times_R}$.


Right Module over Commutative Ring induces Bimodule

Let $\struct {R, +_R, \times_R}$ be a commutative ring. Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$. Let $\circ’ : R \times G \to G$ be the binary operation defined by: :$\forall \lambda \in R: \forall x \in G: \lambda \circ’ x = x \circ \lambda $ Then $\struct{G, +_G, \circ’, \circ}$ is a bimodule over $\struct {R, +_R, \times_R}$.


Diagonals of Kite are Perpendicular

Let $ABCD$ be a kite such that $AC$ and $BD$ are its diagonals. Then $AC$ and $BD$ are perpendicular.


One Diagonal of Kite Bisects the Other

Let $ABCD$ be a kite such that: :$AC$ and $BD$ are its diagonals :$AB = BC$ :$AD = DC$ Then $BD$ is the perpendicular bisector of $AC$.


Theorem of Even Perfect Numbers/Historical Note

The first part of this proof of the '''Theorem of Even Perfect Numbers''' was documented by {{AuthorRef|Euclid}} in {{BookLink|The Elements|Euclid}}: {{EuclidPropLink|book = IX|prop = 36|title = Theorem of Even Perfect Numbers/Sufficient Condition}}. The second part was achieved by {{AuthorRef|Leonhard Paul Euler|Euler}}.


Let $\struct{S, \tau_{_S}}$ be a topological space. Let $\struct{G, *, \tau_{_G}}$ be a topological group. Let $\lambda \in G$. Let $f,g : \struct{S, \tau_{_S}} \to \struct{G, \tau_{_G}}$ be continuous mappings. Then the following results hold: === Product Rule === {{:Combination Theorem for Continuous Mappings/Topological Group/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Topological Group/Multiple Rule}} === Inverse Rule === {{:Combination Theorem for Continuous Mappings/Topological Group/Inverse Rule}}


Combination Theorem for Continuous Mappings/Topological Group/Product Rule

:$f * g : \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.


Combination Theorem for Continuous Mappings/Topological Group/Multiple Rule

:$\lambda * f: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping :$f * \lambda: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.


Combination Theorem for Continuous Mappings/Topological Group/Inverse Rule

:$g^{-1}: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.


Continuous Mapping to Topological Product/Corollary

Let $T = T_1 \times T_2$ be a product space of two topological spaces $T_1$ and $T_2$. Let $T'$ be a topological space. Let $f: T' \to T_1$ be a mapping. Let $g: T' \to T_2$ be a mapping. Let $f \times g : T’ \to T$ be the mapping defined by: :$\forall x \in T’ : \map {\paren {f \times g}} x = \tuple{ \map f x, \map g x}$ Then $f \times g$ is continuous {{iff}} $f$ and $g$ are continuous.


Let $\struct{S, \tau_{_S}}$ be a topological space. Let $\struct{G, *, \tau_{_G}}$ be a topological semigroup. Let $\lambda \in G$. Let $f,g : \struct{S, \tau_{_S}} \to \struct{G, \tau_{_G}}$ be continuous mappings. Then the following results hold: === Product Rule === {{:Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Topological Semigroup/Multiple Rule}}


Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule

:$f * g: \struct{S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.


Combination Theorem for Continuous Mappings/Topological Semigroup/Multiple Rule


Volume of Solid of Revolution/Parametric Form

Let $x: \R \to \R$ and $y: \R \to \R$ be real functions defined on the interval $\closedint a b$. Let $y$ be integrable on the (closed) interval $\closedint a b$. Let $x$ be differentiable on the (open) interval $\openint a b$. Let the points be defined: :$A = \tuple {\map x a, \map y a}$ :$B = \tuple {\map x b, \map y b}$ :$C = \tuple {\map x b, 0}$ :$D = \tuple {\map x a, 0}$ Let the figure $ABCD$ be defined as being bounded by the straight lines $y = 0$, $x = a$, $x = b$ and the curve defined by: :$\set {\tuple {\map x t, \map y t}: a \le t \le b}$ Let the solid of revolution $S$ be generated by rotating $ABCD$ around the $x$-axis (that is, $y = 0$). Then the volume $V$ of $S$ is given by: :$\displaystyle V = \pi \int_a^b \paren {\map y t}^2 \map {x'} t \rd t$


Let $\struct{S, \tau_{_S}}$ be a topological space. Let $\struct{R, +, *, \tau_{_R}}$ be a topological ring. Let $\lambda \in R$. Let $f,g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ be continuous mappings. Then the following results hold: === Sum Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Sum Rule}} === Translation Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Translation Rule}} === Negation Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Negation Rule}} === Product Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Topological Ring/Multiple Rule}}


Let $\struct{S, \tau_{_S}}$ be a topological space. Let $\struct{R, +, *, \tau_{_R}}$ be a topological division ring. 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/Topological Division Ring/Sum Rule}} === Translation Rule === {{:Combination Theorem for Continuous Mappings/Topological Division Ring/Translation Rule}} === Negation Rule === {{:Combination Theorem for Continuous Mappings/Topological  Division Ring/Negation Rule}} === Product Rule === {{:Combination Theorem for Continuous Mappings/Topological Division Ring/Product Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Mappings/Topological  Division Ring/Multiple Rule}} === Inverse Rule === {{:Combination Theorem for Continuous Mappings/Topological Division Ring/Inverse Rule}}


Weight of Body at Earth's Surface

Let $B$ be a body of mass $m$ situated at (or near) the surface of Earth. Then the weight of $B$ is given by: :$W = m g$ where $g$ is the value of the acceleration due to gravity at the surface of Earth.


Combination Theorem for Continuous Mappings/Topological Ring/Sum Rule

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


Combination Theorem for Continuous Mappings/Topological Ring/Negation Rule

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


Combination Theorem for Continuous Mappings/Topological 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/Topological Ring/Product Rule

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


Combination Theorem for Continuous Mappings/Topological Ring/Combined Rule

:$\lambda * f + \mu * g: \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is a continuous mapping :$f * \lambda + g * \mu: \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is a continuous mapping.


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


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


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


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

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


Combination Theorem for Continuous Mappings/Topological Division Ring/Negation Rule

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


Combination Theorem for Continuous Mappings/Topological Ring/Translation Rule

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


Combination Theorem for Continuous Mappings/Topological Division Ring/Translation Rule


Pointwise Operation is Composite of Operation with Mapping to Cartesian Product

Let $S$ be a set. Let $\struct {T, *}$ be an algebraic structure. Let $T^S$ be the set of all mappings from $S$ to $T$. Let the algebraic structure $\struct {T^S, \oplus}$ be the algebraic structure on $T^S$ induced by $*$. Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings. Let $f \times g : S \to T \times T$ be the mapping from $S$ to the cartesian product $T \times T$ Defined by: :$\forall x \in S : \map {\paren {f \times g}} x = \tuple {\map f x, \map g x}$ Then: :$f \oplus g = * \circ \paren {f \times g}$ That is, $f \oplus g$ is the composition of the binary operation $*$ with the mapping $f \times g : S \to T \times T$.


Integral Representation of Bessel Function of the First Kind

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.


Integral Representation of Bessel Function of the First Kind/Integer Order

Let $n \in \Z$ be an integer. Then: :$\displaystyle \map {J_n} x = \dfrac 1 \pi \int_0^\pi \map \cos {n \theta - x \sin \theta} \rd \theta$


Integral Representation of Bessel Function of the First Kind/Non-Integer Order

Let $n \in \Z$ be an integer. Then: :$\displaystyle \map {J_n} x = \dfrac {x^n} {2^n \sqrt \pi \map \Gamma {n + \frac 1 2} } \int_0^\pi \map \cos {x \sin \theta} \cos^{2 n} \theta \rd \theta$


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


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

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


Legendre Transform of Strictly Convex Real Function is Strictly Convex

Let $\map f x$ be a strictly convex real function. Then the function $\map {f^*} p$ acquired through the Legendre Transform is also strictly convex.


Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 1

:$\forall y \in R: \lambda_y = y * I_{_R} \text { and } \rho_y = I_{_R} * y$


Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 2

:$x * I_{_R}$ is a bijection and $x^{-1} * I_{_R}$ is the inverse of $x * I_{_R}$ :$I_{_R} * x$ is a bijection and $I_{_R} * x^{-1}$ is the inverse of $I_{_R} * x$


Open Balls of P-adic Number

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 $a \in \Q_p$. For all $\epsilon \in \R_{>0}$, let $\map {B_\epsilon} a$ denote the open ball of $a$ of radius $\epsilon$.  Then: :$\forall n \in Z : \map {B_{p^{-n} } } a = a + p^{n + 1} \Z_p$


Local Basis of P-adic Number

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Then the set of open balls $\set {\map {B_{p^{-n} } } a : n \in Z}$ is a local basis of $a$ consisting of clopen sets.


Closed Ball of P-adic Number

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ denote the $p$-adic integers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$, let $\map { {B_\epsilon}^-} a$ denote the closed ball of $a$ of radius $\epsilon$.  Then: :$\forall n \in Z : \map {B^-_{p^{-n} } } a = a + p^n \Z_p$ where $a + p^n \Z_p$ denotes the left coset of the principal ideal $p^n \Z_p$ containing $a$ in the subring $\Z_p$. That is, the closed ball $\map { {B_\epsilon}^-} a$ is the set: :$a + p^n \Z_p = \set{a + p^n z : z \in \Z_p}$


Closed Ball is Disjoint Union of Open Balls in P-adic Numbers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$: :let $\map { {B_\epsilon}^-} a$ denote the closed $\epsilon$-ball of $a$.  :let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$.  Then: :$(1): \quad \forall n \in Z : \map {B^-_{p^{-n} } } a = \displaystyle \bigcup_{i \mathop = 0}^{p - 1} \map {B_{p^{-n} } } {a + i p^n}$ :$(2): \quad \forall n \in Z : \set {\map {B_{p^{-n} } } {a + i p^n} : i = 0, \dotsc, p - 1}$ is a set of pairwise disjoint open balls.


Sphere is Disjoint Union of Open Balls in P-adic Numbers

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 $a \in \Q_p$. For all $\epsilon \in \R_{>0}$: :let $\map {S_\epsilon} a$ denote the sphere of $a$ of radius $\epsilon$.  :let $\map {B_\epsilon} a$ denote the open ball of $a$ of radius $\epsilon$.  Then: :$\forall n \in Z : \map {S_{p^{-n} } } a = \displaystyle \bigcup_{i \mathop = 1}^{p - 1} \map {B_{p^{-n} } } {a + i p^n}$


Null Sequence induces Local Basis in Metric Space

Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\sequence{x_n}$ be a real null sequence such that: :$\forall n \in N: x_n > 0$ Let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$ in $M$. Then: :$\mathcal B_{\sequence{x_n}} = \set{\map {B_{x_n}} a : n \in \N}$ is a local basis at $a$.


