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 chemistry, physics, and mathematics, the Boltzmann distribution ( also called the Gibbs Distribution ) is a certain distribution function or probability measure for the distribution of the states of a system.

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative.

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.

Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties.

In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects from a certain collection.

In mathematics, graphs are useful in geometry and certain parts of topology, e. g. Knot Theory.

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 certain areas of mathematics, notably category theory, it is valuable to distinguish between equality on the one hand and isomorphism on the other.

Idempotence ( ) is the property of certain operations in mathematics and computer science, that they can be applied multiple times without changing the result beyond the initial application.

It can refer to certain areas of literature, languages, philosophy, history, mathematics, psychology, and science.

Albert Einstein stated that " as far as the laws of mathematics refer to reality, they are not certain ; and as far as they are certain, they do not refer to reality.

An alternative view is that certain scientific fields ( such as theoretical physics ) are mathematics with axioms that are intended to correspond to reality.

In mathematics, modular arithmetic ( sometimes called clock arithmetic ) is a system of arithmetic for integers, where numbers " wrap around " upon reaching a certain value — the modulus.

Over the last two millennia, physics was a part of natural philosophy along with chemistry, certain branches of mathematics, and biology, but during the Scientific Revolution in the 17th century, the natural sciences emerged as unique research programs in their own right.

However this definition does not allow star polytopes with interior structures, and so is restricted to certain areas of mathematics.

In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups ; they share many properties with their finite quotients.

He or she will usually have a primary qualification in one of several fields: mathematics, the physical sciences ( e. g. chemistry, physics, biology ), engineering ( e. g. mechanical, chemical, Materials Science & Engineering or civil engineering ), medicine, or certain technologies, notably materials or food.

Everett did not draw the conclusion that it was the lack of numbers in their language that prevented them from grasping mathematics, but instead concluded that the Pirahã had a cultural ideology that made them extremely reluctant to adopt new cultural traits, and that this cultural ideology was also the reason that certain linguistic features that were otherwise believed to be universal did not exist in their language.

When theories are studied in mathematics, they are usually expressed in some formal language and their statements are closed under application of certain procedures called rules of inference.

In various branches of mathematics, a useful construction is often viewed as the “ most efficient solution ” to a certain problem.

His dissertation, entitled, " Solutions of the Mathieu equation of period 4 pi and certain related functions ", was beyond the comprehension of the chemistry and physics faculty, and only when some members of the mathematics department, including the chairman, insisted that the work was good enough that they would grant the doctorate if the chemistry department would not, was he granted a Ph. D. in chemistry in 1935.

In electronics, computer science and mathematics, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal.

This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.

In an account that had become standard by the mid-century, Hilbert's problem set was also a kind of manifesto, that opened the way for the development of the formalist school, one of three major schools of mathematics of the 20th century.

In mathematics, the discrete Fourier transform ( DFT ) is a specific kind of discrete transform, used in Fourier analysis.

Game theory is a branch of applied mathematics that considers strategic interactions between agents, one kind of uncertainty.

There is also another very different kind of interpolation in mathematics, namely the " interpolation of operators ".

Galileo maintained strongly that mathematics provided a kind of necessary certainty that could be compared to God's: " With regard to those few mathematical propositions which the human intellect does understand, I believe its knowledge equals the Divine in objective certainty.

In applied mathematics, the delta function is often manipulated as a kind of limit ( a weak limit ) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero.

This kind of extension is used so constantly in contemporary mathematics based on set theory that it can be called an implicit assumption.

His most important work in potential theory is summarised in his 1911 book Researches in Potential Theory ( Potentialtheoretische Untersuchungen ), which received the Jablonowski Society award in Leipzig ( 1500 marks ), and the Richard Lieben award from the University of Vienna ( 2000 crowns ) for the most outstanding work in the field of pure and applied mathematics written by any kind of ' Austrian ' mathematician in the previous three years.

* Block design, a kind of set system in combinatorial mathematics

Having shown how to do science without using numbers, Field proceeded to rehabilitate mathematics as a kind of useful fiction.

Lakatos ' philosophy of mathematics is sometimes regarded as a kind of social constructivism, but this was not his intention.

In mathematics, it is often useful to view or even define subtraction as a kind of addition, the addition of the additive inverse.

In mathematics, a ratio is a relationship between two numbers of the same kind ( e. g., objects, persons, students, spoonfuls, units of whatever identical dimension ), usually expressed as " a to b " or a: b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second ( not necessarily an integer ).

In mathematics, a paraboloid is a quadric surface of special kind.

2 ) " We select the kind of mathematics to use.

" He held that corollarial deduction matches Aristotle's conception of direct demonstration, which Aristotle regarded as the only thoroughly satisfactory demonstration, while theorematic deduction ( A ) is the kind more prized by mathematicians, ( B ) is peculiar to mathematics, and ( C ) involves in its course the introduction of a lemma or at least a definition uncontemplated in the thesis ( the proposition that is to be proved ); in remarkable cases that definition is of an abstraction that " ought to be supported by a proper postulate.

Brouwer then " embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions "; indeed his thesis advisor refused to accept his Chapter II " ' as it stands, ... all interwoven with some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with the foundations of mathematics ' " ( Davis, p. 94 quoting van Stigt, p. 41 ).

* Newman – Shanks – Williams prime, a certain kind of prime number in mathematics

In mathematics, a Voronoi diagram is a special kind of decomposition of a metric space, determined by distances to a specified family of objects ( subsets ) in the space.

Whereas the Taoist philosophical paradigm had promoted scientific and mathematical investigation as a kind of mystical exploration of the workings of the universe, the Confucian paradigm focused far more on social philosophy and morality, which prompted a general lack of further research in mathematics and natural sciences.

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by

In mathematics, a Mahlo cardinal is a certain kind of large cardinal number.