Mathematics. Dictionary

A group in which the group operation is commutative.

The first of the two terms by which a point is referred to, in a system of fixed rectilinear coordinate (Cartesian coordinate) axes.

For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, \sqrt{a^2+b^2}. Denoted by vertical bars | \, |.

The branch of mathematics concerned with algebraic structures, such as groups, rings, and fields.

An angle greater than 0 degrees and less than 90 degrees.

A triangle all of the angles of which are acute.

The arithmetic operation of adding.

Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are inverse problems: given the size of the sumset A + B is small, what can we say about the structures of A and B? In the case of the integers, the classical Freiman's theorem provides a partial answer to this question in terms of multi-dimensional arithmetic progressions. Another typical problem is to find a lower bound for |A + B| in terms of |A| and |B|.…

A unary function that preserves the addition operation; a unary function f such that for any x and y: f(x + y) = f(x) + f(y).

Either of a pair of angles that share a common side

In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to cer…

a vector space having no origin

In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the polynomial ring k [ x 1 , … , x n ] . {\displaystyle k[x_{1},\ldots ,x_{n}].} An affine variety is an affine algebraic set which is not the union of two smaller algebraic sets; algebraically, this means that (the radical of) the ideal generated by the defining polynomials is prime.…

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984.…

In differential geometry, Alexandrov spaces with curvature ≥ k form a generalization of Riemannian manifolds with sectional curvature ≥ k, where k is some real number. By definition, these spaces are locally compact complete length spaces where the lower curvature bound is defined via comparison of geodesic triangles in the space to geodesic triangles in standard constant-curvature Riemannian surfaces. One can show that the Hausdorff dimension of an Alexandrov space with curvature ≥ k is either a non-negative integer or infinite.…

Elementary algebra: A system for representing and manipulating unknown quantities (variables) in equations.

A field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G).

An algebraic variety of one.

A mathematical equation having an algebraic expression on one or both sides, such as 2x + 7y = 3.

A branch of mathematics that studies algebraic varieties (solution sets of polynomial equations) and their generalisations, using techniques from both algebra (chiefly commutative algebra) and geometry.

A complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers; equivalently, a complex number (or element of a number field) that is a root of a monic polynomial whose coefficients are rational numbers.

Any one of the numerous types of mathematical object studied in algebra and especially in universal algebra.

The branch of mathematics that uses tools from abstract algebra to study topological spaces.

A collection of ordered steps that solve a mathematical problem. A precise step-by-step plan for a computational procedure that possibly begins with an input value and yields an output value in a finite number of steps.

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt(n).

An infinite series in the form of \sum_{n=0}^\infty (-1)^n a_n or \sum_{n=0}^\infty (-1)^{n+1} a_n, where a_n > 0 for all n.

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have x ( x y ) = ( x x ) y {\displaystyle x(xy)=(xx)y} ( y x ) x = y ( x x ) {\displaystyle (yx)x=y(xx)} for all x and y in the algebra. Every associative algebra is alternative, given that alternativity is simply a weak form of associativity. However, so too are some strictly non-associative algebras such…

The line perpendicularly connecting a figure’s vertex, especially a triangle’s apex, to the side opposite to the vertex.

In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English, although nowadays German mathematicians use the term "Mittelbare Gruppe") in response to the Banach–Tarski paradox.…

Anabelian geometry is a theory in arithmetic geometry which describes the way in which the algebraic fundamental group of a certain arithmetic variety X, or some related geometric object, can help to recover X. The first results for number fields and their absolute Galois groups were obtained by Jürgen Neukirch, Masatoshi Gündüz Ikeda, Kenkichi Iwasawa, and Kôji Uchida (Neukirch–Uchida theorem, 1969), prior to conjectures made about hyperbolic curves over number fields by Alexander Grothendieck.…

Any smooth (infinitely differentiable) function f, defined on an open set D\subseteq \mathbb{C}\ (\textit{ or }\subseteq\mathbb{R}), whose value in some neighbourhood of any given point x_0 \in D is given by the Taylor series \textstyle \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^{n}.

The subbranch of geometry that investigates geometrical figures via the mathematical properties of their representations in a coordinate system.

A figure formed by two rays which start from a common point (a plane angle) or by three planes that intersect (a solid angle).

A ray that divides an angle into two equal parts.

Rotation opposite to the direction of the hands of a clock; counterclockwise.

A function whose derivative is a given function; an indefinite integral

An imprecise solution or result that is adequate for a defined purpose.

The study of how functions can best be approximated with simpler functions, and how the errors introduced in this way can be quantitatively characterized.

A continuous part of the circumference of a circle (circular arc) or of another curve.

Any of several single-valued or multivalued functions that are inverses of the cosine function.

Any of several single-valued or multivalued functions that are inverses of the sine function. Symbol: arcsin, sin−1

Any of several single-valued or multivalued functions that are inverses of the tangent function. Symbol: arctan, tan-1

A measure of the extent of a surface; it is measured in square units.

A series of propositions organized so that the final proposition is a conclusion which is intended to follow logically from the preceding propositions, which function as premises.

Of, relating to, or using arithmetic; arithmetical.

In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis.

In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example S L 2 ( Z ) . {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} ).} They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. They also give rise to very interesting examples of Riemannian manifolds and hence are objects of interest in differential geometry and topology. Final…

The measure of central tendency of a set of values computed by dividing the sum of the values by their number.

A sequence in which each term except the first is obtained from the previous by adding a constant value, known as the common difference of the arithmetic progression.

In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition.…

A straight line which a curve approaches arbitrarily closely as it goes to infinity. The limit of the curve; its tangent "at infinity".

Any of several methods for describing and analyzing the behaviour of a system at its limits

In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of l 2 {\displaystyle l^{2}} -Betti numbers.

An isomorphism of a mathematical object or system of objects onto itself.

Any measure of central tendency, especially any mean, the median or the mode.

One of the axioms of set theory, equivalent to the statement that an arbitrary direct product of non-empty sets is non-empty; any version of said axiom, for example specifying the cardinality of the number of sets from which choices are made.

A set of axioms from which theorems can be derived.

A fixed one-dimensional figure, such as a line or arc, with an origin and orientation and such that its points are in one-to-one correspondence with a set of numbers; an axis forms part of the basis of a space or is used to position and locate data in a graph (a coordinate axis)

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy ‖ x y ‖ ≤ ‖ x ‖ ‖ y ‖ for all x , y ∈ A . {\displaystyle \|x\,y\|\ \leq \|x\|\,\|y\|\quad {\text{ for all }}x,y\in A.} This ensures that the multiplication operation is continuous with respect to the metric topology.…

The lowest side of a triangle or other polygon, or the lowest face of a cone, pyramid or other polyhedron laid flat.

In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation ∂ w ∂ z ¯ = μ ∂ w ∂ z . {\displaystyle {\frac {\partial w}{\partial {\bar {z}}}}=\mu {\frac {\partial w}{\partial z}}.} for w {\displaystyle w} a complex distribution of the complex variable z {\displaystyle z} in some open set U {\displaystyle U} , with derivatives that are locally L2, and where μ {\displaystyle \mu } is a given complex func…

In mathematics, a Berkovich space, introduced by Berkovich (1990), is a version of an analytic space over a non-Archimedean field (e.g. p-adic field), refining Tate's notion of a rigid analytic space.

In mathematics, Bernstein's theorem may refer to: Bernstein's theorem about the Sato–Bernstein polynomial Bernstein's problem about minimal surfaces Bernstein's theorem on monotone functions Bernstein's theorem (approximation theory) Bernstein's theorem (polynomials) Bernstein's lethargy theorem Bernstein–von Mises theorem Cantor–Bernstein theorem in set theory and order theory Cantor–Bernstein–Schroeder theorem in set theory Bernstein–Kushnirenk…

In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x {\displaystyle x} in a Hilbert space with respect to an orthonormal sequence. The inequality is named for F. W. Bessel, who derived a special case of it in 1828. Conceptually, the inequality is a generalization of the Pythagorean theorem to infinite-dimensional spaces. It states that the "energy" of a vector x {\displaystyle x} , given by ‖ x ‖ 2 {\displaystyle \|x\|^{2}} , is greater than or equal to the sum of the energies of its projections onto a set of perpendicular basis directions.…

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden "qualitative" or topological change in its behavior.…

A function of two arguments from the same vector space which maps onto a field of scalars, which acts like a linear form with respect to either one of its arguments when the other one is held constant.

Billion is a word (in several languages) for a large number with two distinct definitions: 1,000,000,000, i.e. one thousand million, or 109 (ten to the ninth power), as defined on the short scale (also known as a milliard on the long scale). This is now the most common sense of the word in all varieties of English; it has long been established in American English and has since become common in Britain and other English-speaking countries as well. 1,000,000,000,000, i.e.…

a coefficient of any of the terms in the expansion of the binomial (x+y)n, defined by {n \choose k} = \frac{n!}{k!(n-k)!}, read as "n choose k"

The discrete probability distribution of the number of successes in a sequence of n independent trials, each of which yields success with probability p.

