Definition:Die

thumb300pxrightTwo six-sided dice A '''die''' (plural: '''dice''') is a device for generating random numbers, traditionally between $1$ and $6$. A die usually consists of a cube on which each face denotes a different number, usually (again) from $1$ to $6$. Traditionally, opposite faces add to $7$, so: :$1$ is opposite $6$ :$2$ is opposite $5$ :$3$ is opposite $4$. The idea is that when you roll it, or throw it, or otherwise cause it to land upon a flattish surface, one of the sides will be face-up, and (with a good design of die) which side that is can not be precisely predicted. Hence it serves as a generator of a discrete uniform distribution. By using dice with different shapes (usually either platonic solids or prisms), a different probability space can be generated. {{DefinitionWanted|Describe general polyhedral dice, or link to a Wikipedia article on FRP or something.}}


Definition:Equiprobability Space

An '''equiprobability space''' is a finite probability space $\struct {\Omega, \Sigma, \Pr}$ with equiprobable outcomes. That is, for all $\omega_i, \omega_j \in \Omega$: :$\map \Pr {\omega_i} = \map \Pr {\omega_j}$ From Probability Measure on Equiprobable Outcomes, we have that: :$\forall \omega \in \Omega: \map \Pr \omega = \dfrac 1 n$ :$\forall A \subseteq \Omega: \map \Pr A = \dfrac {\card A} n$ Category:Definitions/Probability Theory 4q5j6b2l1h3stji0utitop5kg1lzl5l


Definition:Binary Notation

'''Binary notation''' is the technique of expressing numbers in base $2$. That is, every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 2^j$ where $\forall j \in \Z: r_j \in \set {0, 1}$.


Definition:Hexadecimal Notation

'''Hexadecimal notation''' is the technique of expressing numbers in base $16$. Every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 16^j$ where: :$\forall j \in \Z: r_j \in \left\{ {0, 1, \ldots, 15}\right\}$


Definition:General Logarithm

=== Positive Real Numbers === {{:Definition:General Logarithm/Positive Real}} === Complex Numbers === {{:Definition:General Logarithm/Complex}}


Definition:Conditional Probability

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$. We write the '''conditional probability of $A$ given $B$''' as $\condprob A B$, and define it as: :''the probability that $A$ has occurred, given that $B$ has occurred.''


Definition:Independent Events

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$ such that $\map \Pr A > 0$ and $\map \Pr B > 0$. === Definition 1 === {{:Definition:Independent Events/Definition 1}} === Definition 2 === {{:Definition:Independent Events/Definition 2}}


Definition:Disjoint Events

Let $A$ and $B$ be events in a probability space. Then $A$ and $B$ are '''disjoint''' {{iff}}: :$A \cap B = \O$ It follows by definition of probability measure that $A$ and $B$ are '''disjoint''' {{iff}}: :$\map \Pr {A \cap B} = 0$


Definition:Can't Happen

=== Probability Theory === {{:Definition:Can't Happen/Probability Theory}} === Computer Science === {{:Definition:Can't Happen/Computer Science}} Category:Definitions/Language Definitions 3rytkxheuo3ss7q7wnp18lgbfwcv0im


Definition:Partition (Probability Theory)

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. A '''partition of $\Omega$''' is a family $\left\{{B_i: i \in I}\right\}$ of disjoint events such that $\displaystyle \bigcup_i B_i = \Omega$.


Definition:Increasing Sequence of Events

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $\left \langle {A_n}\right \rangle$ be a sequence of events in $\Sigma$. Then $\left \langle {A_n}\right \rangle$ is described as '''increasing''' iff: :$\forall i \in \N: A_i \subseteq A_{i+1}$


Definition:Limit of Sequence of Events

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. === Increasing Sequence of Events === {{:Definition:Limit of Sequence of Events/Increasing}} === Decreasing Sequence of Events === {{:Definition:Limit of Sequence of Events/Decreasing}} Category:Definitions/Probability Theory 7j4gf0u73j8c1bj99gn7ek0m2kh3jsr


Definition:Decreasing Sequence of Events

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $\left \langle {A_n}\right \rangle$ be a sequence of events in $\Sigma$. Then $\left \langle {A_n}\right \rangle$ is described as '''decreasing''' iff: :$\forall i \in \N: A_{i+1} \subseteq A_i$


Definition:Random Variable/Discrete

Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$. <!-- A '''discrete random variable''' on $\struct {\Omega, \Sigma, \Pr}$ is a random variable on $\struct {\Omega, \Sigma, \Pr}$ such that $\Omega_X$, the image of $X$, is a countable subset of $\R$. --> A '''discrete random variable''' on $\struct {\Omega, \Sigma, \Pr}$ is a mapping $X: \Omega \to \R$ such that: :$(1): \quad$ The image of $X$ is a countable subset of $\R$ :$(2): \quad$ $\forall x \in \R: \set {\omega \in \Omega: \map X \omega = x} \in \Sigma$ Alternatively (and meaning exactly the same thing), the second condition can be written as: :$(2)': \quad$ $\forall x \in \R: \map {X^{-1} } x \in \Sigma$ where $\map {X^{-1} } x$ denotes the preimage of $x$. Note that if $x \in \R$ is not the image of any elementary event $\omega$, then $\map {X^{-1} } x = \O$ and of course by definition of event space as a sigma-algebra, $\O \in \Sigma$. Note that a discrete random variable also fulfils the conditions for it to be a random variable.


Definition:Probability Mass Function

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X: \Omega \to \R$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$. Then the '''(probability) mass function''' of $X$ is the (real-valued) function $p_X: \R \to \closedint 0 1$ defined as: :$\forall x \in \R: \map {p_X} x = \begin{cases} \map \Pr {\set {\omega \in \Omega: \map X \omega = x} } & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end{cases}$ where $\Omega_X$ is defined as $\Img X$, the image of $X$. That is, $\map {p_X} x$ is the probability that the discrete random variable $X$ takes the value $x$.


Definition:Bernoulli Distribution

Let $X$ be a discrete random variable on a probability space. Then $X$ has the '''Bernoulli distribution with parameter $p$''' {{iff}}: : $(1): \quad X$ has exactly two possible values, for example $\Img X = \set {a, b}$ : $(2): \quad \map \Pr {X = a} = p$ : $(3): \quad \map \Pr {X = b} = 1 - p$ where $0 \le p \le 1$.


Definition:Bernoulli Trial

A '''Bernoulli trial''' is an experiment whose sample space has two elements, which can be variously described, for example, as: :'''Success''' and '''failure''' :'''True''' and '''False''' :$1$ and $0$ :the classic '''heads''' and '''tails'''. Formally, a '''Bernoulli trial''' is modelled by a probability space $\struct {\Omega, \Sigma, \Pr}$ such that: :$\Omega = \set {a, b}$ :$\Sigma = \powerset \Omega$ :$\map \Pr a = p, \map \Pr b = 1 - p$ where: :$\powerset \Omega$ denotes the power set of $\Omega$ :$0 \le p \le 1$ That is, $\Pr$ obeys a Bernoulli distribution. === Bernoulli Variable === {{:Definition:Bernoulli Trial/Bernoulli Variable}}


Definition:Binomial Distribution

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Then $X$ has the '''binomial distribution with parameters $n$ and $p$''' {{iff}}: :$\Img X = \set {0, 1, \ldots, n}$ :$\map \Pr {X = k} = \dbinom n k p^k \paren {1 - p}^{n - k}$ where $0 \le p \le 1$.


Definition:Poisson Distribution

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Then $X$ has the '''Poisson distribution with parameter $\lambda$''' (where $\lambda > 0$) {{iff}}: :$\Img X = \set {0, 1, 2, \ldots} = \N$ :$\map \Pr {X = k} = \dfrac 1 {k!} \lambda^k e^{-\lambda}$ It is written: :$X \sim \Poisson \lambda$


Definition:Geometric Distribution

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. $X$ has the '''geometric distribution with parameter $p$''' {{iff}}: :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$ :$\map \Pr {X = k} = \paren {1 - p} p^k$ where $0 < p < 1$.


Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. There are two forms of the '''negative binomial distribution''', as follows: === First Form === {{:Definition:Negative Binomial Distribution/First Form}} === Second Form === {{:Definition:Negative Binomial Distribution/Second Form}}


Definition:Bernoulli Process

A '''Bernoulli process''' is a sequence (either finite or infinite) of Bernoulli trials, each of which has the same parameter $p$. That is, '''Bernoulli process (with parameter $p$)''' is a sequence $\left \langle {X_i}\right \rangle$ (either finite or infinite) such that: * The value of each $X_i$ is one of two values (for example: $a$ or $b$). * The probability that $X_i = a$ is the same for all $i$ (for example: $p$). That is, it is a sequence of experiments, all of which can be modelled by the same Bernoulli distribution. Note: The assumption is that the outcomes of all the Bernoulli trials are independent. {{NamedforDef|Jacob Bernoulli|cat=Bernoulli, Jacob}} Category:Definitions/Probability Theory 6576ce743b258dwovtfzmckcrt9i5kw


Definition:Expectation

The '''expectation''' of a random variable is the arithmetic mean of its values.


Definition:Normalized Weight Function

Let $S = \left \langle {x_1, x_2, \ldots, x_n}\right \rangle$ be a sequence of real numbers. Let $W \left({x}\right)$ be a weight function to be applied to the elements of $S$. Then $W$ is defined as being '''normalized''' if: :$\displaystyle \sum_x W \left({x}\right) = 1$ {{SUBPAGENAME}} {{SUBPAGENAME}} 2r5oz2hi2xbnc4etn6rnpkl1y4pooah


Definition:Variance/Discrete

Let $X$ be a discrete random variable. Then the '''variance of $X$''', written $\var X$, is a measure of how much the values of $X$ varies from the expectation $\expect X$, and is defined as:


Definition:Conditional Expectation

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$. Let $B$ be an event in $\struct {\Omega, \Sigma, \Pr}$ such that $\map \Pr B > 0$. The '''conditional expectation of $X$ given $B$''' is written $\expect {X \mid B}$ and defined as: :$\expect {X \mid B} = \displaystyle \sum_{x \mathop \in \image X} x \, \map \Pr {X = x \mid B}$ where: :$\map \Pr {X = x \mid B}$ denotes the conditional probability that $X = x$ given $B$ whenever this sum converges absolutely.


Definition:Independent Random Variables

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$. Then $X$ and $Y$ are defined as '''independent (of each other)''' {{iff}}: :$\map \Pr {X = x, Y = y} = \map \Pr {X = x} \map \Pr {Y = y}$ where $\map \Pr {X = x, Y = y}$ is the joint probability mass function of $X$ and $Y$.


Definition:Uniform Distribution/Discrete

Let $X$ be a discrete random variable on a probability space. Then $X$ has a '''discrete uniform distribution with parameter $n$''' {{iff}}: :$\Img X = \set {1, 2, \ldots, n}$ :$\map \Pr {X = k} = \dfrac 1 n$ That is, there is a number of outcomes with an equal probability of occurrence.  This is written: :$X \sim \DiscreteUniform n$


Definition:Cumulative Distribution Function

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$. The '''cumulative distribution function''' (or '''c.d.f.''') of $X$ is denoted $\map F X$, and defined as: :$\forall x \in \R: \map {\map F X} x := \map \Pr {X \le x}$


Definition:Random Variable/Continuous

Let $\mathcal E$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$. A '''continuous random variable''' on $\struct {\Omega, \Sigma, \Pr}$ is a random variable $X: \Omega \to \R$ whose cumulative distribution function is continuous for all $x \in \R$.


Definition:Degenerate Distribution

Let $X$ be a discrete random variable on a probability space. Then $X$ has a '''degenerate distribution with parameter $r$''' if: :$\Omega_X = \set r$ :$\map \Pr {X = k} = \begin{cases} 1 & : k = r \\ 0 & : k \ne r \end{cases}$ That is, there is only value that $X$ can take, namely $r$, which it takes with certainty.


Definition:Probability Generating Function

Let $X$ be a discrete random variable whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$. Let $p_X$ be the probability mass function for $X$. The '''probability generating function for $X$''', denoted $\map {\Pi_X} s$, is the formal power series defined by: :$\displaystyle \map {\Pi_X} s := \sum_{n \mathop = 0}^\infty \map {p_X} n s^n \in \R \left[\left[{s}\right]\right]$


Definition:Convergent Series

Let $\left({S, \circ, \tau}\right)$ be a topological semigroup. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$. This series is said to be '''convergent''' {{iff}} its sequence of partial sums $\left \langle {s_N} \right \rangle$ converges in the topological space $\left({S, \tau}\right)$. If $s_N \to s$ as $N \to \infty$, the series '''converges to the sum $s$''', and one writes $\displaystyle \sum_{n \mathop = 1}^\infty a_n = s$. === Convergent Series in a Normed Vector Space (Definition 1) === {{:Definition:Convergent Series/Normed Vector Space/Definition 1}} === Convergent Series in a Normed Vector Space (Definition 2) === {{:Definition:Convergent Series/Normed Vector Space/Definition 2}} === Convergent Series in a Number Field === {{:Definition:Convergent Series/Number Field}}


Definition:Probability Distribution

Let $\tuple {\Omega, \Sigma, \Pr}$ be a probability space. Let $X: \Omega \to \R$ be a random variable on $\tuple {\Omega, \Sigma, \Pr}$. Then the '''probability distribution of $X$''' is the pushforward $X_* \Pr$ of $\Pr$ on $\tuple {\R, \map \BB \R}$, where $\map \BB \R$ denotes the Borel $\sigma$-algebra on $\R$.


Definition:Marginal Probability Mass Function

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $X: \Pr \to \R$ and $Y: \Pr \to \R$ both be discrete random variables on $\left({\Omega, \Sigma, \Pr}\right)$. Let $p_{X, Y}$ be the joint probability mass function of $X$ and $Y$. Then the probability mass functions $p_X$ and $p_Y$ are called the '''marginal (probability) mass functions''' of $X$ and $Y$ respectively. The marginal mass function can be obtained from the joint mass function: {{begin-eqn}} {{eqn | l=p_X \left({x}\right)       | r=\Pr \left({X = x}\right)       | c= }} {{eqn | r=\sum_{y \mathop \in \operatorname{Im} \left({Y}\right)} \Pr \left({X = x, Y = y}\right)       | c= }} {{eqn | r=\sum_y p_{X, Y} \left({x, y}\right)       | c= }} {{end-eqn}}


Definition:Moment of Discrete Random Variable

Let $X$ be a discrete random variable. Then the '''$n$th moment of $X$''' is denoted $\mu'_n$ and defined as: :$\mu'_n = \expect {X^n}$ where $\expect {\, \cdot \,}$ denotes the expectation function.


Definition:Diameter of Quadrilateral

Let $ABCD$ be a quadrilateral: :400px The '''diameters''' of $ABCD$ are the lines $AC$ and $BD$ joining their opposite vertices.


Definition:Matrix Exponential

The '''matrix exponential''', denoted $e^{t \mathbf A}$, of the order $n$ constant square matrix $\mathbf A$, is defined to be the unique solution to the initial value problem: :$X' = \mathbf A X, \ \map X {\mathbf 0} = \mathbf I$ where: :$\mathbf I$ is the unit matrix :$X$ is an order $n$ square matrix which is a function of $t$.


Definition:Stability (Differential Equations)

For first-order autonomous systems, define $\phi \left({t, x_0}\right)$ to be the unique solution with initial condition $x \left({0}\right) = x_0$. {{explain|Make this more precise: "for first-order autonomous systems" should be replaced by a statement specifying something like, "Let $S$ be a first-order autonomous system" (or whatever letter is appropriate for such an object in this context), and then "Let $\phi \left({t, x_0}\right)$ be ... of $S$" or whatever.}} Then a solution with initial condition $x_0$ is '''stable''' on $\left[{0 \,.\,.\, \to}\right)$ if: {{explain|Makes it easier to follow if it is introduced as "Let $\mathbf s$ be a solution ... with initial condition $x_0$.}} :given any $\epsilon > 0$, there exists a $\delta > 0$ such that $\left\Vert{x - x_0}\right\Vert < \delta \implies \left\Vert {\phi \left({t, x}\right) - \phi \left({t, x_0}\right)}\right\Vert < \epsilon$ {{explain|Establish links to define the above notation}} {{refactor|Extract these out into separate pages.}} An equilibrium $x_0$ is '''unstable''' {{iff}} it is not stable. An equilibrium $x_0$ is '''asymptotically stable''' {{iff}}: :For any $x$ in a sufficiently small neighborhood of $x_0$: ::$\displaystyle \lim_{t \to \infty} \phi \left({t, x}\right) = x_0$ {{MissingLinks}} {{definition wanted|This page might be better approached by using the machinery of Convergence.}} Category:Definitions/Differential Equations b042k7x1lii3wuk3yj7ngz836nagvbt


Definition:Nonvanishing

A function $f$ is said to be '''nonvanishing''' if it has no zeroes in its domain. That is, $f$ is '''nonvanishing''' {{iff}}: :$\forall x \in \Dom f: \map f x \ne 0$ In this context, $f$ is (usually) either real-valued or complex-valued. In any case, its codomain needs to contain a zero, so at the very least its codomain needs to be a ring. Category:Definitions/Analysis Category:Definitions/Complex Analysis dyl4eurqrshy64dqvb6662uqmu9sc3q


Definition:Fundamental Matrix

Let $\mathbf x' = A \left({t}\right) \mathbf x$ be a system of $n$ linear first order ODEs. Let $\Phi \left({t}\right)$ be an $n \times n$ matrix function. Then $\Phi \left({t}\right)$ is a '''fundamental matrix''' of the system $\mathbf x' = A \left({t}\right) \mathbf x$ {{iff}}: :it solves the matrix system $\mathbf X'=A(t) \mathbf X$ :$\det \Phi \left({t}\right)$ is nonvanishing


Definition:Lyapunov Function

Let $x_0$ be an equilibrium point of the system $x' = f \left({x}\right)$. Then a function $V$ is a '''Lyapunov function''' of the system on an open set $U$ containing the equilibrium iff: * $V \left({x_0}\right) = 0$ * $V \left({x}\right) > 0$ if $x \in U \setminus \left\{{x_0}\right\}$ * $\nabla V \cdot f \le 0$ for $x \in U$. {{explain|What $\nabla$ means in this context.}} If the inequality is strict except at $x_0$, then $V$ is '''strict'''.


Definition:Continuously Differentiable

A differentiable function $f$ is '''continuously differentiable''' {{iff}} $f$ is of differentiability class $C^1$. That is, if the first order derivative of $f$ (and possibly higher) is continuous.


Definition:Homotopy Class

Let $X$ and $Y$ be topological spaces. Let $K \subseteq X$ be any subset. Let $f : X \to Y$ be a continuous mapping. The '''homotopy class''' or '''$K$-homotopy class''' of $f$ is the equivalence class of $f$ under the equivalence relation defined by homotopy relative to $K$. === Homotopy class of path === {{:Definition:Homotopy Class/Path}}


Definition:Cycle (Periodic Solution)

A '''cycle''', or '''periodic solution''', is a solution of a differential equation which is a periodic function. Category:Definitions/Differential Equations 5k43z9413cja1n5zxw3a74fp5wy6jte


Definition:Differentiability Class

Let $f: \R \to \R$ be a real function. Then $\map f x$ is of '''differentiability class''' $C^k$ {{iff}} $\dfrac {\d^k} {\d x^k} \map f x$ is continuous. That, $f$ is in '''differentiability class $k$''' {{iff}} there exists a $k$th derivative of $f$ which is continuous. If $\dfrac {\d^k} {\d x^k} \map f x$ is continuous for all $k \in \N$, then $\map f x$ is of '''differentiability class''' $C^\infty$.


Definition:Direct Image Mapping

Let $S$ and $T$ be sets. Let $\powerset S$ and $\powerset T$ be their power sets. === Relation === {{:Definition:Direct Image Mapping/Relation}} === Mapping === {{:Definition:Direct Image Mapping/Mapping}}


Definition:Intersection (Geometry)

The '''intersection''' of two lines $AB$ and $CD$ is denoted by $AB \cap CD$. The '''intersection''' of two geometric figures is the set of points shared by both figures. Note that this use of $\cap$ is consistent with that of its more usual context of set intersection. When two lines '''intersect''', they are said to '''cut''' each other.


Definition:Generalized Inverse Gaussian Distribution

The '''generalized inverse Gaussian distribution''' ('''GIG''')  is a three-parameter family of continuous probability distributions with probability density function: :$\displaystyle \forall x > 0: f \left({x}\right) = \frac { \left({a/b}\right)^{p/2} } {2 K_p \left({\sqrt{a b} }\right)} x^{\left({p - 1}\right)} e^{-\left({a x + b / x}\right) / 2}$ where: : $K_p$ is a modified Bessel function of the second kind : $a > 0, b > 0, p$ are real.


Definition:Entropy (Probability Theory)

Let $X$ be a discrete random variable that takes on the values of $\set {x_1, x_2, \ldots, x_n}$ and has a probability mass function of $\map p {x_i}$.  Then the '''entropy''' of $X$ is:  :$\ds \map H X := -\sum_{i \mathop = 1}^n \map p {x_i} \log_2 \map p {x_i}$ and is measured in units of '''bits'''.


Definition:Limit of Real Function/Left

Let $\openint a b$ be an open real interval. Let $f: \openint a b \to \R$ be a real function. Let $L \in \R$. Suppose that: :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: b - \delta < x < b \implies \size {\map f x - L} < \epsilon$ where $\R_{>0}$ denotes the set of strictly positive real numbers. That is, for every real strictly positive $\epsilon$ there exists a real strictly positive $\delta$ such that ''every'' real number in the domain of $f$, less than $b$ but within $\delta$ of $b$, has an image within $\epsilon$ of $L$. :400px Then $\map f x$ is said to '''tend to the limit $L$ as $x$ tends to $b$ from the left''', and we write: :$\map f x \to L$ as $x \to b^-$ or :$\displaystyle \lim_{x \mathop \to b^-} \map f x = L$ This is voiced: :'''the limit of $\map f x$ as $x$ tends to $b$ from the left''' and such an $L$ is called: :'''a limit from the left'''.


Definition:Limit of Real Function/Right

Let $\Bbb I = \openint a b$ be an open real interval. Let $f: \Bbb I \to \R$ be a real function. Let $L \in \R$. Suppose that: :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \Bbb I: a < x < a + \delta \implies \size {\map f x - L} < \epsilon$ where $\R_{>0}$ denotes the set of strictly positive real numbers. That is, for every real strictly positive $\epsilon$ there exists a real strictly positive $\delta$ such that ''every'' real number in the domain of $f$, greater than $a$ but within $\delta$ of $a$, has an image within $\epsilon$ of $L$. :400px Then $\map f x$ is said to '''tend to the limit $L$ as $x$ tends to $a$ from the right''', and we write: :$\map f x \to L$ as $x \to a^+$ or :$\displaystyle \lim_{x \mathop \to a^+} \map f x = L$ This is voiced : '''the limit of $\map f x$ as $x$ tends to $a$ from the right''' and such an $L$ is called: : '''a limit from the right'''.


Definition:Limit of Sequence

=== Topological Space === {{:Definition:Limit Point/Topology/Sequence}} === Metric Space === {{:Definition:Limit of Sequence/Metric Space}} === Normed Division Ring === {{:Definition:Limit of Sequence/Normed Division Ring}} === Normed Vector Space === {{:Definition:Limit of Sequence/Normed Vector Space}}


Definition:Bounded Sequence

A special case of a bounded mapping is a '''bounded sequence''', where the domain of the mapping is $\N$. Let $\struct {T, \preceq}$ be an ordered set. Let $\sequence {x_n}$ be a sequence in $T$. Then $\sequence {x_n}$ is '''bounded''' {{iff}} $\exists m, M \in T$ such that $\forall i \in \N$: :$(1): \quad m \preceq x_i$ :$(2): \quad x_i \preceq M$ That is, {{iff}} it is bounded above and bounded below.


Definition:Bézout Numbers

Let $a, b \in \Z$ such that $a \ne 0$ or $b \ne 0$. Let $d$ be the greatest common divisor of $a$ and $b$. By Bézout's Lemma: :$\exists x, y \in \Z: a x + b y = d$ The numbers $x$ and $y$ are known as '''Bézout numbers''' of $a$ and $b$.


Definition:Parity of Integer

Let $z \in \Z$ be an integer. The '''parity''' of $z$ is whether it is even or odd. That is: :an integer of the form $z = 2 n$, where $n$ is an integer, is of '''even parity'''; :an integer of the form $z = 2 n + 1$, where $n$ is an integer, is of '''odd parity'''. :If $z_1$ and $z_2$ are either both even or both odd, $z_1$ and $z_2$ have '''the same parity'''.  :If $z_1$ is even and $z_2$ is odd, then $z_1$ and $z_2$ have '''opposite parity'''.


