Second-Order Differential Operators in the Limit Circle Case

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.

The Approximation of Laplace-Stieltjes Transforms Concerning Sun’s Type Function

The main aim of this paper is to establish some theorems concerning the error E n F , β , the Sun’s type function U r , and M u σ , F of entire functions defined by Laplace-Stieltjes transforms with infinite order converge in the whole complex plane. Our results exhibit the growth of Laplace-Stieltjes transforms from the point of view of approximation.

Synchronized Lévy queues

We consider a multivariate Lévy process where the first coordinate is a Lévy process with no negative jumps which is not a subordinator and the others are non-decreasing. We determine the Laplace–Stieltjes transform of the steady-state buffer content vector of an associated system of parallel queues. The special structure of this transform allows us to rewrite it as a product of joint Laplace–Stieltjes transforms. We are thus able to interpret the buffer content vector as a sum of independent random vectors.

An Analytical Model of a Corporate Software-Controlled Network Switch

Implementing the almost limitless possibilities of a software-defined network requires additional study of its infrastructure level and assessment of the telecommunications aspect. The aim of this study is to develop an analytical model for analyzing the main quality indicators of modern network switches. Based on the general theory of queuing systems and networks, generated functions and Laplace-Stieltjes transforms, a three-phase model of a network switch was developed. Given that, in this case, the relationship between processing steps is not significant, quality indicators were obtained by taking into account the parameters of single-phase networks. This research identified the dependencies of service latency and service time of incoming network packets on load, as well as equations for finding the volume of a switch’s buffer memory with an acceptable probability for message loss.

The Growth of Entire Functions Defined by Laplace–Stieltjes Transforms Related to Proximate Order and Approximation

In this article, we discuss the growth of entire functions represented by Laplace–Stieltjes transform converges on the whole complex plane and obtain some equivalence conditions about proximate growth of Laplace–Stieltjes transforms with finite order and infinite order. In addition, we also investigate the approximation of Laplace–Stieltjes transform with the proximate order and obtain some results containing the proximate growth order, the error, An∗, and λn, which are the extension and improvement of the previous theorems given by Luo and Kong and Singhal and Srivastava.

Growth and Approximation of Laplace-Stieltjes Transform with p,q-Proximate Order Converges on the Whole Plane

One purpose of this paper is to study the growth of entire functions defined by Laplace-Stieltjes transform converges on the whole complex plane, by introducing the concept of p,q-proximate order, and one equivalence theorem of the p,q-proximate order of Laplace-Stieltjes transforms is obtained. Besides, the second purpose of this paper is to investigate the approximation of entire functions defined by Laplace-Stieltjes transforms with p,q-proximate order, and some results about the p,q-proximate order, the error, and the coefficients of Laplace-Stieltjes transforms are obtained, which are generalization and improvement of the previous theorems given by Luo and Kong, Singhal, and Srivastava.

Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type

In this paper, we introduce the concept of the perfect ϕ \phi -type to describe the growth of the maximal molecule of Laplace-Stieltjes transform by using the more general function than the usual. Based on this concept, we investigate the approximation and growth of analytic functions F ( s ) F(s) defined by Laplace-Stieltjes transforms convergent in the half plane and obtain some results about the necessary and sufficient conditions on analytic functions F ( s ) F(s) defined by Laplace-Stieltjes transforms with perfect ϕ \phi -type, which are some generalizations and improvements of the previous results given by Kong [On generalized orders and types of Laplace-Stieltjes transforms analytic in the right half-plane, Acta Math. Sin. 59A (2016), 91–98], Singhal and Srivastava [On the approximation of an analytic function represented by Laplace-Stieltjes transformations, Anal. Theory and Appl. 31 (2015), 407–420].

ARENS PRODUCTS, ARENS REGULARITY AND RELATED PROBLEMS

We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms.