Resolvent


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Related to Resolvent: resolution, Resolvent kernel

resolvent

[ri′zäl·vənt]
(mathematics)
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.

Resolvent

 

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 ?
In the large N limit, for the Gaussian model, the resolvent obeys the self-consistency equation (also known as the Schwinger-Dyson equation) (see, e.g., [10], Section VII.4):
Nilpotent operators are quasinilpotent of order zero but the converse is not true since a quasinilpotent is nilpotent of order n if and only if the resolvent is a polynomial in 1/[lambda] of order n.
The first step is to analyze the behavior as [absolute value of x - y] [right arrow] [infinity] of the resolvent kernel R([zeta]; K)(x,y).
Researchers have shown that squalene in sebum upon oxidation releases irritating free radicals into the tissues which along with peroxides initiate and maintain the damaging inflammatory pathway.17 The alleviation in the acne lesions with the test formulation appears to be due to resolvent antiseptic astringent detergent and desiccant properties of the ingredients as described in Unani pharmacological literature.
Enamored of the Night for her own sake," Dupin invites his companion to turn away from daily life and plunge with him into nocturnal reveries "seeking, amid the wild lights and shadows of the populous city, that infinity of mental excitement which quiet observation can afford." Dupin's friend is offered the role of the privileged witness and secret sharer of the mastermind's "creative and resolvent" capacities.
Heureusement, quelques milliards de ces disputes se resolvent par un accord.
Factorization of some classes of matrix functions and the resolvent of a Hankel operator.
The plant is also known to have antiseptic, aphrodisiac, astringent, antioxidant, cholagogue, demulcent, digestive, purgative, resolvent, and anti tumor properties [6].
(87) Voir Y P Thomas, << Le droit entre les mots et les choses : Rhetorique et jurisprudence a Rome >> (1978) 23 Archives de philosophie du droit 93 a la p 102 : << La cause est cette "chose" transformee dans le Heu de la controverse : "chose posee dans la dispute des parties et dans la controverse" Telle est la premiere etape d'une mise en forme verbale, par le contrat, de toutes les violences qui se resolvent en proces >> [note omise].