Definition:Parity (Permutation)

Let $n \in \N$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $\rho \in S_n$, that is, let $\rho$ be a permutation of $S_n$. The '''parity''' of $\rho$ is defined as follows: === Even Permutation === {{:Definition:Even Permutation}} === Odd Permutation === {{:Definition:Odd Permutation}} where $\map \sgn \rho$ denotes the sign of $\rho$.


Definition:Linear Operator

A '''linear operator''' is a linear transformation from a module into itself.


Definition:Codomain (Set Theory)

=== Relation === {{:Definition:Codomain (Set Theory)/Relation}} === Mapping === The term '''codomain''' is usually seen when the relation in question is actually a mapping: {{:Definition:Codomain (Set Theory)/Mapping}}


Definition:Kernel of Group Homomorphism

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism. The '''kernel''' of $\phi$ is the subset of the domain of $\phi$ defined as: :$\map \ker \phi := \phi \sqbrk {e_G} = \set {x \in G: \map \phi x = e_H}$ where $e_H$ is the identity of $H$. That is, $\map \ker \phi$ is the subset of $G$ that maps to the identity of $H$.


Definition:Kernel of Ring Homomorphism

Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be rings. Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism. The '''kernel''' of $\phi$ is the subset of the domain of $\phi$ defined as: :$\map \ker \phi = \set {x \in R_1: \map \phi x = 0_{R_2} }$ where $0_{R_2}$ is the zero of $R_2$. That is, $\map \ker \phi$ is the subset of $R_1$ that maps to the zero of $R_2$. From Ring Homomorphism Preserves Zero it follows that $0_{R_1} \in \map \ker \phi$ where $0_{R_1}$ is the zero of $R_1$.


Definition:Range of Sequence

Let $\sequence {x_n}_{n \mathop \in A}$ be a sequence. The '''range of $\sequence {x_n}$''' is the set: :$\set {x_n: n \mathop \in A}$


Definition:Real Sequence

A '''real sequence''' is a sequence (usually infinite) whose codomain is the set of real numbers $\R$.


Definition:Increasing/Sequence

Let $\struct {S, \preceq}$ be a totally ordered set. Let $A$ be a subset of the natural numbers $\N$. Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is '''increasing''' {{iff}}: :$\forall j, k \in A: j < k \implies a_j \preceq a_k$


Definition:Decreasing/Sequence

Let $\struct {S, \preceq}$ be a totally ordered set. Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is '''decreasing''' {{iff}}: :$\forall j, k \in A: j < k \implies a_k \preceq a_j$


Definition:Strictly Increasing/Sequence

Let $\struct {S, \preceq}$ be a totally ordered set. Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is '''strictly increasing''' {{iff}}: :$\forall j, k \in A: j < k \implies a_j \prec a_k$


Definition:Strictly Decreasing/Sequence

Let $\struct {S, \preceq}$ be a totally ordered set. Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is '''strictly decreasing''' {{iff}}: :$\forall j, k \in A: j < k \implies a_k \prec a_j$


Definition:Monotone (Order Theory)/Sequence

Let $\struct {S, \preceq}$ be a totally ordered set. A sequence $\sequence {a_k}_{k \mathop \in A}$ of elements of $S$ is '''monotone''' {{iff}} it is either increasing or decreasing.


Definition:Rational Sequence

A '''rational sequence''' is a sequence (usually infinite) whose codomain is the set of rational numbers $\Q$.


Definition:Self-Map

Let $S$ be a set. A '''self-map on $S$''' is a mapping from $S$ to itself: :$f: S \to S$


Definition:Renaming Mapping

Let $f: S \to T$ be a mapping. The '''renaming mapping''' $r: S / \RR_f \to \Img f$ is defined as: :$r: S / \RR_f \to \Img f: \map r {\eqclass x {\RR_f} } = \map f x$ where: :$\RR_f$ is the equivalence induced by the mapping $f$ :$S / \RR_f$ is the quotient set of $S$ determined by $\RR_f$ :$\eqclass x {\RR_f}$ is the equivalence class of $x$ under $\RR_f$.


Definition:Fixed Element

=== Fixed Point of General Mapping === {{:Definition:Fixed Point}} === Fixed Element of Permutation === The concept is particularly important when studying permutations in the context of group theory: {{:Definition:Fixed Element of Permutation}} Category:Definitions/Mapping Theory Category:Definitions/Group Theory c94dtj5srdlqtg13xmb7pc8mw3q3a5q


Definition:Peano Structure

A '''Peano structure''' $\struct {P, 0, s}$ comprises a set $P$ with a successor mapping $s: P \to P$ and a non-successor element $0$. These three together are required to satisfy Peano's axioms.


Definition:Ordered Set

An '''ordered set''' is a relational structure $\struct {S, \preceq}$ such that the relation $\preceq$ is an ordering. Such a structure may be: :A partially ordered set (poset) :A totally ordered set (toset) :A well-ordered set (woset) depending on whether the ordering $\preceq$ is: :A partial ordering :A total ordering :A well-ordering.


Definition:Hasse Diagram

Let $\struct {S, \preceq}$ be an ordered set. A '''Hasse diagram''' is a method of representing $\struct {S, \preceq}$ as a graph $G$, in which: :$(1):\quad$ The vertices of $G$ represent the elements of $S$ :$(2):\quad$ The edges of $G$ represent the elements of $\preceq$ :$(3):\quad$ If $x, y \in S: x \preceq y$ then the edge representing $x \preceq y$ is drawn so that $x$ is lower down the page than $y$. ::::That is, the edge ascends (usually obliquely) from $x$ to $y$ :$(4):\quad$ If $x \preceq y$ and $y \preceq z$, then as an ordering is transitive it follows that $x \preceq z$. ::::But in a '''Hasse diagram''', the relation $x \preceq z$ is not shown. ::::Transitivity is implicitly expressed by the fact that $z$ is higher up than $x$, and can be reached by tracing a path from $x$ to $z$ completely through ascending edges.


Definition:Conditional/Antecedent

In a conditional $p \implies q$, the statement $p$ is the '''antecedent'''.


Definition:Conditional/Consequent

In a conditional $p \implies q$, the statement $q$ is the '''consequent'''.


Definition:Cardinal

Let $S$ be a set. Associated with $S$ there exists a set $\map \Card S$ called the '''cardinal of $S$'''. It has the properties: : $(1): \quad \map \Card S \sim S$ that is, $\map \Card S$ is (set) equivalent to $S$ : $(2): \quad S \sim T \iff \map \Card S = \map \Card T$


Definition:Philosophical Position

A '''philosophical position''' is, broadly speaking, a belief that a particular statement is either true or false. In the context of mathematics the term is usually reserved for particular fundamental questions whose truth values have not been explicitly determined in any chosen frame of reference or axiom schema. Hence, in order to progress down a particular logical path, one needs to make a decision as to which it is. Usually for such questions, its answer is of far-reaching importance. It having been necessary to take such a position, one then finds that diverging schools of thought arise: one which '''accepts''' and one which '''rejects''' the truth of the question. The answer to the question itself is added to the particular axiom schema for that school to form an expanded schema.


Definition:Set of Sets

A '''set of sets''' is a set, whose elements are themselves all sets. Those elements can themselves be assumed to be subsets of some particular fixed set which is frequently referred to as the universe.


Definition:Well-Orderable Set

Let $S$ be a set. If it is possible to construct an ordering $\preceq$ on $S$ such that $\preceq$ is a well-ordering, then $S$ is defined as being '''well-orderable'''.


Definition:Choice Function

Let $\mathbb S$ be a set of sets such that: :$\forall S \in \mathbb S: S \ne \O$ that is, none of the sets in $\mathbb S$ may be empty. A '''choice function on $\mathbb S$''' is a mapping $f: \mathbb S \to \bigcup \mathbb S$ satisfying: :$\forall S \in \mathbb S: \map f S \in S$. That is, for any set in $\mathbb S$, a '''choice function''' selects an element from that set.   The domain of $f$ is $\mathbb S$.


Definition:Distributive Operation/Right

The operation $\circ$ is '''right distributive''' over the operation $*$ {{iff}}: :$\forall a, b, c \in S: \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c}$


Definition:Distributive Operation/Left

The operation $\circ$ is '''left distributive''' over the operation $*$ {{iff}}: :$\forall a, b, c \in S: a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}$


Definition:Relation Induced by Partition

Let $S$ be a set. Let $\Bbb S$ be a partition of a set $S$. Let $\mathcal R \subseteq S \times S$ be the relation defined as: :$\forall \tuple {x, y} \in S \times S: \tuple {x, y} \in \mathcal R \iff \exists T \in \Bbb S: \set {x, y} \subseteq T$ Then $\mathcal R$ is the '''(equivalence) relation induced by (the partition) $\Bbb S$'''.


Definition:Definition

A '''definition''' lays down the meaning of a concept. It is a statement which tells the reader '''what something is'''. It can be understood as an equation in (usually) natural language.


Definition:Stipulative Definition

A '''stipulative definition''' is a definition which defines how to interpret the meaning of a symbol. It '''stipulates''', or lays down, the meaning of a symbol in terms of previously defined symbols or concepts. The symbol used for a stipulative definition is: :$\text {(the symbol being defined)} := \text {(the meaning of that symbol)}$ This can be written the other way round: :$\text {(a concept being assigned a symbol)} =: \text {(the symbol for it)}$ when it is necessary to emphasise that the symbol has been crafted to abbreviate the notation for the concept.


Definition:Ostensive Definition

An '''ostensive definition''' is a definition which ''shows'' what a symbol is, rather than use words to ''explain'' what it is or what it does. As an example of an '''ostensive definition''', we offer up: :The symbol used for a stipulative definition is $:=$, as in: ::$\text {(the symbol being defined)} := \text {(the meaning of that symbol)}$


Definition:Formal Grammar/Bottom-Up/Extremal Clause

Let $\mathcal F$ be a formal language. Let the rules of formation of $\mathcal F$ be defined in a bottom-up manner. <section notitle="true" name="tc"> The '''extremal clause''' of a bottom-up grammar is the final rule which excludes all collations other than those specified in the formation rules from being well-formed formulas. </section>


Definition:Occurrence (Formal Systems)

Let $\mathcal F$ be a formal language. Let $S, T$ be collations in the alphabet of $\mathcal F$. Each place where $S$ appears in $T$ is called an '''occurrence of $S$ in $T$'''.


Definition:Null String

A '''null string''' (or '''empty string''') is a string with no symbols in it. In particular, the '''null string''' is a word. A null string has a length of $0$.


Definition:Substring

Let $\mathcal L$ be a formal language with alphabet $\mathcal A$. Let $S$ be a string in $\mathcal A$. Let $T$ be a string in $\mathcal A$ such that: :$S = S_1 T S_2$ where: * $S_1$ and $S_2$ are strings in $\mathcal A$ (possibly null); * $S_1 T S_2$ is the concatenation of $S_1$, $T$ and $S_2$. Then $T$ is called a '''substring of $S$'''. It follows from this definition that $S$ is a substring of itself (by considering $S_1$ and $S_2$ as both null).


Definition:Subordinate

Let $\mathbf A$ be a WFF of propositional logic. Let $\circ$ and $\ast$ be connectives. Then $\circ$ is '''subordinate to $\ast$ (in $\mathbf A$)''' iff the scope of $\circ$ is a well-formed part of the scope of $\ast$.


Definition:Venn Diagram

A '''Venn diagram''' is a technique for the graphic depiction of the interrelationship between a small number (usually $3$ or fewer) of sets. The following diagram illustrates the various operations between three sets. :500px The circles represent the sets $S_1$, $S_2$ and $S_3$. The '''white''' surrounding box represents the universal set $\mathbb U$. Each of the areas inside the various circle represents an intersection between the various sets and their complements, as follows: :The '''gray''' area represents $S_1 \cap S_2 \cap S_3$. :The '''purple''' area represents $S_1 \cap S_2 \cap \overline {S_3}$. :The '''orange''' area represents $S_1 \cap \overline {S_2} \cap S_3$. :The '''green''' area represents $\overline {S_1} \cap S_2 \cap S_3$. :The '''red''' area represents $S_1 \cap \overline {S_2} \cap \overline {S_3}$. :The '''blue''' area represents $\overline {S_1} \cap S_2 \cap \overline {S_3}$. :The '''yellow''' area represents $\overline {S_1} \cap \overline {S_2} \cap S_3$. :The surrounding '''white''' area represents $\overline {S_1} \cap \overline {S_2} \cap \overline {S_3}$. The notation $\overline {S_1}$ denotes set complement. If it is required to show on a diagram that a particular intersection is empty, then it is generally shaded '''black'''.


Definition:Order Type

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be totally ordered sets. Then $S$ and $T$ '''have the same (order) type''' {{iff}} they are order isomorphic.


Definition:Order Type of Natural Numbers

The order type of $\struct {\N, \le}$ is denoted $\omega$ ('''omega''').


Definition:Ordered Sum

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be tosets. Let: * the order type of $\left({S, \preceq_1}\right)$ be $\theta_1$ * the order type of $\left({T, \preceq_2}\right)$ be $\theta_2$. Let $S \cup T$ be the union of $S$ and $T$. We define the ordering $\preceq$ on $S$ and $T$ as: :$\forall s \in S, t \in T: a \preceq b \iff \begin{cases} a \preceq_1 b & : a \in S \land b \in S \\ a \preceq_2 b & : \neg \left({a \in S \land b \in S}\right) \land \left({a \in T \land b \in T}\right) \\ & : a \in S, b \in T \end{cases}$ That is: : If $a$ and $b$ are both in $S$, they are ordered as they are in $S$. : If $a$ and $b$ are not both in $S$, but they ''are'' both in $T$, they are ordered as they are in $T$. : Otherwise, that is if $a$ and $b$ are in ''both'' sets, their ordering in $S$ takes priority over that in $T$. The ordered set $\left({S \cup T, \preceq}\right)$ is called the '''ordered sum''' of $S$ and $T$, and is denoted $S + T$.


Definition:Ordered Product

Let $\left({S_1, \preceq_1}\right)$ and $\left({S_2, \preceq_2}\right)$ be tosets. Let: * the order type of $\left({S_1, \preceq_1}\right)$ be $\theta_1$; * the order type of $\left({S_2, \preceq_2}\right)$ be $\theta_2$. Let $T = S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$. Consider the relation $\preceq$ defined on $T$ as follows. Let $a_1$ and $a_2$ be arbitrary elements of $S_1$, and $b_1$ and $b_2$ be arbitrary elements of $S_2$. Then: * $b_1 \prec b_2 \implies \left({a_1, b_1}\right) \prec \left({a_2, b_2}\right)$ * $b_1 = b_2, a_1 \prec a_2 \implies \left({a_1, b_1}\right) \prec \left({a_2, b_2}\right)$ * $b_1 = b_2, a_1 = a_2 \implies \left({a_1, b_1}\right) = \left({a_2, b_2}\right)$ The ordered set $\left({S_1 \times S_2, \preceq}\right)$ is called the '''ordered product''' of $S_1$ and $S_2$, and is denoted $S_1 \cdot S_2$.


Definition:Euler Diagram

An '''Euler diagram''' is a graphical technique for illustrating the relationships between sets. It differs from a Venn diagram in that whereas the latter illustrates all possible intersections between a number of general sets, an '''Euler diagram''' depicts only those which are relevant for the situation being depicted.


Definition:Scope (Logic)

The '''scope''' of a logical connective is defined as the statements that it connects, whether this be simple or compound. In the case of a unary connective, there will be only one such statement. === Connective of Propositional Logic === {{:Definition:Scope (Logic)/Connective}} === Quantifier of Predicate Logic === {{:Definition:Scope (Logic)/Quantifier}}


Definition:Main Connective

In a compound statement, exactly '''one''' of its logical connectives has the largest scope. That connective is called the '''main connective'''. The scope of the '''main connective''' comprises the entire compound statement. {{transclude:Definition:Main Connective/Propositional Logic   |section = def   |title = Propositional Logic   |link = true   |header = 3   |increase = 1 }}


Definition:Codomain of Sequence

The codomain of a sequence can be elements of a set of any objects. If the codomain of a sequence $f$ is $S$, then the sequence is said to be a '''sequence of elements of $S$''', or a '''sequence in $S$'''.


Definition:Length of String

The '''length''' of a finite string in a formal language is the number of symbols it contains. The '''length''' of a string $S$ can be denoted $\map \len S$ or $\size S$.


Definition:Initial Part

A string $T$ is an '''initial part''' of a string $S$ {{iff}} $S$ can be formed by concatenating $T$ with another string $T'$: :$S = TT'$


Definition:Trivial Quotient

Let $\Delta_S$ be the diagonal relation on a set $S$. As $\Delta_S$ is an equivalence, we can form the quotient mapping: :$q_{\Delta_S}: S \to S / \Delta_S$. This quotient mapping is called the '''trivial quotient of $S$'''.


Definition:Conflation

A '''conflation''' is a mistake in which two or more separate but similar ideas become confused with one another.


Definition:Left Inverse Mapping

Let $S, T$ be sets where $S \ne \O$, i.e. $S$ is not empty. Let $f: S \to T$ be a mapping. Let $g: T \to S$ be a mapping such that: :$g \circ f = I_S$ where: :$g \circ f$ denotes the composite mapping $f$ followed by $g$; :$I_S$ is the identity mapping on $S$. Then $g: T \to S$ is called '''a left inverse (mapping)'''.


Definition:Right Inverse Mapping

Let $S, T$ be sets where $S \ne \O$, that is, $S$ is not empty. Let $f: S \to T$ be a mapping. Let $g: T \to S$ be a mapping such that: :$f \circ g = I_T$ where: :$f \circ g$ denotes the composite mapping $g$ followed by $f$ :$I_T$ is the identity mapping on $T$. Then $g: T \to S$ is called '''a right inverse (mapping) of $f$'''.


Definition:Commutative Diagram

A '''commutative diagram''' is a graphical technique designed to illustrate the construction of composite mappings. It is also a widespread tool in category theory, where it deals with morphisms instead of mappings. ::$\begin{xy} \xymatrix@L+2mu@+1em{  S_1 \ar[r]^*{f_1}      \ar@{-->}[rd]_*{f_2 \circ f_1} &  S_2 \ar[d]^*{f_2} \\ &  S_3 }\end{xy}$ It consists of: :$(1): \quad$ A collection of points representing the various domains and codomains of the mappings in question :$(2): \quad$ Arrows representing the mappings themselves. The diagram is properly referred to as '''commutative''' {{iff}} all the various paths from the base of one arrow to the head of another represent equal mappings. A mapping which is uniquely determined by the rest of the diagram may be indicated by a dotted arrow. It is however generally advisable not to use more than one dotted arrow per diagram, so as to avoid confusion.


Definition:Extension of Mapping

As a mapping is, by definition, also a relation, the definition of an '''extension of a mapping''' is the same as that for an extension of a relation: Let: :$f_1 \subseteq X \times Y$ be a mapping on $X \times Y$ :$f_2 \subseteq S \times T$ be a mapping on $S \times T$ :$X \subseteq S$ :$Y \subseteq T$ :$f_2 \restriction_{X \times Y}$ be the restriction of $f_2$ to $X \times Y$. Let $f_2 \restriction_{X \times Y} = f_1$. Then $f_2$ '''extends''' or '''is an extension of''' $f_1$.


Definition:Extension of Operation

Let $\left({S, \circ}\right)$ be a magma. Let $\left({T, \circ \restriction_T}\right)$ be a submagma of $\left({S, \circ}\right)$, where $\circ \restriction_T$ denotes the restriction of $\circ$ to $T$. Then: : '''$\left({S, \circ}\right)$ is an extension of $\left({T, \circ \restriction_T}\right)$''' or : '''$\left({S, \circ}\right)$ extends $\left({T, \circ \restriction_T}\right)$''' We can use the term directly to the operation itself and say: : '''$\circ$ is an extension of $\circ \restriction_T$''' or: : '''$\circ$ extends $\circ \restriction_T$'''


Definition:Union Mapping

Let: :$(1): \quad f_1: S_1 \to T_1$ be a mapping from $S_1$ to $T_1$ :$(2): \quad f_2: S_2 \to T_2$ be a mapping from $S_2$ to $T_2$ Let $f_1$ and $f_2$ be combinable, that is, that they agree on $S_1 \cap S_2$. Then the '''union mapping''' $f = f_1 \cup f_2$ of $f_1$ and $f_2$ is: :$f: S_1 \cup S_2 \to T_1 \cup T_2: \map f s = \begin{cases} \map {f_1} s : & s \in S_1 \\ \map {f_2} s : & s \in S_2 \end{cases}$


Definition:Increasing Union

Let $S_0, S_1, S_2, \ldots, S_i, \ldots$ be a nested sequence of sets, that is: :$S_0 \subseteq S_1 \subseteq S_2 \subseteq \ldots \subseteq S_i \subseteq \ldots$ Let $S$ be the set: :$\displaystyle S = \bigcup_{i \mathop \in \N} S_i$ where $\bigcup$ denotes set union. Then $S$ is called the '''increasing union''' of $S_0, S_1, S_2, \ldots, S_i, \ldots$


Definition:Bound Variable

A '''bound variable''' is a variable which, when it occurs in an expression, can be replaced with another variable without changing the meaning of the statement.


Definition:F-Sigma Set

Let $T = \struct {S, \tau}$ be a topological space. An '''$F_\sigma$ set ($F$-sigma set)''' is a set which can be written as a countable union of closed sets of $T$.


Definition:G-Delta Set

Let $T = \struct {S, \tau}$ be a topological space. A '''$G_\delta$ set ($G$-delta set)''' is a set which can be written as a countable intersection of open sets of $T$.


Definition:Equivalent Topological Bases

Let $X$ be a set. Let $\mathbb S_1$ and $\mathbb S_2$ be subsets of $\powerset X$, the power set of $X$. Let $\mathbb S_1$ and $\mathbb S_2$ be used as a synthetic basis or synthetic sub-basis to generate topologies for $X$. Let $\tau_1$ and $\tau_2$ be the topologies arising from $\mathbb S_1$ and $\mathbb S_2$ respectively. Then $\mathbb S_1$ and $\mathbb S_2$ are '''equivalent''' {{iff}} $\tau_1 = \tau_2$.


Definition:Local Basis

Let $T = \struct {S, \tau}$ be a topological space. Let $x$ be an element of $S$. === Local Basis for Open Sets === {{:Definition:Local Basis/Local Basis for Open Sets}} === Neighborhood Basis of Open Sets === {{:Definition:Local Basis/Neighborhood Basis of Open Sets}}


Definition:Weakly Hereditary Property

Let $\xi$ be a property whose domain is the set of all topological spaces. Then $\xi$ is a '''weakly hereditary property''' {{iff}}: :$\map \xi X \implies \map \xi Y$ where $Y$ is any closed set of $X$ when considered as a subspace.


Definition:Hereditary Property (Topology)

Let $\xi$ be a property whose domain is the set of all topological spaces. Then $\xi$ is a '''hereditary property''' {{iff}}: :$\map \xi X \implies \map \xi Y$ where $Y$ is a subspace of $X$. That is, whenever a topological space has $\xi$, then so does any subspace.


Definition:Adherent Point

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. === Definition from Neighborhood === {{:Definition:Adherent Point/Definition 3}} === Definition from Open Neighborhood === {{:Definition:Adherent Point/Definition 1}} === Definition from Closure === {{:Definition:Adherent Point/Definition 2}}


Definition:Omega-Accumulation Point

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. An '''$\omega$-accumulation point''' of $A$ is a limit point $x$ of $A$ such that every open set containing $x$ also contains an infinite number of points of $A$.


Definition:Condensation Point

Let $T = \struct {X, \tau}$ be a topological space. Let $A \subseteq X$. A '''condensation point''' of $A$ is a limit point $x$ of $A$ such that every open set containing $x$ also contains an uncountable number of points of $A$.


Definition:Limit Point/Topology/Sequence

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. Let $\sequence {x_n}$ be a sequence in $A$. Let $\sequence {x_n}$ converge to a value $\alpha \in A$. Then $\alpha$ is known as a '''limit (point) of $\sequence {x_n}$ (as $n$ tends to infinity)'''.


Definition:Accumulation Point/Sequence

Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $A$. Let $x \in S$. Suppose that: :$\forall U \in \tau: x \in U \implies \set {n \in \N: x_n \in U}$ is infinite Then $x$ is an '''accumulation point''' of $\sequence {x_n}$.


