Definition:Composition of Mappings/Warning

Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$. If $\Dom {f_2} \ne \Cdm {f_1}$, then the '''composite mapping''' $f_2 \circ f_1$ is '''not defined'''. This definition is directly analogous to that of composition of relations owing to the fact that a mapping is a special kind of relation.


Definition:Composition of Mappings/Also known as

In the context of analysis, this is often found referred to as a '''function of a function''', which (according to some sources) makes set theorists wince, as it is technically defined as a '''function on the codomain of a function'''. Some sources call $f_2 \circ f_1$ the '''resultant of $f_1$ and $f_2$''' or the '''product of $f_1$ and $f_2$'''. Some authors write $f_2 \circ f_1$ as $f_2 f_1$. Some use the notation $f_2 \cdot f_1$ or $f_2 . f_1$. Some use the notation $f_2 \bigcirc f_1$. Others, particularly in books having ties with computer science, write $f_1; f_2$ or $f_1 f_2$ (note the reversal of order), which is read as '''(apply) $f_1$, then $f_2$'''.


Definition:Composition of Mappings/Definition 3

The '''composite of $f_1$ and $f_2$''' is defined and denoted as: :$f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \exists y \in S_2: \map {f_1} x = y \land \map {f_2} y = z}$


Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 1

The '''Cartesian space of $S$ indexed by $I$''' is the set of all families $\family {s_i}_{i \mathop \in I}$ with $s_i \in S$ for each $i \in I$: :$S_I := \displaystyle \prod_I S = \set {\family {s_i}_{i \mathop \in I}: s_i \in S}$


Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 2

The '''Cartesian space of $S$ indexed by $I$''' is defined and denoted as: :$\displaystyle S^I := \set {f: \paren {f: I \to S} \land \paren {\forall i \in I: \paren {\map f i \in S} } }$


Definition:Definition/If or Iff

It is a standard convention, when making a definition in mathematics, to use '''if''' to introduce the definiens, when in fact the intent is generally '''iff''', that is: '''{{iff}}'''. This convention is specifically '''not followed''' on {{ProofWiki}}, where the mandatory style is to use '''{{iff}}'''.


Definition:Characteristic Property of Set

Let $S$ be a set which is specified by means of such a propositional function: :$S = \set {a: \map P a}$ The open statement $\map P x$ is known as a '''characteristic property''' of $S$.


Definition:Fresnel Sine Integral Function

The '''Fresnel sine integral function''' is the real function $\R \to \R$ defined by:  :$\displaystyle \map {\operatorname S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$


Definition:Fresnel Cosine Integral Function

The '''Fresnel cosine integral function''' is the real function $\R \to \R$ defined by:  :$\displaystyle \map {\operatorname C} x = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u$


Definition:Ring of Mappings

Let $\struct {R, +, \circ}$ be a ring. Let $S$ be a set. Let $R^S$ be the set of all mappings from $S$ to $R$. The '''ring of mappings''' from $S$ to $R$ is the algebraic structure $\struct {R^S, +', \circ'}$ where $+'$ and $\circ'$ are the (pointwise) operations induced on $R^S$ by $+$ and $\circ$.


Definition:Hypergeometric Differential Equation

A '''hypergeometric differential equation''' is a second order ODE of the form: :$\displaystyle x \paren {1 - x} \frac {\d^2 y} {\d x^2} + \paren {c - \paren {a + b + 1} x} \frac {\d y} {\d x} - a b y = 0$


Definition:Legendre's Associated Differential Equation

'''Legendre's associated differential equation''' is a second order ODE of the form: :$\displaystyle \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - 2 x \frac {\d y} {\d x} + \paren {n \paren {n + 1} - \frac {m^2} {1 - x^2} } y = 0$


Definition:Ring of Mappings/Zero

The zero of the ring of mappings is the constant mapping $f_0 : S \to R$ defined by: :$\quad \forall s \in S : \map {f_0} x = 0$ where $0$ is the zero in $R$


Definition:Ring of Mappings/Additive Inverse

The additive inverse in the ring of mappings is defined by: :$\forall f \in R^S : -f \in R^S : \forall s \in S : \map {\paren {-f} } x = -\map f x$


Definition:Ring of Mappings/Unity

Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1$.


Definition:Ring of Mappings/Units

Let $\struct {R, +, \circ}$ be a ring with unity $1$.


Definition:Ring of Mappings/Commutativity

Let $\struct {R, +, \circ}$ be a commutative ring.


Definition:Coproduct/Sets

Let $S_1$ and $S_2$ be sets. A '''coproduct''' $\struct {C, i_1, i_2}$ of $S_1$ and $S_2$ comprises a set $C$ together with mappings $i_1: S_1 \to C$, $i_2: S_2 \to C$ such that: :for all sets $X$ and mappings $f_1: S_1 \to X$ and $f_2: S_2 \to X$: ::there exists a unique mapping $h: C \to X$ such that: :::$h \circ i_1 = f_1$ :::$h \circ i_2 = f_2$ Hence: ::$\begin{xy} \xymatrix@L+2mu@+1em{ & C \ar@{-->}[dd]_*{h} & \\ S_1 \ar[ru]^*{i_1} \ar[rd]_*{f_1} & & S_2 \ar[lu]_*{i_2} \ar[ld]^*{f_2} \\ & X & }\end{xy}$ :is a commutative diagram.


Definition:Ring of Mappings/Pointwise Addition

Let $\struct {R, +, \circ}$ be a ring. Let $S$ be a set. Let $\struct {R^S, +', \circ'}$ be the ring of mappings from $S$ to $R$. <section notitle=true name=PointwiseAddition> The pointwise operation $+'$ induced by $+$ on the ring of mappings from $S$ to $R$ is called '''pointwise addition''' and is defined as: :$\forall f, g \in R^S: f +’ g \in R^S :$ :::$\forall s \in S : \map {\paren {f +’ g}} x = \map f x + \map g x$ </section>


Definition:Ring of Mappings/Pointwise Multiplication

Let $\struct {R, +, \circ}$ be a ring. Let $S$ be a set. Let $\struct {R^S, +', \circ'}$ be the ring of mappings from $S$ to $R$. <section notitle=true name=PointwiseMultiplication> The pointwise operation $\circ'$ induced by $\circ$ on the ring of mappings from $S$ to $R$ is called '''pointwise multiplication''' and is defined as: :$\forall f, g \in R^S: f \circ’ g \in R^S :$ :::$\forall s \in S : \map {\paren {f \circ’ g}} x = \map f x \circ \map g x$ </section>


Definition:Ring of Sequences/Pointwise Addition

Let $\struct {R, +, \circ}$ be a ring. Let $\struct {R^\N, +', \circ'}$ be the ring of sequences over $R$. <section notitle=true name=PointwiseAddition> The pointwise operation $+'$ induced by $+$ on the ring of sequences is called '''pointwise addition''' and is defined as: :$\forall \sequence {x_n}, \sequence {y_n} \in R^\N: \sequence {x_n} +' \sequence {y_n} = \sequence {x_n + y_n}$ </section>


Definition:Ring of Sequences/Pointwise Multiplication

Let $\struct {R, +, \circ}$ be a ring. Let $\struct {R^\N, +', \circ'}$ be the ring of sequences over $R$. <section notitle=true name=PointwiseMultiplication> The pointwise operation $\circ'$ induced by $\circ$ on the ring of sequences is called '''pointwise multiplication''' and is defined as: :$\forall \sequence {x_n}, \sequence {y_n} \in R^{\N}: \sequence {x_n} \circ' \sequence {y_n} = \sequence {x_n \circ y_n}$ </section>


Definition:Ring of Sequences/Zero

The zero of the ring of sequences is the constant sequence $\tuple {0, 0, 0, \dots}$, where $0$ is the zero of $R$.


Definition:Ring of Sequences/Unity


Definition:Ring of Sequences/Commutativity


Definition:Ring of Sequences/Additive Inverse

The additive inverse in the ring of sequences is defined by: :$\forall \sequence {x_n} \in R^\N: -\sequence {x_n} = \sequence {-x_n}$


Definition:Ring of Sequences/Units


Definition:Euclidean Number

A '''Euclid number''' is a perfect number which is even.


Definition:Friction/Coefficient

Let $B$ be a body at rest on a plane surface $S$ on which friction acts. Let $N$ be the normal reaction of $S$ on $B$. Let a force be applied to $B$ parallel to $S$. === Coefficient of Static Friction === {{:Definition:Friction/Coefficient/Static}} === Coefficient of Kinetic Friction === {{:Definition:Friction/Coefficient/Kinetic}} These '''coefficients of friction''' depend upon the materials out of which $B$ and $S$ are made. It is usual for $\mu_s$ to be greater than $\mu_k$.


Definition:Friction/Coefficient/Static

Let $F$ be the magnitude of that force in the limiting case when $B$ is just about to move. Then the '''coefficient of static friction''' is defined and denoted: :$\mu_s = \dfrac F N$


Definition:Friction/Coefficient/Kinetic

Let $F$ be the magnitude of that force needed to keep $B$ moving at a constant velocity. Then the '''coefficient of kinetic friction''' is defined and denoted: :$\mu_k = \dfrac F N$


Definition:Bimodule

Let $\struct {R, +_R, \times_R}$ and $\struct {S, +_S, \times_S}$ be rings. Let $\struct {G, +_G}$ be an abelian group. Let $\circ_R : R \times G \to G$ and $\circ_S : G \times S \to G$ be binary operations such that: :$(1): \quad \struct {G, +_G, \circ_R}$ is a left module :$(2): \quad \struct {G, +_G, \circ_S}$ is a right module :$(3): \quad \forall \lambda \in R: \forall \mu \in S: \forall x \in G: \paren {\lambda \circ_R x} \circ_S \mu = \lambda \circ_R \paren {x \circ_S \mu}$ Then $\struct {G, +_G, \circ_R, \circ_S}$ is a '''bimodule over $\tuple {R, S}$'''. If $\struct {S, +_S, \times_S} = \struct {R, +_R, \times_R}$ then a '''bimodule over $\tuple {R, R}$''' is simply called a '''bimodule over $R$'''


Definition:Weight (Physics)/Warning

There is a certain amount of confusion in the common mind between weight and mass. The latter is usually determined by measuring its weight. But while the mass of a body is (under normal circumstances) constant, its weight varies according to its position relative to the gravitational field it is in, and so is not a constant property of that body. However, under usual terrestrial conditions the gravitational field is more or less constant (any differences being detectable only by instruments). This means that the weight and mass of a body are commonly considered "the same". Thus a weighing machine, while indicating the mass of a body, does so by measuring its weight.


Definition:Weight (Physics)/Dimension

The dimension of '''weight''' is $M L T^{-2}$: mass times acceleration, that is, a force.


Definition:Whole Number

The phrase '''whole number''' is an ambiguous term which could mean either: :an integer :a natural number :a strictly positive integer. Hence it is recommended that a specific instance of the above is used instead.


Definition:Are

One '''are''' is equal to a square whose side measures $10$ metres. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''are''' }} {{eqn | r = 100       | c = square metres }} {{eqn | o = \approx       | r = 119 \cdot 60       | c = square yards }} {{end-eqn}}


Definition:Finite Difference Operator/Forward Difference

The '''forward difference operator''' on $f$ is defined as: :$\map {\Delta f} x := \map f {x + 1} - \map f x$


Definition:Finite Difference Operator/Backward Difference

The '''backward difference operator''' on $f$ is defined as: :$\map {\nabla f} x := \map f x - \map f {x - 1}$


Definition:Bernoulli Trial/Bernoulli Variable

Let $X$ be a discrete random variable whose sample space is $\Omega$ in such a Bernoulli trial. Then $X$ is known as a '''Bernoulli variable'''.


Definition:Borromean Rings

The '''Borromean rings''' is an arrangement of $3$ interlinked rings such that no $2$ of the rings are themselves linked. That is, if one of them is removed, then the other $2$ are unlinked. :600px


Definition:Basis Representation

Let $b \in \Z$ be an integer such that $b > 1$. Let $n \in \Z$ be an integer such that $n \ne 0$. The '''representation of $n$ to the base $b$''' is the unique string of digits: :$\pm \sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$ where $\pm$ is: :the negative sign $-$ {{iff}} $n < 0$ :the positive sign $+$ (or omitted) {{iff}} $n > 0$ If $n = 0$, then $n$ is represented simply as $0$.


Definition:Lagrangian Mechanics

'''Lagrangian Mechanics''' is a reformulation of mechanics, where the main object of study is the Lagrangian function.


Definition:Action of Physical System

Let $S$ be an integral functional. Suppose that equations of motion of a physical system are to be derived from $S$ by the Principle of Stationary Action. Then $S$ is the '''action''' of a '''physical system'''.


