In mathematics, more specifically in functional analysis, a Banach space ( pronounced ) is a complete normed vector space.

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space.

* Change of any variable quantity, in mathematics and the sciences ( More specifically, the difference operator.

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

In mathematics, and more specifically set theory, the empty set is the unique set having no elements ; its size or cardinality ( count of elements in a set ) is zero.

In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain ( also called a Euclidean ring ) is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean division of the integers.

In mathematics, more specifically algebraic topology, the fundamental group ( defined by Henri Poincaré in his article Analysis Situs, published in 1895 ) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

In mathematics, specifically group theory, a quotient group ( or factor group ) is a group obtained by identifying together elements of a larger group using an equivalence relation.

In pure mathematics, the magnitude of a googolplex could be related to other forms of large-number notation such as tetration, Knuth's up-arrow notation, Steinhaus-Moser notation, or Conway chained arrow notation, though neither googol nor googolplex are anywhere near the largest representable or even specifically named numbers.

In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set.

In mathematics, more specifically in general topology and related branches, a net or Moore – Smith sequence is a generalization of the notion of a sequence.

) Later on this will be corrected to include specifically quantum mathematics, following Feynman.

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold.

In combinatorial mathematics, a Steiner system ( named after Jakob Steiner ) is a type of block design, specifically a t-design with λ = 1 and t ≥ 2.

In mathematics, specifically in ring theory, the simple modules over a ring R are the ( left or right ) modules over R which have no non-zero proper submodules.

In mathematics, specifically in real analysis, the Bolzano – Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space R < sup > n </ sup >.

In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object.

In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects.

In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal ( with respect to set inclusion ) amongst all proper ideals.

In mathematics, specifically in group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup.

In the branch of mathematics known as abstract algebra, a ring is an algebraic concept abstracting and generalizing the algebraic structure of the integers, specifically the two operations of addition and multiplication.

In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.

Combinatorics is an example of a field of mathematics which does not, in general, follow the axiomatic method.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

These connections shed the boundaries between combinatorics and parts of mathematics and theoretical computer science, but at the same time led to a partial fragmentation of the field.

This was a major discovery in an important field of mathematics ; construction problems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philology as a career.

The survey of Hanover fueled Gauss's interest in differential geometry, a field of mathematics dealing with curves and surfaces.

* Computational complexity theory, a field in theoretical computer science and mathematics

The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business accounts.

The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field.

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.

In modern mathematics, the theory of fields ( or field theory ) plays an essential role in number theory and algebraic geometry.

Financial mathematics is a field of applied mathematics, concerned with financial markets.

The field is largely focused on the modelling of derivatives, although other important subfields include insurance mathematics and quantitative portfolio problems.

An example of such a finite field is the ring Z / pZ, which is essentially the set of integers from 0 to p − 1 with integer addition and multiplication modulo p. It is also sometimes denoted Z < sub > p </ sub >, but within some areas of mathematics, particularly number theory, this may cause confusion because the same notation Z < sub > p </ sub > is used for the ring of p-adic integers.

In 1925 Peano switched Chairs unofficially from Infinitesimal Calculus to Complementary Mathematics, a field which better suited his current style of mathematics.

Al-Kindi wrote De Gradibus, in which he demonstrated the application of mathematics to medicine, particularly in the field of pharmacology.

The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, and electrical engineering.

Much of the mathematics behind information theory with events of different probabilities was developed for the field of thermodynamics by Ludwig Boltzmann and J. Willard Gibbs.