Definition:Simple Harmonic Motion

Consider a physical system $S$ whose motion can be expressed in the form of the following equation: :$x = A \map \sin {\omega t + \phi}$ where $A$ and $\phi$ are constants. Then $S$ is in a state of '''simple harmonic motion'''. === Amplitude === {{:Definition:Simple Harmonic Motion/Amplitude}} === Period === {{:Definition:Simple Harmonic Motion/Period}} === Frequency === {{:Definition:Simple Harmonic Motion/Frequency}}


Definition:Simple Harmonic Motion/Period

The '''period''' $T$ of the motion of $S$ is the time required for one complete cycle: :$T = \dfrac {2 \pi} \omega$


Definition:Simple Harmonic Motion/Frequency

The '''frequency''' $\nu$ of the motion of $S$ is the number of complete cycles per unit time: :$\nu = \dfrac 1 T = \dfrac \omega {2 \pi}$


Definition:Periodic Function/Frequency

The '''frequency''' $\nu$ of $f$ is the reciprocal of the period $L$ of $f$: :$\nu = \dfrac 1 L$


Definition:Fluid

A '''fluid''' is a substance which has no defined shape, but which is free to flow according to external forces. === Ideal Fluid === {{:Definition:Fluid/Ideal}} Category:Definitions/Fluid Mechanics Category:Definitions/Ideals in Physics 36by7c8128e3a1nfmb1k1rs3pv2ei7t


Definition:Medium

A '''medium''' is a fluid $F$ through which a body $B$ can pass. Category:Definitions/Mechanics 49mnpwqf8fu8ddljeprdc05kuq7kusi


Definition:Damping Force

Consider a physical system consisting of a body $B$ passing through a medium $M$. In order for $B$ to move, it needs to push $M$ out of the way to make room for it. Hence $M$ exerts a force upon $B$ which opposes the motion of $B$. This force opposing the motion of $B$ is referred to as a '''damping force'''.


Definition:Viscosity

The viscosity of a fluid is a measure of how large a damping force it exerts upon a body moving through it. {{stub|Establish its formal definition: how great the damping force against how large a body at what velocity. Fluid mechanics has not really been started on this site yet. I have a degree semester to infodump.}}


Definition:Fluid/Ideal

An '''ideal fluid''' is a fluid which delivers a damping force whose magnitude is proportional to the velocity of a body passing through it.


Definition:Overdamped

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b > a$, so that the solution of $(1)$ is in the form: :$x = C_1 e^{m_1 t} + C_2 e^{m_1 t}$ for $m_1, m_2 < 0$. Then $S$ is described as being '''overdamped'''. :500px


Definition:Critically Damped

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b = a$, so that the solution of $(1)$ is in the form: :$x = C_1 e^{-a t} + C_2 t e^{-a t}$ Then $S$ is described as being '''critically damped'''. :500px


Definition:Underdamped

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so that the solution of $(1)$ is in the form: :$x = e^{-b t} \paren {C_1 \cos \alpha t + C_2 \sin \alpha t}$ where $\alpha = \sqrt {a^2 - b^2}$. Then $S$ is described as being '''underdamped'''. :600px


Definition:Period of Underdamped Oscillation

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. :600px While the behaviour of $S$ is not strictly speaking periodic, its oscillations can be defined to have a "period" as follows: Let $T$ be the smallest value of $t$ such that: :$x = 0$ :$x' < 0$ Then $T$ is the '''period''' of the oscillation of $S$.


Definition:Amplitude of Underdamped Oscillation

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $(1)$ be subject to the initial conditions: :$x = x_0$ at $t = 0$ :$x' = 0$ at $t = 0$ Let $b < a$, so as to make $S$ underdamped. :600px The '''amplitude''' of the oscillation of $S$ is defined as: :$A = \dfrac {x_0 a} {\sqrt {a^2 - b^2} } e^{-b t}$


Definition:Natural Frequency

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\mathrm d^2 x} {\mathrm d t^2} + 2 b \dfrac {\mathrm d x} {\mathrm d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. :600px The '''natural frequency''' $\nu$ of $S$ is defined as: :$\nu = \dfrac 1 T$ where $T$ is the period of the oscillation of $S$.


Definition:Free Vibration

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the motion of $S$ is referred to as '''free vibration''' of $S$. :600px


Definition:Steady-State/First Order ODE

Consider the Decay Equation: :$\dfrac {\d y} {\d x} = k \paren {y_a - y}$ where: :$k \in \R: k > 0$ :$y = y_0$ at $x = 0$ which has the particular solution: :$(1): \quad y = y_a + \paren {y_0 - y_a} e^{-k x}$ The term $y_a$ is known as the '''steady-state''' component of $(1)$.


Definition:Steady-State/Second Order ODE

Consider the second order ODE: :$\dfrac {\mathrm d^2 y} {\mathrm d x^2} + 2 b \dfrac {\mathrm d y} {\mathrm d x} + a^2 x = K \cos \omega x$ where: :$K \in \R: k > 0$ :$a, b \in \R_{>0}: b < a$ which has the general solution: :$(1): \quad y = e^{-b x} \left({C_1 \cos \alpha x + C_2 \sin \alpha x}\right) + \dfrac K {\sqrt {4 b^2 \omega^2 + \left({a^2 - \omega^2}\right)^2} } \cos \left({\omega x - \phi}\right)$ where: :$\alpha = \sqrt {a^2 - b^2}$ :$\phi = \arctan \left({\dfrac {2 b \omega} {a^2 - \omega^2} }\right)$ The term $\dfrac K {\sqrt {4 b^2 \omega^2 + \left({a^2 - \omega^2}\right)^2} } \cos \left({\omega x - \phi}\right)$ is known as the '''steady-state''' component of $(1)$.


Definition:Resonance

Consider a physical system $S$ whose behaviour is defined by the second order ODE: :$\dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$ where: :$K \in \R: k > 0$ :$a, b \in \R_{>0}: b < a$ which has the general solution: :$(1): \quad y = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \sin \alpha x} + \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}$ where: :$\alpha = \sqrt {a^2 - b^2}$ :$\phi = \map \arctan {\dfrac {2 b \omega} {a^2 - \omega^2} }$ Let $\omega$ be such that the amplitude of the steady-state component of $(1)$ is at a maximum. Then $S$ is said to be in '''resonance'''.


Definition:Forced Vibration

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\mathrm d^2 x} {\mathrm d t^2} + 2 b \dfrac {\mathrm d x} {\mathrm d t} + a^2 x = K \cos \omega x$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the steady-state motion of $S$ is referred to as '''forced vibration''' of $S$. === Forcing Frequency === {{:Definition:Forced Vibration/Forcing Frequency}}


Definition:Forced Vibration/Forcing Frequency

The frequency: :$\dfrac \omega {2 \pi}$ is known as the '''forcing frequency''' of $S$.


Definition:Resonance Frequency

Consider a second order ODE in the form: :$\dfrac {\mathrm d^2 x} {\mathrm d t^2} + 2 b \dfrac {\mathrm d x} {\mathrm d t} + a^2 x = K \cos \omega x$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Let the forcing frequency $\omega$ be such that $S$ is in resonance. Then $\omega$ is known as the '''resonance frequency'''.


Definition:Logarithmic Decrement

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\mathrm d^2 x} {\mathrm d t^2} + 2 b \dfrac {\mathrm d x} {\mathrm d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. :600px Let $T$ be the period of oscillation of $S$. Let $x_1$ and $x_2$ be successive local maxima of $x$. From Ratio of Successive Local Maxima for Underdamped Free Vibration: :$\dfrac {x_1} {x_2} = e^{b T}$ The '''logarithmic decrement''' of $S$ is defined as: :$\ln \left({\dfrac {x_1} {x_2} }\right) = b T$


Definition:Central Force

Consider a particle $p$ of mass $m$ moving in the plane under the influence of a force $\mathbf F$. Let the position of $p$ at time $t$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$. Let $\mathbf F$ be expressed as: :$\mathbf F = F_r \mathbf u_r + F_\theta \mathbf u_\theta$ where: :$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$ :$F_r$ and $F_\theta$ are the magnitudes of the components of $\mathbf F$ in the directions of $\mathbf u_r$ and $\mathbf u_\theta$ respectively. :600px Let $\mathbf F$ have no component perpendicular to $\mathbf u_r$. That is, such that $F_\theta = 0$. Then $\mathbf F$ is referred to as a '''central force'''.


Definition:Polish Space

A '''Polish space''' is a topological space which is separable and metrizable, with a metric that makes it a complete metric space. Category:Definitions/Descriptive Set Theory Category:Definitions/Topology Category:Definitions/Metric Spaces gdlarwvzk7c4gy9pd4q2wo64zxxup1s


Definition:Tetractys

The '''tetractys''' is a central concept in the mystical beliefs of the Pythagoreans. It consists of $10$ points arranged in a triangle: :300px based on the notions that: :$1$ represents the point :$2$ represents the line :$3$ represents the surface :$4$ represents the solid :$1 + 2 + 3 + 4 = 10$ and therefore: :$10$ represents the universe.


Definition:Numerology/Historical Note

The concept of '''numerology''' seems to have begun with {{AuthorRef|Pythagoras of Samos|Pythagoras}} and the Pythagorean school, continuing with the neo-Pythagoreans. {{AuthorRef|Plato}} adopted the ideas, and from there they were passed on through the emerging Christian tradition into the heart of the philosophy of Western civilization, where it still exerts a powerful influence.


Definition:Numerology

'''Numerology''' is a philosophical belief structure, borrowed from the Pythagoreans, that considers numbers to possess mystical properties. At its heart is the pseudo-science of '''gematria''' or '''arithmology''': the process of translating linguistic text (in particular a name) into numerical form, and thence to interpret the significance of that text from the mystical power of that particular number. The techniques of that translation differ between practitioners, and in many cases the same practitioner may use more than one such technique, thereby allowing for multiple different numerical interpretations of any particular given text. Hence a popular criticism of numerology is to accuse practitioners of deciding what they want the answer to be, and then juggling the numbers in order to achieve that particular effect. The practice may have originated from the Greek numerals of the Classical period, in which the natural ordering of the letters of the Greek alphabet were assigned numerical values. It should not be necessary to place much emphasis on the fact that '''numerology''' is not a branch of mathematics.


Definition:Plato's Academy

'''Plato's Academy''' was the second university in the Western world, after the Pythagorean school. It placed great emphasis on the study of mathematics.


Definition:Mapping Preserves Supremum

Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be ordered sets. Let $f: S_1 \to S_2$ be a mapping.


Definition:Mapping Preserves Supremum/Subset

Let $F$ be a subset of $S_1$. $f$ '''preserves supremum of''' $F$ {{iff}} :$F$ admits a supremum in $\left({S_1, \preceq_1}\right)$ implies :: $f^\to \left({F}\right)$ admits a supremum in $\left({S_2, \preceq_2}\right)$ and $\sup \left({f^\to\left({F}\right)}\right) = f \left({\sup F}\right)$


Definition:Mapping Preserves Supremum/All

$f$ '''preserves all suprema''' {{iff}} :for every subset $F$ of $S_1$, $f$ preserves the supremum of $F$.


Definition:Mapping Preserves Supremum/Join

$f$ '''preserves join''' {{iff}} :for every pair of elements $x, y$ of $S_1$, $f$ preserves the supremum of $\left\{ {x, y}\right\}$.


Definition:Mapping Preserves Supremum/Finite

$f$ '''preserves finite suprema''' {{iff}} :for every finite subset $F$ of $S_1$, $f$ preserves the supremum of $F$.


Definition:Mapping Preserves Supremum/Directed

$f$ '''preserves directed suprema''' {{iff}} :for every directed subset $F$ of $S_1$, $f$ preserves the supremum of $F$.


Definition:Mapping Preserves Infimum/Subset

Let $F$ be a subset of $S_1$. $f$ '''preserves infimum of''' $F$ {{iff}} :$F$ admits a infimum in $\left({S_1, \preceq_1}\right)$ implies :: $f^\to \left({F}\right)$ admits an infimum in $\left({S_2, \preceq_2}\right)$ and $\inf \left({f^\to \left({F}\right)}\right) = f \left({\inf F}\right)$


Definition:Mapping Preserves Infimum/All

$f$ '''preserves all infima''' {{iff}} :for every subset $F$ of $S_1$, $f$ preserves the infimum of $F$.


Definition:Mapping Preserves Infimum/Meet

$f$ '''preserves meet''' {{iff}} :for every pair of elements $x, y$ of $S_1$, $f$ preserves the infimum of $\left\{ {x, y}\right\}$.


Definition:Mapping Preserves Infimum/Finite

$f$ '''preserves finite infima''' {{iff}} :for every finite subset $F$ of $S_1$, $f$ preserves the infimum of $F$.


Definition:Mapping Preserves Infimum


Definition:Mapping Preserves Infimum/Filtered

$f$ '''preserves filtered infima''' {{iff}} :for every filtered subset $F$ of $S_1$, $f$ preserves the infimum of $F$.


Definition:Conic Section/Focus-Directrix Property/Historical Note

The focus-directrix definition of a conic section was first documented by {{AuthorRef|Pappus of Alexandria}}. It appears in his {{BookLink|Collection|Pappus of Alexandria}}. As he was scrupulous in documenting his sources, and he gives none for this construction, it can be supposed that it originated with him.


