In mathematics, a bilinear operator is a function combining elements of two vector spaces to yield an element of a third vector space that is linear in each of its arguments.

In mathematics, the orthogonal group of a symmetric bilinear form or quadratic form on a vector space is the group of invertible linear operators on the space which preserve the form: it is a subgroup of the automorphism group of the vector space.

In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product.

In mathematics, a symplectic vector space is a vector space V ( over a field, for example the real numbers R ) equipped with a bilinear form ω: V × V → R that is

In mathematics, the indefinite orthogonal group, O ( p, q ) is the Lie group of all linear transformations of a n = p + q dimensional real vector space which leave invariant a nondegenerate, symmetric bilinear form of signature ( p, q ).

In mathematics, an anyonic Lie algebra is a U ( 1 ) graded vector space L over C equipped with a bilinear operator and linear maps ε: L -> C and Δ: L -> L ⊗ L satisfying

In mathematics, bilinear interpolation is an extension of linear interpolation for interpolating functions of two variables ( e. g., and ) on a regular 2D grid.

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 Schur indicator, named after Issai Schur, or Frobenius – Schur indicator describes what invariant bilinear forms a given irreducible representation of a compact group on a complex vector space has.

In mathematics, a Poisson manifold is a smooth manifold M equipped with a bilinear map

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.

In mathematics, the Weil pairing is a pairing ( bilinear form, though with multiplicative notation ) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity.

In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form.

In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space.

In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories.

In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven dimensional Euclidean space.

In the essay a blind English mathematician named Saunderson argues that since knowledge derives from the senses, then mathematics is the only form of knowledge that both he and a sighted person can agree about.

Political economy was the earlier name for the subject, but economists in the latter 19th century suggested ' economics ' as a shorter term for ' economic science ' that also avoided a narrow political-interest connotation and as similar in form to ' mathematics ', ' ethics ', and so forth.

The polar form simplifies the mathematics when used in multiplication or powers of complex numbers.

In modern mathematics, Euclidean spaces form the prototypes for other, more complicated geometric objects.

Modern mathematics defines an " elliptic integral " as any function which can be expressed in the form

In mathematics, Horner's method ( also known as Horner scheme in the UK or Horner's rule in the U. S .) is either of two things: ( i ) an algorithm for calculating polynomials, which consists in transforming the monomial form into a computationally efficient form ; or ( ii ) a method for approximating the roots of a polynomial.

Modern engineering as it is understood today took form during the scientific revolution, though much of the mathematics and science was built on the work of the Greeks, Egyptians, Mesopotamians, Chinese, Indians and Muslims.

In the 1490s he studied mathematics under Luca Pacioli and prepared a series of drawings of regular solids in a skeletal form to be engraved as plates for Pacioli's book De Divina Proportione, published in 1509.

Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics ( see Problem of induction for more information ).

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element.

In mathematics and computer science, mutual recursion is a form of recursion where two mathematical or computational functions are defined in terms of each other.

This form of mathematics was instrumental in early map-making.

More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle.

In its axiomatic form, mathematical statements about probability theory carry the same sort of epistemological confidence shared by other mathematical statements in the philosophy of mathematics.

Similarly, a form of modern Platonism is found in the predominant philosophy of mathematics, especially regarding the foundations of mathematics.

* Reduction ( mathematics ), the rewriting of an expression into a simpler form

European mathematicians had previously viewed geometry as a more fundamental form of mathematics, serving as the foundation of algebra.

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form.

The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

The precise formulation of what are today recognized as correct statements of the laws of nature did not begin until the 17th century in Europe, with the beginning of accurate experimentation and development of advanced form of mathematics ( see scientific method ).

* Mean Biased Error, a form of calculating of statistical error in mathematics.

In the philosophy of mathematics, the best known form of realism about numbers is Platonic realism, which grants them abstract, immaterial existence.

An extreme form of realism about mathematics is the mathematical multiverse hypothesis advanced by Max Tegmark.

Scientists say that the world and everything in it are based on mathematics.

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

Pythagoras believed that behind the appearance of things, there was the permanent principle of mathematics, and that the forms were based on a transcendental mathematical relation.

Ethics cannot be based on the authoritative certainty given by mathematics and logic, or prescribed directly from the empirical findings of science.

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X

He used his time in Bourg to research mathematics, producing Considérations sur la théorie mathématique de jeu ( 1802 ; “ Considerations on the Mathematical Theory of Games ”), a treatise on mathematical probability that he sent to the Paris Academy of Sciences in 1803.

Two aspects of this attitude deserve to be mentioned: 1 ) he did not only study science from books, as other academics did in his day, but actually observed and experimented with nature ( the rumours starting by those who did not understand this are probably at the source of Albert's supposed connections with alchemy and witchcraft ), 2 ) he took from Aristotle the view that scientific method had to be appropriate to the objects of the scientific discipline at hand ( in discussions with Roger Bacon, who, like many 20th century academics, thought that all science should be based on mathematics ).

The axiom of regularity is arguably the least useful ingredient of Zermelo – Fraenkel set theory, since virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity ( see chapter 3 of ).

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.

" The Four Books on Measurement " were published at Nuremberg in 1525 and was the first book for adults on mathematics in German, as well as being cited later by Galileo and Kepler.

By focusing consciously on an idea, feeling or intention the meditant seeks to arrive at pure thinking, a state exemplified by but not confined to pure mathematics.

On the Infinite was Hilbert ’ s most important paper on the foundations of mathematics, serving as the heart of Hilbert's program to secure the foundation of transfinite numbers by basing them on finite methods.

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.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A.

In mathematics, a binary operation on a set is a calculation involving two elements of the set ( called operands ) and producing another element of the set ( more formally, an operation whose arity is two ).

Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

The first person to describe the mathematics behind Brownian motion was Thorvald N. Thiele in a paper on the method of least squares published in 1880.

Calculus ( Latin, calculus, a small stone used for counting ) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.

On November 29, 1921, the trustees declared it to be the express policy of the Institute to pursue scientific research of the greatest importance and at the same time " to continue to conduct thorough courses in engineering and pure science, basing the work of these courses on exceptionally strong instruction in the fundamental sciences of mathematics, physics, and chemistry ; broadening and enriching the curriculum by a liberal amount of instruction in such subjects as English, history, and economics ; and vitalizing all the work of the Institute by the infusion in generous measure of the spirit of research.

* nLab, a wiki project on mathematics, physics and philosophy with emphasis on the n-categorical point of view