A root-finding method that applies to any continuous functions for which one knows two values with opposite signs, and consists of repeatedly bisecting the interval defined by these values and then selecting the subinterval in which the function changes sign, and therefore must contain a root.

A line or curve that bisects or divides a line segment, angle, or other figure into two equal parts.

In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as blocks, chosen such that number of occurrences of each element satisfies certain conditions making the collection of blocks exhibit symmetry (balance). Block designs have applications in many areas, including experimental design, finite geometry, physical chemistry, software testing, cryptography, and algebraic geometry.…

In geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that a weak, algebraic notion of equivalence (namely, homotopy equivalence) should imply a stronger, topological notion (namely, homeomorphism).

A rectangular object in any number of dimensions.

BPP may refer to:

In mathematics, the braid group on n strands (denoted B n {\displaystyle B_{n}} ), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see § Introduction). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem…

In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer in the context of the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the general linear group in Schur–Weyl duality.

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often itself called "Brownian motion", even in mathematical sources. This motion pattern typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume.…

In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) G = B W B {\displaystyle G=BWB} of certain algebraic groups G = B W B {\displaystyle G=BWB} into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the Schubert cell decomposition of flag varieties: see Weyl group for this.…

In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Buildings were initially introduced by Jacques Tits as a means to understand the structure of isotropic reductive linear algebraic groups over arbitrary fields. The more specialized theory of Bruhat–Tits bu…

The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory, and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants that differ in the additional conditions imposed on the orders of the group elements (see bounded and restricted below).…

Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma, or the orbit-counting theorem, is a result in group theory that is often useful in taking account of symmetry when counting mathematical objects. It was discovered by Augustin Louis Cauchy and Ferdinand Georg Frobenius, and became well known after William Burnside quoted it. The result enumerates orbits of a symmetry group acting on some objects: that…

In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties: A is a topologically closed set in the norm topology of operators. A is closed under the operation of taking adjoints of operators. A…

In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by Eugenio Calabi (1954, 1957). It was proved by Shing-Tung Yau (1977, 1978), who received the Fields Medal and Oswald Veblen Prize in part for his proof. His work, principally an analysis of an elliptic partial differential equation known as the complex Monge–Ampère equation, was an influential early result in the field of geometric analysis.…

Any formal system in which symbolic expressions are manipulated according to fixed rules.

In mathematics, the canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle \,\!\Omega ^{n}=\omega } , which is the n {\displaystyle n} th exterior power of the cotangent bundle Ω {\displaystyle \Omega } on V {\displaystyle V} . Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle T ∗ V {\displaystyle T^{*}V} .…

A generalized kind of number used to denote the size of a set, including infinite sets.

In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.

In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs make them particularly good candida…

In abstract algebra, a cellular algebra is a finite-dimensional associative algebra A with a distinguished cellular basis which is particularly well-adapted to studying the representation theory of A.

The degree Celsius is the unit of temperature on the Celsius temperature scale (originally known as the centigrade scale in English), one of two temperature scales used in the International System of Units (SI), the other being the closely related Kelvin scale. The degree Celsius (symbol: °C) can refer to a specific point on the Celsius temperature scale or to a difference or range between two temperatures. It is named after the Swedish astronomer Anders Celsius (1701–1744), who proposed the first version of it in 1742.…

The point in the interior of any figure of any number of dimensions that has as its coordinates the arithmetic mean of the coordinates of all points on the perimeter of the figure (or of all points in the interior for a center of volume).

Any of various similar theorems.

The point of intersection of the three medians of a given triangle; the point whose (Cartesian) coordinates are the arithmetic mean of the coordinates of the three vertices.

A formula for computing the derivative of the functional composition of two or more functions.

In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves.…

In mathematics, a characteristic class is a way of associating to each principal bundle of a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry.…

The polynomial produced from a given square matrix by first subtracting the appropriate identity matrix multiplied by an indeterminant and then calculating the determinant.

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten invariants. Chern classes were introduced by Shiing-Shen Chern (1946).

A line segment between two points of a curve.

In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via the Landweber exact functor theorem. Typical theories i…

The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to study the four color problem. It was generalised to the Tutte polynomial by Hassler Whitney and W. T. Tutte, linking it to the Potts model of statistical physics.

A two-dimensional geometric figure, a line, consisting of the set of all those points in a plane that are equally distant from a given point (center).

The group of all complex numbers of unit modulus, with the group operation being complex multiplication.

The line that bounds a circle or other two-dimensional figure.

In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The r…

In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing.…

Rotation in the same direction as the hands of a clock.

an interval in the real number line which contains its supremum and infimum.

A set whose complement is open.

A constant by which an algebraic term is multiplied.

The result of a number being divided by one of its factors.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels.…

The coequalizer of f and the zero morphism from X to Y, denoted \operatorname{coker}f.

The vector space generated by all the column vectors of a matrix

a branch of mathematics that studies (usually finite) collections of objects that satisfy specified criteria

Any integer that is a common multiple of the denominators of two or more fractions.

A number that can divide two different numbers, without leaving a remainder.

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations.…

In functional analysis, a branch of mathematics, a compact operator is a linear operator that behaves, in several important respects, like a finite-dimensional operator such as a matrix. In infinite-dimensional spaces, bounded sets are usually not compact, and bounded sequences need not have convergent subsequences. Compact operators partly restore this finite-dimensional behavior by sending bounded sets to sets whose closures are compact, or equivalently, in normed spaces, by sending bounded sequences to sequences with convergent subsequences.…

In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry.

A drafting instrument with two hinged legs, used for drawing circles and arcs and for measuring distances.

An angle which, together with a given angle, makes a right angle.

The branch of mathematics that studies holomorphic functions.

Of a complex number x, the complex number \overline x formed by changing the sign of the imaginary part: The complex conjugate of a + bi is a - bi.

A number of the form a + bi, where a and b are real numbers and i denotes the imaginary unit.

A representation of the space of complex numbers.

A function of one or more independent variables, at least one of which is itself a function of one or more other independent variables; a function of function(s).

A (nonzero) natural number that is expressible as the product of two (or more) natural numbers other than itself and 1.

Interest, as on a loan or a bank account, that is calculated on the total on the principal plus accumulated unpaid interest.

Condensed mathematics is a theory developed by Dustin Clausen and Peter Scholze which replaces a topological space by a certain sheaf of sets, in order to solve some technical problems of doing homological algebra on topological groups. Essentially the same notion was also introduced by Barwick and Haine under the name pyknotic set. The theory aims to unify various mathematical subfields, including topology, complex geometry, and algebraic geometry.…

The probability that an event will take place given the restrictive assumption that another event has taken place, or that a combination of other events has taken place. (Mathematically, the definition would be that the conditional probability of B given A is equal to the joint probability of A and B divided by the probability of A.)

A solid of revolution formed by rotating a triangle around one of its altitudes.

In frequentist inference, a confidence interval (CI) determines lower and upper bounds likely to contain (in repeated sampling) the true value of an unknown statistical parameter, such as a population mean. Rather than reporting a single point estimate (e.g. "the average screen time is 3 hours per day"), a confidence interval provides an interval, such as from 2 hours to 4 hours. A confidence interval is based on a pre-defined coverage probability, called confidence level, typically 95%.…

Having a difference divisible by a modulus.

Any of the four distinct shapes that are the intersections of a cone with a plane, namely the circle, ellipse, parabola and hyperbola.

The transpose of a matrix, after replacing each element with its complex conjugate.

Any topological space which cannot be written as the disjoint union of two or more nonempty open spaces.

In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a parallel and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an affine connection, the most elementary type of connection, gives a means for parallel transport of tangent vectors on a manifold from one point to another along a curve.…

A sequence of notes or chords that results from repeated shifts in pitch of the same interval.

A quantity that remains at a fixed value throughout a given discussion or operation.

A characteristic property of a continuous function.

a function whose value at any point in its domain is equal its limit at the same point

The integral of a function around a contour on the complex plane.

The process or property of approaching some limiting value; typically of an infinite series.

An infinite series whose partial sums converge.

A mathematical function where the line segment between any two points on the graph of the function lies above the graph in a vector space of at least two dimensions.

A set which contains any convex combination of any (finite) subset of its points.

A mathematical operation on two functions that produces a third that expresses how the shape of one is modified by the other; the integral of the product of the two functions after one is reflected about the y-axis and shifted along the x-axis.

In mathematics, Conway polynomial can refer to: the Alexander–Conway polynomial in knot theory the Conway polynomial (finite fields) the polynomial of degree 71 that has Conway's constant as its single positive real root

A number representing the position of a point along a line, arc, or similar one-dimensional figure.

A method of representing the position of any individual point in a space of given dimensionality as a tuple of numerical coordinates.

