In algebra, the ring of polynomial differential forms on the standard n-simplex is the differential graded algebra: Ω poly ∗ ( [ n ] ) = Q [ t 0 , . ...

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The...

condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed p-form on an open ball...

view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras are used for the study...

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets...

algebraic form Schur polynomial Symbol of a differential operator However, as some authors do not make a clear distinction between a polynomial and its...

linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form a 0...

mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of...

idea is named after mathematician Sergei Natanovich Bernstein. Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the...

mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: x y ″ + ( 1...

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)}...

most polynomial growth conditions in ξ under which this integral is well-behaved comprises the pseudo-differential operators. The differential operator...

equations: polynomial equations and, among them, the special case of linear equations. When there is only one variable, polynomial equations have the form P(x) = 0...

if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions. The closed-form problem arises...

orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as...

the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations. Certain...

many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word...

some special cases solutions can be expressed in terms of polynomials called Lamé polynomials. Lamé's equation is d 2 y d x 2 + ( A + B ℘ ( x ) ) y = 0...

(Simpson's rule) and numerical ordinary differential equations (multigrid methods). In computer graphics, polynomials can be used to approximate complicated...

growth condition Multilinear form, which generalises bilinear forms to mappings VN → F Quadratic form, a homogeneous polynomial of degree two in a number...

polynomial equation in the unknown function and its derivatives, its degree of the differential equation is, depending on the context, the polynomial...

In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions...

has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic. The theory...

mathematics, a polynomial p {\displaystyle p} whose Laplacian is zero is termed a harmonic polynomial. The harmonic polynomials form a subspace of the...

In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential...

of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of...

especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally...

defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their...

In differential geometry, an equivariant differential form on a manifold M acted upon by a Lie group G is a polynomial map α : g → Ω ∗ ( M ) {\displaystyle...

generated homology, the Poincaré polynomial is defined as the generating function of its Betti numbers, via the polynomial where the coefficient of x n {\displaystyle...