Definition:Adherent Point/Definition 3

A point $x \in S$ is an '''adherent point of $A$''' {{iff}} every neighborhood $N$ of $x$ satisfies: :$A \cap N \ne \O$


Definition:Boundary (Topology)/Definition 1

The '''boundary of $H$''' consists of all the points in the closure of $H$ which are not in the interior of $H$. Thus, the '''boundary of $H$''' is defined as: :$\partial H := H^- \setminus H^\circ$ where $H^-$ denotes the closure and $H^\circ$ the interior of $H$.


Definition:Boundary (Topology)/Definition 2

$x \in S$ is a '''boundary point''' of $H$ if every neighborhood $N$ of $x$ satisfies: :$H \cap N \ne \O$ and :$\overline H \cap N \ne \O$ where $\overline H$ is the complement of $H$ in $S$. The '''boundary of $H$''' consists of all the '''boundary points''' of $H$.


Definition:Set Meeting Set

Let $\family {S_i}_{i \mathop \in I}$ be an family of sets indexed by some  indexing set $I$. The sets in $\family {S_i}$ are said to '''meet''' {{iff}} their intersection is not empty. That is, {{iff}}: :$\displaystyle \bigcap_{i \mathop \in I} \family {S_i} \ne \O$ That is, {{iff}} $\family {S_i}_{i \mathop \in I}$ is not disjoint.


Definition:Disjoint Sets/Family

Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. Then $\family {S_i}_{i \mathop \in I}$ is '''disjoint''' {{iff}} their intersection is empty: :$\displaystyle \bigcap_{i \mathop \in I} S_i = \O$


Definition:Negative Vector

The inverse of a vector $\mathbf x \in G$ is usually denoted $-\mathbf x$, and called the '''negative of $\mathbf x$'''.


Definition:Vector Subtraction

Let $\struct {F, +_F, \times_F}$ be a field. Let $\struct {G, +_G}$ be an abelian group. Let $V := \struct {G, +_G, \circ}_R$ be the corresponding '''vector space over $F$'''. Let $\mathbf x$ and $\mathbf y$ be vectors of $V$. Then the operation of '''(vector) subtraction''' on $\mathbf x$ and $\mathbf y$ is defined as: :$\mathbf x - \mathbf y := \mathbf x + \paren {-\mathbf y}$ where $-\mathbf y$ is the negative of $\mathbf y$. The $+$ on the {{RHS}} is vector addition. === Arrow Representation === {{:Definition:Vector Subtraction/Arrow Representation}}


Definition:Bounded Variation

Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a real function.  For each finite subdivision $P$ of $\closedint a b$, write:  :$P = \set {x_0, x_1, \ldots, x_n}$ with: :$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$ Also write: :$\displaystyle \map {V_f} P = \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } }$ We say $f$ is of '''bounded variation''' if there exists a $M \in \R$ such that: :$\map {V_f} P \le M$ for all finite subdivisions $P$.


Definition:Total Variation

Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a function of bounded variation. Let $X$ be the set of finite subdivisions of $\closedint a b$. For each $P \in X$, write:  :$P = \set {x_0, x_1, \ldots, x_{\size P - 1} }$ with: :$a = x_0 < x_1 < x_2 < \cdots < x_{\size P - 2} < x_{\size P - 1} = b$ Also write: :$\displaystyle \map {V_f} P = \sum_{i \mathop = 1}^{\size P - 1} \size {\map f {x_i} - \map f {x_{i - 1} } }$ We define the '''total variation''' $V_f$ of $f$ on $\closedint a b$ by:  :$\displaystyle V_f = \sup_{P \in X} \paren {\map {V_f} P}$ This supremum is finite as, since $f$ is of bounded variation, there exists $M \in \R$ with: :$\map {V_f} P \le M$ for all $P \in X$.


Definition:Truncated Real-Valued Function

Let $f: \R \to \R$ be a real function. Let $m \in \Z_{>0}$ be a (positive) integer constant. Let $f_m: \R \to \R$ be the real function defined as: :$\forall x \in \Dom f: \map {f_m} x = \begin {cases} \map f x & : \map f x \le m \\ m & : \map f x > m \end {cases}$ Then $\map {f_m} x$ is called '''(the function) $f$ truncated by $m$'''.


Definition:Integrable Function/Unbounded Above

Let $f: \R \to \R$ be a positive real function. Let $f$ be unbounded above on the open interval $\openint a b$. Let $f_m$ denote the function $f$ truncated by $m$ for $m \in \Z_{>0}$. Suppose that $f_m$ is Darboux integrable on $\openint a b$ for all $m \in \Z_{>0}$. Suppose also that the following limit exists: :$\displaystyle \lim_{m \mathop \to \infty} \int_a^b \map {f_m} x \rd x$ Then $f$ is '''integrable on $\openint a b$''' and can be expressed in the conventional notation of the Darboux integral: :$\displaystyle \int_a^b \map f x \rd x$


Definition:Integrable Function/Unbounded

Let $f: \R \to \R$ be a real function. Let $f$ be unbounded on the open interval $\openint a b$. Let: :$f^+$ denote the positive part of $f$ :$f^-$ denote the negative part of $f$. Let $f^+$ and $-f^-$ both be integrable on $\openint a b$. Then $f$ is '''integrable on $\openint a b$ and its (definite) integral is understood to be: :$\displaystyle \int_a^b \map f x \rd x := \int_a^b \map {f^+} x \rd x - \int_a^b \paren {-\map {f^-} x} \rd x$


Definition:Integrable Function/Measure Space

Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f \in \MM_{\overline \R}, f: X \to \overline \R$ be a measurable function. Then $f$ is said to be '''$\mu$-integrable''' {{iff}}: :$\displaystyle \int f^+ \rd \mu < +\infty$ and :$\displaystyle \int f^- \rd \mu < +\infty$ where $f^+$, $f^-$ are the positive and negative parts of $f$, respectively. The integral signs denote $\mu$-integration of positive measurable functions.


Definition:Baire Category

The '''Baire category''' of a topological space $T$ is a way of specifying how dense $T$ is: === First Category (Meager) === {{:Definition:Meager Space}} === Second Category (Non-Meager) === {{:Definition:Non-Meager Space}} {{NamedforDef|René-Louis Baire|cat = Baire}}


Definition:Homeomorphism/Terminology

Let a '''homeomorphism''' exist between $T_\alpha$ and $T_\beta$. Then $T_\alpha$ and $T_\beta$ are said to be '''homeomorphic'''. The symbolism $T_\alpha \sim T_\beta$ is often seen to denote that $T_\alpha$ is '''homeomorphic''' to $T_\beta$.


Definition:Filter Basis/Definition 1

Let $\BB \subset \powerset S$ such that $\O \notin \BB$ and $\BB \ne \O$. Then $\FF := \set {V \subseteq S: \exists U \in \BB: U \subseteq V}$ is a filter on $S$ {{iff}}: :$\forall V_1, V_2 \in \BB: \exists U \in \BB: U \subseteq V_1 \cap V_2$ Such a $\BB$ is called a '''filter basis''' of $\FF$.


Definition:Filter Basis/Generated Filter

$\FF$ is said to be '''generated by $\BB$'''.


Definition:Filter Basis/Definition 2

Let $\BB$ be a subset of a filter $\FF$ on $S$ such that $\BB \ne \O$. Then $\BB$ is a '''filter basis''' of $\FF$ {{iff}}: :$\forall U \in \FF: \exists V \in \BB: V \subseteq U$


Definition:Fleenor-Heronian Triangle

A '''Fleenor-Heronian triangle''' is a '''Heronian triangle''' whose sides have lengths form a set of $3$ consecutive integers.


Definition:Fleenor-Heronian Triangle/Sequence

The sequence of Fleenor-Heronian triangles begins: :$\paren {1, 2, 3}$ :$\paren {3, 4, 5}$ :$\paren {13, 14, 15}$ :$\paren {51, 52, 53}$ :$\paren {193, 194, 195}$ :$\paren {723, 724, 725}$ :$\paren {2701, 2702, 2703}$ :$\paren {10 \, 083, 10 \, 084, 10 \, 085}$ :$\paren {37 \, 633, 37 \, 634, 37 \, 635}$


Definition:Countable Topological Space

A '''countable topological space''' is a '''topological space''' whose underlying set is '''countably infinite'''.


Definition:Uncountable Topological Space

An '''uncountable topological space''' is a '''topological space''' whose underlying set is '''uncountable'''.


Definition:Finite Topological Space

A '''finite topological space''' is a '''topological space''' whose underlying set is '''finite'''.


Definition:Uncountable Discrete Ordinal Space

Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Let $S$ be the set of points of $\hointr 0 \Omega$ of the form $\alpha + 1$, where $\alpha$ is a limit ordinal. Let $\tau$ be the subspace topology induced by the order topology on $\hointr 0 \Omega$. $\struct {S, \tau}$ is known as the '''uncountable discrete ordinal space'''.


Definition:Long Line

Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Let $L$ be the set constructed as follows. Between each ordinal $\alpha \in \hointr 0 \Omega$ and its successor $\alpha + 1$, let a copy of the open (real) unit interval $\openint 0 1$ be inserted. Let a total ordering $\preccurlyeq$ be applied to $L$ according to the betweenness described above. Let the order topology $\tau$ be applied to the ordered structure $\struct {L, \preccurlyeq}$. The resulting topological space $\struct {L, \preccurlyeq, \tau}$ is called the '''long line'''. Informally it can be seen that $L$ is of the form: :$0, \openint 0 1, 1, \openint 0 1, 2, \openint 0 1, \ldots, \openint 0 1, \alpha, \openint 0 1, \alpha + 1, \openint 0 1, \ldots, \openint 0 1, \Omega - 1, \openint 0 1$


Definition:Extended Long Line

Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the open ordinal space on $\Omega$. Let $L^*$ be the set constructed as follows. Between each ordinal $\alpha \in \closedint 0 \Omega$ and its successor $\alpha + 1$, let a copy of the open (real) unit interval $\openint 0 1$ be inserted. Let a total ordering $\preccurlyeq$ be applied to $L^*$ according to the betweenness described above. Let the order topology $\tau$ be applied to the ordered structure $\struct {L^*, \preccurlyeq}$. The resulting topological space $\struct {L^*, \preccurlyeq, \tau}$ is called the '''extended long line'''. Informally it can be seen that $L$ is of the form: :$0, \openint 0 1, 1, \openint 0 1, 2, \openint 0 1, \ldots, \openint 0 1, \alpha, \openint 0 1, \alpha + 1, \openint 0 1, \ldots, \openint 0 1, \Omega - 1, \openint 0 1, \Omega$


Definition:Altered Long Line

Let $\struct {L, \preccurlyeq, \tau}$ denote the '''long line'''. Let $p$ be a point which is not an element of $L$. Consider the set $S := L \cup \set p$. Let $T = \struct {S, \tau_p}$ be the topological space whose topology $\tau_p$ is defined as follows: The open sets of $T$ are the open sets of $L$ together with those generated by the following neighborhoods of $P$: :$\map {U_s} p := \set p \cup \set {\displaystyle \bigcup_{\alpha \mathop = \beta}^\Omega \openint \alpha {\alpha + 1}: 1 \le \beta < \Omega}$ where $\Omega$ denotes the first uncountable ordinal. $\map {U_s} p$ is therefore a right-hand ray without the ordinals. The topological space $T = \struct {S, \tau_p}$ is known as the '''altered long line'''.


Definition:Altered Long Line/Greatest Element

$p$ is the greatest element of $S = L \cup \set p$.


Definition:Unit Square under Lexicographic Ordering

Let $S$ be the unit square in the (real) Cartesian plane: {{begin-eqn}} {{eqn | l = S       | r = \set {\tuple {x, y}: x, y \in \R, 0 \le x \le 1, 0 \le y \le 1}       | c =  }} {{eqn | r = \closedint 0 1 \times \closedint 0 1       | c = where $\closedint 0 1$ is the closed unit interval }} {{end-eqn}} Let $\preccurlyeq$ be the lexicographical ordering applied to $S$: :$\forall \tuple {x_1, y_1}, \tuple {x_2, y_2} \in S: \tuple {x_1, y_1} \preccurlyeq \tuple {x_2, y_2} \iff \begin {cases} x_1 < x_2 \\ x_1 = x_2 \land y_1 \le y_2 \end {cases}$ Let $\tau$ be the order topology be applied to the ordered structure $\struct {S, \preccurlyeq}$. Then the topological space $\struct {S, \preccurlyeq, \tau}$ is known as the '''unit square under lexicographical ordering'''.


Definition:Right Order Topology

Let $\struct {S, \preccurlyeq}$ be a totally ordered set. Let $\tau$ be the topology on $S$ generated by the basis sets of the form: :$S_a = \set {x: a \prec x}$ for $a \in S$. Then the topological space $\struct {S, \preccurlyeq, \tau}$ is known as the '''right order topology on $S$'''.


Definition:Left Order Topology

Let $\struct {S, \preccurlyeq}$ be a totally ordered set. Let $\tau$ be the topology on $S$ generated by the basis sets of the form: :$S_a = \set {x: x \prec a}$ for $a \in S$. Then the topological space $\struct {S, \preccurlyeq, \tau}$ is known as the '''left order topology on $S$'''.


Definition:Nested Interval Topology

Let $S = \openint 0 1$ denote the open real interval: :$\openint 0 1 = \set {x \in \R: 0 < x < 1}$ Let $\tau$ be the topology defined on $\openint 0 1$ by defining the open sets $U_n$ as: :$\forall n \in \N_{>0}: U_n := \openint 0 {1 - \dfrac 1 n}$ together with $\O$ and $S$ itself. Then $\tau$ is referred to as the '''nested interval topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''nested interval space'''.


Definition:Overlapping Interval Topology

Let $S = \closedint {-1} 1$ denote the open real interval: :$\closedint {-1} 1 = \set {x \in \R: -1 \le x \le 1}$ Let $\BB$ be the set: :$\BB = \set {\hointl a 1: -1 < a < 0} \cup \set {\hointr {-1} b: 0 < b < 1}$ where: :$\hointl a 1$ is the half-open interval $\set {x \in \R: a < x \le 1}$. :$\hointr {-1} b$ is the half-open interval $\set {x \in \R: -1 \le x < b}$. Then $\BB$ is the basis for a topology $\tau$ on $\R$. Thus the sets of the form $\openint a b$ such that $a < 0 < b$ are open sets in $S$. $\tau$ is referred to as the '''overlapping interval topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''overlapping interval space'''.


Definition:Hjalmar Ekdal Topology

Let $\tau$ be the topology defined on the (strictly) positive integers $\Z_{>0}$ as follows: A subset $H$ of $\Z_{>0}$ is in $\tau$ {{iff}} $H$ contains the (immediate) successor of every odd integer that is also in $H$: :$\tau := \set {H \subseteq \Z_{>0}: 2 n - 1 \in H \implies 2 n \in H}$ Then $\tau$ is referred to as the '''Hjalmar Ekdal topology'''. The topological space $T = \struct {\Z_{>0}, \tau}$ is referred to as the '''Hjalmar Ekdal space'''.


Definition:Prime Ideal Topology

Let $S$ be the set of all prime ideals $P$ of the integers $\Z$. Let $\BB$ be the set of all sets $V_x$ defined as: :$V_x = \set {P \in S: x \notin P}$ for all $x \in \Z_{>0}$. Then $\BB$ is the basis for a topology $\tau$ on $S$. $\tau$ is referred to as the '''prime ideal topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''prime ideal space'''.


Definition:Divisor Topology

Let $S = \set {x \in \Z: x \ge 2}$ denote the set of integers greater than $1$. Let $\BB$ be the set of all sets $U_n$ defined for all $n \ge 2$ as: :$U_n = \set {x \in \Z_{>0}: x \divides n}$ where $\divides$ denotes the divisor relation. Then $\BB$ is the basis for a topology $\tau$ on $S$. Then $\tau$ is referred to as the '''divisor topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''divisor space'''.


Definition:Evenly Spaced Integer Topology

Let $\Z$ denote the set of integers. Let $\BB$ be the set of sets defined as: :$\BB = \set {a + k \Z: a, k \in \Z, k \ne 0}$ where $a + k \Z := \set {a + k \lambda: \lambda \in \Z}$. Then $\BB$ is the basis for a topology $\tau$ on $S$. Then $\tau$ is referred to as the '''evenly spaced integer topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''evenly spaced integer space'''.


Definition:P-adic Topology

Let $\Z$ denote the set of integers. Let $p \in \Z$ be a fixed prime number. Let $\BB$ be the set of all sets $\map {U_\alpha} n$ defined as: :$\map {U_\alpha} n = \set {n + \lambda p^\alpha: \lambda \in \Z}$ Then $\BB$ is the basis for a topology $\tau$ on $S$. $\tau$ is referred to as the '''$p$-adic topology (on $\Z$)'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''$p$-adic (topological) space'''.


Definition:Relatively Prime Integer Topology

Let $\Z_{>0}$ denote the set of (strictly) positive integers. Let $\BB$ be the set of sets $\set {\map {U_a} b: a, b \in \Z_{>0} }$ where: :$\map {U_a} b = \set {b + n a \in \Z_{>0}: \gcd \set {a, b} = 1}$ where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$. Then $\BB$ is the basis for a topology $\tau$ on $\Z_{>0}$. $\tau$ is then referred to as the '''relatively prime integer topology'''. The topological space $T = \struct {\Z_{>0}, \tau}$ is referred to as the '''relatively prime integer space'''.


Definition:Smirnov's Deleted Sequence Topology

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $A$ denote the set defined as: :$A := \set {\dfrac 1 n: n \in \Z_{>0} }$ Let $\tau$ be the topology defined as: :$\tau = \set {H: \exists U \in \tau_d, B \subseteq A: H = U \setminus B}$ That is, $\tau$ consists of the open sets of $\struct {\R, \tau_d}$ which have had any number of the set of the reciprocals of the positive integers removed. $\tau$ is then referred to as '''Smirnov's deleted sequence topology on $\R$'''.


Definition:Indiscrete Extension of Reals

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $D$ be an everywhere dense subset of $\struct {\R, \tau_d}$ with an everywhere dense complement in $\R$. Let $\BB$ be the set of sets: :$\BB := \set {H: \exists U \in \tau_d: H = U \cap D}$ Let $\tau^*$ be the topology generated from $\tau_d$ by the addition of all sets of $\BB$. :$\tau^* = \tau_d \cup \BB$ $\tau^*$ is then referred to as an '''indiscrete extension of $\R$'''.


Definition:Pointed Extension of Reals

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $D$ be an everywhere dense subset of $\struct {\R, \tau_d}$ with an everywhere dense complement in $\R$. Let $\BB$ be the set of sets defined as: :$\BB = \set {\set x \cup \paren {U \cap D}: x \in U \in \tau_d}$ Let $\tau'$ be the topology generated from $\BB$. $\tau'$ is referred to as a '''pointed extension of $\R$'''.


Definition:Indiscrete Extension of Reals/Rational

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\Q$ denote the set of rational numbers. Let $\tau^*$ be the indiscrete extension of $\struct {\R, \tau_d}$: :$\tau^* = \tau_d \cup \set {H: \exists U \in \tau_d: H = U \cap \Q}$ $\tau^*$ is then referred to as the '''indiscrete rational extension of $\R$'''.


Definition:Pointed Extension of Reals/Irrational

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\Bbb I := \R \setminus \Q$ denote the set of irrational numbers. Let $\BB$ be the set of sets defined as: :$\BB = \set {\set x \cup \paren {U \cap \Bbb I}: x \in U \in \tau_d}$ Let $\tau'$ be the topology generated from $\BB$. $\tau'$ is referred to as '''pointed irrational extension of $\R$'''.


Definition:Pointed Extension of Reals/Rational

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\Q$ denote the set of rational numbers. Let $\BB$ be the set of sets defined as: :$\BB = \set {\set x \cup \paren {U \cap \Q}: x \in U \in \tau_d}$ Let $\tau'$ be the topology generated from $\BB$. $\tau'$ is referred to as the '''pointed rational extension of $\R$'''.


Definition:Discrete Extension of Reals

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $D$ be an everywhere dense subset of $\struct {\R, \tau_d}$ with an everywhere dense complement in $\R$. Let $\BB$ be the set of sets defined as: :$\BB = \tau_d \cup \set {\set x: x \in D}$ Let $\tau*$ be the topology generated from $\BB$. $\tau^*$ is referred to as a '''discrete extension of $\R$'''.


Definition:Discrete Extension of Reals/Rational

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\Q$ denote the set of rational numbers. Let $\BB$ be the set of sets defined as: :$\BB = \tau_d \cup \set {\set x: x \in \Q}$ Let $\tau*$ be the topology generated from $\BB$. $\tau^*$ is referred to as the '''discrete rational extension of $\R$'''.


Definition:Discrete Extension of Reals/Irrational

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\Bbb I := \R \setminus \Q$ denote the set of irrational numbers. Let $\BB$ be the set of sets defined as: :$\BB = \tau_d \cup \set {\set x: x \in \Bbb I}$ Let $\tau*$ be the topology generated from $\BB$. $\tau^*$ is referred to as the '''discrete irrational extension of $\R$'''.


Definition:Telophase Topology

Let $S = \closedint 0 1 \cup \set {1^*}$ where: :$\closedint 0 1$ is the closed unit interval $\set {x \in \R: 0 \le x \le 1}$ :$1^*$ is a second right hand endpoint of $\closedint 0 1$. Let $\BB$ be a local basis defined as: :$\BB = \set {\openint a 1 \cup \set {1^*}: a \in \closedint 0 1}$ Let $\tau$ be the topology generated from $\BB$. $\tau$ is referred to as the '''telophase topology'''.


Definition:Simple Matroid

Let $M = \struct{S, \mathscr I}$ be a matroid. $M$ is called a '''simple matroid''' if it has no loops or parallel elements.


Definition:Irrational Slope Topology

Let $S = \set {\tuple {x, y}: y \ge 0, x, y \in \Q}$. Let $\theta \in \R \setminus \Q$ be some fixed irrational number. :500pxrightthumb$\epsilon$-neighborhood $\map {N_\epsilon} {x, y}$: intervals are of [[Definition:Length of Real Intervallength $\epsilon$]] Let $\BB$ be the set of $\epsilon$-neighborhoods of the form: :$\map {N_\epsilon} {x, y} = \set {\tuple {x, y} } \cup \map {B_\epsilon} {x + \dfrac y \theta} \cup \map {B_\epsilon} {x - \dfrac y \theta}$ where: :$\map {B_\epsilon} \zeta := \set {r \in \Q: \size {r - \zeta} < \epsilon}$ that is, the open $\epsilon$-ball at $\zeta$ in $\Q^2$. Hence each $\map {N_\epsilon} {x, y}$ consists of: :the singleton $\set {\tuple {x, y} }$ together with: :two open intervals of length $\epsilon$ in the rational numbers whose midpoints are at the irrational points $x \pm \dfrac y \theta$ The lines joining $\tuple {x, y}$ to $\tuple {x \pm \dfrac y \theta, 0}$ have slope $\pm \theta$ and are not part of $\map {N_\epsilon} {x, y}$. Let $\tau$ be the topology generated from $\BB$. $\tau$ is referred to as the '''irrational slope topology'''.


Definition:Deleted Diameter Topology

Let $\struct {\R^2, \tau_d}$ be the real number plane with the usual (Euclidean) topology induced by the Euclidean metric $d$. Let $\BB$ be the sub-basis of $\tau_d$ defined as the set of all open balls of $\struct {\R^2, d}$ with the horizontal diameters apart from the centers excluded. Let $\sigma$ be the topology generated from $\BB$. $\sigma$ is referred to as the '''deleted diameter topology'''.


Definition:Deleted Radius Topology

Let $\struct {\R^2, \tau_d}$ be the real number plane with the usual (Euclidean) topology induced by the Euclidean metric $d$. Let $\BB$ be the sub-basis of $\tau_d$ defined as the set of all open balls of $\struct {\R^2, d}$ with the horizontal radii to the right of the centers excluded. Let $\tau$ be the topology generated from $\BB$. $\tau$ is referred to as the '''deleted radius topology'''.


Definition:Half-Plane/Left

=== Open Left Half-Plane === {{:Definition:Half-Plane/Open/Left}} === Closed Left Half-Plane === {{:Definition:Half-Plane/Closed/Left}}


Definition:Half-Plane/Right

=== Open Right Half-Plane === {{:Definition:Half-Plane/Open/Right}} === Closed Right Half-Plane === {{:Definition:Half-Plane/Closed/Right}}


Definition:Half-Plane/Upper

=== Open Upper Half-Plane === {{:Definition:Half-Plane/Open/Upper}} === Closed Upper Half-Plane === {{:Definition:Half-Plane/Closed/Upper}}


Definition:Half-Plane/Lower

=== Open Lower Half-Plane === {{:Definition:Half-Plane/Open/Lower}} === Closed Lower Half-Plane === {{:Definition:Half-Plane/Closed/Lower}}


Definition:Half-Plane/Open/Left

The '''open left half-plane''' $H_{\text {OL} }$ is the area of $P$ on the left of $L$. That is, where $x < 0$: :$H_{\text {OL} } := \set {\tuple {x, y}: x \in \R_{<0} }$


Definition:Half-Plane/Closed/Left

The '''closed left half-plane''' $H_{\text {CL} }$ is the area of $P$ on the left of and including $L$. That is, where $x \le 0$: :$H_{\text {CL} } := \set {\tuple {x, y}: x \in \R_{\le 0} }$


Definition:Half-Plane/Closed/Right

The '''closed right half-plane''' $H_{\text {CR} }$ is the area of $P$ on the right of and including $L$. That is, where $x \ge 0$: :$H_{\text {CR} } := \set {\tuple {x, y}: x \in \R_{\ge 0} }$


Definition:Half-Plane/Closed/Upper

The '''closed upper half-plane''' $H_{\text {CU} }$ is the area of $P$ above and including $L$. That is, where $y \ge 0$: :$H_{\text {CU} } := \set {\tuple {x, y}: y \in \R_{\ge 0} }$


Definition:Half-Plane/Closed/Lower

The '''closed lower half-plane''' $H_{\text {CL} }$ is the area of $P$ below and including $L$. That is, where $y \le 0$: :$H_{\text {CL} } := \set {\tuple {x, y}: y \in \R_{\le 0} }$


Definition:Half-Plane/Open/Lower

The '''open lower half-plane''' $H_{\text {OL} }$ is the area of $P$ below $L$. That is, where $y < 0$: :$H_{\text {OL} } := \set {\tuple {x, y}: y \in \R_{< 0} }$


Definition:Half-Plane/Open/Upper

The '''open upper half-plane''' $H_{\text {OU} }$ is the area of $P$ above $L$. That is, where $y > 0$: :$H_{\text {OU} } := \set {\tuple {x, y}: y \in \R_{> 0} }$


Definition:Half-Plane/Open/Right

The '''open right half-plane''' $H_{\text {OR} }$ is the area of $P$ on the right of $L$. That is, where $x > 0$: :$H_{\text {OR} } := \set {\tuple {x, y}: x \in \R_{>0} }$


Definition:Half-Disc Topology

Let $P = \set {\tuple {x, y}: x \in \R, y \in \R_{>0} }$ be the open upper half-plane.  Let $\struct {P, \tau_d}$ be the open upper half-plane with the Euclidean topology. Let $L$ denote the $x$-axis Let $\BB$ be the set of sets of the form: :$\set x \cup \paren {U \cap P}$ where: :$x \in L$ :$U$ is a Euclidean neighborhood of $x$. Let $\tau^*$ be the topology generated from $\BB$. $\tau^*$ is referred to as the '''half-disc topology'''.


Definition:Irregular Lattice Topology

Let $S$ be the subset of the lattice points of the Cartesian plane defined as: :$S := \set {\tuple {i, j}: i, j \in \Z_{>0} } \cup \set {\tuple {i, 0}: i \in \Z_{\ge 0} }$ Let each point of $S$ be defined as being open. Let each point of $S$ of the form $\tuple {i, 0}$ such that $i \ne 0$ have as a local basis sets $U_n$ of the form: :$\map {U_n} {i, 0} := \set {\tuple {i, k}: k = 0 \text { or } k \ge n}$ Let the point $\tuple {0, 0}$ have as a local basis a set $V_n$ of the form: :$V_n := \set {\tuple {i, k}: i = k = 0 \text { or } i, k \ge n}$ Let $\tau$ be the topology generated from all these $U_n$ and $V_n$. $\tau$ is referred to as the '''irregular lattice topology'''.


Definition:Lattice Point/Rational

$P$ is a '''rational lattice point''' of $\CC$ {{iff}} $a_1, a_2, \ldots, a_n$ are all rational numbers.


Definition:Arens Square

Let $A$ be the subset of the unit square defined as: :$A := \set {\tuple {i, j}: 0 < i < 1, 0 < j < 1, i, j \in \Q} \setminus \set {\tuple {i, j}: i = \dfrac 1 2}$ That is: :$A := \openint 0 1^2 \cup \Q \times \Q \setminus \set {\tuple {\dfrac 1 2}, j: j \in \Q}$ That is, the set of rational points in the interior of the unit square from which are excluded the points whose $x$ coordinates are equal to $\dfrac 1 2$. :rightthumb300pxArens square with example local bases marked Let $S$ be the set defined as: :$S = A \cup \set {\tuple {0, 0} } \cup \set {\tuple {1, 0} } \cup \set {\tuple {\dfrac 1 2, r \sqrt 2}: r \in \Q, 0 < r \sqrt 2 < 1}$ Let $\BB$ be the basis for a topology generated on $S$ be defined by granting: :to each point of $A$ the local basis of open sets inherited by $A$ from the Euclidean topology on the unit square; :to the other points of $S$ the following local bases: {{begin-eqn}} {{eqn | l = \map {U_n} {0, 0}       | o = :=       | r = \set {\tuple {x, y}: 0 < x < \dfrac 1 4, 0 < y < \dfrac 1 n} \cup \set {\tuple {0, 0} } }} {{eqn | l = \map {U_n} {1, 0}       | o = :=       | r = \set {\tuple {x, y}: \dfrac 3 4 < x < 1, 0 < y < \dfrac 1 n} \cup \set {\tuple {1, 0} } }} {{eqn | l = \map {U_n} {\tuple {\dfrac 1 2, r \sqrt 2} }       | o = :=       | r = \set {\tuple {x, y}: \dfrac 1 4 < x < \dfrac 3 4, \size {y - r \sqrt 2} < \dfrac 1 n} }} {{end-eqn}} Let $\tau$ be the topology generated from $\BB$. $\struct {S, \tau}$ is referred to as the '''Arens square'''.


Definition:Simplified Arens Square

Let $A$ be the set of points in the interior of the unit square: :$A := \set {\tuple {i, j}: 0 < i < 1, 0 < j < 1, i, j \in \R} = \openint 0 1^2$ :rightthumb300pxSimplified Arens square with example local bases marked Let $S$ be the set defined as: :$S = A \cup \set {\tuple {0, 0} } \cup \set {\tuple {1, 0} }$ Let $\BB$ be the basis for a topology generated on $S$ be defined by granting: :to each point of $A$ the local basis of open sets inherited by $A$ from the Euclidean topology on the unit square; :to the other points of $S$ the following local bases: {{begin-eqn}} {{eqn | l = \map {U_n} {0, 0}       | o = :=       | r = \set {\tuple {x, y}: 0 < x < \dfrac 1 2, 0 < y < \dfrac 1 n} \cup \set {\tuple {0, 0} } }} {{eqn | l = \map {U_m} {1, 0}       | o = :=       | r = \set {\tuple {x, y}: \dfrac 1 2 < x < 1, 0 < y < \dfrac 1 m} \cup \set {\tuple {1, 0} } }} {{end-eqn}} Let $\tau$ be the topology generated from $\BB$. $\struct {S, \tau}$ is referred to as the '''simplified Arens square'''.


Definition:Metrizable Tangent Disc Topology

Let $A$ be a countable subset of the $x$-axis in the Cartesian plane $\R^2$. Let $P$ denote the open upper half-plane in $\R^2$. Let $S := A \cup P$. Let $\struct {S, \tau}$ be the topological subspace of the Niemytzki plane. $\struct {S, \tau}$ is referred to as the '''metrizable tangent disc topology'''.


Definition:Sorgenfrey's Half-Open Square Topology

Let $A = \struct {\R, \sigma}$ be a Sorgenfrey line. Let $S = A \times A$ denote the product space of $A$ with itself, whose topology $\tau$ is such that: :the neighborhood of a point $\tuple {p, q}$ is a rectangle in $\R^2$ of the form: ::$\set {\set {x, y}: p \le x < p + \epsilon_1, q \le y < q + \epsilon_2}$ :for some $\epsilon_1, \epsilon_2 \in \R_{>0}$. $\struct {S, \tau}$ is referred to as '''Sorgenfrey's half-open square topology'''.


Definition:Michael's Product Topology

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\Bbb I := \R \setminus \Q$ denote the set of irrational numbers. Let $\struct {S, \sigma} := \struct {\R, \tau^*} \times \struct {\Bbb I, \tau'}$, where: :$\tau^*$ is the discrete irrational extension of $\tau_d$ by $\Bbb I$ :$\tau'$ is the subspace topology on $\Bbb I$ induced by $\tau_d$. $\struct {S, \sigma}$ is referred to as '''Michael's product topology'''.


Definition:Tychonoff Plank

Let $\omega$ be the first transfinite ordinal. Let $\Omega$ be the first uncountable ordinal. Let $\closedint 0 \Omega$ and $\closedint 0 \omega$ be closed ordinal spaces which have both been given the interval topology. The '''Tychonoff plank''' is the topological space defined as: :$T = \closedint 0 \Omega \times \closedint 0 \omega$


Definition:Tychonoff Plank/Deleted

Let $S = \closedint 0 \Omega$ and $\closedint 0 \omega$ be closed ordinal spaces which have both been given the interval topology. Hence let $T = \struct {S, \tau}$ denote the Tychonoff plank. The '''deleted Tychonoff plank''' is the topological subspace defined as: :$T_\infty = \struct {S \setminus \set {\tuple {\Omega, \omega} }, \tau}$


Definition:Alexandroff Plank

:rightthumb600px Let $\Omega$ be the first uncountable ordinal. Let the closed ordinal space $\closedint 0 \Omega$ be given the interval topology. Let the closed real interval $\closedint {-1} 1$ be given the interval topology. Let $\struct {S, \tau}$ be the product space of $\closedint 0 \Omega$ with $\closedint {-1} 1$ Consider the point $p := \tuple {\Omega, 0} \in S$ Let $\sigma$ be the expansion of $\tau$ generated by adding to $\tau$ the sets of the form: :$\map U {\alpha, n} := \set p \cup \hointl \alpha \Omega \times \openint 0 {\dfrac 1 n}$ Then the topological space $T = \struct {S, \sigma}$ is known as the '''Alexandroff plank'''.


Definition:Dieudonné Plank

:rightthumb600px Let $\omega$ be the first transfinite ordinal. Let $\Omega$ be the first uncountable ordinal. Let $S = \paren {\closedint 0 \Omega \times \closedint 0 \omega} \setminus \set {\tuple {\Omega, \omega} }$ be the underlying set of the deleted Tychonoff plank. Let the topology $\tau$ be generated by declaring as open: :each point of $\hointr 0 \Omega \times \hointr 0 \omega$ together with the sets defined as: {{begin-eqn}} {{eqn | l = \map {U_\alpha} \beta       | r = \set {\tuple {\beta, \gamma}: \alpha < \gamma \le \omega} }} {{eqn | l = \map {V_\alpha} \beta       | r = \set {\tuple {\gamma, \beta}: \alpha < \gamma \le \Omega} }} {{end-eqn}} The topological space $\struct {S, \tau}$ is known as the '''Dieudonné plank'''.


Definition:Tychonoff Corkscrew

Let $\omega$ be the first transfinite ordinal. Let $\Omega$ be the first uncountable ordinal. For each ordinal $\alpha$, let $A_\alpha$ denote the totally ordered set defined as: :$A_\alpha := \set {-0, -1, -2, \ldots, \alpha, \ldots, 2, 1, 0}$ with the order topology. {{explain|What is the meaning of the above ordering?}} :thumbright600pxOne instance of the plane $P^*$ Let $P$ be the product space defined as: :$P := A_\Omega \times A_\omega$ Let $P^*$ be the topological subspace of $P$: :$P^* := P \setminus \set {\tuple {\Omega, \omega} }$ Consider an infinite stack of copies of $P^*$. Let a rectangular corkscrew lattice $C$ be formed such that it spirals in both directions by: :$(1): \quad$ cutting each $P^*$ immediately below the positive $A_\Omega$ axis $\tuple {\Omega, \omega}$ to $\tuple {0, \omega}$ :$(2): \quad$ joining the fourth quadrant of one $P^*$ to the first quadrant of the $P^*$ immediately below it, along the positive $A_\Omega$ axis. Let $\sequence {\map { {A_\Omega}^+} i}_{i \mathop = -\infty}^{+ \infty}$ be the family of positive $A_\Omega$ axes. Then $i$ will be referred to as the '''level''' of $A_\Omega$. By convention, the points of $C$ which lie above $\map { {A_\Omega}^+} i$ are at a level '''greater''' than $i$. If $x$ is such a point, this is denoted $\map L x > i$. We create $S$ by adding to $C$ two distinguished points $a^+$ and $a^-$ which can be considered as being infinity points at the top and bottom of the axis of the corkscrew. Basis neighborhoods of $a^+$ consist of all points of $S$ which lie above a certain level. Basis neighborhoods of $a^-$ consist of all points of $S$ which lie below a certain level. The topological space $\struct {S, \tau}$ so generated is referred to as the '''Tychonoff corkscrew'''.


Definition:Tychonoff Corkscrew/Deleted

Let $T = \struct {S, \tau}$ denote the Tychonoff corkscrew. Let $a^-$ denote the distinguished point of $S$ considered as being the infinity point at the bottom of $T$. The '''deleted Tychonoff corkscrew''' is the topological subspace defined as: :$T_\infty = \struct {S \setminus \set {a^-}, \tau}$


Definition:Thomas's Plank/Definition 1

Let $L_n$ be lines embedded in the Cartesian plane $\R^2$ defined as: :$\forall n \in \N: L_n = \begin {cases} \set {\tuple {x, 0}: x \in \openint 0 1} & : n = 0 \\ \set {\tuple {x, \dfrac 1 n}: x \in \hointr 0 1} & : n > 0 \end {cases}$ Let $S = \displaystyle \bigcup {n \mathop \in \N}$. Let a topology $\tau$ be applied to $S$, defined as follows: :For $n \ge 1$, each point of $L_n$ except for $\tuple {0, \dfrac 1 n}$ is open. :neighborhood bases of $\tuple {0, \dfrac 1 n}$ are subsets of $L_n$ with finite complements. :neighborhood bases of $\tuple {x, 0}$ are the sets $\map {U_i} {x, 0}$ defined as: ::$\map {U_i} {x, 0} := \set {\tuple{x, \dfrac 1 n}: n > i}$ '''Thomas's plank''' is the topological space $\struct {S, \tau}$.


Definition:Thomas's Plank/Definition 2

Let $L := \openint 0 1$ denote the open unit interval. Let $S_1 := L \cup \set p$ denote the Alexandroff extension of $L$. Let $S_2 := \Z_{>0} \cup \set q$ denote the Alexandroff extension of the (strictly) positive integers $\Z_{>0}$. Let $\struct {S, \tau} := \paren {S_1 \times S_2} \setminus \set {\tuple {p, q} }$ be the subspace of the product space $S_1 \times S_2$ with $\set {\tuple {p, q} }$ removed. '''Thomas's plank''' is the topological space $\struct {S, \tau}$.


Definition:Thomas's Corkscrew

Let copies of Thomas's plank be used to construct a corkscrew in the same manner as the Tychonoff plank is used to construct the Tychonoff corkscrew. '''Thomas's corkscrew''' is the topological space that results.


Definition:Parallel Line Topology/Strong

Let $\BB$ be the set of sets of the form: {{begin-eqn}} {{eqn | l = \map V {a, b}       | r = \set {\paren {x, 1}: a \le x < b} }} {{eqn | l = \map U {a, b}       | r = \set {\paren {x, 0}: a < x \le b} \cup \set {\paren {x, 1}: a < x \le b} }} {{end-eqn}} that is: :the left half-open real intervals on $B$ and: :the right half-open real intervals on $A$ together with the interior of their projection onto $B$. $\BB$ is then taken to be the basis for a topology $\sigma$ on $S$. Thus $\sigma$ is referred to as the '''strong parallel line topology'''. The topological space $T = \struct {S, \sigma}$ is referred to as the '''strong parallel line space'''.


Definition:Parallel Line Topology/Weak

Let $\BB$ be the set of sets of the form: {{begin-eqn}} {{eqn | l = \map U {a, b}       | r = \set {\paren {x, 0}: a < x \le b} \cup \set {\paren {x, 1}: a < x \le b} }} {{eqn | l = \map W {a, b}       | r = \set {\paren {x, 0}: a < x < b} \cup \set {\paren {x, 1}: a \le x < b} }} {{end-eqn}} that is: :the left half-open real intervals on $B$ together with the interior of their projection onto $A$ and: :the right half-open real intervals on $A$ together with the interior of their projection onto $B$. $\BB$ is then taken to be the basis for a topology $\tau$ on $S$. Thus $\tau$ is referred to as the '''weak parallel line topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''weak parallel line space'''.


Definition:Concentric Circle Topology

:rightthumb280pxOpen Set of $T$ Let $C_1$ and $C_2$ be concentric circles in the Cartesian plane $\R^2$ such that $C_1$ is inside $C_2$. Let $S = C_1 \cup C_2$. Let $\BB$ be the set of sets consisting of: :all singleton sets of $C_2$ :all open intervals on $C_1$ each together with its projection from the center of the circles onto $C_2$ except for the midpoint. $\BB$ is then taken to be the sub-basis for a topology $\tau$ on $S$. Thus $\tau$ is referred to as the '''concentric circle topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''concentric circle space'''.


Definition:Appert Space

Let $\Z_{>0}$ denote the (strictly) positive integers. For a given subset $H$ of $\Z_{>0}$ and element $n$ of $\Z_{>0}$, let $\map N {n, H}$ denote the number of integers in $E$ which are less than or equal to $n$: :$\forall n \in \Z_{>0}, H \subseteq \Z_{>0}: \map N {n, H} = \card {\set {x \in H: x \le n} }$ Let $\tau$ be the topology defined as: :$1 \notin H \implies H \in \tau$ :$1 \in H$ and $\displaystyle \lim_{n \mathop \to \infty} \dfrac {\map N {n, H} } n = 1 \implies H \in \tau$ for $H \subseteq \Z_{>0}$. $\tau$ is referred to as the '''Appert topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''Appert space'''.


Definition:Maximal Compact Topology

Let $X = \Z_{>0} \times \Z_{>0}$ be the set of all lattice points $\tuple {i, j}$ of the Cartesian plane where $i$ and $j$ are both (strictly) positive integers. Let $S = X \cup \set {x, y}$ where $x$ and $y$ are two new elements of $X$. Let $\tau$ be the topology defined on $S$ as follows: :Each lattice point of $S$ is an open point. :The open neighborhoods of $x$ are of the form $S \setminus A$ where $A$ is any set of lattice points with at most finitely many points on each row :The open neighborhoods of $y$ are of the form $S \setminus B$ where $B$ is any set of lattice points selected from at most finitely many rows. $\tau$ is referred to as the '''maximal compact topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''maximal compact space'''.


Definition:Minimal Hausdorff Topology

:thumbright420pxExamples of [[Definition:Neighborhood Basisneighborhood bases of $a$ and $-a$]] Let $\omega$ be the first transfinite ordinal. Let $\struct {A, \tau_1}$ denote the totally ordered set defined as: :$A := \set {1, 2, 3, \ldots, \omega, \ldots, -3, -2, -1}$ with the interval topology $\tau_1$. Let $\struct {\Z_{>0}, \tau_2}$ denote the (strictly) positive integers with the discrete topology $\tau_2$. Let $S$ be defined as: :$S = A \times \Z_{>0} \cup \set {a, -a}$ where $a$ and $-a$ are new elements of $S$. Let $\tau$ be the topology defined on $S$ as: :the product topology on $\struct {A, \tau_1} \times \struct {\Z_{>0}, \tau_2}$ together with neighborhood bases of $a$ and $-a$ defined as: {{begin-eqn}} {{eqn | l = \map { {M_n}^+} a       | o = :=       | r = \set a \cup \set {\tuple {i, j}: i < \omega, j > n} }} {{eqn | l = \map { {M_n}^-} {-a}       | o = :=       | r = \set {-a} \cup \set {\tuple {i, j}: i > \omega, j > n} }} {{end-eqn}} $\tau$ is referred to as the '''minimal Hausdorff topology'''. The topological space $T = \struct {S, \tau}$ is referred to as the '''minimal Hausdorff space'''.


Definition:Complex Vector Space

Let $\C$ be the set of complex numbers. Then the $\C$-module $\C^n$ is called the '''complex ($n$-dimensional) vector space'''.


Definition:Linearly Independent/Set/Complex Vector Space

Let $\tuple {\C^n, +, \cdot}_\C$ be a complex vector space. Let $S \subseteq \C^n$. Then $S$ is a '''linearly independent set of complex vectors''' if every finite sequence of distinct terms in $S$ is a linearly independent sequence. That is, such that: :$\displaystyle \forall \set {\lambda_k: 1 \le k \le n} \subseteq \C: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0 \implies \lambda_1 = \lambda_2 = \cdots = \lambda_n = 0$ where $\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n$ are distinct elements of $S$.


Definition:Linearly Dependent/Set/Complex Vector Space

Let $\struct {\C^n, +, \cdot}_\C$ be a complex vector space. Let $S \subseteq \C^n$. Then $S$ is a '''linearly dependent set''' if there exists a sequence of distinct terms in $S$ which is a linearly dependent sequence. That is, such that: :$\displaystyle \exists \set {\lambda_k: 1 \le k \le n} \subseteq \C: \sum_{k \mathop = 1}^n \lambda_k \mathbf v_k = \mathbf 0$ where $\set {\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n} \subseteq S$, and such that at least one of $\lambda_k$ is not equal to $0$.


Definition:Pointwise Operation/Induced Structure

The algebraic structure $\struct {T^S, \oplus}$ is called the '''algebraic structure on $T^S$ induced by $\circ$'''.


Definition:Pointwise Addition of Linear Operators

Let $V$ be a vector space. Let $\map \LL V$ denote the set of linear operators on $V$. :$+: \map \LL V \times \map \LL V \to \map \LL V: \forall S, T \in \map \LL V:$ ::$\forall u \in V: \map {\paren {S + T} } u := \map S u + \map T u$ where $+$ on the {{RHS}} is vector addition.


Definition:Pointwise Addition on Complex Vector Space

Let $\C^n$ be a complex vector space. Let $S$ and $T$ be linear operators on $\C^n$. Then the '''pointwise sum of $S$ and $T$''' is defined as: :$S + T: \C^n \to \C^n:$ ::$\forall u \in \C^n: \map {\paren {S + T} } u := \map S u + \map T u$ where $+$ on the {{RHS}} is complex vector addition.


Definition:Matrix Space/Real

Let $\R$ denote the set of real numbers. The '''$m \times n$ matrix space over $\R$''' is referred to as the '''real matrix space''', and can be denoted $\map {\MM_\R} {m, n}$.


Definition:Matrix Space/Complex

Let $\C$ denote the set of complex numbers. The '''$m \times n$ matrix space over $\C$''' is referred to as the '''complex matrix space''', and can be denoted $\map {\MM_\C} {m, n}$.


Definition:Ones Matrix/Square

A '''square ones matrix of order $n$''' is a square matrix of order $n$ all of whose elements are $1$. It is often denoted $\mathbf J_n$.


Definition:Matrix Entrywise Addition/Ring

Let $\struct {R, +, \cdot}$ be a ring. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$. Let $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$. Then the '''matrix entrywise sum of $\mathbf A$ and $\mathbf B$''' is written $\mathbf A + \mathbf B$, and is defined as follows. Let $\mathbf A + \mathbf B = \mathbf C = \sqbrk c_{m n}$. Then: :$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = a_{i j} + b_{i j}$ Thus $\sqbrk c_{m n}$ is the $m \times n$ matrix whose entries are made by performing the (ring) addition operation $+$ on corresponding entries of $\mathbf A$ and $\mathbf B$.


Definition:Limit of Sequence/Normed Vector Space

Let $L \in X$. Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$. Let $\sequence {x_n}_{n \mathop \in \N}$ converge to $L$.  Then $L$ is a '''limit of  $\sequence {x_n}_{n \mathop \in \N}$ as $n$ tends to infinity''' which is usually written: :$\displaystyle L = \lim_{n \mathop \to \infty} x_n$


Definition:Zero Matrix/Ring

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\map {\MM_R} {m, n}$ be an $m \times n$ matrix space over $R$. The '''zero matrix of $\map {\MM_R} {m, n}$''', denoted $\mathbf 0_R$, is the $m \times n$ matrix whose elements are all $0_R$, and can be written $\sqbrk {0_R}_{m n}$.


Definition:Complex Matrix

A '''complex matrix''' is a matrix whose elements are all complex numbers.


Definition:Zero Matrix

Let $\Bbb F$ be one of the standard number system $\N$, $\Z$, $\Q$, $\R$ and $\C$. Let $\map \MM {m, n}$ be an $m \times n$ matrix space over $\Bbb F$. The '''zero matrix of $\map \MM {m, n}$''', denoted $\mathbf 0$, is the $m \times n$ matrix whose elements are all zero, and can be written $\sqbrk 0_{m n}$.


Definition:Matrix Scalar Product/Ring

Let $\struct {R, +, \circ}$ be a ring. Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $\struct {R, +, \circ}$. Let $\lambda \in R$ be any element of $R$. The '''scalar product of $\lambda$ and $\mathbf A$''' is defined as follows. Let $\lambda \circ \mathbf A = \mathbf C$. Then: :$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = \lambda \circ a_{i j}$ Thus $\sqbrk c_{m n}$ is the $m \times n$ matrix composed of the product of $\lambda$ with the corresponding elements of $\mathbf A$.


Definition:Negative Matrix/Ring

Let $\struct {R, +, \circ}$ be a ring. Let $\map {\MM_R} {m, n}$ denote the $m \times n$ matrix space over $\struct {R, +, \circ}$. Let $\mathbf A = \sqbrk a_{m n}$ be an element of $\map {\MM_R} {m, n}$. Then the '''negative (matrix) of $\mathbf A$''' is denoted and defined as: :$-\mathbf A := \sqbrk {-a}_{m n}$ where $-a$ is the ring negative of $a$.


Definition:Inverse Matrix/Right

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix of order $m \times n$. Let $\mathbf B = \sqbrk b_{n m}$ be a matrix of order $n \times m$ such that: :$\mathbf A \mathbf B = I_m$ where $I_m$ denotes the unit matrix of order $m$. Then $\mathbf B$ is known as a '''right inverse (matrix)''' of $\mathbf A$.


Definition:Inverse Matrix/Left

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix of order $m \times n$. Let $\mathbf B = \sqbrk b_{n m}$ be a matrix of order $n \times m$ such that: :$\mathbf B \mathbf A = I_n$ where $I_n$ denotes the unit matrix of order $n$. Then $\mathbf B$ is known as a '''left inverse (matrix)''' of $\mathbf A$.


Definition:Row Operation

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$. A '''row operation''' on $\mathbf A$ is a sequence of '''elementary row operations''' performed on $\mathbf A$ in turn.