Copula is a Latinate term meaning 'link' which may refer to:

A proposition which follows easily from the statement or proof of another proposition.

In a right triangle, the reciprocal of the sine of an angle. Symbols: cosec, csc

In a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse. Symbol: cos

In mathematics, a Costas array can be regarded geometrically as a set of n points, each at the center of a square in an n×n square tiling such that each row or column contains only one point, and all of the n(n − 1)/2 displacement vectors between each pair of dots are distinct. This results in an ideal "thumbtack" auto-ambiguity function, making the arrays useful in applications such as sonar and radar. Costas arrays can be regarded as two-dimensional cousins of the one-dimensional Golomb ruler construction, and, as well as being of mathematical interest, have similar applications in experimental design and phased array radar engineering.…

In a right triangle, the reciprocal of the tangent of an angle. Symbols: cot, ctg or ctn

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian…

An example that counters a general rule; an exception to a general rule; a specific instance of the falsity of (and falsifiability of) any proposed universal applicability of a rule or statement.

A statistical measure defined as \scriptstyle\operatorname{Cov}(X, Y) = \operatorname{E}((X - \mu) (Y - \nu)) given two real-valued random variables X and Y, with expected values \scriptstyle E(X)\,=\,\mu and \scriptstyle E(Y)\,=\,\nu.

In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If p : X ~ → X {\displaystyle p:{\tilde {X}}\to X} is a covering, ( X ~ , p ) {\displaystyle ({\tilde {X}},p)} is said to be a covering space or cover of X {\displaystyle X} , and X {\displaystyle X} is said to be the base of the covering, or simply the base.…

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections.…

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-sides of the equations. It is named after Gabriel Cramer, who published the rule fo…

A point at which the derivative of a function is zero or undefined, such as a minimum, a maximum, a point of inflection or, in multivariate calculus, a saddle point.

The third power of a number, value, term or expression.

Of a number or expression, a number or expression the cube of which is equal to the given number or expression.

Relating to polynomials of the form ax^3 + bx^2 + cx + d .

The extent to which a subspace is curved within a metric space.

In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.

A one-dimensional figure of non-zero length; the graph of a continuous map from a one-dimensional space.

In mathematics, the curve complex is a simplicial complex C(S) associated to a finite-type surface S, which encodes the combinatorics of simple closed curves on S. The curve complex turned out to be a fundamental tool in the study of the geometry of the Teichmüller space, of mapping class groups and of Kleinian groups. It was introduced by W.J.Harvey in 1978.

In differential geometry, the cut locus of a point p on a manifold is the closure of the set of all other points on the manifold that are connected to p by two or more distinct shortest geodesics. More generally, the cut locus of a closed set X on the manifold is the closure of the set of all other points on the manifold connected to X by two or more distinct shortest geodesics.

In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called cells) of different dimensions in specific ways. The notion generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. CW complexes have better categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation (often with a much smaller complex).…

A surface created by projecting a closed two-dimensional curve along an axis intersecting the plane of the curve.

In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations. Since around 1970, D-module theory has been built up, mainly as a response to the ideas of Mikio Sato on algebraic analysis, and expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara index theorem of Masaki Kashiwara.…

In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold.…

A polygon with ten sides and ten angles.

Concerning numbers expressed in decimal or mathematical calculations performed using decimal.

A dot separating the whole part of a decimal number from its fractional part.

A reduction in the number of stitches, usually accomplished by suspending the stitch to be decreased from another existing stitch or by knitting it together with another stitch. See .

The integral of a function between an upper and lower limit

A unit of measurement of angle equal to of a circle's circumference.

The number or expression written below the line in a fraction (such as 2 in ½).

In an equation, the variable whose value depends on one or more variables in the equation.

In mathematics, the derived category D(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes.…

A scalar that encodes certain characteristics of a given transformation matrix; the unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value 1 for the unit matrix; abbreviated as: det.

A line joining non-adjacent vertices of a polygon.

A matrix in which the entries outside the main diagonal are all zero.

Any straight line between two points on the circumference of a circle that passes through the center of the circle; a chord that passes through the center of the circle.

The quality or attribute which is added to those of the genus to constitute a species; a differentia.

A mathematical quantity of the form \frac{f(a+h)-f(a)}{h} for some function f(x), having two main interpretations

An equation involving the derivatives of a function.

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f ( x ) d x {\displaystyle f(x)\,dx} is an example of a 1-form, and can be integrated over an interval [ a , b ] {\displaystyle [a,b]} contained in the domain of f {\displaystyle f} : ∫ a b f ( x ) d x .…

The study of geometry, especially geometric structures on differentiable manifolds, using techniques from calculus, linear algebra and multilinear algebra.

The process of applying the derivative operator to a function; of calculating a function's derivative.

A position in a sequence of numerals representing a place value in a positional number system.

A digital signature is a mathematical scheme for verifying the authenticity of digital messages or documents. A valid digital signature on a message gives any recipient confidence that the message came from a sender known to the recipient. In contrast, a message authentication code only gives confidence to a specific recipient that the message came from a known sender. Digital signatures are a type of public-key cryptography, and are commonly used for software distribution, financial transactions, contract management software, and in other cases where it is important to detect forgery or tampering.…

The number of independent coordinates needed to specify uniquely the location of a point in a space; also, any of such independent coordinates.

In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects. The direct limit of the objects A i {\displaystyle A_{i}} , where i {\displaystyle i} ranges over some directed set I {\displaystyle I} , is denoted by lim → ⁡ A i {\displaystyle \varinjlim A_{i}} .…

In mathematics, more specifically in algebra, the direct sum of a collection of abelian groups is an abelian group constructed by combining the given groups as described below. If the input abelian groups have additional structure (for example, are vector spaces, modules, or topological abelian groups), then the direct sum usually maintains that structure (as an example of the opposite, the direct sum of fields is not a field as it contains zero divisors).…

A graph in which the edges are ordered pairs, so that, if the edge (a, b) is in the graph, the edge (b, a) need not be in the graph and is distinct from (a, b) if it is.

A reduction from the regular price.

Collectively, most discrete or computer science-related branches of mathematics, such as graph theory and combinatorics.

An expression that gives information about the roots of a polynomial; for example, the expression D = b2 - 4ac determines whether the roots of the quadratic equation ax2 + bx + c = 0 are real and distinct (D > 0), real and equal (D = 0) or complex (D < 0).

The length of the shortest path between two points.

In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x ⋅ ( y + z ) = x ⋅ y + x ⋅ z {\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z} is always true in elementary algebra. For example, in elementary arithmetic, one has 2 ⋅ ( 1 + 3 ) = ( 2 ⋅ 1 ) + ( 2 ⋅ 3 ) . {\displaystyle 2\cdot (1+3)=(2\cdot 1)+(2\cdot 3).} Therefore, one would say that multiplication distributes over addition.…

The operator which maps a function F=(F1, ... F'n) from a n-dimensional vector space to itself to the function \sum_{i=1}^n \frac{\partial F_i}{\partial x_i}.

An infinite series whose partial sums are divergent.

A number or expression that is to be divided by another.

The process of dividing a number by another.

In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse; that is, an element usually denoted a–1, such that a a–1 = a–1 a = 1. So, (right) division may be defined as a / b = a b–1, but this notation is avoided, as one may have a b–1 ≠ b–1 a. A commutative division ring is a field.…

In an expression involving division, the number by which another number is being divided.

A polyhedron with twelve faces; the regular dodecahedron has regular pentagons as faces and is one of the Platonic solids.

The set of input (argument) values for which a function is defined.

In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricting the possible quadratic forms on the second cohomology group of a compact simply connected 4-manifold. Important consequences of this theorem include the existence of an exotic R4 and the failure of the smooth h-cobordism theorem in 4 dimensions.…

Twice as large; a number multiplied by two.

In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca) is a sort of generalization of a Drinfeld module, consisting roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it.…

The vector space which comprises the set of linear functionals of a given vector space.

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured.…

A one-dimensional face of a polytope. In particular, the joining line between two vertices of a polygon; the place where two faces of a polyhedron meet.

A scalar \lambda, such that there exists a non-zero vector \mathrm{x} (a corresponding eigenvector) for which the image of \mathrm{x} under a given linear transformation A is equal to the image of \mathrm{x} under multiplication by \lambda; i.e. A \mathrm{x} = \lambda \mathrm{x}.

A vector that is only scaled (not rotated out of its span) under a particular linear transformation; a left or right eigenvector depending on context; given a linear transformation A, a vector x such that Ax=λx [or xA=λx] for some scalar λ (called the eigenvalue).

Any function that is composed of algebraic functions, trigonometric functions, exponential functions and/or logarithmic functions, combined using addition, subtraction, multiplication and/or division

A closed curve, the locus of a point such that the sum of the distances from that point to two other fixed points (called the foci of the ellipse) is constant; equivalently, the conic section that is the intersection of a cone with a plane that does not intersect the base of the cone.