Definition:Regular Open Set

Let $T$ be a topological space. Let $A \subseteq T$. Then $A$ is '''regular open in $T$''' {{iff}}: :$A = A^{- \circ}$


Definition:Regular Closed Set

Let $T$ be a topological space. Let $A \subseteq T$. Then $A$ is '''regular closed in $T$''' {{iff}}: :$A = A^{\circ -}$


Definition:Exterior (Topology)

Let $T$ be a topological space. Let $H \subseteq T$. === Definition 1 === {{:Definition:Exterior (Topology)/Definition 1}} === Definition 2 === {{:Definition:Exterior (Topology)/Definition 2}}


Definition:Dense-in-itself

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Then $H$ is dense-in-itself {{iff}} it contains no isolated points.


Definition:Meager Space

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. $A$ is '''meager in $T$''' {{iff}} it is a countable union of subsets of $S$ which are nowhere dense in $T$.


Definition:Separable Space

A topological space $T = \struct {S, \tau}$ is '''separable''' {{iff}} there exists a countable subset of $S$ which is everywhere dense in $T$.


Definition:Second-Countable Space

A topological space $T = \struct {S, \tau}$ is '''second-countable''' or '''satisfies the Second Axiom of Countability''' {{iff}} its topology has a countable basis.


Definition:First-Countable Space

A topological space $T = \struct {S, \tau}$ is '''first-countable''' or '''satisfies the First Axiom of Countability''' {{iff}} every point in $S$ has a countable local basis.


Definition:Neighborhood Basis

Let $\struct{X, \tau}$ be a topological space. Let $x \in X$. Let $\BB$ be a set of neighborhoods of $x$. Then $\BB$ is a '''neighborhood basis''' at $x$ {{iff}}: :For each neighborhood $N$ of $x$, there is an $M \in \BB$ such that $M \subseteq N$.


Definition:Closed Mapping

Let $X, Y$ be topological spaces. Let $f: X \to Y$ be a mapping.  If, for any closed set $V \subseteq X$, the image $\map f V$ is closed in $Y$, then $f$ is referred to as a '''closed mapping'''.


Definition:Continuity/Metric Subspace

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$. Let $Y \subseteq A_1$. By definition, $\left({Y, d_Y}\right)$ is a metric subspace of $A_1$. Let $a \in Y$ be a point in $Y$. Then $f$ is '''$\left({d_Y, d_2}\right)$-continuous at $a$''' {{iff}}: : $\forall \epsilon > 0: \exists \delta > 0: d_Y \left({x, a}\right) < \delta \implies d_2 \left({f \left({x}\right), f \left({a}\right)}\right) < \epsilon$ Similarly, $f$ is '''$\left({d_Y, d_2}\right)$-continuous''' {{iff}}: : $\forall a \in Y: f$ is $\left({d_Y, d_2}\right)$-continuous at $a$


Definition:Open Set/Topology

Let $T = \left({S, \tau}\right)$ be a topological space. Then the elements of $\tau$ are called the '''open sets of $T$'''. Thus: : '''$U \in \tau$''' and: : '''$U$ is open in $T$''' are equivalent statements.


Definition:Open Set/Metric Space

Let $M = \struct {A, d}$ be a metric space. Let $U \subseteq A$. Then $U$ is an '''open set in $M$''' {{iff}} it is a neighborhood of each of its points. That is: :$\forall y \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} y \subseteq U$ where $\map {B_\epsilon} y$ is the open $\epsilon$-ball of $y$.


Definition:Open Set/Complex Analysis

{{:Definition:Open Set/Complex Analysis/Definition 1}}


Definition:Open Set/Real Analysis

=== Real Numbers === {{:Definition:Open Set/Real Analysis/Real Numbers}} === Real Euclidean Space === {{:Definition:Open Set/Real Analysis/Real Euclidean Space}}


Definition:Open Region

=== Complex Analysis === {{:Definition:Open Region/Complex}} === Open Region in the Plane === {{:Definition:Open Region/Plane}}


Definition:Neighborhood (Topology)

Let $T = \struct {S, \tau}$ be a topological space. === Neighborhood of a Set === {{:Definition:Neighborhood (Topology)/Set}} === Neighborhood of a Point === The set $A$ can be a singleton, in which case the definition is of the '''neighborhood of a point'''. {{:Definition:Neighborhood (Topology)/Point}}


Definition:Open Ball

Let $M = \struct {A, d}$ be a metric space or pseudometric space. Let $a \in A$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''open $\epsilon$-ball of $a$ in $M$''' is defined as: :$\map {B_\epsilon} a := \set {x \in A: \map d {x, a} < \epsilon}$ If it is necessary to show the metric or pseudometric itself, then the notation $\map {B_\epsilon} {a; d}$ can be used.


Definition:Neighborhood (Complex Analysis)

Let $z_0 \in \C$ be a complex number. Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number. The '''$\epsilon$-neighborhood''' of $z_0$ is defined as: :$\map {N_\epsilon} {z_0} := \set {z \in \C: \cmod {z - z_0} < \epsilon}$


Definition:Neighborhood (Real Analysis)/Epsilon

On the real number line with the usual metric, the '''$\epsilon$-neighborhood''' of $\alpha$ is defined as the open interval: :$\map {N_\epsilon} \alpha := \openint {\alpha - \epsilon} {\alpha + \epsilon}$ where $\epsilon \in \R_{>0}$ is a (strictly) positive real number.


Definition:Limit of Real Function

Let $\openint a b$ be an open real interval. Let $c \in \openint a b$. Let $f: \openint a b \setminus \set c \to \R$ be a real function. Let $L \in \R$. Suppose that: :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$ where $\R_{>0}$ denotes the set of strictly positive real numbers. That is: :For every (strictly) positive real number $\epsilon$, there exists a (strictly) positive real number $\delta$ such that ''every'' real number $x \ne c$ in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of $L$. $\epsilon$ is usually considered as having the connotation of being "small" in magnitude, but this is a misunderstanding of its intent: the point is that (in this context) $\epsilon$ ''can be made'' arbitrarily small. :400px Then $\map f x$ is said to '''tend to the limit $L$ as $x$ tends to $c$''', and we write: :$\map f x \to L$ as $x \to c$ or :$\displaystyle \lim_{x \mathop \to c} \map f x = L$ This is voiced: :'''the limit of $\map f x$ as $x$ tends to $c$'''. It can directly be seen that this definition is the same as that for a general metric space.


Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $c$ be a limit point of $M_1$. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ ''except possibly'' at $c$. Let $L \in M_2$. $f \left({x}\right)$ is said to '''tend to the limit $L$ as $x$ tends to $c$''' and is written: :$f \left({x}\right) \to L$ as $x \to c$ or :$\displaystyle \lim_{x \mathop \to c} f \left({x}\right) = L$ {{iff}} the following equivalent conditions hold: === $\epsilon$-$\delta$ Condition === {{:Definition:Limit of Function (Metric Space)/Epsilon-Delta Condition}} === $\epsilon$-Ball Condition === {{:Definition:Limit of Function (Metric Space)/Epsilon-Ball Condition}}


Definition:Limit of Complex Function

The definition for the limit of a complex function is exactly the same as that for the general metric space.


Definition:Limit of Sequence/Metric Space

Let $M = \left({A, d}\right)$ be a metric space or pseudometric space. Let $\left \langle {x_n} \right \rangle$ be a sequence in $M$. Let $\left \langle {x_n} \right \rangle$ converge to a value $l \in A$. Then $l$ is a '''limit of $\left \langle {x_n} \right \rangle$ as $n$ tends to infinity'''. If $M$ is a metric space, this is usually written: :$\displaystyle l = \lim_{n \mathop \to \infty} x_n$


Definition:Limit of Sequence (Number Field)

=== Real Numbers === {{:Definition:Limit of Sequence/Real Numbers}} === Rational Numbers === {{:Definition:Limit of Sequence/Rational Numbers}} === Complex Numbers === {{:Definition:Limit of Sequence/Complex Numbers}}


Definition:Divergent Sequence

A sequence which is not convergent is '''divergent'''.


Definition:Divergent Series

A series which is not convergent is '''divergent'''.


Definition:Divergent Function

A function which is not convergent is '''divergent'''.


Definition:Convergent Mapping

=== Metric Space === {{:Definition:Convergent Mapping/Metric Space}} === Real Function === As the real number line $\R$ under the usual (Euclidean) metric forms a metric space, the definition also holds for real functions: {{:Definition:Convergent Mapping/Real Function}} === Complex Function === As the complex plane $\C$ under the usual (Euclidean) metric forms a metric space, the definition also holds for complex functions: {{:Definition:Convergent Mapping/Complex Function}}


Definition:Filter

Let $\left({S, \preccurlyeq}\right)$ be an ordered set. A '''filter of $\left({S, \preccurlyeq}\right)$''' (or a '''filter on $\left({S, \preccurlyeq}\right)$''') is a subset $\mathcal F \subseteq S$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = 1         | m = \mathcal F \ne \varnothing }} {{axiom | n = 2         | m = x, y \in \mathcal F \implies \exists z \in \mathcal F: z \preccurlyeq x, z \preccurlyeq y }} {{axiom | n = 3         | m = \forall x \in \mathcal F: \forall y \in S: x \preccurlyeq y \implies y \in \mathcal F }} {{end-axiom}} === Proper Filter === {{:Definition:Filter/Proper Filter}}


Definition:Convergent Filter

Let $\struct {S, \tau}$ be a topological space. Let $\FF$ be a filter on $S$. Then $\FF$ '''converges''' to a point $x \in S$ {{iff}}: :$\forall N_x \subseteq S: N_x \in \FF$ where $N_x$ is a neighborhood of $x$. That is, a filter '''converges''' to a point $x$ {{iff}} every neighborhood of $x$ is an element of that filter. If there is a point $x \in S$ such that $\FF$ '''converges''' to $x$, then $\FF$ is '''convergent'''.


Definition:Standard Number Field

The '''standard number fields''' are the following sets of numbers: :The rational numbers: $\Q = \set {p / q: p, q \in \Z, q \ne 0}$ :The real numbers: $\R = \set {x: x = \sequence {s_n} }$ where $\sequence {s_n}$ is a Cauchy sequence in $\Q$ :The complex numbers: $\C = \set {a + i b: a, b \in \R, i^2 = -1}$. These sets are indeed fields: :$\struct {\Q, +, \times, \le}$ is an ordered field, and also a metric space. :$\struct {\R, +, \times, \le}$ is an ordered field, and also a complete metric space. :$\struct {\C, +, \times}$ is a field, but cannot be ordered compatibly with $+$ and $\times$. However, it can be treated as a metric space.


Definition:Continuous Invariant

Let $P$ be a property whose domain is the set of all topological spaces. Suppose that whenever $\map P T$ holds, then so does $\map P {T'}$, where: :$T$ and $T'$ are topological spaces :$\phi: T \to T'$ is a continuous mapping from $T$ to $T'$ :$\phi \sqbrk T = T'$, where $\phi \sqbrk T$ denotes the image of $\phi$. Then $P$ is a '''continuous invariant'''.


Definition:Image (Set Theory)/Relation/Subset

Let $X \subseteq S$ be a subset of $S$. Then the '''image set (of $X$ by $\RR$)''' is defined as: :$\RR \sqbrk X := \set {t \in T: \exists s \in X: \tuple {s, t} \in \RR}$


Definition:Open Invariant

Let $P$ be a property whose domain is the set of all topological spaces. Suppose that whenever $\map P T$ holds, then so does $\map P {T'}$, where: :$T$ and $T'$ are topological spaces :$\phi: T \to T'$ is a mapping from $T$ to $T'$ :$\phi \sqbrk T = T'$, where $\phi \sqbrk T$ denotes the image of $\phi$ :$T'$ is an open set. Then $P$ is an '''open invariant'''.


Definition:Closed Invariant

Let $P$ be a property whose domain is the set of all topological spaces. Suppose that whenever $\map P T$ holds, then so does $\map P {T'}$, where: :$T$ and $T'$ are topological spaces :$\phi: T \to T'$ is a mapping from $T$ to $T'$ :$\phi \sqbrk T = T'$, where $\phi \sqbrk T$ denotes the image of $\phi$ :$T'$ is a closed set. Then $P$ is a '''closed invariant'''.


Definition:Product Space (Topology)

Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be topological spaces. Let $S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$. Let $\tau$ be the Tychonoff topology on $S_1 \times S_2$. From Natural Basis of Tychonoff Topology of Finite Product, $\tau$ is the topology generated by the natural basis: :$\BB = \set {U_1 \times U_2: U_1 \in \tau_1, U_2 \in \tau_2}$ The topological space $\struct {S_1 \times S_2, \tau}$ is called  the '''product space''' of $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$.