Null Sequence induces Local Basis in Metric Space/Sequence of Reciprocals

:$\mathcal B = \set {\map {B_{1/n}} a : n \in \N}$ is a local basis at $a$.


Consecutive Integers which are Powers of 2 or 3

The only pairs of consecutive positive integers which are powers of $2$ or $3$ are: :$\tuple {1, 2}$, $\tuple {2, 3}$, $\tuple {3, 4}$, $\tuple {8, 9}$


Altitudes of Triangle Bisect Angles of Orthic Triangle

Let $\triangle ABC$ be a triangle. Let $\triangle DEF$ be its orthic triangle. The altitudes of $\triangle ABC$ are the angle bisectors of $\triangle DEF$.


Open Ball in Normed Division Ring is Open Ball in Induced Metric

Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring. Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {B_\epsilon} {a; \norm {\,\cdot\,} }$ denote the open ball in the normed division ring $\struct {R, \norm {\,\cdot\,} }$. Let $\map {B_\epsilon} {a; d }$ denote the open ball in the metric space $\struct {R, d}$. Then: :$\map {B_\epsilon} {a; \norm {\,\cdot\,} }$ = $\map {B_\epsilon} {a; d }$


Closed Ball in Normed Division Ring is Closed Ball in Induced Metric

Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring. Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ denote the closed ball in the normed division ring $\struct {R, \norm {\,\cdot\,} }$. Let $\map {{B_\epsilon}^-} {a; d }$ denote the closed ball in the metric space $\struct {R, d}$. Then: :$\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ = $\map {{B_\epsilon}^-} {a; d }$


Sphere in Normed Division Ring is Sphere in Induced Metric

Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring. Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ denote the sphere in the normed division ring $\struct {R, \norm {\,\cdot\,} }$. Let $\map {S_\epsilon} {a; d }$ denote the sphere in the metric space $\struct {R, d}$. Then: :$\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ = $\map {S_\epsilon} {a; d }$


Sphere is Set Difference of Closed Ball with Open Ball

Let $M = \struct{A, d}$ be a metric space or pseudometric space. Let $a \in A$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {{B_\epsilon}^-} {a; d}$ denote the $\epsilon$-closed ball of $a$ in $M$. Let $\map {B_\epsilon} {a; d}$ denote the $\epsilon$-open ball of $a$ in $M$. Let $\map {S_\epsilon} {a; d}$ denote the $\epsilon$-sphere of $a$ in $M$. Then: :$\map {S_\epsilon} {a; d} = \map {{B_\epsilon}^-} {a; d} \setminus \map {B_\epsilon} {a; d}$


Sphere is Set Difference of Closed Ball with Open Ball/Normed Division Ring

Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {{B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-closed ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$. Let $\map {B_\epsilon} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-open ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$. Let $\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ denote the $\epsilon$-sphere of $a$ in $\struct {R, \norm {\,\cdot\,} }$. Then: :$\map {S_\epsilon} {a; \norm {\,\cdot\,} } = \map { {B_\epsilon}^-} {a; \norm {\,\cdot\,} } \setminus \map {B_\epsilon} {a; \norm {\,\cdot\,} }$


Conservation of Momentum

Let $P$ be a physical system. Let it have the action $S$: :$\displaystyle S = \int_{t_0}^{t_1} L \rd t$ where $L$ is the standard Lagrangian, and $t$ is time. Suppose $L$ does not depend on one of the coordinates explicitly: :$\dfrac {\partial L} {\partial x_j} = 0$ Then the total momentum of $P$ along the axis $x_j$ is conserved.


Conservation of Angular Momentum (Lagrangian Mechanics)

Let $P$ be a physical system composed of finite number of particles. Let it have the action $S$: :$\displaystyle S = \int_{t_0}^{t_1} L \rd t$ where $L$ is the standard Lagrangian, and $t$ is time. Suppose $L$ is invariant {{WRT}} rotation around $z$-axis. Then the total angular momentum of $P$ along $z$-axis is conserved.


Leibniz's Law for Sets

Let $S$ be an arbitrary set. Then: :$x = y \dashv \vdash x \in S \iff y \in S$ for all $S$ in the universe of discourse. This is therefore the justification behind the notion of the definition of set equality.


Area of Annulus

Let $A$ be an annulus whose inner radius is $r$ and whose outer radius is $R$. The area of $A$ is given by: :$\map \Area A = \pi \paren {R^2 - r^2}$


Area of Annulus as Area of Rectangle

Let $A$ be an annulus whose inner radius is $r$ and whose outer radius is $R$. The area of $A$ is given by: :$\map \Area A = 2 \pi \paren {r + \dfrac w 2} \times w$ where $w$ denotes the width of $A$. That is, it is the area of the rectangle contained by: :the width of $A$ :the circle midway in radius between the inner radius and outer radius of $A$.


Characterization of Closed Ball in P-adic Numbers

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. For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {{B_\epsilon}^-} a$ denote the closed ball of center $a$ of radius $\epsilon$.  Let $x, y \in \Q_p$. Let $n \in Z$. {{TFAE}} ::$(1)\quad x \in \map {B^{\,-}_{p^{-n}}} y$ ::$(2)\quad \norm{x -y}_p \le p^{-n}$ ::$(3)\quad \map {B^{\,-}_{p^{-n}}} x = \map {B^{\,-}_{p^{-n}}} y$ ::$(4)\quad x - y \in p^n \Z_p$ ::$(5)\quad x + p^n \Z_p = y + p^n \Z_p$


Characterization of Open Ball in P-adic Numbers

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. For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon} a$ denote the closed ball of center $a$ of radius $\epsilon$.  Let $n \in Z$. Let $x, y \in \Q_p$. {{TFAE}}: :$(1): \quad x \in \map {B_{p^{-n} } } y$ :$(2): \quad \norm{x - y}_p < p^{-n}$ :$(3): \quad \map {B_{p^{-n} } } x = \map {B_{p^{-n} } } y$ :$(4): \quad x - y \in p^{n + 1} \Z_p$ :$(5): \quad x + p^{n + 1} \Z_p = y + p^{n + 1} \Z_p$


Complete Archimedean Valued Field is Real or Complex Numbers

Let $\struct{k, \norm{\,\cdot\,}}$ be a complete valued field with Archimedean norm $\norm{\,\cdot\,}$. Then either: :$k$ is isomorphic to the real numbers $\R$ and $\norm{\,\cdot\,}$ is equivalent to the absolute value $\size{\,\cdot\,}$ on $\R$ or :$k$ is isomorphic to the complex numbers $\C$ and $\norm{\,\cdot\,}$ is equivalent to the complex modulus $\size{\,\cdot\,}$ on $\C$


Equivalence of Definitions of Convergent P-adic Sequence

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 $\Q_p$. {{TFAE}} === Definition 1 === {{:Definition:Convergent Sequence/P-adic Numbers/Definition 1|Definition 1}} === Definition 2 === {{:Definition:Convergent Sequence/P-adic Numbers/Definition 2}} === Definition 3 === {{:Definition:Convergent Sequence/P-adic Numbers/Definition 3}} === Definition 4 === {{:Definition:Convergent Sequence/P-adic Numbers/Definition 4}}


Equivalence of Definitions of Convergence in Normed Division Rings

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a sequence in $R$. {{TFAE|def = Convergent Sequence in Normed Division Ring}} === Definition 1 === {{:Definition:Convergent Sequence/Normed Division Ring/Definition 1}} === Definition 2 === {{:Definition:Convergent Sequence/Normed Division Ring/Definition 2}} === Definition 3 === {{:Definition:Convergent Sequence/Normed Division Ring/Definition 3}}


Finite Complement Topology is not Metrizable

Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$. Then $T$ is not a metrizable space.


Poisson Brackets of Classical Particle in Radial Potential on Plane

Let $P$ be a classical particle embedded in a 2-dimensional Euclidean manifold. Let the real-valued functions $\map r t$, $\map \theta t$ denote the position of $P$ in polar coordinates, where $t$ is time. Suppose, the potential energy of $P$ depends only on $r$. Then $P$ has the following Poisson brackets: {{begin-eqn}} {{eqn| l = \sqbrk {r, p_r}       | r = 1 }} {{eqn| l = \sqbrk {\theta, p_\theta}      | r = 1 }} {{eqn| l = \sqbrk {r, H}      | r = \dfrac {p_r} m }} {{eqn| l = \sqbrk {\theta, H}       | r = \dfrac {p_\theta} {m r^2} }} {{eqn| l = \sqbrk {p_r, H}       | r = -\dfrac {\partial U} {\partial r} }} {{eqn| l = \sqbrk {p_\theta, H}       | r = 0 }} {{end-eqn}}


Multiplication by Power of 10 by Moving Decimal Point

Let $n \in \R$ be a real number. Let $n$ be expressed in decimal notation. Let $10^d$ denote a power of $10$ for some integer $d$ Then $n \times 10^d$ can be expressed in decimal notation by shifting the decimal point $d$ places to the right. Thus, if $d$ is negative, and so $10^d = 10^{-e}$ for some $e \in \Z_{>0}$, $n \times 10^d$ can be expressed in decimal notation by shifting the decimal point $e$ places to the left.


Number of Significant Figures in Result of Multiplication

Let $m$ and $n$ be numbers which are presented to $d_m$ and $d_n$ significant figures respectively. Then the most significant figures that $m \times n$ can have is $\min \set {d_m, d_n}$.


Number of Significant Figures in Result of Division

Let $m$ and $n$ be numbers which are presented to $d_m$ and $d_n$ significant figures respectively. Then the most significant figures that $\dfrac m n$ can have is $\min \set {d_m, d_n}$.


Number of Significant Figures in Result of Square Root

Let $m$ be a numbers which is presented to $d$ significant figures. Then the most significant figures that $\sqrt m$ can have is also $d$.


Number of Significant Figures in Result of Addition or Subtraction

Let $m$ and $n$ be numbers. Let $d_m$ and $d_n$ be the position of the least significant digit of $m$ and $n$ respectively. Then the least significant digit in either $m + n$ or $m - n$ is in the position corresponding to the greater significant digit of $d_m$ and $d_n$.