In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K2, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions (x, y) for: y 2 = x 3 + a x + b {\displaystyle y^{2}=x^{3}+ax+b} for some coefficients a and b in K.…

The unique set that contains no elements, denoted ∅ or }}.

Either of the two nodes of a graph of degree 1.

The set of all points lying on or above the function's graph.

Exactly identical, having the same value.

An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; in mathematical problems, equations describe various essential aspects of the problem, each of which contributes to the resolution of the problem in part.

A triangle having all three sides equal.

Any one of the subsets into which an equivalence relation partitions a set, each of these subsets containing all the elements of the set that are equivalent under the equivalence relation.

In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelate…

A binary relation that is reflexive, symmetric and transitive.

Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc.…

A sigmoid function giving the probability that a measurement, under the influence of normally distributed errors with standard deviation, is within a certain distance from the mean value.

A rough calculation or assessment of the value, size, or cost of something.

In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements.…

Euler's formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. It is named after Leonhard Euler, who mentioned the formula in his influential 1748 treatise Introductio in analysin infinitorum. Euler's formula states that, for any real number x, one has e i x = cos ⁡ x + i sin ⁡ x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively.…

In number theory, Euler's totient function counts the positive integers up to a given integer n {\displaystyle n} that are relatively prime to n {\displaystyle n} . It is written using the Greek letter phi as φ ( n ) {\displaystyle \varphi (n)} or ϕ ( n ) {\displaystyle \phi (n)} , and may also be called Euler's phi function. In other words, it is the number of integers k {\displaystyle k} in the range 1 ≤ k ≤ n {\displaystyle 1\leq k\leq n} for which the greatest common divisor gcd ( n , k ) {\displaystyle \gcd(n,k)} is equal to 1.…

Any function whose value is unchanged if the independent variable changes sign i.e. f(x) = f(-x)

An integer that is divisible by two.

In mathematics, an exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.

In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space X {\displaystyle X} and subspaces A {\displaystyle A} and U {\displaystyle U} such that U {\displaystyle U} is also a subspace of A {\displaystyle A} , the theorem says that under certain circumstances, we can cut out (excise) U {\displaystyle U} from both spaces such that the relative homologies of the pairs ( X ∖ U , A ∖ U ) {\displaystyle (X\setminus U,A\setminus U)} into ( X , A ) {\displaystyle (X,A)} are isomorphic.…

The weighted average of outcome values, using probability as the weighting function.

The number by which a value (called the base) is said to be raised to a power in exponentiation: for example, the 3 in 2^3=8.

Any function that has an exponent as an independent variable.

Any of a class of continuous probability distributions used to model the time between events that occur independently at a constant average rate.

Any function in which an independent variable is in the form of an exponent; they are the inverse functions of logarithms

A formal power series with one indeterminate, whose coefficients are fractions with factorial denominators (of index corresponding to the power of the indeterminate) and numerators which represent a sequence of numbers that is to be studied.

In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis. Important special cases include: exponential map (Riemannian geometry) for a manifold with a Riemannian metric, exponential map (Lie theory) from a Lie algebra to a Lie group, More generally, in a manifold with an affine connection, X ↦ γ X ( 1 ) {\displaystyle X\mapsto \gamma _{X}(1)} , where γ X {\displaystyle \gamma _{X}} is a geodesic with initial velocity X, is sometimes also called the exponential map.…

An arrangement of symbols denoting values, operations performed on them, and grouping symbols.

In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle V,} which has a product, called exterior product or wedge product and denoted with ∧ {\displaystyle \wedge } , such that v ∧ v = 0 {\displaystyle v\wedge v=0} for every vector v {\displaystyle v} in V . {\displaystyle V.} The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol ∧ {\displaystyle \wedge } and the fact that the product of two elements of V {\displaystyle V} is "outside" V .…

An angle formed between one side of a polygon and an extension of an adjacent side.

Extreme value theory or extreme value analysis (EVA) is the study of extremes in statistical distributions. It is widely used in many disciplines, such as structural engineering, finance, actuarial science, economics, earth sciences, traffic prediction, and geological engineering. For example, EVA might be used in the field of hydrology to estimate the probability of an unusually large flooding event, such as the 100-year flood. Similarly, for th…

Any of the flat bounding surfaces of a polyhedron; more generally, any of the bounding pieces of a polytope of any dimension.

Any of various objects multiplied together to form some whole.

The result of multiplying a given number of consecutive integers from 1 to the given number. In equations, it is symbolized by an exclamation mark (!). For example, 5! = 1 × 2 × 3 × 4 × 5 = 120.

The process of creating a list of factors.

In algebraic topology and category theory, factorization homology is a variant of topological chiral homology, motivated by an application to topological quantum field theory and cobordism hypothesis in particular. It was introduced by David Ayala, John Francis, and Nick Rozenblyum.

The Fahrenheit scale () is a temperature scale based on one proposed in 1724 by the physicist Daniel Gabriel Fahrenheit (1686–1736). It uses the degree Fahrenheit (symbol: °F) as the unit. Several accounts of how he originally defined his scale exist, but the original paper suggests the lower defining point, 0 °F, was established as the freezing temperature of a solution of brine made from a mixture of water, ice, and ammonium chloride (a salt).…

In number theory, Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as a p ≡ a ( mod p ) . {\displaystyle a^{p}\equiv a{\pmod {p}}.} For example, if a = 2 and p = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p; that is, if a is coprime to p, then Fermat's little theorem is equivalent to the statement that ap − 1 − 1 is an integer multiple of p, or in symbols: a p − 1 ≡ 1 ( mod p ) .…

In mathematics, and particularly topology, a fiber bundle (Commonwealth English: fibre bundle) is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space E {\displaystyle E} and a product space B × F {\displaystyle B\times F} is defined using a continuous surjective map, π : E → B , {\displaystyle \pi :E\to B,} that in small regions of E {\displaystyle E} behaves just like a projection from corresponding regions of B × F {\displaystyle B\times F} to B .…

The form of a syllogism with respect to the relative position of the middle term.

A difference between the value of a function evaluated at a number, and the value of the same function evaluated at a different number, a fixed distance from the first.

A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity.…

In algebraic geometry, a finite morphism between two affine varieties X , Y {\displaystyle X,Y} is a dense regular map which induces isomorphic inclusion k [ Y ] ↪ k [ X ] {\displaystyle k\left[Y\right]\hookrightarrow k\left[X\right]} between their coordinate rings, such that k [ X ] {\displaystyle k\left[X\right]} is integral over k [ Y ] {\displaystyle k\left[Y\right]} . This definition can be extended to the quasi-projective varieties, such th…

In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable. When G is not solvable, a similar role is played by the generalized Fitting subgroup F*, which is generated by the Fitting subgroup and the components of G.…

In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is valid for all abelian categories, as well as the category of groups. The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual to each other.

A value which is unchanged by a function or other mapping. Formally: a value x for which f(x) = x.

In mathematics, in particular in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e., f P : O Y , f ( P ) → O X , P {\displaystyle f_{P}\colon {\mathcal {O}}_{Y,f(P)}\to {\mathcal {O}}_{X,P}} is a flat map for all P in X. A map of rings A → B {\displaystyle A\to B} is called flat if it is a homomorphism that makes B a flat A-module. A morphism of schemes is called faithfully flat if it is both surjective and flat.…

A function that maps a real number to the largest integer that is not greater than it.

In mathematics, a p-dimensional foliation is a partition of a manifold into submanifolds, all of the same dimension p, locally modeled on the decomposition of Rn into the p-dimensional planes cut out by the equations x p + 1 = a p + 1 , … , x n = a n {\displaystyle x_{p+1}=a_{p+1},\ldots ,x_{n}=a_{n}} . The submanifolds are called the leaves of the foliation. The 3-sphere has a famous codimension-1 foliation called the Reeb foliation. The submanifolds are required to be connected and injectively immersed, but they are not required to be embedded.…

Any mathematical rule expressed symbolically.

A mathematical set that has a non-integer and constant Hausdorff dimension, corresponding to a geometric figure or object that is self-similar at arbitrarily small scales and thus has infinite complexity.

A ratio of two numbers (numerator and denominator), usually written one above the other and separated by a horizontal bar called the vinculum or, alternatively, in sequence on the same line and separated by a solidus (diagonal bar).

In linear algebra, a frame of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent. In the terminology of signal processing, a frame provides a redundant, stable way of representing a signal. Frames are used in error detection and correction and the design and analysis of filter banks and more generally in applied mathematics, computer science, and engineering.

