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Let be a Schrödinger operator on the Heisenberg group n, where Δn is the sublaplacian on n and the nonnegative potential V belongs to the reverse Hölder class Bq with q[Q/2,+). Here  Q=2n+2 is the homogeneous dimension of n. Assume that {esL}s>0 is the heat semigroup generated byL. The Lusin area integral SL;α and the Littlewood–Paley–Stein function gλ,L* associated with the Schrödinger operator L are defined, respectively, by

SL;α(f)(u):=Γα(u)sddsesLf(υ)2dυdssQ/2+11/2,

where

Γα(u):={(υ,s)n×(0,+):u1υ<αs},

and

gλ,L*(f)(u):=0nss+u1υ2λsddsesLf(υ)2dυdssQ/2+11/2 ,

Where λ(0,+) is a parameter. In this article, the author shows that there is a relationship between SL;α and the operator gλ,L* and for any 1p<, the following inequality holds true:

SL;2j(f)LpnC2jQ/2+2jQ/psL(f)Lp(n).

Based on this inequality and known results for the Lusin area integral SL;1, the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function gλ,L* on Lp(n). In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator L on n. By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator gλ,L* acting on the Morrey spaces for an appropriate choice of λ>0. It can be shown that the same conclusions hold for the operator gλ,L* on generalized Morrey spaces as well.

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In this paper, a relationship between the zeros and critical points of a polynomial p(z) is established. The relationship is used to prove Sendov’s conjecture in some special cases.

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A fluid queueing system in which the fluid flow in to the buffer is regulated by the state of the background queueing process is considered. In this model, the arrival and service rates follow chain sequence rates and are controlled by an exponential timer. The buffer content distribution along with averages are found using continued fraction methodology. Numerical results are illustrated to analyze the trend of the average buffer content for the model under consideration. It is interesting to note that the stationary solution of a fluid queue driven by a queue with chain sequence rates does not exist in the absence of exponential timer.

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In this paper, we define an orthonormal basis for 2-*-inner product space and obtain some useful results. Moreover, we introduce a 2-norm on a dense subset of a 2-*-inner product space. Finally, we obtain a version of the Selberg, Buzano’s and Bessel inequality and its results in an A-2-inner product space.

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We provide a Maltsev characterization of congruence distributive varieties by showing that a variety 𝓥 is congruence distributive if and only if the congruence identity α(βγβ)_αβγαβγ … (k factors) holds in 𝓥, for some natural number k.

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Fix 2 < n < ω and let CAn denote the class of cyindric algebras of dimension n. Roughly CAn is the algebraic counterpart of the proof theory of first order logic restricted to the first n variables which we denote by Ln. The variety RCAn of representable CAns reflects algebraically the semantics of L n. Members of RCAn are concrete algebras consisting of genuine n-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CAn has a finite equational axiomatization, RCAn is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CAn substantially richer than that of Boolean algebras, just as much as Lω,ω is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCAn are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever 𝔄 V is atomic, then its Dedekind-MacNeille completion, sometimes referred to as its minimal completion, is also in V. From our hitherto obtained algebraic results we show, employing the powerful machinery of algebraic logic, that the celebrated Henkin-Orey omitting types theorem, which is one of the classical first (historically) cornerstones of model theory of L ω,ω, fails dramatically for L n even if we allow certain generalized models that are only locallly clasfsical. It is also shown that any class K such that NrnCAωCRCAn¯K¯ScNrnCAn+3 , where CRCAn is the class of completely representable CAns, and Sc denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that SdRaCAω¯K¯ScRaCA5 is not elementary, where Sd denotes the operation of forming dense subalgebra.

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Let 𝔄 be a unital Banach algebra and its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ is finite dimensional. We also prove that a C *-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

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We give two new simple characterizations of the Cauchy distribution by using the Möbius and Mellin transforms. They also yield characterizations of the circular Cauchy distribution and the mixture Cauchy model.

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In this paper we present different variants of the well-known Hermite–Hadamard inequality, in a generalized context. We consider general fractional integral operators for h-convex and r-convex functions.

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In this study, a normalized form of regular Coulomb wave function is considered. By using the differential subordinations method due to Miller and Mocanu, we determine some conditions on the parameters such that the normalized regular Coulomb wave function is lemniscate starlike and exponential starlike in the open unit disk, respectively. In additon, by using the relationship between the regular Coulomb wave function and the Bessel function of the first kind we give some conditions for which the classical Bessel function of the first kind is lemniscate and exponential starlike in the unit disk 𝔻.

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