Definition:Conic Section/Intersection with Cone/Tilting Angle

The inclination $\phi$ of the slicing plane with the axis of $C$ is known as the '''tilting angle''' of $D$ for $K$.


Definition:Conic Section/Intersection with Cone/Slicing Plane

The plane $D$ is known as the '''slicing plane''' of $C$ for $K$.


Definition:Vertical Line/Definition 1

A line is '''vertical''' {{iff}} it is aligned perpendicular to the horizon.


Definition:Vertical Line/Definition 2

A line is '''vertical''' {{iff}} it is parallel to the path taken by a body released from rest in a constant gravitational field.


Definition:Vertical

=== Vertical Line === {{:Definition:Vertical Line/Definition 2}} Hence the adjective '''vertically''', meaning '''in a vertical line'''. === Vertical Plane === {{:Definition:Vertical Plane}}


Definition:Vertical Plane

A plane is '''vertical''' {{iff}} it contains a vertical line.


Definition:Horizontal

=== Horizontal Line === {{:Definition:Horizontal Line/Definition 2}} Hence the adjective '''horizontally''', meaning '''in a horizontal line'''. === Horizontal Plane === {{:Definition:Horizontal Plane}}


Definition:Horizontal Line/Definition 2

A line is '''horizontal''' {{iff}} it is aligned parallel to the horizon:


Definition:Horizontal Line/Definition 1

A line is '''horizontal''' {{iff}} it is perpendicular to a given vertical line.


Definition:Horizontal Line/Definition 3

A line is '''horizontal''' {{iff}} it can be embedded within a horizontal plane.


Definition:Horizontal Plane

A plane is '''horizontal''' {{iff}} it is perpendicular to a given vertical line.


Definition:Conic Section/Focus-Directrix Property/Circle

:400px It is not possible to define the circle using the focus-directrix property. This is because as the eccentricity $e$ tends to $0$, the distance $p$ from $P$ to $D$ tends to infinity. Thus a circle can in a sense be considered to be a degenerate ellipse whose foci are at the same point, that is, the center of the circle.


Definition:Circle/Focus

As a circle cannot be specified using the focus-directrix property, the focus cannot be defined using that technique. Consider an ellipse $K$ with eccentricity $e$. Let $e$ tend to zero. Then as $e$ becomes smaller, the foci of $K$ become closer together. In the limit, the foci coincide. Thus the focus of the circle can be understood as being its center.


Definition:Acoustics

'''Acoustics''' is the branch of physics which studies the properties of sound.


Definition:Optics

'''Optics''' is the branch of physics which studies the properties of light.


Definition:Light

'''Light''' is electromagnetic radiation whose wavelength is between approximately $700 \, \text{nm}$ and $390 \, \text{nm}$. It is characterised by its ability to stimulate the visual senses of humans. Category:Definitions/Optics Category:Definitions/Physics 4rebzuewkr183shrxfw4160ir9ltfv2


Definition:Sound (Physics)

'''Sound''' is the vibration of an elastic medium whose frequency ranges between approximately $20 \, \text {Hz}$ and $20 \, \text {kHz}$. It is characterised by the ability of humans to detect it with their aural sensory systems.


Definition:Filter in Ordered Set

Let $\left({S, \preceq}\right)$ be a preordered set. Let $F$ be a subset of $S$. $F$ is '''filter''' in $\left({S, \preceq}\right)$ {{iff}} :$F$ is non-empty filtered and upper.


Definition:Ideal in Ordered Set

Let $\left({S, \preceq}\right)$ be a preordered set. Let $I$ be a subset of $S$. $I$ is '''ideal''' in $\left({S, \preceq}\right)$ {{iff}}: :$I$ is non-empty directed and lower.


Definition:Logarithmic Spiral

The '''logarithmic spiral''' is the locus of the equation expressed in Polar coordinates as: :$r = a e^{b \theta}$ :600px


Definition:Spiral

A '''spiral''' is a plane curve, or part of a plane curve, which can be expressed in polar coordinates in the form: :$r = \map f \theta$ where $f$ is either (strictly) increasing or (strictly) decreasing. Hence a '''spiral''' is a plane curve which emanates from a central point, getting progressively farther away (or closer in) as it revolves around that point.


Definition:Gabriel's Horn

'''Gabriel's Horn''' is the name given to the solid of revolution formed by rotating about the $x$-axis the curve: :$y = \dfrac 1 x$ :400px


Definition:Hydrostatics

'''Hydrostatics''' is the branch of physics concerned with the study of forces on fluids at rest.


Definition:Tautochrone

Let $A$ and $B$ be points in space that are neither on the same horizontal line nor on the same vertical line. A '''tautochrone''' is the curve into which a wire is to be shaped so that a bead sliding down it (without friction) takes the same time to reach the lowest point independently of where on that wire it is released from rest.


Definition:Evolute

Consider a curve $C$ embedded in a plane. The '''evolute''' of $C$ is the locus of the centers of curvature of each point on $C$.


Definition:Involute

Consider a curve $C$ embedded in a plane. Imagine an ideal (zero thickness) cord $K$ wound round $C$. The '''involute''' of $C$ is the locus of the end of $K$ as it is unwound from $C$.


Definition:Abscissa

Consider the graph $y = \map f x$ of a real function $f$ embedded in a Cartesian plane. :500px The $x$ coordinate of a point $P = \tuple {x, y}$ on $f$ is known as the '''abscissa''' of $P$.


Definition:Ordinate

Consider the graph $y = \map f x$ of a real function $f$ embedded in a Cartesian plane. :500px The $y$ coordinate of a point $P = \tuple {x, y}$ on $f$ is known as the '''ordinate''' of $P$.


Definition:Lemniscate of Bernoulli/Linguistic Note

The word '''lemniscate''' comes from the Latin word '''lemniscus''', which means '''pendant ribbon'''. The word may ultimately derive from the Latin '''lēmniscātus''', which means '''decorated with ribbons'''. This may in turn come from the ancient Greek island of '''Lemnos''' where ribbons were worn as decorations.


Definition:Integer Partition/Part

In a '''partition''' of a (strictly) positive integer, one of the summands in that partition is referred to as a '''part'''.


Definition:Integer Partition

A '''partition''' of a (strictly) positive integer $n$ is a way of writing $n$ as a sum of (strictly) positive integers.


Definition:Integer Partition/Partition Function

The '''partition function''' $p: \Z_{>0} \to \Z_{>0}$ is defined as: :$\forall n \in \Z_{>0}: \map p n =$ the number of partitions of the (strictly) positive integer $n$.


Definition:Genus of Surface

Let $S$ be a surface. Let $G = \left({V, E}\right)$ be a graph which is embedded in $S$. Let $G$ be such that each of its faces is a simple closed curve. Let $\chi \left({G}\right) = v - e + f = 2 - 2 p$ be the Euler characteristic of $G$ where: : $v = \left|{V}\right|$ is the number of vertices : $e = \left|{E}\right|$ is the number of edges : $f$ is the number of faces. Then $p$ is known as the '''genus''' of $S$.


Definition:Plane Geometry/Historical Note

The first definitive work on Plane Geometry was {{ElementsLink}}.


Definition:Hydrodynamics

'''Hydrodynamics''' is the branch of physics concerned with the study of fluids in motion.


Definition:Analytical Mechanics

'''Analytical mechanics''' is a subfield of mathematical physics which uses techniques of analysis, in particular the calculus of variations, to solve problems in mechanics. Hence, instead of solving equations in vector quantities, it involves solutions of differential equations of scalar quantities.


Definition:Metric System

The '''metric system''' is the colloquial term for the system of measurement based on the metre. Its main characteristic is that its units are constructed on a decimal system.


Definition:Decimal System

A '''decimal system''' is a system of measurement in which the standard multiples and fractions of the units of measurement are powers of $10$.


Definition:Up-Complete

Let $\left({S, \precsim}\right)$ be a preordered set. Then $\left({S, \precsim}\right)$ is '''up-complete''' {{iff}}: :every directed subset of $S$ admits a supremum in $\left({S, \precsim}\right)$.


Definition:Definite Integral/Integrand

In the expression for the definite integral: :$\displaystyle \int_a^b f \left({x}\right) \, \mathrm d x$ the function $f$ is called the '''integrand'''.


Definition:Method of Least Squares (Approximation Theory)

Let there be a set of points $\set {\tuple {x_k, y_k}: k \in \set {1, 2, \ldots, n} }$ plotted on a Cartesian $x y$ plane which correspond to measurements of a physical system. Let it be required that a straight line is to be fitted to the points. The '''method of least squares''' is a technique of producing a straight line of the form $y = m x + c$ such that: :the points $\set {\tuple {x_k', y_k'}: k \in \set {1, 2, \ldots, n} }$ are on the line $y = m x + c$ :$\forall k \in \set {1, 2, \ldots, n}: y_k' = y_k$ :$\displaystyle \sum_n \paren {x_k' = x_k}^2$ is minimised.


Definition:Gaussian Distribution

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Then $X$ has a '''Gaussian distribution''' {{iff}} the probability density function of $X$ is: :$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$ for $\mu \in \R, \sigma \in \R_{> 0}$. This is written:  :$X \sim \Gaussian \mu {\sigma^2}$


Definition:Stirling Numbers/Karamata Notation

The notation $\displaystyle {n \brack k}$ and $\displaystyle {n \brace k}$ for  '''Stirling numbers''' is known as '''Karamata notation'''.


Definition:Stirling Numbers of the First Kind/Notation

The notation $\displaystyle {n \brack k}$ for the unsigned Stirling numbers of the first kind is that proposed by {{AuthorRef|Jovan Karamata}} and publicised by {{AuthorRef|Donald Ervin Knuth|Donald E. Knuth}}. The notation $\map s {n, k}$ for the signed Stirling numbers of the first kind is similar to variants of that sometimes given for the unsigned. Usage is inconsistent in the literature.


{{:Definition:Stirling Numbers of the First Kind/Unsigned/Definition 1}}


{{:Definition:Stirling Numbers of the First Kind/Signed/Definition 1}}


Definition:Stirling Numbers of the Second Kind/Notation

The notation $\displaystyle {n \brace k}$ for Stirling numbers of the second kind is that proposed by {{AuthorRef|Jovan Karamata}} and publicised by {{AuthorRef|Donald Ervin Knuth|Donald E. Knuth}}. Other notations exist. Usage is inconsistent in the literature.


Definition:Integer Sequence

An '''integer sequence''' is a sequence (usually infinite) whose codomain is the set of integers $\Z$.


Definition:Hypergeometric Series

A '''hypergeometric series''' is a power series: :$\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_{n \mathop \ge 0} \beta_n z^n$ in which the ratio of successive coefficients is a rational function of $n$: :$\dfrac {\beta_{n + 1} } {\beta_n} = \dfrac {\map A n} {\map B n}$ where $\map A n$ and $\map B n$ are polynomials in $n$. {{expand|Needs to include a derivation of the notation ${}_p F_q \tuple {a_1, \ldots, a_p; b_1, \ldots, b_q; z}$}} {{questionable|{{BookLink|The Penguin Dictionary of Mathematics|David Nelson|ed = 4th|edpage = Fourth Edition}} defines this where the coefficient $\beta_n$ is specifically $\dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} n!}$}}


Definition:Geodesy

'''Geodesy''' is the branch of geometry which studies the shape of the surface of the Earth.


Definition:First Fundamental Form

Let $S$ be a surface of a three-dimensional euclidean space. Let $p$ be a point of $S$ and $T_pS$ be the tangent space to $S$ at the point $p$. The '''first fundamental form''' is the bilinear form: :$\operatorname I: T_p S \times T_p S \longrightarrow \R$ induced from the dot product of $\R^3$:  :$\map {\operatorname I} {x, y} = \innerprod x y$ The '''first fundamental form''' is a way to calculate the length of a given line $C \subset S$ or the area of a given bounded region $R \subset S$. {{Disambiguate|Definition:Region}} {{expand}}


Definition:Gaussian Curvature

The '''Gaussian curvature''' $\Kappa$ of a surface at a point is the product of the principal curvatures, $\kappa_1$ and $\kappa_2$, at the given point: :$\Kappa = \kappa_1 \kappa_2$


Definition:Geodesic Curve

Let $A$ and $B$ be points on a surface $S$. A '''geodesic curve''' between $A$ and $B$ is the line between $A$ and $B$ completely embedded in $S$ whose length is the smallest of all lines between $A$ and $B$.


Definition:Integral Curvature

Let $ABC$ be a geodesic triangle on a surface $S$. The '''integral curvature''' of $ABC$ is given by: :$\displaystyle \int_{ABC} \Kappa \, \mathrm d a$ where $\Kappa$ is the Gaussian curvature at each point on $ABC$.


Definition:Geodesic Triangle

Let $A$, $B$ and $C$ be points on a surface $S$. Let $AB$, $BC$ and $CA$ be geodesic curves between $A$ and $B$, $B$ and $C$, and $C$ and $A$ respectively. The figure enclosed by $AB$, $BC$ and $CA$ is known as a '''geodesic triangle''', and can be denoted $ABC$.