Range of Common Logarithm of Number between 1 and 10

Let $x \in \R$ be a real number such that: :$1 \le x < 10$ Then: :$0 \le \log_{10} x \le 1$ where $\log_{10}$ denotes the common logarithm function.


Common Logarithm of Number in Scientific Notation

Let $n$ be a positive real number which is presented (possibly approximated) in scientific notation as: :$n = a \times 10^d$ where: :$1 \le a < 10$ :$d \in \Z$ is an integer. Then: :$\log_{10} n = \log_{10} a + d$ where: :$0 \le \log_{10} a < 1$


Characteristic of Common Logarithm of Number Greater than 1

Let $x \in \R_{>1}$ be a (strictly) positive real number greater than $1$. The characteristic of its common logarithm $\log_{10} x$ is equal to one less than the number of digits to the left of the decimal point of $x$.


Characteristic of Common Logarithm of Number Less than 1

Let $x \in \R_{>0}$ be a (strictly) positive real number such that $x < 1$. The characteristic of its common logarithm $\log_{10} x$ is equal to one less than the number of zero digits to the immediate right of the decimal point of $x$.


P-adic Norm satisfies Non-Archimedean Norm Axioms

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers with $p$-adic norm $\norm {\,\cdot\,}_p : \Q_p \times \Q_p \to \R_{\ge 0}$. Then $\norm {\,\cdot\,}_p$ satisfies the non-Archimedean norm axioms: {{begin-axiom}} {{axiom | n = \text N 1         | lc= Positive Definiteness:         | q = \forall x \in \Q_p         | ml= \norm x_p = 0         | mo= \iff         | mr= x = 0 }} {{axiom | n = \text N 2         | lc= Multiplicativity:         | q = \forall x, y \in \Q_p         | ml= \norm {x \cdot y}_p         | mo= =         | mr= \norm x_p \times \norm y_p }} {{axiom | n = \text N 4         | lc= Ultrametric Inequality:         | q = \forall x, y \in Q_p         | ml= \norm {x + y}_p         | mo= \le         | mr= \max \set {\norm x_p, \norm y_p} }} {{end-axiom}}


Circles with Same Poles are Parallel

Let $S$ be a sphere. Let $C$ and $D$ be circles on $S$ (either great circles or small circles). Let $C$ and $D$ both have the same pair of poles. Then $C$ and $D$ are parallel.


Three Points on Sphere in Same Hemisphere

Let $S$ be a sphere. Let $A$, $B$ and $C$ be points on $S$ which do not all lie on the same great circle. Then it is possible to divide $S$ into two hemispheres such that $A$, $B$ and $C$ all lie on the same hemisphere.


Side of Spherical Triangle is Less than 2 Right Angles

Let $ABC$ be a spherical triangle on a sphere $S$. Let $AB$ be a side of $ABC$. The '''length''' of $AB$ is less than $2$ right angles.


Center is Element of Closed Ball

Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\epsilon \in \R_{>0}$ be a positive real number. Let $\map { {B_\epsilon}^-} a$ be the closed $\epsilon$-ball of $a$ in $M$. Then: :$a \in \map {{B_\epsilon}^-} a$


Center is Element of Closed Ball/Normed Division Ring

Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map { {B_\epsilon}^-} a$ be the closed $\epsilon$-ball of $a$ in $\struct{R, \norm {\,\cdot\,} }$. Then: :$a \in \map { {B_\epsilon}^-} a$


Center is Element of Closed Ball/P-adic Numbers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map { {B_\epsilon}^-} a$ be the closed $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$. Then: :$a \in \map { {B_\epsilon}^-} a$


Length of Arc of Small Circle

Let $S$ be a sphere. Let $\bigcirc FCD$ be a small circle on $S$. Let $C$ and $D$ be the points on $\bigcirc FCD$ such that $CD$ is the arc of $\bigcirc FCD$ whose length is to be determined.


Definition:Geographical Coordinates

Let $J$ be a point on Earth's surface. The '''geographical coordinates''' of $J$ are the definition of the position of $J$ with respect to the equator and the principal meridian.


Two Parallel Lines lie in Same Plane

Let $L_1$ and $L_2$ be two lines which are parallel. Then $L_1$ and $L_2$ both lie in the same plane.


Center is Element of Open Ball

Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\epsilon \in \R_{>0}$ be a positive real number. Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$ in $M$. Then: :$a \in \map {B_\epsilon} a$


Center is Element of Open Ball/Normed Division Ring

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$ in $\struct{R, \norm {\,\cdot\,} }$. Then: :$a \in \map {B_\epsilon} a$


Center is Element of Open Ball/P-adic Numbers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$. Then: :$a \in \map {B_\epsilon} a$


Equivalence of Definitions of Non-Archimedean Division Ring Norm

Let $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$. {{TFAE}} === Definition 1 === {{:Definition:Non-Archimedean/Norm (Division Ring)/Definition 1}} === Definition 2 === {{:Definition:Non-Archimedean/Norm (Division Ring)/Definition 2}}


Sine of Half Angle for Spherical Triangles

:$\sin \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin b \sin c} }$ where $s = \dfrac {a + b + c} 2$.


Open Ball in P-adic Numbers is Closed Ball

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$:  :Let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$ :Let $\map {B^-_\epsilon} a$ denote the closed $\epsilon$-ball of $a$.  Then: :$\forall n \in Z : \map {B_{p^{-n} } } a = \map {B^-_{p^{-\paren {n + 1} } } } a$


Tangent of Half Angle for Spherical Triangles

:$\tan \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin s \, \map \sin {s - a} } }$ where $s = \dfrac {a + b + c} 2$.


Countable Basis for P-adic Numbers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$. For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$.   Then: :$\BB_p = \set {\map {B_{p^{-n} } } q : q \in \Q, n \in \Z}$ is a countable basis for $\struct{\Q_p, \tau_p}$.


P-adic Numbers is Second Countable Topological Space

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$. Then the topological space $\struct {\Q_p, \tau_p}$ is second-countable.


P-adic Numbers is Totally Disconnected Topological Space

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$. Then the topological space $\struct {\Q_p, \tau_p}$ is totally disconnected.


P-adic Numbers is Hausdorff Topological Space

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$. Then the topological space $\struct{\Q_p, \tau_p}$ is Hausdorff.


P-adic Numbers is Locally Compact Topological Space

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$. Then the topological space $\struct {\Q_p, \tau_p}$ is locally compact.


Open and Closed Balls in P-adic Numbers are Compact Subspaces

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $n \in \Z$. Then the open ball $\map {B_{p^{-n} } } a$ and closed ball $\map {B^-_{p^{-n}}} a$ are compact. {{disambiguate|Definition:Compact}}


Ambiguous Case

Let $\triangle ABC$ be a triangle. Let the sides $a, b, c$ of $\triangle ABC$ be opposite $A, B, C$ respectively. Let the sides $a$ and $b$ be known. Let the angle $\angle B$ also be known. Then it may not be possible to know the value of $\angle A$. This is known as the '''ambiguous case'''.


Analogue Formula for Spherical Law of Cosines

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: {{begin-eqn}} {{eqn | l = \sin a \cos B       | r = \cos b \sin c - \sin b \cos c \cos A }} {{eqn | l = \sin a \cos C       | r = \cos c \sin b - \sin c \cos b \cos A }} {{end-eqn}}


Four-Parts Formula

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. We have: :$\cos a \cos C = \sin a \cot b - \sin C \cot B$ That is: :$\map \cos {\text {inner side} } \cdot \map \cos {\text {inner angle} } = \map \sin {\text {inner side} } \cdot \map \cot {\text {other side} } - \map \sin {\text {inner angle} } \cdot \map \cot {\text {other angle} }$ This is known as the '''four-parts formula''', as it defines the relationship between each of four consecutive parts of $\triangle ABC$.


Countable Basis for P-adic Numbers/Closed Balls

For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon^-} a$ denote the closed $\epsilon$-ball of $a$.   Then: :$\BB_p = \set {\map {B^-_{p^{-n} } } q : q \in \Q, n \in \Z}$ is a countable basis for $\struct{\Q_p, \tau_p}$.


Countable Basis for P-adic Numbers/Cosets

Let $\Z_p$ be the $p$-adic integers. Then: :$\BB_p = \set {q + p^n \Z_p : q \in \Q, n \in \Z}$ is a countable basis for $\struct{\Q_p, \tau_p}$.


Napier's Tangent Rule for Right Spherical Triangles

Let $\triangle ABC$ be a right spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let the angle $\sphericalangle C$ be a right angle. Let the remaining parts of $\triangle ABC$ be arranged according to the '''interior''' of this circle, where the symbol $\Box$ denotes a right angle. :410px Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Then the sine of the '''middle part''' equals the product of the tangents of the '''adjacent parts'''.


Napier's Cosine Rule for Quadrantal Triangles

Let $\triangle ABC$ be a quadrantal triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let the side $c$ be a right angle. Let the remaining parts of $\triangle ABC$ be arranged according to the '''exterior''' of this circle, where the symbol $\Box$ denotes a right angle. :410px Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Then the sine of the '''middle part''' equals the product of the cosine of the '''opposite parts'''.


'''Napier's Rules for Right Spherical Triangles''' are the special cases of the Spherical Law of Cosines for a spherical triangle one of whose angles is a right angle. :410px Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let angle $\angle C$ be a right angle. Let the remaining parts of $\triangle ABC$ be arranged in a circle as the '''interior''' of the above, where the symbol $\Box$ denotes a right angle. Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Let the remaining two parts be called '''opposite parts'''. Then: :The sine of the '''middle part''' equals the product of the tangents of the '''adjacent parts'''. :The sine of the '''middle part''' equals the product of the cosines of the '''opposite parts'''.


Napier's Rules for Quadrantal Triangles

'''Napier's Rules for Quadrantal Triangles''' are the special cases of the Spherical Law of Cosines for a spherical triangle one of whose sides is a right angle. :410px Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let side $c$ be a right angle. Let the remaining parts of $\triangle ABC$ be arranged in a circle as the '''exterior''' of the above, where the symbol $\Box$ denotes a right angle. Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Let the remaining two parts be called '''opposite parts'''. Then: :The sine of the '''middle part''' equals the product of the tangents of the '''adjacent parts'''. :The sine of the '''middle part''' equals the product of the cosines of the '''opposite parts'''.


Spherical Triangle is Polar Triangle of its Polar Triangle

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$. Then $\triangle ABC$ is the polar triangle of $\triangle A'B'C'$.


