Definition:Induced Norm on Quotient of Cauchy Sequences

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\mathcal C$ be the ring of Cauchy sequences over $R$ Let $\mathcal N$ be the set of null sequences. For all $\sequence {x_n} \in \mathcal C$, let $\eqclass {x_n} {}$ denote the left coset $\sequence {x_n} + \mathcal N$ Let $\norm {\, \cdot \,}_1: \mathcal C \,\big / \mathcal N \to \R_{\ge 0}$ be defined by: :$\displaystyle \forall \eqclass {x_n} {} \in \mathcal C \,\big / \mathcal N: \norm {\eqclass {x_n} {} }_1 = \lim_{n \mathop \to \infty} \norm {x_n}$ $\norm {\, \cdot \,}_1$ is called the '''induced norm on the quotient ring of Cauchy sequences'''.


Definition:Linear Differential Operator

A '''linear differential operator''' is a differential operator $\mathscr L$ with the property that: :$\map {\mathscr L} {\alpha \phi_1 + \beta \phi_2} = \alpha \map {\mathscr L} {\phi_1} + \beta \map {\mathscr L} {\phi_2}$ Thus if $\phi_1$ and $\phi_2$ are solutions to the differential equation $\map {\mathscr L} {\phi_i} = 0$, then so is any linear combination of $\phi_1$ and $\phi_2$.


Definition:Linear Integral Operator

A '''linear integral operator''' is a integral operator $\mathscr L$ with the property that: :$\map {\mathscr L} {\alpha \phi_1 + \beta \phi_2} = \alpha \map {\mathscr L} {\phi_1} + \beta \map {\mathscr L} {\phi_2}$ Thus if $\phi_1$ and $\phi_2$ are solutions to the integral equation $\map {\mathscr L} {\phi_i} = 0$, then so is any linear combination of $\phi_1$ and $\phi_2$.


Definition:Set/Distinction between Element and Set

It is important to distinguish between an element, for example $a$, and a singleton containing it, that is, $\set a$. That is $a$ and $\set a$ are ''not'' the same thing. While it is true that: :$a \in \set a$ it is not true that: :$a = \set a$ neither is it true that: :$a \in a$


Definition:Experiment/Informal Definition

An '''experiment''' is defined as: : ''a course of action whose consequence is not predetermined.''


Definition:Experiment/Formal Definition

An '''experiment''', which can conveniently be denoted $\EE$, is a measure space $\struct {\Omega, \Sigma, \Pr}$ such that $\map \Pr \Omega = 1$.


Definition:Sample Space/Discrete

If $\Omega$ is a countable set, whether finite or infinite, then it is known as a '''discrete sample space'''.


Definition:Event/Simple Event

A '''simple event''' in $\EE$ is an event in $\EE$ which consists of exactly $1$ elementary event. That is, it is a singleton subset of the sample space $\Omega$ of $\EE$.


Definition:Complementary Event

Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$. Let $A \in \Sigma$ be an event in $\EE$. The '''complementary event''' to $A$ is defined as $\relcomp \Omega A$. That is, it is the subset of the sample space of $\EE$ consisting of all the elementary events of $\EE$ that are not in $A$.


Definition:Trivial Event

Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$. The empty set $\O$ is a subset of $\Sigma$, by Empty Set is Subset of All Sets, and so is an event in $\EE$. It can be referred to as the '''the trivial event''' of $\EE$.


Definition:Probability Measure/Definition 1

Let $\EE$ be defined as a measure space $\struct {\Omega, \Sigma, \Pr}$. Then $\Pr$ is a measure on $\EE$ such that $\map \Pr \Omega = 1$.


Definition:Probability Measure/Definition 2

Let $\Omega$ be the sample space on $\EE$. Let $\Sigma$ be the event space of $\mathcal E$. A '''probability measure on $\EE$''' is a mapping $\Pr: \Sigma \to \R$ which fulfils the Kolmogorov axioms: {{:Axiom:Kolmogorov Axioms}}


Definition:Probability Measure/Definition 3

Let $\Omega$ be the sample space on $\EE$. Let $\Sigma$ be the event space of $\mathcal E$. A '''probability measure on $\EE$''' is a mapping $\Pr: \Sigma \to \R$ which fulfils the following axioms: {{begin-axiom}} {{axiom | n = \text I         | q = \forall A \in \Sigma         | ml= \map \Pr A         | mo= \ge         | mr= 0 }} {{axiom | n = \text {II}         | ml= \map \Pr \Omega         | mo= =         | mr= 1 }} {{axiom | n = \text {III}         | q = \forall A \in \Sigma         | ml= \map \Pr A         | mo= =         | mr= \sum_{\bigcup \set e \mathop = A} \map \Pr {\set e}         | rc= where $e$ denotes the elementary events of $\EE$ }} {{end-axiom}}


Definition:Probability Measure/Definition 4

Let $\Omega$ be the sample space on $\EE$. Let $\Sigma$ be the event space of $\mathcal E$. A '''probability measure on $\EE$''' is an additive function $\Pr: \Sigma \to \R$ which fulfils the following axioms: {{begin-axiom}} {{axiom | n = 1         | q = \forall A, B \in \Sigma: A \cap B = \O         | ml= \map \Pr {A \cup B}         | mo= =         | mr= \map \Pr A + \map \Pr B }} {{axiom | n = 2         | ml= \map \Pr \Omega         | mo= =         | mr= 1 }} {{end-axiom}}


Definition:Decimal

=== Decimal System === {{:Definition:Decimal System}} === Decimal Notation === {{:Definition:Decimal Notation}} === Decimal Expansion === {{:Definition:Decimal Expansion}} === Decimal Point === {{:Definition:Decimal Point|}} === Decimal Place === {{:Definition:Decimal Place}} === Decimal Part === Erroneously used to mean fractional part: {{:Definition:Decimal Part}} Category:Definitions/Number Systems cfaggi6xrg4msoheotg59i1d97p96lw


Definition:Decimal Expansion/Decimal Place

Let the '''decimal expansion''' of $x$ be: :$x = \sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_{10}$ Then $d_k$ is defined as being the digit in the $k$th '''decimal place'''.


Definition:Number Names

This page gathers together instances of names of numbers in various languages and contexts.


Definition:Number Names/Welsh

These are the names of the numbers in Welsh: {{begin-axiom}} {{axiom | n = 1         | lc = '''un''' }} {{axiom | n = 2         | lc = '''dau''', '''dwy''' }} {{axiom | n = 3         | lc = '''toi''', '''tair''' }} {{axiom | n = 4         | lc = '''pedwar''', '''pedair''' }} {{axiom | n = 5         | lc = '''pump''' (or '''pum''') }} {{axiom | n = 6         | lc = '''chwech''' (or '''chwe''') }} {{axiom | n = 7         | lc = '''saith''' }} {{axiom | n = 8         | lc = '''wyth''' }} {{axiom | n = 9         | lc = '''naw''' }} {{axiom | n = 10         | lc = '''deg''' }} {{axiom | n = 11         | lc = '''un ar ddeg''' }} {{axiom | n = 12         | lc = '''deuddeg''' }} {{axiom | n = 13         | lc = '''toi ar ddeg''', '''tair ar ddeg''' }} {{axiom | n = 14         | lc = '''pedwar ar ddeg''', '''pedair ar ddeg''' }} {{axiom | n = 15         | lc = '''pymtheg''' }} {{axiom | n = 16         | lc = '''un ar bymtheg''' }} {{axiom | n = 17         | lc = '''dau ar bymtheg''', '''dwy ar bymtheg''' }} {{axiom | n = 18         | lc = '''deunaw''' }} {{axiom | n = 19         | lc = '''pedwar ar bymtheg''', '''pedair ar bymtheg''' }} {{axiom | n = 20         | lc = '''ugain''' }} {{axiom | n = 30         | lc = '''deg ar hugain''' }} {{axiom | n = 40         | lc = '''dugain''' }} {{axiom | n = 50         | lc = '''deg ar deugain''' or '''hanner cant''' }} {{axiom | n = 60         | lc = '''trigain''' }} {{axiom | n = 70         | lc = '''trigain a deg''' or '''deg ar thrigain''' }} {{axiom | n = 80         | lc = '''pedwar ugain''' }} {{axiom | n = 90         | lc = '''pedwar ugain a deg''' or '''deg a phedwar ugain''' }} {{axiom | n = 100         | lc = '''cant''' }} {{end-axiom}} The different forms for $2$, $3$, $4$ and their derivatives are for masculine and feminine. The different forms for $5$ and $6$ are elisions for use in front of certain letters. The different forms for $50$, $70$ and $90$ are separate ways of saying the numbers. Note: :there are no distinct words for $11$ and $12$, indicating that there has been less influence of duodecimal systems on Welsh :there are clear indications of base $5$ and vigesimal influences as well as decimal :the interesting forms for $18$ (expressed as $2 \times 9$) and $50$ (expressed both as $10 + 40$ and $\dfrac 1 2 \times 100$).


Definition:Continuous Real Function/Point/Definition by Epsilon-Delta

Then '''$f$ is continuous at $x$''' {{iff}} the limit $\displaystyle \lim_{y \mathop \to x} \map f y$ exists and: :$\displaystyle \lim_{y \mathop \to x} \, \map f y = \map f x$


Definition:Continuous Real Function/Point/Definition by Neighborhood

Then '''$f$ is continuous at $x$''' {{iff}} the limit $\displaystyle \lim_{y \mathop \to x} \map f y$ exists and: :$\displaystyle \lim_{y \mathop \to x} \, \map f y = \map f x$ :for every $\epsilon$-neighborhood $N_\epsilon$ of $\map f x$ in $\R$, there exists a $\delta$-neighborhood $N_\delta$ of $x$ in $A$ such that $\map f x \in N_\epsilon$ whenever $x \in N_\delta$.


Definition:Approximation

An '''approximation''' is an estimate of a quantity. It is usually the case that there exists some knowledge about the accuracy of the estimate. The notation: :$a \approx b$ indicates that $b$ is an '''approximation''' to $a$.


Definition:Cut (Analysis)

Let $\alpha \subset \Q$ be a subset of the set of rational numbers $\Q$ which has the following properties: :$(1): \quad \alpha \ne \O$ and $\alpha \ne \Q$, that is: $\alpha$ contains at least one rational number but not all rational numbers :$(2): \quad$ If $p \in \Q$ and $q \in \Q$ such that $q < p$, then $q \in \Q$ :$(3): \quad \alpha$ does not contain a greatest element. Then $\alpha$ is called a '''cut'''.


Definition:Cut (Analysis)/Lower Number

Let $\alpha$ be a cut. Let $p \in \alpha$. Then $p$ is referred to as a '''lower number''' of $\alpha$.


Definition:Cut (Analysis)/Upper Number

Let $\alpha$ be a cut. Let $q \in \Q$ such that $q \notin \alpha$. Then $p$ is referred to as an '''upper number''' of $\alpha$.


Definition:Cut (Analysis)/Rational

Let $r \in \Q$ be a rational number. Let $\alpha$ be the cut consisting of all rational numbers $p$ such that $p < r$. Then $\alpha$ is referred to as a '''rational cut'''.


Definition:Equality of Cuts

Let $\alpha$ and $\beta$ be cuts such that: :$p \in \alpha \implies p \in \beta$ :$q \in \beta \implies q \in \alpha$ Then $\alpha$ and $\beta$ are '''equal'''.


Definition:Ordering of Cuts/Strict

:$\alpha$ '''is less than ''' $\beta$, denoted $\alpha < \beta$ {{iff}}: :there exists a rational number $p \in \Q$ such that $p \in \alpha$ but $p \notin \beta$.


Definition:Ordering of Cuts

Let $\alpha$ and $\beta$ be cuts. $\alpha$ and $\beta$ conventionally have the following ordering imposed on them, as follows: :$\alpha$ '''is less than or equal to''' $\beta$, denoted $\alpha \le \beta$ {{iff}}: :$\alpha = \beta$ or $\alpha < \beta$ where $<$ denotes the strict ordering relation between $\alpha$ and $\beta$. This can also be expressed as $\beta \ge \alpha$. === Strict Ordering === {{:Definition:Ordering of Cuts/Strict}}


Definition:Positive Cut/Strictly Positive

Let $\alpha > 0^*$. Then $\alpha$ can be referred to as a '''strictly positive cut'''.


Definition:Positive Cut

Let $\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$. Let $\alpha \ge 0^*$. Then $\alpha$ can be referred to as a '''positive cut'''. === Strictly Positive Cut === {{:Definition:Positive Cut/Strictly Positive}}


Definition:Negative Cut

Let $\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$. Let $\alpha \le 0^*$. Then $\alpha$ can be referred to as a '''negative cut'''. === Strictly Negative Cut === {{:Definition:Negative Cut/Strictly Negative}}


Definition:Negative Cut/Strictly Negative

Let $\alpha < 0^*$. Then $\alpha$ can be referred to as a '''strictly negative cut'''.