Definition:Lagrangian

Let $P$ be a physical system composed of $n \in \N$ particles. Let the real variable $t$ be the time of $P$. $\forall i \le n$ let $\map { {\mathbf x}_i} t$ be the position of the $i$th particle. Suppose, the action $S$ of $P$ is of the following form: :$\displaystyle S = \int_{t_1}^{t_2} L \rd t$ where $L$ is a mapping of (possibly) $t$, $\map {{\mathbf x}_i} t$ and their derivatives. Then $L$ is the '''Lagrangian''' of $P$. {{NamedforDef|Joseph Louis Lagrange|cat = Lagrange}} Category:Definitions/Lagrangian Mechanics Category:Definitions/Physics Category:Definitions/Applied Mathematics k0yf3wthx912zivh4vfjf544vgbcowt


Definition:Free Group/Definition 1

A group $G$ is a '''free group''' {{iff}} it is isomorphic to the free group on some set.


Definition:Free Group/Definition 2

A group $G$ is a '''free group''' {{iff}} it has a presentation of the form $\gen S$, where $S$ is a set. That is, it has a presentation without relators. In this context, '''free''' means '''free of non-trivial relations'''.


Definition:Standard Lagrangian

Let $P$ be a physical system composed of countable number of particles. Let $L$ be a Lagrangian of $P$. Let: :$L = T - V$ where $T$ is the kinetic energy and $V$ the potential energy of $P$. Then $L$ is the '''Standard Lagrangian''' of $P$. {{NamedforDef|Joseph Louis Lagrange|cat = Lagrange}}


Definition:Kinetic Energy of Classical Particle

Let $\MM$ be an $n$-dimensional Euclidean manifold. Let $P$ be a particle with an inertial mass $m_i$. Let $t$ be the time variable of $P$. Suppose the position of $P$ is a real differentiable $n$-dimensional vector-valued mapping $\mathbf x = \map {\mathbf x} t$. Then the '''kinetic energy''' of a '''classical particle''' $P$ is: :$T = \dfrac {m_i} 2 \paren {\dfrac {\d \mathbf x} {\d t} }^2$ where $\paren {\dfrac {\d \mathbf x} {\d t} }^2$ is the dot product of the vector $\dfrac {\d \mathbf x} {\d t}$ with itself.


Definition:Pedal Triangle

=== Pedal Triangle of Point with respect to Triangle === {{:Definition:Pedal Triangle/Point}} === Orthic (Pedal) Triangle === {{:Definition:Orthic Triangle}} {{:Definition:Orthic Triangle/Also known as}}


Definition:Orthic Triangle

Let $\triangle ABC$ be a triangle. Let $\triangle DEF$ be the triangle formed by the feet of the altitudes $AD$, $BC$ and $ED$ of $\triangle ABC$. :400px $\triangle DEF$ is known as the '''orthic triangle of $\triangle ABC$'''. That is, the '''orthic triangle of $\triangle ABC$''' is the pedal triangle of its orthocenter.


Definition:Pedal Triangle/Point

Let $\triangle ABC$ be a triangle. Let $P$ be a point in the plane of $\triangle ABC$. Let $PD$, $PE$ and $PF$ be perpendiculars dropped from $P$ to $BC$, $AC$ and $AB$ respectively. Let $\triangle DEF$ be the triangle formed by the feet of the perpendiculars $PD$, $PE$ and $PF$. :400px $\qquad$ 400px $\triangle DEF$ is known as the '''pedal triangle of $P$ with respect to $\triangle ABC$'''.


Definition:Total Energy of Particles

Let $P$ be a physical system composed of a countable number of particles. Suppose the kinetic and potential energy of $P$ is respectively $T$ and $U$. Denote: :$E = T + U$ Then $E$ is known as the '''total energy''' of $P$.


Definition:Annulus (Geometry)

An '''annulus''' is a plane figure whose boundary consists of $2$ concentric circles: :400px In the above diagram, the '''annulus''' is the colored area.


Definition:Annulus (Geometry)/Center

The '''center''' of an '''annulus''' is the common center of the $2$ concentric circles that form its boundary. In the above diagram, the '''center''' is the point $O$.


Definition:Annulus (Geometry)/Inner Radius

The '''inner radius''' of an '''annulus''' is the radius of the smaller of the $2$ concentric circles that form its boundary. In the above diagram, the '''inner radius''' is denoted $r$.


Definition:Annulus (Geometry)/Outer Radius

The '''outer radius''' of an '''annulus''' is the radius of the larger of the $2$ concentric circles that form its boundary. In the above diagram, the '''outer radius''' is denoted $R$.


Definition:Power of Element/Ring

Let $\struct {R, \circ, +}$ be a ring. Let $r \in R$. Let $n \in \Z_{>0}$ be the set of strictly positive integers. The '''$n$th power of $r$ in $R$''' is defined as the $n$th power of $r$ with respect to the semigroup $\struct {R, \circ}$: :$\forall n \in \Z_{>0}: r^n = \begin {cases} r & : n = 1 \\ r^{n - 1} \circ r & : n > 1  \end {cases}$ If $R$ is a ring with unity where $1_R$ is that unity, the definition extends to $n \in \Z_{\ge 0}$: :$\forall n \in \Z_{\ge 0}: r^n = \begin {cases} 1_R & : n = 0 \\ r^{n - 1} \circ r & : n > 0 \end {cases}$


Definition:Half-Range Fourier Series

=== Half-Range Fourier Cosine Series === {{:Definition:Half-Range Fourier Cosine Series/Formulation 1}} === Half-Range Fourier Sine Series === {{:Definition:Half-Range Fourier Sine Series/Formulation 1}}


Definition:Half-Plane/Edge

Let $H$ denote one of the half-planes into which $L$ divides $P$. Then $L$ is called the '''edge''' of $H$.


Definition:Complex Half-Plane

Let $\C$ denote the complex plane. A '''complex half-plane''' is a '''half-plane''' of $\C$ whose edge is an infinite straight line of $\C$. Category:Definitions/Complex Analysis 72g45o95hyx7v1pet7o7s9denayhta5


Definition:New Moon

The '''new moon''' is the astronomical phenomenon of the Moon in its orbit coming to its closest point to the Sun: :800px Note that the above diagram is not to scale. At this point in its orbit, the Moon is between Earth and the Sun. Thus the hemisphere of the Moon which is illuminated by the Sun is facing directly away from Earth. Thus for an observer of the Moon from Earth, unless an actual solar eclipse is in progress, at this point it cannot be seen, because: :$(1): \quad$ It is dark :$(2): \quad$ The Sun is more or less behind it, outshining everything else in the sky. Category:Definitions/Astronomy ts9gid2sqz6h6f9db2q271okr3mkbfs


Definition:Zero Mapping

Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$. Let $S$ be a set. Let $f_0: S \to \mathbb A$ denote the constant mapping: :$\forall x \in S: \map {f_0} x = 0$ Then $f_0$ is referred to as the '''zero mapping'''.


Definition:Non-Trivial Ring

A '''non-trivial ring''' is a ring which is not trivial. That is, a ring $R$ such that: :$\exists x, y \in R: x \circ y \ne 0_R$ where $0_R$ denotes the zero of $R$.


Definition:Convergent Sequence/P-adic Numbers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\sequence {x_n} $ be a sequence in $\Q_p$.


Definition:Phase of Moon

The '''phase of the Moon''' is its apparent shape when viewed in the night sky of Earth. This shape is dependent upon the position of the Moon with respect to the Sun and Earth. :800px When the Moon is between the Sun and Earth, the sunlit side faces more or less away from Earth, and so less than half of it can be seen lit up. When Earth is between the Sun and the Moon, the sunlit side faces more or less away towards Earth, and so more than half of it can be seen lit up. When the Moon is directly between the Sun and Earth, the sunlit side is facing completely away from Earth, and in normal conditions cannot be seen. This is the '''phase''' known as '''new moon'''. When Earth is directly between the Sun and the Moon, the sunlit side is facing completely towards Earth, and in normal conditions appears as a brign disk. This is the '''phase''' known as '''full moon'''. At the point half way between new moon and full moon, the '''phase''' is known as '''first quarter'''. At the point half way between full moon and new moon, the '''phase''' is known as '''last quarter'''. Category:Definitions/Astronomy ixuonqcy1kirih2w49f13upg59nu45c


Definition:Safe Prime/Sequence

The sequence of safe primes begins: :$5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, \ldots$


Definition:Metric System/Scaling Prefixes/yocto-

'''yocto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-24}$. === Symbol === {{:Symbols:Y/yocto-}}


Definition:Metric System/Scaling Prefixes/zepto-

'''zepto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-21}$. === Symbol === {{:Symbols:Z/zepto-}}


Definition:Metric System/Scaling Prefixes/atto-

'''atto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-18}$. === Symbol === {{:Symbols:A/atto-}}


Definition:Metric System/Scaling Prefixes/femto-

'''femto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-15}$. === Symbol === {{:Symbols:F/femto-}}


Definition:Metric System/Scaling Prefixes/pico-

'''pico-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-12}$. === Symbol === {{:Symbols:P/pico-}}


Definition:Metric System/Scaling Prefixes/nano-

'''nano-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-9}$. === Symbol === {{:Symbols:N/nano-}}


Definition:Metric System/Scaling Prefixes/micro-

'''micro-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-6}$. === Symbol === {{:Symbols:Greek/Mu/micro-}}


Definition:Metric System/Scaling Prefixes/milli-

'''milli-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-3}$. === Symbol === {{:Symbols:M/milli-}}


Definition:Metric System/Scaling Prefixes/centi-

'''centi-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-2}$. === Symbol === {{:Symbols:C/centi-}}


Definition:Metric System/Scaling Prefixes/deci-

'''deci-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{-1}$. === Symbol === {{:Symbols:D/deci-}}


Definition:Harmonic Potential Energy

Let $P$ be a physical particle. Let its position $\map x t$ be a real function, where $t$ is time. Let $k > 0$. Then the potential energy of the form :$\map U x = \frac 1 2 k x^2$ is called the '''harmonic potential energy'''.


Definition:Metric System/Scaling Prefixes/deka-

'''deka-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^1$. === Symbol === {{:Symbols:D/deka-}}


Definition:Harmonic Oscillator

Let $P$ be a physical particle. Let the potential energy of $P$ be that of the harmonic potential. Then $P$ is called the '''harmonic oscillator'''.


Definition:Metric System/Scaling Prefixes/hecto-

'''hecto-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^2$. === Symbol === {{:Symbols:H/hecto-}}


Definition:Metric System/Scaling Prefixes/kilo-

'''kilo-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^3$. === Symbol === {{:Symbols:K/kilo-}}


Definition:Metric System/Scaling Prefixes/mega-

'''mega-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^6$. === Symbol === {{:Symbols:M/mega-}}


Definition:Metric System/Scaling Prefixes/giga-

'''giga-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^9$. === Symbol === {{:Symbols:G/giga-}}


Definition:Metric System/Scaling Prefixes/tera-

'''tera-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{12}$. === Symbol === {{:Symbols:T/tera-}}


Definition:Metric System/Scaling Prefixes/peta-

'''peta-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{15}$. === Symbol === {{:Symbols:P/peta-}}


Definition:Metric System/Scaling Prefixes/exa-

'''exa-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{18}$. === Symbol === {{:Symbols:E/exa-}}


Definition:Metric System/Scaling Prefixes/zetta-

'''zetta-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{21}$. === Symbol === {{:Symbols:Z/zetta-}}


Definition:Metric System/Scaling Prefixes/yotta-

'''yotta-''' is the  Système Internationale d'Unités metric scaling prefix denoting a multiplier of $10^{24}$. === Symbol === {{:Symbols:Y/yotta-}}


Definition:Convergent Sequence/P-adic Numbers/Definition 1

The sequence $\sequence {x_n}$ '''converges to $x \in \Q_p$''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x}_p < \epsilon$


Definition:Convergent Sequence/P-adic Numbers/Definition 3

The sequence $\sequence {x_n}$ '''converges to $x \in \Q_p$''' {{iff}}: :$\sequence {x_n}$ converges to $x$ in the $p$-adic metric


Definition:Convergent Sequence/P-adic Numbers/Definition 4

The sequence $\sequence {x_n}$ '''converges to $x \in \Q_p$''' {{iff}}: :the real sequence $\sequence {\norm {x_n - x}_p }$ converges to $0$ in the reals $\R$


Definition:Limit of Sequence/P-adic Numbers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\sequence {x_n} $ be a sequence in $\Q_p$. <section notitle="true" name="LimitDefinition"> Let $\sequence {x_n}$ converge to $x \in \Q_p$  Then $x$ is a '''limit of $\sequence {x_n}$ as $n$ tends to infinity''' which is usually written: :$\displaystyle x = \lim_{n \mathop \to \infty} x_n$ </section>