Side of Spherical Triangle is Supplement of Angle of Polar Triangle

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$. Then $A'$ is the supplement of $a$. That is: :$A' = \pi - a$ and it follows by symmetry that: :$B' = \pi - b$ :$C' = \pi - c$


Integers are Dense in P-adic Integers/Unit Ball

The integers $\Z$ are dense in the closed ball $\map {B^-_1} 0$.


Analogue Formula for Spherical Law of Cosines/Corollary

{{begin-eqn}} {{eqn | l = \sin A \cos b       | r = \cos B \sin C + \sin B \cos C \cos a }} {{eqn | l = \sin A \cos c       | r = \cos C \sin B + \sin C \cos B \cos a }} {{end-eqn}}


Four-Parts Formula/Corollary

:$\cos A \cos c = \sin A \cot B - \sin c \cot b$


Cosine in Terms of Haversine

:$\cos \theta = 1 - 2 \hav \theta$ where $\cos$ denotes cosine and $\hav$ denotes haversine.


Spherical Law of Haversines

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: :$\hav a = \map \hav {b - c} + \sin b \sin c \hav A$ where $\hav$ denotes haversine.


Sign of Haversine

The haversine is non-negative for all $\theta \in \R$.


Haversine Function is Even

The haversine is an even function: :$\forall \theta \in \R: \map \hav {-\theta} = \hav \theta$


Delambre's Analogies

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.


Napier's Analogies


Half Angle Formulas for Spherical Triangles


Half Side Formulas for Spherical Triangles


Tangent of Half Side for Spherical Triangles

:$\tan \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\map \cos {S - B} \, \map \cos {S - C} } }$ where $S = \dfrac {a + b + c} 2$.


Zenith Distance is Complement of Celestial Altitude

Let $X$ be the position of a star (or other celestial body) on the celestial sphere. The zenith distance $z$ of $X$ is the complement of the altitude $a$ of $X$: :$z = 90 \degrees - a$


P-adic Metric on P-adic Numbers is Non-Archimedean Metric/Corollary 1

Then: :$\forall x, y, z \in R: \norm {x - y}_p \le \max \set {\norm {x - z}_p, \norm {y - z}_p}$


Zenith Distance of North Celestial Pole equals Colatitude of Observer

Let $O$ be an observer of the celestial sphere. Let $P$ be the position of the north celestial pole with respect to $O$. Let $z$ denote the zenith distance of $P$. Let $\psi$ denote the (terrestrial) colatitude of $O$. Then: :$z = \psi$


Altitude of North Celestial Pole equals Latitude of Observer

Let $O$ be an observer of the celestial sphere. Let $P$ be the position of the north celestial pole with respect to $O$. Let $a$ denote the altitude of $P$. Let $\phi$ denote the (terrestrial) latitude of $O$. Then: :$a = \phi$


Celestial Body moves along Parallel of Declination

Consider the celestial sphere $C$ whose observer is $O$. Let $B$ be a celestial body on $C$. Let $\delta$ be the declination of $B$. Let $P_\delta$ be the parallel of declination whose declination is $\delta$. Then $B$ appears to move along the path of $P_\delta$, in the direction from the north to east to south to west. It takes $1$ sidereal day for $B$ to travel around $P_\delta$.


Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$, let $\map { {B_\epsilon}^-} a$ denote the closed $\epsilon$-ball of $a$.  Let $n, m \in Z$, such that $n < m$. Then: :$(1) \quad \map {B^-_{p^{-n}}} a = \displaystyle \bigcup_{i \mathop = 0}^{p^\paren {m - n} - 1} \map {B^-_{p^{-m} } } {a + i p^n}$ :$(2) \quad \set {\map {B^-_{p^{-m} } } {a + i p^n} : i = 0, \dots, p^\paren {m - n} - 1}$ is a set of pairwise disjoint closed balls


Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Union of Closed Balls

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$, let $\map {{B_\epsilon}^-} a$ denote the closed $\epsilon$-ball of $a$.  Let $n, m \in Z$, such that $n < m$. Then: :$\map {B^-_{p^{-n} } } a = \displaystyle \bigcup_{i \mathop = 0}^{p^\paren {m - n} - 1} \map {B^-_{p^{-m} } } {a + i p^\paren {m - n} }$


Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Disjoint Closed Balls

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. For all $\epsilon \in \R_{>0}$, let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$.  Then: :$\forall n \in Z : \set{\map {B^-_{p^{-m} } } {a + i p^n} : i = 0, \dotsc, p^\paren {m - n} - 1}$ is a set of pairwise disjoint open balls.


Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1

:$\forall y \in \Q_p: \norm y_p \le p^{-n}$ {{iff}} there exists $i \in \Z$ such that: ::$(1)\quad 0 \le i \le p^\paren {m - n} - 1$ ::$(2)\quad \norm {y - i p^n}_p \le p^{-m}$


Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Necessary Condition

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $n, m \in Z$, such that $n < m$. Let $y \in \Q_p$. Let $\norm{y}_p \le p^{-n}$. Then there exists $i \in \Z$ such that: ::$(1) \quad 0 \le i \le p^\paren {m - n} - 1$ ::$(2) \quad \norm {y - i p^n}_p \le p^{-m}$


Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Sufficient Condition

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $y \in \Q_p$ Let $n, m \in Z$, such that $n < m$. Let there exist $i \in \Z$: :$(1): \quad 0 \le i \le p^\paren {m - n} - 1$ :$(2): \quad \norm {y - i p^n}_p \le p^{-m}$ Then: :$\norm y_p \le p^{-n}$


Open and Closed Balls in P-adic Numbers are Totally Bounded

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $n \in \Z$. Then the open ball $\map {B_{p^{-n} } } a$ and closed ball $\map {B^-_{p^{-n} } } a$ are totally bounded subspaces.


Cofactor Sum Identity

Let $J_n$ be the $n \times n$ matrix of all ones. Let $A$ be an $n \times n$ matrix. Let $A_{ij}$ denote the cofactor of element $\paren {i,j}$ in $\det\paren A$, $1\le i,j \le n$.                  Then: :$\displaystyle \det \paren {A -J_n} = \det \paren A - \sum_{i \mathop  = 1}^n \sum_{j \mathop = 1}^n A_{ij} $


Sum of Elements of Invertible Matrix

Let $\mathbf J_n$ be the $n \times n$ square ones matrix. Let $\mathbf B$ be an $n\times n$ invertible matrix with entries $b_{i j}$, $1 \le i, j \le n$. Then: :$\displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n b_{i j} = 1 -  \map \det {\mathbf B} \map \det {\mathbf B^{-1} - \mathbf J_n}$


Open and Closed Balls in P-adic Numbers are Clopen in P-adic Metric

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $n \in \Z$. Then the open ball $\map {B_{p^{-n}}} a$ and closed ball $\map {B^-_{p^{-n}}} a$ are clopen in the $p$-adic metric.


Closed Balls Centered on P-adic Number is Countable

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Then the set of all closed balls centered on $a$ is the countable set: :$\BB^- = \set {\map {B^-_{p^{-n} } } a: n \in \Z}$


Closed Balls Centered on P-adic Number is Countable/Open Balls

Then the set of all open balls centered on $a$ is the countable set: :$\BB = \set {\map {B_{p^{-n} } } a : n \in \Z}$


Closed Balls Centered on P-adic Number is Countable/Lemma

:$\exists n \in \Z : p^{-n} \le \epsilon < p^{-\paren {n - 1} }$


Closed Balls Centered on P-adic Number is Countable/Open Balls/Lemma

:$\exists n \in \Z : p^{-\paren {n + 1} } < \epsilon \le p^{-n}$


Open Ball contains Smaller Open Ball

Let $M = \struct{A, d}$ be a metric space. Let $a \in A$. Let $\epsilon, \delta \in \R_{>0}$ such that $\epsilon \le \delta$. Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball on $a$. Let $\map {B_\delta} a$ be the open $\delta$-ball on $a$. Then: :$\map {B_\epsilon} a \subseteq \map {B_\delta} a$


Closed Ball contains Smaller Closed Ball

Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\epsilon, \delta \in \R_{> 0}$ such that $\epsilon \le \delta$. Let $\map {B^-_\epsilon} a$ be the closed $\epsilon$-ball on $a$. Let $\map {B^-_\delta} a$ be the closed $\delta$-ball on $a$. Then: :$\map {B^-_\epsilon} a \subseteq \map {B^-_\delta} a$


Open Ball contains Strictly Smaller Closed Ball

Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\epsilon, \delta \in \R_{>0}$ such that $\epsilon < \delta$. Let $\map {B^-_\epsilon} a$ be the closed $\epsilon$-ball on $a$. Let $\map {B_\delta} a$ be the open $\delta$-ball on $a$. Then: :$\map {B^-_\epsilon} a \subseteq \map {B_\delta} a$


Closed Ball contains Smaller Open Ball

Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\epsilon, \delta \in \R_{> 0}$ such that $\epsilon \le \delta$. Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball on $a$. Let $\map {B^-_\delta} a$ be the closed $\delta$-ball on $a$. Then: :$\map {B_\epsilon} a \subseteq \map {B^-_\delta} a$


Open and Closed Balls in P-adic Numbers are Compact Subspaces/P-adic Integers

Then the set of $p$-adic integers $\Z_p$ is compact.


Open Balls form Local Basis for Point of Metric Space

Let $M = \struct {A, d}$ be a metric space. Let $x \in A$. Let $\BB_x$ be the set of all open balls of $M$ centered on $x$. That is: :$\BB_x = \set {\map {B_\epsilon} x : \epsilon \in \R_{>0}}$ Then $\BB$ is a local basis of $x$.


Local Basis of P-adic Number/Cosets

Let $\Z_p$ be the $p$-adic integers. Then the set $\set {a + p^n \Z_p: n \in Z}$ is a local basis of $a$ consisting of clopen sets.


Local Basis of P-adic Number/Closed Balls

Then the set of closed balls $\set {\map {B^-_{p^{-n} } } a: n \in Z}$ is a local basis of $a$ consisting of clopen sets.


Measurement of Terrestrial Longitude

To measure the (terrestrial) longitude of a position: :$(1): \quad$ Identify the exact moment of noon at the location in question :$(2): \quad$ Work out the precise time of day $t$ that this happens. Let $N$ be the time of noon at the principal meridian on the day in question. Let $x$ be the number of hours before $N$ at the principal meridian. If $t$ is after $N$, then $x$ is treated as negative. Then the (terrestrial) longitude is $15 x \degrees$. If $x$ is positive, the (terrestrial) longitude is east. If $x$ is negative, the (terrestrial) longitude is west.


Coherent Sequence is Partial Sum of P-adic Expansion

Let $p$ be a prime number. Let $\sequence{\alpha_n}$ be a coherent sequence. Then there exists a unique $p$-adic expansion of the form: :$\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$ such that: :$\forall n \in \N: \alpha_n = \displaystyle \sum_{i \mathop = 0}^n d_i p^i$


Difference of Consecutive terms of Coherent Sequence

Let $p$ be a prime number. Let $\sequence {\alpha_n}$ be a coherent sequence. Then: :for all $n \in \N_{>0}$ there exists $c_n \in \N$ such that: ::$0 \le c_n < p$ ::$\alpha_n - \alpha_{n - 1} = c_n p^n$


Vandermonde Matrix Identity for Cauchy Matrix

Assume values $\set { x_1,\ldots,x_n,y_1,\ldots,y_n }$ are distinct in matrix {{begin-eqn}} {{eqn   | l = C    | r = \paren {\begin{smallmatrix}         \dfrac {1} {x_1 - y_1} & \dfrac {1} {x_1 - y_2} & \cdots & \dfrac {1} {x_1 - y_n} \\         \dfrac {1} {x_2 - y_1} & \dfrac 1 {x_2 - y_2}   & \cdots & \dfrac {1} {x_2 - y_n} \\         \vdots                 & \vdots                 & \cdots & \vdots \\         \dfrac {1} {x_n - y_1} & \dfrac {1} {x_n - y_2} & \cdots & \dfrac {1} {x_n - y_n} \\ \end{smallmatrix} }   | c = Cauchy matrix of order $n$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn   | l = C   | r = -P V_x^{-1} V_y Q^{-1}   | c = Vandermonde matrix identity for a Cauchy matrix  }} {{end-eqn}}


Function that Satisfies Axioms of Uncertainty

Let $n \in \N$ be a natural number. Let $p_1, p_2, \dotsc, p_n$ be real numbers such that: :$\forall i \in \set {1, 2, \dotsc, n}: p_i \ge 0$ :$\displaystyle \sum_{i \mathop = 1}^n p_i = 1$ Let $\map H {p_1, p_2, \ldots, p_n}$ be a real-valued function which satisfies the axioms of uncertainty. Then: :$\map H {p_1, p_2, \ldots, p_n} = \displaystyle -\lambda \sum_{i \mathop = 1}^n p_i \log_b p_i$ where: :$\lambda \in \R_{>0}$ :$b \in \R_{>1}$


Uncertainty Function satisfies Axioms of Uncertainty

Let $X$ be a random variable. Let $X$ take a finite number of values with probabilities $p_1, p_2, \dotsc, p_n$. Let $\map H X$ be the '''uncertainty function''' of $X$: :$\map H X = \displaystyle -\sum_k p_k \lg p_k$ where: :$\lg$ denotes logarithm base $2$ :the summation is over those $k$ where $p_k > 0$. Then the uncertainty function satisfies the Axioms of Uncertainty.


Number of k-Cycles in Symmetric Group

Let $n \in \N$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $k \in N$ such that $k \le n$. The number of elements $m$ of $S_n$ which are $k$-cycles is given by: :$m = \paren {k - 1}! \dbinom n k = \dfrac {n!} {k \paren {n - k}!}$


Number of k-Cycles on Set of k Elements

Let $k \in \N$ be a natural number. Let $S_k$ denote the symmetric group on $k$ letters. The number of elements of $S_k$ which are $k$-cycles is $\paren {k - 1}!$.


Zero Padded Basis Representation

Let $b \in \Z: b > 1$. Let $m \in \Z_{> 0}$. For every $n \in \Z_{\ge 0}$ such that $n < b^m$, there exists one and only one sequence $\sequence {r_j}_{0 \mathop \le j \mathop \le m - 1}$ such that: : $(1): \quad \displaystyle n = \sum_{j \mathop = 0}^{m - 1} r_j b^j$ : $(2): \quad \displaystyle \forall j \in \closedint 0 {m - 1}: r_j \in \N_b$


Sum of Elements of Inverse of Matrix with Column of Ones

Let $\mathbf B = \sqbrk b_n$ denote the inverse of a square matrix $\mathbf A$ of order $n$. Let $\mathbf A$ be such that it has a row or column of all ones. Then the sum of elements in $\mathbf B$ is one: :$\displaystyle \sum_{i \mathop  = 1}^n \sum_{j \mathop = 1}^n b_{ij}  = 1$


Coherent Sequence is Partial Sum of P-adic Expansion/Lemma

:$\forall n \in \N: \alpha_n = \displaystyle \sum_{i \mathop = 0}^n b_{i, i} p^i$


Limit to Infinity of Number of p-Groups of Order p^m

Let $p$ be a prime number. Let $m \in \N$ be a natural number Let $\map \nu {p^n}$ denote the $\nu$ function of $p^n$: the number of group types of order $p^m$. Then: :$\map \nu {p^m} = p^{A m^3}$ where: :$\displaystyle \lim_{m \mathop \to \infty} A = \dfrac 2 {27}$


Copeland-Erdős Constant is Normal

The Copeland-Erdős constant whose decimal expansion is formed by concatenating the prime numbers in ascending order: :$0 \cdotp 23571 \, 11317 \, 1923 \ldots$ is normal with respect to base $10$.


Koebe Quarter Theorem

Let $f: \C \to \C$ be a schlicht function, that is, a univalent complex function such that $\map f 0 = 0$ and $\map {f'} 0 = 1$. Then the image of the unit disk contains the closed disk of radius $\dfrac 1 4$. Hence for any $w \in f \sqbrk {\Bbb D}$ we have that $\cmod w \le \dfrac 1 4$. The constant $\dfrac 1 4$ is sharp and so cannot be improved.


Limit to Infinity of Summation of Euler Phi Function over Square

:$\displaystyle \lim_{n \mathop \to \infty} \dfrac {\map \Phi n} {n^2} = \dfrac 3 {\pi^2}$ where: :$\map \Phi n = \displaystyle \sum_{k \mathop = 1}^n \map \phi k$ :$\map \phi k$ is the Euler $\phi$ function of $k$. Numerically, this evaluates to: :$\dfrac 3 {\pi^2} \approx 0 \cdotp 30396 35509 \ldots$ {{OEIS|A104141}}


Blaschke's Theorem

Let $C$ be a closed convex curve. Let the minimum width $w$ of $C$ be such that $w \ge 1$. Then $C$ can contain a circle whose radius $\dfrac 1 3$.


At Least One Third of Zeros of Riemann Zeta Function on Critical Line

At least $\dfrac 1 3$ of the nontrivial zeroes of the Riemann $\zeta$ function lie on the critical line.


Complex Sine Function is Entire

Let $\sin: \C \to \C$ be the complex sine function.  Then $\sin$ is entire.


Summation of Reciprocal of Zero of Order 1 Bessel Function by Order 0 Bessel Function of it

:$\displaystyle \sum_{n \mathop = 1}^\infty \dfrac 1 {x_n \map {J_0 } {x_n} } = 0 \cdotp 38479 \ldots$ where: :$x_n$ is the $n$th zero of the order $1$ Bessel function of the first kind :$\map {J_0 } {x_n}$ is the order $0$ Bessel function of the first kind of $x_n$.


Bloch's Theorem

Let $f: \C \to \C$ be a holomorphic function in the unit disk $\cmod z \le 1$. Let $\cmod {\map {f'} 0} = 1$. Then there exists: :a disk $D$ of radius $B$ :an analytic function $\phi$ in $D$ such that $\map f {\map \phi z} = z$ for all $z$ in $D$ where $B > \dfrac 1 {72}$ is an absolute constant.


Complex Cosine Function is Entire

Let $\cos: \C \to \C$ be the complex cosine function.  Then $\cos$ is entire.


Arccosecant Logarithmic Formulation

Let $x$ be a real number.  Let $x \in \hointl {-\infty} {-1} \cup \hointr 1 {\infty}$. Then:  :$\displaystyle \arccsc x = -i \map \Ln {\sqrt {1 - \frac 1 {x^2} } + \frac i x}$ where: :$\arccsc$ is the arccosecant function :$\Ln$ is the principal branch of the complex logarithm whose imaginary part lies in $\hointl {-\pi} \pi$.


Landau's Theorem

Let $f: \C \to \C$ be a holomorphic function in the unit disk $\cmod z \le 1$. Let $\cmod {\map {f'} 0} = 1$. Then there exists: :a disk $D$ of radius $B$ :an analytic function $\phi$ in $D$ such that $\map f {\map \phi z} = z$ for all $z$ in $D$ such that $L$ is an absolute constant where: :$L > B$ where $B$ is Bloch's constant.


Lower Bound of Bloch's Constant

Bloch's constant has a lower bound as follows: :$\dfrac {\sqrt 3} 4 + \dfrac 2 {10 \, 000} \le B$


Upper Bound of Bloch's Constant

Bloch's constant has an upper bound as follows: :$B \le \sqrt {\dfrac {\sqrt 3 -1} 2} \times \dfrac {\map \Gamma {\frac 1 3} \map \Gamma {\frac {11} {12} } } {\map \Gamma {\frac 1 4} }$


Limit of Difference between Consecutive Prime Numbers

The Prime Number Theorem indicates that the average value of the difference between two consecutive prime numbers is of the order of $\log p_n$. Let $E = \displaystyle \liminf_{n \mathop \to \infty} \dfrac {p_{n + 1} - p_n} {\log p_n}$. If there are infinitely many twin primes, then $E = 0$. If not, then it is not known what the value of $E$ is.


Distribution of Numbers with More than 2 Prime Factors

For sufficiently large $x$, there always exists an integer with more than $2$ prime factors between $\paren {x - x^\alpha}$ and $x$, where: :$\alpha \ge 0 \cdotp 477 \ldots$


All Nontrivial Zeroes of Riemann Zeta Function are on Critical Strip

All of the nontrivial zeroes of the Riemann $\zeta$ function lie on the critical strip.


Nontrivial Zeroes of Riemann Zeta Function are Symmetrical with respect to Critical Line

The nontrivial zeroes of the Riemann $\zeta$ function are distributed symmetrically with respect to the  critical line. That is, suppose $s_1 = \sigma_1 + i t$ is a nontrivial zero of $\zeta$. Then there exists another nontrivial zero $s_2$ of $\zeta$ such that: :$s_2 = 1 - \sigma_1 + i t$


Critical Line Theorem

There exist an infinite number of nontrivial zeroes of the Riemann $\zeta$ function on the critical line.