Definition:Echelon Matrix/Echelon Form/Non-Unity Variant/Definition 1

$\mathbf A$ is in '''non-unity echelon form''' {{iff}}: :$(1): \quad$ Each row (except perhaps row $1$) starts with a sequence of zeroes :$(2): \quad$ Except when for row $k$ and row $k + 1$ are zero rows, the number of zeroes in this initial sequence in row $k + 1$ is strictly greater than the number of zeroes in this initial sequence in row $k$ :$(3): \quad$ The non-zero rows appear before any zero rows.


Definition:Echelon Matrix/Echelon Form/Non-Unity Variant/Definition 2

$\mathbf A$ is in '''non-unity echelon form''' {{iff}} it contains no adjacent rows of the form: :$\begin {pmatrix} 0 & 0 & \cdots & 0 & x_1 & x_2 & \cdots \\ 0 & 0 & \cdots & 0 & y_1 & y_2 & \cdots \\ \end {pmatrix}$ where: :$(1): \quad y_1 \ne 0$ :$(2): \quad x_1$ can be any value at all, including $0$.


Definition:Echelon Matrix/Echelon Form

$\mathbf A$ is in '''echelon form''' {{iff}}: :$(1): \quad$ The leading coefficient in each non-zero row is $1$ :$(2): \quad$ The leading $1$ in any non-zero row occurs to the right of the leading $1$ in any previous row :$(3): \quad$ The non-zero rows appear before any zero rows.


Definition:Transposition/Adjacent

Let $S_n$ denote the symmetric group on $n$ letters. An '''adjacent transposition''' is a transposition that exchanges two consecutive integers $j$ and $j + 1$, where $1 \le j < n$. In cycle notation, they are denoted: :$\begin {pmatrix} j & j + 1 \end {pmatrix}$


Definition:Rank/Matrix/Definition 1

Let $K$ be a field. Let $\mathbf A$ be an $m \times n$ matrix over $K$. Then the '''rank''' of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is the dimension of the subspace of $K^m$ generated by the columns of $\mathbf A$. That is, it is the dimension of the column space of $\mathbf A$.


Definition:Gaussian Elimination

Let $\mathbf A$ be a matrix over a field $K$. Let $\mathbf E$ be a reduced echelon matrix which is row equivalent to $\mathbf A$. The '''Gaussian elimination method''' is a technique for converting $\mathbf A$ into $\mathbf E$ by means of a sequence of elementary row operations.


Definition:Simultaneous Linear Equations/Matrix Representation/Matrix of Coefficients

The matrix $\mathbf A$ is known as the '''matrix of coeffficients''' of the system.


Definition:Taxicab Norm

Let $\mathbf v = \tuple {v_1, v_2, \ldots, v_n}$ be a vector in $\R^n$. The '''taxicab norm''' of $\mathbf v$ is defined as: :$\displaystyle \norm {\mathbf v}_1 = \sum_{k \mathop = 1}^n \size {v_k}$


Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matrix

Let $\begin {bmatrix} \mathbf A & \mathbf b \end {bmatrix}$ be the block matrix formed from $\mathbf A$ and $\mathbf b$. Then $\begin {bmatrix} \mathbf A & \mathbf b \end {bmatrix}$ is known as the '''augmented matrix''' of the system.


Definition:Augmented Matrix

Let $\mathbf A$ be a matrix of order $n \times m$. Let $\mathbf B$ be a matrix of order $n \times k$. The '''augmented matrix''' of $\mathbf A$ and $\mathbf B$ is the block matrix $\begin {pmatrix} \mathbf A & \mathbf B \end {pmatrix}$ of order $n \times \paren {m + k}$. === Augmented Matrix of Simultaneous Equations === {{:Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matrix}}


Definition:Simultaneous Equations/Linear Equations/Solution

Let $\tuple {x_1, x_2, \ldots, x_n}$ satisfy each of the equations in $\displaystyle \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$. Then $\tuple {x_1, x_2, \ldots, x_n}$ is referred to as a '''solution to the '''system of simultaneous linear equations'''


Definition:Equivalent Systems of Simultaneous Linear Equations

Let $S_1$ and $S_2$ be two systems of simultaneous linear equations. Then $S_1$ and $S_2$ are '''equivalent''' {{iff}}: :every solution to $S_1$ is also a solution to $S_2$ and: :every solution to $S_2$ is also a solution to $S_1$.


Definition:Homogeneous Simultaneous Linear Equations

A '''system of simultaneous linear equations''': :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ is referred to as '''homogeneous''' {{iff}}: :$\forall i \in \set {1, 2, \ldots, m}: \beta_i = 0$ That is: {{begin-eqn}} {{eqn | r = 0       | l = \alpha_{1 1} x_1 + \alpha_{1 2} x_2 + \cdots + \alpha_{1 n} x_n }} {{eqn | r = 0       | l = \alpha_{2 1} x_1 + \alpha_{2 2} x_2 + \cdots + \alpha_{2 n} x_n }} {{eqn | o = \cdots}} {{eqn | r = 0       | l = \alpha_{m 1} x_1 + \alpha_{m 2} x_2 + \cdots + \alpha_{m n} x_n }} {{end-eqn}}


Definition:Consistent Simultaneous Linear Equations

A '''system of simultaneous linear equations''': :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ is referred to as '''consistent''' {{iff}} it has at least one solution.


Definition:Trivial Solution to Homogeneous Simultaneous Linear Equations

Let $S$ be a '''system of homogeneous simultaneous linear equations''': :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$ The solution: :$\tuple {x_1, x_2, \ldots, x_n}$ such that: :$\forall j \in \set {1, 2, \ldots, n}: x_j = 0$ is known as the '''trivial solution''' to $S$.


Definition:Scalar

=== R-Algebraic Structure === Let $\struct {R, +_R, \times_R}$ be the scalar ring of an $R$-algebraic structure $\struct {S, *_1, *_2, \ldots, *_n, \circ}_R$. {{:Definition:Scalar/R-Algebraic Structure}} === Module === Let $\struct {R, +_R, \times_R}$ be the scalar ring of a module $\struct {G, +_G, \circ}_R$. {{:Definition:Scalar/Module}} === Vector Space === Let $\struct {K, +_K, \times_K}$ be the scalar field of a vector space $\struct {G, +_G, \circ}_K$. {{:Definition:Scalar/Vector Space}} === Scalar (Matrix Theory) === {{:Definition:Scalar (Matrix Theory)}} === Scalar Quantity === {{:Definition:Scalar Quantity}}


Definition:Scalar (Matrix Theory)

Let $\map \MM {m, n}$ be a matrix space of order $m \times n$ over an underlying structure $R$. The elements of $R$ are referred to as '''scalars'''.


Definition:Matrix Scalar Product/Scalar

The element $\lambda$ of the underlying structure of $\map \MM {m, n}$ is known as a '''scalar'''.


Definition:Equality of Matrices

Let $\mathbf A$ and $\mathbf B$ be matrices over an underlying structure $R$. Then $\mathbf A$ is equal to $\mathbf B$ {{iff}}: :$(1): \quad$ the order of $\mathbf A$ equals the order of $\mathbf B$: $m \times n$, say :$(2): \quad$ for all $i \in \set {1, 2, \ldots, m}$ and $j \in \set {1, 2, \ldots, n}$, we have that $a_{i j} = b_{i j}$.


Definition:Matrix Entrywise Addition/General

Let $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_k$ be matrices all of order of $m \times n$. Then the '''matrix entrywise sum of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_k$''' is written $\mathbf A_1 + \mathbf A_2 + \ldots + \mathbf A_k$, and is defined as follows: Then: :$\forall i \in \closedint 1 m, j \in \closedint 1 n: K_{i j} = \paren {a_1}_{i j} + \paren {a_2}_{i j} + \cdots + \paren {a_k}_{i j}$ where $\paren {a_l}_{i j}$ is the element of $\mathbf A_l$ whose indices are $\tuple {i, j}$. Thus $\mathbf K = \sqbrk k_{m n}$ is the $m \times n$ matrix whose entries are made by performing the adding corresponding entries of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_k$. This can be expressed in summation form as: :$\displaystyle K = \sum_{j \mathop = 1}^k \mathbf A_j$


Definition:Rank/Matrix/Definition 2

Let $K$ be a field. Let $\mathbf A$ be an $m \times n$ matrix over $K$. Let $\mathbf A$ be converted to echelon form $\mathbf B$. Let $\mathbf B$ have exactly $k$ non-zero rows. Then the '''rank''' of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is $k$.


Definition:Perfect Field/Definition 1

$F$ is a '''perfect field ''' {{iff}} $F$ has no inseparable extensions.


Definition:Perfect Field/Definition 2

$F$ is a '''perfect field''' {{iff}} one of the following holds: :$\Char F = 0$ :$\Char F = p$ with $p$ prime and $\Frob$ is an automorphism of $F$ where: :$\Char F$ denotes the characteristic of $F$ :$\Frob$ denotes the Frobenius endomorphism on $F$


Definition:Elementary Operation/Column

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$. The '''elementary column operations''' on $\mathbf A$ are operations which act upon the columns of $\mathbf A$ as follows. For some $i, j \in \closedint 1 n: i \ne j$: {{begin-axiom}} {{axiom | n = \text {ECO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply column $i$ by $\lambda$         | m = \kappa_i \to \lambda \kappa_i }} {{axiom | n = \text {ECO} 2         | t = For some $\lambda \in K$, add $\lambda$ times column $j$ to column $i$         | m = \kappa_i \to \kappa_i + \lambda \kappa_j }} {{axiom | n = \text {ECO} 3         | t = Interchange columns $i$ and $j$         | m = \kappa_i \leftrightarrow \kappa_j }} {{end-axiom}}


Definition:Column Operation

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$. A '''column operation''' on $\mathbf A$ is a sequence of '''elementary column operations''' performed on $\mathbf A$ in turn.


Definition:Column Equivalence

Two matrices $\mathbf A = \sqbrk a_{m n}, \mathbf B = \sqbrk b_{m n}$ are '''column equivalent''' if one can be obtained from the other by a finite sequence of elementary column operations. This relationship can be denoted $\mathbf A \sim \mathbf B$.


Definition:Determinant/Matrix/In Full

When written out in full, the determinant of $\mathbf A$ is denoted: :$\map \det {\mathbf A} = \begin {vmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \\ \end {vmatrix}$


Definition:Determinant/Matrix/Definition 1

Let $\lambda: \N_{> 0} \to \N_{> 0}$ be a permutation on $\N_{>0}$. The '''determinant''' of $\mathbf A$ is defined as: :$\displaystyle \map \det {\mathbf A} := \sum_{\lambda} \paren {\map \sgn \lambda \prod_{k \mathop = 1}^n a_{k \map \lambda k} } = \sum_\lambda \map \sgn \lambda a_{1 \map \lambda 1} a_{2 \map \lambda 2} \cdots a_{n \map \lambda n}$ where: :the summation $\displaystyle \sum_\lambda$ goes over all the $n!$ permutations of $\set {1, 2, \ldots, n}$ :$\map \sgn \lambda$ is the sign of the permutation $\lambda$.


Definition:Determinant/Matrix/Definition 2

The '''determinant''' of $\mathbf A$ is defined as follows: For $n = 1$, the order $1$ determinant is defined as: {{:Definition:Determinant/Matrix/Order 1}} For $n > 1$, the determinant of order $n$ is defined recursively as: :$\displaystyle \map \det {\mathbf A} := \begin {vmatrix} a_{1 1} & a_{1 2} & a_{1 3} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & a_{2 3} & \cdots & a_{2 n} \\ a_{3 1} & a_{3 2} & a_{3 3} & \cdots & a_{3 n} \\  \vdots &  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 1} & a_{n 2} & a_{n 3} & \cdots & a_{n n} \\ \end {vmatrix} = a_{1 1} \begin {vmatrix} a_{2 2} & a_{2 3} & \cdots & a_{2 n} \\ a_{3 2} & a_{3 3} & \cdots & a_{3 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 2} & a_{n 3} & \cdots & a_{n n} \\ \end {vmatrix} - a_{1 2} \begin {vmatrix} a_{2 1} & a_{2 3} & \cdots & a_{2 n} \\ a_{3 1} & a_{3 3} & \cdots & a_{3 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 1} & a_{n 3} & \cdots & a_{n n} \\ \end {vmatrix} + \cdots + \paren {-1}^{n + 1} a_{1 n} \begin {vmatrix} a_{2 1} & a_{2 2} & \cdots & a_{2, n - 1} \\ a_{3 1} & a_{3 3} & \cdots & a_{3, n - 1} \\  \vdots &  \vdots & \ddots &       \vdots \\ a_{n 1} & a_{n 3} & \cdots & a_{n, n - 1} \\ \end {vmatrix}$


Definition:Polygamma Function

The $n$th '''polygamma function''', $\psi_n$, is defined, for $z \in \C \setminus \Z_{\le 0}$, by the $n$th derivative of the digamma function:  :$\map {\psi_n} z = \dfrac {\d^n} {\d z^n} \map \psi z$ where $\psi$ is the digamma function.


Definition:Elementary Matrix/Row Operation

Let $\mathbf E$ be a unit matrix on which exactly one elementary row operation $e$ has been performed. Then $\mathbf E$ is called the '''elementary row matrix''' for the elementary row operation $e$.


