hcju/proofwikips huggingface dataset:
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.
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''' $F$  :$F$ 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'''  :for every subset $F$ of $S_1$, $f$ preserves the supremum of $F$.
Definition:Mapping Preserves Supremum/Join
$f$ '''preserves join'''  :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'''  :for every finite subset $F$ of $S_1$, $f$ preserves the supremum of $F$.
Definition:Mapping Preserves Supremum/Directed
$f$ '''preserves directed suprema'''  :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''' $F$  :$F$ 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'''  :for every subset $F$ of $S_1$, $f$ preserves the infimum of $F$.
Definition:Mapping Preserves Infimum/Meet
$f$ '''preserves meet'''  :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'''  :for every finite subset $F$ of $S_1$, $f$ preserves the infimum of $F$.
Definition:Mapping Preserves Infimum
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 Infimum/Filtered
$f$ '''preserves filtered infima'''  :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'''  it is aligned perpendicular to the horizon.
Definition:Vertical Line/Definition 2
A line is '''vertical'''  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'''  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'''  it is aligned parallel to the horizon:
Definition:Horizontal Line/Definition 1
A line is '''horizontal'''  it is perpendicular to a given vertical line.
Definition:Horizontal Line/Definition 3
A line is '''horizontal'''  it can be embedded within a horizontal plane.
Definition:Horizontal Plane
A plane is '''horizontal'''  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.
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 $\left({S, \preceq}\right)$  :$F$ 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 $\left({S, \preceq}\right)$ : :$I$ 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 .
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''' : :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'''  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:Stirling Numbers of the First Kind/Unsigned/Definition 1}}
Definition:Stirling Numbers of the First Kind/Signed
{{: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}}
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 $p$'''  $x^4 \equiv a \pmod p$ has a solution. That is, : : $\exists x \in \Z: x^4 \equiv a \pmod p$
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: :$\Delta f = 0$