Prime-Counting Function in terms of Eulerian Logarithmic Integral

Let $\map \pi x$ denote the prime-counting function of a number $x$. Let $\map \Li x$ denote the Eulerian logarithmic integral of $x$: :$\map \Li x := \displaystyle \int_2^x \dfrac {\d t} {\ln t}$ Then: :$\map \pi x = \map \Li x + \map {\mathcal O} {x \, \map \exp {-c \sqrt {\ln x} } }$ where: :$\mathcal O$ is the big-O notation :$c$ is some constant.


Mertens' Third Theorem

:$\displaystyle \lim_{x \mathop \to \infty} \ln x \prod_{\substack {p \mathop \le x \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 p} = e^{-\gamma}$ where $\gamma$ denotes the Euler-Mascheroni constant.


Abi-Khuzam Inequality

Let $\triangle ABC$ be a triangle. Then: :$\sin A \cdot \sin B \cdot \sin C \le k A \cdot B \cdot C$ where: :$A, B, C$ are measured in radians :$k = \paren {\dfrac {3 \sqrt 3} {2 \pi} }^3 \approx 0 \cdotp 56559 \, 56245 \ldots$


Limit to 1 of Zeta of s minus Reciprocal of s-1

:$\displaystyle \lim_{s \mathop \to 1} \paren {\map \zeta s - \dfrac 1 {s - 1} } = \gamma$ where: :$\zeta$ denotes the Riemann $\zeta$ (zeta) function :$\gamma$ denotes the Euler-Mascheroni constant.


Limit to Infinity of x minus Gamma of Reciprocal of x

:$\displaystyle \lim_{x \mathop \to \infty} \paren {x - \map \Gamma {\dfrac 1 x} } = \gamma$ where: :$\Gamma$ denotes the $\Gamma$ (Gamma) function :$\gamma$ denotes the Euler-Mascheroni constant.


Jung's Theorem

Let $S \subseteq \R^n$ be a compact subspace of an $n$-dimensional Euclidean space. Let $d = \displaystyle \max_{x, y \mathop \in S} \map d {x, y}$ be the diameter of $S$. Then there exists a closed ball ${B_r}^-$ with radius $r$ such that: :$r = d \sqrt {\dfrac n {2 \paren {n + 1} } }$ such that $S \subseteq {B_r}^-$.


Jung's Theorem in the Plane

Let $S \subseteq \R^2$ be a compact region in a Euclidean plane. Let $d$ be the diameter of $S$. Then there exists a circle $C$ with radius $r$ such that: :$r = d \dfrac {\sqrt 3} 3$ such that $S \subseteq C$.


Alternating Sum and Difference of Factorials to Infinity

According to {{AuthorRef|Leonhard Paul Euler}}: {{begin-eqn}} {{eqn | l = \sum_{n \mathop = 0}^\infty \paren {-1}^n n!       | r = \int_0^\infty \dfrac {e^{-u} } {1 + u} \rd u       | c =  }} {{eqn | r = G       | c = the Euler-Gompertz constant }} {{eqn | o = \approx       | r = 0 \cdotp 59634 \, 73623 \, 23194 \, 07434 \, 10784 \, 99369 \, 27937 \, 6074 \ldots       | c =  }} {{end-eqn}} {{explain|Clarify meaning of this equality. Using naive manipulations (carelessly swapping integration/sum and using geometric series results) it's quite straightforward to see why someone might think this is true but we should establish the sense in which it is, since the sum on the LHS diverges.<br/>What we need to do is go back to Euler's original statement of this and see what he meant. He did lots of stuff like this, plugging values into formulae that they weren't applicable to, like e.g. $1 + 2 + 3 + \ldots {{=}} \dfrac 1 {12}$, I presume it's like one of those.}}


Arcsine Function in terms of Gaussian Hypergeometric Function

:$\displaystyle \arcsin x = x \, {}_2 \map {F_1} {\frac 1 2, \frac 1 2; \frac 3 2; x^2}$


Mean Distance between Two Random Points in Cuboid

Let $B$ be a cuboid in the Cartesian $3$-space $\R^3$ as: :$\size x \le a$, $\size y \le b$, $\size z \le c$ Let $E$ denote the mean distance $D$ between $2$ points chosen at random from the interior of $B$. Then: {{begin-eqn}} {{eqn | l = E       | r = \dfrac {2 r} {15} - \dfrac 7 {45} \paren {\paren {r - r_1} \paren {\dfrac {r_1} a}^2 + \paren {r - r_2} \paren {\dfrac {r_2} b}^2 + \paren {r - r_3} \paren {\dfrac {r_3} c}^2}       | c =  }} {{eqn | o =       | ro= +       | r = \dfrac 8 {315 a^2 b^2 c^2} \paren {a^7 + b^7 + c^7 - {r_1}^7 - {r_2}^7 - {r_3}^7 + r^7}       | c =  }} {{eqn | o =       | ro= +       | r = \dfrac 1 {15 a b^2 c^2} \paren {b^6 \sinh^{-1} \dfrac a b + c^6 \sinh^{-1} \dfrac a c - {r_1}^2 \paren { {r_1}^4 - 8 b^2 c^2} \sinh^{-1} \dfrac a {r_1} }       | c =  }} {{eqn | o =       | ro= +       | r = \dfrac 1 {15 a^2 b c^2} \paren {c^6 \sinh^{-1} \dfrac b c + a^6 \sinh^{-1} \dfrac b a - {r_2}^2 \paren { {r_2}^4 - 8 c^2 a^2} \sinh^{-1} \dfrac b {r_2} }       | c =  }} {{eqn | o =       | ro= +       | r = \dfrac 1 {15 a^2 b^2 c} \paren {a^6 \sinh^{-1} \dfrac c a + b^6 \sinh^{-1} \dfrac c b - {r_3}^2 \paren { {r_3}^4 - 8 a^2 b^2} \sinh^{-1} \dfrac c {r_3} }       | c =  }} {{eqn | o =       | ro= -       | r = \dfrac 4 {15 a b c} \paren {a^4 \arcsin \dfrac {b c} {r_2 r_3} + b^4 \arcsin \dfrac {a c} {r_3 r_1} + c^4 \arcsin \dfrac {a b} {r_1 r_2} }       | c =  }} {{end-eqn}} where: {{begin-eqn}} {{eqn | l = r       | r = \sqrt {a^2 + b^2 + c^2} }} {{eqn | l = r_1       | r = \sqrt {b^2 + c^2} }} {{eqn | l = r_2       | r = \sqrt {a^2 + c^2} }} {{eqn | l = r_3       | r = \sqrt {a^2 + b^2} }} {{end-eqn}}


Mean Distance between Two Random Points in Unit Cube

The mean distance $R$ between $2$ points chosen at random from the interior of a unit cube is given by: {{begin-eqn}} {{eqn | l = R       | r = \frac {4 + 17 \sqrt 2 - 6 \sqrt3 - 7 \pi} {105} + \frac {\map \ln {1 + \sqrt 2 } } 5 + \frac {2 \, \map \ln {2 + \sqrt 3} } 5 }} {{eqn | o = \approx       | r = 0 \cdotp 66170 \, 71822 \, 67176 \, 23515 \, 582 \ldots       | c =  }} {{end-eqn}} The value $R$ is known as the Robbins constant.


Uniform Matroid is Matroid

Let $S$ be a finite set of cardinality $n$. Let $0 \le k \le n$. Let $U_{k,n} = \struct{S, \mathscr I}$ be the uniform matroid of rank $k$. Then $U_{k,n}$ is a matroid.


Matroid Induced by Linear Independence in Vector Space is Matroid

Let $V$ be a vector space. Let $S$ be a finite subset of $V$. Let $\struct{S, \mathscr I}$ be the matroid induced on $S$ by linear independence in $V$. That is, $\mathscr I$ is the set of linearly independent subsets of $S$. Then $\struct{S, \mathscr I}$ is a matroid.


Matroid Induced by Algebraic Independence is Matroid

Let $L / K$ be a field extension. Let $S \subseteq L$ be a finite subset of $L$. Let $\struct{S, \mathscr I}$ be the matroid induced by algebraic independence over $K$ on $S$. That is, $\mathscr I$ is the set of algebraically independent subsets of $S$. Then $\struct{S, \mathscr I}$ is a matroid.


Matroid Induced by Affine Independence is Matroid

Let $\R^n$ be the $n$-dimensional real Euclidean space. Let $S = \set{x_1, \dots, x_r}$ be a finite subset of $\R^n$. Let $\struct{S, \mathscr I}$ be the matroid induced by affine independence on $S$. That is, $\mathscr I$ is the set of affinely independent subsets of $S$. Then $\struct{S, \mathscr I}$ is a matroid.


Matroid Induced by Linear Independence in Abelian Group is Matroid

Let $\struct{G, +}$ be a torsion-free Abelian group. Let $\struct{G, +, \times}$ be the $\Z$-module associated with $G$. Let $S$ be a finite subset of $G$. Let $\struct{S, \mathscr I}$ be the matroid induced by linear independence in $G$ on $S$. That is, $\mathscr I$ is the set of linearly independent subsets of $S$. Then $\struct{S, \mathscr I}$ is a matroid.


Largest Mutually Coprime Subset of Initial Segment of Natural Numbers

Let $n \in \N$ be a natural number. Consider the set $\N_n$ defined as: :$\N_n = \closedint 1 n = \set {1, 2, \ldots n}$ Let $Q_n$ be the largest subset of $\N_n$ such that no element of $Q_n$ is the divisor of another element of $Q_n$. Let $\map f n$ be the cardinality of $Q_n$. Then for sufficiently large $n$: :$0 \cdotp 6725 \ldots \le \dfrac {\map f n} n \le 0 \cdotp 6736 \ldots$


Independent Set can be Augmented by Larger Independent Set

Let $M = \struct{S, \mathscr I}$ be a matroid. Let $X, Y \in \mathscr I$ such that: :$\size X < \size Y$ Then there exists non-empty $Z \subseteq Y \setminus X$ such that: :$X \cup Z \in \mathscr I$ :$\size {X \cup Z} = \size Y$


All Bases of Matroid have same Cardinality

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\rho: \powerset S \to \Z$ be the rank function of $M$. Let $B$ be a base of $M$. Then: :$\size B = \map \rho S$ That is, all bases of $M$ have the same cardinality, which is the rank of $M$.


Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom

Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. Then $\mathscr B$ is the set of bases of a matroid on $S$ {{iff}} $\mathscr B$ satisfies the base axiom: {{:Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 1}}


Number of Integer Partitions into Sum of Consecutive Primes

Let $n$ be a natural number. Let $\map f n$ denote the number of integer partitions of $n$ where the parts are consecutive prime numbers. For example: :$\map f {41} = 3$ because: :$41 = 11 + 13 + 17 = 2 + 3 + 5 + 7 + 11 + 13$ Then: :$\displaystyle \lim_{x \mathop \to \infty} \dfrac 1 x \sum_{n \mathop = 1}^x \map f n = \ln 2$


Dixon's Theorem (Group Theory)

Let $P_1$ and $P_2$ be distinct elements of the symmetric group on $n$ letters. The probability that $\set {P_1, P_2}$ forms a generator of $S_n$ approaches $\dfrac 3 4$ as $n$ tends to infinity.


P-adic Integer is Limit of Unique P-adic Expansion

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 $x$ is the limit of a unique $p$-adic expansion of the form: :$\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$


P-adic Number is Limit of Unique P-adic Expansion

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $x \in \Q_p$. Then $x$ is the limit of a unique $p$-adic expansion.


Number of Natural Numbers Less than x which are Squares or Sums of Two Squares

Let $x$ be a real number. The number of natural numbers smaller than $x$ which are either square or the sum of $2$ squares is given by the expression: :$\map N x \approx \dfrac {k x} {\sqrt {\ln x} }$ where $k$ is given by: :$k = \sqrt {\dfrac 1 2 \displaystyle \prod_{\substack {r \mathop = 4 n \mathop + 3 \\ \text {$r$ prime} } } \paren {1 - \dfrac 1 {r^2} }^{-1} }$ The number $k$ is known as the Landau-Ramanujan constant: {{:Definition:Landau-Ramanujan Constant}}


Finite Set Contains Subset of Smaller Cardinality

Let $S$ be a finite sets. Let :$\size S = n$ where $\size {\, \cdot \,}$ denotes cardinality. Let $0 \le m \le n$. Then there exists a subset $X \subseteq S$ such that: :$\size X = m$


Bounds on Number of Odd Terms in Pascal's Triangle

Let $P_n$ be the number of odd elements in the first $n$ rows of Pascal's triangle. Then: :$0 \cdotp 812 \ldots < \dfrac {P_n} {n^{\lg 3} } < 1$ where $\lg 3$ denotes logarithm base $2$ of $3$. The lower bound $0 \cdotp 812 \ldots$ is known as the Stolarsky-Harborth constant.


Integral from 0 to 1 of Complete Elliptic Integral of First Kind

Let $G$ denote Catalan's constant. Then: :$2 G = \displaystyle \int_0^1 \map K k \rd k$ where $\map K k$ denotes the complete elliptic integral of the first kind: :$\map K k = \displaystyle \int \limits_0^{\pi / 2} \dfrac {\d \phi} {\sqrt {1 - k^2 \sin^2 \phi} }$


Trivial Group is Smallest Group

Let $G = \struct {\set e, \circ}$ be a trivial group. Then $G$ is the smallest group possible, in that there exists no set with lower cardinality which is the underlying set of a group.


Definition:Salem Constant

The '''Salem constant''' is the greatest real root of Lehmer's polynomial: :$x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1 = 0$ Its value is approximately: :$1 \cdotp 17628 \, 08182 \, 599 \ldots$ {{OEIS|A073011}}


Partial Sums of P-adic Expansion forms Coherent Sequence

Let $p$ be a prime number. Let $\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$ be a $p$-adic expansion. Let $\sequence{\alpha_n}$ be the sequence of partial sums;  that is: :$\forall n \in \N :\alpha_n = \displaystyle \sum_{i \mathop = 0}^n d_i p^i$. Then $\sequence{\alpha_n}$ is a coherent sequence.


Gibbs Phenomenon

The Fourier series overshoots at a jump discontinuity, and adding more terms to the sum does not cause this overshoot to die out. :500px :500px {{finish|Anyone care to flesh this out? I've got bored with it.}}


Mergelyan-Wesler Theorem

Let $P = \sequence {D_1, D_2, \dotsc}$ be an infinite sequence of disjoint open disks whose union is the unit disk $D$ except for a set of measure zero. Let $r_n$ be the radius of $D_n$. Then: :$\displaystyle \sum_{k \mathop = 1}^\infty r_k = +\infty$


Definition:Solid-Packing Constant for Circles

Let $P = \sequence {D_1, D_2, \dotsc}$ be an infinite sequence of disjoint open disks whose union is the unit disk $D$ except for a set of measure zero. Let $r_n$ be the radius of $D_n$. Let $x \in \R_{>0}$ be a (strictly) positive real number. Let $\map {M_x} P$ be defined as: :$\map {M_x} P = \displaystyle \sum_{k \mathop = 1}^\infty {r_k}^x$ For each $P$, there exists a (real) number $\map e P$ such that: :$\map {M_x} P$ is divergent for $x < \map e P$ :$\map {M_x} P$ is convergent for $x > \map e P$ From the Mergelyan-Wesler Theorem: :$1 < \map e P < 2$ for all $P$. The constant $S$ such that: :$S < \map e P$ is known as the '''solid-packing constant for circles'''. It can be interpreted as the fractal dimension of the set of points of $P$ which are not covered by the $D_n$ open disks.


Value of Solid-Packing Constant for Circles

The value of the solid-packing constant for circles is estimated as being: :$S \approx 1 \cdotp 306951 \ldots$


Integer Arbitrarily Close to Rational in Valuation Ring of P-adic Norm

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$. Let $x \in \Q$ such that $\norm x_p \le 1$. Then for all $i \in \N$ there exists $\alpha \in \Z$ such that: :$\norm{x - \alpha}_p \le p^{-i}$


Equivalence Class in P-adic Integers Contains Unique Coherent Sequence

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\mathbf a$ be an equivalence class in $\Q_p$ such that $\norm{\mathbf a}_p \le 1$. Then $\mathbf a$ has exactly one representative that is a coherent sequence.


Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/P-adic Expansion

Then $\mathbf a$ has exactly one representative that is a $p$-adic expansion of the form: :$\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$


Equivalence Class in P-adic Numbers Contains Unique P-adic Expansion

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\mathbf a$ be an equivalence class in $\Q_p$. Then $\mathbf a$ has exactly one representative that is a $p$-adic expansion.


Value of Plastic Constant

The plastic constant $P$ is evaluated as: {{begin-eqn}} {{eqn | l = P       | r = \sqrt [3] {\frac {9 + \sqrt {69} } {18} } + \sqrt [3] {\frac {9 - \sqrt {69} } {18} }       | c =  }} {{eqn | r = 1 \cdotp 32471 \, 79572 \, 44746 \, 02596 \, 09088 \, 54 \ldots       | c =  }} {{end-eqn}}


Plastic Constant is Smallest Pisot-Vijayaraghavan Number

The plastic constant is the smallest Pisot-Vijayaraghavan number.


Unique Integer Close to Rational in Valuation Ring of P-adic Norm

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$. Let $x \in \Q$ such that $\norm{x}_p \le 1$. Then for all $i \in \N$ there exists a unique $\alpha \in \Z$ such that: :$(1): \quad \norm {x - \alpha}_p \le p^{-i}$ :$(2): \quad  0 \le \alpha \le p^i - 1$


Hermite Constant for Dimension 2

The Hermite constant for dimension $2$ is: :$\gamma_2 = \dfrac 2 {\sqrt 3}$ or, as it is often presented: :$\paren {\gamma_2}^2 = \dfrac 4 3$


Upper Bound of Hermite Constant

Let $\gamma_n$ be the Hermite constant of dimension $n$. Then: :$\gamma_n \le \dfrac {\paren {1 + \epsilon_n} n} {\pi e}$ where $e_n \to 0$


Angle between Straight Lines in Plane

Let $L_1$ and $L_2$ be straight lines embedded in a cartesian plane, given by the equations: {{begin-eqn}} {{eqn | ll= L_1:       | l = y       | r = m_1 x + c_1 }} {{eqn | ll= L_2:       | l = y       | r = m_2 x + c_2 }} {{end-eqn}} Then the angle $\psi$ between $L_1$ and $L_2$ is given by: :$\psi = \arctan \dfrac {m_1 - m_2} {1 + m_1 m_2}$


Parallel Straight Lines have Same Slope

Let $L_1$ and $L_2$ be straight lines in the plane. Let $L_1$ and $L_2$ have slopes of $m_1$ and $m_2$ respectively. Then $L_1$ and $L_2$ are parallel {{iff}} $m_1 = m_2$.


Product of Slopes of Perpendicular Lines is Minus 1

Let $L_1$ and $L_2$ be straight lines in the plane. Let $L_1$ and $L_2$ have slopes of $m_1$ and $m_2$ respectively. Then $L_1$ and $L_2$ are perpendicular {{iff}} $m_1 m_2 = -1$.


Slope of Tangent to Lemniscate at Origin

Consider the lemniscate of Bernoulli $M$ embedded in a Cartesian plane such that its foci are at $\tuple {a, 0}$ and $\tuple {-a, 0}$ respectively. Let $O$ denote the origin. The tangents to $M$ at $O$ are at an angle of $45 \degrees = \dfrac \pi 4$ to the $x$-axis. :630px


Area of Lobe of Lemniscate of Bernoulli

Consider the lemniscate of Bernoulli $M$ embedded in a Cartesian plane such that its foci are at $\tuple {a, 0}$ and $\tuple {-a, 0}$ respectively. Let $O$ denote the origin. The area of one lobe of $M$ is $a^2$.


Equation of Cardioid/Polar

Let $C$ be a cardioid embedded in a polar coordinate plane such that: :its stator of radius $a$ is positioned with its center at $\polar {a, 0}$ :there is a cusp at the origin. The polar equation of $C$ is: :$r = 2 a \paren {1 + \cos \theta}$


Length of Perimeter of Cardioid

Consider the cardioid $C$ embedded in a polar plane given by its polar equation: :$r = 2 a \paren {1 + \cos \theta}$ where $a > 0$. The length of the perimeter of $C$ is $16 a$.