Definition:Elementary Matrix/Column Operation

Let $\mathbf E$ be a unit matrix on which exactly one elementary column operation $e$ has been performed. Then $\mathbf E$ is called the '''elementary column matrix''' for the elementary column operation $e$.


Definition:Elementary Matrix Operation

=== Elementary Row Operation === {{:Definition:Elementary Row Operation}} === Elementary Column Operation === {{:Definition:Elementary Column Operation}}


Definition:Elementary Operation/Row

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$. The '''elementary row operations''' on $\mathbf A$ are operations which act upon the rows of $\mathbf A$ as follows. For some $i, j \in \closedint 1 m: i \ne j$: {{begin-axiom}} {{axiom | n = \text {ERO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply row $i$ by $\lambda$         | m = r_i \to \lambda r_i }} {{axiom | n = \text {ERO} 2         | t = For some $\lambda \in K$, add $\lambda$ times row $j$ to row $i$         | m = r_i \to r_i + \lambda r_j }} {{axiom | n = \text {ERO} 3         | t = Exchange rows $i$ and $j$         | m = r_i \leftrightarrow r_j }} {{end-axiom}}


Definition:Matrix Product (Conventional)/Pre-Multiplication

Let $\mathbf A \mathbf B$ be the product of $\mathbf A$ with $\mathbf B$. Then $\mathbf B$ is '''pre-multiplied''' by $\mathbf A$.


Definition:Matrix Product (Conventional)/Post-Multiplication

Let $\mathbf A \mathbf B$ be the product of $\mathbf A$ with $\mathbf B$. Then $\mathbf A$ is '''post-multiplied''' by $\mathbf B$.


Definition:Distance/Points/Normed Vector Space

Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $x, y \in X$ . Then the function $\norm {\, \cdot \,} : X \times X \to \R$: :$\map d {x, y} = \norm {x - y}$ is called the '''distance between $x$ and $y$'''.


Definition:Inverse of Elementary Row Operation

Let $e$ be an elementary row operation which transforms a matrix $\mathbf A$ to another matrix $\mathbf B$. Let $e'$ be an elementary row operation which transforms $\mathbf B$ back to $\mathbf A$. Then $e'$ is the '''inverse of the elementary row operation $e$'''.


Definition:Inverse of Elementary Column Operation

Let $e$ be an elementary column operation which transforms a matrix $\mathbf A$ to another matrix $\mathbf B$. Let $e'$ be an elementary column operation which transforms $\mathbf B$ back to $\mathbf A$. Then $e'$ is the '''inverse of the elementary column operation $e$'''.


Definition:Extended Weight Function

Let $S$ be a set. Let $\mathscr F$ be the set of all finite subsets of $S$. Let $w: S \to \R$ be a weight function. The '''extended weight function''' of $w$ is the function $w^+: \mathscr F \to \R$ defined by: :$\forall A \in \mathscr F : \map {w^+} A = \displaystyle \sum_{a \mathop \in A} \map w a$


Definition:Matrix of Minors

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $a_{i j}$ denote the element whose indices are $\tuple {i, j}$: :$\mathbf A = \begin {pmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \\ \end {pmatrix}$ For each $a_{i j}$, let $b_{i j}$ denote the minor of $a_{i j}$: :$b_{i j} = \begin {vmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1, i - 1} & a_{1, i + 1} & \cdots & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2, i - 1} & a_{2, i + 1} & \cdots & \cdots & a_{2 n} \\  \vdots &  \vdots & \ddots & \vdots &  \vdots & \ddots & \vdots \\ a_{j - 1, 1} & a_{j - 1, 2} & \cdots & a_{j - 1, i - 1} & a_{j - 1, i + 1} & \cdots & \cdots & a_{j - 1, n} \\ a_{j + 1, 1} & a_{j + 1, 2} & \cdots & a_{j + 1, i - 1} & a_{j + 1, i + 1} & \cdots & \cdots & a_{j + 1, n} \\  \vdots &  \vdots & \ddots & \vdots &  \vdots & \ddots & \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n, i - 1} & a_{n, i + 1} & \cdots & \cdots & a_{n n} \\ \end {vmatrix}$ Then $\mathbf B = \sqbrk b_n$ is called the '''matrix of minors''' of $\mathbf A$. :$\mathbf B = \begin {pmatrix} b_{1 1} & b_{1 2} & \cdots & b_{1 n} \\ b_{2 1} & b_{2 2} & \cdots & b_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ b_{n 1} & b_{n 2} & \cdots & b_{n n} \\ \end {pmatrix}$


Definition:Pythagorean Quadrilateral

A '''Pythagorean quadrilateral''' is a convex quadrilateral whose diagonals intersect at right angles, and is formed by fitting four right triangles with integer-valued sides. The interest is finding solutions where no two triangles are similar. The smallest Pythagorean quadrilateral is formed by the triangles with side lengths: :$\tuple {25, 60, 65}, \tuple {91, 60, 109}, \tuple {91, 312, 325}, \tuple {25, 312, 313}$ The smallest primitive Pythagorean quadrilateral, where each Pythagorean triple is primitive is: :$\tuple {28435, 20292, 34933}, \tuple {284795, 20292, 285517}, \tuple {284795, 181908, 337933}, \tuple {28435, 181908, 184117}$ The smallest anti-primitive Pythagorean quadrilateral, where no Pythagorean triples are primitive is: :$\tuple {1209, 6188, 6305}, \tuple {10659, 6188, 12325}, \tuple {10659, 23560, 25859}, \tuple {1209, 23560, 23591}$ with common divisors: :$13, 17, 19, 31$


Definition:EPORN

An '''EPORN''' is a natural number which can be expressed as the product of a number and its reversal in two different ways.


Definition:Dudeney Number

A '''Dudeney number''' is a natural number which is equal to the sum of the digits of its cube when expressed in decimal notation.


Definition:Supremum Norm/Continuous on Closed Interval Real-Valued Function

Let $I = \closedint a b$ be a closed real interval. Let $\map \CC I$ be the space of real-valued functions, continuous on $I$. Let $f \in \map \CC I$. Let $\size {\, \cdot \,}$ denote the absolute value. Suppose $\sup$ denotes the supremum of real-valued functions. Then the '''supremum norm''' over $\map \CC I$ is defined as :$\displaystyle \norm {f}_\infty := \sup_{x \mathop \in I} \size {\map f x}$


Definition:Think of a Number

A '''think of a number''' puzzle is usually in the form of a game between two players. Player '''A''' asks player '''B''' to: :''Think of a number'' perhaps with constraints. Let this number be referred to as $n$. Player '''A''' asks player '''B''' to perform certain arithmetical manipulations on $n$. As a result, player '''B''' is left with another number, which we will refer to as $m$. The game now goes one of $2$ ways: :$(1): \quad$ Player '''A''' announces: ::::''The number you have been left with is $m$.'' :$(2): \quad$ Player '''A''' asks what $m$ is, and on learning what it is, instantaneously replies: ::::''The number you first thought of was $n$.''


Definition:Ancient Egyptian Papyrus

Most of our knowledge of the mathematics of {{WP|Ancient_Egypt|ancient Egypt}} has come to us through a number of documents written on {{WP|Papyrus|papyrus}}.


Definition:Babylonian Mathematics

'''Babylonian mathematics''' is the name given to the mathematics of the {{WP|First_Babylonian_Dynasty|ancient Babylon}}. Its content was arithmetical and algebraical, and considerably in advance of the mathematics of the approximately contemporaneous {{WP|Ancient_Egypt|ancient Egypt}}.


Definition:Loculus of Archimedes

The '''loculus of Archimedes''' is a dissection of the square into $14$ pieces: :500px


Definition:Rusty Compass

A '''rusty compass''' is (despite what its name may make it seem) an ideal tool for drawing circles. It differs from a normal compass in that, once it has been opened, it is too stiff and rusty to be changed. Hence it can draw circles with only one radius.


Definition:Historic Measures

Various historic systems of measurement that are no longer in common use are gathered here.


Definition:Historic Measures/Length

Various historic systems of measurement of length that are no longer in common use are gathered here.


Definition:Historic Measures/Length/Leuca

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''leuca''' }} {{eqn | r = 1500       | c = paces }} {{eqn | r = 7500       | c = feet }} {{eqn | r = 90 \, 000       | c = inches }} {{end-eqn}}


Definition:Heap Problem

A '''heap problem''' is one where: :an unknown $x$ is established :$x$ is expressed as the sum of one or more fractions of $x$ and one or more other whole number. The object is to find $x$.


Definition:Sterling

'''Sterling''' is the system of monetary units that England (later Britain and the United Kingdom) inherited from the Roman system. Until $1971$ its main units were pounds, shillings and pence (that is, "pennies"). From $15$ February $1971$ its main units have been pounds and (new) pence.


Definition:Money

'''Money''' is a measure of purchasing power.


Definition:Penny

A '''penny''' is one of two coins of the Sterling monetary system: === New Penny === {{:Definition:Sterling/Penny/New}} === Old Penny === {{:Definition:Sterling/Penny/Old}}


Definition:Donkey and Mule Problem

A '''Donkey and Mule problem''' is of the form: :$2$ participants are comparing their quantity of goods they own relative to the other. :Participant $1$ suggests transferring some of the goods of one of them to the other, and remarks on their relative quantities after that exchange. :Participant $2$ then suggests the same (or a similar) thing, but transferring the goods in the other direction, and again remarks on their relative quantities after that exchange. From these remarks, $2$ simultaneous linear equations can be set up and (assuming the question is well-crafted) solved. The name comes from the original setting of this problem concerning a '''donkey''' and a '''mule''' each carrying a number of sacks.


Definition:Troy/Talent

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''talent''' }} {{eqn | r = 75       | c = troy pounds }} {{eqn | r = 900       | c = troy ounces }} {{end-eqn}}


Definition:Box-Jenkins Model

The '''Box-Jenkins model''' is a very general mathematical model for time series analysis in forecasting and prediction. === ARMA (Autoregressive Moving Average) === {{:Definition:Box-Jenkins Model/ARMA}} === ARIMA (Autoregressive Integrated Moving Average) === {{:Definition:Box-Jenkins Model/ARIMA}}


Definition:Recreational Chess

'''Recreational chess''' is a subgenre of recreational mathematics which explores the geometry of the chessboard and the moves of its pieces outside of the context of an actual game of chess itself. The name amusingly belies the fact chess is itself a recreation.


Definition:Heronian Triangle/Definition 2

A '''Heronian triangle''' is a triangle whose side lengths and altitudes are all rational numbers.


Definition:Proper Heronian Triangle

A '''proper Heronian triangle''' is a Heronian triangle which is specifically not right-angled.


Definition:Integer Heronian Triangle

An '''integer Heronian triangle''' is a Heronian triangle whose sides are integers.


Definition:Independence System

Let $S$ be a finite set. Let $\mathscr F$ be a set of subsets of $S$ satisfying the independence system axioms: {{:Definition:Independence System Axioms}} The ordered pair $I = \struct{S, \mathscr F}$ is called an '''independence system on $S$''', or simply an '''independence system''' when the context is obvious.


Definition:Independence System Axioms

{{begin-axiom}} {{axiom | n = \text I 1         | mr= \O \in \mathscr F  }} {{axiom | n = \text I 2         | q = \forall X \in \mathscr F: \forall Y \subseteq S         | mr= Y \subseteq X \implies Y \in \mathscr F }} {{end-axiom}}


Definition:Matroid Axioms/Axioms 1

{{begin-axiom}} {{axiom | n = \text I 1         | mr= \O \in \mathscr I  }} {{axiom | n = \text I 2         | q = \forall X \in \mathscr I: \forall Y \subseteq S         | mr= Y \subseteq X \implies Y \in \mathscr I }} {{axiom | n = \text I 3         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}}


Definition:Matroid Axioms/Axioms 2

{{begin-axiom}} {{axiom | n = \text I 1         | mr= \O \in \mathscr I  }} {{axiom | n = \text I 2         | q = \forall X \in \mathscr I: \forall Y \subseteq S         | mr= Y \subseteq X \implies Y \in \mathscr I }} {{axiom | n = \text I 3'         | q = \forall U, V \in \mathscr I         | mr= \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I }} {{end-axiom}}


Definition:Matroid Axioms/Axioms 3

{{begin-axiom}} {{axiom | n = \text I 1         | mr= \O \in \mathscr I  }} {{axiom | n = \text I 2         | q = \forall X \in \mathscr I: \forall Y \subseteq S         | mr= Y \subseteq X \implies Y \in \mathscr I }} {{axiom | n = \text I 3''         | q = \forall U, V \in \mathscr I         | mr= \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren{V \cup Z \in \mathscr I} \land \paren{\size{V \cup Z} = \size U} }} {{end-axiom}}


Definition:Matroid Axioms/Axioms 4

{{begin-axiom}} {{axiom | n = \text I 1         | mr= \O \in \mathscr I  }} {{axiom | n = \text I 2         | q = \forall X \in \mathscr I: \forall Y \subseteq S         | mr= Y \subseteq X \implies Y \in \mathscr I }} {{axiom | n = \text I 3'''         | q = \forall A \subseteq S         | mr= \text{ all maximal subsets } Y \subseteq A \text{ with } Y \in \mathscr I \text{ have the same cardinality} }} {{end-axiom}}


Definition:P-adic Norm/Definition 1

Let $p \in \N$ be a prime. Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on $\Q$. The '''$p$-adic norm''' on $\Q$ is the mapping $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ defined as: :$\forall q \in \Q: \norm q_p := \begin{cases} 0 & : q = 0 \\ p^{-\map {\nu_p} q} & : q \ne 0 \end{cases}$


Definition:P-adic Norm/Definition 2

Let $p \in \N$ be a prime. Let $k, m, n \in \Z : p \nmid m, n$. Let $\displaystyle r := p^k \frac m n$. The '''$p$-adic norm''' on $\Q$ is the mapping $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ defined as: :$\forall r \in \Q: \norm r_p := \begin{cases} 0 & : r = 0 \\ p^{-k} & : r \ne 0 \end{cases}$


Definition:Meridian

=== Terrestrial Meridian === {{:Definition:Meridian (Terrestrial)}} === Celestial Meridian === {{:Definition:Celestial Meridian}} Category:Definitions/Geography Category:Definitions/Spherical Astronomy mg2cgyg0lbb4qelfbv45cd272nq6nwq


Definition:Geography Puzzle

A '''geography puzzle''' is a puzzle whose solution is dependent upon some particular aspect of geography or geodesy.