Definition:Convergent Sequence/P-adic Numbers/Definition 2

The sequence $\sequence {x_n}$ '''converges to $x \in \Q_p$''' {{iff}}: :$\sequence {x_n}$ converges to $x$ in the $p$-adic norm


Definition:Convergent Sequence/Normed Division Ring/Definition 1

The sequence $\sequence {x_n}$ '''converges to $x \in R$ in the norm $\norm {\, \cdot \,}$''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x} < \epsilon$


Definition:Convergent Sequence/Normed Division Ring/Definition 2

The sequence $\sequence {x_n}$ '''converges to $x \in R$ in the norm $\norm {\, \cdot \,}$''' {{iff}}: :$\sequence {x_n}$ converges to $x$ in the metric induced by the norm $\norm {\, \cdot \,}$


Definition:Convergent Sequence/Normed Division Ring/Definition 3

The sequence $\sequence {x_n}$ '''converges to $x \in R$ in the norm $\norm {\, \cdot \,}$''' {{iff}}: :the real sequence $\sequence {\norm {x_n - x} }$ converges to $0$ in the reals $\R$


Definition:Rounding/Treatment of Half/Round Up

The '''round up''' convention is that the '''larger''' of those two integers is used: :$y = 10^n \floor {\dfrac x {10^n} + \dfrac 1 2}$


Definition:Rounding/Treatment of Half/Round Down

The '''round down''' convention is that the '''smaller''' of those two integers is used: :$y = 10^n \ceiling {\dfrac x {10^n} - \dfrac 1 2}$


Definition:Rounding/Treatment of Half/Round to Even

The '''round to even''' convention is that the '''nearest even integer''' to $\dfrac x {10^n}$ is used: :$y = \begin {cases} 10^n \floor {\dfrac x {10^n} + \dfrac 1 2} & : \text {$\floor {\dfrac x {10^n} + \dfrac 1 2}$ even} \\ 10^n \ceiling {\dfrac x {10^n} - \dfrac 1 2} & : \text {$\floor {\dfrac x {10^n} + \dfrac 1 2}$ odd} \end {cases}$


Definition:Cumulative Rounding Error

Let $S$ be a set of continuous data which is to be rounded to the nearest $n$th power of $10$. The '''cumulative rounding error''' of $S$ is defined as: :$R = \displaystyle \sum_{x \mathop \in S} \paren {x - \overline x}$ where $\overline x$ denotes the rounded value of a given $x$.


Definition:Scientific Notation/Exponent

The number $e$ is known as the '''exponent'''.


Definition:Significant Figures

Let $n$ be a number expressed in decimal notation. The number of digits to which $n$ is rounded, apart from any digits needed to locate the decimal point, are referred to as the '''significant figures''' of $n$.


Definition:Graph (Statistics)

In the context of statistics, a '''graph''' is a word that loosely means a pictorial or diagrammatic presentation of a set of data.


Definition:Bar Graph

A '''bar graph''' is a form of graph which consists of a finite set of (usually) vertical  bars whose length determines the statistic being communicated.


Definition:Pie Graph

A '''pie graph''' is a form of graph which consists of a circle divided into sectors whose areas represent the proportion of the corresponding statistic relative to the whole.


Definition:Pictograph

A '''pictograph''' is a variant of a bar graph whose  bars consist of a number of icons or other pictorial symbols arranged in a (usually) line.


Definition:Inequality/Member

Let $\RR$ be an inequality. Let $a \mathrel \RR b$. Then both $a$ and $b$ are referred to as '''members''' of the inequality $\RR$.


Definition:General Logarithm/Common/Notation for Negative Logarithm

Let $n \in \R$ be a real number such that $0 < n < 1$. Let $n$ be presented (possibly approximated) in scientific notation as: :$a \times 10^{-d}$ where $d \in \Z_{>0}$ is a (strictly) positive integer. Let $\log_{10} n$ denote the common logarithm of $n$. Then it is the standard convention to express $\log_{10} n$ in the form: :$\log_{10} n = \overline d \cdotp m$ where $m := \log_{10} a$ is the mantissa of $\log_{10} n$.


Definition:General Logarithm/Common/Mantissa

Let $\log_{10} n$ be expressed in the form: :$\log_{10} n = \begin {cases} c \cdotp m & : d \ge 0 \\ \overline c \cdotp m & : d < 0 \end {cases}$ where: : $c = \size d$ is the absolute value of $d$ : $m := \log_{10} a$ $\log_{10} a$ is the '''mantissa''' of $\log_{10} n$.


Definition:General Logarithm/Common/Characteristic

$c$ is the '''characteristic''' of $\log_{10} n$.


Definition:Antilogarithm

Let $x \in \R_{>0}$ be a strictly positive real number. Let $b \in \R_{>1}$ be a real number which is greater than $1$. Let $y = \log_b x$ be the logarithm of $x$ base $b$. Then the '''antilogarithm of $y$ base $b$''' is: :$\operatorname {alog}_b y = x$


Definition:Sample Statistic/Raw

'''Raw data''' are collected data which have not been organized or analysed in any way.


Definition:Spherical Astronomy

'''Spherical astronomy''' is a branch of astronomy which studies the angular positions of the celestial bodies, with no concern for their distance from Earth. Its mathematical aspects can be considered a subfield of spherical geometry.


Definition:Great Circle

Let $S$ be a sphere. A '''great circle''' of $S$ is defined as the intersection of $S$ with a plane passing through the center of $S$. :400px


Definition:Small Circle

Let $S$ be a sphere. A '''small circle''' on $S$ is defined as the intersection of $S$ with a plane which does not pass through the center of $S$. :400px


Definition:Pole of Great Circle

Let $C$ be a great circle of $S$. Let $AB$ be the diameter of $S$ situated perpendicular to the plane of $C$. :700px The points $A$ and $B$, where the diameter intersects $S$, are the '''poles''' of the great circle $C$.


Definition:Pole of Circle

Let $S$ be a sphere whose center is $O$. Let $AB$ be a diameter of $S$, such that $A$ and $B$ are the points of intersection $S$ Let $C$ be a (small) circle of $S$ embedded in a plane perpendicular to $AB$.   :700px The points $A$ and $B$ are the '''poles''' of the circle $C$. === Special Case: Pole of Great Circle === {{:Definition:Pole of Great Circle}}


Definition:Geographical Equator

The '''(geographical) equator''' is the great circle described on the surface of Earth whose plane is perpendicular to Earth's axis of rotation. :500px


Definition:Earth's Poles

Let it be assumed to be established that planet Earth rotates in space about its axis. The '''poles''' of Earth are the endpoints of Earth's axis, where it intersects the surface of Earth. That is, the '''poles''' of Earth are the poles of Earth's equator. :500px


Definition:Earth's Poles/North

Let an observer $A$ be situated on Earth's equator. Let $A$ be facing the direction of Earth's rotation in space about its axis. Then the '''North Pole''' is the particular one of Earth's poles which is on the left of $A$.


Definition:Earth's Poles/South

Let an observer $A$ be situated on Earth's equator. Let $A$ be facing the direction of Earth's rotation in space about its axis. Then the '''South Pole''' is the particular one of Earth's poles which is on the right of $A$.


Definition:Earth's Axis

Let it be assumed to be established that planet Earth rotates in space about a more-or-less fixed diameter $D$. Then $D$ is known as '''Earth's axis'''. :500px


Definition:Closed Ball/P-adic Numbers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''closed $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p }$''' is defined as: :$\map { {B_\epsilon}^-} a = \set {x \in R: \norm {x - a}_p \le \epsilon}$


Definition:Closed Ball/P-adic Numbers/Radius

In $\map { {B_\epsilon}^-} a$, the value $\epsilon$ is referred to as the '''radius''' of the closed $\epsilon$-ball.


Definition:Closed Ball/P-adic Numbers/Center

In $\map { {B_\epsilon}^-} a$, the value $a$ is referred to as the '''center''' of the closed $\epsilon$-ball.


Definition:Open Ball/P-adic Numbers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''open $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$''' is defined as: :$\map {B_\epsilon} a = \set {x \in \Q_p: \norm{x - a}_p < \epsilon}$


Definition:Open Ball/P-adic Numbers/Radius

In $\map {B_\epsilon} a$, the value $\epsilon$ is referred to as the '''radius''' of the open $\epsilon$-ball.


Definition:Open Ball/P-adic Numbers/Center

In $\map {B_\epsilon} a$, the value $a$ is referred to as the '''center''' of the open $\epsilon$-ball.


Definition:Meridian (Terrestrial)

A '''(terrestrial) meridian''' is a semi-great circle on Earth's surface whose endpoints are Earth's poles. :600px


Definition:Meridian (Terrestrial)/Principal

The '''principal meridian''' is the meridian of Earth which passes through the {{WP|Royal_Observatory,_Greenwich|Royal Observatory}} in {{WP|Greenwich|Greenwich}}, {{WP|Greater_London|Greater London}}, {{WP|England|England}}, which is used as the internationally-recognised $0 \degrees$ reference longitude meridian.


Definition:North (Terrestrial)

'''North''' is the direction on (or near) Earth's surface along the meridian directly towards the North Pole.


Definition:South (Terrestrial)

'''South''' is the direction on (or near) Earth's surface along the meridian directly towards the South Pole.


Definition:East (Terrestrial)

'''East''' is the direction on (or near) Earth's surface along the small circle in the direction of Earth's rotation in space about its axis.


Definition:West (Terrestrial)

'''West''' is the direction on (or near) Earth's surface along the small circle in the opposite direction from Earth's rotation in space about its axis.


Definition:Cardinal Directions (Terrestrial)

The $4$ '''cardinal directions''' are the directions on (or near) Earth's surface defined as follows: === East === {{:Definition:East (Terrestrial)}} === West === {{:Definition:West (Terrestrial)}} === North === {{:Definition:North (Terrestrial)}} === South === {{:Definition:South (Terrestrial)}} :600px


Definition:Cardinal Directions

The $4$ cardinal directions: :North :South :East :West are the basis of navigation and geographical coordinates.


Definition:East

=== East (Terrestrial) === {{:Definition:East (Terrestrial)}} === East (Celestial) === {{:Definition:East (Celestial)}}


Definition:West

=== West (Terrestrial) === {{:Definition:West (Terrestrial)}} === West (Celestial) === {{:Definition:West (Celestial)}}


Definition:South

=== South (Terrestrial) === {{:Definition:South (Terrestrial)}} === South (Celestial) === {{:Definition:South (Celestial)}}


Definition:North

=== North (Terrestrial) === {{:Definition:North (Terrestrial)}} === North (Celestial) === {{:Definition:North (Celestial)}}


Definition:Intercardinal Directions

The $4$ '''intercardinal directions''' are the directions on (or near) Earth's surface halfway between the cardinal directions: === Northeast === {{:Definition:Northeast}} === Northwest === {{:Definition:Northwest}} === Southeast === {{:Definition:Southeast}} === Southwest === {{:Definition:Southwest}}


Definition:Northeast

'''Northeast''' is the direction on (or near) Earth's surface halfway between '''north''' and '''east'''.


Definition:Pole of Arc

Let $S$ be a sphere whose center is $O$. Let $C$ be the arc of a circle on $S$, either a small circle or a great circle. The '''poles''' of $C$ are the poles of the circle on $S$ of which $C$ is a part.


Definition:Spherical Angle

Let $S$ be a sphere. Let $PQ$ be a diameter of $S$. Let $PAQ$ and $PBQ$ be great circles on $S$ both passing through the points $P$ and $Q$. Let tangents $S$ and $T$ to $PAQ$ and $PBQ$ respectively be drawn through $P$. :500px Then the angle between $S$ and $T$ is known as the '''spherical angle''' between $PAQ$ and $PBQ$. Thus a '''spherical angle''' is defined with respect only to $2$ great circles.


Definition:Hemisphere

Let $S$ be a sphere. Let $S$ be bisected by a plane which passes through the center of $S$. Each of the halves of $S$ into which $S$ is divided is called a '''hemisphere'''.


Definition:Spherical Triangle/Side/Length

Let $ABC$ be a spherical triangle on a sphere $S$. Let $AB$ be a side of $ABC$. The '''length''' of $AB$ is defined as the angle subtended at the center of $S$ by $AB$.


Definition:Sphere/P-adic Numbers

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $a \in \Q_p$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''$\epsilon$-sphere of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$''' is defined as: :$\map {S_\epsilon} a = \set {x \in \Q_p: \norm {x - a} = \epsilon}$


Definition:Sphere/P-adic Numbers/Radius

In $\map {S_\epsilon} a$, the value $\epsilon$ is referred to as the '''radius''' of the $\epsilon$-sphere.