Definition:Biquadratic Residue

Let $p$ be an odd prime. Let $a \in \Z$ be an integer such that $a \not \equiv 0 \pmod p$. Then $a$ is a '''biquadratic residue of $p$''' {{iff}} $x^4 \equiv a \pmod p$ has a solution. That is, {{iff}}: : $\exists x \in \Z: x^4 \equiv a \pmod p$


Definition:Gaussian Prime

A '''Gaussian prime''' is a Gaussian integer which has exactly $4$ divisors which are themselves Gaussian integers.


Definition:Harmonic Function

A '''harmonic function''' is a is a twice continuously differentiable function $f: U \to \R$ (where $U$ is an open set of $\R^n$) which satisfies Laplace's equation: :$\dfrac {\partial^2 f} {\partial {x_1}^2} + \dfrac {\partial^2 f} {\partial {x_2}^2} + \cdots + \dfrac {\partial^2 f} {\partial {x_n}^2} = 0$ everywhere on $U$. This is usually written as: :$\nabla^2 f = 0$ or: :$\Delta f = 0$


Definition:Galois Connection

Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered sets. Let $g: S \to T$, $d: T \to S$ be mappings. Then $\struct {g, d}$ is a '''Galois connection''' {{iff}}: :$g$ and $d$ are increasing mappings and ::$\forall s \in S, t \in T: t \precsim \map g s \iff \map d t \preceq s$ $g$ is '''upper adjoint''' and $d$ is '''lower adjoint''' of a Galois connection. {{definition wanted|monotone, antitone, closure operator, closed}} {{NamedforDef|Évariste Galois|cat = Galois}}


Definition:Taylor Series/Remainder

Consider the Taylor series expansion $\map T {\map f \xi}$ of $f$ about the point $\xi$: :$\displaystyle \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$ Let $\map {T_n} {\map f \xi}$ be the Taylor polynomial: :$\displaystyle \sum_{n \mathop = 0}^n \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$ for some $n \in \N$. The difference: :$\displaystyle \map {R_n} x = \map f x - \map {T_n} {\map f \xi} = \int_\xi^x \map {f^{\paren {n + 1} } } t \dfrac {\paren {x - t}^n} {n!} \rd t$ is known as the '''remainder of $\map T {\map f \xi}$''' at $x$.


Definition:Taylor Series/Remainder/Lagrange Form

The '''Lagrange form''' of the remainder $R_n$ is given by: :$R_n = \dfrac {\map {f^{\paren {n + 1} } } {x^*} } {\paren {n + 1}!} \paren {x - \xi}^{n + 1}$ where $x^* \in \openint \xi x$.


Definition:Taylor Series/Remainder/Cauchy Form

The '''Cauchy form''' of the remainder $R_n$ is given by: :$R_n = \dfrac {\paren {x - \eta}^n} {n!} \paren {x - \xi} \map {f^{\paren {n + 1} } } \eta$ where $\eta \in \closedint \xi x$.


Definition:Pentagonal Number/Examples

The first few pentagonal numbers are as follows: :700px === Sequence of Pentagonal Numbers === {{:Definition:Pentagonal Number/Sequence}}


Definition:Polygonal Number/Degenerate Case

Consider the polygonal number $P \left({2, n}\right)$ when $k = 2$. In this case, the polygon degenerates into a straight line, and the recurrence formula becomes: :$P \left({2, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({2, n-1}\right) + 0 \times \left({n-1}\right) + 1 & : n > 0 \end{cases}$ Hence: :$P \left({2, n}\right) = P \left({2, n-1}\right) + 1$ and the sequence goes: :$0, 1, 2, 3, \ldots$ which is of course the natural numbers.


Definition:Triangular Number/Definition 2

:$\displaystyle T_n = \sum_{i \mathop = 1}^n i = 1 + 2 + \cdots + \left({n-1}\right) + n$


Definition:Triangular Number/Definition 1

:$T_n = \begin{cases} 0 & : n = 0 \\ n + T_{n-1} & : n > 0 \end{cases}$


Definition:Triangular Number/Definition 3

:$\forall n \in \N: T_n = P \left({3, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({3, n - 1}\right) + \left({n - 1}\right) + 1 & : n > 0 \end{cases}$ where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.


Definition:Polygonal Number/Examples

=== Triangular Numbers === When $k = 3$, the recurrence relation is: {{:Definition:Triangular Number/Definition 3}} See Triangular Number. === Square Numbers === When $k = 4$, the recurrence relation is: {{:Definition:Square Number/Definition 4}} See Square Number. Square numbers are of course better known as: :$S_n = n^2$ === Pentagonal Numbers === When $k = 5$, the recurrence relation is: {{:Definition:Pentagonal Number/Definition 3}} See Pentagonal Number. === Hexagonal Numbers === When $k = 6$, the recurrence relation is: {{:Definition:Hexagonal Number/Definition 3}} See Hexagonal Number. === Heptagonal Numbers === When $k = 7$, the recurrence relation is: {{:Definition:Heptagonal Number/Definition 3}} See Heptagonal Number. === Octagonal Numbers === When $k = 8$, the recurrence relation is: {{:Definition:Octagonal Number/Definition 3}} See Octagonal Number.


Definition:Pentagonal Number/Definition 1

:$P_n = \begin{cases} 0 & : n = 0 \\ P_{n-1} + 3 n - 2 & : n > 0 \end{cases}$


Definition:Pentagonal Number/Definition 2

:$\displaystyle P_n = \sum_{i \mathop = 1}^n \left({3 i - 2}\right) = 1 + 4 + \cdots + \left({3 \left({n - 1}\right) - 2}\right) + \left({3 n - 2}\right)$


Definition:Pentagonal Number/Definition 3

:$\forall n \in \N: P_n = \map P {5, n} = \begin {cases} 0 & : n = 0 \\ \map P {5, n - 1} + 3 \paren {n - 1} + 1 & : n > 0 \end {cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.


Definition:Hexagonal Number/Definition 1

:$H_n = \begin{cases} 0 & : n = 0 \\ H_{n-1} + 4 \left({n-1}\right) + 1 & : n > 0 \end{cases}$


Definition:Hexagonal Number/Definition 2

:$\displaystyle H_n = \sum_{i \mathop = 1}^n \left({4 \left({i - 1}\right) + 1}\right) = 1 + 5 + \cdots + \left({4 \left({n-2}\right) + 1}\right) + \left({4 \left({n - 1}\right) + 1}\right)$


Definition:Hexagonal Number/Definition 3

:$\forall n \in \N: H_n = P \left({6, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({6, n - 1}\right) + 4 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$ where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.


Definition:Square Number/Definition 2

:$S_n = \begin {cases} 0 & : n = 0 \\ S_{n - 1} + 2 n - 1 & : n > 0 \end {cases}$


Definition:Square Number/Definition 1

An integer $n$ is classified as a '''square number''' {{iff}}: :$\exists m \in \Z: n = m^2$ where $m^2$ denotes the integer square function. ==== Euclid's Definition ==== {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book VII/18 - Square Number}}'' {{EuclidDefRefNocat|VII|18|Square Number}}


Definition:Square Number/Definition 3

:$\displaystyle S_n = \sum_{i \mathop = 1}^n \paren {2 i - 1} = 1 + 3 + 5 + \cdots + \paren {2 n - 1}$


Definition:Square Number/Definition 4

:$\forall n \in \N: S_n = \map P {4, n} = \begin{cases} 0 & : n = 0 \\ \map P {4, n - 1} + 2 \paren {n - 1} + 1 & : n > 0 \end{cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.


Definition:Triangular Number/Sequence

The sequence of triangular numbers, for $n \in \Z_{\ge 0}$, begins: :$0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, \ldots$


Definition:Pentagonal Number/Sequence

The sequence of pentagonal numbers, for $n \in \Z_{\ge 0}$, begins: :$0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, \ldots$


Definition:Hexagonal Number/Sequence

The sequence of hexagonal numbers, for $n \in \Z_{\ge 0}$, begins: :$0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, \ldots$


Definition:Odd Integer/Definition 1

An integer $n \in \Z$ is '''odd''' {{iff}} it is not divisible by $2$. That is, {{iff}} it is not even.


Definition:Odd Integer/Definition 2

An integer $n \in \Z$ is '''odd''' {{iff}}: :$\exists m \in \Z: n = 2 m + 1$


Definition:Convex Polyhedron

Let $P$ be a polyhedron. $P$ is a '''convex polyhedron''' {{iff}}: :For all points $A$ and $B$ located inside $P$, the line $AB$ is also inside $P$.


Definition:Neusis Construction/Historical Note

Many of the constructions performed by {{AuthorRef|Archimedes of Syracuse|Archimedes}} were '''neusis constructions'''. Some sources suggest that {{AuthorRef|Oenopides of Chios}} (c. $440$ BCE) was the first to rank '''compass and straightedge constructions''' above '''neusis constructions''' in importance. It may have been {{AuthorRef|Hippocrates of Chios}} (c. $430$ BCE) who publicised the principle to use a '''compass and straightedge construction''' in preference to a '''neusis construction''' whenever possible. {{ElementsLink}} uses '''compass and straightedge constructions''' exclusively. Under {{AuthorRef|Plato}}, '''neusis''' was considered a last resort when all other methods of construction had failed. By the time of {{AuthorRef|Pappus of Alexandria}}, to use a '''neusis construction''' when other methods would work was considered a serious error.


Definition:Center of Gravity

Let $B$ be a body. The '''center of gravity''' of $B$ is that point through which the resultant of the system of parallel forces formed by the weights of all the particles constituting $B$ passes for all positions of $B$.


Definition:Archimedean Spiral/Archimedes' Definition

:''If a straight line of which one extremity remains fixed be made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line revolves, a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.''


Definition:Information Theory

'''Information theory''' is that branch of mathematics concerned with the flow of information.


Definition:Special Function

'''Special functions''' are particular functions which have more or less established names and notations due to their importance in, for example,  analysis, functional analysis or physics. There is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special.


Definition:Stochastic Process

A '''stochastic process''' is a collection of random variables representing the evolution of some real-world system over time.


Definition:Non-Linear System

A '''non-linear system''' is a system of differential equations which are non-linear.


Definition:Catastrophe Theory

'''Catastrophe theory''' is a branch of bifurcation theory in the study of dynamical systems. It is also a particular special case of more general singularity theory in geometry.


Definition:Chaos Theory

'''Chaos theory''' is the branch of non-linear systems that studies the behavior and condition of dynamical systems that are highly sensitive to initial conditions.


Definition:Atomic Physics

'''Atomic physics''' is the branch of physics concerned with the structure and behaviour of the atom.


Definition:Wave Mechanics

'''Wave mechanics''' is the branch of physics concerned with the physical behaviour of waves.


Definition:Oscillation/Oscillation on Set

Let $A \subseteq X$ be any non-empty subset $A$ of $X$. The '''oscillation of $f$ on (or over) $A$ with respect to $d$''', denoted $\omega_f \left({A; d}\right)$, is defined as the diameter of $f \left({A}\right)$: :$\displaystyle \omega_f \left({A; d}\right) := \operatorname{diam} \left({f \left({A}\right)}\right) = \sup_{x,y \mathop \in A} d \left({f \left({x}\right), f \left({y}\right)}\right)$ where the supremum is taken in the extended real numbers $\overline \R$. The metric $d$ is often suppressed from the notation if it is clear from context, in which case one would simply write $\omega_f \left({A}\right)$. Similarly, one would speak of the '''oscillation of $f$ on $A$''' in this case.


Definition:Oscillation/Oscillation at Point

Let $x \in X$. Let $X$ be a topological space. Denote with $\mathcal N_x$ the set of neighborhoods of $x$. The '''oscillation of $f$ at $x$ with respect to $d$''', denoted by $\omega_f \left({x; d}\right)$, is defined as: :$\displaystyle \omega_f \left({x; d}\right) := \inf_{U \mathop \in \mathcal N_x} \omega_f \left({U; d}\right)$ where $\omega_f \left({U; d}\right)$ denotes the oscillation of $f$ on $U$. The metric $d$ is often suppressed from the notation if it is clear from context, in which case one would simply write $\omega_f \left({x}\right)$. Similarly, one would speak of the '''oscillation of $f$ at $x$''' in this case.


Definition:Point of Inflection

Let $f$ be a real function which is differentiable on an interval $\Bbb I \subseteq \R$. Let $\xi \in \Bbb I$. === Definition 1 === {{:Definition:Point of Inflection/Definition 1}} === Definition 2 === {{:Definition:Point of Inflection/Definition 2}}


Definition:Point of Inflection/Definition 1

$f$ has a '''point of inflection at $\xi$''' {{iff}} $\xi$ is a point on $f$ at which $f$ changes from being concave to convex, or vice versa.


Definition:Point of Inflection/Definition 2

$f$ has a '''point of inflection at $\xi$''' {{iff}} the derivative $f'$ of $f$ has either a local maximum or a local minimum at $\xi$.


Definition:Ordering on Mappings

Let $S$ be a set. Let $\left({T, \preceq}\right)$ be an ordered set. Let $f, g: S \to T$ be mappings. Then '''ordering on mappings''' $f$ and $g$ denoted $f \preceq g$ is defined by :$\forall s \in S: f\left({s}\right) \preceq g\left({s}\right)$


Definition:Euler-Lagrange Equation

The '''Euler–Lagrange equation''' is an equation satisfied by a function $\mathbf q$ of a real argument $t$, which is a stationary point of the functional: :$\ds \map S {\mathbf q} = \int_a^b \map L {t, \map {\mathbf q} t, \map {\mathbf q'} t} \rd t$ where: :$\mathbf q$ is the function to be found: :$\mathbf q: \closedint a b \subset \R \to X : t \mapsto x = \map {\mathbf q} t$ such that: ::$\mathbf q$ is differentiable ::$\map {\mathbf q} a = \mathbf x_a$ ::$\map {\mathbf q} b = \mathbf x_b$ :$\mathbf q'$ is the derivative of $\mathbf q$: ::$\mathbf q': \closedint a b \to T_{\map {\mathbf q} t} X: t \mapsto v = \map {\mathbf q'} t$ :$T_{\map {\mathbf q} t} X$ denotes the tangent space to $X$ at the point $\map {\mathbf q} t$ :$L$ is a real-valued function with continuous first partial derivatives: ::$L: \closedint a b \times T X \to \R: \tuple {t, x, v} \mapsto \map L {t, x, v}$ where: :$T X$ is the tangent bundle of $X$ defined by: ::$T X = \bigcup_{x \mathop \in X} \set x \times T_x X$ The ''Euler–Lagrange equation'', then, is given by: :$\displaystyle \map {L_x} {t, \map {\mathbf q} t, \map {\mathbf q'} t} - \dfrac \d {\d t} \map {L_v} {t, \map {\mathbf q} t, \map {\mathbf q'} t} = 0$ where: :$L_x$ and $L_v$ denote the partial derivatives of $L$ with respect to the second and third arguments respectively.


Definition:Oscillation/Real Space/Oscillation at Point/Infimum

Let $\mathcal N_x$ be the set of open subset neighborhoods of $x$. The '''oscillation of $f$ at $x$''' is defined as: :$\displaystyle \omega_f \left({x}\right) := \inf_{U \mathop \in \mathcal N_x} \omega_f \left({U \cap X}\right)$ where $\omega_f \left({U \cap X}\right)$ denotes the oscillation of $f$ on $U \cap X$.


Definition:Oscillation/Real Space/Oscillation at Point/Limit

The '''oscillation of $f$ at $x$''' is defined as: :$\displaystyle \omega_f \left({x}\right) := \lim_{h \to 0^+} \omega_f \left({\left({x - h \,.\,.\, x + h}\right) \cap X}\right)$ where $\omega_f \left({\left({x - h \,.\,.\, x + h}\right) \cap X}\right)$ denotes the oscillation of $f$ on $\left({x - h \,.\,.\, x + h}\right) \cap X$.


Definition:Language of Predicate Logic

In order to define $\LL_1$, it is necessary to specify: * An alphabet $\AA$ * A collation system with the unique readability property for $\AA$ * A formal grammar (which determines the WFFs of $\LL_1$)


Definition:Abel's Integral Equation

'''Abel's integral equation''' is an integral equation whose purpose is to solve Abel's mechanical problem, which finds how long it will take a bead to slide down a wire. The purpose of '''Abel's integral equation''' is to find the shape of the curve into which the wire is bent in order to yield that result: Let $T \left({y}\right)$ be a function which specifies the total time of descent for a given starting height. :$\displaystyle T \left({y_0}\right) = \int_{y \mathop = y_0}^{y \mathop = 0} \, \mathrm d t = \frac 1 {\sqrt {2 g} } \int_0^{y_0} \frac 1 {\sqrt {y_0 - y} } \frac {\mathrm d s} {\mathrm d y} \, \mathrm d y$ where: :$y$ is the height of the bead at time $t$ :$y_0$ is the height from which the bead is released :$g$ is Acceleration Due to Gravity :$s \left({y}\right)$ is the distance along the curve as a function of height.


Definition:Abelian Integral

An '''Abelian integral''' is a complex Riemann integral of the form :$\displaystyle \int_{z_0}^z R \left({x, w}\right) \mathrm d x$ where $R \left({x, w}\right)$ is an arbitrary rational function of the two variables $x$ and $w$. These variables are related by the equation: :$F \left({x, w}\right) = 0$  where $F \left({x, w}\right)$ is an irreducible polynomial in $w$: :$F \left ({x, w}\right) \equiv \phi_n \left({x}\right) w^n + \cdots + \phi_1 \left({x}\right) w + \phi_0 \left({x}\right)$ whose coefficients $\phi_j \left({x}\right), j = 0, 1, \ldots, n$ are rational functions of $x$.  {{NamedforDef|Niels Henrik Abel|cat = Abel}}


Definition:Abelian Function

An '''Abelian function''' is an inverse function of an Abelian integral. {{NamedforDef|Niels Henrik Abel|cat = Abel}}


Definition:Dirichlet Function

A '''Dirichlet function''' $D: \R \to \R$ is a real function defined as: :$\forall x \in \R: \map D x = \begin {cases} c & : x \in \Q \\ d & : x \notin \Q \end {cases}$ for $c, d \in \R$ such that $c \ne d$. The canonical example of this has $c = 1$ and $d = 0$. {{NamedforDef|Johann Peter Gustav Lejeune Dirichlet|cat = Dirichlet}}


Definition:Liouville's Constant

'''Liouville's constant''' is the real number defined as: {{begin-eqn}} {{eqn | l = \sum_{n \mathop \ge 1} \dfrac 1 {10^{n!} }       | r = \frac 1 {10^1} + \frac 1 {10^2} + \frac 1 {10^6} +  \frac 1 {10^{24} } + \cdots       | c =  }} {{eqn | r = 0 \cdotp 11000 \, 10000 \, 00000 \, 00000 \, 00010 \, 00 \ldots       | c =  }} {{end-eqn}} {{OEIS|A012245}}


Definition:Riemann Surface/Elliptic/Definition 2

A '''Riemann surface''' is '''elliptic''' {{iff}} it admits a metric of constant positive curvature.


Definition:Riemann Surface/Elliptic

{{:Definition:Riemann Surface/Elliptic/Definition 1}}


Definition:Riemann Surface/Parabolic

{{:Definition:Riemann Surface/Parabolic/Definition 1}}


Definition:Riemannian Geometry (Mathematical Branch)

'''Riemannian geometry''' is the branch of differential geometry that studies Riemannian manifolds.


Definition:Riemannian Manifold

A '''Riemannian manifold''' is a smooth manifold on the real space $\R^n$ upon which a Riemannian metric has been imposed. === Dimension === {{:Definition:Riemannian Manifold/Dimension}} {{NamedforDef|Bernhard Riemann|cat = Riemann}}


Definition:Riemannian Metric

Consider a smooth manifold $\mathcal M$ on the real space $\R^n$. A '''Riemannian metric''' on $\mathcal M$ is a metric $\mathrm d s$ between nearby points $\left({x_1, x_2, \ldots, x_n}\right)$ and $\left({x_1 + \mathrm d x_1, x_2 + \mathrm d x_2, \ldots, x_n + \mathrm d x_n}\right)$ by means of the quadratic differential form: :$\displaystyle \mathrm d s^2 = \sum_{i, j \mathop = 1}^n g_{i j} \, \mathrm d x_i \, \mathrm d x_j$ where each $g_{i j}$ is a suitable real-valued function of $x_1, \ldots, x_n$. Different instances of $g_{i j}$ define different Riemannian geometries on the manifold under discussion. A manifold with such a '''Riemannian metric''' applied is known as a '''Riemannian manifold'''. {{NamedforDef|Bernhard Riemann|cat = Riemann}}


Definition:Riemann-Christoffel Tensor

A '''Riemann-Christoffel tensor''' is a tensor field which expresses the curvature of a Riemannian manifold. The '''Riemann-Christoffel tensor''' is given in terms of the Levi-Civita connection $\nabla$ by: :$R \left({u, v}\right) w = \nabla_u \nabla_v w - \nabla_v \nabla_u w - \nabla_{\left[{u, v}\right]} w$ where $\left[{u, v}\right]$ is the Lie bracket of vector fields. It measures the extent to which the metric tensor is not locally isometric to that of Euclidean space.


Definition:Riemannian Manifold/Dimension

The '''dimension''' of the '''Riemannian manifold''' on $\R^n$ is $n$.


Definition:Riemannian Geometry

A '''Riemannian geometry''' is a Riemannian manifold $\mathcal M$ together with the Riemannian metric that has been applied to $M$.


Definition:Riemann Zeta Function/Critical Line

Let $s = \sigma + i t$. The line defined by the equation $\sigma = \dfrac 1 2$ is known as the '''critical line'''. Hence the popular form of the statement of the Riemann Hypothesis: :''All the nontrivial zeroes of the Riemann zeta function lie on the '''critical line'''.''


Definition:Weierstrass Function

A '''Weierstrass function''' is a real function defined on a closed real interval $I$ which is: :$(1):\quad$ continuous on $I$ :$(2):\quad$ nowhere differentiable on $I$.


Definition:Entire Function/Transcendental

Let $f$ be an '''entire function''' that has an essential singularity at $\infty$. Then $f$ is a '''transcendental entire function'''.


Definition:Dedekind Cut/Definition 1

A '''Dedekind cut''' of $\struct {S, \preceq}$ is a non-empty proper subset $L \subsetneq S$ such that: :$(1): \quad \forall x \in L: \forall y \in S: y \prec x \implies y \in L$ ($L$ is a lower set in $S$) :$(2): \quad \forall x \in L: \exists y \in L: x \prec y$


Definition:Dedekind Cut/Definition 2

A '''Dedekind cut''' of $\struct {S, \preceq}$ is an ordered pair $\tuple {L, R}$ such that: :$(1): \quad \set {L, R}$ is a partition of $S$. :$(2): \quad L$ does not have a greatest element. :$(3): \quad \forall x \in L: \forall y \in R: x \prec y$.


Definition:Cartesian Product of Ordered Sets

Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be ordered sets. '''Cartesian product of (ordered sets)''' $\left({S_1, \preceq_1}\right)$ and $\left({S_2, \preceq_2}\right)$ is a pair $\left({S_1 \times S_2, \preceq}\right)$ where :${\preceq} \subseteq \left({S_1 \times S_2}\right) \times \left({S_1 \times S_2}\right)$ such that :$\forall s_1, t_1 \in S_1, s_2, t_2 \in S_2: \left({s_1,s_2}\right) \preceq \left({t_1,t_2}\right) \iff s_1 \preceq_1 t_1 \land s_2 \preceq_2 t_2$


Definition:Prime Number/Definition 7

A '''prime number''' $p$ is an integer greater than $1$ which cannot be written in the form: :$p = a b$ where $a$ and $b$ are both positive integers less than $p$.


Definition:Perfect Number/Definition 1

A '''perfect number''' is a (strictly) positive integer equal to its aliquot sum.


Definition:Perfect Number/Definition 2

A '''perfect number''' $n$ is a (strictly) positive integer such that: : $\sigma \left({n}\right) = 2 n$ where $\sigma: \Z_{>0} \to \Z_{>0}$ is the sigma function.


Definition:Momentum

=== Linear Momentum === {{:Definition:Linear Momentum}} === Angular Momentum === {{:Definition:Angular Momentum}}


Definition:Linear Momentum/Relativistic Model

A more accurate model for the linear momentum of a body is given by: :$\mathbf p = \gamma m \mathbf v$ where $\gamma$ is the Lorentz Factor: :$\gamma = \dfrac c {\sqrt {c^2 - v^2} } = \dfrac 1 {\sqrt {1 - v^2 / c^2} }$ where: : $c$ is the speed of light in vacuum : $v$ is the magnitude of $\mathbf v$: $v = \size {\mathbf v}$ It is clear $\gamma \approx 1$ (and thus that $\mathbf p \approx m \mathbf v$) for values of $v$ much less than $c$.


Definition:Thrust

The '''thrust''' of a rocket is defined as the product of the exhaust velocity and the rate at which its fuel is being consumed: :$\mathbf t = \mathbf b \paren {\dfrac {\d m} {\d t} }$ where: :$\mathbf b$ is the exhaust velocity :$m$ is the mass of the rocket as a function of time $t$.