Definition:Addition of Cuts

Let $\alpha$ and $\beta$ be cuts. Let the operation of '''addition''' be defined on $\alpha$ and $\beta$ as: :$\gamma := \alpha + \beta$ where $\gamma$ is the set of all rational numbers $r$ such that: :$\exists p \in \alpha, q \in \beta: r = p + q$ In this context, $\gamma$ is known as the '''sum of $\alpha$ and $\beta$'''.


Definition:Negative of Cut

Let $\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$: :$0^* = \set {r \in \Q: r < 0}$ Let $\beta$ be the unique cut such that: :$\alpha + \beta = 0^*$ where $+$ denotes the operation of addition of cuts. Then $\beta$ is referred to as the '''negative''' of $\alpha$. It is usually denoted $-\alpha$.


Definition:Subtraction of Cuts

Let $\alpha$ and $\beta$ be cuts. Let $\beta - \alpha$ denote the operation on $\alpha$ and $\beta$ defined as $\beta + \paren {-\alpha}$. Then the operator $-$ is known as '''subtraction''', and $\beta - \alpha$ is itself called the '''difference''' between $\beta$ and $\alpha$.


Definition:Multiplication of Positive Cuts

Let $0^*$ denote the rational cut associated with the (rational) number $0$. Let $\alpha$ and $\beta$ be positive cuts, that is, cuts such that $\alpha \ge 0^*$ and $\beta \ge 0^*$, where $\ge$ denotes the ordering on cuts. Let the operation of '''multiplication''' be defined on $\alpha$ and $\beta$ as: :$\gamma := \alpha \beta$ where $\gamma$ is the set of all rational numbers $r$ such that either: :$r < 0$ or :$\exists p \in \alpha, q \in \beta: r = p q$ where $p \ge 0$ and $q \ge 0$. In this context, $\gamma$ is known as the '''product of $\alpha$ and $\beta$'''.


Definition:Multiplication of Cuts

Let $0^*$ denote the rational cut associated with the (rational) number $0$. Let $\alpha$ and $\beta$ be cuts. The operation of '''multiplication''' is defined on $\alpha$ and $\beta$ as: :$\alpha \beta := \begin {cases} \size \alpha \, \size \beta & : \alpha \ge 0^*, \beta \ge 0^* \\ -\paren {\size \alpha \, \size \beta} & : \alpha < 0^*, \beta \ge 0^* \\ -\paren {\size \alpha \, \size \beta} & : \alpha \ge 0^*, \beta < 0^* \\ \size \alpha \, \size \beta & : \alpha < 0^*, \beta < 0^* \end {cases}$ where: :$\size \alpha$ denotes the absolute value of $\alpha$ :$\size \alpha \, \size \beta$ is defined as in Multiplication of Positive Cuts :$\ge$ denotes the ordering on cuts. In this context, $\alpha \beta$ is known as the '''product of $\alpha$ and $\beta$'''.


Definition:Absolute Value of Cut

Let $\alpha$ be a cut. The '''absolute value of $\alpha$''' is denoted and defined as: :$\size \alpha = \begin {cases} \alpha & : \alpha \ge 0^* \\ -\alpha & : \alpha < 0^* \end {cases}$ where: :$0^*$ denotes the rational cut associated with the (rational) number $0$ :$\ge$ denotes the ordering on cuts.


Definition:Ordered Field Isomorphism

Let $\struct {S, +, \circ, \preceq}$ and $\struct {T, \oplus, *, \preccurlyeq}$ be ordered fields. An '''ordered field isomorphism''' from $\struct {S, +, \circ, \preceq}$ to $\struct {T, \oplus, *, \preccurlyeq}$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ An ordered group isomorphism from the ordered group $\struct {S, +, \preceq}$ to the ordered group $\struct {T, \oplus, \preccurlyeq}$ :$(2): \quad$ A group isomorphism from the group $\struct {S_{\ne 0}, \circ}$ to the semigroup $\struct {T_{\ne 0}, *}$ where $S_{\ne 0}$ and $T_{\ne 0}$ denote the sets $S$ and $T$ without the zeros of $S$ and $T$ respectively.


Definition:Real Number/Also known as

When the term '''number''' is used in general discourse, it is often tacitly understood as meaning '''real number'''. They are sometimes referred to in the pedagogical context as '''ordinary numbers''', so as to distinguish them from '''complex numbers''' However, depending on the context, the word '''number''' may also be taken to mean '''integer''' or '''natural number'''. Hence it is wise to be specific.


Definition:Real Number/Number Line Definition

A '''real number''' is defined as a '''number''' which is identified with a point on the '''real number line'''. ==== Real Number Line ==== {{:Definition:Real Number/Real Number Line}}


Definition:Real Number/Cauchy Sequences

Consider the set of rational numbers $\Q$. For any two Cauchy sequences of rational numbers $X = \sequence {x_n}, Y = \sequence {y_n}$, define an equivalence relation between the two as: :$X \equiv Y \iff \forall \epsilon \in \Q_{>0}: \exists n \in \N: \forall i, j > n: \size {x_i - y_j} < \epsilon$ A '''real number''' is an equivalence class $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.


Definition:Real Number/Digit Sequence

Let $b \in \N_{>1}$ be a given natural number which is greater than  $1$. The set of '''real numbers''' can be expressed as the set of all sequences of digits: :$z = \sqbrk {a_n a_{n - 1} \dotsm a_2 a_1 a_0 \cdotp d_1 d_2 \dotsm d_{m - 1} d_m d_{m + 1} \dotsm}$ such that: :$0 \le a_j < b$ and $0 \le d_k < b$ for all $j$ and $k$ :$\displaystyle z = \sum_{0 \mathop \le j \le n} a_j b^j + \sum_{k \mathop \ge 1} d_k b^{-k}$ It is usual for $b$ to be $10$.


Definition:Final Topology/Definition 1

The '''final topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$''' is defined as: :$\tau = \set{U \subseteq X: \forall i \in I: \map {f_i^{-1}} U \in \tau_i} \subseteq \powerset X$


Definition:Final Topology/Definition 2

Let $\tau$ be the finest topology on $X$ such that each $f_i: Y_i \to X$ is $\tuple{\tau_i, \tau}$-continuous. Then $\tau$ is known as the '''final topology on $X$ with respect to $\family{f_i}_{i \mathop \in I}$'''.


Definition:Subfamily

Let $I$ and $S$ be sets. Let $x: I \to S$ be an indexing function for $S$. Let $\family {x_i}_{i \mathop \in I}$ be a family indexed by $I$. Let $J \subseteq I$ be a subset of $I$. The family $\family {x_j}_{j \mathop \in J}$ indexed by $J$ is known as a '''subfamily''' of $\family {x_i}_{i \mathop \in I}$.


Definition:Accumulation Point

Let $\paren { S, \tau }$ be a topological space. Let $A \subseteq S$. === Accumulation Point of Sequence === {{:Definition:Accumulation Point/Sequence}} === Accumulation Point of Set === {{:Definition:Accumulation Point/Set}}


Definition:Strictly Smaller Set

Let $S$ and $T$ be sets. $S$ is defined as being '''strictly smaller than''' $T$ {{iff}}: :$(1): \quad$ There exists a bijection from $S$ to a subset of $T$ :$(2): \quad$ There exists no such bijection from $T$ to a subset of $S$. '''$S$ is strictly smaller than $T$''' can be denoted: :$S < T$


Definition:Smaller Set

Let $S$ and $T$ be sets. $S$ is defined as being '''smaller than''' $T$ {{iff}} there exists a bijection from $S$ to a subset of $T$. '''$S$ is smaller than $T$''' can be denoted: :$S \le T$


Definition:Strictly Larger Set

Let $S$ and $T$ be sets. $S$ is defined as being '''strictly larger than''' $T$ {{iff}}: :$(1): \quad$ There exists a bijection from $T$ to a subset of $S$ :$(2): \quad$ There exists no such bijection from $S$ to a subset of $T$. '''$S$ is strictly larger than $T$''' can be denoted: :$S > T$


Definition:Larger Set

Let $S$ and $T$ be sets. $S$ is defined as being ''' larger than''' $T$ {{iff}} there exists a bijection from $T$ to a subset of $S$. '''$S$ is larger than $T$''' can be denoted: :$S \ge T$


Definition:Comparable Sets/Cardinality

Let $S$ and $T$ be sets. Then $S$ and $T$ are '''comparable (in size)''' {{iff}} either: :$S$ can be put into one-to-one correspondence with a subset of $T$ or: :$T$ can be put into one-to-one correspondence with a subset of $S$ or both. That is, if either $S$ is smaller than $T$ or $T$ is smaller than $S$.


Definition:First Order Logic with Identity

A system of '''first order logic with identity''' is a system of predicate logic with the following elements in its alphabet: ==== Connectives ==== {{begin-axiom}} {{axiom | ml= \land          | mo= :         | t = the conjunction sign }} {{axiom | ml= \lor         | mo= :         | t = the disjunction sign }} {{axiom | ml= \implies         | mo= :         | t = the conditional sign }} {{axiom | ml= \iff         | mo= :         | t = the biconditional sign }} {{axiom | ml= \neg         | mo= :         | t = the negation sign }} {{end-axiom}} ==== Quantifiers ==== {{begin-axiom}} {{axiom | ml= \exists         | mo= :         | t = the existential quantifier sign }} {{axiom | ml= \forall         | mo= :         | t = the universal quantifier sign }} {{end-axiom}} ==== Identity ==== {{begin-axiom}} {{axiom | ml= =         | mo= :         | t = the equality sign }} {{end-axiom}}


Definition:Frege Set Theory

The '''Frege system of set theory''' is a system of axiomatic set theory which has as its sole axiom the comprehension principle: {{:Axiom:Comprehension Principle}} In support of this, the various logical axioms supporting predicate logic also hold.


Definition:Zermelo Set Theory

'''Zermelo set theory''' is a system of axiomatic set theory. Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the following axioms: === The Axiom of Extension === {{:Axiom:Axiom of Extension/Set Theory}} === The Axiom of the Empty Set === {{:Axiom:Axiom of Empty Set/Set Theory}} === The Axiom of Pairing === {{:Axiom:Axiom of Pairing/Set Theory}} === The Axiom of Specification === {{:Axiom:Axiom of Specification/Set Theory}} === The Axiom of Unions === {{:Axiom:Axiom of Unions/Set Theory}} === The Axiom of Powers === {{:Axiom:Axiom of Powers/Set Theory}} === The Axiom of Infinity === {{:Axiom:Axiom of Infinity}}


Definition:Zermelo-Fraenkel-Skolem Set Theory

'''Zermelo-Fraenkel-Skolem set theory''' is a system of axiomatic set theory. It consists of Zermelo-Fraenkel set theory with the inclusion of a system of first order formulas to specify the existence of properties of sets. {{NamedforDef|Ernst Friedrich Ferdinand Zermelo|name2 = Abraham Halevi Fraenkel|name3 = Thoralf Albert Skolem|cat = Zermelo|cat2 = Fraenkel|cat3 = Skolem}}


Definition:Morse-Kelley Set Theory

'''Morse-Kelley set theory''' is a system of axiomatic set theory. It is a stronger form of Zermelo-Fraenkel-Skolem set theory which allows not only for first order formulas to specify the existence of properties of sets, but also defines properties by quantifying over properties as well as over sets. {{stub}} {{NamedforDef|Anthony Perry Morse|name2 = John Leroy Kelley|cat = Morse|cat2 = Kelley}}


Definition:Von Neumann-Bernays-Gödel Set Theory

'''Von Neumann-Bernays-Gödel set theory''' is a system of axiomatic set theory. Its main feature is that it classifies collections of objects into: :sets, whose construction is strictly controlled and: :classes, which have fewer restrictions on how they may be generated. All sets are classes, but not all classes are sets.


Definition:Model of Class-Set Theory

Let $V$ be a collection of objects. Then $V$ is a '''model of class-set theory''' {{iff}} it satisfies the axioms of Von Neumann-Bernays-Gödel set theory. Thus:  :The elements of $V$ are then known as sets :The subcollections of $V$ are known as classes.


Definition:Collection

A '''collection''' is an aggregation of objects whose nature is arbitrary. The precise nature and structure of the '''collection''' itself is left unspecified. Hence the word '''collection''' can be conveniently used for an aggregation which may be either a '''set''' or a '''class''', and such that whatever is being said about it applies to both.


Definition:Element/Notation

The symbol universally used in modern mainstream mathematics to mean '''$x$ is an element of $S$''' is: :$x \in S$ Similarly, $x \notin S$ means '''$x$ is ''not'' an element of $S$'''. The symbol can be reversed: :$S \ni x$ means '''$S$ has $x$ as an element''', that is, $x$ is an element of $S$ but this is rarely seen. Some texts (usually older ones) use $x \mathop {\overline \in} S$ or $x \mathop {\in'} S$ instead of $x \notin S$.


Definition:Element/Also known as

The term '''member''' is sometimes used as a synonym for element (probably more for the sake of linguistic variation than anything else). In the contexts of geometry and topology, '''elements''' of a set are often called '''points''', in particular when they ''are'' (geometric) points. $x \in S$ can also be read as: * '''$x$ is in $S$''' * '''$x$ belongs to $S$''' * '''$S$ includes $x$''' * '''$x$ is included in $S$''' * '''$S$ contains $x$''' However, '''beware''' of this latter usage: '''$S$ contains $x$''' can also be interpreted as '''$x$ is a subset of $S$'''. Such is the scope for misinterpretation that it is '''mandatory''' that further explanation is added to make it clear whether you mean subset or element.


Definition:Element/Class

Let $S$ be a class. An '''element of $S$''' is a member of $S$.


Definition:Class Equality/Definition 1

$A$ and $B$ are '''equal''', denoted $A = B$, {{iff}}: :$\forall x: \paren {x \in A \iff x \in B}$ where $\in$ denotes class membership.


Definition:Class Equality/Definition 2

$A$ and $B$ are '''equal''', denoted $A = B$, {{iff}}: :$A \subseteq B$ and $B \subseteq A$ where $\subseteq$ denotes the subclass relation.


Definition:First-Order Property of Sets

A '''first-order property of sets''' is a property defined by a logical formula whose domain is entirely over sets. Thus the quantifiers $\forall x$ and $\exists x$ are valid when $x$ is a set, but $\forall A$ and $\exists A$ are ''not'' valid when $A$ is a class. Hence we allow, for example: :$\map \phi {A_1, A_2, \ldots, A_n, x}$ to be a logical formula whose variables $A_1, A_2, \ldots, A_n$ represent classes, and are all free, and $x$ is the only free variable representing a set.


Definition:Swelled Class

Let $A$ denote a class. Then $A$ is a '''swelled class''' {{iff}} every subclass of every element of $A$ is also an element of $A$.


Definition:Supercomplete Class

Let $A$ denote a class. Then $A$ is a '''supercomplete class''' {{iff}} both: :$A$ is transitive and: :$A$ is swelled.


Definition:Von Neumann-Bernays-Gödel Axioms

=== The Axiom of Extension === {{:Axiom:Axiom of Extension/Class Theory}} === The Axiom of Specification === {{:Axiom:Axiom of Specification/Class Theory}} {{finish}}


Definition:Basic Universe

A '''basic universe''' $V$ is a universal class which satisfies the following axioms: {{:Definition:Basic Universe Axioms}}


Definition:Basic Universe Axioms

=== $\text A 1$: Axiom of Transitivity === {{:Axiom:Axiom of Transitivity}} === $\text A 2$: Axiom of Swelledness === {{:Axiom:Axiom of Swelledness}} === $\text A 3$: Axiom of the Empty Set === {{:Axiom:Axiom of Empty Set/Class Theory}} === $\text A 4$: Axiom of Pairing === {{:Axiom:Axiom of Pairing/Class Theory}} === $\text A 5$: Axiom of Unions === {{:Axiom:Axiom of Unions/Class Theory}} === $\text A 6$: Axiom of Powers === {{:Axiom:Axiom of Powers/Class Theory}}


Definition:Zermelo-Fraenkel Universe

A '''Zermelo-Fraenkel universe''' is a basic universe which also satisfies the axiom of infinity and the axiom of substitution: {{:Definition:Basic Universe Axioms}} === $\text A 7$: Axiom of Infinity === {{:Axiom:Axiom of Infinity/Class Theory}} === $\text A 8$: Axiom of Substitution === {{:Axiom:Axiom of Substitution}}


Definition:Zermelo Universe

A '''Zermelo universe''' is a basic universe which also satisfies the axiom of infinity: {{:Definition:Basic Universe Axioms}} === $\text A 7$: Axiom of Infinity === {{:Axiom:Axiom of Infinity/Class Theory}}


Definition:Empty Class (Class Theory)

A class is defined as being '''empty''' {{iff}} it has no elements. That is: :$\forall x: x \notin A$ or: :$\neg \exists x: x \in A$ The '''empty class''' is usually denoted $\O$ or $\emptyset$.


Definition:Singleton Class

Let $a$ be a set. The class $\set a$ is a '''singleton (class)'''. It can be referred to as '''singleton $a$'''.


Definition:Ordered Pair/Wiener Formalization

The concept of an ordered pair can be formalized by the definition: :$\tuple {a, b} := \set {\set {\O, \set a}, \set {\set b} }$


Definition:Ordered Pair/Empty Set Formalization

The concept of an ordered pair can be formalized by the definition: :$\tuple {a, b} := \set {\set {\O, a}, \set {\set \O, b} }$


Definition:Set Intersection/Class

Let $A$ be a class. The '''intersection of $A$''' is: :$\displaystyle \bigcap A := \set {x: \forall y \in A: x \in y}$ That is, the class of all elements which belong to all the elements of $A$.


Definition:Class Union

Let $A$ and $B$ be two classes. The '''(class) union''' $A \cup B$ of $A$ and $B$ is defined as the class of all sets $x$ such that either $x \in A$ or $x \in B$ or both: :$x \in A \cup B \iff x \in A \lor x \in A$ or: :$A \cup B = \set {x: x \in A \lor x \in B}$


Definition:Class Intersection

Let $A$ and $B$ be two classes. The '''(class) intersection''' $A \cap B$ of $A$ and $B$ is defined as the class of all sets $x$ such that $x \in A$ and $x \in B$: :$x \in A \cap B \iff x \in A \land x \in A$ or: :$A \cap B = \set {x: x \in A \land x \in B}$


Definition:Power Set/Class Theory

The '''power set''' of a set $S$ is the class of all the subsets of $S$: :$\powerset S := \set {T: T \subseteq S}$


Definition:Polygon/Adjacent/Side to Vertex

Each vertex of a polygon is formed by the intersection of two sides. The two sides that form a particular vertex are referred to as the '''adjacents''' of that vertex, or described as '''adjacent to''' that vertex.


Definition:Polygon/Adjacent/Vertex to Side

Each side of a polygon intersects two other sides, and so is terminated at either endpoint by two vertices. The two vertices that terminate a particular side are referred to as the '''adjacents''' of that side, or described as '''adjacent to''' that side.


Definition:Polygon/Adjacent/Vertices

Those two vertices are described as '''adjacent to each other'''.


Definition:Polygon/Adjacent/Sides

Two sides of a polygon that meet at the same vertex are '''adjacent''' to each other.


Definition:Diophantine Analysis

'''Diophantine analysis''' is the branch of number theory which is concerned with the solution of Diophantine equations.


Definition:Generatrix/Also known as

A '''generatrix''' of a surface can also be referred to as a '''generator''' or '''generator element'''.


Definition:Vector Product

=== Dot Product === {{:Definition:Dot Product}} === Cross Product === {{:Definition:Vector Cross Product}} === Outer Product === {{:Definition:Outer Product}}


Definition:Stochastic Calculus

'''Stochastic calculus''' is a generalization of calculus to the subject of stochastic processes.


Definition:QEF

The initials of '''Quod Erat Faciendum''', which is {{WP|Latin|Latin}} for '''which was to be done'''. These initials were traditionally added to the end of a geometric construction.


Definition:QEI

The initials of '''Quod Erat Inveniendum''', which is {{WP|Latin|Latin}} for '''which was to be found'''. These initials were traditionally added after the completion of a calculation (either arithmetic or algebraic).


Definition:Quintal Notation

'''Quintal notation''' is the technique of expressing numbers in base $5$. That is, every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 5^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, 2, 3, 4}$


Definition:Random

=== Random Error === {{:Definition:Random Error}} === Random Number === {{:Definition:Random Number}} === Random Sample === {{:Definition:Random Sample}} === Random Selection === {{:Definition:Random Selection}} === Random Variable === {{:Definition:Random Variable}} === Random Vector === {{:Definition:Random Vector}} === Random Walk === {{:Definition:Random Walk}}


Definition:Decade

A '''decade''' is a set or ordered tuple of $10$ numbers. Its most common usage is as a period of time: === Decade (Time Period) === {{:Definition:Decade (Time)}}


Definition:Parastrophe

=== $(1-3)$ Parastrophe === {{:Definition:(1-3) Parastrophe}} === $(2-3)$ Parastrophe === {{:Definition:(2-3) Parastrophe}} === $(1-2)$ Parastrophe === {{:Definition:(1-2) Parastrophe}}


Let $M_1 = \struct{X_1, \norm {\,\cdot\,}_{X_1} }$ and $M_2 = \struct{X_2, \norm {\,\cdot\,}_{X_2} }$ be normed vector spaces. Let $f: X_1 \to X_2$ be a mapping from $X_1$ to $X_2$. Let $a \in X_1$ be a point in $X_1$. === Continuous at a Point === {{Definition:Continuous Mapping (Normed Vector Space)/Point}} === Continuous on a Space === {{Definition:Continuous Mapping (Normed Vector Space)/Space}}


{{Definition:Continuous Mapping (Normed Vector Space)/Point/Definition 1}}


Definition:Continuous Mapping (Normed Vector Space)/Point/Definition 1

'''$f$ is continuous at (the point) $a$ (with respect to the norms $\norm {\,\cdot\,}_{X_1}$ and $\norm {\,\cdot\,}_{X_2}$)''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in X_1: \norm {x - a}_{X_1} < \delta \implies \norm {\map f x - \map f a}_{X_2} < \epsilon$ where $\R_{>0}$ denotes the set of all strictly positive real numbers.


{{Definition:Continuous Mapping (Normed Vector Space)/Space/Definition 1}}


Definition:Continuous Mapping (Normed Vector Space)/Space/Definition 1

$f$ is '''continuous from $\struct{X_1, \norm {\,\cdot\,}_{X_1} }$ to $\struct{X_2, \norm {\,\cdot\,}_{X_2} }$''' {{iff}} it is continuous at every point $x \in X_1$.


Definition:Statics

'''Statics''' is the branch of mechanics investigating the behaviour of forces of bodies at rest.


Definition:Limit of Function (Normed Vector Space)/Epsilon-Delta Condition

:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < \norm {x - c}_{X_1} < \delta \implies \norm {\map f x - L}_{X_2} < \epsilon$ That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that ''every'' point in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of some point $L$ in the codomain of $f$.


Definition:Limit of Function (Normed Vector Space)/Epsilon-Ball Condition

:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \map f {B_\delta \left({c; \norm {\,\cdot\,}_{X_1}}\right) \setminus \set c} \subseteq \map {B_\epsilon} {L; \norm {\,\cdot\,}_{X_2}}$. where: : $\map {B_\delta} {c; \norm {\,\cdot\,}_{X_1}} \setminus \set c$ is the deleted $\delta $-neighborhood of $c$ in $M_1$ : $B_\epsilon \left({L; \norm {\,\cdot\,}_{X_2}}\right)$ is the open $\epsilon$-ball of $L$ in $M_2$. That is, for every open $\epsilon$-ball of $L$ in $M_2$, there exists a deleted $\delta $-neighborhood of $c$ in $M_1$ whose image is a subset of that open $\epsilon$-ball.


Definition:Body/Rigid

A '''rigid body''' is a body $B$ with the property that the distance between any two points of $B$ remains constant regardless of any external forces.


Definition:Motion

A body is defined as being '''in motion''' if it is not stationary with respect to the given frame of reference. Thus '''motion''' is the state of '''changing position'''.


Definition:Ceteris Paribus

'''Ceteris paribus''' is a Latin phrase meaning '''all other things being equal'''. It is also translated as '''in the absence of any other change''' (in the conditions of a theorem, for example). It is often used in logical arguments in natural language when expressing a general comparison between two effects, and is a plea to the hearer to suspend counterarguments based on differences in the general situations.


Definition:Natural Place

According to the ancient Greek philosophers, everything had a position in the universe where it was most comfortable. That position was referred to as its '''natural place'''. Hence, when displaced from its '''natural place''', it would eagerly try to return to that '''natural place'''. Hence: :a stone, being made of earth, when released from a hand holding it in midair, will push downwards in order to return as soon as possible to the centre of the universe, that is, Earth. :tongues of fire will shoot upwards, so as to rise to its '''natural place''' above the sphere of air :bubbles of air will rise up through water to its '''natural place''' above the sphere of water.


Definition:Vacuum

A '''vacuum''' is the physical state of being completely empty of matter. It is a physical ideal, and has never been (and probably can never be) fully realised.


Definition:Inclined Plane

An '''inclined plane''' is a plane surface which is neither horizontal nor vertical.


Definition:Gravity

'''Gravity''' is the tendency of bodies with mass to attract each other.


Definition:Gravity/Gravitational Force

The '''gravitational force''' on a body $B$ is the force which is exerted on $B$ as a result of the gravitational field whose influence it is under.


Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space. Let $Y \subseteq X$ be a subset of $X$. === Limit Point of Set === {{:Definition:Limit Point/Normed Vector Space/Set}} === Limit Point of Sequence === {{:Definition:Limit Point/Normed Vector Space/Sequence}}


Definition:Displacement/Dimension

The dimension of '''displacement''' is length $\mathsf L$.


Definition:Mass/Units

The units of measurement of '''mass''' are as follows: * The SI unit of '''mass''' is the kilogram $\mathrm {kg}$. * The CGS unit of '''mass''' is the gram $\mathrm g$. * The FPS unit of '''mass''' is the pound $\mathrm {lb}$. Thus: :$1 \ \mathrm {kg} = 10^3 \ \mathrm g$ :$1 \ \mathrm {lb} = 0.453 \, 592 \, 37 \ \mathrm{kg}$


Definition:Gram/Symbol

The symbol for the '''gram''' is $\mathrm g$.


Definition:CGS/Base Units

{| class="wikitable" style="margin:1em auto 1em auto" |+ CGS base units |- !Name ! Unit symbol ! Dimension ! Symbol |- ! centimetre | $\mathrm{cm}$ | Length | $l$ |- ! gram | $\mathrm g$ | Mass | $m$ |- ! second | $\mathrm s$ | Time | $t$ |- |}


Definition:CGS/Derived Units/Dyne

The '''dyne''' is the '''CGS''' unit of force: :$1 \ \mathrm {dyn} = 1 \ \mathrm g \ \mathrm{cm} \ \mathrm s^{-2}$


Definition:MKS/Derived Units

=== Newton === {{:Definition:Newton (Unit)}}


Definition:Expectation/Discrete

Let $X$ be a discrete random variable. The '''expectation of $X$''' is written $\expect X$, and is defined as: :$\expect X := \displaystyle \sum_{x \mathop \in \image X} x \, \map \Pr {X = x}$ whenever the sum is absolutely convergent, that is, when: :$\displaystyle \sum_{x \mathop \in \image X} \size {x \, \map \Pr {X = x} } < \infty$


Definition:Limit Point/Normed Vector Space/Sequence

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


Definition:Artificial Intelligence

'''Artificial intelligence''' is a branch of computer science which studies intelligent behavior, and attempts to model it via computing techniques.


Definition:Deduction

'''Deduction''' is the process in logic which derives new true statements from a set of premises, themselves assumed true.


Definition:Temporal Logic

'''Temporal logic''' is a subcategory of modal logic which introduces time, by defining the concepts: :sometimes :always.


Definition:Limit Point/Normed Vector Space/Set

Let $\alpha \in X$.  Then $\alpha$ is a '''limit point of $Y$''' {{iff}} ''every'' deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $Y$: :$\forall \epsilon \in \R_{>0}: \map {B_\epsilon} \alpha \setminus \set \alpha \cap Y \ne \O$ that is: :$\forall \epsilon \in \R_{>0}: \set {x \in Y: 0 < \norm {x - \alpha} < \epsilon} \ne \O$ Note that $\alpha$ does not have to be an element of $A$ to be a '''limit point'''.


Definition:Abel Summation Method

{{Help|It is difficult finding a concise and complete definition of exactly what the Abel Summation Method actually is. All and any advice as to how to implement this adequately is requested of anyone. This is what is said in the Spring encyclopedia on the page "Abel summation method":}} The series: :$\displaystyle \sum a_n$ can be summed by the Abel method ($A$-method) to the number $S$ if, for any real $x$ such that $0 < x < 1$, the series: :$\displaystyle \sum_{k \mathop = 0}^\infty a_k x^k$ is convergent and: :$\displaystyle \lim_{x \mathop \to 1^-} \sum_{k \mathop =0}^\infty a_k x^k = S$ {{help|This is what we have on Wikipedia page {{WP|Divergent_series|Divergent series}}: }} :$\displaystyle \map f x = \sum_{n \mathop = 0}^\infty a_n e^{-n x} = \sum_{n \mathop = 0}^\infty a_n z^n$ where $z = \map \exp {−x}$. Then the limit of $\map f x$ as $x$ approaches $0$ through positive reals is the limit of the power series for $\map f z$ as $z$ approaches $1$ from below through positive reals. The '''Abel sum''' $\map A s$ is defined as: :$\displaystyle \map A s = \lim_{z \mathop \to 1^-} \sum_{n \mathop = 0}^\infty a_n z^n$ {{NamedforDef|Niels Henrik Abel|cat = Abel}}


Definition:Above

In the context of real numbers, '''above''' means '''greater than'''.


=== Absolute Maximum === {{:Definition:Maximum Value of Real Function/Absolute}} === Local Maximum === {{:Definition:Maximum Value of Real Function/Local}}


=== Absolute Minimum === {{:Definition:Minimum Value of Real Function/Absolute}} === Local Minimum === {{:Definition:Minimum Value of Real Function/Local}}


Definition:Topology Generated by Synthetic Basis/Definition 3

The '''topology on $S$ generated by $\BB$''' is defined as: :$\tau = \set {U \subseteq S: \forall x \in U: \exists B \in \BB: x \in B \subseteq U}$


Definition:Compact Space/Normed Vector Space/Subspace

Let $M = \struct{X, \norm {\,\cdot\,}}$ be a normed vector space. Let $K \subseteq X$ be a subset of $X$. The normed vector subspace $M_K = \struct {K, \norm {\,\cdot\,}_K}$ is '''compact in $M$''' {{iff}} $M_K$ is itself a compact normed vector space.


Definition:Relation/Notation

If $\tuple {x, y}$ is an ordered pair such that $\tuple {x, y} \in \RR$, we use the notation: :$s \mathrel \RR t$ or: :$\map \RR {s, t}$ and can say: :'''$s$ bears $\RR$ to $t$''' :'''$s$ stands in the relation $\RR$ to $t$''' If $\tuple {s, t} \notin \RR$, we can write: $s \not \mathrel \RR t$, that is, by drawing a line through the relation symbol. See Complement of Relation.


Definition:Relation/Class Theory

Let $V$ be a basic universe. In the context of class theory, a '''relation''' $\RR$ is a subclass of the Cartesian product $V \times V$.


Definition:Braces

'''Braces''' is the name given to the pair of '''curly brackets''': :$\set {\cdots}$ These are conventionally used mostly for: :$(1): \quad$ Set delimiters, to membership of sets and classes. :$(2): \quad$ the fractional part of a real number: ::::$\fractpart x := x - \floor x$ :::where $\floor x$ denotes the floor of $x$. :$(3): \quad$ Karamata notation for the Stirling numbers of the second kind: ::::$\displaystyle {n \brace k}$ On {{ProofWiki}}, which implements the $\LaTeX$ mathematical markup language, '''braces''' are also used to delimit arguments to $\LaTeX$ commands. Category:Definitions/Language Definitions mo8a908hi9dpxipe6mapqibiyhtquki


Definition:Non-Limit Ordinal

An ordinal $\lambda$ is a '''non-limit ordinal''' {{iff}} it is not a limit ordinal. That is, either the zero ordinal or a successor ordinal.


Definition:Natural Numbers/Inductive Set Definition

Let $x$ be a set which is an element of every inductive set. Then $x$ is a natural number.


Definition:Distinguished Element

Let $S$ be a set. A '''distinguished element of $S$''' is an element which is distinguished in some manner, by the imposition of some property or properties on $S$, from the other elements of $S$.


Definition:Inductive Set

Let $S$ be a set of sets. Then $S$ is '''inductive''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | lc= $S$ contains the empty set:         | q =          | m = \quad \O \in S }} {{axiom | n = 2         | lc= $S$ is closed under the successor mapping:         | q = \forall x         | m = \paren {x \in S \implies x^+ \in S}         | rc= where $x^+$ is the successor of $x$ }} {{axiom | rc= That is, where $x^+ = x \cup \set x$ }} {{end-axiom}}


Definition:Roster Notation

'''Roster notation''' is the technique of specifying the elements in a set by listing them between a pair of braces $\set{}$. === Explicit Definition === {{:Definition:Explicit Set Definition}} If there are many elements in a set, then it becomes tedious and impractical to list them all in one big long explicit definition. Fortunately, however, there are other techniques for listing sets. === Implicit Definition === {{:Definition:Implicit Set Definition}} Category:Definitions/Set Theory j6vc5f7i8000i4wxyug8pnf47osksua


Definition:Inductive Class/General

Let $A$ be a class. Let $g: A \to A$ be a mapping on $A$. Then $A$ is '''inductive under $g$''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | lc= $A$ contains the empty set:         | q =          | m = \quad \O \in A }} {{axiom | n = 2         | lc= $A$ is closed under $g$:         | q = \forall x         | m = \paren {x \in A \implies \map g x \in A} }} {{end-axiom}}


Definition:Minimally Inductive Class under General Mapping/Definition 1

$A$ is '''minimally inductive under $g$''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | t = $A$ is inductive under $g$ }} {{axiom | n = 2         | t = No proper subclass of $A$ is inductive under $g$. }} {{end-axiom}}


Definition:Minimally Inductive Class under General Mapping/Definition 2

$A$ is '''minimally inductive under $g$''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | t = $A$ is inductive under $g$ }} {{axiom | n = 2         | t = Every subclass of $A$ which is inductive under $g$ contains all the elements of $A$. }} {{end-axiom}}


Let $A$ be a class. Let $g: A \to A$ be a mapping. === Definition 1 === {{:Definition:Minimally Inductive Class under General Mapping/Definition 1}} === Definition 2 === {{:Definition:Minimally Inductive Class under General Mapping/Definition 2}} === Definition 3 === {{:Definition:Minimally Inductive Class under General Mapping/Definition 3}}


Definition:Left Normal Element of Relation

Let $A$ be a class. Let $\RR$ be a relation on $A$. An element $x$ of $A$ is '''left normal''' with respect to $\RR$ {{iff}}: :$\forall y \in A: \map \RR {x, y}$ holds.


Definition:Right Normal Element of Relation

Let $A$ be a class. Let $\RR$ be a relation on $A$. An element $x$ of $A$ is '''right normal''' with respect to $\RR$ {{iff}}: :$\forall y \in A: \map \RR {y, x}$ holds.


Definition:Comparable Sets

=== Comparable in Size === {{:Definition:Comparable Sets/Cardinality}} === Comparable by Subset Ordering === {{:Definition:Comparable Sets/Subset Ordering}}


Definition:Comparable Sets/Subset Ordering

Let $S$ and $T$ be sets. Then $S$ and $T$ are '''comparable (with respect to the subset ordering)''' {{iff}} either: :$S \subseteq T$ or: :$T \subseteq S$ or both.


Definition:Nest/Class Theory

Let $C$ be a class. $C$ is a '''nest''' {{iff}}: :$\forall x, y \in C: x \subseteq y$ or $y \subseteq x$


Definition:Well-Ordered Class under Inclusion

Let $A$ be a class which is also a nest. Let $A$ have the property that every non-empty subclass of $A$ has a smallest element under the inclusion relation. Then $A$ is said to be '''well-ordered under inclusion'''.


Definition:Bounded Class

Let $B$ be a class. === Bounded by Set === {{:Definition:Bounded Class/Bounded by Set}} === Bounded Subset of Class === {{:Definition:Bounded Class/Bounded Subset of Class}}


Definition:Bounded Class/Bounded Subset of Class

Let $B$ be a subclass of a class $A$. Then $B$ is a '''bounded subset of $A$''' {{iff}}: :there exists a set $x \in A$ such that $B$ is '''bounded by $x$'''


Definition:Absolute Difference

Let $a$ and $b$ be real numbers. The '''absolute difference''' between $a$ and $b$ is defined and denoted as: :$\size {a - b}$ where $\size {\, \cdot \,}$ is the absolute value function.


Definition:Arbitrary Constant

An '''arbitrary constant''' is a symbol used to represent an object which is neither a specific number nor a variable. It is used to represent a general object (usually a number, but not necessarily) whose value can be assigned when the expression is instantiated. === In the context of Calculus === {{:Definition:Arbitrary Constant (Calculus)}}


Definition:Congruence (Number Theory)/Modulus

The number $m$ in this congruence is known as the '''modulus''' of the congruence.


Definition:Cubic (Geometry)

'''Cubic''' is an adjective which means '''in the shape of a cube'''.


Definition:Interior (Topology)/Definition 3

The '''interior''' of $H$ is the set of all interior points of $H$.


Definition:Interior Point (Topology)/Definition 3

Let $h \in H$. $h$ is an '''interior point''' of $H$ {{iff}} $h$ is an element of an open set of the subspace topology of $H$.


Definition:Weighted Mean/Normalized

Let the weights be normalized. Then the '''weighted mean''' of $S$ can be expressed in the form: :$\displaystyle \bar x := \sum_{i \mathop = 1}^n \map W {x_i} x_i$ as by definition of normalized weight function all the weights add up to $1$.


Definition:Operation/Operator

An '''operator''' is a symbol used to identify an '''operation'''.


Definition:Rate of Change/Dimension

Let a physical quantity $P$ have dimension $\mathsf D$. Then the '''rate of change''' of $P$ has dimension $\mathsf {D T}^{-1}$.


Definition:Section (Geometry)

=== Section of Line by Line === {{:Definition:Section of Line by Line}} === Definition:Section of Solid by Plane === {{:Definition:Section of Solid by Plane}} Category:Definitions/Geometry 10vp9dywstfn2857na6umnnx396075o


Definition:Absolute Number

An '''absolute number''' is a number in an expression which has a single value. It is either expressed using actual figures, in an agreed number system, or by a symbol which is understood to represent that specific number. === Examples === :$2$, $\sqrt 5$, $1.976$, $\dfrac 2 3$, $\pi$ (pi), $e$ (Euler's number)


Definition:Long Period Prime

A '''long period prime''' is a prime number $p$ whose reciprocal has the maximum period $p - 1$.