In mathematics, a frame bundle is a principal fiber bundle F ( E ) {\displaystyle F(E)} associated with any vector bundle E {\displaystyle E} . The fiber of F ( E ) {\displaystyle F(E)} over a point x {\displaystyle x} is the set of all ordered bases, or frames, for E x {\displaystyle E_{x}} . The general linear group acts naturally on F ( E ) {\displaystyle F(E)} via a change of basis, giving the frame bundle the structure of a principal G L ( k , R ) {\displaystyle \mathrm {GL} (k,\mathbb {R} )} -bundle (where k is the rank of E {\displaystyle E} ).…

In mathematics, particularly in group theory, the Frattini subgroup Φ ( G ) {\displaystyle \Phi (G)} of a group G is the intersection of all maximal subgroups of G. For the case that G has no maximal subgroups, for example the trivial group {e} or a Prüfer group, it is defined by Φ ( G ) = G {\displaystyle \Phi (G)=G} . It is analogous to the Jacobson radical in the theory of rings, and intuitively can be thought of as the subgroup of "small elements" (see the "non-generator" characterization below).…

In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator T : X → Y between two Banach spaces with finite-dimensional kernel ker ⁡ T {\displaystyle \ker T} and finite-dimensional (algebraic) cokernel coker ⁡ T = Y / ran ⁡ T {\displaystyle \operatorname {coker} T=Y/\operatorname {ran} T} , and with closed range ran ⁡ T {\displaystyle \operatorname {ran} T} .…

In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance, the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1, 0) and (0, 1) as its basis.…

In mathematics, the free factor complex (sometimes also called the complex of free factors) is a free group counterpart of the notion of the curve complex of a finite type surface. The free factor complex was originally introduced in a 1998 paper of Allen Hatcher and Karen Vogtmann. Like the curve complex, the free factor complex is known to be Gromov-hyperbolic. The free factor complex plays a significant role in the study of large-scale geometr…

In mathematics, the free group F S {\displaystyle F_{S}} over a given set S {\displaystyle S} consists of all words that can be built from members of S {\displaystyle S} , considering two words to be different unless their equality follows from the group axioms (e.g. s t = s u u − 1 t {\displaystyle st=suu^{-1}t} but s ≠ t − 1 {\displaystyle s\neq t^{-1}} for s , t , u ∈ S {\displaystyle s,t,u\in S} ). The members of S {\displaystyle S} are called generators of F S {\displaystyle F_{S}} , and the number of generators is the rank of the free group.…

In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the behavior of simultaneously taking suspensions and increasing the index of the homotopy groups of the space in question. It was proved in 1937 by Hans Freudenthal. The theorem is a corollary of the homotopy…

In the mathematical field of differential geometry, a Frobenius manifold, introduced by Dubrovin, is a flat Riemannian manifold with a certain compatible multiplicative structure on the tangent space. The concept generalizes the notion of Frobenius algebra to tangent bundles. Frobenius manifolds occur naturally in the subject of symplectic topology, more specifically quantum cohomology. The broadest definition is in the category of Riemannian supermanifolds.…

In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducing. It can be used to leverage knowledge about representations of a subgroup to find and classify representations of "large" groups that contain them. It is named for Ferdinand Georg Frobenius, the inventor of the representation theory of finite groups.

A relation in which each element of the domain is associated with exactly one element of the codomain.

The branch of mathematics dealing with infinite-dimensional vector spaces, whose elements are actually functions, as well as generalizations such as Banach spaces and Hilbert spaces.

A category homomorphism; a morphism from a source category to a target category which maps objects to objects and arrows to arrows (either covariantly or contravariantly), in such a way as to preserve morphism composition and identities.

For a specified topological space, the group whose elements are homotopy classes of loops (images of some arbitrary closed interval whose endpoints are both mapped to a designated point) and whose group operation is concatenation.

In Galois theory, a branch of abstract algebra, the Galois group of a certain type of field extension is a symmetry group characterizing how it extends the base field. Each element of the Galois group is a transformation of the field extension which leaves each element of the base field fixed. This connection between fields and groups, given by the fundamental theorem of Galois theory, allows for group-theoretic tools to be used on problems in field theory, such as describing the solutions to quintic polynomials.…

A meromorphic function which generalizes the notion of factorial to complex numbers and has singularities at the nonpositive integers; any of certain generalizations or analogues of said function, such as extend the factorial to domains other than the complex numbers.

A method of reducing an augmented matrix to row echelon form.

In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized n-gons encompass as special cases projective planes (generalized triangles, n = 3) and generalized quadrangles (n = 4). Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the Moufang property have been completely classified by Tits and Weiss.…

A formal power series with one indeterminate, whose coefficients encode a sequence that can be studied by algebraic manipulation of the series; any one of several generalizations, such as to encode more than one sequence or use more than one indeterminate.

A natural number representing any of several related measures of the complexity of a given manifold or graph.

A course allowing the parallel-transport of vectors along a course that causes tangent vectors to remain tangent vectors throughout that course .

Either of two slightly different discrete probability distributions, each based on repetitions of a trial with "success" probability p: (1) the number X of trials required to obtain one success, or (2) the number Y = X − 1 of failed trials before the first success.

Geometric group theory is an area in mathematics devoted to the study of finitely generated groups by exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups can act non-trivially (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). An important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects.…

A measure of central tendency of a set of n values computed by extracting the nth root of the product of the values.

A sequence in which each term except the first is obtained from the previous by multiplying it by a constant value, known as the common ratio of the geometric progression.

Any infinite series whose terms are in geometric progression; any series of the form \textstyle a+ a r+ a r^2+ a r^3 \cdots = a \sum_{i=0}^\infty r^i, where a \neq 0, r \neq 0, and r \neq 1.

In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological space.…

The branch of mathematics dealing with spatial relationships.

In mathematics, a Golomb ruler is a set of marks at integer positions along a ruler such that no two pairs of marks are the same distance apart. The number of marks on the ruler is its order, and the largest distance between two of its marks is its length. Translation and reflection of a Golomb ruler are considered trivial, so the smallest mark is customarily put at 0 and the next mark at the smaller of its two possible values. Golomb rulers can be viewed as a one-dimensional special case of Costas arrays.…

In mathematical representation theory, a good filtration is a filtration of a representation of a reductive algebraic group G such that the subquotients are isomorphic to the spaces of sections F(λ) of line bundles λ over G/B for a Borel subgroup B. In characteristic 0 this is automatically true as the irreducible modules are all of the form F(λ), but this is not usually true in positive characteristic. Mathieu (1990) showed that the tensor product of two modules F(λ)⊗F(μ) has a good filtration, completing the results of Donkin (1985) who proved it in most cases and Wang (1982) who proved it in large characteristic.…

A differential operator that maps each point of a scalar field to a vector pointed in the direction of the greatest rate of change of the scalar. Notation for a scalar field φ: ∇&phi

A first-order iterative optimization algorithm for finding a local minimum of a differentiable function.

A data chart (graphical representation of data) intended to illustrate the relationship between a set (or sets) of numbers (quantities, measurements or indicative numbers) and a reference set, whose elements are indexed to those of the former set(s) and may or may not be numbers.

The branch of mathematics dealing with the properties of graphs (networks of vertices and edges).

The relation >, indicating that one value is larger than another.

The largest positive integer (respectively polynomial, element of a given ring) that is a divisor of each of a given set of integers (respectively polynomials, elements of a given ring).

In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in R 2 {\displaystyle \mathbb {R} ^{2}} ) bounded by C. It is the two-dimensional special case of Stokes' theorem (surface in R 3 {\displaystyle \mathbb {R} ^{3}} ). In one dimension, it is equivalent to the fundamental theorem of calculus. In two dimensions, it is equivalent to the divergence theorem. I…

Gromov's compactness theorem can refer to either of two mathematical theorems: Gromov's compactness theorem (geometry) stating that certain sets of Riemannian manifolds are relatively compact in the Gromov-Hausdorff metric Gromov's compactness theorem (topology) on the existence of limits of sequences of pseudoholomorphic curves

A mapping of a mathematical group to the permutations of an object that is compatible with the group operation.

The mathematical theory of groups.

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representa…

In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal basis. The Haar sequence is now recognised as the first known wavelet basis and is extensively used as a teaching example. The Haar sequence was proposed in 1909 by Alfréd Haar.…

A study of the representation of functions or signals as the superposition of basic waves, involving the notions of harmonic functions, trigonometric series, Fourier series, Fourier transforms, almost periodic functions, and others.

A function of two, three or n variables which is a solution to any of Laplace's equations

The divergent series whose terms are the reciprocals of the positive integers; the series \textstyle\sum_{n=1}^\infty \frac 1 n.

Any half-sphere, formed by a plane intersecting the center of a sphere. .

A polygon with seven sides and seven angles.

A polygon with six sides and six angles.

A graphical display of numerical data in the form of upright bars, with the area of each bar representing frequency.

a continuous bijection from one topological space to another, with continuous inverse.

specifically, such relationship in the context of the geometry of perspective.

A continuous deformation of one continuous function or map to another.

The number 100; ten tens.

One of a hundred equal parts of a whole.

A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone. The function y(x) = 1/x draws a hyperbola.

Any function that is a solution of the hypergeometric equation; usually an expansion of an hypergeometric series

The side of a right triangle opposite the right angle.

A regular icosahedron: one of the Platonic solids, all of whose faces are regular (equilateral) triangles

An equation which always holds true regardless of the choice of input variables.

A diagonal matrix all of whose diagonal elements are equal to 1.

A number of the form bi, where b is any real number and i denotes the imaginary unit.

A function defined by a (multivariable) implicit equation when one of the variables is regarded as the value of the function, especially where said equation is such that the value is not directly calculable from the other variables.

A vulgar fraction of which the numerator has a greater magnitude than the denominator, such as 3/2.

An integral where at least one of the endpoints is taken as a limit, either to a specific number or to infinity.

Any function of a real variable whose value increases (or is constant) as the variable increases.

The set of all functions whose derivative is a given function; the set of all antiderivatives of a given function.

In an equation, any variable whose value is not dependent on any other in the equation.

A statement which might be true for any natural number. In mathematical induction, such a statement is proven true for all natural numbers by first proving it for the base case and then proving that each case implies the next.

A statement that of two quantities one is specifically less than (or greater than) another. Symbol: or \leq or > or \geq or \ne, as appropriate.

Greater than any positive quantity or magnitude; limitless.

Any expression that represents the addition a countably infinite number of ordered summands, often explicitly indexed by the positive or nonnegative integers.

A number that has an infinite numerical value that cannot be counted.

The problem of solving an ordinary differential equation together with an initial condition (a set of specified values for the unknown function or its derivatives at some point).

A vector space which is additionally equipped with an inner product.

A number that is not a fraction; an element of the infinite and numerable set {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Any function that possesses a finite integral

A method of integration directly related to the rule for differentiation of products; can be written as \int u\ dv = uv - \int v\ du.

A point where a graph crosses an axis.

The price paid for obtaining, or price received for providing, money or goods in a credit transaction, calculated as a fraction of the amount or value of what was borrowed.

The inner angle between two sides of a polygon.

A theorem that states for each value between the least upper bound and greatest lower bound of the image of a continuous function there is a corresponding point in its domain that the function maps to that value.

The process of estimating the value of a function at a point from its values at nearby points.

Of two sets, to have at least one element in common.

The branch of algebra concerned with actions of groups on algebraic varieties from the point of view of their effect on functions.

The result of a circle inversion; the set of all such points; the curve described by such a set.

For a given function f, another function, denoted f−1, that reverses the mapping action of f; given a function f: X\rightarrow Y, a function g: Y\rightarrow X such that, \forall x\in X,\ f(x) = y \implies g(y)=x.

Of a matrix A, another matrix B such that A multiplied by B and B multiplied by A both equal the identity matrix.

Any real number that cannot be expressed as a ratio of two integers.

A morphism which has a two-sided inverse; the composition of the morphism and such an inverse yields either one of two identity morphisms (depending on the order of composition).

A triangle having at least two sides equal.

In multivariable calculus, the result of applying integrals to a function of more than one variable in such a way that each of the integrals considers some of the variables as given constants.

The probability distribution of a random variable whose sample space is the Cartesian product of the sample spaces of its (at least two) component random variables.

A branch of topology dealing with knots.

Any of several similar sets of conjectures applicable to restricted domains of number theory and/or geometry.

The smallest positive integer which is divisible by (equivalently, is an integer multiple of) each of a specified finite set of integers.

A method of fitting a straight line through a series of points such that the sum of the squares of the residuals is minimized

A proposition proved or accepted for immediate use in the proof of some other proposition.

The relation <, indicating that one value is smaller than another.

The set of values x for which a real-valued function f(x) is equal to a given constant.

algebraic expressions whose literal coefficients represent the same value regardless of the numbers assigned to them.

A value to which a sequence converges. Equivalently, the common value of the upper limit and the lower limit of a sequence: if the upper and lower limits are different, then the sequence has no limit (i.e., does not converge).

A test for determining whether an infinite series converges, which involves calculating the limit of the ratio between the series being tested and one known to converge or diverge.

An infinitely extending one-dimensional figure that has no curvature; one that has length but not breadth or thickness.

An integral whose integrand's domain is a curve.

A part of a straight line bounded by two points.

The branch of mathematics that deals with vectors, vector spaces, linear transformations and systems of linear equations.

a sum, each of whose summands is an appropriate vector times an appropriate scalar (or ring element)

The state of being linearly dependent

A polynomial equation of the first degree (such as x = 2y - 7)

Any function whose graph is a straight line: f(x)=ax+b

A linear mapping from a vector space to its field of scalars.

The branch of mathematics concerned with the minimization or maximization of a linear function of several variables and inequalities; used in many branches of industry to minimize costs or maximize production.

A linear equation for deriving a single predicted value from one or more known explanatory values.

A map between vector spaces which preserves the operations of vector addition and scalar multiplication.

A maximum within a restricted domain; a point on a function whose value is greater than the values of all other points near it, especially if another local maximum of greater value is known to exist.

A point on a graph (or its associated function) whose value is less than all other points near it.

The set of all points whose coordinates satisfy a given equation or condition.

For a number x, the exponent by which a given base number must be raised in order to obtain the power x. Written \log_b x. For example, \log_{10} 1000 = 3 because 10^3 = 1000 and \log_2 16 = 4 because 2^4 = 16.

A value that is less than or equal to every element of a given set of values.

A Hausdorff topological space that looks locally like the "ordinary" Euclidean space \mathbb{R}^n.

The group of self-homeomorphisms of a topological space, modulo homotopy.

The projection of a joint probability distribution onto one of its component random variables.

A measure of the inertia of a mass of matter, one of four fundamental properties of matter. SI unit of mass: kilogram.

A method of proof which, in terms of a predicate P, could be stated as: if P(0) is true and if for any natural number n \ge 0, P(n) implies P(n + 1), then P(n) is true for any natural number n.

A subfield of logic and mathematics consisting of both the mathematical study of logic and the application of this study to other areas of mathematics, exemplified by questions on the expressive power of formal logics and the deductive power of formal proof systems.

An abstract mathematical representation of a process, device or concept using a number of variables to represent inputs, outputs and internal states, and sets of equations and inequalities to describe their interaction.

A process by which a matrix (a rectangular table of numbers or abstract quantities that can be added and multiplied) is broken down into simpler numerical building blocks.

The greatest value of a set or other mathematical structure, especially the global maximum or a local maximum of a function.

Any function of multiple variables that satisfies certain properties and yields a number representative of its arguments; or, the number so yielded; a measure of central tendency.

Any of various theorems that saliently concern mean values.

A branch of mathematical analysis, concerned with the theory of integration, that generalizes the intuitive notions of length, area and volume.

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex, and they all intersect at the triangle's centroid. In the case of isosceles and equilateral triangles, a median bisects any angle at a vertex whose two adjacent sides are equal in length. The concept of a median extends to tetrahedra.

The base unit of length in the metric system, equal to 100 centimeters.

Any set on which a metric is defined, giving a distance between any two elements.

A point which divides a line segment into two lines of equal length.

For a given square matrix M over some field K, the smallest-degree monic polynomial over K which, when applied to M, yields the zero matrix.

Any surface that has the least possible surface area of those into which it could transform; typically the shapes of soap bubbles that span wire frames

The smallest member of a batch or sample or the lower bound of a probability distribution.

A minor arc may refer to: An arc that is smaller than a semicircle A term that arises in connection with the Hardy–Littlewood circle method

The sign −, indicating subtraction or a negative number.

A relationship between Calabi-Yau manifolds where two such manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.

A number consisting of a whole number and a fraction, such as 11½.

Any formal system that attempts to deal with modalities, such as possibility and necessity, but also obligation and permission.

A state of a system that is represented by an eigenfunction of that system.

Any system of arithmetic for integers which, for some given positive integer n, is equivalent to the set of integers being mapped onto the finite set {0, ... n} according to congruence modulo n, and in which addition and multiplication are defined consistently with the results of ordinary arithmetic being so mapped.

The absolute value of a complex number.

A set which is closed under an associative binary operation, and which contains an element which is an identity for the operation.

A function that either never decreases or never increases as its independent variable increases.

(formally) An arrow in a category; (less formally) an abstraction that generalises a map from one mathematical object to another and is structure-preserving in a way that depends on the branch of mathematics from which it arises.

A whole number that is obtained by multiplying a specific whole number by another whole number.

The process of computing the sum of an addition with one and the same number a specified number of times (i.e. 4 times 5 = [0 +] 5 + 5 + 5 + 5 = 20) or any other analogous binary operation that combines other mathematical objects.

The number of values for which a given condition holds.

The logarithm in base : either the function that given returns such that = , or the value of given by the function.

A positive integer {1, 2, 3, ...}.

In mathematics, a negative number is the opposite of a positive real number. Equivalently, a negative number is a real number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative.…

Any set of polygons joined edge to edge that, when folded along the edges between adjoining polygons so that the outer edges touch, form a given polyhedron.