Definition:P-Product Metric

Let $M_{1'} = \struct {A_{1'}, d_{1'} }$ and $M_{2'} = \struct {A_{2'}, d_{2'} }$ be metric spaces. Let $A_{1'} \times A_{2'}$ be the cartesian product of $A_{1'}$ and $A_{2'}$. Let $p \in \R_{\ge 1}$. The '''$p$-product metric''' on $A_{1'} \times A_{2'}$ is defined as: :$\map {d_p} {x, y} := \paren {\paren {\map {d_{1'} } {x_1, y_1} }^p + \paren {\map {d_{2'} } {x_2, y_2} }^p}^{1/p}$ where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_{1'} \times A_{2'}$. The metric space $\mathcal M_p := \struct {A_{1'} \times A_{2'}, d_p}$ is the '''$p$-product (space)''' of $M_{1'}$ and $M_{2'}$.


Definition:Identification Topology

Let $\struct {S_1, \tau_1}$ be a topological space. Let $S_2$ be a set. Let $f: S_1 \to S_2$ be a mapping. The '''identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$''' is defined as: :$\tau_2 = \set {V \in \powerset {S_2}: f^{-1} \sqbrk V \in \tau_1}$


Definition:Topological Sum

Let $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ be topological spaces. The '''topological sum''' $\struct {Z, \tau_3}$ of $X$ and $Y$ is defined as: :$Z = X \sqcup Y$ where: :$X \sqcup Y$ denotes the disjoint union of $X$ and $Y$ :$\tau_3$ is the topology generated by $\tau_1$ and $\tau_2$.


Definition:Finite Intersection Property

Let $\Bbb S$ be a set of sets. Let $\Bbb S$ have the property that: :the intersection of any finite number of sets in $\Bbb S$ is not empty. Then $\Bbb S$ satisfies the '''finite intersection property'''.


Definition:Filter Sub-Basis

Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. Let $\BB \subset \powerset S$ be a set of subsets of $\powerset S$ which satisfies the finite intersection property. That is, the intersection of any finite number of sets in $\BB$ is not empty. Then $\BB$, together with the finite intersections of all its elements, is a basis for a filter $\FF$ on $S$. Thus $\BB$ is a '''sub-basis''' for $\FF$.


Definition:Cluster Point of Filter

Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. Let $\FF \subset \powerset X$ be a filter on $S$. Let $x \in S$ be an element of every set in $\FF$: :$x \in X: \forall U \in \FF: x \in U$ Then $x$ is a '''cluster point of $\FF$'''.


Definition:Principal Ultrafilter

Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. Let $\FF \subset \powerset S$ be an ultrafilter on $S$ with a cluster point. Then $\FF$ is a '''principal ultrafilter on $S$'''.


Definition:Neighborhood Filter

Let $T = \left({S, \tau}\right)$ be a topological space. === Set === {{:Definition:Neighborhood Filter/Set}} === Point === {{:Definition:Neighborhood Filter/Point}} Category:Definitions/Neighborhoods Category:Definitions/Filter Theory i2b1xtpqf0z3iaiel1kxg70lobnic8w


Definition:Group Automorphism

Let $\struct {G, \circ}$ be a group. Let $\phi: G \to G$ be a (group) isomorphism from $G$ to itself. Then $\phi$ is a group automorphism.


Definition:Scheme of Abbreviation

When translating from a series of sentences in natural language into a collection of propositional formulas in symbolic logic, it is necessary to replace each simple statement with a statement label. As stated in the definition of statement label, it is imperative that each sentence be replaced with a different label. A '''scheme of abbreviation''' is the list of simple statements, together with their associated statement labels, for reference and unambiguous key in order to translate any conclusion reached back into natural language.


Definition:Negation Normal Form

A propositional formula $P$ is in '''negation normal form''' ('''NNF''') {{iff}}: * The only logical connectives connecting substatements of $P$ are Not, And and Or, that is, elements of the set $\left\{{\neg, \land, \lor}\right\}$; * The Not sign $\neg$ appears only in front of atomic statements. That is $P$ is in '''negation normal form''' iff it consists of literals, conjunctions and disjunctions.


Definition:Finite Group

A '''finite group''' is a group of finite order. That is, a group $\struct {G, \circ}$ is a '''finite group''' {{iff}} its underlying set $G$ is finite.


Definition:Cayley Table

A '''Cayley table''' is a technique for describing an algebraic structure (usually a finite group) by putting all the products in a square array: :$\begin {array} {c|cccc} \circ & a & b & c & d \\ \hline a & a & a & b & a \\ b & b & c & a & d \\ c & d & e & f & a \\ d & c & d & a & b \\ \end {array}$


Definition:Latin Square

Let $n \in \Z_{>0}$ be some given (strictly) positive integer $n$. A '''Latin square''' of order $n$ is a square array of size $n \times n$ containing $n$ different symbols, such that every row and column contains '''exactly one''' of each symbol. That is, each row and column is a permutation of the same $n$ symbols.


Definition:Infinite Group

A group which is not finite is an '''infinite group'''.


Definition:General Linear Group

Let $K$ be a field. The set of all invertible order-$n$ square matrices over $K$ is a group under (conventional) matrix multiplication. This group is called the '''general linear group (of degree $n$)''' and is denoted $\GL {n, K}$, or $\GL n$ if the field is implicit. The field itself is usually $\R$, $\Q$ or $\C$, but can be ''any'' field.


Definition:Noetherian Ring

=== Definition 1 === {{Definition:Noetherian Ring/Definition 1}} === Definition 2 === {{Definition:Noetherian Ring/Definition 2}} === Definition 3 === {{Definition:Noetherian Ring/Definition 3}} === Definition 4 === {{Definition:Noetherian Ring/Definition 4}}


Definition:Product (Abstract Algebra)

=== General Operation === {{:Definition:Operation/Binary Operation/Product}} === Group Product === {{:Definition:Product (Abstract Algebra)/Group}} === Ring Product === {{:Definition:Product (Abstract Algebra)/Ring}} === Field Product === {{:Definition:Product (Abstract Algebra)/Field}} Category:Definitions/Operations Category:Definitions/Abstract Algebra Category:Definitions/Multiplication bnffxibu0958x3dqn2zydtradw911qk


Definition:Product Inverse

The inverse of $x \in U_R$ by $\circ$ is called the '''(ring) product inverse of $x$'''. The usual means of denoting the product inverse of an element $x$ is by $x^{-1}$. Thus it is distinguished from the additive inverse of $x$, that is, the (ring) negative of $x$, which is usually denoted $-x$.


Definition:Affine Algebraic Variety

Let $k$ be a field. Then a subset $X \subseteq k^n$ is an '''affine algebraic variety''' if the following hold: * $X$ is an affine algebraic set  * $X$ is irreducible


Definition:Zariski Topology

=== On an Affine Space === {{:Definition:Zariski Topology/Affine Space}} === On the spectrum of a ring === {{:Definition:Zariski Topology/Spectrum of Ring}}


Definition:Radical of Integer

The '''radical of an integer''' $n \in \Z$ is the product of the individual prime factors of $n$. The radicals of the first few integers are given here: {| |- ! align="right" | $n$ !! ! align="right" | Decomposition !! ! align="right" | $\map {\operatorname {rad} } n$ |- | align="right" | $1$ || | align="right" | $1$ || | align="right" | $1$ |- | align="right" | $2$ || | align="right" | $2$ || | align="right" | $2$ |- | align="right" | $3$ || | align="right" | $3$ || | align="right" | $3$ |- | align="right" | $4$ || | align="right" | $2^2$ || | align="right" | $2$ |- | align="right" | $5$ || | align="right" | $5$ || | align="right" | $5$ |- | align="right" | $6$ || | align="right" | $2 \times 3$ || | align="right" | $6$ |- | align="right" | $7$ || | align="right" | $7$ || | align="right" | $7$ |- | align="right" | $8$ || | align="right" | $2^3$ || | align="right" | $2$ |- | align="right" | $9$ || | align="right" | $3^2$ || | align="right" | $3$ |- | align="right" | $10$ || | align="right" | $2 \times 5$ || | align="right" | $10$ |- | align="right" | $11$ || | align="right" | $11$ || | align="right" | $11$ |- | align="right" | $12$ || | align="right" | $2^2 \times 3$ || | align="right" | $6$ |- | align="right" | $13$ || | align="right" | $13$ || | align="right" | $13$ |- | align="right" | $14$ || | align="right" | $2 \times 7$ || | align="right" | $14$ |- | align="right" | $15$ || | align="right" | $3 \times 5$ || | align="right" | $15$ |- | align="right" | $16$ || | align="right" | $2^4$ || | align="right" | $2$ |} {{OEIS|A007947}} The radical of $n$ can alternatively be described as the largest square-free integer which divides $n$.


Definition:Generator of Subgroup

Let $\struct {G, \circ}$ be a group. Let $S \subseteq G$. Let $H$ be the subgroup generated by $S$. Then '''$S$ is a generator of $H$''', denoted $H = \gen S$, {{iff}} $H$ is the subgroup generated by $S$.


Definition:Join of Subgroups

Let $\struct {G, \circ}$ be a group. Let $A$ and $B$ be subgroups of $G$. The '''join''' of $A$ and $B$ is written and defined as: :$A \vee B := \gen {A \cup B}$ where $\gen {A \cup B}$ is the subgroup generated by $A \cup B$. By the definition of subgroup generator, this can alternatively be written: :$\displaystyle A \vee B := \bigcap \set {T: T \text { is a subgroup of } G: A \cup B \subseteq T}$ === General Definition === {{:Definition:Join of Subgroups/General Definition}}


Definition:Argand Diagram

An '''Argand diagram''' is a graphical representation of a set of complex numbers on the complex plane: :400px


Definition:Topological Manifold/Smooth Manifold

Let $M$ be a second-countable locally Euclidean space of dimension $d$.  Let $\mathscr F$ be a smooth differentiable structure on $M$. Then $\left({M, \mathscr F}\right)$ is called a '''smooth manifold of dimension $d$'''.


Definition:Kampyle of Eudoxus

:300px {{refactor|Separate pages for Cartesian and Polar equations}} The '''Kampyle of Eudoxus''' has the Cartesian equation: :$x^4 = x^2 + y^2$ where the point $x = y = 0$ is specifically excluded. In polar coordinates, it is described by the equation: :$r = \sec^2 \theta$ {{stub}} {{NamedforDef|Eudoxus of Cnidus|cat = Eudoxus}}


Definition:Cube Number

A '''cube number''' (or just '''cube''') is a number which can be expressed as the third power of an integer.


Definition:One

The immediate successor element of zero in the set of natural numbers $\N$ is called '''one''' and has the symbol $1$. === Naturally Ordered Semigroup === {{:Definition:Unit (One)/Naturally Ordered Semigroup}}


Definition:Ordered Tuple as Ordered Set

The rigorous definition of an ordered tuple is as a finite sequence whose domain is $\N^*_n$. However, it is possible to treat an ordered tuple as an extension of the concept of an ordered pair. === Ordered Triple === {{:Definition:Ordered Tuple as Ordered Set/Ordered Triple}} === Ordered Quadruple === {{:Definition:Ordered Tuple as Ordered Set/Ordered Quadruple}} === Ordered Tuple === {{:Definition:Ordered Tuple as Ordered Set/Ordered Tuple}} === Ordered Singleton === {{:Definition:Ordered Tuple as Ordered Set/Ordered Singleton}} Category:Definitions/Ordered Tuples 8watrb6opfg28e0ug7ndutke9mj0gcg


Definition:Zero Element

An element $z \in S$ is called a '''two-sided zero element''' (or simply '''zero element''' or '''zero''') {{iff}} it is both a '''left zero''' and a '''right zero''': :$\forall x \in S: x \circ z = z = z \circ x$


Definition:Invertible Operation

The operation $\circ$ is '''invertible''' {{iff}}: :$\forall a, b \in S: \exists r, s \in S: a \circ r = b = s \circ a$


Definition:Mathematical System

A '''mathematical system''' is a set $\SS = \struct {E, O, A}$ where: :$E$ is a non-empty set of elements :$O$ is a set of relations and operations on the elements of $E$ :$A$ is a set of axioms concerning the elements of $E$ and $O$. === Abstract System === A '''mathematical system''' $\SS = \struct {E, O, A}$ is classed as '''abstract''' if the elements of $E$ and $O$ are defined only by their properties as specified in $A$. === Concrete System === A '''mathematical system''' $\SS = \struct {E, O, A}$ is classed as '''concrete''' if the elements of $E$ and $O$ are understood as objects independently of their existence in $\SS$ itself. The distinction between '''abstract''' and '''concrete''' is of questionable value from a modern standpoint, as it is a moot point, for example, as to whether the natural numbers exist independently of Peano's axioms or are specifically '''defined''' by them. === Algebraic System === A '''mathematical system''' $\SS = \struct {E, O, A}$ is classed as '''algebraic''' if it has many of the properties of the set of integers. This is usually because such a system is itself an abstraction of certain properties of the integers. The axioms are usually not considered as separate entities from the operations, as their nature is implicit in the operations themselves. Specifically, an algebraic system can be defined as follows: {{:Definition:Algebraic System}}


Definition:Scalar Field

Let $\struct {G, +_G, \circ}_K$ be a vector space, where: :$\struct {K, +_K, \times_K}$ is a field :$\struct {G, +_G}$ is an abelian group $\struct {G, +_G}$ :$\circ: K \times G \to G$ is a binary operation. Then the field $\struct {K, +_K, \times_K}$ is called the '''scalar field''' of $\struct {G, +_G, \circ}_K$.


Definition:Bilinear Mapping

Let $\left({R, +_R, \times_R}\right)$ be a commutative ring. Let $\left({A_1, +_1, \circ_1}\right)_R, \left({A_2, +_2, \circ_2}\right)_R, \left({A_3, +_3, \circ_3}\right)_R$ be $R$-modules. Let $\oplus: A_1 \times A_2 \to A_3$ be a binary operator with the property that: $\forall \left({a_1, a_2}\right) \in A_1 \times A_2$: : $a_1 \mapsto a_1 \oplus a_2$ is a linear transformation from $A_1$ to $A_3$ : $a_2 \mapsto a_1 \oplus a_2$ is a linear transformation from $A_2$ to $A_3$ Then $\oplus$ is a '''bilinear mapping'''. That is, $\forall a, b \in R, \forall x, y \in A_2, z \in A_3$: : $\left({\left({a \circ_1 x}\right) +_1 \left({y \circ_1 b}\right)}\right) \oplus z = \left({a \circ_3 \left({x \oplus z}\right)}\right) +_3 \left({\left({y \oplus z}\right) \circ_3 b}\right)$ and for all $z \in A_1, x,y \in A_2$: : $z \oplus \left({\left({a \circ_2 x}\right) +_2 \left({y \circ_2 b}\right)}\right) = \left({a \circ_3 \left({z \oplus x}\right)}\right) +_3 \left({\left({z \oplus y}\right) \circ_3 b}\right)$ Equivalently, this can be expressed: : $\left({x +_1 y}\right) \oplus z = \left({x \oplus z}\right) +_3 \left({y \oplus z}\right)$ : $z \oplus \left({x +_2 y}\right) = \left({z \oplus x}\right) +_3 \left({z \oplus y}\right)$ : $\left({a \circ_1 x}\right) \oplus z = a \circ_3 \left({x \oplus z}\right)$ : $z \oplus \left({y \circ_2 b}\right) = \left({z \oplus y}\right) \circ_3 b$ If $\left({A, +, \circ}\right)_R = A_1 = A_2 = A_3$, the notation simplifies considerably: : $\left({\left({a \circ x}\right) + \left({b \circ y}\right)}\right) \oplus z = \left({a \circ \left({x \oplus z}\right)}\right) + \left({b \circ \left({y \oplus z}\right)}\right)$ : $z \oplus \left({\left({a \circ x}\right) + \left({y \circ b}\right)}\right) = \left({a \circ \left({z \oplus x}\right)}\right) + \left({\left({z \oplus y}\right) \circ b}\right)$ or equivalently, more easily digested: : $\left({x + y}\right) \oplus z = \left({x \oplus z}\right) + \left({y \oplus z}\right)$ : $z \oplus \left({x + y}\right) = \left({z \oplus x}\right) + \left({z \oplus y}\right)$ : $\left({a \circ x}\right) \oplus z = a \circ \left({x \oplus z}\right)$ : $z \oplus \left({y \circ b}\right) = \left({z \oplus y}\right) \circ b$ === Non-Commutative Ring === {{:Definition:Bilinear Mapping/Non-Commutative Ring}}


Definition:Algebra over Field

Let $F$ be a field. An '''algebra over $F$''' is an ordered pair $\struct {A, *}$ where: :$A$ is a vector space over $F$ :$* : A^2 \to A$ is a bilinear mapping That is, it is an algebra $\struct {A, *}$ over the ring $F$ where: :$F$ is a field :the $F$-module $A$ is a vector space.


Definition:Algebra over Ring

Let $R$ be a commutative ring. An '''algebra over $R$''' is an ordered pair $\left({A, *}\right)$ where: :$A$ is an $R$-module :$*: A^2 \to A$ is an $R$-bilinear mapping


Definition:Second Order Ordinary Differential Equation

A '''second order ordinary differential equation''' is an ordinary differential equation in which any derivatives with respect to the independent variable have order no greater than $2$.


Definition:Physical Universe

The '''physical universe''', or usually just '''universe''', is commonly defined as, and understood to be, '''{{WP|Universe|the totality of everything that exists}}'''. === Real-World === Used to describe a phenomenon or object that has a genuine existence in the physical universe.


Definition:Body

In the context of physics and applied mathematics, a '''body''' is an object (possibly idealized) which is imagined to have a material existence in the physical universe. A '''body''' can be considered as an aggregate of particles.


Definition:Physical Law

A '''physical law''' is a statement about the physical universe whose truth value stems from observations.


Definition:Force

A '''force''' is an influence which causes a body to undergo a change in velocity. '''Force''' is a vector quantity.


Definition:Velocity

The '''velocity''' $\mathbf v$ of a body $M$ is defined as the first derivative of the displacement $\mathbf s$ of $M$ from a given point of reference {{WRT|Differentiation}} time $t$: :$\mathbf v = \dfrac {\d \mathbf s} {\d t}$


Definition:Frame of Reference

A '''frame of reference''' is a coordinate system which is more or less arbitrarily or conveniently disposed in a model of the physical universe. === Point of Reference === {{:Definition:Frame of Reference/Point of Reference}}


Definition:Mathematical Model

A '''mathematical model''' is an equation, or a system of equations, whose purpose is to provide an approximation to the behavior of a real-world phenomenon.


Definition:Displacement

The '''(physical) displacement''' of a body is a measure of its position relative to a given point of reference in a particular frame of reference. '''Displacement''' is a vector quantity, so it specifies a magnitude and direction from the point of reference.


Definition:Scalar Quantity

A '''scalar quantity''' is a real-world concept that needs for its model a mathematical object which contains only one (usually numeric) component.


Definition:Magnitude

The '''magnitude''' (or '''size''') of a quantity (either vector or scalar) is a measure of how big it is. It is usually encountered explicitly in the context of vectors: If $\mathbf v$ is the vector quantity in question, then its '''magnitude''' is denoted: :$\size {\mathbf v}$ or :$v$


Definition:Direction

The '''direction''' of a vector quantity is a measure of which way it is pointing, relative to a particular frame of reference.


Definition:Speed

The '''speed''' of a body is a measure of the magnitude of its velocity, taking no account of its direction. It is, therefore, a scalar quantity.


Definition:Acceleration

The '''acceleration''' $\mathbf a$ of a body $M$ is defined as the first derivative of the velocity $\mathbf v$ of $M$ relative to a given point of reference {{WRT|Differentiation}} time $t$: :$\mathbf a = \dfrac {\d \mathbf v} {\d t}$


Definition:Linear Momentum

The '''linear momentum''' of a body is its mass multiplied by its velocity. :$\mathbf p = m \mathbf v$


Definition:Mass

The '''mass''' of a body is a measure of how much matter it contains. '''Mass''' is equivalent to inertia. '''Mass''' also determines the degree to which a body creates or is affected by a gravitational field. It is a scalar quantity.


Definition:Second Order Partial Differential Equation

A '''second order partial differential equation''' is a partial differential equation in which any derivatives with respect to the independent variable have order no greater than $2$. Category:Definitions/Differential Equations dj1g50oyvq5f2s9tfxf8qo3ebx5n9fl


Definition:Proportion

Two real variables $x$ and $y$ are '''proportional''' {{iff}} one is a constant multiple of the other: :$\forall x, y \in \R: x \propto y \iff \exists k \in \R, k \ne 0: x = k y$


Definition:Aggregate

An '''aggregate''' is an archaic word for an '''infinite set''', as used by {{AuthorRef|Georg Cantor}}. It is used in current parlance to mean a general accumulation of objects, but the concept is vague.


Definition:Ideal (Physics)

An '''ideal''' (or '''idealized''') object is one in which certain attributes are approximated to zero or infinity.


Definition:Particle

A '''particle''' is a representation of an object in the physical university which is idealized as having no magnitude. That is, it is modelled as being a mass concentrated at a single point.


Consider the implicit function $\map f {x, y, c} = 0$ in the $\tuple {x, y}$-plane where $c$ is a constant. For each value of $c$, we have that $\map f {x, y, c} = 0$ defines a relation between $x$ and $y$ which can be graphed in the cartesian plane. Thus, each value of $c$ defines a particular curve. The complete set of all these curve for each value of $c$ is called a '''one-parameter family of curves'''. === Parameter === {{:Definition:One-Parameter Family of Curves/Parameter}}


Definition:Orthogonal Trajectories

Let $\map f {x, y, c}$ define a one-parameter family of curves $F$. Let $\map g {x, y, c}$ also define a one-parameter family of curves $G$, with the property that: :Every curve in $F$ is orthogonal to every curve in $G$. Then $F$ is a '''family of (reciprocal) orthogonal trajectories''' of $G$, and contrariwise.


Definition:Orthogonal (Analytic Geometry)

Two curves are '''orthogonal''' if they intersect at right angles. The term perpendicular can also be used, but the latter term is usual when the intersecting lines are straight.


Definition:Quadrature

'''Quadrature''' is an archaic term meaning '''the process of finding area'''.


Definition:Lune

A '''lune''' is a geometric figure formed by the boundary of two intersecting circles: :300px The area coloured red is a '''lune'''. Category:Definitions/Plane Geometry 3jqvkjclp95b23vs1zi0n1flb3p6fsh


Definition:Physical Quantity

A '''physical quantity''' is a physical property which can be quantified.


Definition:Rest Mass

The '''rest mass''' of a body is its '''mass''' when it is '''at rest'''.


Definition:Physical Property

A '''physical property''' is any measurable property whose describes the state of a physical system.


Definition:Measurable Property

A '''measurable property''' is a property of a physical system which can be compared against a standard unit of measurement to provide a numerical value defining the quantity of that property.


Definition:Physical System

A '''physical system''' is a portion of the physical universe which has been chosen for investigation for a particular purpose.


Definition:Quantitative Property

A '''quantitative property''' is a property of a physical system which exists in a range of values corresponding with a number, such that the system can be described as having a particular number of a unit of measurement of that property.


Definition:Unit of Measurement

A '''unit of measurement''' is a specified magnitude of a given physical quantity, defined by convention. It is used as a standard for measurement of that physical quantity. Any other value of the physical quantity can be expressed as a multiple of that '''unit of measurement'''.


Definition:Inertia

'''Inertia''' is the tendency of a body to maintain the same velocity in the absence of an external force, in accordance with Newton's First Law of Motion. Equivalently put, '''inertia''' is the resistance of a body to a change in its motion. '''Inertia''' is equivalent to mass.


Definition:Friction

'''Friction''' is a force whose tendency is to reduce the velocity of a body relative to its surroundings. It is caused by interaction of the matter in the body with that of the environment in which it is traveling. '''Friction''' is (at a fundamental level) an electromagnetic force. That is, bits of the medium in or on which the body is traveling get in the way (on a molecular level) of bits of the body. The less friction, the less the reduction in velocity. If there were no '''friction''', then motion would continue for ever.


Definition:Force of Gravity

The '''force of gravity''' or '''gravitational force''' is the force on a body as a result of Newton's Law of Universal Gravitation. When used in an unqualified sense, it is usual for this to mean the force on a body at the surface of the Earth. From Gravity at Earth's Surface, this is approximately $9.8 \ \mathrm N \ \mathrm{kg}^{-1}$ The force of gravity varies across the earth's surface, and therefore it makes little sense to use it as a standard. However, the {{WP|General_Conference_on_Weights_and_Measures|CGPM}} adopted a standard acceleration of gravity of $9.806 \, 65  \ \mathrm N \ \mathrm{kg}^{-1}$ in 1901. Category:Definitions/Gravity 0qw8377izdwvjztb4f6nhkuaquyo923


Definition:Dimension (Measurement)

Every physical quantity has a '''dimension''' associated with it. No attempt is made here to provide an abstract definition of this term. Instead, it will be defined by example.


Definition:Rate

The '''rate''' of a physical process is its (first) derivative with respect to time. Loosely speaking, it means '''how fast something progresses''', with a wider scope than change in physical displacement. === Dimension === {{:Definition:Rate/Dimension}} Category:Definitions/Dimensions of Measurement tt43pngp41nnkp4tbffuxe13olzp1e3


Definition:Physical Process

A '''physical process''' is a physical system whose importance is in the way it changes over time.


Definition:Periodic Process

A '''periodic process''' is a physical process whose state can be modeled by a periodic function. === Example === The canonical example of such a process is the motion of a pendulum. Category:Definitions/Physics Category:Definitions/Applied Mathematics 0s0oo3vz3ydmblst61a9dup16j9frke


Definition:Time

A true definition of what '''time''' actually ''is'' has baffled philosophers from the dawn of, er, time. Therefore it will be left as an exercise for the reader. It is usually treated as a scalar quantity.


Definition:Time Period

The '''time period''' of a periodic process is the period of the periodic function which models that process. Category:Definitions/Physics Category:Definitions/Applied Mathematics Category:Definitions/Time m4o5wsjh9xnfpa78bufkhus4rnf4jh7


Definition:Fundamental Unit (Physics)

A '''fundamental unit''' a unit of measurement for a measurable physical property from which every other unit for that quantity can be derived. The '''fundamental unit''' for a particular measurable quantity is chosen by convention. They are the '''units''' of the fundamental dimensions of physics.


Definition:SI Units

The '''SI Units''' are the elements of the [http://en.wikipedia.org/wiki/International_System_of_Units International System of Units].


Definition:Time/Unit/Second

The '''second''' is the SI base unit of time, and also therefore of the MKS system. It is also the base unit of time for the FPS and CGS systems. The '''second''' is defined as: :the duration of $9 \ 192 \ 631 \ 770$ periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium $133$ atom at rest at $0 \ \mathrm K$.


Definition:Kilogram

The '''kilogram''' is the SI base unit of mass. It is defined as being equal to: :The fixed numerical value of the Planck constant $h$ to be $6 \cdotp 62607015 \times 10^{-34}$ when expressed in the unit Joule seconds. The Joule second is equal to $1 \ \mathrm {kg} \mathrm m^2 \mathrm s^{−1}$, where the metre and the second are defined in terms of: :the speed of light $c$ :the time of transition between the two hyperfine levels of the ground state of the caesium $133$ atom at rest at $0 \ \mathrm K$.


Definition:Gram

The '''gram''' is the CGS base unit of mass. It is defined as the mass of a cubic centimetre water at the temperature of melting ice.


Definition:Temperature

'''Temperature''' is a physical property of matter that quantifies how hot or cold a body is. It is a scalar quantity which can be mapped directly to the real number line.


Definition:Kelvin

The '''kelvin''' is the SI base unit of temperature. Its two reference points are: :$0 \ \mathrm K$, which is set at absolute zero, the temperature at which all thermal motion stops :$273 \cdotp 16 \ \mathrm K$, which is set at the {{WP|Triple_point|triple point}} of water.


Definition:Absolute Zero

'''Absolute zero''' is the lowest temperature which can theoretically be achieved. It is the temperature where all motion due to thermal effects stops. Before that temperature can be reached, quantum effects come into play.


Definition:Electric Charge

'''Electric charge''' is a physical quantity of matter which causes it to experience a force when near other electrically charged matter. It is a scalar quantity.


Definition:Electric Current

'''Electric current''' is the physical process caused by the flow of electrically charged particles, usually electrons. Thus it is defined as the rate of flow of electric charge: :$I = \dfrac {\d Q} {\d t}$


Definition:Ampere (Unit)

The '''ampere''' is the SI base unit of electric current. It is defined as being: :The constant current which will produce a force of attraction whose value is $2 \times 10^{–7}$ newtons per metre of length between two straight, parallel conductors of infinite length and negligible circular cross section placed one metre apart in a vacuum. This arises from Ampère's Force Law, which states that such an attractive force exists. === Symbol === {{:Symbols:A/Ampere}} {{NamedforDef|André-Marie Ampère|cat = Ampère}}


Definition:Newton (Unit)

The '''newton''' is the SI unit of force. It is defined as being: :The amount of force required to accelerate a mass of one kilogram at a rate of one metre per second squared.


Definition:Coulomb (Unit)

The '''coulomb''' is the SI unit of electric charge. It is defined as being: :The amount of electric charge carried in one second by a constant current of one ampere.


Definition:Cube/Geometry

A '''cube''' is a hexahedron whose $6$ faces are all congruent squares.


Definition:CGS

'''CGS''' is the centimetre-gram-second standard system of units of measurement. This system is rarely used nowadays, the SI units having largely taken over.


Definition:MKS

'''MKS''' is the metre-kilogram-second standard system of units of measurement.


Definition:FPS

'''FPS''' is the foot-pound-second standard system of units of measurement. It is derived from the imperial system, and was used throughout the British Empire until the mid-20th century.


Definition:Imperial

The '''imperial system''' is a system of measurement based on traditional established folk measures, adopted by the British empire and used in Britain until the mid-20th century. Its base units can be understood as being the FPS Base Units


Definition:Free Fall

A body $B$ influenced by a gravitational field $M$ is in '''free fall''' {{iff}} the force on it caused by $M$ is the only force on $B$.


Definition:Gravitational Field

Every body which has mass influences every other body which has mass, according to Newton's Law of Universal Gravitation. Thus any body can be considered as being surrounded by a field given rise to by its mass, called a '''gravitational field'''. Its value $\mathbf g$ at any point is given by: :$\mathbf g = \dfrac {G M} {d^3} \mathbf d$ where: :$G$ is the gravitational constant; :$M$ is the mass of the body; :$\mathbf d$ is the displacement vector from the point to the center of gravity of the body, whose magnitude is $d$. Category:Definitions/Gravity a6klacxrfb4shh9tbts35rik9o2b3gs


Definition:Field (Physics)

A '''field''', in the context of physics, is a physical quantity associated with every point of spacetime. The physical quantity in question may be either in vector or scalar form. Category:Definitions/Physics tcg1synkvpwd2bx0r2lh1zhpm5qoiku


Definition:Spacetime

'''Spacetime''' (or''' space–time''', or '''space/time''') is a mathematical model that combines space and time into a single object. The form this model takes is often a four-dimensional vector space: three dimensions for space and one for time. More complicated models are used for more specialized applications. {{SUBPAGENAME}} ibycmme5pk6pp2aeslpiylu7v6owaax


Definition:Cube Root

{{:Definition:Cube Root/Real}}


Definition:Unit Vector

Let $\mathbf v$ be a vector quantity. The '''unit vector''' in the direction of $\mathbf v$ is defined and denoted as: :$\mathbf {\hat v} = \dfrac {\mathbf v} {\size {\mathbf v} }$ where $\size {\mathbf v}$ is the magnitude of $\mathbf v$.


Definition:Spring

A '''spring''' is a mechanical device for storing elastic potential energy. It works by resisting a force which acts so as to deform it. The greater the deformation, the greater the force. === Ideal Spring === {{:Definition:Spring/Ideal}} === Equilibrium Position === {{:Definition:Spring/Equilibrium Position}}


Definition:Density (Physics)

'''Density''' is a physical quantity. The '''density''' of a body is its mass per unit volume.


Definition:Volume

'''Volume''' is the measure of the extent of a body. It has three dimensions and is specified in units of length cubed.


Definition:Homogeneous (Physics)

A body is said to be '''homogeneous''' {{iff}} the substance of any part of it is indistinguishable from any other part.


Definition:Weight (Physics)

The '''weight''' of a body is the magnitude of the force exerted on it by the influence of a gravitational field. The context is that the gravitational field in question is usually that of the Earth.


Definition:Chain (Physics)

A '''chain''' is an inelastic thread whose stiffness and width are approximated to zero. The mass of a '''chain''' is usually defined in terms of linear density.


Definition:Cord (Physics)

A '''cord''' is an inelastic thread whose mass, stiffness and width are approximated to zero. That is, it is a chain whose mass is zero.


Definition:Cone

A '''cone''' is a three-dimensional geometric figure which consists of the set of all straight lines joining the boundary of a plane figure $PQR$ to a point $A$ not in the same plane of $PQR$: :300px


Definition:Right Circular Cone

A '''right circular cone''' is a cone: : whose base is a circle : in which there is a line perpendicular to the base through its center which passes through the apex of the cone: : which is made by having a right-angled triangle turning along one of the sides that form the right angle. :300px {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book XI/18 - Cone}}'' {{EuclidDefRefNocat|XI|18|Cone}}


Definition:Cross-Section

A '''cross-section''' of a 3-dimensional figure $F$ is the intersection of $F$ with a plane. {{SUBPAGENAME}} ogr1fkh0dxg0lfa72ibeihamehx0jxp


Definition:Power of Mapping

Let $f: S \to S$ be a mapping from $S$ to itself. Because the domain of $f$ is equal to the codomain of $f$ (both are $S$), the composite mapping $f \circ f$ is defined. We define the '''$n$th power of $f$''' as: :$\forall n \in \N: f^n = \begin{cases} I_S & : n = 0 \\ f \circ f^{n - 1} & : n > 0 \end{cases}$ where $I_S$ is the identity mapping.


=== Zero-Based === {{:Definition:Initial Segment of Natural Numbers/Zero-Based}} === One-Based === {{:Definition:Initial Segment of Natural Numbers/One-Based}}


Definition:Count

To '''count''' a set $S$ is to establish a bijection between $S$ and an initial segment $\N_n$ of the natural numbers $\N$. If $S \sim \N_n$ (where $\sim$ denotes set equivalence) then we have '''counted''' $S$ and found it has $n$ elements. If $S \sim \N$ then $S$ is infinite but countable.


Definition:Multiplicative Notation

:$x y$ is used to indicate the result of the operation on $x$ and $y$. There is no symbol used to define the operation itself. :$e$ or $1$ is used for the identity element. :$x^{-1}$ is used for the inverse element. :$x^n$ is used to indicate the $n$th power of $x$.


Definition:Additive Notation

:$x + y$ is used to indicate the result of the operation $+$ on $x$ and $y$. :$e$ or $0$ is used for the identity element. Note that in this context, $0$ is '''not''' a zero element. :$-x$ is used for the inverse element. :$n x$ is used to indicate the $n$th power of $x$.


Definition:Set of Integer Multiples

The set $n \Z$ is defined as: :$\set {x \in \Z: n \divides x}$ for some $n \in \Z_{>0}$. That is, it is the set of all integers which are divisible by $n$, that is, the '''set of integer multiples of $n$'''. Thus we have: :$n \Z = \set {\ldots, -3 n, -2 n, -n, 0, n, 2 n, 3 n, \ldots}$


Definition:Transversal (Group Theory)

Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $S \subseteq G$ be a subset of $G$. === Left Transversal === {{:Definition:Transversal (Group Theory)/Left Transversal}} === Right Transversal === {{:Definition:Transversal (Group Theory)/Right Transversal}} === Transversal === A '''transversal for $H$ in $G$''' is either a left transversal or a right transversal.


Definition:Product of Differences

Let $n \in \Z_{> 0}$ be a strictly positive integer. Let $\tuple {x_1, x_2, \ldots, x_n}$ be an ordered $n$-tuple of real numbers. The '''product of differences''' of $\tuple {x_1, x_2, \ldots, x_n}$ is defined and denoted as: :$\map {\Delta_n} {x_1, x_2, \ldots, x_n} = \displaystyle \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j}$ When the underlying ordered $n$-tuple is understood, the notation is often abbreviated to $\Delta_n$. Thus $\Delta_n$ is the product of the difference of all ordered pairs of $\tuple {x_1, x_2, \ldots, x_n}$ where the index of the first is less than the index of the second.


Definition:Ordinary Space

'''Ordinary space''' (or just '''space''') is a word used to mean '''the universe we live in'''. The intuitive belief is that space is $3$-dimensional and therefore isomorphic to the real vector space $\R^3$. Hence '''ordinary space''' is usually taken as an alternative term for Euclidean $3$-dimensional space.