Definition:Minimally Closed Class

Let $A$ be a class. Let $g: A \to A$ be a mapping. === Definition 1 === {{:Definition:Minimally Closed Class/Definition 1}} === Definition 2 === {{:Definition:Minimally Closed Class/Definition 2}}


Definition:Minimally Closed Class/Definition 1

$A$ is '''minimally closed under $g$ with respect to $b$''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | t = $A$ is closed under $g$ }} {{axiom | n = 2         | t = There exists $b \in A$ such that no proper subclass of $A$ containing $b$ is closed under $g$. }} {{end-axiom}}


Definition:Minimally Closed Class/Definition 2

$A$ is '''minimally closed under $g$ with respect to $b$''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | t = $A$ is closed under $g$ }} {{axiom | n = 2         | t = There exists $b \in A$ such that every subclass of $A$ containing $b$ which is closed under $g$ contains all the elements of $A$. }} {{end-axiom}}


Definition:Minimally Inductive Class under General Mapping/Definition 3

$A$ is '''minimally inductive under $g$''' {{iff}} $A$ is minimally closed under $g$ with respect to $\O$.


Definition:Abstract Space

An '''abstract space''' is: :a set of objects together with: :a set of axioms which define operations on and relations between those objects. === Metric Space === {{:Definition:Metric Space}} === Topological Space === {{:Definition:Topological Space}} === Vector Space === {{:Definition:Vector Space}}


Definition:Rectangle Function

The '''rectangle function''' is the real function $\Pi: \R \to \R$ defined as: :$\forall x \in \R: \map \Pi x := \begin {cases} 1 : & \size x \le \dfrac 1 2 \\ 0 : & \size x > \dfrac 1 2 \end {cases}$


Definition:Triangle Function

The '''triangle function''' is the real function $\Lambda: \R \to \R$ defined as: :$\forall x \in \R: \map \Lambda x := \begin {cases} 1 - \size x : & \size x \le 1 \\ 0 : & \size x > 1 \end {cases}$ where $\size x$ denotes the absolute value function.


Definition:Rectangle Function/Graph

The graph of the rectangle function is illustrated below: :400px


Definition:Triangle Function/Graph

The graph of the triangle function is illustrated below: :400px


Definition:Heaviside Step Function/Graph

The graph of the Heaviside step function is illustrated below: :600px


Definition:Signum Function/Graph

The graph of the signum function is illustrated below: :400px


Definition:Dirac Delta Function/Graph

The graph of the construction of the Dirac delta function is illustrated below: :400px In the limit, the graph can be approximated as follows, where it is understood that the blue arrow represents a ray from $0$ up the $y$-axis: :400px


Definition:Lituus

The '''lituus''' is the locus of the equation expressed in Polar coordinates as: :$r^2 \theta = a^2$ :600px


Definition:Reciprocal Spiral

The '''reciprocal spiral''' is the locus of the equation expressed in Polar coordinates as: :$r = \dfrac a \theta$ :600px


Definition:Fermat's Spiral

'''Fermat's spiral''' is the locus of the equation expressed in Polar coordinates as: :$r^2 = \dfrac a \theta$ :600px


Definition:Cornu Spiral

The '''Cornu spiral''' is the locus $C$ of the equation expressed in intrinsic coordinates as: :$s = a^2 \kappa$ where: :$s$ denotes the length of arc at a point of $C$ from the origin :$\kappa$ denotes the curvature of $C$ at that point. :600px


Definition:Sampling Function

The '''sampling function''' is the real function $\operatorname {III}: \R \to \R$ defined as: :$\forall x \in \R: \map {\operatorname {III} } x := \displaystyle \sum_{n \mathop \in \Z} \map \delta {x - n}$ where $\delta$ denotes the Dirac delta function.


Definition:Sampling Function/Graph

The graph of the sampling function is illustrated below: :600px It is to be understood that the blue arrows represent rays from the $x$-axis for constant $n \in \Z$.


Definition:Even Impulse Pair Function

The '''even impulse pair function''' is the real function $\operatorname {II}: \R \to \R$ defined as: :$\forall x \in \R: \map {\operatorname {II} } x := \dfrac 1 2 \map \delta {x + \dfrac 1 2} + \dfrac 1 2 \map \delta {x - \dfrac 1 2}$ where $\delta$ denotes the Dirac delta function.


Definition:Even Impulse Pair Function/Graph

The graph of the even impulse pair function is illustrated below: :400px It is to be understood that the blue arrows represent rays from the $x$-axis for constant $n \in \set {-\dfrac 1 2, \dfrac 1 2}$.


Definition:Odd Impulse Pair Function

The '''odd impulse pair function''' is the real function $\operatorname {I_I}: \R \to \R$ defined as: :$\forall x \in \R: \map {\operatorname {I_I} } x := \dfrac 1 2 \map \delta {x + \dfrac 1 2} - \dfrac 1 2 \map \delta {x - \dfrac 1 2}$ where $\delta$ denotes the Dirac delta function.


Definition:Odd Impulse Pair Function/Graph

The graph of the odd impulse pair function is illustrated below: :300px It is to be understood that the blue arrows represent rays from the $x$-axis for constant $n \in \set {-\dfrac 1 2, \dfrac 1 2}$.


Definition:Filtering Function

The '''filtering function''' is the real function $\operatorname {sinc}: \R \to \R$ defined as: :$\forall x \in \R: \map {\operatorname {sinc} } x := \dfrac {\sin \pi x} {\pi x}$ where $\sin$ denotes the (real) sine function.


Definition:Filtering Function/Graph

The graph of the filtering function is illustrated below: :500px


Definition:Convolution Integral/Positive Real Domain

Let $f$ and $g$ be supported on the positive real numbers $\R_{\ge 0}$ only. The '''convolution integral''' of $f$ and $g$ may be defined as: :$\displaystyle \map f t * \map g t := \int_0^t \map f u \map g {t - u} \rd u$


Definition:Convolution of Real Sequences

Let $\sequence f$ and $\sequence g$ be real sequences. The '''convolution''' of $f$ and $g$ is defined as: :$\displaystyle \sequence {f_i} * \sequence {g_i} := \sum_{j \mathop \in \Z_{\ge 0} } f_i g_{i - j}$


Definition:Even Impulse Pair Function/2 Dimensional

Let $\operatorname {II}: \R \to \R$ denote the '''even impulse pair function'''. The $2$-dimensional form of $\operatorname {II}$ is defined and denoted: :$\forall x, y \in \R: \map {\operatorname { {}^2 II} } {x, y} := \map {\operatorname {II} } x \map {\operatorname {II} } y$


Definition:Dirac Delta Function/2 Dimensional

Let $\delta: \R \to \R$ denote the '''Dirac delta function'''. The $2$-dimensional form of $\delta$ is defined and denoted: :$\forall x, y \in \R: \map { {}^2 \delta} {x, y} := \map \delta x \map \delta y$


Definition:Sampling Function/2 Dimensional

Let $\operatorname {III}: \R \to \R$ denote the '''sampling function'''. The $2$-dimensional form of $\operatorname {III}$ is defined and denoted: :$\forall x, y \in \R: \map {\operatorname { {}^2 III} } {x, y} := \map {\operatorname {III} } x \map {\operatorname {III} } y$


Definition:Filtering Function/2 Dimensional

Let $\operatorname {sinc}: \R \to \R$ denote the '''filtering function'''. The $2$-dimensional form of $\operatorname {sinc}$ is defined and denoted: :$\forall x, y \in \R: \map {\operatorname { {}^2 sinc} } {x, y} := \map {\operatorname {sinc} } x \map {\operatorname {sinc} } y$


Definition:Sub-Gaussian Distribution

The distribution of a random variable $X$ with expectation $\mu = \expect X$ is called '''sub-Gaussian''' if there exists a $\sigma \in \R_{>0}$ such that: :$\expect {e^{\lambda \paren {X - \mu} } } \le e^{\sigma^2 \lambda^2 / 2}$ for all $\lambda \in \R$. Category:Definitions/Probability Distributions 51lpgxg6t77wxtvk5u4xkl7z40jpaa9


Definition:Sampling Function/Linguistic Note

The name '''shah''' for the '''sampling function''' derives from its similarity in shape and appearance to the Russian '''Ш''', whose name is itself pronounced '''shah'''.


{{:Definition:Fourier Transform/Real Function/Formulation 1}}


Definition:Sub-Exponential Distribution

The distribution of a random variable $X$ with expectation $\mu = \expect X$ is called '''sub-exponential''' if there exists $\nu\in \R_{> 0}, \alpha \in \R_{\ge 0}$ such that: :$\expect {e^{\lambda \paren {X - \mu} } } \le e^{\nu^2 \lambda^2 / 2}$ for all $\size \lambda < \dfrac 1 \alpha$. {{explain|Where does $\alpha$ come into it?}}


Definition:Progression

The word '''progression''' is a term used for either '''sequence''' or '''series''' in the following specific contexts. Because its usage is in general ambiguous, it is deprecated on {{ProofWiki}}.


Definition:Arithmetic Progression

The term '''arithmetic progression''' is used to mean one of the following: === Arithmetic Sequence === {{:Definition:Arithmetic Sequence}} === Arithmetic Series === {{:Definition:Arithmetic Series}}


Definition:Geometric Progression

The term '''geometric progression''' is used to mean one of the following: === Geometric Sequence === {{:Definition:Geometric Sequence}} === Geometric Series === {{:Definition:Geometric Series}}


Definition:Arithmetic-Geometric Progression

The term '''arithmetic-geometric progression''' is used to mean one of the following: === Arithmetic-Geometric Sequence === {{:Definition:Arithmetic-Geometric Sequence}} === Arithmetic-Geometric Series === {{:Definition:Arithmetic-Geometric Series}}


Definition:Harmonic Progression

The term '''harmonic progression''' is used to mean one of the following: === Harmonic Sequence === {{:Definition:Harmonic Sequence}} === Harmonic Series === {{:Definition:Harmonic Series}}


Definition:Harmonic Series/General

Let $\sequence {x_n}$ be a sequence of numbers such that $\sequence {\size {x_n} }$ is a harmonic sequence. Then the series defined as: :$\displaystyle \sum_{n \mathop = 1}^\infty x_n$ is a '''harmonic series'''.


Definition:Square Wave/Points of Discontinuity

The points $\alpha + n \lambda$, for $n \in \Z$, are jump discontinuities. The values $\map S {\alpha + n \lambda}$ may or may not be explicitly defined. It is a common approach to include either endpoint of the intervals from $\alpha$ to $\alpha + \lambda$, and from $\alpha - \lambda$ to $\alpha$, in order to ensure that the domain of $S$ is simply defined, for example: :$\forall x \in \R: \map S x = \begin {cases} \delta + \gamma & : x \in \hointr \alpha {\alpha + \lambda} \\ \delta - \gamma & : x \in \hointr {\alpha - \lambda} \alpha \\ \map S {x + 2 \lambda} & : x < \alpha - \lambda \\ \map S {x - \lambda l} & : x \ge \alpha + \lambda \end {cases}$ Another approach is to make $\map S {\alpha + n \lambda} = \delta$ for all $n \in \Z$. The precise treatment of the discontinuities is often irrelevant or immaterial.


Definition:Square Wave

A '''square wave''' is a periodic real function $S: \R \to \R$ defined as follows: :$\forall x \in \R: \map S x = \begin {cases} \delta + \gamma & : x \in \openint \alpha {\alpha + \lambda} \\ \delta - \gamma & : x \in \openint {\alpha - \lambda} \alpha \\ \map S {x + 2 \lambda} & : x < \alpha - \lambda \\ \map S {x - 2 \lambda} & : x > \alpha + \lambda \end {cases}$ where: :$\alpha$, $\lambda$, $\gamma$ and $\delta$ are given real constants.


Definition:Square Wave/Graph

The graph of $S$ is given below: :800px


Definition:Given

A '''given''' is an object whose properties and existence are stipulated at the beginning of an exposition.


Definition:Finite Discontinuity

Let $A \subseteq \R$ be a subset of the real numbers. Let $f: A \to \R$ be a real function. Let $f$ be discontinuous at $a \in A$. The point $a$ is a '''finite discontinuity''' of $f$ {{iff}} either: :$(1): \quad a$ is a removable discontinuity or: :$(2): \quad a$ is a jump discontinuity. Category:Definitions/Continuity 9hs3d2ni3o3srwdp37y6gv50znysm6k


Definition:Triangle Wave

A '''triangle wave''' is a periodic real function $T: \R \to \R$ defined as follows: :$\forall x \in \R: \map T x = \begin {cases} \size x & : x \in \closedint {-\lambda} \lambda \\ \map T {x + 2 \lambda} & : x < -\lambda \\ \map T {x - 2 \lambda} & : x > +\lambda \end {cases}$ where: :$\lambda$ is a given real constant :$\size x$ denotes the absolute value of $x$.