Any of a family of continuous probability distributions such that the probability density function is the Gaussian function

A hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. When used, the null hypothesis is presumed true until statistical evidence in the form of a hypothesis test indicates otherwise. Therefore, the null and the alternative hypothesis must be mutually exclusive and exhaustive.

A line that graphically represents the real numbers as a series of points whose distance from an origin is proportional to their value

The branch of pure mathematics concerned with the properties of integers.

The number or expression written above the line in a fraction (such as 1 in ½).

The study of algorithms to solve mathematical problems concerning continuous sets of values (such as the real numbers, complex numbers or vector spaces).

The study of numerical analysis algorithms to obtain approximate values of definite integrals.

An angle that is greater than 90 degrees and less than 180 degrees.

A triangle one of whose angles is obtuse.

A polygon with eight sides and eight angles.

Any function whose value changes sign if the independent variable changes sign i.e. f(-x) = -f(x)

Any integer that produces a noninteger when divided by two.

An interval in the real number line which does not contain its supremum and infimum. If specified by a pair of real numbers, then it consists of all the points on the real line whose values lie strictly between those two real numbers.

Informally, a set such that the target point of a movement by a small amount in any direction from any point in the set is still in the set; exemplified by a full circle without its boundary.

A branch of functional analysis, being the study of linear operators on function spaces.

The sequence in which the various operations in a mathematical expression are to be evaluated.

An object containing exactly two elements in a fixed order, so that, when the elements are different, exchanging them gives a different object. Notation: (a, b) or \langle a, b\rangle.

A natural number used to denote position in a sequence.

An equation involving the derivatives of a function of only one independent variable.

Of a figure: having all its sides and angles equal.

The point at which the axes of a coordinate system intersect.

The set of all vectors which are orthogonal to a given set of vectors.

A square matrix whose columns, considered as vectors, are orthonormal to each other. This implies that the transpose of such a matrix is also its inverse.

In statistical significance testing, the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true.

Lines or planes that never meet, staying the same distance apart at all points.

A convex quadrilateral in which each pair of opposite edges are parallel and of equal length.

Any measured quantity of a statistical population that summarises or describes an aspect of the population.

A set of equations that defines the coordinates of the dependent variables (x, y and z) of a curve or surface in terms of one or more independent variables or parameters.

A derivative with respect to one variable of a function of several variables with the other variables held constant.

A differential equation that involves the partial derivatives of a function of several variables.

The sum of the first n terms of an infinite series, where n is a finite number.

A polygon with five sides and five angles.

Any of the ninety-nine points that divide an ordered distribution into one hundred parts, each containing one per cent of the population.

A number that is the sum of all of its divisors except itself.

The sum of the distance of all the lengths of the sides of an object.

Any function whose value repeats after the regular addition of a period to its independent variable; i.e., a function ƒ such that, for some positive constant p, ƒ(x + p) = ƒ(x) for all x.

A one-to-one mapping from a finite set to itself.

A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition.

At or forming a right angle (to something).

A line that divides a line segment into two equal parts while also being perpendicular to said line segment.

An irrational and transcendental constant representing the ratio of the circumference of a Euclidean circle to its diameter; approximately 3.14159265.

The theorem which states that any partition of a finite set of n elements into m (< n) subsets (allowing empty subsets) must include a subset with two or more elements; any of certain reformulations concerning the partition of infinite sets where the cardinality of the unpartitioned set exceeds that of the partition (so there is no one-to-one correspondence).

The value of a digit of numbers. For example, ones, tens, hundreds in the decimal counting; ones, sixes, thirty-sixes in the senary counting.

A flat surface extending infinitely in all directions (e.g. horizontal or vertical plane); a bounded portion thereof.

The locus of points in the Euclidean plane that satisfies some geometric or algebraic definition; a curve in a plane (as opposed to higher-dimensional space).

The sign +, indicating addition or a positive number.

A decimal point (now especially when reading decimal fractions aloud).

The general field of topology, not restricting attention to specific classes of spaces, and not using algebraic topology.

The coordinates of a point in a plane, measured as its Cartesian distance from the origin and the angle measured counterclockwise from the x-axis to a line joining the point to the origin.

A plane figure bounded by edges that are all straight lines.

A solid figure with many flat faces and straight edges.

An expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0.

A ring (which is also a commutative algebra), denoted K[X], formed from the set of polynomial (usually of one variable, in a given set, X), with coefficients in a given ring (often a field), K.

The mean (usually arithmetic, but possibly geometric) of all members of a population or all possible values from which a sample can be taken, as opposed to a sample mean.

The probability that a statistical test will reject the null hypothesis when the alternative hypothesis is true.

Any infinite series of the general form \sum_{i=0}^\infty a_i (x-c)^i.

The generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic.

The factorization of a positive integer into its constituent prime numbers.

Any natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

Any theorem that concerns the distribution of prime numbers.

For a given modulus n, a number g such that for every a coprime to n there exists an integer k such that g'k ≡ a (mod n); a generator (or primitive element) of the multiplicative group, modulo n, of integers relatively prime to n.

Chosen or assumed among a branch of possible values of a multi-valued function so that the function is single-valued.

A polyhedron with parallel ends of the same polygonal shape and size, the other faces being parallelogram-shaped sides.

A number, between 0 and 1, expressing the precise likelihood of an event happening.

Any function whose integral over a set gives the probability that a random variable has a value in that set.

A distribution of all possible values of a random variable together with an indication of their probabilities.

A measurable space having a unit measure

A quantity obtained by multiplication of two or more numbers.

The branch of mathematics that investigates those properties of figures that are invariant under homographies, special kinds of spaces called projective spaces, and the geometric structures that inhabit them.

A plane in which any pair of distinct point determine a line and any pair of distinct lines intersect at a point.

A sequence of statements consisting of axioms, assumptions, statements already demonstrated in another proof, and statements that logically follow from previous statements in the sequence, and which concludes with a statement that is the object of the proof.

The proof of a statement adduced by deriving a contradiction from the statement's negation.

A vulgar fraction in which the magnitude of the numerator is less than or equal to that of the denominator, such as 2/3.

A statement of equality between two ratios.

A formal deductive system in which formulae representing propositions can be formed by combining atomic propositions using logical connectives.

An instrument for measuring angles, usually a marked semicircle.

A surface generated by rotating a tractrix about its asymptote; a tractricoid

A solid with triangular lateral faces and a polygonal (often square or rectangular) base.

In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation: a 2 + b 2 = c 2 .…

One fourth of a circle or disc; a sector with an angle of 90°.

A quadratic polynomial, function or equation.

A polynomial equation of the second degree; an equation that can be rearranged in standard form as ax^2 + bx + c = 0, where x is an unknown value, a, b and c are known numbers, and where a does not equal zero.

A homogeneous polynomial of degree 2 in a given number of variables.

The mathematical theorem which states that, for given odd prime numbers p and q, the question of whether p is a square modulo q is equivalent to the question of whether q is a square modulo p.

A polygon with four sides.

One of the class of values of a variate which divides the members of a batch or sample into equal-sized subgroups of adjacent values or a probability distribution into distributions of equal probability.

The number (either a fraction or an integer) resulting from the division of one number by another.

A space obtained from another by identification of points that are equivalent to one another in some equivalence relation.

A line segment between any point of a circle or sphere and its center.

A quantity whose value is random and to which a probability distribution is assigned, such as the possible outcome of a roll of a dice.

In descriptive statistics, the range of a set of data is the size or width of the narrowest interval which contains all the data. It is calculated as the difference between the largest and smallest values (also known as the sample maximum and minimum). It is expressed in the same units as the data. The range provides an indication of statistical dispersion. Robust measures of range include the interdecile range and the interquartile range.

A ratio comparing two quantities expressed in different units.

The relative magnitudes of two quantities (usually expressed as a quotient).

Any function expressible as the quotient of two (coprime) polynomials (and which thus has poles at a finite, discrete set of points which are the roots of the denominator).

A number that can be expressed as the ratio of two integers.

A line extending indefinitely in one direction from a point.

The branch of mathematics dealing with the real numbers and related structures.

The limit of a convergent sequence of rational numbers, whether the limit is a rational number such as 2, -5, or 2/7 or whether the limit is an irrational number such as the square root of two or the circumference of the circle whose radius is one.

The number obtained by dividing 1 by another given number; the result of exchanging the numerator and the denominator of a fraction.

Any quadrilateral having opposing sides parallel and four right angles.

an equation that recursively defines a sequence; each term of the sequence is defined as a function of the preceding terms.

An angle greater than 180 degrees but less than 360 degrees.

A polygon which is both equiangular and equilateral (i.e. having all sides the same length and all interior angles the same).

The number of data points satisfying something divided by the total number of data points.

The number left over after a simple subtraction

A parallelogram having all sides of equal length.

Half of the angle formed by a single straight line, equivalent to 90 degrees.

A triangle having a right angle as one of its interior angles.

A common mnemonic for understanding notation conventions for vectors in three dimensions.