Definition:Chess/Chessboard/Square/White

A '''white square''' of a chessboard is one of the squares which is of the lighter colour.


Definition:Chess/Chessboard/Square/Black

A '''black square''' of a chessboard is one of the squares which is of the darker colour.


Definition:Formation of Ordinary Differential Equation by Elimination

Let $\map f {x, y, C_1, C_2, \ldots, C_n} = 0$ be an equation: :whose dependent variable is $y$ :whose independent variable is $x$ :$C_1, C_2, \ldots, C_n$ are constants which are deemed to be arbitrary. A '''differential equation''' may be '''formed''' from $f$ by: :differentiating $n$ times {{WRT|Differentiation}} $x$ to obtain $n$ equations in $x$ and $\dfrac {\d^k y} {\d x^k}$, for $k \in \set {1, 2, \ldots, n}$ :'''eliminating''' $C_k$ from these $n$ equations, for $k \in \set {1, 2, \ldots, n}$.


Definition:Real Number/Equality

Two real numbers are defined as being '''equal''' {{iff}} they correspond to the same point on the real number line.


Definition:Nonzero Integer

A '''nonzero integer''' is an element of the set: :$\Z \setminus \set 0$ where: :$\Z$ denotes the set of integers :$\setminus$ denotes set difference. Thus: :$\Z \setminus \set 0 = \set {\ldots, -3, -2, -1, 1, 2, 3, \ldots}$ and can be denoted $\Z_{\ne 0}$.


Definition:Basis Expansion/Termination

Let the basis expansion of $x$ in base $b$ be: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$ Let it be the case that: :$\exists m \in \N: \forall k \ge m: d_k = 0$ That is, every digit of $x$ in base $b$ after a certain point is zero. Then $x$ is said to '''terminate'''.


Definition:Basis Expansion/Recurrence

Let the basis expansion of $x$ in base $b$ be: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$ Let there be a finite sequence of $p$ digits of $x$: :$\tuple {d_{r + 1} d_{r + 1} \ldots d_{r + p} }$ such that for all $k \in \Z_{\ge 0}$ and for all $j \in \set {1, 2, \ldots, p}$: :$d_{r + j + k p} = d_{r + j}$ where $p$ is the smallest $p$ to have this property. That is, let $x$ be of the form: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} \ldots}_b$ That is, $\tuple {d_{r + 1} d_{r + 2} \ldots d_{r + p} }$ repeats from then on, or '''recurs'''. Then $x$ is said to '''recur'''.


Definition:Basis Expansion/Recurrence/Notation

Let the basis expansion of $x$ in base $b$ be: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$ such that $x$ is recurring. Let the non-recurring part of $x$ be: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r}_b$ Let the recurring part of $x$ be: :$\sqbrk {\ldots d_{r + 1} d_{r + 2} \ldots d_{r + p} \ldots}_b$ Then $x$ is denoted: :$x = s.d_1 d_2 d_3 \ldots d_r \dot d_{r + 1} d_{r + 2} \ldots \dot d_{r + p}$ That is, a dot is placed over the first and last digit of the first instance of the recurring part.


Definition:Basis Expansion/Recurrence/Non-Recurring Part

Let the basis expansion of $x$ in base $b$ be recurring: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p } d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} \ldots}_b$ The '''non-recurring part''' of $x$  is: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r}$


Definition:Closed Ball/Metric Space

Let $M = \struct {A, d}$ be a metric space. Let $a \in A$. Let $\epsilon \in \R_{>0}$ be a positive real number. The '''closed $\epsilon$-ball of $a$ in $M$''' is defined as: :$\map { {B_\epsilon}^-} a := \set {x \in A: \map d {x, a} \le \epsilon}$ where $B^-$ recalls the notation of topological closure. If it is necessary to show the metric itself, then the notation $\map { {B_\epsilon}^-} {a; d}$ can be used.


Definition:Multiplicative Inverse/Number

Let $\Bbb F$ be one of the standard number fields: $\Q$, $\R$, $\C$. Let $a \in \Bbb F$ be any arbitrary number. The '''multiplicative inverse''' of $a$ is its inverse under addition and can be denoted: $a^{-1}$, $\dfrac 1 a$, $1 / a$, and so on. :$a \times a^{-1} = 1$


Definition:Instant of Time

An '''instant''' is a specific point in time. Category:Definitions/Time gqd2hk6fo21a7y0qbew88gqj2dafg19


Definition:Time Series

A '''time series''' is a sequence of observations taken at a sequence of instants of time.


Definition:Time Series Analysis

'''Time series analysis''' is the discipline of analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Its main activity is concerned with techniques for determining the nature of the dependence of adjacent observations of such time series.


Definition:Observation

An '''observation''' is an item of raw data that has been obtained as the result of measurement or counting. Hence it is a particular value taken by a variable in statistics.


Definition:Torsion Element

=== Torsion Element of Group === Let $G$ be a group. {{:Definition:Torsion Element of Group}} === Torsion Element of Module === {{:Definition:Torsion Element of Module}} Category:Definitions/Abstract Algebra 49qe6r916v0d0x19ufjdzfby1gb0xwo


Definition:Time Series/Adjacent Observations

Two observations of a '''time series''' are '''adjacent''' {{iff}} the index of one of them is the immediate predecessor of the other (and the other is the immediate successor of the one).


Definition:Time Series/Dependence of Adjacent Observations

A defining characteristic of a '''time series''' is that every pair of adjacent observations is dependent.


Definition:Time Series/Timestamp

A '''timestamp''' of an '''observation''' $x$ of a '''time series''' is the instant with which $x$ is associated.


Definition:Forecasting

In the context of the discipline of '''time series analysis''', '''forecasting''' is the estimation of future observations of a given time series based on knowledge of current and past such observations.


Definition:Time Series/Current Value

The '''current value''' of a '''time series''' $T$ is an '''observation''' $x$ of $T$ whose timestamp is the '''current time'''. That is, it is the most recent observation.


Definition:Past

The '''past''' is the portion of time previous to the present. That is, the '''past''' is what has already happened.


Definition:Present

The '''present''' is the time at this point '''now'''.


Definition:Future

The '''future''' is the portion of time subsequent to the present. That is, the '''future''' is what has not happened yet.


Definition:Transfer Function (Time Series Analysis)

A '''transfer function''', in the context of time series analysis, is a function of time which theoretically models the future output for each possible input.


Definition:Intervention Event

An '''intervention event''' is an event whose occurrence is not characteristic of a particular time series such that it affects the behaviour of that time series. As a result, forecasting future values based upon the past values is likely to be compromised.


Definition:Control Scheme

A '''control scheme''' is a system whereby potential deviations of future values from a desired target may be compensated for, by adjustment of input values.


Definition:Lead Time

When forecasting on a time series, the '''lead time''' is the future time over which forecasts are needed. The '''lead time''' depends upon the problem for which the forecasting is being performed.


Definition:Time Series/Discrete

A '''discrete time series''' is such that the timestamps of the observations occur at well-defined instants, separated one from another by a time interval.


Definition:Time Series/Equispaced

A '''time series''' is '''equispaced''' {{iff}} the time intervals between the timestamps is equal for all pairs of adjacent observations.


Definition:Time Series/Origin

The '''origin''' of a '''time series''' is an arbitrary timestamp of an observation which is chosen in order that all other timestamps can be measured from that '''origin'''.


Definition:Forecast Function

Let $T$ be a time series. Let $S$ denote the range of $T$. Let $L$ denote the set of lead times of $T$. Let $t$ denote the origin of $T$. A '''forecast function''' $\hat z_t : L \to S$ is a function which provides forecasts of the future values of $T$ at the lead times $L$. The objective is to make the mean squares of the deviations $z_{t + l} - \map {\hat z_t} l$ as small as possible for each lead time $l$. In the above: :$\map {\hat z_t} l$ denotes the forecast value of the observation at the timestamp of lead time $l$ :$z_{t + l}$ denotes the actual value of the observation at the timestamp of $l$.


Definition:Time Series/Forecast Value

A '''forecast value''' of a time series $T$ is an estimate of a future value at some lead time $t + l$.


Definition:Time Series/Actual Value

An '''actual value''' of a time series $T$ is the result of a measurement of an observation at some time $t$. The term is usually made in reference to a forecast value made after its lead time has elapsed, and its timestamp is now the current time.


Definition:Hyperbolic Sine/Real

The '''real hyperbolic sine''' function is defined on the real numbers as: :$\sinh: \R \to \R$: :$\forall x \in \R: \sinh x := \dfrac {e^x - e^{-x} } 2$


Definition:Hyperbolic Cosine/Real

The '''real hyperbolic cosine''' function is defined on the real numbers as: :$\cosh: \R \to \R$: :$\forall x \in \R: \cosh x := \dfrac {e^x + e^{-x} } 2$


Definition:Real Hyperbolic Tangent/Definition 2

The '''real hyperbolic tangent''' function is defined on the real numbers as: :$\tanh: \R \to \R$: :$\forall x \in \R: \tanh x := \dfrac {\sinh x} {\cosh x}$ where: :$\sinh$ is the real hyperbolic sine :$\cosh$ is the real hyperbolic cosine


Definition:Real Hyperbolic Tangent/Definition 1

The '''real hyperbolic tangent''' function is defined on the real numbers as: :$\tanh: \R \to \R$: :$\forall x \in \R: \tanh x := \dfrac {e^z - e^{-x} } {e^z + e^{-x} }$


Definition:Real Hyperbolic Cotangent/Definition 1

The '''real hyperbolic cotangent''' function is defined on the real numbers as: :$\coth: \R_{\ne 0} \to \R$: :$\forall x \in \R_{\ne 0}: \coth x := \dfrac {e^x + e^{-x} } {e^x - e^{-x} }$ where it is noted that at $x = 0$: :$e^x - e^{-x} = 0$ and so $\coth x$ is not defined at that point.


Definition:Real Hyperbolic Cotangent/Definition 2

The '''real hyperbolic cotangent''' function is defined on the real numbers as: :$\coth: \R_{\ne 0} \to \R$: :$\forall x \in \R_{\ne 0}: \coth x := \dfrac {\cosh x} {\sinh x}$ where: :$\sinh$ is the real hyperbolic sine :$\cosh$ is the real hyperbolic cosine It is noted that at $x = 0$ we have that $\sinh x = 0$, and so $\coth x$ is not defined at that point.


Definition:Real Hyperbolic Cosecant/Definition 2

The '''real hyperbolic cosecant''' function is defined on the real numbers as: :$\csch: \R_{\ne 0} \to \C$: :$\forall x \in \R_{\ne 0}: \csch x := \dfrac 1 {\sinh x}$ where $\sinh$ is the real hyperbolic sine. It is noted that at $x = 0$ we have that $\sinh x = 0$, and so $\csch x$ is not defined at that point.


Definition:Real Hyperbolic Cosecant/Definition 1

The '''real hyperbolic cosecant''' function is defined on the real numbers as: :$\csch: \R_{\ne 0} \to \R$: :$\forall x \in \R_{\ne 0}: \csch x := \dfrac 2 {e^x - e^{-x} }$ where it is noted that at $x = 0$: :$e^x - e^{-x} = 0$ and so $\csch x$ is not defined at that point.


Definition:Mean Square

Let $S$ be a set of numbers. The '''mean square''' of $S$ is the arithmetic mean of the squares of the elements of $S$: :$s^2 = \dfrac 1 n \displaystyle \sum_{i \mathop = 1}^n {x_i}^2$ where $S = \set {x_1, x_2, \ldots, x_n}$.


Definition:Deviation

Let $S$ be a set of observations. Let $x \in S$. The '''deviation''' of $x$ is the difference between $x$ and some other value, whose nature depends upon the context.