Definition:Triangle Wave/Graph

The graph of $T$ is given below: :1000px


Definition:Sawtooth Wave

A '''sawtooth wave''' is a periodic real function $S: \R \to \R$ defined as follows: :$\forall x \in \R: \map S x = \begin {cases} x & : x \in \openint {-\lambda} \lambda \\ \map S {x + 2 \lambda} & : x < -\lambda \\ \map S {x - 2 \lambda} & : x > +\lambda \end {cases}$ where $\lambda$ is a given real constant


Definition:Sawtooth Wave/Points of Discontinuity

The points $\paren {2 r + 1} \lambda$, for $r \in \Z$, are jump discontinuities. The values $\map S {\paren {2 r + 1} \lambda}$ may or may not be explicitly defined. It is a common approach to include one of the endpoints of the interval from $-\lambda$ to $\lambda$, in order to ensure that the domain of $S$ is simply defined. For the sawtooth wave for example: :$\forall x \in \R: \map S x = \begin {cases} x & : x \in \hointr {-\lambda} \lambda \\ \map S {x + 2 \lambda} & : x < -\lambda \\ \map S {x - 2 \lambda} & : x \ge +\lambda \end {cases}$ and, for the inverse sawtooth wave: :$\forall x \in \R: \map S x = \begin {cases} -x & : x \in \hointr {-\lambda} \lambda \\ \map S {x + 2 \lambda} & : x < -\lambda \\ \map S {x - 2 \lambda} & : x \ge +\lambda \end {cases}$ Another approach is to make $\map S {\paren {2 r + 1} \lambda} = 0$ for all $r \in \Z$. The precise treatment of the discontinuities is often irrelevant or immaterial.


Definition:Sawtooth Wave/Graph

The graph of $S$ is given below: :800px


Definition:Sawtooth Wave/Inverse/Graph

The graph of $S$ is given below: :800px


Definition:Set/Point Set

A '''set''' whose elements are all (geometric) points is often called a '''point set'''. In particular, the Cartesian plane and complex plane can each be seen referred to as a '''two-dimensional point set'''.


Definition:Definite Integral/Limits of Integration/Upper Limit

The limit $b$ is referred to as the '''upper limit''' of the integral'''.


Definition:Definite Integral/Limits of Integration/Lower Limit

The limit $a$ is referred to as the '''lower limit''' of the integral'''.


Definition:Fourier Series/Range 2 Pi

Let $\alpha \in \R$ be a real number. Let $f: \R \to \R$ be a function such that $\displaystyle \int_\alpha^{\alpha + 2 \pi} \map f x \rd x$ converges absolutely. Let: {{begin-eqn}} {{eqn | l = a_n       | r = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x }} {{eqn | l = b_n       | r = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} \map f x \sin n x \rd x }} {{end-eqn}} Then: :$\dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$ is called the '''Fourier Series''' for $f$.


Definition:Half-Range Fourier Cosine Series/Formulation 1

Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$. Then the ''' half-range Fourier cosine series''' of $\map f x$ over $\openint 0 \lambda$ is the series: :$\map f x \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$ where for all $n \in \Z_{\ge 0}$: :$a_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \cos \frac {n \pi x} \lambda \rd x$


Definition:Half-Range Fourier Sine Series/Formulation 1

Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$. Then the '''half-range Fourier sine series''' of $\map f x$ over $\openint 0 \lambda$ is the series: :$\map f x \sim \displaystyle \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$ where for all $n \in \Z_{> 0}$: :$b_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x$


Definition:Half-Range Fourier Series/Range Pi

=== Half-Range Fourier Cosine Series === {{:Definition:Half-Range Fourier Cosine Series/Range Pi}} === Half-Range Fourier Sine Series === {{:Definition:Half-Range Fourier Sine Series/Range Pi}}


Definition:Half-Range Fourier Cosine Series/Range Pi

Let $\map f x$ be a real function defined on the interval $\openint 0 \pi$. Then the ''' half-range Fourier cosine series''' of $\map f x$ over $\openint 0 \pi$ is the series: :$\map f x \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty a_n \cos n x$ where for all $n \in \Z_{\ge 0}$: :$a_n = \displaystyle \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x$


Definition:Half-Range Fourier Sine Series/Range Pi

Let $\map f x$ be a real function defined on the interval $\openint 0 \pi$. Then the ''' half-range Fourier sine series''' of $\map f x$ over $\openint 0 \pi$ is the series: :$\map f x \sim \displaystyle \sum_{n \mathop = 1}^\infty b_n \sin n x$ where for all $n \in \Z_{> 0}$: :$b_n = \displaystyle \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$


Definition:Everywhere Dense/Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $Y \subseteq X$ be a subset of $X$. Suppose: :$\forall x \in X : \forall \epsilon \in \R : \epsilon > 0 : \exists y \in Y : \norm {x - y} < \epsilon$ Then $Y$ is '''(everywhere) dense''' in $X$.


Definition:Separable Space/Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $Y \subseteq X$ be a subset of $X$. Let $Y$ be dense in $X$. Let $Y$ be a countable set: :$Y = \set {y_1, y_2, \ldots}$ Suppose: :$\forall x \in X : \forall \epsilon \in \R : \epsilon > 0 : \exists y_{n \mathop \in \N} \in Y : \norm {y_n - x} < \epsilon$ Then $X$ is '''separable'''.


Definition:Conjugate Point/Definition 3

Let $y = \map y x$ and $y = \map {\tilde y} x$ be extremal functions. Let: :$M = \paren {a, \map y a}$ :$\tilde M = \paren {\tilde a, \map y {\tilde a} }$ Let both $y = \map y x$ and $y = \map {\tilde y} x$ pass through the point $M$. Let  :$\displaystyle \lim_{\norm {\map y x - \map {\tilde y} x}_{1, \infty} \to 0} \sqbrk {\paren {x, \map y x}: \map y x - \map {\tilde y} x = 0} = \tilde M$ In other words, let $\tilde M$ be the limit points of intersection of $y = \map y x$ and $y = \map {\tilde y} x$ as $\norm {\map y x - \map {\tilde y} x}_{1, \infty} \to 0$. {{explain|the notation $\norm {\map y x - \map {\tilde y} x}_{1, \infty}$ with particular regard to the subscript This is most likely the standard norm of Sobolev space, also know as np-norm. Original source never calls it by word; only by special symbols}} Then $\tilde M$ is '''conjugate''' to $M$.


Definition:Separated by Neighborhoods/Sets/Neighborhoods

Let $A, B \subseteq S$ such that: :$\exists N_A, N_B \subseteq S: \exists U, V \in \tau: A \subseteq U \subseteq N_A, B \subseteq V \subseteq N_B: N_A \cap N_B = \O$ That is, that $A$ and $B$ both have neighborhoods in $T$ which are disjoint. Then $A$ and $B$ are described as '''separated by neighborhoods'''.


Definition:Separated by Neighborhoods/Points/Neighborhoods

Let $x, y \in S$ such that: :$\exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \in U \subseteq N_x, y \in V \subseteq N_y: N_x \cap N_y = \O$ That is, that $x$ and $y$ both have neighborhoods in $T$ which are disjoint. Then $x$ and $y$ are described as '''separated by neighborhoods'''.


Definition:Separated by Neighborhoods/Points/Open Sets

Let $x, y \in S$ such that: :$\exists U, V \in \tau: x \in U, y \in V: U \cap V = \O$ That is, that $x$ and $y$ both have open neighborhoods in $T$ which are disjoint. Then $x$ and $y$ are described as '''separated by (open) neighborhoods'''.


Definition:Separated by Neighborhoods/Sets/Open Sets

Let $A, B \subseteq S$ such that: :$\exists U, V \in \tau: A \subseteq U, B \subseteq V: U \cap V = \O$ That is, that $A$ and $B$ both have open neighborhoods in $T$ which are disjoint. Then $A$ and $B$ are described as '''separated by (open) neighborhoods'''.


Definition:Positive Definite Functional

Let $f : S \to T$ be a mapping. Let $f \in \MM$, where $\MM$ stands for some function space. Let $F \sqbrk f : \MM \to \R$ be a real-valued functional. Suppose: :$\forall f \in \MM: F \sqbrk f > 0$ Then $F$ is a '''positive-definite functional'''. Category:Definitions/Functional Analysis kdrak7sbetlc0p4syfabm44plyku69w


Definition:Earthworm Sequence

The '''earthworm sequence''' is the integer sequence $\sequence {h_n}$ defined as follows. Let $h: \Z \to \Z$ be the mapping defined as: :$\forall a \in \Z: \map h a = 2 a \pmod {100}$ Then: :$h_n = \begin {cases} x: x \in \set {10, 11, \dotsc, 99} & : n = 0 \\ \map h {n - 1} & : n > 0 \end {cases}$ That is: :$h_0$ is an arbitrary integer between $10$ and $99$ :$h_n$ is the result of multiplying the previous term in the sequence by $2$ and keeping only the last two digits.


Definition:Bounded Sequence/Normed Vector Space

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space. Let $\sequence {x_n}$ be a sequence in $X$. Then $\sequence {x_n}$ is '''bounded''' {{iff}}: :$\exists K \in \R$ such that $\forall n \in \N: \norm {x_n} \le K$


Definition:Bounded Sequence/Normed Vector Space/Unbounded

$\sequence {x_n}$ is '''unbounded''' {{iff}} it is not bounded.


Definition:Quality of Integer Triple

Let $\tuple {a, b, c}$ be an ordered triple of (strictly) positive integers. The '''quality''' of $\tuple {a, b, c}$ is defined and denoted as: :$\map q {a, b, c} := \dfrac {\ln c} {\map \ln {\map {\operatorname {rad} } {abc} } }$ where $\operatorname {rad}$ denotes the radical of an integer. Category:Definitions/Number Theory rp83swjw9ou6vt9w5luv52on617nlyg


Definition:Continuous Function Space

Let $X, Y$ be topological spaces. Let $f : X \to Y$ be a continuous mapping. Then the set of all such mappings $f$ is known as '''continuous function space''' and is denoted by $\CC \paren {X,Y}$: :$\CC \paren {X, Y} = \set {f : X \to Y}$


Definition:Bourbaki Dangerous Bend Symbol

The '''Bourbaki dangerous bend symbol''' is a symbol invented by the {{AuthorRef|Nicolas Bourbaki|Bourbaki group}} to warn the reader that there exists the serious possibility of making a common mistake while interpreting or utilising the material presented. It was originally presented by {{AuthorRef|Nicolas Bourbaki|Bourbaki}}, as the pre-$1949$ French traffic road sign: :20px Variants exist, notably {{AuthorRef|Donald Ervin Knuth}}'s version: :40px {{NamedforDef|Nicolas Bourbaki|cat = Bourbaki}}


Definition:Continuous Real-Valued Function Space

Let $X$ be a topological space. Let $f : X \to \R$ be a continuous real valued mapping. Then the set of all such mappings $f$ is known as '''continuous real-valued function space''' and is denoted by $\map \CC X$: :$\map \CC X := \map \CC {X, \R} = \set {f : X \to \R}$


Definition:Space of Continuous on Closed Interval Real-Valued Functions

Let $f : \closedint a b \to \R$ be a continuous real valued function. Then the set of all such mappings $f$ is known as '''continuous on closed interval real-valued function space''' and is denoted by $\CC \closedint a b$: :$\CC \closedint a b := \CC \paren {\closedint a b, \R} = \set {f : \closedint a b \to \R}$


Definition:Space of Continuous Functions of Differentiability Class k

Let $X, Y$ be normed vector spaces. Let $f : X \to Y$ be a mapping of a differentiability class $k \in \N_{>0}$ in the sense of Frechet. Then the set of all such mappings $f$ is known as '''continuous function space of differentiability class k''' and is denoted by $\CC^k \paren {X,Y}$: :$\CC^k \paren {X, Y} = \set {f : X \to Y}$


Definition:Real-Valued Periodic Function Space

Let $I = \closedint 0 T$ be a closed real interval. Let $\map \CC I$ be the space of real-valued functions, continuous on  $I$. Let the elements of $\map \CC I$ be real periodic functions with the period $T$:  :$f \in \map \CC I: \map f T = \map f 0$ Then the set of all such mappings $f$ is known as '''real-valued periodic (over $I$) function space''' and is denoted by $\map {\CC_{per}} I$: :$\map {\CC_{per}} I:= \CC_{per} \paren {I, \R} = \set {f : I \to \R : \map f T = \map f 0}$


Definition:Space of Real Polynomials of Degree n

Let $\map \CC \R$ be the space of real-valued continuous functions. Let $f \in \map \CC \R$ be a polynomial over real numbers of degree not greater than $n$. Then the set of all such mappings $f$ is known as '''space of real polynomials of degree $\le n$''' and is denoted by $\PP_n \paren {\R}$: :$\PP_n \paren {\R} := \set {f : \R \to \R : \map \deg f \le n}$


Definition:Space of Real Polynomials

Let $\PP_n$ be the space of real polynomials of degree not greater than n. Then the union of all such spaces is called the '''space of real polynomials''' and is denoted by $\PP$: :$\displaystyle \PP := \bigcup _{n \mathop = 0}^{\infty}\PP_n$


Definition:Jaina Mathematics

'''Jainism''' is a religion and philosophy which was founded in India around the $6$th century BCE. '''Jaina mathematics''' is mathematics done by those following Jainism. This has been practised on the Indian subcontinent from the founding of Jainism up to modern times.


Definition:Complete Lattice/Definition 1

Let $\struct {S, \preceq}$ be a lattice. Then $\struct {S, \preceq}$ is a '''complete lattice''' {{iff}}: :$\forall T \subseteq S: T$ admits both a supremum and an infimum.


Definition:Completely Hausdorff Space/Definition 1

$\struct {S, \tau}$ is a '''completely Hausdorff space''' or '''$T_{2 \frac 1 2}$ space''' {{iff}}: :$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U^- \cap V^- = \O$  That is, for any two distinct elements $x, y \in S$ there exist open sets $U, V \in \tau$ containing $x$ and $y$ respectively whose closures are disjoint.


Definition:Completely Hausdorff Space/Definition 2

$\struct {S, \tau}$ is a '''completely Hausdorff space''' or '''$T_{2 \frac 1 2}$ space''' {{iff}}: :$\forall x, y \in S, x \ne y : \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x^- \cap N_y^- = \O$ That is: :$\struct {S, \tau}$ is a '''$T_{2 \frac 1 2}$ space''' {{iff}} every two points in $S$ are separated by closed neighborhoods.


Definition:Polyomino/Free

A '''free polyomino''' is a polyomino which is not distinguished from its image under a reflection. That is, it is considered '''free''' of the plane in which it is embedded, and can be "lifted up and turned over".


Definition:Ordering on Natural Numbers/Von Neumann Construction

Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction. The '''strict ordering''' of $\omega$ is the relation $<$ defined by: :$\forall m, n \in \omega: m < n \iff m \subsetneq n$ The '''(weak) ordering''' of $\omega$ is the relation $\le$ defined by: :$\forall m, n \in \omega: m \le n \iff m \subseteq n$


Definition:Cardinality of Finite Class/Definition 1

Let $A$ be such that: :there exists a bijection $\phi$ from $A$ to $n$ where $n$ is a natural number as defined by the von Neumann construction. Then $A$ has '''cardinality $n$'''.


Definition:Cardinality of Finite Class/Definition 2

Let $A$ be such that: :there exists a bijection $\phi$ from $A$ to the set $\set {1, 2, \dotsc, n} = n^+ \setminus \set 0$ where: :$n$ is a natural number as defined by the von Neumann construction :$n^+$ is the successor of $n$. Then $A$ has '''cardinality $n$'''.


Definition:Finite Class

Let $A$ be a class. Let there exist a natural number $n$ as defined by the von Neumann construction such that there exists a bijection $\phi$ from $A$ to $n$. Then $A$ is a '''finite class'''.


Definition:Infinite Class

Let $A$ be a class. $A$ is an '''infinite class''' {{iff}} $A$ is not a finite class. </onlyinclude>


Definition:Statistic

A '''statistic''' is broadly defined as '''a quantity which is calculated from sample data.


Definition:Dispersion (Statistics)

Let $S$ be a sample of a population in the context of statistics. The '''dispersion''' of $S$ is a general term meaning how much the data describing the sample are spread out. The word can also be applied to a random variable.


Definition:Denumerable Class

Let $A$ be a class. $A$ is '''denumerable''' {{iff}} there exists a bijection: :$f: S \to \N$ where $\N$ is the set of natural numbers.


Definition:Non-Denumerable Class

A class which is neither denumerable nor finite is described as '''non-denumerable'''.


Definition:Machine (Mechanics)

A '''machine''', in the context of mechanics, is a physical artefact which takes energy in a particular form and converts it into energy in another form for a specific purpose.


Definition:Mach Number

'''Mach number''' is a unit of measurement of speed of a body $B$ in a fluid $F$. It is defined as the ratio of the speed of $B$ to the speed of sound in $F$. {{NamedforDef|Ernst Waldfried Josef Wenzel Mach|cat = Mach}}


Definition:Supersonic

The speed of a body $B$ is defined as '''supersonic''' {{iff}} its Mach number is greater than $1$. That is, if $B$ is moving faster than the speed of sound in the fluid in which it is moving.


Definition:Rest

=== At Rest === The term '''at rest''' is a synonym for '''stationary''': {{:Definition:At Rest}} === Rest Mass === {{:Definition:Rest Mass}} Category:Definitions/Physics 91a1fkbupc2a1maip7g8fnbkrtui9l7


Definition:Electric Charge/Dimension

The dimension of measurement of '''electric charge''' is $\mathsf {I T}$.


Definition:Electric Charge/Units

The SI unit of '''electric charge''' is the coulomb $\mathrm C$.


Definition:Electric Field Strength/Units

The SI unit for '''electric field strength''' can be given either as: :volt per metre: $\mathrm V / \mathrm m$ :newton per coulomb: $\mathrm N / \mathrm C$


Definition:Electric Field Strength/Dimension

'''Electric field strength''' has the dimension $\mathsf {M L T}^{-3} \mathsf I^{-1}$.


Definition:Electric Field Strength

'''Electric field strength''' is the measure of the intensity of an electric field.


Definition:Electrostatics

'''Electrostatics''' is the branch of physics concerned with the study of stationary electrically charged bodies, and the forces and fields they give rise to.


Definition:Electric Charge/Sign

The '''sign''' of an '''electric charge''' can be one of $2$ types: === Positive Electric Charge === {{:Definition:Electric Charge/Sign/Positive}} === Negative Electric Charge === {{:Definition:Electric Charge/Sign/Negative}}


Definition:Electric Charge/Sign/Like

If $2$ electric charges are of the same sign, they are referred to as being '''like (electric) charges'''.


Definition:Electric Charge/Sign/Unlike

If $2$ electric charges are of opposite sign, they are referred to as being '''unlike (electric) charges'''.


Definition:Electric Charge/Sign/Positive

A '''positive electric charge''' is an electric charge which is of the same sign as the electric charge on a proton. When it is necessary to assign a value to a '''positive electric charge''', a $+$ (plus) sign is used, and the value assigned is a positive number.


Definition:Electric Charge/Sign/Negative

A '''negative electric charge''' is an electric charge which is of the same sign as the electric charge on an electron. When it is necessary to assign a value to a '''negative electric charge''', a $-$ (minus) sign is used, and the value assigned is a negative number.


Definition:Electric Charge/Sign/Convention

It appears at first glance that the definition of the sign of an electric charge is both arbitrary and circular. That is: :a positive electric charge is the type of electric charge that can be found on a proton while: :a proton is defined as carrying a positive electric charge. Early in the development of the theory of electrostatics, it was noticed that there were two types of electric charge. What they were called was at that point arbitrary; one type was called '''positive''', and the other type was called '''negative'''. When the proton and electron were discovered, the signs of their electric charges were identified, and subsequently used to ''define'' the sign of an arbitrary electric charge. Hence the fact that a proton is positive and an electron is negative is simply the result of a convention.


Definition:Coulomb (Unit)/Base Units

The SI base units of the '''coulomb''' are: :$\mathrm C := \mathrm A \, \mathrm s$ where: :$\mathrm A$ denotes amperes :$\mathrm s$ denotes seconds (of time).


Definition:Bounded Mapping/Real-Valued/Definition 3

$f$ is '''bounded on $S$''' {{iff}}: :$\exists a, b \in \R_{\ge 0}: \forall x \in S: \map f x \in \closedint a b$ where $\closedint a b$ denotes the (closed) real interval from $a$ to $b$.


Definition:Negation Function

The '''negation function''' is the function defined on the various standard number systems as follows: === Integer Negation Function === {{:Definition:Negation Function/Integer}} === Rational Negation Function === {{:Definition:Negation Function/Rational}} === Real Negation Function === {{:Definition:Negation Function/Real}} === Complex Negation Function === {{:Definition:Negation Function/Complex}}


Definition:Negation Function/Integer

The negation function $h: \Z \to \Z$ is defined on the set of integers as: :$\forall n \in \Z: \map h n = -n$


Definition:Union Mapping/Finite Set

Let $S = \set {f_1, f_2, \ldots, f_n}$ denote a finite set of mappings. The '''union mapping''' $f$ of $S$ is defined when: :$\forall i, j \in \set {1, 2, \ldots, n}: f_i$ and $f_j$ are combinable and is defined as: :$\forall x \in \displaystyle \bigcup \set {\Dom {f_i}: i \in \set {1, 2, \ldots, n} } x \in \Dom {f_i} \implies f = \map {f_i} x$


Definition:Union Mapping/Family

Let $I$ be an indexing set. Let $F = \family {f_i}_{i \mathop \in I}$ be a family of mappings indexed by $I$ The '''union mapping''' $f$ of $F$ is defined when: :$\forall i, j \in I: f_i$ and $f_j$ are combinable and is defined as: :$\forall x \in \displaystyle \bigcup \set {\Dom {f_i}: i \in I} x \in \Dom {f_i} \implies f = \map {f_i} x$


Definition:Set Product/Projection

The mappings $\phi_i$ are the '''projections''' of $P$.


Definition:Group Product/Group Law

The operation $\circ$ can be referred to as the '''group law'''.


Definition:Group Product/Product Element

Let $a, b \in G$ such that $ = a \circ b$. Then $g$ is known as the '''product of $a$ and $b$'''.


Definition:Closed under Inversion

Let $\struct {G, \circ}$ be an group with an identity element $e$. Let $H \subseteq G$ be a subset of $G$. Then $H$ is '''closed under inversion''' {{iff}}: :$\forall h \in H: h^{-1} \in H$ That is, {{iff}} the inverse of every element of $H$ is itself in $H$.


Definition:Torus Group

The '''torus group''' is the direct product of the circle group $C$ with itself: :$T := \struct {K, \times} \times \struct {K, \times}$ where: :$K = \set {z \in \C: \cmod z = 1}$


Definition:Closed Set/Normed Vector Space/Definition 1

'''$F$ is closed in $V$''' {{iff}} its complement $X \setminus F$ is open in $V$.


Definition:Closed Set/Normed Vector Space/Definition 2

'''$F$ is closed (in $V$)''' {{iff}} every limit point of $F$ is also a point of $F$. That is: {{iff}} $F$ contains all its limit points.


Definition:Simple Graph/Formal Definition

Let $V$ be a set. Let $\RR$ be an endorelation on $V$ which is antireflexive and symmetric. Let $E$ be the set whose elements of the form: :$\set {\tuple {v_a, v_b}, \tuple {v_b, v_a} }$. where $\tuple {v_a, v_b}$ and $\tuple {v_b, v_a}$ are elements of $\RR$  A '''simple graph''' is an ordered pair $G = \struct {V, E}$, where $V$ and $E$ are defined as above. $V$ is called the vertex set. $E$ is called the edge set.


Definition:Directed Graph/Formal Definition

A '''directed graph''' or '''digraph''' $D$ is a non-empty set $V$ together with an antireflexive relation $E$ on $V$. The elements of $E$ are the arcs.


Definition:Directed Graph/Simple Digraph

If the relation $E$ in $D$ is also specifically asymmetric, then $D$ is called a '''simple digraph'''. That is, in a '''simple digraph''' there are no pairs of arcs (like there are between $v_1$ and $v_4$ in the diagram above) which go in both directions between two vertices.


Definition:Star Graph

The '''star graph of order $n$''', denoted $S_n$ is a simple graph with $n$ vertices with the following properties: :One distinguished vertex is of degree $n - 1$ :The remaining vertices are all of degree $1$ and are adjacent only to the distinguished vertex. :700px


Definition:Loop-Graph/Formal Definition

A '''loop-graph''' $G$ is a non-empty set $V$ together with a symmetric relation $E$ on $G$. Thus it can be seen that a loop-graph is a simple graph with the stipulation that the relation $E$ does not need to be antireflexive.


Definition:Bounded Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $M' = \struct {Y, \norm {\, \cdot \,}_Y}$ be a normed vector subspace of $M$. === Definition 1 === {{:Definition:Bounded Normed Vector Space/Definition 1}} === Definition 2 === {{:Definition:Bounded Normed Vector Space/Definition 2}}


Definition:Bounded Normed Vector Space/Definition 1

'''$M'$ is bounded (in $M$)''' {{iff}}: :$\exists x \in X, C \in \R: \forall y \in Y: \norm {x - y} \le C$


Definition:Bounded Normed Vector Space/Definition 2

'''$M'$ is bounded (in $M$)''' {{iff}}: :$\exists \epsilon \in \R_{>0} : \exists x \in X : Y \subseteq \map {B_\epsilon^-} x$ where $\map {B_\epsilon^-} x$ is a closed ball in $M$.


Definition:Loop-Graph/Loop-Digraph/Formal Definition

A '''loop-digraph''' $D$ is a non-empty set $V$ together with a relation $E$ on $D$. Thus it can be seen that a '''loop-digraph''' is a directed graph with the stipulation that the relation $E$ does not need to be antireflexive.


Definition:Loop-Graph/Loop-Digraph/Loop

Such an arc is called a loop.