Definition:Sphere/P-adic Numbers/Center

In $\map {S_\epsilon} a$, the value $a$ is referred to as the '''center''' of the $\epsilon$-sphere.


Definition:Earth/Shape/First Approximation

To a first approximation, the shape of '''Earth''' can be considered to be spherical.


Definition:Earth/Shape/Second Approximation

To a second approximation, '''Earth''' is an oblate spheroid.


Definition:Spheroid/Oblate

An '''oblate spheroid''' is an ellipsoid which is the solid of revolution formed by rotating an ellipse about its minor axis.


Definition:Spheroid/Prolate

A '''prolate spheroid''' is an ellipsoid which is the solid of revolution formed by rotating an ellipse about its major axis.


Definition:Spheroid

A '''spheroid''' is an ellipsoid which is the solid of revolution formed by rotating an ellipse about one of its axes. === Oblate Spheroid === {{:Definition:Oblate Spheroid}} === Prolate Spheroid === {{:Definition:Prolate Spheroid}}


Definition:Ellipsoid

An '''ellipsoid''' is a solid figure whose plane sections are circles or ellipses.


Definition:Ellipse/Axis

Let $K$ be an ellipse whose foci are at $F_1$ and $F_2$. === Major Axis === {{:Definition:Ellipse/Major Axis}} === Minor Axis === {{:Definition:Ellipse/Minor Axis}} Category:Definitions/Ellipses j11v3s7kuz7fer16mgjjoejkzdqyncn


Definition:Longitude (Terrestrial)

Let $J$ be a point on Earth's surface that is not one of the two poles $N$ and $S$. Let $\bigcirc NJS$ be a meridian passing through $J$, whose endpoints are by definition $N$ and $S$. The '''longitude''' of $J$, and of the meridian $\bigcirc NJS$ itself, is the (spherical) angle that $\bigcirc NJS$ makes with the principal meridian $\bigcirc NKS$. :500px If $\bigcirc NJS$ is on the eastern hemisphere, the '''longitude''' is defined as '''longitude $n \degrees$ east''', where $n \degrees$ denotes $n$ degrees (of arc), written $n \degrees \, \mathrm E$. If $\bigcirc NJS$ is on the western hemisphere, the '''longitude''' is defined as '''longitude $n \degrees$ west''', written $n \degrees \, \mathrm W$. If $\bigcirc NJS$ is the principal meridian itself, the '''longitude''' is defined as '''$0 \degrees$ longitude'''. If $\bigcirc NJS$ is the other half of the great circle that contains the principal meridian, the '''longitude''' is defined as '''$180 \degrees$ longitude'''.


Definition:Longitude

=== Terrestrial Longitude === {{:Definition:Longitude (Terrestrial)}} === Celestial Longitude === {{:Definition:Longitude (Celestial)}} === Galactic Longitude === {{:Definition:Longitude (Galactic)}}


Definition:Latitude (Terrestrial)

Let $J$ be a point on Earth's surface that is not one of the two poles $N$ and $S$. Let $\bigcirc NJS$ be a meridian passing through $J$, whose endpoints are by definition $N$ and $S$. Let $\bigcirc NJS$ pass through the equator at $L$. The '''latitude''' of $J$ is the (spherical) angle $\sphericalangle LOJ$ , where $O$ is the center of Earth. :500px If $J$ is in the northern hemisphere of Earth, the '''latitude''' is defined as '''latitude $n \degrees$ north''', where $n \degrees$ denotes $n$ degrees (of arc), written $n \degrees \, \mathrm N$. If $J$ is in the southern hemisphere of Earth, the '''latitude''' is defined as '''latitude $n \degrees$ south''', written $n \degrees \, \mathrm S$. At the North Pole, the '''latitude''' is $90 \degrees \, \mathrm N$. At the South Pole, the '''latitude''' is $90 \degrees \, \mathrm S$.


Definition:Semi-Great Circle

Let $S$ be a sphere. A '''semi-great circle''' on $S$ is a semicircle consisting of half of a great circle on $S$.


Definition:Northern Hemisphere

The '''northern hemisphere''' of Earth is the hemisphere between the equator and the North Pole. Points in the northern hemisphere have latitude between $0 \degrees \, \mathrm N$ (the equator itself) and $90 \degrees \, \mathrm N$ (the North Pole).


Definition:Southern Hemisphere

The '''southern hemisphere''' of Earth is the hemisphere between the equator and the South Pole. Points in the southern hemisphere have latitude between $0 \degrees \, \mathrm S$ (the equator itself) and $90 \degrees \, \mathrm S$ (the South Pole).


Definition:Eastern Hemisphere

The '''eastern hemisphere''' of Earth is the hemisphere immediately to the east of the principal meridian. Points in the eastern hemisphere have longitude between $0 \degrees \, \mathrm E$ (the principal meridian itself) and $180 \degrees \, \mathrm E$.


Definition:Western Hemisphere

The '''western hemisphere''' of Earth is the hemisphere immediately to the west of the principal meridian. Points in the western hemisphere have longitude between $0 \degrees \, \mathrm W$ (the principal meridian itself) and $180 \degrees \, \mathrm W$.


Definition:Latitude

=== Terrestrial Latitude === {{:Definition:Latitude (Terrestrial)}} === Celestial Latitude === {{:Definition:Latitude (Celestial)}} Category:Definitions/Geographical Coordinates Category:Definitions/Spherical Astronomy jbsqzvied2ezel4hr69qms34of08u5l


Definition:Terrestrial Hemisphere

=== Northern Hemisphere === {{:Definition:Northern Hemisphere}} === Southern Hemisphere === {{:Definition:Southern Hemisphere}} === Eastern Hemisphere === {{:Definition:Eastern Hemisphere}} === Western Hemisphere === {{:Definition:Western Hemisphere}} Category:Definitions/Earth alp6k1su22746wvaev2c4nuli2fp2mv


Definition:Colatitude (Terrestrial)

Let $J$ be a point on Earth's surface that is not one of the two poles $N$ and $S$. Let $\phi$ denote the latitude of $J$. The '''colatitude''' of $J$ is the (spherical) angle $90 \degrees - \phi$, that is: :if $J$ is in the northern hemisphere of Earth, the '''colatitude''' is the (spherical) angle $\sphericalangle NOJ$ :if $J$ is in the southern hemisphere of Earth, the '''colatitude''' is the (spherical) angle $\sphericalangle SOJ$. :500px


Definition:Parallel of Latitude

Let $J$ be a point on Earth's surface that is not one of the two poles $N$ and $S$. Let $C$ be the small circle through $J$ parallel to the plane containing the equator. Then $C$ is known as the '''parallel of latitude''' of $J$. Thus $C$ consists of all the points on Earth's surface which have the same latitude as $J$. :500px


Definition:Imperial/Length/Mil

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''mil''' }} {{eqn | r = \dfrac 1 {1000}       | c = inch }} {{eqn | r = 25 \cdotp 4       | c = microns }} {{eqn | r = 0 \cdotp 0254       | c = millimetres }} {{eqn | r = 2 \cdotp 54 \times 10^{-5}       | c = metres }} {{end-eqn}}


Definition:Metric System/Length

Various metric units of length which have specific names are as follows: === Micrometre === {{Definition:Metric System/Length/Micrometre}} === Millimetre === {{Definition:Metric System/Length/Millimetre}} === Centimetre === {{Definition:Metric System/Length/Centimetre}} === Decimetre === {{Definition:Metric System/Length/Decimetre}} === Metre === {{Definition:Metric System/Length/Metre}} === Hectometre === {{Definition:Metric System/Length/Hectometre}} === Kilometre === {{Definition:Metric System/Length/Kilometre}}


Definition:Non-Archimedean Norm Axioms

{{begin-axiom}} {{axiom | n = \text N 1         | lc= Positive Definiteness:         | q = \forall x \in R         | ml= \norm x = 0         | mo= \iff         | mr= x = 0_R }} {{axiom | n = \text N 2         | lc= Multiplicativity:         | q = \forall x, y \in R         | ml= \norm {x \circ y}         | mo= =         | mr= \norm x \times \norm y }} {{axiom | n = \text N 4         | lc= Ultrametric Inequality:         | q = \forall x, y \in R         | ml= \norm {x + y}         | mo= \le         | mr= \max \set {\norm x, \norm y} }} {{end-axiom}}


Definition:Non-Archimedean/Norm (Division Ring)/Definition 1

A norm $\norm {\, \cdot \,}$ on $R$ is '''non-Archimedean''' {{iff}} $\norm {\, \cdot \,}$ satisfies the axiom: {{begin-axiom}} {{axiom | n = \text N 4         | lc= Ultrametric Inequality:         | q = \forall x, y \in R         | ml= \norm {x + y}         | mo= \le         | mr= \max \set {\norm x, \norm y} }} {{end-axiom}}


Definition:Non-Archimedean/Norm (Division Ring)/Definition 2

A '''non-Archimedean norm''' on $R$ is a mapping from $R$ to the non-negative reals: :$\norm {\, \cdot \,}: R \to \R_{\ge 0}$ satisfying the non-Archimedean norm axioms: {{:Definition:Non-Archimedean Norm Axioms}}


Definition:Kilometre/Symbol

The symbol for the '''kilometre''' is $\mathrm {km}$: :$\mathrm k$ for kilo :$\mathrm m$ for metre.


Definition:Nautical Mile

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''nautical mile''' }} {{eqn | r = 1852       | c = metres (by definition) }} {{eqn | r = 10       | c = cables }} {{eqn | o = \approx       | r = 6076       | c = feet }} {{eqn | o = \approx       | r = 1 \cdotp 151       | c = (international) mile }} {{eqn | o = \approx       | r = 0 \cdotp 999 \, 363       | c = admiralty (UK nautical) mile }} {{end-eqn}}


Definition:Admiralty Mile

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''admiralty mile''' }} {{eqn | r = 6080       | c = feet (by definition) }} {{eqn | r = 1853 \cdotp 184       | c = metres }} {{eqn | o = \approx       | r = 1 \cdotp 1515       | c = (international) mile }} {{end-eqn}}


Definition:Inner Angle of Spherical Triangle

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Consider the $4$ consecutive parts of $\triangle ABC$: :$B, a, C, b$ Then $C$ is known as the '''inner angle''' of $B, a, C, b$


Definition:Inner Side of Spherical Triangle

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Consider the $4$ consecutive parts of $\triangle ABC$: :$B, a, C, b$ Then $a$ is known as the '''inner side''' of $B, a, C, b$


Definition:Quadrantal Triangle

Let $\triangle ABC$ be a spherical triangle. Let one of the sides of $\triangle ABC$ be a right angle: $\dfrac \pi 2$. :500px Then $\triangle ABC$ is known as a '''quadrantal (spherical) triangle'''.


Definition:Right Spherical Triangle

Let $\triangle ABC$ be a spherical triangle. Let one of the angles of $\triangle ABC$ be a right angle: $\dfrac \pi 2$. :500px Then $\triangle ABC$ is known as a '''right spherical triangle'''.


Definition:Spherical Excess of Spherical Triangle

Let $\triangle ABC$ be a spherical triangle. Let $E$ be the sum of the angles of $\triangle ABC$. The '''spherical excess''' of $\triangle ABC$ is how much $E$ is greater than $\pi$ radians.


Definition:Polar Triangle

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let $A'$, $B'$ and $C'$ be the poles of the sides $BC$, $AC$ and $AB$ respectively which are in the same hemisphere as the points $A$, $B$ and $C$ respectively. :400px Then the spherical triangle $\triangle A'B'C'$ is the '''polar triangle''' of $\triangle ABC$.


Definition:Length of Arc of Great Circle

Let $E$ be a great circle on a sphere $S$ whose center is $O$. Let $AB$ be an arc of $E$. The '''length''' of $AB$ is defined as the angle subtended at $O$ by $AB$.


Definition:Haversine

The '''haversine''' of an angle is defined as: {{begin-eqn}} {{eqn | l = \hav \theta       | r = \dfrac 1 2 \paren {1 - \cos \theta}       | c =  }} {{eqn | r = \dfrac 1 2 \vers \theta       | c = where $\vers \theta$ denotes the versed sine of $\theta$ }} {{eqn | r = \sin^2 \dfrac {\theta} 2       | c = Double Angle Formula for Cosine: Corollary 2 }} {{end-eqn}}


Definition:Versed Sine

The '''versed sine''' of an angle is defined as: :$\vers \theta = 1 - \cos \theta$


Definition:Knot (Unit of Measurement)

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''knot''' }} {{eqn | r = 1 \, 852       | c = metres per hour (exactly) }} {{eqn | o = \approx       | r = 0 \cdotp 51444       | c = metres per second }} {{eqn | o = \approx       | r = 1 \cdotp 15078       | c = (international) miles per hour }} {{end-eqn}}


Definition:Zenith

Let an observer $O$ on the surface of Earth be assumed to be the center of the celestial sphere $C$. The '''zenith''' $Z$ is the point on $C$ which is vertically overhead.


