resolvent

re·sol·vent

R0178800 (rĭ-zŏl′vənt)adj. Causing or able to cause separation into constituents; solvent.n. A resolvent substance, especially a medicine that reduces inflammation or swelling.

resolvent

(rɪˈzɒlvənt) adjserving to dissolve or separate something into its elements; resolvingn1. something that resolves; solvent2. (Medicine) a drug or agent able to reduce swelling or inflammation

Thesaurus

Noun

resolvent - a liquid substance capable of dissolving other substances; "the solvent does not change its state in forming a solution"dissolvent, dissolver, dissolving agent, solventacetone, dimethyl ketone, propanone - the simplest ketone; a highly inflammable liquid widely used as an organic solvent and as material for making plasticschlorobenzene - a colorless volatile flammable liquid with an almond odor that is made from chlorine and benzene; used as a solvent and in the production of phenol and DDT and other organic compoundscarbolic acid, hydroxybenzene, oxybenzene, phenylic acid, phenol - a toxic white soluble crystalline acidic derivative of benzene; used in manufacturing and as a disinfectant and antiseptic; poisonous if taken internallycarbon tet, carbon tetrachloride, perchloromethane, tetrachloromethane - a colorless nonflammable liquid used as a solvent for fats and oils; because of its toxicity its use as a cleaning fluid or fire extinguisher has declinedmedium - an intervening substance through which something is achieved; "the dissolving medium is called a solvent"hexane - a colorless flammable liquid alkane derived from petroleum and used as a solventmenstruum - (archaic) a solventnaphtha - any of various volatile flammable liquid hydrocarbon mixtures; used chiefly as solventsremover - a solvent that removes a substance (usually from a surface); "paint remover"; "rust remover"; "hair remover"alcahest, alkahest, universal solvent - hypothetical universal solvent once sought by alchemistsmethylbenzene, toluene - a colorless flammable liquid obtained from petroleum or coal tar; used as a solvent for gums and lacquers and in high-octane fuelsxylene, xylol - a colorless flammable volatile liquid hydrocarbon used as a solvent

Resolvent

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\\α_Λ - Aα₂ α_Λ + 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 - v_k σ + a₄ = 0, k = 1, 2, 3. Thus, if x_n and n_k are the roots of these quadratic equations, x₁x₂= x₃x₄ = n, x₁x₃ = x₂x₂x₄ = x₂x₂x₄ = n₃, = ξ₁x₂/n_3, 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 R_λ = (A - λ E)^-1 where the complex parameter λ takes on any values outside the spectrum of A.

resolvent

[re-zol´vent] 1. promoting resolution or the dissipation of a pathologic growth.2. an agent that promotes resolution.

re·sol·vent

(rē-zol'vĕnt), 1. Causing resolution. 2. An agent that arrests an inflammatory process or causes the absorption of a neoplasm.

resolvent

(rĭ-zŏl′vənt)adj. Causing or able to cause separation into constituents; solvent.n. A resolvent substance, especially a medicine that reduces inflammation or swelling.

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resolvent

Related to resolvent: resolution, Resolvent kernel

noun

Synonyms for resolvent

noun a liquid substance capable of dissolving other substances

Synonyms

dissolvent
dissolver
dissolving agent
solvent

Related Words

acetone
dimethyl ketone
propanone
chlorobenzene
carbolic acid
hydroxybenzene
oxybenzene
phenylic acid
phenol
carbon tet
carbon tetrachloride
perchloromethane
tetrachloromethane
medium
hexane
menstruum
naphtha
remover
alcahest
alkahest
universal solvent
methylbenzene
toluene
xylene
xylol