Definition:Radiometric Dating

'''Radiometric dating''' is a technique whose purpose is to work out the age $T$ of a physical object $B$. The pieces of information are known: :$(1): \quad$ The ratio $r_0$ of a radioactive isotope $E_R$ of a chemical element $E$ to its stable isotope $E_S$ in $B$ at the time it came into being :$(2): \quad$ The ratio $r$ of $E_R$ to $E_S$ at the time now :$(3): \quad$ The half-life of the radioactive isotope $E_R$ It is known from scientific investigation what $r_0$ is when a physical object is created. It is also known from scientific investigation what the rate of radioactive decay of $E_R$ is. Hence it can be worked out by use of the First-Order Reaction how long it would take for the ratio of $E_R$ to $E_S$ to reach its current ratio $r$.


Definition:Radiometric Dating/Radiocarbon Dating

'''Radiocarbon dating''' is a specific application of '''radiometric dating''' which is used to determine how long a piece of organic matter has been dead. Because of bombardment by cosmic rays, the ratio of (radioactive) carbon-14 to (stable) carbon-12 in the atmosphere of Earth is fairly constant. This ratio is known. The ratio of carbon-14 to carbon-12 in a living organism is the same as it is in Earth's atmosphere, by biological respiration. However, when the organism dies, it no longer respires, and the carbon in its body stays where it was at the time of its death. As time passes, the carbon-14 decays to nitrogen-14 via the process of beta decay. The half-life of carbon-14 is known to be $5700 \pm 40$ years.


Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm/Representative

Each Cauchy sequence $\sequence {y_n}$ of the left coset $\sequence {x_n} + \NN$ is called a '''representative''' of the $p$-adic number $\sequence {x_n} + \NN$.


Definition:Real Function/Definition 1

Let $S \subseteq \R$ be a subset of the set of real numbers $\R$. Suppose that, for each value of the independent variable $x$ of $S$, there exists a corresponding value of the dependent variable $y$. Then the dependent variable $y$ is a '''(real) function''' the independent variable $x$.


Definition:Real Function/Range

Let $f: S \to \R$ be a real function. The range of $f$ is the set of values that the dependent variable can take.


Definition:Real Function/Definition 2

A '''(real) function''' is correspondence between a domain set $D$ and a set range set $R$ that assigns to each element of $D$ a unique element of $R$.


Definition:Implicit Function/General

Let: :$f: \R^{n + 1} \to \R, \tuple {x_1, x_2, \ldots, x_n, z} \mapsto \map f {x_1, x_2, \ldots, x_n, z}$ be a real-valued function on $\R^{n + 1}$, where: :$\tuple {x_1, x_2, \ldots, x_n} \in \R^n, z \in \R$ Let a relation between $x_1, x_2, \ldots, x_n$ and $z$ be expressed in the form: :$\map f {x_1, x_2, \ldots, x_n, z} = 0$ defined on some subset $S \subseteq \R^n$. If there exists a function $g: S \to \R$ such that: :$\forall \tuple {x_1, x_2, \ldots, x_n} \in S: z = \map g {x_1, x_2, \ldots, x_n} \iff \map f {x_1, x_2, \ldots, x_n, z} = 0$ then the relation $\map f {x_1, x_2, \ldots, x_n, z} = 0$ defines $z$ as an '''implicitly defined function''' of $x_1, x_2, \ldots, x_n$.


Definition:Real Function/Two Variables/Substitution for y

Let $f: S \times T \to \R$ be a (real) function of two variables: :$z = \map f {x, y}$ Then: :$\map f {x, a}$ means the real function of $x$ obtained by replacing the independent variable $y$ with $a$. In this context, $a$ can be: :a real constant such that $a \in T$ :a real function $\map g x$ whose range is a subset of $T$.


Definition:Bounded Region of Plane

Let $D \subseteq \R^2$ be a subset of the plane. $D$ is '''bounded''' {{iff}} there exists a circle in the plane which completely encloses $D$.


Definition:Semilattice Homomorphism

Let $\struct {S, \circ}$ and $\struct {T, *}$ be semilattices. Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$. That is, $\forall a, b \in S$: :$\map \phi {a \circ b} = \map \phi a * \map \phi b$ Then $\phi: \struct {S, \circ} \to \struct {T, *}$ is a semilattice homomorphism.


Definition:Subsemilattice

Let $\struct {R, \circ}$ be an algebraic structure with a binary operation. A '''subsemilattice of $\struct {R, \circ}$''' is a subset $S$ of $R$ such that $\struct {S, \circ_S}$ is a semilattice.


Definition:Atom (Physics)/Diameter

The diameter of an '''atom''' (considered approximately spherical) is between about $1$ and $5$ angstroms.


Definition:Atom (Physics)/Classical

The classical model of the '''atom''' is as a hard spherical body which is electrically neutral. In this model, '''atoms''' exert no forces on one another until they are brought into contact, at which point they cannot be brought any closer together, no matter how hard they are pushed to do so.


Definition:Gram-Equivalent

The '''gram-equivalent''' of a chemical element $E$ in a particular compound is the ratio of the mass of $E$ to another, with the mass of hydrogen taken as $1$.


Definition:Ion

An '''ion''' is an atom or a fragment of a molecule which has an electric charge on it through having either more or fewer electrons than protons.


Definition:Faraday (Unit of Charge)

The '''Faraday''' is defined as the amount of electric charge needed to release one gram-equivalent of a chemical element. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''Faraday''' }} {{eqn | r = 96 \, 485       | c = coulombs per mole: $\mathrm C \, \mathrm {mol}^{-1}$ }} {{end-eqn}} </onlyinclude> === Symbol === {{:Definition:Faraday (Unit of Charge)/Symbol}} {{NamedforDef|Michael Faraday|cat = Faraday}}


Definition:Water

'''Water''' is the common term for the chemical substance whose molecules are composed of $2$ atoms of hydrogen and $1$ atom of oxygen. It can be expressed using the chemical formula $\mathrm H_2 \mathrm O$.


Definition:Homogeneous (Physics)/Warning

Just to specify that a body ''is made of the same substance throughout'' is '''not''' an adequate definition of '''homogeneous'''. For example, a column of air in the atmosphere is denser at the bottom than at the top. The fact that it is "all made of air" is one thing, but the air at the bottom can be distinguished from that higher up because the densities are different. Thus a column of air is '''not''' '''homogeneous'''.


Definition:Molecular Weight

Let $S$ be a substance. Let $S$ be homogeneous. The '''molecular weight''' of $S$ is defined as: :$W_S = m_S \times \dfrac {12} {m_C}$ where: :$m_S$ is the (arithmetic) mean mass of one molecule of $S$ :$m_C$ is the mass of one molecule of carbon-12. It is understood that the atoms that compose the molecules of $S$ may be of different isotopes according to their relative natural concentrations at the source of wherever $S$ came from. Hence the use of the (arithmetic) mean mass in the definition of $m_S$.


Definition:Avogadro's Number

'''Avogadro's number''' is defined as the number of particles in a mole. Its value is defined as: :$6 \cdotp 02214 076 \times 10^{23}$ and is usually denoted either $N_A$ or $L$. Thus it is the constant of proportion for converting the mass of one mole of a substance to the (arithmetic) mean mass of one molecule of that substance. {{NamedforDef|Lorenzo Romano Amedeo Carlo Avogadro|cat = Avogadro}}


Definition:Fiber of Truth/Solution

Let $P: X \to \set {\T, \F}$ be a propositional function defined on a domain $X$. Let $S = \set {x \in X: \map P x = \T}$ be the '''fiber of truth (under $P$)'''. Then an element of $S$ is known as a '''solution''' of $P$.


Definition:Partial Derivative/Value at Point

Let $\map f {x_1, x_2, \ldots, x_n}$ be a real function of $n$ variables Let $f_i = \dfrac {\partial f} {\partial x_i}$ be the partial derivative of $f$ {{WRT|Differentiation}} $x_i$. Then the value of $f_i$ at $x = \tuple {a_1, a_2, \ldots, a_n}$ can be denoted: :$\valueat {\dfrac {\partial f} {\partial x_i} } {x_1 \mathop = a_1, x_2 \mathop = a_2, \mathop \ldots, x_n \mathop = a_n}$ or: :$\valueat {\dfrac {\partial f} {\partial x_i} } {a_1, a_2, \mathop \ldots, a_n}$ or: :$\map {f_i} {a_1, a_2, \mathop \ldots, a_n}$ and so on. Hence we can express: :$\map {f_i} {a_1, a_2, \mathop \ldots, a_n} = \valueat {\dfrac \partial {\partial x_i} \map f {a_1, a_2, \mathop \ldots, a_{i - 1}, x_i, a_{i + i}, \mathop \ldots, a_n} } {x_i \mathop = a_i}$ according to what may be needed as appropriate.


Definition:Closure (Topology)/Definition 6

The '''closure of $H$ (in $T$)''', denoted $H^-$, is the set of all adherent points of $H$.


Definition:Partial Derivative/Higher Derivative/Third Derivative

Let $u = \map f {x, y, z}$ be a function of the $3$ independent variables $x$, $y$ and $z$. The following is an example of one of the $3$rd derivatives of $f$: :$\dfrac {\partial^3 u} {\partial z^2 \partial y} := \map {\dfrac \partial {\partial z} } {\dfrac {\partial^2 u} {\partial z \partial y} } =: \map {f_{2 3 3} } {x, y, z}$


Definition:Partial Derivative/Higher Derivative/Fourth Derivative

Let $u = \map f {x, y, z}$ be a function of the $3$ independent variables $x$, $y$ and $z$. The following is an example of one of the $4$th derivatives of $f$: :$\dfrac {\partial^4 u} { \partial x \partial y \partial z^2} := \map {\dfrac \partial {\partial x} } {\dfrac {\partial^3 u} {\partial y \partial z^2} } =: \map {f_{3 3 2 1} } {x, y, z}$


Definition:Grandi's Series

'''Grandi's series''' is the divergent series: :$\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n$ {{NamedforDef|Luigi Guido Grandi|cat = Grandi}} Category:Definitions/Convergence Category:Definitions/Series 2gl0szzqyunwl361wc70zcwwk5gczpy


Definition:Partial Derivative/Order

$u = \map f {x_1, x_2, \ldots, x_n}$ be a function of the $n$ independent variables $x_1, x_2, \ldots, x_n$. The '''order''' of a partial derivative of $u$ is the '''number of times it has been (partially) differentiated''' by at least one of $x_1, x_2, \ldots, x_n$. For example: :a second partial derivative of $u$ is of '''second order''', or '''order $2$''' :a third partial derivative of $u$ is of '''third order''', or '''order $3$''' and so on.


Definition:Event/Occurrence/Impossibility

Let $A \in \Sigma$ be an event of $\EE$ whose probability of occurring is equal to $0$. Then $A$ is described as '''impossible'''. That is, it is an '''impossibility''' for $A$ to occur.


Definition:Event/Occurrence/Certainty

Let $A \in \Sigma$ be an event of $\EE$ whose probability of occurring is equal to $1$. Then $A$ is described as '''certain'''. That is, it is a '''certainty''' that $A$ occurs.


Definition:Event/Occurrence/Equality

Let $A, B \in \Sigma$ be events of $\EE$ such that $A = B$. Then: :the occurrence of $A$ inevitably brings about the occurrence of $B$ and: :the occurrence of $B$ inevitably brings about the occurrence of $A$.


Definition:Event/Occurrence/Intersection

Let $\omega \in A \cap B$, where $A \cap B$ denotes the intersection of $A$ and $B$. Then '''both $A$ and $B$ occur'''.


Definition:Event/Occurrence/Union

Let $\omega \in A \cup B$, where $A \cup B$ denotes the union of $A$ and $B$. Then '''either $A$ or $B$ occur'''.


Definition:Event/Occurrence/Difference

Let $\omega \in A \setminus B$, where $A \setminus B$ denotes the difference of $A$ and $B$. Then '''$A$ occurs but $B$ does not occur'''.


Definition:Event/Occurrence/Symmetric Difference

Let $\omega \in A * B$, where $A * B$ denotes the symmetric difference of $A$ and $B$. Then '''either $A$ occurs or $B$ occurs, but not both'''.


Definition:Uniform Cauchy Criterion

Let $S \subseteq \mathbb R$. Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.  We say that $\sequence {f_n}$ satisfies the '''uniform Cauchy criterion''' or is '''uniformly Cauchy''' on $S$ if for all $\varepsilon \in \R_{> 0}$, there exists $N \in \N$ such that:  :$\size {\map {f_n} x - \map {f_m} x} < \varepsilon$ for all $x \in S$ and $n, m > N$. By Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent, this criterion gives a necessary and sufficient condition for a sequence of real functions to be uniformly convergent.


Definition:Complete Set of Events

Let $I$ be an indexing set. Let $\family {A_i}_{i \mathop \in I}$ be a family of events in a probability space indexed by $I$. $\family {A_i}_{i \mathop \in I}$ is a '''complete set of events''' {{iff}}: :$\displaystyle \map \Pr {\bigcup_{i \mathop \in I} A_i} = 1$