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

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.

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

Furthermore, in mathematics, the letter alpha is used to denote the area underneath a normal curve in statistics to denote significance level when proving null and alternative hypotheses.

In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

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

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.

C *- algebras ( pronounced " C-star ") are an important area of research in functional analysis, a branch of mathematics.

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated.

As he began to understand physics and how physicists were using mathematics, he developed a coherent mathematical theory for what he found, most importantly in the area of integral equations.

The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and

Indeed, the usefulness of this area of mathematics to linguistics has borne organizations such as TextGraphs, as well as various ' Net ' projects, such as WordNet, VerbNet, and others.

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, Archimedes used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi.

The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.

* Integration, in mathematics, a fundamental concept of calculus — the operation of calculating the area between the curve of a function on the x-axis or y-axis

Knots have been the subject of interest for their ancient origins, their common uses, and the area of mathematics known as knot theory.

Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts ; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics.

Furthermore, a mathematician is not someone who merely manipulates formulas, numbers or equations — the diversity of mathematics allows for research concerning how concepts in one area of mathematics can be used in other areas too.

# proof theory and constructive mathematics ( considered as parts of a single area ).

In a similar way to proof theory, model theory is situated in an area of interdisciplinarity between mathematics, philosophy, and computer science.

Many of these structures are drawn from functional analysis, a research area within pure mathematics that was influenced in part by the needs of quantum mechanics.