Definition:Celestial Sphere

The '''celestial sphere''' is an imaginary sphere, an indeterminate distance from Earth, which intersects the straight lines from the celestial bodies to the observer. :480px It is then convenient to identify the directions of the celestial bodies with their position on the '''celestial sphere'''.


Definition:Celestial

In the context of astronomy, the word '''celestial''' is an adjective whose meaning is loosely '''of the sky'''. It is used to define bodies and phenomena vaguely defined as being outside Earth's atmosphere, hence does not apply to meteorological phenomena. It is often used as an antithesis for the word '''terrestrial''', which means '''of Earth'''.


Definition:Terrestrial

The word '''terrestrial''' is an adjective whose meaning is loosely '''of Earth'''. It is usually used in the context of astronomy and geodesy to refer to concepts which need to be distinguished from their celestial counterparts.


Definition:Celestial Sphere/Observer

In the context of astronomy, in particular spherical astronomy, the '''observer''' is considered to be a reference point located at the center of the celestial sphere. In practice, of course, an actual '''observer''' is physically located on the surface of Earth. However, the celestial sphere is considered to be so much larger than the radius of Earth that the '''observer''' is considered to be at the center of Earth.


Definition:Celestial Horizon

The '''celestial horizon''' is the great circle on the celestial sphere whose plane is perpendicular to the line from the zenith to the observer. Thus the '''celestial horizon''' is the edge of the visible part of the celestial sphere. The rest of it is hidden by Earth itself. :480px


Definition:Visible Hemisphere

The '''visible hemisphere''' is the hemisphere of the celestial sphere between the celestial horizon and the zenith. That is, it is the half of the sky that can be seen from Earth at an arbitrary time. The other hemisphere of the celestial sphere is hidden from an observer by Earth itself. :480px


Definition:Vertical Circle

Let $X$ be the position of a star (or other celestial body) on the celestial sphere. The '''vertical circle''' through $X$ is defined as the great circle passing through both $X$ and the zenith $Z$.


Definition:Celestial Altitude

Let $X$ be a point on the celestial sphere. The '''(celestial) altitude''' of $X$ is defined as the angle subtended by the the arc of the vertical circle through $X$ between the celestial horizon and $X$ itself.


Definition:Zenith Distance

Let $X$ be the position of a star (or other celestial body) on the celestial sphere. The '''zenith distance''' of $X$ is defined as the angle subtended by the the arc of the vertical circle through $X$ between $X$ and the zenith.


Definition:Parallel of Altitude

Let $X$ be a point on the celestial sphere. Let $LXM$ be a small circle through $X$ which is parallel to the celestial horizon. Then $LXM$ is a '''parallel of altitude''', and is such that all celestial bodies on it have the same altitude.


Definition:North Celestial Pole

Let $O$ be a celestial observer situated at the surface of Earth in the Northern hemisphere. Let $OP$ be a straight line parallel to Earth's axis. Let $P$ be the point in the visible hemisphere at which $OP$ meets the celestial sphere. Then $P$ is the '''north celestial pole'''. That is, the '''north celestial pole''' is the pole of the celestial equator which is the zenith of the North Pole.


Definition:Polaris

'''Polaris''' is a star whose position in the celestial sphere is very nearly exactly at the north celestial pole. Hence it appears not to move through the sky during the course of a night.


Definition:Principal Vertical Circle

The vertical circle on the celestial sphere which passes through the north celestial pole (and south celestial pole) is called the '''principal vertical circle'''. :480px


Definition:North Point of Horizon

The point on the celestial horizon through which the principal vertical circle passes is called the '''north point of the horizon'''.


Definition:South Point of Horizon

The point on the celestial horizon exactly opposite the '''north point of the horizon''' is called the '''south point of the horizon'''.


Definition:West Point of Horizon

Let an observer $O$ be facing the '''north point of the horizon''' $N$, facing directly away from the '''south point of the horizon''' $S$. Let the celestial equator meet the celestial horizon at the points $E$ and $W$. Let $W$ be the point on the left of $O$. This is called the '''west point of the horizon'''.


Definition:East Point of Horizon

Let an observer $O$ be facing the '''north point of the horizon''' $N$, facing directly away from the '''south point of the horizon''' $S$. Let the celestial equator meet the celestial horizon at the points $E$ and $W$. Let $E$ be the point on the right of $O$. This is called the '''east point of the horizon'''.


Definition:Cardinal Points (Celestial)

The '''cardinal points''' of the celestial sphere are the '''north point''', the '''sourth point''', the '''east point''' and the '''west point''': === North Point of Horizon === {{:Definition:North Point of Horizon}} === South Point of Horizon === {{:Definition:South Point of Horizon}} === East Point of Horizon === {{:Definition:East Point of Horizon}} === West Point of Horizon === {{:Definition:West Point of Horizon}}


Definition:Azimuth (Astronomy)

Let $X$ be a point on the celestial sphere. The spherical angle between the principal vertical circle and the vertical circle on which $X$ lies is the '''azimuth''' of $X$. The '''azimuth''' is usually measured in degrees, $0 \degrees$ to $180 \degrees$ either west or east, depending on whether $X$ lies on the eastern or western hemisphere of the celestial sphere.


Definition:Prime Vertical

The vertical circle on the celestial sphere which passes through both the east point and the west point is called the '''prime vertical'''.


Definition:Celestial Equator

Consider the celestial sphere with observer $O$. Let $P$ be the north celestial pole. The great circle whose plane is perpendicular to $OP$ is known as the '''celestial equator'''.


Definition:Nadir

Consider the celestial sphere $C$ with an observer $O$ at the center. Let $Z$ be the zenith of $C$ Let  the straight line $ZO$ be produced from $O$ to meet $C$ at $R$. The point $R$ is known as the '''nadir''' of $C$.


Definition:South Celestial Pole

Let $O$ be a celestial observer situated at the surface of Earth in the Southern hemisphere. Let $OQ$ be a straight line parallel to Earth's axis. Let $Q$ be the point in the visible hemisphere at which $OQ$ meets the celestial sphere. Then $Q$ is the '''south celestial pole'''. That is, the '''south celestial pole''' is the pole of the celestial equator which is the zenith of the South Pole.


Definition:Celestial Axis

Consider the celestial sphere with observer $O$. The '''celestial axis''' is the line which joins the north celestial pole and the south celestial pole.


Definition:Celestial Pole

Consider the celestial sphere $C$. The '''celestial poles''' of $C$ are the north celestial pole and the south celestial pole: === North Celestial Pole === {{:Definition:North Celestial Pole}} === South Celestial Pole === {{:Definition:South Celestial Pole}}


Definition:Declination

Consider the celestial sphere $C$ with observer $O$. Let $P$ and $Q$ be the north celestial pole and south celestial pole respectively. Let $X$ be a point on $C$. Let $PXQ$ be the vertical circle through $X$. Let $D$ be the point where the celestial equator intersects $PXQ$. The length of the arc $DX$ of $PXQ$ is known as the '''declination''' of $X$.


Definition:Declination/North

If $X$ is in the northern celestial hemisphere, $DX$ is '''north declination'''.


Definition:Declination/South

If $X$ is in the southern celestial hemisphere, $DX$ is '''south declination'''.


Definition:North Polar Distance

Consider the celestial sphere $C$ with observer $O$. Let $X$ be a point on $C$. Let $X$ have declination of $\delta$. The '''north polar distance''' of $X$, denoted $\operatorname {N.P.D.} x$, is defined as: :$\operatorname {N.P.D.} x = 90 \degrees - \delta$ That is, the '''north polar distance''' of $X$ length of the arc from the north celestial pole $P$ along the vertical circle through $X$. :500px


Definition:Observer's Meridian

Consider the celestial sphere whose observer is $O$. Let $P$ and $Q$ be the north celestial pole and south celestial pole respectively. Let $Z$ be the zenith. Let $S$ be the south point of the horizon. The '''observer's meridian''' is the semi-great circle $PZSQ$.


Definition:Babylonian Number System

The number system as used in the {{WP|First_Babylonian_Dynasty|Old Babylonian empire}} was a positional numeral system where the number base was a combination of decimal (base $10$) and sexagesimal (base $60$). The characters were written in {{WP|Cuneiform|cuneiform}} by a combination of: :a thin vertical wedge shape, to indicate the digit $1$ :a fat horizontal wedge shape, to indicate the digit $10$ arranged in groups to indicate the digits $2$ to $9$ and $20$ to $50$. :600px At $59$ the pattern stops, and the number $60$ is represented by the digit $1$ once again. Thus these groupings were placed side by side: :the rightmost grouping would indicate a number from $1$ to $59$ :the one to the left of that would indicate a number from $60 \times 1$ to $60 \times 59$ and so on, each grouping further to the left indicating another multiplication by $60$ For fractional numbers there was no actual radix point. Instead, the distinction was inferred by context. The fact that they had no symbol to indicate the zero digit means that this was not a true positional numeral system as such. For informal everyday arithmetic, they used a decimal system which was the decimal part of the full sexagesimal system.


Definition:Number System

A '''number system''' is a technique for representing numbers.


Definition:Numeral System

A '''numeral system''' is: :a set of symbols that is used to represent a specific subset of the set of numbers (usually natural numbers), referred to as numerals :a set of rules which define how to combine the numerals so as to be able to express other numbers.


Definition:Numeral

A '''numeral''' is a symbol which is used to identify a particular number (usually a natural number). A '''numeral''' is one of a small subset of the numbers. Other numbers can be created by combining '''numerals''' using specific well-defined rules.


Definition:Positional Numeral System

A '''positional number system''' is a number system with the following properties: :It has a set of numerals which represent a subset of the numbers. :The number being represented is written as a string of these numerals, which represent a different value according to their position in the numerals. The design of the '''positional number system''' is such that all numbers can be represented by such a string, which may or may not be infinite in length.


Definition:Musical Interval

A '''musical interval''' is a ratio of frequencies of musical notes which is recognised by convention as being particularly aesthetically pleasing or meaningful. === Octave === {{:Definition:Octave}} === Fifth === {{:Definition:Fifth (Music)}} === Fourth === {{:Definition:Fourth (Music)}} === Tone === {{:Definition:Tone (Music)}} === Semitone === {{:Definition:Semitone}}


Definition:Neusis

=== Neusis Ruler === {{:Definition:Neusis Ruler}} === Neusis Construction === {{:Definition:Neusis Construction}}


Definition:Decimal Expansion/Historical Note

The idea of representing fractional values by extending the decimal notation to the right appears to have been invented by {{AuthorRef|Simon Stevin}}, who published the influential book {{BookLink|De Thiende|Simon Stevin}}. The idea was borrowed from the Babylonian number system, but streamlined to base $10$ from the cumbersome sexagesimal. However, his notation was cumbersome: he would write, for example, $25 \bigcirc \! \! \! \! \! \! 0 \ \, 3 \bigcirc \! \! \! \! \! \! 1 \ \, 7 \bigcirc \! \! \! \! \! \! 2 \ \, 9 \bigcirc \! \! \! \! \! \! 3$ for what we now give as $25 \cdotp 379$. {{AuthorRef|John Napier}}, in the early $17$th century, appears to have been the first into print with the contemporary notation, although {{AuthorRef|Walter William Rouse Ball}} suggests that credit for this ought to be due to {{AuthorRef|Henry Briggs}}. It was not until a century later, however, that the decimal point came into general use.


Definition:Trigonometry/Linguistic Note

The word '''trigonometry''' derives from the Greek for '''measurement of triangles'''.


Definition:Epicycle/Historical Note

{{AuthorRef|Claudius Ptolemy}}'s {{BookLink|Almagest|Claudius Ptolemy}} modelled the orbits of the planets of the solar system as a system of epicycles.


Definition:Sextant

A '''sextant''' is a device for measuring the celestial altitude of a celestial body. It consists of a telescope and a system of adjustable reflectors for presenting the image of the celestial body to the eye so as to be next to an image of the horizon. The resulting reading on the scale of the '''sextant''' gives an indication of the altitude of the body.


Definition:Astrolabe

An '''astrolabe''' is a device for measuring the celestial altitude of a celestial body. It consists of a series of disks on which are shown the positions of prominent stars. These are aligned with their positions on what can be seen of the celestial sphere, and hence obtain a reading of their altitudes.