Definition:Deviation from Forecast

Let $T$ be a time series. Let $S$ denote the range of $T$. Let $L$ denote the set of lead times of $T$. Let $\hat z_t$ be a forecast function on $L$. Let $\map {\hat z_t} l$ denote the forecast value of the observation at the timestamp of lead time $l$. Let $z_{t + l}$ denote the actual value of the observation at the timestamp of $l$. The '''deviation (from forecast)''' is the difference between $\map {\hat z_t} l$ and $z_{t + l}$: :$\Delta_l := z_{t + l} - \map {\hat z_t} l$


Definition:Probability Limit/Upper

Let $\Delta_p$ be a probability limit for $\map {\hat z_t} l$. The '''upper probability limit''' of $\map {\hat z_t} l$ is the value $\map {\hat z_t} l + \Delta_p$.


Definition:Probability Limit

Let $T$ be a time series. Let $S$ denote the range of $T$. Let $L$ denote the set of lead times of $T$. Let $\hat z_t$ be a forecast function on $L$. Let $\map {\hat z_t} l$ denote the forecast value of the observation at the timestamp of lead time $l$. A '''probability limit''', for a given probability $p$, is the deviation $\Delta_p$, either positive or negative, from $\map {\hat z_t} l$ such that the probability that the actual value lies within $\Delta_p$ of $\map {\hat z_t} l$ is greater than $p$. That is, the actual value of the time series at $l$, when it occurs, will be within those '''probability limits''' within that stated probability $p$.


Definition:Probability Limit/Lower

Let $\Delta_p$ be a probability limit for $\map {\hat z_t} l$. The '''lower probability limit''' of $\map {\hat z_t} l$ is the value $\map {\hat z_t} l - \Delta_p$.


Definition:Stochastic Model

A '''stochastic model''' is a mathematical model which incorporates stochastic (random) processes in its definition.


Definition:Measure of Central Tendency

A '''measure of central tendency''' is a central or typical value for a probability distribution or set of sample data.


Definition:Median (Statistics)

Let $S$ be a set of quantitative data. Let $S$ be arranged in order of size. The '''median''' is the element of $S$ that is in the '''middle''' of that ordered set. Suppose there are an odd number of element of $S$ such that $S$ has cardinality $2 n - 1$. The '''median''' of $S$ in that case is the $n$th element of $S$. Suppose there are an even number of element of $S$ such that $S$ has cardinality $2 n$. Then the '''middle''' of $S$ is not well-defined, and so the '''median''' of $S$ in that case is the arithmetic mean of the $n$th and $n + 1$th elements of $S$.


Definition:Mode

Let $S$ be a set of quantitative data. The '''mode''' of $S$ is the element of $S$ which occurs most often in $S$. If there is more than one such element of $S$ which occurs equally most often, it is then understood that each of these is a '''mode'''. If there is no element of $S$ which occurs more often (in the extreme case, all elements are equal) then $S$ has '''no mode'''


Definition:Least Significant Digit

Let $b \in \Z: b \ge 2$ be a number base Let $n$ be a number which is reported to $r$ significant figures, to base $b$, that is: :$n = d_1 \times b^k + d_2 \times b^{k - 1} + \dotsb + d_{r - 1} \times b^{k - r + 2} + d_r \times b^{k - r + 1}$ where: :$d_1, d_2, \dotsc, d_r$ are the significant figures of $n$ :$b^k$ is the largest power of $b$ less than or equal to $n$. Then the digit $d_r$ is known as the '''least significant digit''' of $n$. Note that the usual situation is when $b = 10$, but in the field of computer science, binary is usual.


Definition:Most Significant Digit

Let $b \in \Z: b \ge 2$ be a number base Let $n$ be a number which is reported to $r$ significant figures, to base $b$, that is: :$n = d_1 \times b^k + d_2 \times b^{k - 1} + \dotsb + d_{r - 1} \times b^{k - r + 2} + d_r \times b^{k - r + 1}$ where: :$d_1, d_2, \dotsc, d_r$ are the significant figures of $n$ :$b^k$ is the largest power of $b$ less than or equal to $n$. Then the digit $d_1$ is known as the '''most significant digit''' of $n$. Note that the usual situation is when $b = 10$, but in the field of computer science, binary is usual.


Definition:Left (Direction)

The direction '''left''' is '''that way''': :$\gets$


Definition:Right (Direction)

The direction '''right''' is '''that way''': :$\to$


Definition:Time Series/Continuous

A '''continuous time series''' is one in which the set of timestamps of the observations forms a continuous function.


Definition:Backward Shift Operator

Let $T = \sequence {z_t}$ be a discrete time series. The '''backward shift operator''' $B$ is defined as: :$\forall t: \map B {z_t} = z_{t - 1}$


Definition:Forward Shift Operator

Let $T = \sequence {z_t}$ be a discrete time series. The '''forward shift operator''' $F$ is defined as: :$\forall t: \map F {z_t} = z_{t + 1}$ === Iterated === {{:Definition:Forward Shift Operator/Iterated}}


Definition:Forward Shift Operator/Iterated

$F$ can be '''iterated''' on $\sequence {z_t}$ as follows: :$\map {F^m} {z_t} := z_{t + m}$


Definition:Limit of Real Function/Limit at Infinity/Positive

$L$ is the '''limit of $f$ at infinity''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists c \in \R: \forall x > c : \size {\map f x - L} < \epsilon$ This is denoted as: :$\ds \lim_{x \mathop \to \infty} \map f x = L$


Definition:Limit of Real Function/Limit at Infinity/Negative

$L$ is the '''limit of $f$ at minus infinity''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists c \in \R: \forall x < c: \size {\map f x - L} < \epsilon$ This is denoted as: :$\displaystyle \lim_{x \mathop \to - \infty} \map f x = L$


Definition:Limit of Real Function/Limit at Infinity/Positive/Increasing Without Bound

Suppose that: :$\forall M \in \R_{>0}: \exists N \in \R_{>0}: \forall x > N : \map f x > M$ for $M$ sufficiently large. Then we write: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty$ or :$\map f x \to +\infty$ as $x \to +\infty$ That is, $\map f x$ can be made arbitrarily large by making $x$ sufficiently large. This is voiced: :'''$\map f x$ increases without bound as $x$ increases without bound'''. or: :'''$\map f x$ tends to (plus) infinity as $x$ tends to (plus) infinity'''.


Definition:Limit of Real Function/Limit at Infinity/Negative/Increasing Without Bound

Suppose that: :$\forall M \in \R_{>0}: \exists N \in \R_{<0}: \forall x < N : \map f x > M$ for $M$ sufficiently large. Then we write: :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty$ or :$\map f x \to +\infty$ as $x \to -\infty$ This is voiced: :'''$\map f x$ increases without bound as $x$ decreases without bound'''. or: :'''$\map f x$ tends to (plus) infinity as $x$ tends to minus infinity'''.


Definition:Limit of Real Function/Limit at Infinity/Positive/Decreasing Without Bound

Suppose that: :$\forall M \in \R_{<0}: \exists N \in \R_{>0}: x > N \implies \map f x < M$ for $M$ sufficiently large in magnitude. Then we write: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty$ or :$\map f x \to -\infty$ as $x \to +\infty$ That is, $-\map f x$ can be made arbitrarily large by making $x$ sufficiently large. This is voiced: :'''$\map f x$ decreases without bound as $x$ increases without bound'''. or: :'''$\map f x$ tends to minus infinity as $x$ tends to (plus) infinity'''.


Definition:Limit of Real Function/Limit at Infinity/Negative/Decreasing Without Bound

Suppose that: :$\forall M \in \R_{<0}: \exists N \in \R_{<0}: x < N \implies \map f x < M$ for $M$ sufficiently large in magnitude. Then we write: :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty$ or :$\map f x \to +\infty \ \text{as} \ x \to -\infty$ This is voiced: :'''$\map f x$ decreases without bound as $x$ decreases without bound'''. or: :'''$\map f x$ tends to minus infinity as $x$ tends to minus infinity'''.


Definition:Deterministic Model

A '''deterministic model''' is a mathematical model based on physical laws which are possible to be calculated exactly (or nearly exactly) for known inputs.


Definition:Stationary Model

A '''stationary model''' is a stochastic model for describing a time series which assumes that the underlying stochastic process remains in equilibrium about a constant mean level. That is, it is a stochastic model with an underlying stochastic process which is itself stationary.


Definition:Statistical Equilibrium

Let $S$ be a stochastic process. Suppose that the observations of the time series to which $S$ gives rise have a constant mean level. Then $S$ is said to be in '''(statistical) equilibrium'''.


Definition:Stationary Stochastic Process

A '''stationary stochastic process''' is stochastic process which remains in equilibrium about a constant mean level.


Definition:Constant Mean Level

Let $S$ be a stochastic process giving rise to a time series $T$. $T$ has a '''constant mean level''' {{iff}} the mean of $S$ does not change as time progresses.


Definition:Non-Stationary Stochastic Process

A '''non-stationary stochastic process''' is a stochastic process which does not remain in equilibrium about a constant mean level. That is, it is a stochastic process which is not a '''stationary stochastic process'''.


Definition:Non-Stationary Model

A '''non-stationary model''' is a stochastic model for describing a time series which assumes that the underlying stochastic process does not remain in equilibrium about a constant mean level. That is, it is a stochastic model with an underlying stochastic process which is itself non-stationary.


Definition:White Noise Process

Let $S$ be a stochastic process consisting of a time series $\map {z_r} t$ such that: :the terms of the sequence $\sequence {z_r}$ are independent random variables :the terms of $\sequence {z_r}$ are governed by a normal distribution with zero expectation and a given variance $\sigma^2$. Then $S$ is known as a '''white noise process'''.


Definition:Independent Shocks

Let $M$ be a stochastic model which describes a time series in which adjacent observations are highly dependent. Then $M$ may be able to be modelled by a time series whose elements are of the form: :$\map z t = \map {z_d} t + \map {z_r} t$ where: :$\map {z_d} t$ has a deterministic model :$\map {z_r} t$ has a stochastic model which consists of a sequence of independent random variables from a specified probability distribution (usually a white noise process). The terms of the sequence $\sequence {z_r}$ are known as '''independent shocks'''.


Definition:Linear Filter

Let $S$ be a stationary stochastic process governed by a white noise process: :$\map z t = \mu + a_t$ where: :$\mu$ is a constant mean level :$a_t$ is an independent shock at timestamp $t$. A '''linear filter''' takes the terms of $S$, and uses a weight function $\psi$ to apply a weighted sum of the past values so that: {{begin-eqn}} {{eqn | l = \map z t       | r = \mu + a_t + \psi_1 a_{t - 1} + \psi_2 a_{t - 2} + \cdots       | c =  }} {{eqn | r = \mu + \map \psi B a_t       | c =  }} {{end-eqn}} where $B$ denotes the backward shift operator, hence: :$\map \psi B := 1 + \psi_1 B + \psi_2 B^2 + \cdots$ === Transfer Function === {{:Definition:Linear Filter/Transfer Function}} === Stable === {{:Definition:Linear Filter/Stable}}


Definition:Linear Filter/Transfer Function

The operator: :$\map \psi B := 1 + \psi_1 B + \psi_2 B^2 + \cdots$ is the transfer function of $L$.


Definition:Ferrier's Prime

'''Ferrier's prime''' is the prime number: :$\dfrac {2^{148} + 1} {17} = 20 \, 988 \, 936 \, 657 \, 440 \, 586 \,486 \, 151 \, 264 \, 256 \, 610 \, 222 \, 593 \, 863 \, 921$ {{NamedforDef|Aimé Ferrier|cat = Ferrier}}


Definition:Weighted Sum

Let $S = \sequence {x_1, x_2, \ldots, x_n}$ be a sequence of real numbers. Let $\map W x$ be a weight function to be applied to the terms of $S$. The '''weighted sum''' of $S$ is defined as: :$\bar x := \displaystyle \sum_{i \mathop = 1}^n \map W {x_i} x_i$ This means that elements of $S$ with a larger weight contribute more to the '''weighted sum''' than those with a smaller weight.


Definition:Linear Filter/Stable

Consider the sequence $\sequence {\psi_k}$ formed by the weight function $\psi$ of $L$. Suppose that: :$\displaystyle \sum_k \size {\psi_k} < \infty$ Then $L$ is said to be '''stable''', and the model for $S$ is stationary. Hence $\mu$ is the mean about which $S$ varies.


Definition:Autoregressive Model

Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$ Let $M$ be a model where the current value of $S$ is expressed as a finite linear aggregate of the past values along with a shock: :$\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t$ $M$ is known as an '''autoregressive (AR) process of order $p$'''.


Definition:Metrical Geometry

'''Metrical geometry''' is the study of points, lines, surfaces and volumes in real space, and is based on the concepts of distance and angle. It is what is usually understood in the everyday world as '''geometry''' proper.


Definition:Regression Model

Let $S$ be a stochastic process based on an equispaced time series. Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$ Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$: :$\tilde z_t = z_t - \mu$ Let $M$ be a linear filter model defined as :$\tilde z_t = \phi_1 \tilde x_1 + \phi_2 \tilde x_2 + \dotsb + \phi_p \tilde x_p + a$ which relates a dependent variable $z$ to a set of independent variables $x_1, x_2, \dotsc, x_p$ plus an error term $a$. $M$ is known as a '''regression model'''.