Definition:Meet-Continuous Lattice

Let $\left({S, \preceq}\right)$ be a meet semilattice. Then $\left({S, \preceq}\right)$ is '''meet-continuous''' {{iff}} :$\left({S, \preceq}\right)$ is up-complete and ::(MC): for every an element $x \in S$ and a directed subset $D$ of $S$: $x \wedge \sup D = \sup \left\{ {x \wedge d: d \in D}\right\}$


Definition:Twin Primes/Sequence

The sequence of twin primes begins: :$3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, \ldots$


Definition:Prime-Counting Function/Approximations

A table of some of the values of the prime-counting ($\pi$) function compared with $\dfrac x {\ln x}$ and the Eulerian logarithmic integral $\displaystyle \map \Li x = \int_2^x \frac {\d t} {\map \ln t}$: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $\map \pi n$  ! align="right" style = "padding: 2px 10px" | $\dfrac x {\ln x}$ ! align="right" style = "padding: 2px 10px" | $\map \Li x$ |- | align="right" style = "padding: 2px 10px" | $1 \, 000$ | align="right" style = "padding: 2px 10px" | $168$ | align="right" style = "padding: 2px 10px" | $145$ | align="right" style = "padding: 2px 10px" | $178$ |- | align="right" style = "padding: 2px 10px" | $10 \, 000$ | align="right" style = "padding: 2px 10px" | $1 \, 229$ | align="right" style = "padding: 2px 10px" | $1 \, 068$ | align="right" style = "padding: 2px 10px" | $1 \, 246$ |- | align="right" style = "padding: 2px 10px" | $100 \, 000$ | align="right" style = "padding: 2px 10px" | $9 \, 596$ | align="right" style = "padding: 2px 10px" | $8 \, 686$ | align="right" style = "padding: 2px 10px" | $9 \, 630$ |- | align="right" style = "padding: 2px 10px" | $1 \, 000 \, 000$ | align="right" style = "padding: 2px 10px" | $78 \, 498$ | align="right" style = "padding: 2px 10px" | $72 \, 382$ | align="right" style = "padding: 2px 10px" | $78 \, 628$ |- | align="right" style = "padding: 2px 10px" | $10 \, 000 \, 000$ | align="right" style = "padding: 2px 10px" | $664 \, 579$ | align="right" style = "padding: 2px 10px" | $620 \, 421$ | align="right" style = "padding: 2px 10px" | $664 \, 918$ |}


Definition:Rational Point in Plane

Consider the Cartesian plane $\R^2$. The points $\tuple {x, y} \in \R^2$ such that $x, y \in \Q$ are called the '''rational points (of $\R^2$)'''.


Definition:Linear Polynomial

A '''linear polynomial''' is a polynomial whose degree is $1$. Category:Definitions/Polynomial Theory 8ffuktvxhlxm4sq1yfifr3z2ugp5i91


Definition:Algebraic Number/Degree

Let $\alpha$ be an algebraic number. By definition, $\alpha$ is the root of at least one polynomial $P_n$ with rational coefficients. The '''degree''' of $\alpha$ is the degree of the minimal polynomial $P_n$ whose coefficients are all in $\Q$.


Definition:Gelfond-Schneider Constant

The '''Gelfond-Schneider constant''' is defined as $2$ raised to the power of the square root of $2$: :$2^{\sqrt 2}$ Its decimal expansion begins: :$2^{\sqrt 2} \approx 2 \cdotp 66514 \, 4$ {{OEIS|A007507}} {{NamedforDef|Alexander Osipovich Gelfond|name2 = Theodor Schneider|cat = Gelfond|cat2 = Schneider}}


Definition:Bernoulli Numbers/Sequence

The sequence of '''Bernoulli numbers''' begins: {{begin-eqn}} {{eqn | l = B_0       | r = 1       | c =  }} {{eqn | l = B_1       | ro= -       | r = \dfrac 1 2       | c =  }} {{eqn | l = B_2       | r = \dfrac 1 6       | c =  }} {{eqn | l = B_4       | ro= -       | r = \dfrac 1 {30}       | c =  }} {{eqn | l = B_6       | r = \dfrac 1 {42}       | c =  }} {{eqn | l = B_8       | ro= -       | r = \dfrac 1 {30}       | c =  }} {{eqn | l = B_{10}       | r = \dfrac 5 {66}       | c =  }} {{eqn | l = B_{12}       | ro= -       | r = \dfrac {691} {2730}       | c =  }} {{end-eqn}} The odd index Bernoulli numbers, apart from $B_1$, are all equal to $0$.


Definition:Partial Fractions Expansion

Let $\map R x = \dfrac {\map P x} {\map Q x}$ be a rational function, where $\map P x$ and $\map Q x$ are expressible as polynomial functions. Let $\map Q x$ be expressible as: :$\map Q x = \displaystyle \prod_{k \mathop = 1}^n \map {q_k} x$ where the $\map {q_k} x$ are themselves polynomial functions of degree at least $1$. Let $\map R x$ be expressible as: :$\map R x = \map r x \displaystyle \sum_{k \mathop = 0}^n \dfrac {\map {p_k} x} {\map {q_k} x}$ where: :$\map r x$ is a polynomial function which may or may not be the null polynomial, or be of degree $0$ (that is, a constant) :each of the $\map {p_k} x$ are polynomial functions :the degree of $\map {p_k} x$ is strictly less than the degree of $\map {q_k} x$ for all $k$. Then $\map r x \displaystyle \sum_{k \mathop = 0}^n \dfrac {\map {p_k} x} {\map {q_k} x}$ is a '''partial fractions expansion''' of $\map R x$.


Definition:Hypocycloid

Let a circle $C_1$ roll around the inside of another (larger) circle $C_2$. The locus of a given point on the circumference of $C_1$ is known as a '''hypocycloid'''.


Definition:Epicycloid

Let a circle $C_1$ roll around the outside of another circle $C_2$. The locus of a given point on the circumference of $C_1$ is known as an '''epicycloid'''.


Definition:Astroid

An '''astroid''' is a hypocycloid with $4$ cusps. :400px


Definition:Circular Error of Pendulum

The period of a pendulum $P$ is dependent upon the angle from the vertical from which the bob is released. That period of $P$ is described by Motion of Pendulum as being: :$T = 2 \sqrt {\dfrac a g} \, \map K {\map \sin {\dfrac \alpha 2} }$ where: :$a$ is the length of the rod of $P$ :$\map K k$ is the complete elliptic integral of the first kind. The '''circular error''' of $P$ is the difference between $T$ and its approximate period as described by period Approximate Motion of Pendulum: :$T' = 2 \pi \sqrt {\dfrac a g}$


Definition:Tusi Couple

A '''Tusi couple''' is a hypocycloid with $2$ cusps. :400px


Definition:Hypocycloid/Generator

The circles $C_1$ and $C_2$ can be referred to as the '''generators''' of the hypocycloid. === Stator === {{Definition:Hypocycloid/Generator/Stator}} === Rotor === {{Definition:Hypocycloid/Generator/Rotor}}


Definition:Hypocycloid/Generator/Rotor

The circle $C_1$ can be referred to as the '''rotor''' of the hypocycloid.


Definition:Hypocycloid/Generator/Stator

The circle $C_2$ can be referred to as the '''stator''' of the hypocycloid.


Definition:Epicycloid/Generator

The circles $C_1$ and $C_2$ can be referred to as the '''generators''' of the epicycloid. === Stator === {{Definition:Epicycloid/Generator/Stator}} === Rotor === {{Definition:Epicycloid/Generator/Rotor}}


Definition:Epicycloid/Generator/Stator

The circle $C_2$ can be referred to as the '''stator''' of the epicycloid.


Definition:Epicycloid/Generator/Rotor

The circle $C_1$ can be referred to as the '''rotor''' of the epicycloid.


Definition:Deltoid

A '''deltoid''' is a hypocycloid with $3$ cusps. :400px


Definition:Oscillation/Real Space/Oscillation on Set

Let $A \subseteq X$ be any non-empty subset $A$ of $X$. The '''oscillation of $f$ on (or over) $A$''' is defined as: :$\displaystyle \map {\omega_f} A := \sup_{x, y \mathop \in A} \size {\map f x - \map f y}$ where the supremum is taken in the extended real numbers $\overline \R$.


Let $X$ and $Y$ be real sets. Let $f: X \to Y$ be a real function. <section notitle="true" name="definition"> Let $x \in X$. === Definition 1 === {{:Definition:Oscillation/Real Space/Oscillation at Point/Infimum}} === Definition 2 === {{:Definition:Oscillation/Real Space/Oscillation at Point/Epsilon}} === Definition 3 === {{:Definition:Oscillation/Real Space/Oscillation at Point/Limit}} </section>


Definition:Oscillation/Real Space

Let $X$ and $Y$ be real sets. Let $f: X \to Y$ be a real function. === Oscillation on a Set === {{:Definition:Oscillation/Real Space/Oscillation on Set}} {{transclude:Definition:Oscillation/Real Space/Oscillation at Point   |section  = definition   |increase = 1   |title    = Oscillation at a Point   |header   = 3   |link     = true }}


Definition:Nephroid

A '''nephroid''' is an epicycloid with $2$ cusps. :600px


Definition:Curvature

Let $C$ be a curve defined by a real function which is twice differentiable. The '''curvature''' of a $C$ is the reciprocal of the radius of the osculating circle to $C$ and is often denoted $\kappa$ (Greek '''kappa''').


Definition:Radius of Curvature

The '''radius of curvature''' of a curve $C$ at a point $P$ is defined as the reciprocal of the absolute value of its curvature: :$\rho = \dfrac 1 {\size k}$


Definition:Normal to Curve

Let $C$ be a curve embedded in the plane. The '''normal to $C$''' at a point $P$ is defined as the straight line which lies perpendicular to the tangent at $P$ and in the same plane as $P$.


Definition:Center of Curvature

Let $C$ be a curve embedded in the plane. Let $N$ be the normal to $C$ at a point $P$. Let $\rho$ be the radius of curvature of $C$ at $P$. Let $Q$ be the point on $N$ which is at a distance $\rho$ from $P$ on the concave side of $C$. $Q$ is known as the '''center of curvature''' of $C$ at $P$.


Definition:Curvature/Cartesian Form

Let $C$ be embedded in a cartesian plane. The '''curvature''' $\kappa$ of $C$ at a point $P = \tuple {x, y}$ is given by: :$\kappa = \dfrac {y''} {\paren {1 + y'^2}^{3/2} }$ where: :$y' = \dfrac {\d y} {\d x}$ is the derivative of $y$ {{WRT|Differentiation}} $x$ at $P$ :$y'' = \dfrac {\d^2 y} {\d x^2}$ is the second derivative of $y$ {{WRT|Differentiation}} $x$ at $P$.


Definition:Intrinsic Equation

Let $C$ be a curve. An '''intrinsic equation''' of $C$ is an equation that defines $C$ using a relation between the intrinsic properties of $C$. Thus an '''intrinsic equation''' defines the shape of $C$ without specifying its position relative to an arbitrarily defined coordinate system. === Natural Equation === {{:Definition:Intrinsic Equation/Natural Equation}} === Cesàro Equation === {{:Definition:Intrinsic Equation/Cesàro Equation}} === Whewell Equation === {{:Definition:Intrinsic Equation/Whewell Equation}}


Definition:Intrinsic Property

Let $C$ be a curve. The '''intrinsic properties''' of $C$ are those properties that do not depend on the nature of the embedding of $C$ in a coordinate system.


Definition:Intrinsic Property/Examples

Examples of the intrinsic properties of a curve include: :Arc length $s$ :Curvature $\kappa$ :Turning angle $\psi$ :Torsion $\tau$ :Radius of curvature $\rho$


Definition:Turning Angle

Let $C$ be a curve. The '''turning angle''' $\psi$ of $C$ at a point $P$ is defined by the differential equation: :$\kappa = \dfrac {\d \psi} {\d s}$ where: :$\kappa$ is the curvature at $P$ :$\dfrac {\d \psi} {\d s}$ is the derivative of $\psi$ {{WRT|Differentiation}} the arc length $s$. That is, the '''turning angle''' is a measure of how sharply a curve changes direction. The '''turning angle''' is an intrinsic property of $C$.


Definition:Intrinsic Equation/Whewell Equation

A '''Whewell equation''' is an intrinsic equation for a curve $C$ such that $C$ is expressed in terms of its intrinsic properties: :Arc length $s$ :Turning angle $\psi$


Definition:Curvature/Whewell Form

The '''curvature''' $\kappa$ of $C$ at a point $P$ can be expressed in the form of a Whewell equation as: :$\kappa = \dfrac {\d \psi} {\d s}$ where: :$\psi$ is the turning angle of $C$ :$s$ is the arc length of $C$.


Definition:Curvature/Parametric Form

Let $C$ be embedded in a cartesian plane and defined by the parametric equations: :$\begin{cases} x = \map x t \\ y = \map y t \end{cases}$ The '''curvature''' $\kappa$ of $C$ at a point $P = \tuple {x, y}$ is given by: :$\kappa = \dfrac {x' y'' - y' x''} {\tuple {x'^2 + y'^2}^{3/2} }$ where: :$x' = \dfrac {\d x} {\d t}$ is the derivative of $x$ {{WRT|Differentiation}} $t$ at $P$ :$y' = \dfrac {\d y} {\d t}$ is the derivative of $y$ {{WRT|Differentiation}} $t$ at $P$ :$x''$ and $y''$ are the second derivatives of $x$ and $y$ {{WRT|Differentiation}} $t$ at $P$.


Definition:Astronomical Unit/Approximate Values

The '''astronomical unit''' is approximately $150 \, 000 \, 000 \, \mathrm {k m}$, or $93$ million (international) miles.