Definition:GPS

'''GPS''', or '''Global Positioning System''', is a technique using signals from satellites to identify the geographical coordinates of an appropriately configured receiver.


Definition:Noon

'''Noon''' is the point in time during the course of a day that the Sun reaches its highest celestial altitude. Category:Definitions/Navigation Category:Definitions/Geography Category:Definitions/Astronomy pj5i99obzjmns4nlyj39e74w9laz4xn


Definition:Unknown

An '''unknown''' is an indeterminate variable in a system of equations or inequalities whose value the solver is invited to determine.


Definition:Diophantine Problem

A '''Diophantine problem''' is a system of Diophantine equations which has more unknowns than it has equations.


Definition:Binary Tuple

A '''binary tuple''' is an ordered tuple whose domain is the set $\set {0, 1}$. That is, it is an ordered $n$-tuple $\tuple {x_1, x_2, \ldots, x_n}$ such that each $x_i$ is either $0$ or $1$. The set of all binary $n$-tuples can often be seen denoted $V_n$: :$V_n := \set {\tuple {x_1, x_2, \ldots, x_n}: \forall i \in \set {1, 2, \ldots n}: x_i \in \set {0, 1} }$


Definition:Set of Strings of Given Length

Let $n \in \N$ be a natural number. Let $\Sigma$ be an alphabet. The set of all strings from $\Sigma$ of length $n$ is denoted $\Sigma^{\paren n}$.


Definition:Set of Finite Strings

Let $\Sigma$ be an alphabet. The set of all finite strings from $\Sigma$ is denoted $\Sigma^*$.


Definition:Symmetric Mapping

=== Mapping Theory === {{:Definition:Symmetric Mapping (Mapping Theory)}} === Linear Algebra === {{:Definition:Symmetric Mapping (Linear Algebra)}} Category:Definitions/Mapping Theory Category:Definitions/Linear Algebra g3wpcxdc1eztpzylzbb1npci2h88ha7


Definition:Symmetric Mapping (Mapping Theory)

Let $n \in \N$ be a natural number. Let $S^n$ be an $n$-dimensional cartesian space on a set $S$. Let $f: S^n \to T$ be a mapping from $S^n$ to a set $T$. Then $f$ is a '''symmetric mapping''' {{iff}}: :$\map f {x_1, x_2, \dotsc, x_n} = \map f {x_{\map \pi 1}, x_{\map \pi 2}, \dotsc x_{\map \pi n} }$ for all permutations $\pi$ on $\set {1, 2, \dotsc n}$. That is, a '''symmetric mapping''' is a mapping defined on a cartesian space whose values are preserved under permutation of its arguments.


Definition:Uncertainty

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


Definition:Chebyshev-Sylvester Constant

The '''Chebyshev-Sylvester constant''' is a mathematical constant $\alpha$ which was shown by {{AuthorRef|Pafnuty Lvovich Chebyshev}} and {{AuthorRef|James Joseph Sylvester}} that for sufficiently large $x$, there exists at least one prime number $p$ satisfying: :$x < p < \paren {1 + \alpha} x$ The number $\alpha$ was demonstrated to be $0 \cdotp 092 \ldots$ In $1896$, {{AuthorRef|Jacques Salomon Hadamard}} and {{AuthorRef|Charles de la Vallée Poussin}} independently proved the Prime Number Theorem. This demonstrated that the above inequality is true for all $\alpha > 0$ for sufficiently large $x$,. Hence this constant is now only of historical interest. {{NamedforDef|Pafnuty Lvovich Chebyshev|name2 = James Joseph Sylvester|cat = Chebyshev|cat2 = Sylvester}}


Definition:Du Bois-Reymond Constants

The '''du Bois-Reymond constants''' are the constants $C_n$ where: :$C_n = \displaystyle \int_0^\infty \size {\map {\dfrac \d {\d t} } {\dfrac {\sin t} t}^n} \rd t - 1$


Definition:Copeland-Erdős Constant

The '''Copeland-Erdős constant''' is the real number whose decimal expansion is formed by concatenating the prime numbers in ascending order: :$C_{10} = 0 \cdotp 23571 \, 11317 \, 19232 \, 93137 \, 4143 \ldots$ {{OEIS|A033308}}


Definition:Koebe Constant

The Koebe Quarter Theorem states: {{:Koebe Quarter Theorem}} That constant $\dfrac 1 4$ by which $\size {\map {f'} 0}$ is bounded below can be found referred to as the '''Koebe constant'''. {{NamedforDef|Paul Koebe|cat = Koebe}}


Definition:Meissel-Mertens Constant

Consider the expression: :$\displaystyle M = \map {\lim_{n \mathop \to \infty} } {\sum_{\substack {p \mathop \le n \\ \text {$p$ prime} } } \dfrac 1 p - \ln \ln p}$ Then: :$M \approx 0 \cdotp 26149 \, 72128 \, 47642 \, 78375 \, 54268 \, 38608 \, 69585 \, 90516 \ldots$ {{OEIS|A077761}} The constant $M$ is known as the '''Meissel-Mertens Constant'''.


Definition:Matroid Theory

'''Matroid Theory''' is the branch of mathematics which concerns the role which matroids play in disparate branches of combinatorial theory and algebra such as graph theory, lattice theory, combinatorial optimization, and linear algebra.


Definition:Kakeya's Constant

'''Kakeya's constant''' is defined as the area of the smallest simple convex domain in which one can put a line segment of length $1$ which will coincide with itself when rotated $180 \degrees$: :$K = \dfrac {\paren {5 - 2 \sqrt 2} \pi} {24} \approx 0 \cdotp 28425 \, 82246 \ldots$ {{OEIS|A093823}} {{expand|Needs considerable work done here by someone who understands exactly what is going on here. The case of the equilateral triangle is well known; so is the case of the Perron tree; I also remember a piece by Martin Gardner on the subject which demonstrates that a star-shaped area derived from the deltoid can be made arbitrarily small; and so on. Exactly what is meant here by simple convex domain needs rigorous clarification.}}


Definition:Matroid

Let $M = \struct{S,\mathscr I}$ be an independence system.


Definition:Matroid Axioms

<onlyinclude> Let $S$ be a finite set. Let $\mathscr I$ be a set of subsets of $S$. The '''matroid axioms''' are the conditions on $S$ and $\mathscr I$ in order for the ordered pair $\struct {S, \mathscr I}$ to be a matroid:  === Axioms 1 === {{:Definition:Matroid Axioms/Axioms 1}} === Axioms 2 === {{:Definition:Matroid Axioms/Axioms 2}} === Axioms 3 === {{:Definition:Matroid Axioms/Axioms 3}} === Axioms 4 === {{:Definition:Matroid Axioms/Axioms 4}}


Definition:Matroid/Independent Set

An element of $\mathscr I$ is called an '''independent set''' of $M$.


Definition:Matroid/Dependent Set

A subset of $S$ that is not an element of $\mathscr I$ is called a '''dependent set''' of $M$.


Definition:Base of Matroid

A '''base''' of $M$ is a maximal independent subset of $S$.


Definition:Rank Function (Matroid)

The '''rank function''' of $M$ is the mapping $\rho : \powerset S \to \Z$ from the power set of $S$ into the integers defined by: :$\forall A \subseteq S : \map \rho A = \max \set {\size X : X \subseteq A \land X \in \mathscr I}$ where $\size A$ denotes the cardinality of $A$.


Definition:Rank (Matroid)

The '''rank''' of $M$, denoted by $\map \rho M$, is the image of $S$ under $\rho$, that is: :$\map \rho M = \map \rho S$.


Definition:Flat (Matroid)

A subset $A \subseteq S$ is a '''flat''' of $M$ {{iff}}: :$\forall x \in S \setminus A : \map \rho {A \cup \set x} = \map \rho A + 1$


Definition:Depends Relation (Matroid)

For $x \in S$ and $A \subseteq S$, $x$ is said to '''depend''' on $A$, denoted as $x \sim A$, {{iff}}: :$\map \rho {A \cup \set x} = \map \rho A$


Definition:Closure Operator (Matroid)

The '''closure operator''' of the matroid $M$ is the mapping $\sigma : \powerset S \to \powerset S$ defined by: :$\map \sigma A$ is the set of elements of $S$ which depend on $A$


Definition:Spanning Subset (Matroid)

A subset $X \subseteq S$ is '''spanning''' in $M$ {{iff}}: :there exists a base $B$ contained in $X$


Definition:Circuit (Matroid)

A '''circuit''' of $M$ is a minimal dependent subset of $S$.


Definition:Base Axiom (Matroid)

{{begin-axiom}} {{axiom | n = \text B 1         | q = \forall B_1, B_2 \in \mathscr B         | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \cup \set y} \setminus \set x \in \mathscr B }} {{end-axiom}}


Definition:Rank Axioms (Matroid)/Definition 1

{{begin-axiom}} {{axiom | n = \text R 1         | mr= \map \rho \O = 0 }} {{axiom | n = \text R 2         | q = \forall X \in \powerset S \land y \in S         | mr= \map \rho X \le \map \rho {X \cup \set y} \le \map \rho X + 1 }} {{axiom | n = \text R 3         | q = \forall X \in \powerset S \land y, z \in S         | mr= \map \rho {X \cup \set y} = \map \rho {X \cup \set z} = \map \rho X \implies \map \rho {X \cup \set{y,z} } = \map \rho X }} {{end-axiom}}


Definition:Rank Axioms (Matroid)/Definition 2

{{begin-axiom}} {{axiom | n = \text R 1'         | q = \forall X \in \powerset S         | mr = 0 \le \map \rho X \le \size X }} {{axiom | n = \text R 2'         | q = \forall X, Y \in \powerset S         | mr = X \subseteq Y \implies \map \rho X \le \map \rho Y }} {{axiom | n = \text R 3'         | q = \forall X, Y \in \powerset S         | mr = \map \rho {X \cup Y} + \map \rho {X \cap Y} \le \map \rho X + \map \rho Y }} {{end-axiom}}


Definition:Rank Axioms (Matroid)

Let $S$ be a finite set. Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ into the integers === Definition 1 === $\rho$ is said to satisfy the '''rank axioms''' {{iff}}: {{:Definition:Rank Axioms (Matroid)/Definition 1}} === Definition 2 === $\rho$ is said to satisfy the '''rank axioms''' {{iff}}: {{:Definition:Rank Axioms (Matroid)/Definition 2}} Category:Definitions/Matroid Theory hchpgizbnsrgwofz4fawrvgwwsrjclf


Definition:Isomorphism (Matroid)

Let $M_1 = \struct{S_1, \mathscr I_1}$ and $M_2 = \struct{S_2, \mathscr I_2}$ be matroids. Let $f : S_1 \to S_2$ be a bijection. Then $f$ is called an '''isomorphism''' if: :$\forall X \subseteq S : X \in \mathscr I_1 \iff \map f X \in \mathscr I_2$ If $f$ is an '''isomorphism''' then $M_1$ is said to be '''isomorphic''' to $M_2$.


Definition:Bloch's Constant

Recall Bloch's Theorem: {{:Bloch's Theorem}} The lower bound of $B$ is known as '''Bloch's constant'''.


Definition:Landau's Constant

Recall Landau's Theorem: {{:Landau's Theorem}} The constant of $L$ is known as '''Landau's constant'''.


Definition:Uniform Matroid

Let $S$ be a finite set of cardinality $n$. Let $\mathscr I_{k,n}$ be the set of all subsets of $S$ of cardinality less than or equal to $k$.  Then the ordered pair $\struct {S, \mathscr I_{k, n} }$ is called the '''uniform matroid of rank $k$''' and is denoted $U_{k,n}$.


Definition:Free Matroid

Let $S$ be a finite set. Let $\mathscr I = \powerset S$ be the power set of $S$. That is, let $\mathscr I$ be the set of all subsets of $S$: :$\mathscr I := \set {X: X \subseteq S}$ Then the ordered pair $\struct{S, \mathscr I}$ is called the '''free matroid of $S$'''.


Definition:Matroid Induced by Linear Independence/Vector Space

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


Definition:Cycle Matroid

Let $G$ be a graph. Let $E$ be the edge set of $G$. Let $\mathscr I$ be the set of edge sets of subgraphs of $G$ that contain no cycles. Then the ordered pair $\struct{E, \mathscr I}$ is called the '''cycle matroid''' of the graph $G$.


