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.

* 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.

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

The most general setting in which these words have meaning is an abstract branch of mathematics called category theory.

In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.

Though respected for their contributions to various academic disciplines ( respectively mathematics, linguistics, and literature ), the three men became known to the general public only by making often-controversial and disputed pronouncements on politics and public policy that would not be regarded as noteworthy if offered by a medical doctor or skilled tradesman.

In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right.

A term dating from the 1940s, " general abstract nonsense ", refers to its high level of abstraction, compared to more classical branches of mathematics.

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.

There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory ; this article gives an overview of the available techniques and some of their general properties, while the specific algorithms are described in subsidiary articles linked below.

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

In general, mathematical models may include logical models, as far as logic is taken as a part of mathematics.

Tegmark writes that " abstract mathematics is so general that any Theory Of Everything ( TOE ) that is definable in purely formal terms ( independent of vague human terminology ) is also a mathematical structure.

Numerical analysis is the study of algorithms that use numerical approximation ( as opposed to general symbolic manipulations ) for the problems of mathematical analysis ( as distinguished from discrete mathematics ).

In general, Hans got better grades than Niels ; however, a new mathematics teacher, Bernt Michael Holmboe, was appointed in 1818.

Enrollment in computer-related degrees in U. S. has dropped recently due to lack of general interests in science and mathematics and also out of an apparent fear that programming will be subject to the same pressures as manufacturing and agriculture careers.

Although Descartes did not pursue the subject, he preceded Leibniz in envisioning a more general science of algebra or " universal mathematics ," as a precursor to symbolic logic, that could encompass logical principles and methods symbolically, and mechanize general reasoning.

Rolf Nevanlinna's grandfather Edavard Engelbert Neovius ( 1823 – 88 ), a major general in the Czar's army, taught mathematics in the Hamina Cadet School, Nevanlinnas father Otto Neovius-Nevanlinna ( 1867 – 1927 ) was a prominent mathematics teacher while one of his uncles was a mathematics professor and another a mathematics teacher.

The difference is that whereas SWEBOK defines the software engineering knowledge that practitioners should have after four years of practice, SE2004 defines the knowledge that an undergraduate software engineering student should possess upon graduation ( including knowledge of mathematics, general engineering principles, and other related areas ).

Different paths exist whose programmes are mostly decided at national level, but a general basic education is provided to every pupil, similarly to the liceo, but less focused on the Humanities ( no philosophy or arts, but more mathematics than in the non-scientific liceo ); the main ones are:

Finally, the mathematics of general relativity appeared to be incomprehensibly dense.