Definition:Astronomical Unit

The '''astronomical unit''' is a derived unit of length. It is defined as being $149 \, 597 \, 870 \, 700$ metres. The '''astronomical unit''' is the standard unit of measurement used by astronomers when discussing distances within the solar system. It is derived as the approximate mean distance from the Earth to the Sun. === Approximate Values === {{:Definition:Astronomical Unit/Approximate Values}} === Symbol === {{:Definition:Astronomical Unit/Symbol}}


Definition:Division Algebra/Definition 1

$\left({A_F, \oplus}\right)$ is a '''division algebra''' {{iff}}: : $\forall a, b \in A_F, b \ne \mathbf 0_A: \exists_1 x \in A_F, y \in A_F: a = b \oplus x, a = y \oplus b$


Definition:Division Algebra/Definition 2

$A$ is a division algebra {{iff}} it has no zero divisors: :$\forall a, b \in A_F: a \oplus b = \mathbf 0_A \implies a = \mathbf 0_A \lor b = \mathbf 0_A$


Definition:Conjugate Quaternion/Matrix Form

Let $\mathbf x$ be a quaternion defined in matrix form as: :$\mathbf x = \begin{bmatrix} a + bi & c + di \\ -c + di & a - bi \end{bmatrix}$ The conjugate quaternion of $\mathbf x$ is defined as: :$\overline {\mathbf x} = \begin{bmatrix} a - bi & -c - di \\ c - di & a + bi \end{bmatrix}$


Definition:Conjugate Quaternion/Ordered Pair

Let $\mathbf x$ be a quaternion defined as an ordered pair $\left({a, b}\right)$  of complex numbers. The conjugate quaternion of $\mathbf x$ is defined as: :$\overline {\mathbf x} = \overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$


Definition:Field Norm of Quaternion

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$. The '''norm''' of $\mathbf x$ is the real number defined as: :$n \left({\mathbf x}\right) := \left\vert{\mathbf x \overline {\mathbf x} }\right\vert = \left\vert{\overline {\mathbf x} \mathbf x }\right\vert = a^2 + b^2 + c^2 + d^2$


Definition:Positive Definite (Ring)

Let $\struct {R, +, \times}$ be a ring whose zero is denoted $0_R$. Let $f: R \to \R$ be a (real-valued) function on $R$. Then $f$ is '''positive definite''' {{iff}}: :$\forall x \in R: \begin {cases} \map f x = 0 & : x = 0_R \\ \map f x > 0 & : x \ne 0_R \end {cases}$ Category:Definitions/Ring Theory p3oa3bgzq9l393gfnrxy925ltk362pq


Definition:Nonassociative Algebra

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring. Then $\left({A_R, \oplus}\right)$ is a nonassociative algebra {{iff}} it is not necessarily the case that $\oplus$ is associative.


Definition:Lie Algebra

Let $\struct {A_R, \oplus}$ be an algebra over a ring. Then $\struct {A_R, \oplus}$ is a Lie algebra {{iff}}: :$\forall a \in A_R: a^2 = 0$ :$\forall a, b, c \in A_R: a \paren {b c} + b \paren {c a} + c \paren {a b} = 0$


Definition:Lie Group

A '''Lie group''' is a group which is also a smooth manifold.


Definition:Cayley Algebra

The '''Cayley algebra''' is the nonassociative division algebra formed by the Octonions under octonion multiplication.


Definition:Normal Family

Let $X = \left({M_1, d_1}\right)$ and $Y = \left({M_2, d_2}\right)$ be complete metric spaces. Let $\mathcal F = \left \langle{f_i}\right \rangle_{i \mathop \in I}$ be a family of continuous mappings $f_i: X \to Y$. Then $\mathcal F$ is a '''normal family''' {{iff}}: : every sequence of mappings in $\mathcal F$ contains a subsequence which converges uniformly on compact subsets of $X$ to a continuous function $f: X \to Y$.


Definition:Locally Bounded/Mapping

Let $M = \left({A, d}\right)$ be a metric space. Let $f$ be a mapping defined on $M$. Then $f$ is said to be '''locally bounded''' {{iff}}: :for all $x \in A$, there is some neighbourhood $N$ of $x$ such that $f \left[{N}\right]$ is bounded.


Definition:Uniformly Bounded

Let $X$ be a  set and let $Y = \left({A, d}\right)$ be a metric space. Let $\mathcal F = \left \langle{f_i}\right \rangle_{i \mathop \in I}$ be a family of mappings $f_i: X \to Y$. Then $\mathcal F$ is said to be '''uniformly bounded''' if every mapping $f \in \mathcal F$ can be bounded by the same constant. That is, if there exists some $M \in \R$ such that: :$\forall x, y \in X, i \in I : d \left({f_i \left({x}\right), f_i \left({y}\right)}\right) \le M$ Category:Definitions/Metric Spaces Category:Definitions/Boundedness 9v1jbgvwtmrcsgrniksvg1s7rlg2f48


Definition:Locally Bounded/Family of Mappings

Let $M = \left({A, d}\right)$ be a metric space. Let $\mathcal F = \left \langle{f_i}\right \rangle_{i \mathop \in I}$ be a family of mappings defined on $M$. Then $\mathcal F$ is said to be '''locally bounded''' {{iff}}: : for all $x \in A$, there is some neighbourhood $N$ of $x$ such that $\mathcal F$ is uniformly bounded on $N$.


Definition:Locally Bounded

=== Locally Bounded Mapping === {{:Definition:Locally Bounded/Mapping}} === Locally Bounded Family of Mappings === {{:Definition:Locally Bounded/Family of Mappings}} Category:Definitions/Metric Spaces Category:Definitions/Boundedness 2nyk2o70hbyhpuzn2alte3406z76g5w


Definition:Axiom/Also known as

An '''axiom''' is also known as a '''postulate'''. Among ancient Greek philosophers, the term '''axiom''' was used for a general truth that was common to everybody (see Euclid's "common notions"), while  '''postulate''' had a specific application to the subject under discussion. For most authors, the distinction is no longer used, and the terms are generally used interchangeably. This is the position of {{ProofWiki}}. However, some believe there is a difference significant enough to matter: :''... we shall use "postulate" instead of "axiom" hereafter, as "axiom" has a pernicious historical association of "self-evident, necessary truth", which "postulate" does not have; a postulate is an arbitrary assumption laid down by the mathematician himself and not by God Almighty.'' :::: -- {{BookReference|Men of Mathematics|1937|Eric Temple Bell}}: Chapter $\text{II}$: Modern Minds in Ancient Bodies


Definition:Bounded Sequence/Complex

Let $\sequence {z_n}$ be a complex sequence. Then $\sequence {z_n}$ is '''bounded''' {{iff}}: :$\exists M \in \R$ such that $\forall i \in \N: \cmod {z_i} \le M$ where $\cmod {z_i}$ denotes the complex modulus of $z_i$.


Definition:Bounded Sequence/Complex/Unbounded

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


Definition:Mechanical Curve

A '''mechanical curve''' is a curve which cannot be constructed using a compass and straightedge. Note that while it is not possible to construct such a curve in this manner, it may well be possible to construct as many points on it as required.


Definition:Conic Section/Intersection with Cone/Degenerate Hyperbola

Let $\phi < \theta$, that is: so as to make $K$ a hyperbola. However, let $D$ pass through the apex of $C$. Then $K$ degenerates into a pair of intersecting straight lines.


Definition:Geometric Curve

A '''geometric curve''' is a curve which can be constructed using a compass and straightedge.


Definition:Number Theory/Also known as

Older sources refer to '''number theory''' variously as: :'''theory of numbers''' :'''the higher arithmetic''' :'''arithmetic'''. {{AuthorRef|Carl Friedrich Gauss}} himself used the term '''arithmetic''', after the manner of the ancient Greeks. '''Number theory''' can also be seen described by the fanciful name '''the Queen of Mathematics''' for various whimsical reasons.


Definition:Ore Graph

Let $G = \struct {V, E}$ be an undirected simple graph. Then $G$ is an '''Ore graph''' {{iff}}: :the sum of the degrees of every pair of non-adjacent vertices is greater than or equal to the order of $G$. That is, {{iff}}: :$\forall u, v \in V: \set {u, v} \notin E \implies \map {\deg_G} u + \map {\deg_G} v \ge \card V$ {{NamedforDef|Øystein Ore|cat = Ore}}


Keith.U/Whatever/Definition:Exponential/Real

For all definitions of the '''real exponential function''': : The domain of $\exp$ is $\R$ : The codomain of $\exp$ is $\R_{>0}$ For $x \in \R$, the real number $\exp x$ is called the '''exponential of $x$'''. === As a Sum of a Series === {{:Definition:Exponential Function/Real/Sum of Series}} === As a Limit of a Sequence === {{:Definition:Exponential Function/Real/Limit of Sequence}} === As an Extension of the Rational Exponential === {{:Definition:Exponential Function/Real/Extension of Rational Exponential}} === As the Inverse to the Natural Logarithm === {{:Definition:Exponential Function/Real/Inverse of Natural Logarithm}} === As the Solution of a Differential Equation === {{:Definition:Exponential Function/Real/Differential Equation}}


Definition:Exponential Function/Real


Definition:Boundary Value Problem

A '''boundary value problem''' is a differential equation for which a particular solution is to be found, such that the particular solution must satisfy certain boundary conditions.


Definition:Boundary Condition

Let $\Phi$ be a differential equation to which a particular solution is to be found. A '''boundary condition''' is an equation relating particular values of the dependent and independent variables which the particular solution to $\Phi$ must satisfy. It is usual for such '''boundary conditions''' to correspond to the physical extremities of bodies, aspects of whose internal nature is being modelled by means of $\Phi$.


Definition:Subdivision (Graph Theory)

Let $G = \struct {V, E}$ be a graph. === Edge Subdivision === {{:Definition:Subdivision (Graph Theory)/Edge}} === Graph Subdivision === {{:Definition:Subdivision (Graph Theory)/Graph}}


Definition:Isoperimetrical Problem

An '''isoperimetrical problem''' is a mathematical problem that asks what figure has the greatest area of all those figures which have a perimeter of a given length.


Definition:Subdivision (Graph Theory)/Edge

The '''edge subdivision operation''' for an edge $\set {u, v} \in E$ is the deletion of $\set {u, v}$ from $G$ and the addition of two edges $\set {u, w}$ and $\set {w, v}$ along with the new vertex $w$.  This operation generates a new graph $H$: :$H = \struct {V \cup \set w, \paren {E \setminus \set {u, v} } \cup \set {\set {u, w}, \set {w, v} } }$


Definition:Subdivision (Graph Theory)/Graph

A graph which has been derived from $G$ by a sequence of edge subdivision operations is called a '''subdivision of $G$'''.


Definition:Algorism/Algorist

During the middle ages, an '''algorist''' was an arithmetician who used algorism, as opposed to an  abacist who calculated using an abacus. By the time of {{AuthorRef|Leonhard Paul Euler}}, the term had evolved to mean a mathematician who devised algorithms for more complicated operations than those of the classical algorithms.


Definition:Abacism/Abacist

An '''abacist''' is an arithmetician who uses an abacus to do arithmetic, as opposed to an  algorist who calculates using algorism.


Definition:Riesel Number

Let $k$ be an odd positive integer. Then $k$ is a '''Riesel number''' {{iff}}: :For all positive integers $n$, the number $k \, 2^n - 1$ is composite.


Definition:Real Interval/Unit Interval/Open

The open interval between $0$ and $1$ is referred to as the '''open unit interval'''. :$\left({0 \,.\,.\, 1}\right) = \left\{ {x \in \R: 0 < x < 1}\right\}$


Definition:Real Interval/Unit Interval/Closed

The closed interval from $0$ to $1$ is denoted $\mathbb I$ (or a variant) by some authors: :$\mathbb I := \closedint 0 1 = \set {x \in \R: 0 \le x \le 1}$ This is often referred to as the '''closed unit interval'''.


Definition:Power (Algebra)/Real Number/Definition 2

Let $f : \Q \to \R$ be the real-valued function defined as: :$f \left({ q }\right) = x^q$ where $a^q$ denotes $a$ to the power of $q$. Then we define $x^r$ as the unique continuous extension of $f$ to $\R$.


Definition:Power (Algebra)/Real Number/Definition 1

We define $x^r$ as: :$x^r := \map \exp {r \ln x}$ where $\exp$ denotes the exponential function.


Definition:Nonnumerical Analysis

'''Nonnumerical analysis''' is a branch of discrete mathematics that studies subjects emerging from the evolution of computer science, and includes such topics as: :information processing :analysis of algorithms and so on.


Definition:Replacement Operation

The '''replacement operation''' on two variables $a$ and $b$ is denoted: :$a \gets b$ and is interpreted as: :the value of variable $a$ is to be replaced by the current value of variable $b$, while leaving the value of variable $b$ the same.


Definition:Exchange Operation

The '''exchange operation''' on two variables $a$ and $b$ is denoted: :$a \leftrightarrow b$ and is interpreted as: :the value of variable $a$ is to be replaced by the current value of variable $b$ and at the same time: :the value of variable $b$ is to be replaced by the current value of variable $a$.


Definition:Computational Method/Computational Sequence

Each $x \in I$ defines a '''computational sequence''' $x_0, x_1, x_2, \ldots$ as follows: : $x_0 = x$ : $\forall k \ge 0: x_{k+1} = f \left({x_k}\right)$


Definition:Algorithm/Step

An algorithm consists of a finite set of '''steps''', uniquely identified by means of a label, conventionally numeric. A '''step''' of an algorithm consists of: :an '''operation''' :an '''instruction''' as to what the algorithm is to do next, which will be one of the following: ::$(1): \quad$ By default: to move onto the next '''step''' in the sequence ::$(2): \quad$ Based on the result of a '''condition''', the specific '''step''' to perform next ::$(3): \quad$ To '''terminate'''.


Definition:Flow Chart

A '''flow chart''' is a graphical depiction of an algorithm in which the steps are depicted in the form of boxes connected together by arrows. Conventionally, the shape of the box representing a step is dependent upon the type of operation encapsulated within the step: :Rectangular for an action :A different shape, conventionally a diamond, for a condition. On {{ProofWiki}}, the preferred shape for condition boxes is rectangular with rounded corners.  This is to maximise ease and neatness of presentation: configuring a description inside a diamond shaped boxes in order for it to be aesthetically pleasing can be challenging and tedious. Also on {{ProofWiki}}, it is part of the accepted style to implement the start and end points of the algorithm using a box of a particular style, in this case with a double border.


Definition:Algorithm/Action

In an algorithm, an '''action''' is part of a step of that algorithm. It is an operation which changes the state of the algorithm in some manner.


Definition:Algorithm/Condition

In an algorithm, a '''condition''' is part of a step of that algorithm. It is an operation which determines the outcome of a decision as to which step to execute next,


Definition:Element is Way Below

Let $\left({S, \preceq}\right)$ be an ordered set. Let $x, y \in S$. Then $x$ '''is way below''' $y$, denoted $x \ll y$, {{iff}} :for every directed subset $D$ of $S$ ::if $D$ admits a supremum and $y \preceq \sup D$ ::then there exists $d \in D$ such that $x \preceq d$


Definition:Programming Language

A '''programming language''' is a special purpose language used to implement algorithms for a computer to execute.


Definition:Program

A '''program''' is an algorithm which has been written for a computer to execute.


Definition:Algorithm/Formal Specification

An '''algorithm''' can be implemented formally as a computational method $\left({Q, I, \Omega, f}\right)$ as follows: Let $A$ be a finite set of symbols. Let $A^*$ be the set of all collations on $A$: :$\left\{ {x_1 x_2 \cdots x_n: n \ge 0, \forall j: 1 \le j \le n: x_j \in A}\right\}$ The states of the computation are encoded so as to be represented by elements of $A^*$. Let $N \in \Z_{\ge 0}$. Let $Q$ be the set of all ordered pairs $\left({\sigma, j}\right)$ where $\sigma \in A^*, j \in \Z: 0 \le j \le N$. Let $I \subseteq Q$ such that $j = 0$. Let $\Omega \subseteq Q$ such that $j = N$. Let $\theta, \sigma \in A^*$. Then $\theta$ '''occurs in $\sigma$''' {{iff}} $\sigma$ has the form: :$\alpha \theta \omega$ where $\alpha, \omega \in A^*$. Let $f$ be a mapping of the following type: :$f \left({\left({\sigma, j}\right)}\right) = \begin{cases}\left({\sigma, a_j}\right) : & \sigma_j \text { does not occur in } \sigma \\ \left({\alpha \theta_j \omega, b_j}\right) : & \alpha \text { is the shortest element of $A^*$ such that } \sigma  = \alpha \theta_j \omega\end{cases}$ :$f \left({\left({\sigma, N}\right)}\right) = \left({\sigma, N}\right)$


Definition:Golden Mean/Geometrical Interpretation

Let $\Box ADEB$ be a square. Let $\Box ADFC$ be a rectangle such that: :$AC : AD = AD : BC$ where $AC : AD$ denotes the ratio of $AC$ to $AD$. :200px Then if you remove $\Box ADEB$ from $\Box ADFC$, the sides of the remaining rectangle have the same ratio as the sides of the original one. Thus if $AC = \phi$ and $AD = 1$ we see that this reduces to: :$\phi : 1 = 1 : \phi - 1$ where $\phi$ is the golden mean.


Definition:Golden Mean/Definition 1

Let a line segment $AB$ be divided at $C$ such that: :$AB : AC = AC : BC$ Then the '''golden mean''' $\phi$ is defined as: :$\phi := \dfrac {AB} {AC}$


Definition:Golden Mean/One Minus Golden Mean

The number: :$1 - \phi$ is often denoted $\hat \phi$.


Definition:Golden Mean/Definition 2

The '''golden mean''' is the unique positive real number $\phi$ satisfying: :$\phi = \dfrac {1 + \sqrt 5} 2$


Definition:Golden Mean/Definition 3

The '''golden mean''' is the unique positive real number $\phi$ satisfying: :$\phi = \dfrac 1 {\phi - 1}$


Definition:Basis Expansion/Positive Real Numbers

Let $x \in \R$ be a real number such that $x \ge 0$. Let $b \in \N: b \ge 2$. Let us define the recursive sequence: :$\forall n \in \N: n \ge 1: \sequence {f_n} = \begin {cases} b \paren {x - \floor x} & : n = 1 \\ b \paren {f_{n - 1} - \floor {f_{n - 1} } } & : n > 1 \end{cases}$ Then we define: :$\forall n \in \N: n \ge 1: \sequence {d_n} = \floor {f_n}$ It follows from the method of construction and the definition of the floor function that: :$\forall n: 0 \le f_n < b$ and hence $\forall n: 0 \le d_n \le b - 1$ :$\forall n: f_n = 0 \implies f_{n + 1} = 0$ and hence $d_{n + 1} = 0$. Hence we can express $x = \floor x + \displaystyle \sum_{j \mathop \ge 1} \frac {d_j} {b^j}$ as: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$ where: :$s = \floor x$ :it is not the case that there exists $m \in \N$ such that $d_M = b - 1$ for all $M \ge m$. (That is, the sequence of digits does not end with an infinite sequence of $b - 1$.) This is called the '''expansion of $x$ in base $b$'''. The generic term for such an expansion is a '''basis expansion'''. It follows from the Division Theorem that for a given $b$ and $x$ this expansion is unique.


Definition:Basis Expansion/Negative Real Numbers

Let $x \in \R: x < 0$. We take the absolute value $y$ of $x$, that is: :$y = \size x$ Then we take the '''expansion of $y$ in base $b$''': :$\size {s . d_1 d_2 d_3 \ldots}_b$ where $s = \floor y$. Finally, the '''expansion of $x$ in base $b$''' is defined as: :$-\sqbrk {s . d_1 d_2 d_3 \ldots}_b$


Definition:Less Than (Real Numbers)

Let $\R_{>0}$ denote the set of  strictly positive real numbers. Let $x, y \in \R$. Then we write $x < y$ {{iff}}: :$ y - x \in \R_{>0}$ and we say that $x$ is '''less than''' $y$.


Definition:Strictly Positive/Real Number/Definition 2

The '''strictly positive real numbers''', written $R_{>0}$, is the subset of $\R$ that satisfies the following: {{begin-axiom}} {{axiom | n = \R_{>0} 1         | lc= Closure under addition         | q = \forall x, y \in \R_{>0}         | m = x + y \in \R_{>0} }} {{axiom | n = \R_{>0} 2         | lc= Closure under multiplication         | q = \forall x, y \in \R_{>0}         | m = xy \in \R_{>0} }} {{axiom | n = \R_{>0} 3         | lc=  Trichotomy         | q = \forall x \in \R         | m = x \in \R_{>0} \lor x = 0 \lor -x \in \R_{>0} }} {{end-axiom}}


Definition:Strictly Positive/Real Number/Definition 1

The '''strictly positive real numbers''' are the set defined as: :$\R_{>0} := \set {x \in \R: x > 0}$ That is, all the real numbers that are strictly greater than zero.


Definition:Decimal Expansion/Decimal Point

The dot that separates the integer part from the fractional part of $x$ is called the '''decimal point'''. That is, it is the radix point when used specifically for a base $10$ representation.


Definition:Number Base/Integers

Let $n \in \Z$ be an integer. Let $b$ be any integer such that $b > 1$. By the Basis Representation Theorem, $n$ can be expressed uniquely in the form: :$\displaystyle n = \sum_{j \mathop = 0}^m r_j b^j$ where: :$m$ is such that $b^m \le n < b^{m + 1}$ :all the $r_j$ are such that $0 \le r_j < b$. <noinclude> {{MissingLinks|The bounds on $n$ are not stated as part of the Basis Representation Theorem. Is there some other link to these bounds?}}</noinclude> <noinclude>{{Improve|The definition is incomplete as the Basis Representation Theorem is only stated for strictly positive integers}}</noinclude> The number $b$ is known as the '''number base''' to which $n$ is represented. $n$ is thus described as being '''(written) in base $b$'''. Thus we can write $\displaystyle n = \sum_{j \mathop = 0}^m {r_j b^j}$ as: :$\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$ or, if the context is clear: :${r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$


Definition:Number Base/Real Numbers

Let $x \in \R$ be a real number such that $x \ge 0$. Let $b \in \N: b \ge 2$. See the definition of Basis Expansion for how we can express $x$ in the form: :$x = \sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$ Then we express $m$ as for integers, and arrive at: :$x = \sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0 \cdotp d_1 d_2 d_3 \ldots}_b$ or, if the context is clear: :$r_m r_{m - 1} \ldots r_2 r_1 r_0 \cdotp d_1 d_2 d_3 \ldots_b$


Definition:Number Base/Integer Part

In the basis expansion: :$x = \left[{r_m r_{m-1} \ldots r_2 r_1 r_0 . d_1 d_2 d_3 \ldots}\right]_b$ the part $r_m r_{m-1} \ldots r_2 r_1 r_0$ is known as the '''integer part'''.


Definition:Number Base/Fractional Part

In the basis expansion: :$x = \sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0 . d_1 d_2 d_3 \ldots}_b$ the part $.d_1 d_2 d_3 \ldots$ is known as the '''fractional part'''.


Definition:Number Base/Radix Point

In the basis expansion: :$x = \sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0 \cdotp d_1 d_2 d_3 \ldots}_b$ the dot that separates the integer part from the fractional part is called the '''radix point'''.


Definition:Natural Logarithm/Positive Real/Definition 3

Let $x \in \R$ be a real number such that $x > 0$. The '''(natural) logarithm''' of $x$ is defined as: :$ \displaystyle \ln x := \lim_{n \mathop \to \infty} n \paren {\sqrt [n] x - 1}$


Definition:General Logarithm/Binary

Logarithms base $2$ are becoming increasingly important in computer science. They are often referred to as '''binary logarithms'''.


Definition:Pointwise Equicontinuous

Let $X = \left({A, d}\right)$ and $Y = \left({B, \rho}\right)$ be metric spaces. Let $\left\langle{f_i}\right\rangle_{i \mathop \in I}$ be a family of mappings $f_i: X \to Y$. Then $\left\langle{f_i}\right\rangle_{i \mathop \in I}$ is said to be '''pointwise equicontinuous''' at $x_0 \in A$ {{iff}}: :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall i \in I: \forall x \in A: d \left({x, x_0}\right) < \delta \implies \rho \left({f_i \left({x}\right), f_i \left({x_0}\right)}\right) < \epsilon$


Definition:Uniformly Equicontinuous

Let $X = \left({A, d}\right)$ and $Y = \left({B, \rho}\right)$ be metric spaces. Let $\left\langle{f_i}\right\rangle_{i \mathop \in I} $ be a family of mappings $f_i: X \to Y$. Then $\left\langle{f_i}\right\rangle_{i \mathop \in I}$ is said to be '''uniformly equicontinuous''' on $S \subseteq A$ {{iff}}: :$\forall \epsilon \in \R_{>0} : \exists \delta \in \R_{>0}: \forall i \in I: \forall x, y \in S : d \left({x, y}\right) < \delta \implies \rho \left({f_i \left({x}\right), f_i \left({y}\right) }\right) < \epsilon$


Definition:Discontinuity of the First Kind

Let $X$ be an open subset of $\R$. Let $f: X \to Y$ be a real function. Let $f$ be discontinuous at some point $c \in X$. Then $c$ is called a '''discontinuity of the first kind''' of $f$ {{iff}}: :$\displaystyle \lim_{x \mathop \to c^-} \map f x$ and $\displaystyle \lim_{x \mathop \to c^+} \map f x$ exist where $\displaystyle \lim_{x \mathop \to c^-} \map f x$ and $\displaystyle \lim_{x \mathop \to c^+} \map f x$ denote the limit from the left and limit from the right at $c$ respectively.