Definition:Riemann Zeta Function/Zero

The '''zeroes''' of the Riemann zeta function are, according to the conventional definition of the zero of a (complex) function, the points $s \in \C$ such that: :$\map \zeta s = 0$ === Trivial Zeroes === {{:Definition:Riemann Zeta Function/Zero/Trivial}} === Nontrivial Zeroes === {{:Definition:Riemann Zeta Function/Zero/Nontrivial}}


Definition:Riemann Zeta Function/Zero/Trivial

The '''trivial zeroes''' of the Riemann zeta function $\zeta$ are the strictly negative even integers : :$\set {n \in \Z: n = -2 \times k: k \in \N_{\ne 0} } = \set {-2, -4, -6, \ldots}$


Definition:Brocard Points

:700px Let $\triangle ABC$ be a triangle such that $A \to B \to C \to A$ travels anticlockwise around $\triangle ABC$. === First Brocard Point === {{:Definition:Brocard Points/First}} === Second Brocard Point === {{:Definition:Brocard Points/Second}}


Definition:Brocard Points/First

Let the point $P$ be constructed such that: :$\angle PAB = \angle PBC = \angle PCA$ Then $P$ is the '''first Brocard point''' of $\triangle ABC$.


Definition:Brocard Points/Second

Let the point $Q$ be constructed such that: :$\angle QAC = \angle QCB = \angle QBA$ Then $Q$ is the '''second Brocard point''' of $\triangle ABC$.


Definition:Brocard Angle

:700px Let $\triangle ABC$ be a triangle such that $A \to B \to C \to A$ travels anticlockwise around $\triangle ABC$. Let $P$ and $Q$ be the first and second Brocard points of $\triangle ABC$ respectively. Let: {{begin-eqn}} {{eqn | l = \omega       | r = \angle PAB = \angle PBC = \angle PCA }} {{eqn | l = \omega'       | r = \angle QAC = \angle QCB = \angle QBA }} {{end-eqn}} From Brocard Angle is Unique: :$\omega = \omega'$ This angle $\omega = \omega'$ is the '''Brocard angle''' of $\triangle ABC$.


Definition:Lehmer's Constant

Let the sequence $\sequence {b_k}_{k \mathop \in \N}$ be defined by the recurrence relation: {{begin-eqn}} {{eqn | l = b_k       | r = \begin {cases} 0 & : k = 0 \\ {b_{k - 1} }^2 + b_{k - 1} + 1 & : \text {otherwise} \end{cases} }} {{eqn | r = \sequence {0, 1, 3, 13, 183, 33973 \ldots} }} {{end-eqn}} {{OEIS|A002065}} '''Lehmer's constant''' is the real number $\xi$ defined as: {{begin-eqn}} {{eqn | l = \xi       | r = \map \cot {\sum_{k \mathop = 0}^\infty \paren {-1}^k \arccot b_k} }} {{eqn | r = \map \cot {\arccot 0 - \arccot 1 + \arccot 3 - \arccot 13 + \arccot 183 - \arccot 33973 + \dotsb} }} {{eqn | o = \approx       | r = 0 \cdotp 59263 \, 27182 \ldots }} {{end-eqn}} {{OEIS|A030125}}


Definition:Vectorial Matroid

Let $V$ be a vector space. Let $S$ be a finite subset of $V$. Let $\struct{S, \mathscr I}$ be the matroid induced by linear independence in $V$ on $S$. From Matroid Induced by Linear Independence in Vector Space is Matroid, $\struct{S, \mathscr I}$ is a matroid. Then any matroid isomorphic to $\struct{S, \mathscr I}$ is called a '''vectorial matroid'''.


Definition:Matroid Induced by Algebraic Independence

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


Definition:Algebraic Matroid

Let $L / K$ be a field extension. Let $S \subseteq L$ be a finite subset of $L$. Let $\struct{S, \mathscr I}$ be the matroid induced by Algebraic Independence over $K$ on $S$. From Matroid Induced by Algebraic Independence is Matroid, $\struct{S, \mathscr I}$ is a matroid. Then any matroid isomorphic to $\struct{S, \mathscr I}$ is called an '''algebraic matroid'''.


Definition:Affinely Dependent

Let $\R^n$ be the $n$-dimensional real Euclidean space. Let $S = \set{x_1, \dots, x_r}$ be a finite subset of $\R^n$. An element $x \in \R^n$ is '''affinely dependent''' on $S$ if there exist real numbers $\set{\lambda_i: 1 \le i \le r}$ such that: :$(1) \quad x = \displaystyle \sum_{i = 1}^r \lambda_i x_i$  :$(2) \quad \displaystyle \sum_{i = 1}^r \lambda_i = 1$


Definition:Affinely Dependent/Independent

Let $\R^n$ be the $n$-dimensional real Euclidean space. Let $X = \set{x_1, \dots, x_r}$ be a finite subset of $\R^n$. The subset $X$ is '''affinely independent''' if no element $x \in X$ is affinely dependent on $X \setminus \set x$


Definition:Euler-Gompertz Constant

=== Integral Form === {{:Definition:Euler-Gompertz Constant/Integral Form}} === As a Continued Fraction === {{:Definition:Euler-Gompertz Constant/Continued Fraction}}


Definition:Torsion-Free Group

Let $G$ be a group. Then $G$ is a '''torsion-free group''' {{iff}} the only torsion element of $G$ is the group identity element.


Definition:Matroid Induced by Affine Independence

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


Definition:Matroid Induced by Linear Independence/Abelian Group

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


Definition:Euler-Gompertz Constant/Integral Form

The '''Euler-Gompertz constant''' is the real number $G$ defined as: :$G = \displaystyle \int_0^\infty \dfrac {e^{-u} } {1 + u} \rd u$


Definition:Twin Primes Constant

The '''twin primes constant''' is the real number: {{begin-eqn}} {{eqn | l = \Pi_2       | o = :=       | r = \displaystyle \prod_{\substack {p \mathop \ge 3 \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 {\paren {p - 1}^2} } }} {{eqn | o = \approx       | r = 0 \cdotp 66016 \, 18       | c =  }} {{end-eqn}} {{OEIS|A005597}}


Definition:Robbins Constant

The '''Robbins constant''' $R$ is defined as the mean distance $D$ between $2$ points chosen at random from the interior of a unit cube: {{begin-eqn}} {{eqn | l = R       | r = \frac {4 + 17 \sqrt 2 - 6 \sqrt3 - 7 \pi} {105} + \frac {\map \ln {1 + \sqrt 2 } } 5 + \frac {2 \, \map \ln {2 + \sqrt 3} } 5 }} {{eqn | o = \approx       | r = 0 \cdotp 66170 \, 71822 \, 67176 \, 23515 \, 582 \ldots       | c =  }} {{end-eqn}} {{OEIS|A073012}}


=== Vector Space === {{:Definition:Matroid Induced by Linear Independence/Vector Space}} === Abelian Group === {{:Definition:Matroid Induced by Linear Independence/Abelian Group}} Category:Examples of Matroids ggj6ymjjxk3r856h4pw8v4ka19ao229


Definition:Loop (Matroid)

A '''loop''' of $M$ is an element $x$ of $S$ such that $\set x$ is a dependent subset of $S$. That is, $x \in S$ is a '''loop''' {{iff}} $\set x \not \in \mathscr I$.


Definition:Parallel (Matroid)

Two elements $x, y \in S$ are said to be '''parallel''' in $M$ {{iff}} they are not loops but $\set {x, y}$ is a dependent subset of $S$. That is, $x, y \in S$ are '''parallel''' {{iff}}: :$\set x, \set y \in \mathscr I$ and $\set {x, y} \notin \mathscr I$.


Definition:Central Binomial Coefficient

Let $n \in \N$ be a natural number. The binomial coefficient: :$\dbinom {2 n} n$ is known as a '''central binomial coefficient'''.


Definition:Landau-Ramanujan Constant

{{begin-eqn}} {{eqn | l = k       | r = \sqrt {\dfrac 1 2 \displaystyle \prod_{\substack {r \mathop = 4 n \mathop + 3 \\ \text {$r$ prime} } } \paren {1 - \dfrac 1 {r^2} }^{-1} } }} {{eqn | o = \approx       | r = 0 \cdotp 76422 \, 3653 \ldots }} {{end-eqn}}


Definition:Stolarsky-Harborth Constant

The '''Stolarsky-Harborth constant''' is the lower bound for the number $\beta$ defined as: :$\beta > \dfrac {P_n} {n^{\lg 3} }$ where: :$P_n$ is the number of odd elements in the first $n$ rows of Pascal's triangle :$\lg 3$ denotes the logarithm base $2$ of $3$. Its value is given by: :$\beta \approx 0 \cdotp 81255 \, 65590 \, 160063 \, 8769 \ldots$ {{OEIS|A077464}}


Definition:Lehmer's Polynomial

'''Lehmer's polynomial''' is the expression: :$x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$


Definition:Salem Number

A '''Salem number''' is a real algebraic integer $\alpha$ such that: :$\alpha > 1$ : its conjugate roots all have absolute value no greater than $1$, at least one of which has absolute value exactly $1$.


Definition:Wilbraham-Gibbs Constant

The '''Wilbraham-Gibbs constant''' is the real number defined as: {{begin-eqn}} {{eqn | l = G'       | r = \int_0^\pi \dfrac {\sin u} u \rd u       | c =  }} {{eqn | o = \approx       | r = 1 \cdotp 85193 \, 7052 \ldots       | c =  }} {{end-eqn}} {{OEIS|036792}}


Definition:Covertex

=== Covertex of Ellipse === Let $K$ be an ellipse. {{:Definition:Covertex of Ellipse}} === Covertex of Hyperbola === Let $K$ be a hyperbola. {{:Definition:Covertex of Hyperbola}}


Definition:Kepler's Equation

Consider a system consisting of two bodies $B_1$ and $B_2$ in elliptical orbit around each other. Let $e$ denote the eccentricity of that orbit. '''Kepler's equation''' is the equation that describes the relation between the mean anomaly and eccentric anomaly of $B_1$ with respect to $B_2$: :$M = E - e \sin E$ where: :$M$ is the mean anomaly :$E$ is the eccentric anomaly. {{NamedforDef|Johannes Kepler|cat = Kepler}}


Definition:Lemniscate of Bernoulli/Geometric Definition

Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$. The '''lemniscate of Bernoulli''' is the locus of points $M$ in the plane such that: :$P_1 M \times P_2 M = a^2$


Definition:Lemniscate of Bernoulli/Cartesian Definition

The '''lemniscate of Bernoulli''' is the curve defined by the Cartesian equation: :$\paren {x^2 + y^2}^2 = 2 a^2 \paren {x^2 - y^2}$


Definition:Lemniscate of Bernoulli/Polar Definition

The '''lemniscate of Bernoulli''' is the curve defined by the polar equation: :$r^2 = 2 a^2 \cos 2 \theta$


Definition:Lemniscate of Bernoulli/Parametric Definition

The '''lemniscate of Bernoulli''' is the curve defined by the parametric equation: :$\begin{cases} x = \dfrac {a \sqrt 2 \cos t} {\sin^2 t + 1} \\ y = \dfrac {a \sqrt 2 \cos t \sin t} {\sin^2 t + 1} \end{cases}$


Definition:Lemniscate of Bernoulli/Cartesian Definition/Also defined as

Some sources define the Cartesian equation for the '''lemniscate of Bernoulli''' as: :$\paren {x^2 + y^2}^2 = a^2 \paren {x^2 - y^2}$ which is the same but for a scale factor: :620px


Definition:Lemniscate of Bernoulli/Polar Definition/Also defined as

Some sources define the polar equation for the '''lemniscate of Bernoulli''' as: :$r^2 = a^2 \cos 2 \theta$ which is the same but for a scale factor: :620px


Definition:Twin Primes Constant/Also defined as

{{AuthorRef|François Le Lionnais}} and {{AuthorRef|Jean Brette}}, in their {{BookLink|Les Nombres Remarquables|François Le Lionnais}} of $1983$, define the '''twin primes constant''' as: {{begin-eqn}} {{eqn | l = \Pi_2       | o = :=       | r = 2 \displaystyle \prod_{\substack {p \mathop \ge 3 \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 {\paren {p - 1}^2} } }} {{eqn | o = \approx       | r = 1 \cdotp 32032 \, 36316 \ldots       | c =  }} {{end-eqn}}


Definition:Pisot-Vijayaraghavan Number

Let $\alpha$ be a real algebraic integer greater than $1$. Let $\alpha$ be such that the absolute values of its Galois conjugates are all less than $1$. Then $\alpha$ is a '''Pisot-Vijayaraghavan number'''.