Arc Length for Polar Curve

Let $a$ and $b$ be real numbers. Let $\mathcal C$ be a simple curve continuous on $\closedint a b$ and continuously differentiable on $\openint a b$. Let $\mathcal C$ be described by the parametric equations: :$\begin {cases} x & = r \cos \theta \\ y & = r \sin \theta \end {cases}$ where: :$r$ is a function of $\theta$ :$\theta \in \closedint a b$. Then the length $s$ of $\mathcal C$ is given by: :$\displaystyle s = \int_a^b \sqrt {r^2 + \paren {\frac {\d r} {\d \theta} }^2} \rd \theta$


Number of Petals of Odd Index Rhodonea Curve

Let $n$ be an odd positive integer. Let $R$ be a '''rhodonea curve''' defined by one of the polar equations: {{begin-eqn}} {{eqn | l = r       | r = a \cos n \theta }} {{eqn | l = r       | r = a \sin n \theta }} {{end-eqn}} Then $R$ has $n$ petals.


Number of Petals of Even Index Rhodonea Curve

Let $n$ be an even (strictly) positive integer. Let $R$ be a '''rhodonea curve''' defined by one of the polar equations: {{begin-eqn}} {{eqn | l = r       | r = a \cos n \theta }} {{eqn | l = r       | r = a \sin n \theta }} {{end-eqn}} Then $R$ has $2 n$ petals.


Equation of Tractrix

Let $S$ be a cord of length $a$ situated as a (straight) line segment whose endpoints are $P$ and $T$. Let $S$ be aligned along the $x$-axis of a cartesian plane with $T$ at the origin and $P$ therefore at the point $\tuple {a, 0}$. Let $T$ be dragged along the $y$-axis. :File:Tractrix.png


Equation of Witch of Agnesi/Parametric Form

The equation of the Witch of Agnesi can be presented in paremetric form as: :$\begin {cases} x = 2 a \cot \theta \\ y = a \paren {1 - \cos 2 \theta} \end {cases}$


Area of Loop of Folium of Descartes

Consider the folium of Descartes $F$, given in parametric form as: :$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$ The area $\AA$ of the loop of $F$ is given as: :$\AA = \dfrac {3 a^2} 2$


Asymptote to Folium of Descartes

Consider the folium of Descartes $F$, given in parametric form as: :$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$ The straight line whose equation is given by: :$x + y + a = 0$ is an asymptote to $F$.


Behaviour of Parametric Equations for Folium of Descartes according to Parameter

Consider the folium of Descartes $F$, given in parametric form as: :$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$ Then: :$F$ has a discontinuity at $t = -1$. :For $t < -1$, the section in the $4$th quadrant is generated :For $-1 < t \le 0$, the section in the $2$nd quadrant is generated  :For $0 \le t$, the section in the $1$st quadrant is generated.


Evolute of Ellipse/Cartesian Form

The evolute of $E$ is given by the Cartesian equation: :$\paren {a x}^{2 / 3} + \paren {b y}^{2 / 3} = \paren {a^2 - b^2}^{2 / 3}$


Evolute of Ellipse/Parametric Form

The evolute of $E$ can be expressed using the parametric equation: :$\begin {cases} a x = \paren {a^2 - b^2} \cos^3 \theta \\ b y = \paren {a^2 - b^2} \sin^3 \theta \end {cases}$


Evolute of Ellipse

Let $E$ be an ellipse embedded in a Cartesian plane with the equation: :$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$


Equation of Ovals of Cassini

Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$. Let $b$ be a real constant.


Equation of Ovals of Cassini/Cartesian Form

The Cartesian equation: :$\paren {x^2 + y^2 + a^2}^2 - 4 a^2 x^2 = b^4$ describes the '''ovals of Cassini'''.


Equation of Ovals of Cassini/Polar Form

The polar equation: :$r^4 + a^4 - 2 a^2 r^2 \cos 2 \theta = b^4$ describes the '''ovals of Cassini'''.


Lemniscate of Bernoulli is Special Case of Ovals of Cassini

The lemniscate of Bernoulli is a special case of the ovals of Cassini.


Cramer's Rule

Let $n \in \N$. Let $b_1, b_2, \dots, b_n$ be real numbers. Let $\mathbf b = \tuple {b_1, b_2, \dots, b_n}^T$. Let $x_1, x_2, \dots, x_n$ be real numbers. Let $\mathbf x = \tuple {x_1, x_2, \dots, x_n}^T$. Let $A$ be an invertible $n \times n$ matrix with coefficients in $\R$. For each $i \in \set {1, \dots, n}$, let $A_i$ be the matrix obtained by replacing the $i$th column with $\mathbf b$. Let: :$A \mathbf x = \mathbf b$ Then: :$\mathbf x_i = \dfrac {\map \det {A_i} } {\map \det A}$ for each $i \in \set {1, \dots, n}$.


Equation of Limaçon of Pascal

Let $C$ be a circle of diameter $a$ whose circumference passes through the origin $O$ and whose diameter through $O$ lies on the horizontal. Let $b$ be a real constant.


Equation of Limaçon of Pascal/Polar Form

The limaçon of Pascal can be defined by the polar equation: :$r = b + a \cos \theta$


Equation of Cissoid of Diocles

Let $C$ be a circle of radius $a$ whose circumference passes through the origin $O$ and whose diameter through $O$ lies on the horizontal.


Equation of Cissoid of Diocles/Polar Form

The cissoid of Diocles can be defined by the polar equation: :$r = 2 a \sin \theta \tan \theta$


Equation of Cissoid of Diocles/Parametric Form

The cissoid of Diocles can be defined by the parametric equation: :$\begin {cases} x = 2 a \sin^2 \theta \\ y = \dfrac {2 a \sin^3 \theta} {\cos \theta} \end {cases}$


Primitive of One plus x Squared over One plus Fourth Power of x

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


Primitive of Minus One plus x Squared over One plus Fourth Power of x

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


Dissection of Square into 8 Acute Triangles

A square can be dissected into $8$ acute triangles.


Dissection of Square into 9 Acute Triangles

A square can be dissected into $9$ acute triangles.


Length of Chord Projected from Point on Intersecting Circle

Let $C_1$ and $C_2$ be two circles which intersect at $A$ and $B$. Let $T$ be a point on $C_1$. Let $P$ and $Q$ be the points $TA$ and $TB$ intersect $C_2$. :400px Then $PQ$ is constant, wherever $T$ is positioned on $C_1$.


First Projection on Ordered Pair of Sets

Let $a$ and $b$ be sets. Let $w = \tuple {a, b}$ denote the ordered pair of $a$ and $b$. Let $\map {\pr_1} w$ denote the first projection on $w$. Then: :$\displaystyle \map {\pr_1} w = \bigcup \bigcap w$ where $\displaystyle \bigcup$ and $\displaystyle \bigcap$ denote union and intersection respectively.


Intersection of Doubleton

Let $\set {x, y}$ be a doubleton. Then: :$\displaystyle \bigcap \set {x, y} = x \cup y$


Second Projection on Ordered Pair of Sets

Let $a$ and $b$ be sets. Let $w = \tuple {a, b}$ denote the ordered pair of $a$ and $b$. Let $\map {\pr_2} w$ denote the second projection on $w$. Then: :$\displaystyle \map {\pr_2} w = \begin {cases} \displaystyle \map \bigcup {\bigcup w \setminus \bigcap w} & : \displaystyle \bigcup w \ne \bigcap w \\ \displaystyle \bigcup \bigcup w & : \displaystyle \bigcup w = \bigcap w \end {cases}$ where: :$\displaystyle \bigcup$ and $\displaystyle \bigcap$ denote union and intersection respectively. :$\setminus$ denotes the set difference operator.


Order Isomorphism between Tosets is not necessarily Unique

Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be tosets. Let $\struct {S_1, \preccurlyeq_1} \cong \struct {S_2, \preccurlyeq_2}$, that is, let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be order isomorphic. Then it is not necessarily the case that there is exactly one mapping $f: S_1 \to S_2$ such that $f$ is an order isomorphism.


Pi as Sum of Sequence of Reciprocal of Product of Three Consecutive Integers/Lemma

:$\displaystyle \iiint \dfrac x {x^2 + 1} \rd x \rd x \rd x = x \map \arctan x + \dfrac {\paren {x^2 - 1} \map \ln {x^2 + 1} - 3 x^2} 4$ with all integration constants at $0$.


Supremum of Set Equals Maximum of Suprema of Subsets

Let $S$ be a non-empty real set. Let $\set {S_i: i \in \set {1, 2, \ldots, n} }$, $n \in \N_{>0}$, be a set of non-empty subsets of $S$. Let $S = \bigcup S_i$. Then: :$S_i$ has a supremum for every $i$ in $\set {1, 2, \ldots, n}$ {{iff}}: :$S$ has a supremum and, in either case: :$\sup S = \max \set {\sup S_1, \sup S_2, \ldots, \sup S_n}$


Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 1

:$\forall j \in \N: \norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren {j + 1} }$


Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 2

:$\forall j \in \N: \norm {\alpha_{j + 1} - \alpha_j }_p \le p^{-\paren {j + 1} }$


Differential Equation defining Confocal Conics

Consider the equation: :$(1): \quad \dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$ where $a^2 > b^2$ and $-\lambda < a^2$. defining the set of confocal conics whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$. The differential equation defining these confocal conics is: :$x y \paren {\paren {y'}^2 - 1} + \paren {x^2 - y^2 - a^2 + b^2} y' = 0$


Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 3

:$\sequence {\alpha_n}$ and $\sequence {\beta_n}$ are representatives of the same equivalence class in $\Q_p$.


Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 4

:$\sequence {\alpha_j}$ is the only coherent sequence that represents $\mathbf a$.


Elimination of Constants by Partial Differentiation

Let $x_1, x_2, \dotsc, x_m$ be independent variables. Let $c_1, c_2, \dotsc, c_n$ be arbitrary constants. Let this equation: :$(1): \quad \map f {x_1, x_2, \dotsc, x_m, z, c_1, c_2, \dotsc, c_n} = 0$ define a dependent variable $z$ via the implicit function $f$. Then it may be possible to eliminate the constants by successive partial differentiation of $(1)$.


Representatives of same P-adic Number iff Difference is Null Sequence

Let $p$ be a prime number. Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rational numbers $\Q$. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\sequence{\alpha_n}$ and $\sequence{\beta_n}$ be Cauchy sequences in $\struct {Q, \norm {\,\cdot\,}_p}$. Then:  :$\sequence {\alpha_n}$ and $\sequence {\beta_n}$ are representatives of the same equivalence class in $\Q_p$ {{iff}}: :the sequence $\sequence {\alpha_n - \beta_n}$ is a null sequence.