Mathematics. Dictionary

Galois group · G

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.…

Gamma function · G

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.

Gaussian elimination · G

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

Generalized polygon · G

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.…

Generating function · G

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.

Genus · G

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

Geodesic · G

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

Geometric distribution · G

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 · G

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.…

Geometric series · G

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.

Geometrization conjecture · G

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.…

Golomb ruler · G

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.…

Good filtration · G

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.…

Gradient descent · G

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

Graph theory · G

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

Greatest common divisor · G

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).

Green's theorem · G

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 · G

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

Group action · G

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

Group theory · G

The mathematical theory of groups.