In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm.

This was particularly important because it shows that Cotton Mather had influence in mathematics during the time of Puritan New England.

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.

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 telecommunication industry has also motivated advances in discrete mathematics, particularly in graph theory and information theory.

Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and combinatorics, are important in addressing the challenging bioinformatics problems associated with understanding the tree of life.

Another definition of the GCD is helpful in advanced mathematics, particularly ring theory.

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.

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

Hutton's mother-Sarah Balfour-insisted on his education at the High School of Edinburgh where he was particularly interested in mathematics and chemistry, then when he was 14 he attended the University of Edinburgh as a " student of humanity " i. e. Classics ( Latin and Greek ).

Gödel attended the Evangelische Volksschule, a Lutheran school in Brünn from 1912 to 1916, and was enrolled in the Deutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages and religion.

The development of algebraic geometry from its classical to modern forms is a particularly striking example of the way an area of mathematics can change radically in its viewpoint, without making what was correctly proved before in any way incorrect ; of course mathematical progress clarifies gaps in previous proofs, often by exposing hidden assumptions, which progress has revealed worth conceptualizing.

The Fourier transform is useful in applied mathematics, particularly physics and signal processing.

This makes Prolog ( and other logic programming languages ) particularly useful for database, symbolic mathematics, and language parsing applications.

In mathematics, particularly in differential geometry and global analysis, spinors have since found broad applications to algebraic and differential topology, symplectic geometry, gauge theory, complex algebraic geometry, index theory, and special holonomy.

Meanwhile, however, significant progress in geometry, mathematics, and astronomy was made in the medieval era, particularly in the Islamic world as well as Europe.

This was particularly appropriate because Onsager, like Willard Gibbs, had been involved primarily in the application of mathematics to problems in physics and chemistry and, in a sense, could be considered to be continuing in the same areas Gibbs had pioneered.

In 1868 after a short stint in Kherson gymnasium worked as a gymnasium teacher of physics and mathematics at gymnasiums of Taganrog, particularly the Chekhov Gymnasium.

The status of abstract entities, particularly numbers, is a topic of discussion in mathematics.

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces ; as such, they form a natural context for the theory of integration.

He is particularly famous for his association with the concept of the embodied mind, which he has written about in relation to mathematics.

He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics.

In mathematics, particularly topology, one describes

The Ionian School of philosophers were the first natural philosophers ( φυσιολόγοι: physiologoi ) who tried to explain phenomena according to non-supernatural laws, and Pythagoras introduced the abstract mathematical-relations which formed the basis of the science of mathematics.

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

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

He is also noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation.

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.

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

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows ( also called morphisms, although this term also has a specific, non category-theoretical meaning ), where these collections satisfy some basic conditions.

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

It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and mathematical analysis.

High-dimensional spaces occur in mathematics and the sciences for many reasons, frequently as configuration spaces such as in Lagrangian or Hamiltonian mechanics ; these are abstract spaces, independent of the physical space we live in.

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.

Reacting against authors such as J. S. Mill, Sigwart and his own former teacher Brentano, Husserl criticised their psychologism in mathematics and logic, i. e. their conception of these abstract and a-priori sciences as having an essentially empirical foundation and a prescriptive or descriptive nature.

In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation.

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.

* Function ( mathematics ), an abstract entity that associates an input to a corresponding output according to some rule

However, his abstract mathematics was denounced as " Jewish ", useless, and " un-German " and he lost his position in 1935.

In the abstract, a coordinate system as used in mathematics and geodesy is, e. g., in ISO terminology, referred to as a coordinate system.

The term " applied mathematics " also describes the professional specialty in which mathematicians work on problems, often concrete but sometimes abstract.

This exemplifies the abstract nature of mathematics and how it is not restricted to questions one may ask in daily life.

Category theory, another field within " foundational mathematics ", is rooted on the abstract axiomatization of the definition of a " class of mathematical structures ", referred to as a " category ".

Intuitionistic logic substitutes constructability for abstract truth and is associated with a transition from the proof to model theory of abstract truth in modern mathematics.

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.