of abstract mathematics: in particular to algebraic geometry, invariant theory, and representation theory. Polarization and related techniques form the...

Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables Polarization identity...

polynomial Diagonal form Graded algebra Hilbert series and Hilbert polynomial Multilinear form Multilinear map Polarization of an algebraic form Schur polynomial...

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors...

theory, polarization is the maximal totally isotropic subspace of a certain skew-symmetric bilinear form on a Lie algebra. The notion of polarization plays...

known as the half-form correction) is a technical modification of the above procedure that is necessary in the case of real polarizations and often convenient...

examples of a *-ring are fields of complex numbers and algebraic numbers with complex conjugation as the involution. One can define a sesquilinear form over...

Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished...

linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory...

not unique, and a choice of subspace V is called a polarization. The subspaces that give such an isomorphism are called Lagrangian subspaces or simply...

products represent the quadratic form. Through the polarization identity the quadratic form is related to a symmetric bilinear form B(u, v) = ⁠1/4⁠(q(u + v) −...

is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For...

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space...

mathematics, algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall, with L being used as the letter after K. Algebraic L-theory...

variability of the pole of a wave, as noted in studies of polarization, might view elliptical polarization as an exhibit of the elliptical shape of a wave...

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as...

non-zero vectors of V. An indefinite quadratic form takes on both positive and negative values and is called an isotropic quadratic form. More generally...

polarized") light has an equal amount of power in each of two linear polarizations. The s polarization refers to polarization of a wave's electric field...

interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for...

equations of the model into algebraic equations suitable for efficient implementation on a computer. The use of basis sets is equivalent to the use of an approximate...

examples where the power of an odd prime dividing the order is odd. If the abelian variety has a principal polarization then the form on Ш is skew symmetric...

split-octonions. Every composition algebra has an associated bilinear form B(x,y) constructed with the norm N and a polarization identity: B ( x , y )   =  ...

attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper. Some aspects of the exterior...

The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation. They were defined by George Gabriel Stokes...

dense algebraic span, in the inner product norm, follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous...

known as the Bloch sphere picture of qubit state space. An example of pure and mixed states is light polarization. An individual photon can be described...

variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity...

{\displaystyle C} be a non-singular algebraic curve of genus g {\displaystyle g} over Q {\displaystyle \mathbb {Q} } . Then the set of rational points on C {\displaystyle...

by a Kramers-Kronig transform. The polarization of the reflected light depends on the symmetry of the arrangement of the incident probing light with respect...

algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of...