Definition:Plastic Constant

The '''plastic constant''' is the real root of the cubic: :$x^3 - x - 1 = 0$ {{explain|There are $3$ roots of this equation -- needs to be established exactly which one we are talking about here.}} Its value is approximately: :$P = 1 \cdotp 32471 \, 795 \ldots$ {{OEIS|A060006}}


Definition:Hermite Constant

Let $n \in \N$ be a natural number. Let $L$ be an integer lattice in Euclidean space $R^n$ unit covolume. That is: :$\map {\operatorname {vol} } {\dfrac {R^n} L} = 1$ Let $\map {\lambda_1} L$ denote the least length of a nonzero element of $L$. Let $\sqrt {\gamma_n}$ be the maximum of $\map {\lambda_1} L$ over all such integer lattices $L$. The '''Hermite constant of dimension $n$''' is the constant $\gamma_n$. {{NamedforDef|Charles Hermite|cat = Hermite}}


Definition:Landau's Problems

'''Landau's problems''' are a set of $4$ (still) unsolved problems about prime numbers listed by {{AuthorRef|Edmund Georg Hermann Landau}} in an address at the $1912$ International Congress of Mathematicians. They are as follows: === Goldbach's Conjecture === {{:Goldbach's Conjecture}} === Twin Prime Conjecture === {{:Twin Prime Conjecture}} === Legendre's Conjecture === {{:Legendre's Conjecture}} === Is there an Infinite Number of Primes of Form $n^2 + 1$? === {{:Is there an Infinite Number of Primes of Form n^2 + 1?}}


Definition:Polar Equation

A '''polar equation''' is an equation defining the locus of a set of points in the polar coordinate plane. Such an equation is generally presented in terms of the variables: :$r$: the radial coordinate :$\theta$: the angular coordinate


Definition:Cartesian Equation

A '''Cartesian equation''' is an equation defining the locus of a set of points in a Cartesian space. Such an equation is generally presented in terms of the variables: :$x$ and $y$ for a $2$-dimensional space :$x$, $y$ and $z$ for a $3$-dimensional space :$x_1, x_2, \ldots, x_n$ for a general $n$-dimensional space


Definition:Lemniscate of Bernoulli/Focus

Each of the two points $P_1$ and $P_2$ can be referred to as a '''focus''' of the lemniscate.


Definition:Cardioid

A '''cardioid''' is an epicycloid with $1$ cusp. :500px


Definition:Trifolium/Cosine Form

The '''trifolium''' can be defined by the polar equation in the form: :$r = a \cos 3 \theta$ where: :$0 \le \theta \le \pi$ :400px


Definition:Trifolium/Sine Form

The '''trifolium''' can be defined by the polar equation in the form: :$r = a \sin 3 \theta$ where: :$0 \le \theta \le \pi$ :400px


Definition:Rhodonea Curve

A '''rhodonea curve''' is a curve defined by either one of the polar equations: {{begin-eqn}} {{eqn | l = r       | r = a \cos n \theta }} {{eqn | l = r       | r = a \sin n \theta }} {{end-eqn}}


Definition:Rhodonea Curve/Petal

Each of the loops of a '''rhodonea curve''' is referred to as a '''petal'''.


Definition:Quadrifolium/Cosine Form

The '''quadrifolium''' can be defined by the polar equation in the form: :$r = a \cos 2 \theta$ where: :$0 \le \theta \le 2 \pi$ :400px


Definition:Quadrifolium/Sine Form

The '''quadrifolium''' can be defined by the polar equation in the form: :$r = a \sin 2 \theta$ where: :$0 \le \theta \le 2 \pi$ :400px


Definition:Trochoid

Consider a circle $C$ rolling without slipping along a straight line. Consider a point $P$ on the line of a radius of $C$ at a distance $b$ from the center of $C$. The curve traced out by $P$ is called a '''trochoid'''.


Definition:Trochoid/Curtate

Let $b < a$. The curve traced out by $P$ is called a '''curtate trochoid'''. :500px


Definition:Trochoid/Prolate

Let $b > a$. The curve traced out by $P$ is called a '''prolate trochoid'''. :500px


Definition:Folium of Descartes/Cartesian Form

The '''folium of Descartes''' is the locus of the equation: :$x^3 + y^3 - 3 a x y = 0$


Definition:Omega Constant/Decimal Expansion

The decimal expansion of the omega constant $\Omega$ starts: :$0 \cdotp 56714 \, 32904 \, 097838 \, 72999 \, 96866 \, 22 \ldots$


Definition:Folium of Descartes/Parametric Form

The '''folium of Descartes''' is the locus of the equation given in parametric form as: :$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$


Definition:Ovals of Cassini

Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$. The '''ovals of Cassini''' are the loci of points $M$ in the plane such that: :$P_1 M \times P_2 M = b^2$ for a real constant $b$.


Definition:Ovals of Cassini/Shape

When $b > a$, $M$ is in one continuous piece, either oval or bone-shaped. When $b < a$, $M$ is in two separate pieces, each surrounding one of the foci of $M$. When $b = a$, $M$ degenerates into the lemniscate of Bernoulli.


Definition:Ovals of Cassini/Focus

Each of the two points $P_1$ and $P_2$ can be referred to as a '''focus''' of the ovals.


Definition:Sigma-Algebra Generated by Collection of Subsets/Generator

One says that $\GG$ is a '''generator''' for $\map \sigma {\GG}$. Also, elements $G$ of $\GG$ may be called '''generators'''.


Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 1

The '''$\sigma$-algebra generated by $\GG$''', $\map \sigma \GG$, is the smallest $\sigma$-algebra on $X$ that contains $\GG$. That is, $\map \sigma \GG$ is subject to: :$(1): \quad \GG \subseteq \map \sigma \GG$ :$(2): \quad \GG \subseteq \Sigma \implies \map \sigma \GG \subseteq \Sigma$ for any $\sigma$-algebra $\Sigma$ on $X$


Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 2

The '''$\sigma$-algebra generated by $\GG$''', $\map \sigma \GG$, is the intersection of all $\sigma$-algebras on $X$ that contain $\GG$.


Definition:Limaçon of Pascal

Let $C$ be a circle of diameter $a$ with a distinguished point $O$ on the circumference. Let $OQ$ be a chord of $C$. The '''limaçons of Pascal''' are the loci of points $P$ in the plane such that: :$PQ = b$ where: :$OPQ$ is a straight line :$b$ is a real constant.


Definition:Limaçon of Pascal/Shape

Let $L$ denote a limaçon of Pascal. Depending on the value of $b$, the shape of $L$ is as follows: :For $b \ge 2 a$, $L$ is wholly convex. :For $a < b < 2 a$, $L$ has a concavity. :For $b = a$, $L$ degenerates to a cardioid. :For $0 < b < a$, $L$ has a loop inside its generating circle. :For $b = \dfrac a 2$, the internal loop of $L$ passes through the center of the generating circle. :For $b = 0$, $L$ degenerates to a circle. :For $b < 0$, $L$ is the same curve as for $-b$. :200px 160px 125px 108px 96px 90px 74px 70px


Definition:Differential Equation/Solution/Particular Solution

Let $S$ denote the solution set of $\Phi$. A '''particular solution''' of $\Phi$ is the element of $S$, or subset of $S$, which satisfies a particular boundary condition of $\Phi$.


Definition:Convergent Sequence/Normed Vector Space

Let $\tuple {X, \norm {\,\cdot \,}}$ be a normed vector space. Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$. Let $L \in X$.  The sequence $\sequence {x_n}_{n \mathop \in \N}$ '''converges to the limit $L \in X$''' {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall n \in \N: n > N \implies \norm {x_n - L} < \epsilon$


Definition:Compact Space/Normed Vector Space

Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space.  Let $K \subseteq X$. Then $K$ is '''compact''' {{iff}} every sequence in $K$ has a convergent subsequence with limit $L \in K$. That is, if: :$\sequence {x_n}_{n \mathop \in \N} :\forall n \in \N : x_n \in K \implies \exists \sequence {x_{n_k} }_{k \mathop \in \N} : \exists L \in K: \displaystyle  \lim_{k \mathop \to \infty} x_{n_k} = L$


Definition:Producer of Dedekind Cut

Let $\struct {S, \preceq}$ be a totally ordered set. Let $S' \subseteq S$. Let $\tuple {L, R}$ be a Dedekind cut of $S'$. An $\alpha \in S$ is referred to as a '''producer''' of $\tuple {L, R}$ {{iff}}: :$l \preceq \alpha$ for all $l \in L$ :$\alpha \preceq r$ for all $r \in R$.


Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients

A '''linear first order ordinary differential equation with constant coefficients''' is a linear first order ordinary differential equation which is in (or can be manipulated into) the form: :$\dfrac {\d y} {\d x} + a y = \map Q x$ where: :$\map Q x$ is a function of $x$ :$a$ is a constant.


Definition:Complementary Function of Linear First Order ODE With Constant Coefficients

Consider the linear first order ODE with constant coefficients: :$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$ The general solution to the reduced equation: :$\dfrac {\d y} {\d x} + a y = 0$ is the '''complementary function''' of $(1)$. From First Order ODE: $\dfrac {\d y} {\d x} = k y$, the '''complementary function''' of $(1)$ is $C e^{-a x}$.


Definition:Reduced Equation of Linear ODE with Constant Coefficients/First Order

Consider the linear first order ODE with constant coefficients: :$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$ The equation: :$\dfrac {\d y} {\d x} + a y = 0$ is the '''reduced equation''' of $(1)$.


Consider the linear $n$th order ODE with constant coefficients: :$(1): \quad \displaystyle \sum_{k \mathop = 0}^n a_k \dfrac {\d^k y} {d x^k} = \map R x$ The equation: :$\displaystyle \sum_{k \mathop = 0}^n a_k \dfrac {\d^k y} {d x^k} = 0$ is the '''reduced equation''' of $(1)$. === First Order Linear ODE === {{:Definition:Reduced Equation of Linear ODE with Constant Coefficients/First Order}} === Second Order Linear ODE === {{:Definition:Reduced Equation of Linear ODE with Constant Coefficients/Second Order}}


Definition:Reduced Equation of Linear ODE with Constant Coefficients/Second Order

Consider the linear second order ODE with constant coefficients: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + p \dfrac {\d y} {\d x} + q y = \map R x$ The equation: :$\dfrac {\d^2 y} {\d x^2} + p \dfrac {\d y} {\d x} + q y = 0$ is the '''reduced equation''' of $(1)$.


Definition:Linear Second Order ODE with Constant Coefficients

A '''linear second order ODE with constant coefficients''' is a second order ODE which can be manipulated into the form: :$y'' + p y' + q y = \map R x$ where: :$p$ and $q$ are real constants :$\map R x$ is a function of $x$. Thus it is a linear second order ODE: :$y'' + \map P x y' + \map Q x y = \map R x$ where $\map P x$ and $\map Q x$ are constant functions.


Definition:Open Ball/Normed Vector Space

Let $\struct{X, \norm {\,\cdot\,} }$ be a normed vector space. Let $x \in X$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''open $\epsilon$-ball of $x$ in $\struct {X, \norm {\,\cdot\,} }$''' is defined as: :$\map {B_\epsilon} x = \set {y \in X: \norm{x - y} < \epsilon}$


Definition:Open Set/Normed Vector Space

Let $V = \struct{X, \norm {\,\cdot\,} }$ be a normed vector space. Let $U \subseteq X$. Then $U$ is an '''open set in $V$''' {{iff}}: :$\forall x \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} x \subseteq U$ where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$.


Let $V = \struct {X, \norm {\,\cdot\,} }$ be a normed vector space. Let $F \subset X$. ==== Definition 1 ==== {{:Definition:Closed Set/Normed Vector Space/Definition 1}} ==== Definition 2 ==== {{:Definition:Closed Set/Normed Vector Space/Definition 2}}


Definition:P-adic Unit

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ denote the $p$-adic integers. The set of '''$p$-adic units''', denoted $\Z_p^\times$, is the set of invertible elements of $\Z_p$. From P-adic Unit has Norm Equal to One, the set of '''$p$-adic units''' is: :$\Z_p^\times = \set {x \in \Q_p: \norm x_p = 1}$


Definition:Linear nth Order ODE with Constant Coefficients

Let $n \in \Z_{>0}$ be a (strictly) positive integer. A '''linear $n$th order ODE with constant coefficient''' is an ordinary differential equation of order $n$ which can be manipulated into the form: :$\displaystyle \sum_{k \mathop = 0}^n a_k \dfrac {\d^k y} {d x^k} = \map R x$ where $a_k$ is a real constant for all $0 \le k \le n$. That is: :$a_n \dfrac {\d^n y} {d x^n} + a_{n - 1} \dfrac {\d^{n - 1} y} {d x^{n - 1} } + \dotsb + a_1 \dfrac {\d y} {d x} + a_0 y = \map R x$


Definition:Lie Theory

'''Lie theory''' is the study of Lie groups and Lie algebras.