Of a number or expression, a number which, when raised to a specified power, yields the specified number or expression.

The stepped appearance of a matrix that has undergone with the result that the leading coefficient or pivot (that is, the first nonzero number from the left) of a nonzero row is to the right of the pivot of the row above it.

The vector space generated by all the row vectors of a matrix.

A (usually rigid), flat, rectangular measuring or drawing device with graduations in units of measurement; a straightedge with markings.

Given a random sample \textstyle \mathbf{x}_{1},\ldots,\mathbf{x}_{N} from an \textstyle n-dimensional random variable \textstyle \mathbf{X}, a mean defined as

A function that assigns a scalar value to every point in space.

Multiplication of a vector by a scalar (which belongs to the vector space's field of scalars).

The dot product of one of the three vectors with the cross product of the other two.

An ordered, usually numerical sequence used for measurement; means of assigning a magnitude.

A triangle having each of its three sides of different lengths.

A type of display using Cartesian coordinates to display values for two variables for a set of data.

a method of writing, or of displaying real numbers as a decimal number between 1 and 10 followed by an integer power of 10

A unit of time equal to 1/60 of a minute; also a unit of angular measure.

Part of a circle, extending to the center; circular sector.

The part of a circle between its circumference and a chord (usually other than the diameter).

Half of a circle, bounded by a diameter and half the circumference.

An ordered list of objects, typically indexed with natural numbers.

A collection of zero or more objects, possibly infinite in size, and disregarding any order or repetition of the objects which may be contained within it.

The mathematical theory of sets.

A set of general techniques for computing or estimating the size of sifted sets of integers.

A measure of how likely it is to draw a false conclusion in a statistical test, when the results are really just random variations.

Having the same shape, in particular, having corresponding angles equal and corresponding line segments proportional.

A generalization of a triangle or tetrahedron to an arbitrary dimension; more accurately, the convex hull of n+1 affinely independent points in n-dimensional space.

In a right triangle, the ratio of the length of the side opposite an angle to the length of the hypotenuse.

A particular type of factorisation of a matrix into a product of three matrices, of which the second is a diagonal matrix that has as the entries on its diagonal the singular values of the original matrix.

A square matrix equal to the negative of its transpose.

The distance from any point on the circle of a right circular cone to its apex.

The ratio of the vertical and horizontal distances between two points lying on the line.

A function that has derivatives of all finite orders everywhere in its domain.

A three-dimensional figure (as opposed to a surface, an area, or a curve).

A value that satisfies an equation.

A group G which is part of a finite chain Gi (0 ≤ i ≤ n) of subgroups, where G0 is the trivial subgroup and Gn = G, such that each intermediate subgroup Gi is a normal subgroup of the next in the chain (i.e., Gi ◁ Gi+1) and each quotient Gi+1/Gi is a cyclic group.

Any curve in three-dimensional space; includes any plane curve as a subset.

A theorem providing conditions under which an operator or matrix can be diagonalized.

The set of all points in three-dimensional Euclidean space (or n-dimensional space, in topology) that are a fixed distance from a fixed point .

The arranging of non-overlapping spheres in space

A polygon with four straight sides of equal length and four right angles; an equilateral rectangle; a regular quadrilateral.

Any integer which is the product of some integer times itself: any integer whose square root is also an integer.

The number which, when squared, yields the original number.

A measure of how spread out data values are around the mean, defined as the square root of the variance. Represented with the Greek letter σ.

The normal distribution with a mean of zero and a variance of one.

A point on a curve where the gradient is zero. This point can be a maximum, a minimum, or a point of inflection.

A function that maps elements of an index set to elements of a collection of random variables; historically, a collection of random variables indexed by a set of numbers usually regarded as points in time.

An angle that is equal to 180° (pi radians or two right angles).

A matrix formed by selecting certain rows and columns from a larger matrix.

The topology of a subset S of a topological space X which is obtained by considering any subset of S to be an open set if it corresponds to the intersection of S with some open set of X.

Replacing a variable with a value or another expression.

The process of subtracting a number from another.

An arithmetic computation, especially one posed to a student as an exercise (not necessarily limited to addition).

The total area on the surface of an object.

An integral over a 2-dimensional domain

A formal system of deductive logic in which aspects and relationships of natural language are represented by a system of symbols.

The set that contains all elements belonging to either of given two sets but not to the both; the relative complement of the intersection in the union of given two sets.

A group whose elements are precisely all of the bijections of some set with itself and whose operation is composition of those bijections.

A square matrix that is its own transpose, and is thereby symmetric about the main diagonal

A group whose elements are all the transformations under which a given object remains invariant and whose group operation is function composition.

a set of equations required to be satisfied simultaneously or considered as a whole

A fiber bundle for which the base space is a differentiable manifold and each fiber over a point of that manifold is the tangent space of that point.

An n-dimensional vector space that represents the set of all vectors tangent to given n-dimensional M at point x.

A mathematical object that describes linear relations on scalars, vectors, matrices and other algebraic objects, and is represented as a multidimensional array.

The most general bilinear operation in various contexts (as with vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, modules, and so on), denoted by ⊗.

One of ten equal parts of a whole.

Any value (variable or constant) or expression separated from another term by a space or an appropriate character, in an overall expression or table.

A polyhedron with four faces; the regular tetrahedron, the faces of which are equal equilateral triangles, is one of the Platonic solids.

A mathematical statement of some importance that has been proven to be true. Minor theorems are often called propositions. Theorems which are not very interesting in themselves but are an essential part of a bigger theorem's proof are called lemmas.

The branch of mathematics dealing with those properties of a geometrical object (of arbitrary dimensionality) that are unchanged by continuous deformations (such as stretching, bending, etc., without tearing or gluing).

Subgroup of an abelian group whose generators have finite order.

any irrational number that is not an algebraic number

The replacement of the variables in an algebraic expression by their values in terms of another set of variables; a mapping of one space onto another or onto itself; a function that changes the position or direction of the axes of a coordinate system.

A matrix (of dimension n×m) that represents some linear transformation from ℝm→ℝn.

A square matrix whose rows consist of nonnegative real numbers, with each row summing to 1. Used to describe the transitions of a Markov chain; its element in the ith row and jth column describes the probability of moving from state i to state j in one time step.

A relation between two mathematical figures such as a straight line where the coordinates of each point in one figure is a constant added to the coordinates of a corresponding point in the other figure.

The matrix derived from performing a transpose operation on a given matrix.

A line intersecting a pair of parallel lines.

A (convex) quadrilateral with two (non-adjacent) parallel sides.

A method of finding an approximate value for a definite integral, based on dividing the area under the graph of the function into a series of strips, each of which is approximately a trapezoid, and summing the area of all the trapezoids to approximate the area.

A polygon with three sides and three angles.

The inequality that states that the magnitude of the sum of two vectors is less than or equal to the sum of the magnitudes of the vectors.

The branch of mathematics that deals with the relationships between the sides and angles of (in particular) right-angled triangles, as represented by the trigonometric functions, and with calculations based on said relationships.

The number 1,000,000,000,000; a thousand billions.

A sequence of three elements or 3-tuple.

The removal of the least significant digits from a decimal number.

A table showing all possible truth values for an expression, derived from the truth values of its components.

Either of a pair of prime numbers that differ by two.

The property of being uniformly continuous.

A type of convergence of a sequence of functions { f'n }, in which the speed of convergence of f'n(x) to f(x) does not depend on x.

A symmetric probability distribution wherein every outcome is equally likely to occur at any point in the distribution.

A unique factorization ring which is also an integral domain.

Oneness, singularity, seen as a component of a whole number; a magnitude of one.

The circle of radius 1 with centre at the origin, used in defining trigonometric functions.

A fraction whose numerator is 1 and whose denominator is a positive integer; for example, \frac12 or \frac17.

A vector with the length of 1, defined as \mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}.

A variable (usually x, y or z) whose value is to be found.

A value that is greater than or equal to every element of a given set of values.

A quantity that may assume any one of a set of values.

Any member of a (generalized) vector space.

The branch of mathematics that covers the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space.

a function mapping a subset of \mathbb{R}^n into a subset of \mathbb{R}^n

A set of elements called vectors, together with some field and operations called addition (mapping two vectors to a vector) and scalar multiplication (mapping a vector and an element in the field to a vector), satisfying a list of constraints; equivalently, a module over a field.

An angular point of a polygon, polyhedron or higher order polytope.

The sum of the degrees of a set of vertices.

A second-order linear partial differential equation for the description of waves or standing wave fields.

The measurement of the extent of something from side to side.

A question of whether an element of a certain group (or monoid or the like) is the identity, given an obscure representation of that element.

The axis on a graph that is usually drawn left to right and usually shows the range of values of an independent variable.

The axis on a graph that is usually drawn from bottom to top and usually shows the range of values of variable dependent on one other variable, or the second of two independent variables.

A value of the independent variables of a function, for which the function is equal to zero.