Note on a Stieltjes Transform in terms of the Lerch Function

In this work the authors derive the Stieltjes transform of the logarithmic function in terms of the Lerch function. This transform is used to derive closed form solutions involving fundamental constants and special functions. Specifically we derive the definite integral given by\[\int_{0}^{\infty} \frac{(1-b x)^m \log ^k(c (1-b x))+(b x+1)^m \log ^k(c (b x+1))}{a+x^2}dx\]where $a,b,c,m$ and $k$ are general complex numbers subject to the restrictions given in connection with the formulas.

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.

Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F(x)=exp{∫0Tθ(t,x(t))dt} successfully exist under the certain condition, where θ(t,u)=∫Rexp{iuv}dσt(v) is a Fourier–Stieltjes transform of a complex Borel measure σt∈M(R) and M(R) is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of F(x) sucessfully holds on the Wiener space.

Empirical Analysis of Duality Spaces Connecting Distinct Optical form Transforms

This paper presents a novel technique of construction a precise functional frame in presence of the new proposed constraints during the planning straightforward extension of excessive considerable dimensional generalizations using a empirical relationship of two absolutely distinct transforms having diverse kernels transform for the Laplace Stieltjes spaces consisting of analytical signals from two dimensions at any point heavily affecting the successful development for the view of the Gelfand Shilov techniques a subspace of a Schwartz space simple objective function along with their duals implies continuity having functional analyst approach under many classical conventional transforms arise naturally as Laplace Stieltjes transform of certain distributions extensively used in many applications like magnetic field theory follows from the belongings of strong continuity at origin lean heavily in constructing multidimensional S type spaces based on the testing function spaces upto some desired order for infinitely differentiable functions t, x with Gelfand Shilov concept under one umbrella.

Model of Functioning Data-Transfer Systems Special Purposes Taking into Account the Influence of Cyber Attack

The article proposes an analytical model of the functioning of the data transmission system, taking into account the influence of short-term interruptions in messages and the recovery of information distorted by cyber-attacks. The model uses the Laplace-Stieltjes transform and the Lagrange multiplier method. This helps determine the performance of switching nodes, the throughput of communication channels and their total cost in a communication system.

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.