In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group.

Note that congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angle s. The latter sort of properties are called invariant ( mathematics ) | invariant s and studying them is the essence of geometry.

Bombieri is also known for his pro bono service on behalf of the mathematics profession, e. g. for serving on external review boards and for peer-reviewing extraordinarily complicated manuscripts ( like the papers of John Nash on embedding Riemannian manifolds and of Per Enflo on the invariant subspace problem ).

The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics.

Note that congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angle s. The unchanged properties are called invariant ( mathematics ) | invariant s.

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic ( or Euler – Poincaré characteristic ) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.

From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century ; from the initial idea of homology as a topologically invariant relation on chains, the range of applications of homology and cohomology theories has spread out over geometry and abstract algebra.

In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations.

Several major strands of more abstract mathematics ( including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme leading to the study of the classical groups ) built on projective geometry.

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems.

In mathematics, the indefinite orthogonal group, O ( p, q ) is the Lie group of all linear transformations of a n = p + q dimensional real vector space which leave invariant a nondegenerate, symmetric bilinear form of signature ( p, q ).

In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space.

* invariant ( mathematics )

In mathematics, a fixed point ( sometimes shortened to fixpoint, also known as an invariant point ) of a function is a point that is mapped to itself by the function.

In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space.

* In mathematics, a notation for the Arf invariant for quadratic forms over the 2-element field.

In mathematics, a Casimir invariant or Casimir operator is a distinguished element of the centre of the universal enveloping algebra of a Lie algebra.

In mathematics, the Schwarzian derivative, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all linear fractional transformations.

In mathematics, the symbolic method in invariant theory is an algorithm developed by,,, and in the 19th century for computing invariants of algebraic forms.

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.

In the field of mathematics known as functional analysis, the invariant subspace problem for a complex Banach space H of dimension > 1 is the question whether

In mathematics, the Lyusternik – Schnirelmann category ( or, Lusternik – Schnirelmann category, LS-category, or simply, category ) of a topological space is the homotopical invariant defined to be the smallest integer number such that there is an open covering of with the property that each inclusion map is nullhomotopic.

The concept of a linear subspace ( or vector subspace ) is important in linear algebra and related fields of mathematics.

In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space V. Here " increasing " means each is a proper subspace of the next ( see filtration ):

In mathematics, a relatively compact subspace ( or relatively compact subset ) Y of a topological space X is a subset whose closure is compact.

In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower dimensional subspace, called the Poincaré section, transversal to the flow of the system.

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology ( or the relative topology, or the induced topology, or the trace topology ).

In mathematics, an eigenplane is a two-dimensional invariant subspace in a given vector space.

In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanned by such curves ( up to linear equivalence ) has a one-dimensional subspace on which it is positive definite ( not uniquely determined ), and decomposes as a direct sum of some such one-dimensional subspace, and a complementary subspace on which it is negative definite.

In algebraic topology, a branch of mathematics, the excision theorem is a useful theorem about relative homology — given topological spaces X and subspaces A and U such that U is also a subspace of A, the theorem says that under certain circumstances, we can cut out ( excise ) U from both spaces such that the relative homologies of the pairs ( X, A ) and ( X

In algebraic topology, a branch of mathematics, the ( singular ) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces.

In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups.

In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace.

In functional analysis and related areas of mathematics, the beta-dual or-dual is a certain linear subspace of the algebraic dual of a sequence space.

In topology, a branch of mathematics, a retraction, as the name suggests, " retracts " an entire space into a subspace.

Binary relations are used in many branches of mathematics to model concepts like " is greater than ", " is equal to ", and " divides " in arithmetic, " is congruent to " in geometry, " is adjacent to " in graph theory, " is orthogonal to " in linear algebra and many more.

In mathematics, a bilinear operator is a function combining elements of two vector spaces to yield an element of a third vector space that is linear in each of its arguments.

It can also be used to denote abstract vectors and linear functionals in mathematics.

Mathematics used in rendering includes: linear algebra, calculus, numerical mathematics, signal processing, and Monte Carlo methods.

Students of control engineering may start with a linear control system course dealing with the time and complex-s domain, which requires a thorough background in elementary mathematics and Laplace transform ( called classical control theory ).

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.

In mathematics, any vector space, V, has a corresponding dual vector space ( or just dual space for short ) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors.

However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics.

* Gramian matrix, used in mathematics to test for linear independence of functions

* Lambda indicates an eigenvalue in the mathematics of linear algebra.

In mathematics, a linear map, linear mapping, linear transformation, or linear operator ( in some contexts also called linear function ) is a function between two modules ( including vector spaces ) that preserves the operations of module ( or vector ) addition and scalar multiplication.

Linear algebra is the branch of mathematics concerning vector spaces, often finite or countably infinite dimensional, as well as linear mappings between such spaces.

The use of matrices in quantum mechanics, special relativity, and statistics helped spread the subject of linear algebra beyond pure mathematics.

In system analysis ( a subfield of mathematics ), linear prediction can be viewed as a part of mathematical modelling or optimization.

Operators are of critical importance to both linear algebra and functional analysis, and they find application in many other fields of pure and applied mathematics.

In applied mathematics, semigroups are fundamental models for linear time-invariant systems.

* In mathematics, a linear system of divisors of dimension 3

In mathematics, the Cauchy – Schwarz inequality ( also known as the Bunyakovsky inequality, the Schwarz inequality, or the Cauchy – Bunyakovsky – Schwarz inequality, or Cauchy – Bunyakovsky inequality ), is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas.

In mathematics, a topological vector space ( also called a linear topological space ) is one of the basic structures investigated in functional analysis.

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results ( e. g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants ).