In mathematics and theoretical physics, a large diffeomorphism is a diffeomorphism that cannot be continuously connected to the identity diffeomorphism ( because it is topologically non-trivial ).

In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a function between smooth manifolds that preserves the local differentiable structure.

In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism.

In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface.

In mathematics, a quasi-invariant measure μ with respect to a transformation T, from a measure space X to itself, is a measure which, roughly speaking, is multiplied by a numerical function of T. An important class of examples occurs when X is a smooth manifold M, T is a diffeomorphism of M, and μ is any measure that locally is a measure with base the Lebesgue measure on Euclidean space.

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.

Next September, after receiving a degree from Yale's Master of Arts in Teaching Program, I will be teaching somewhere -- that much is guaranteed by the present shortage of mathematics teachers.

But because science is based on mathematics doesn't mean that a hot rodder must necessarily be a mathematician.

Like primitive numbers in mathematics, the entire axiological framework is taken to rest upon its operational worth.

In the new situation, philosophy is able to provide the social sciences with the same guidance that mathematics offers the physical sciences, a reservoir of logical relations that can be used in framing hypotheses having explanatory and predictive value.

So, too, is the mathematical competence of a college graduate who has majored in mathematics.

The principal of the school announced that -- despite the help of private tutors in Hollywood and Philadelphia -- Fabian is a 10-o'clock scholar in English and mathematics.

In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.

The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.

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.

The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation.

In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.

The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.

:" A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.

Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.

There is no prize awarded for mathematics, but see Abel Prize.

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

Isomorphisms are studied in mathematics in order to extend insights from one phenomenon to others: if two objects are isomorphic, then any property that is preserved by an isomorphism and that is true of one of the objects, is also true of the other.

If an isomorphism can be found from a relatively unknown part of mathematics into some well studied division of mathematics, where many theorems are already proved, and many methods are already available to find answers, then the function can be used to map whole problems out of unfamiliar territory over to " solid ground " where the problem is easier to understand and work with.

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

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

In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds.

In mathematics, a bicategory is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not ( strictly ) associative, but only associative up to an isomorphism.

In mathematics, specifically linear algebra, a degenerate bilinear form ƒ ( x, y ) on a vector space V is one such that the map from to ( the dual space of ) given by is not an isomorphism.

In mathematics, the statement that " Property P characterizes object X " means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as " Property Q characterises Y up to isomorphism ".

In mathematics, there are up to isomorphism exactly two hyperfinite type II factors ; one infinite and one finite.

In mathematics, the derived category D ( C ) of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C. The construction proceeds on the basis that the objects of D ( C ) should be chain complexes in C, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes.

* First isomorphism theorem, that describe the relationship between quotients, homomorphisms, and subobjects ( mathematics )

In mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem or the correspondence theorem, states that if is a normal subgroup of a group, then there exists a bijection from the set of all subgroups of such that contains, onto the set of all subgroups of the quotient group.

In mathematics, a commutativity constraint on a monoidal category ' is a natural isomorphism from to, where is the category with the opposite tensor product.

In mathematics, the Picard group of a ringed space X, denoted by Pic ( X ), is the group of isomorphism classes of invertible sheaves ( or line bundles ) on X, with the group operation being tensor product.

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

Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later ; it is now applied throughout mathematics.

These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of, constructive mathematics.

* Timeline of category theory and related mathematics

In category theory, a branch of mathematics, a functor is a special type of mapping between categories.

In mathematics, especially in category theory and homotopy theory, a groupoid ( less often Brandt groupoid or virtual group ) generalises the notion of group in several equivalent ways.

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space.

Other branches of mathematics, however, such as logic, number theory, category theory or set theory are accepted as part of pure mathematics, though they find application in other sciences ( predominantly computer science and physics ).

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 ".

Because of its applicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed category theory as a foundational system for mathematics, independent of set theory.

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

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.

* Core of a triangulated category in mathematics

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure ( i. e. the composition of morphisms ) of the categories involved.

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.

As an alternative to set theory, others have argued for category theory as a foundation for certain aspects of mathematics.

In category theory, a branch of mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets.

In mathematics, a category is an algebraic structure that comprises " objects " that are linked by " arrows ".