Definition:Jump Discontinuity

Let $X$ be an open subset of $\R$. Let $f: X \to Y$ be a real function. Let $f$ be discontinuous at some point $c \in X$. Then $c$ is called a '''jump discontinuity''' of $f$ {{iff}}: :$\displaystyle \lim_{x \mathop \to c^-} \map f x$ and $\displaystyle \lim_{x \mathop \to c^+} \map f x$ exist and are not equal where $\displaystyle \lim_{x \mathop \to c^-} \map f x$ and $\displaystyle \lim_{x \mathop \to c^+} \map f x$ denote the limit from the left and limit from the right at $c$ respectively. === Jump === {{:Definition:Jump Discontinuity/Jump}}


Definition:Summation/Indexed

The composite is called the '''summation''' of $\tuple {a_1, a_2, \ldots, a_n}$, and is written: :$\displaystyle \sum_{j \mathop = 1}^n a_j = \paren {a_1 + a_2 + \cdots + a_n}$


Definition:Summation/Inequality

The '''summation''' of $\left({a_1, a_2, \ldots, a_n}\right)$ can be written: :$\displaystyle \sum_{1 \mathop \le j \mathop \le n} a_j = \left({a_1 + a_2 + \cdots + a_n}\right)$


Definition:Summation/Propositional Function

Let $R \left({j}\right)$ be a propositional function of $j$. Then we can write the '''summation''' as: :$\displaystyle \sum_{R \left({j}\right)} a_j = \text{ The sum of all $a_j$ such that $R \left({j}\right)$ holds}$. If more than one propositional function is written under the summation sign, they must ''all'' hold.


Definition:Summation/Index Variable

Consider the '''summation''', in either of the three forms: :$\displaystyle \sum_{j \mathop = 1}^n a_j \qquad \sum_{1 \mathop \le j \mathop \le n} a_j \qquad \sum_{\map R j} a_j$ The variable $j$, an example of a bound variable, is known as the '''index variable''' of the summation.


Definition:Summation/Infinite

Let an infinite number of values of $j$ satisfy the propositional function $R \left({j}\right)$. Then the precise meaning of $\displaystyle \sum_{R \left({j}\right)} a_j$ is: :$\displaystyle \sum_{R \left({j}\right)} a_j = \left({\lim_{n \mathop \to \infty} \sum_{\substack {R \left({j}\right) \\ -n \mathop \le j \mathop < 0}} a_j}\right) + \left({\lim_{n \mathop \to \infty} \sum_{\substack {R \left({j}\right) \\ 0 \mathop \le j \mathop \le n} } a_j}\right)$ provided that both limits exist. If either limit ''does'' fail to exist, then the '''infinite summation''' does not exist.


Definition:Real Interval/Bounded

Let $I$ be an interval. Let $I$ be either open, half-open or closed. Then $I$ is said to be a '''bounded (real) interval'''.


Definition:Fiber of Truth

Let $P: X \to \set {\mathrm T, \mathrm F}$ be a propositional function defined on a domain $X$. The '''fiber of truth (under $P$)''' is the preimage, or '''fiber''', of $\mathrm T$ under $P$: :$\set {x \in X: \map P x = \mathrm T}$ That is, the elements of $X$ whose image under $P$ is $\mathrm T$.


Definition:Way Below Closure

Let $\struct {S, \preceq}$ be an ordered set. Let $x \in S$. The '''way below closure''' of $x$, denoted by $x^\ll$, is defined by :$x^\ll := \set {y \in S: y \ll x}$ where $y \ll x$ denotes that $y$ is way below $x$.


Definition:Axiom of Approximation

Let $\left({S, \preceq}\right)$ be an ordered set. '''Axiom of approximation''' says  :$\forall x \in S: x = \sup \left({x^\ll}\right)$ where $x^\ll$ denotes the way below closure of $x$.


Definition:Continuous Ordered Set

Let $\left({S, \preceq}\right)$ be an ordered set. {{explain|$x^\ll$}} Then $\left({S, \preceq}\right)$ is '''continuous''' {{iff}} :(for all elements $x$ of $S$: $x^\ll$ is directed) and :$\left({S, \preceq}\right)$ is up-complete and satisfies axiom of approximation.


Definition:Euler's Number/Base of Exponential

There is a number $x \in \R$ such that: :$\displaystyle \lim_{h \to 0} \frac{ x^{h} - 1 }{ h } = 1$ This number is called '''Euler's Number''' and is denoted $e$.


Definition:Midpoint-Convex

Let $f$ be a real function defined on a real interval $I$. $f$ is '''midpoint-convex''' {{iff}}: :$\forall x, y \in I: f \left({\dfrac {x + y} 2}\right) \le \dfrac {f \left({x}\right) + f \left({y}\right)} 2$


Definition:Midpoint-Concave

Let $f$ be a real function defined on a real interval $I$. $f$ is '''midpoint-concave''' {{iff}}: :$\forall x, y \in I: f \left({\dfrac {x + y} 2}\right) \ge \dfrac {f \left({x}\right) + f \left({y}\right)} 2$


Definition:Strictly Midpoint-Convex

Let $f$ be a real function defined on a real interval $I$. $f$ is '''strictly midpoint-convex''' {{iff}}: :$\forall x, y \in I : f \left({\dfrac {x + y} 2}\right) < \dfrac {f \left({x}\right) + f \left({y}\right)} 2$


Definition:Strictly Midpoint-Concave

Let $f$ be a real function defined on a real interval $I$. $f$ is '''strictly midpoint-concave''' {{iff}}: :$\forall x, y \in I : f \left({\dfrac {x + y} 2}\right) > \dfrac {f \left({x}\right) + f \left({y}\right)} 2$


Definition:Summation/Propositional Function/Iverson's Convention

Let $\displaystyle \sum_{R \left({j}\right)} a_j$ be the summation over all $a_j$ such that $j$ satisfies $R$. This can also be expressed: :$\displaystyle \sum_{j \mathop \in \Z} a_j \left[{R \left({j}\right)}\right]$ where $\left[{R \left({j}\right)}\right]$ is Iverson's convention.


Definition:Product Notation (Algebra)/Index

The composite is called the '''product''' of $\tuple {a_1, a_2, \ldots, a_n}$, and is written: :$\displaystyle \prod_{j \mathop = 1}^n a_j = \paren {a_1 \times a_2 \times \cdots \times a_n}$


Definition:Product Notation (Algebra)/Inequality

The '''product''' of $\left({a_1, a_2, \ldots, a_n}\right)$ can be written: :$\displaystyle \prod_{1 \mathop \le j \mathop \le n} a_j = \left({a_1 \times a_2 \times \cdots \times a_n}\right)$


Definition:Product Notation (Algebra)/Propositional Function

Let $R \left({j}\right)$ be a propositional function of $j$. Then we can write: :$\displaystyle \prod_{R \left({j}\right)} a_j = \text{ The product of all } a_j \text{ such that } R \left({j}\right) \text{ holds}$. If more than one propositional function is written under the product sign, they must ''all'' hold.


Definition:Product Notation (Algebra)/Infinite

Let an infinite number of values of $j$ satisfy the propositional function $\map R j$. Then the precise meaning of $\displaystyle \prod_{\map R j} a_j$ is: :$\displaystyle \prod_{\map R j} a_j = \paren {\lim_{n \mathop \to \infty} \prod_{\substack {\map R j \\ -n \mathop \le j \mathop < 0} } a_j} \times \paren {\lim_{n \mathop \to \infty} \prod_{\substack {\map R j \\ 0 \mathop \le j \mathop \le n} } a_j}$ provided that both limits exist. If either limit ''does'' fail to exist, then the '''infinite product''' does not exist.


Definition:Product Notation (Algebra)/Index Variable

Consider the '''product''', in either of the three forms: :$\displaystyle \prod_{j \mathop = 1}^n a_j \qquad \prod_{1 \mathop \le j \mathop \le n} a_j \qquad \prod_{R \left({j}\right)} a_j$ The variable $j$, an example of a bound variable, is known as the '''index variable''' of the product.


Definition:Summation/Inequality/Multiple Indices

Let $\displaystyle \sum_{0 \mathop \le j \mathop \le n} a_j$ denote the '''summation''' of $\left({a_0, a_1, a_2, \ldots, a_n}\right)$. Summands with multiple indices can be denoted by propositional functions in several variables, for example: :$\displaystyle \sum_{0 \mathop \le i \mathop \le n} \left({\sum_{0 \mathop \le j \mathop \le n} a_{i j} }\right) = \sum_{0 \mathop \le i, j \mathop \le n} a_{i j}$ :$\displaystyle \sum_{0 \mathop \le i \mathop \le n} \left({\sum_{0 \mathop \le j \mathop \le i} a_{i j} }\right) = \sum_{0 \mathop \le j \mathop \le i \mathop \le n} a_{i j}$


Definition:Telescoping Series

A '''telescoping series''' is a series whose partial sums eventually only have a fixed number of terms after cancellation through algebraic manipulation.


Definition:Norm of Subdivision

Let $\closedint a b$ be a closed interval of the set of real numbers $\R$. Let $\Delta = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a finite subdivision of $\closedint a b$. Then the '''norm''' of $\Delta$ is defined as: :$\max \set {x_1 - x_0, x_2 - x_1, \ldots, x_n - x_{n - 1} }$ and is denoted $\norm \Delta$.


Definition:Continuous Extension/Real Function

Let $A$, $B \subseteq \R$ be subsets of the real numbers such that $A \subseteq B$. Let $f: A \to \R$ and $g: B \to \R$ be continuous real functions. Then $g$ is a '''continuous extension''' of $f$ {{iff}}: :$\forall x \in A : f \left({x}\right) = g \left({x}\right)$


Definition:Definite Integral/Riemann

Let $\Delta$ be a finite subdivision of $\closedint a b$, $\Delta = \set {x_0, \ldots, x_n}$, $x_0 = a$ and $x_n = b$. Let there for $\Delta$ be a corresponding sequence $C$ of sample points $c_i$, $C = \tuple {c_1, \ldots, c_n}$, where $c_i \in \closedint {x_{i - 1} } {x_i}$ for every $i \in \set {1, \ldots, n}$. Let $\map S {f; \Delta, C}$ denote the Riemann sum of $f$ for the subdivision $\Delta$ and the sample point sequence $C$. Then $f$ is said to be '''(properly) Riemann integrable''' on $\closedint a b$ {{iff}}: :$\exists L \in \R: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall$ finite subdivisions $\Delta$ of $\closedint a b: \forall$ sample point sequences $C$ of $\Delta: \norm \Delta < \delta \implies \size {\map S {f; \Delta, C} - L} < \epsilon$ where $\norm \Delta$ denotes the norm of $\Delta$. The real number $L$ is called the '''Riemann integral''' of $f$ over $\closedint a b$ and is denoted: :$\displaystyle \int_a^b \map f x \rd x$


Definition:Definite Integral/Darboux

Let $f$ be bounded on $\closedint a b$. Suppose that: :$\displaystyle \underline {\int_a^b} \map f x \rd x = \overline {\int_a^b} \map f x \rd x$ where $\displaystyle \underline {\int_a^b}$ and $\displaystyle \overline {\int_a^b}$ denote the lower integral and upper integral, respectively. Then the '''definite (Darboux) integral of $f$ over $\closedint a b$''' is defined as: :$\displaystyle \int_a^b \map f x \rd x = \underline {\int_a^b} \map f x \rd x = \overline {\int_a^b} \map f x \rd x$ $f$ is formally defined as '''(properly) integrable over $\closedint a b$ in the sense of Darboux''', or '''(properly) Darboux integrable over $\closedint a b$'''. More usually (and informally), we say: :'''$f$ is (Darboux) integrable over $\closedint a b$.'''


Definition:Auxiliary Relation

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $\mathcal R \subseteq S \times S$ be a relation on $S$. Then $\mathcal R$ is an '''auxiliary relation''' {{iff}} {{begin-axiom}} {{axiom | n = 1         | q = \forall x, y \in S         | m = \left({x, y}\right) \in \mathcal R \implies x \preceq y }} {{axiom | n = 2         | q = \forall x, y, z, u \in S         | m = x \preceq y \land \left({y, z}\right) \in \mathcal R \land z \preceq u \implies \left({x, u}\right) \in \mathcal R }} {{axiom | n = 3         | q = \forall x, y, z \in S         | m = \left({x, z}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x \vee y, z}\right) \in \mathcal R }} {{axiom | n = 4         | q = \forall x \in S         | m = \left({\bot, x}\right) \in \mathcal R }} {{end-axiom}}


Definition:Sign of Ordered Tuple

Let $n \in \N$ be a natural number such that $n > 1$. Let $\tuple {x_1, x_2, \ldots, x_n}$ be an ordered $n$-tuple of real numbers. Let $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be the product of differences of $\tuple {x_1, x_2, \ldots, x_n}$: :$\displaystyle \map {\Delta_n} {x_1, x_2, \ldots, x_n} = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j}$ The '''sign of $\tuple {x_1, x_2, \ldots, x_n}$''' is defined and denoted as: :$\map \epsilon {x_1, x_2, \ldots, x_n} := \map \sgn {\Delta_n}$ where $\sgn$ denotes the signum function. That is: :$\displaystyle \map \epsilon {x_1, x_2, \ldots, x_n} := \map \sgn {\prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j} }$ where: :$\map \sgn \pi := \sqbrk {x > 0} - \sqbrk {x < 0}$ :$\sqbrk {x > 0}$ etc. is Iverson's convention.


Definition:Floor Function/Definition 1

The '''floor function of $x$''' is defined as the supremum of the set of integers no greater than $x$: :$\floor x := \sup \set {m \in \Z: m \le x}$ where $\le$ is the usual ordering on the real numbers.