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Related to Resolvent: resolution, Resolvent kernel


For a linear operator T on a Banach space, the function, defined on the complement of the spectrum of T given by (T- λ I)-1 for each λ in this complement, where I is the identity operator; this enables a study of T relative to its eigenvalues.



a mathematical term with various meanings. We speak, for example, of resolvent equations, resolvent kernels, and resolvent operators.

In algebra, the term “resolvent” is used in several senses. Thus, by the resolvent of the algebraic equation f(x) = 0 of degree n we mean an algebraic equation g(x) = 0 such that its coefficients are rational functions of the coefficients of f(x) and, if the roots of this equation are known, it is possible to find the roots of f(x) = 0 by solving simpler equations of degree at most n. For example, the equation

ν3 - a2v2 + (a1a3 - 4α4) ν - (α\αΛ - 2 αΛ + a]) = 0

is one of the cubic resolvents of the fourth-degree equation

(1) x4 + a1x3 + a2x2 + a3x + a4 = 0

If ν1, v2, and v3 are the roots of the resolvent equation, the roots x1, x2, x3, and x4 of equation (1) can be found by solving the quadratic equations σ2 - vk σ + a4 = 0, k = 1, 2, 3. Thus, if xn and nk are the roots of these quadratic equations, x1x2= x3x4 = n, x1x3 = x2x2x4 = x2x2x4 = n3, Resolvent = ξ1x2/n3, and so on. A Galois resolvent of the equation f(x) is an algebraic equation g(x) = 0 irreducible over a given field (seeGALOIS THEORY) such that when one of its roots is adjoined to the field, there results a field containing all the roots of the equation f(x) = 0.

The term “resolvent” is used in a somewhat different sense in what is known as the Hilbert-Chebotarev resolvent problem.

In the theory of integral equations, the resolvent of the equation

is a function Γ (s, t; λ) of the variables s and t and the parameter λ such that the solution of equation (2) can be represented in the form

provided that λ is not an eigenvalue of equation (2). For example, the resolvent of the kernel K(s, t) = S + t is the function

In the theory of linear operators, the resolvent of the operator A is the family of operators Rλ = (A - λ E)-1 where the complex parameter λ takes on any values outside the spectrum of A.

References in periodicals archive ?
The third step is to construct the resolvent kernel of the operator [H.
A matrix function which calculates subgraph centrality is the matrix resolvent
However, more accurate methods were developed recently, based on superior-order Taylor expansion, that reduce algebraic equations to polynomial ones, named resolvent equations.
4]M are the same literal M and the index 2 < index 4, so the resolvent of (1) and (2) will be (4) and is called a lock resolvent of clause (1) and clause (2).
An operator T is called polaroid if iso [sigma](T) [subset or equal to] [pi](T), where [pi](T) is the set of poles of the resolvent of T and iso[sigma](T) is the set of all isolated points of [sigma](T).
where A is the infinitesimal generator of a compact, analytic resolvent operator R(t),t > 0 in a Banach space X, [x.
Cet argument est illustre par le protocole d'un dialogue entre deux etudiants universitaires qui resolvent un probleme mathematique avec CAS.
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The significance of the resolvent in this context is that it allows Y to be written purely in terms of the perturbation, according to the variation of parameters formula
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Dans le texte du plan d'urbanisme, les contradictions du developpement durable se resolvent au sein d'un projet de compacite, compacite souhaitable parce qu'elle permet de faire des economies en plus de limiter les besoins de deplacement et de favoriser le transport collectif.