Stieltjes Transform Research Articles

A Family of Mexican Hat Wavelet Stieltjes Transform for Unbounded Non-decreasing Functions

A Family of Mexican Hat Wavelet Stieltjes Transform for Unbounded Non-decreasing Functions

A Note on Some Novel Laplace and Stieltjes Transforms Associated with the Relaxation Modulus of the Andrade Model

In the framework of linear viscoelasticity, the authors have previously calculated a novel inverse Laplace transform involving the Mittag–Leffler function in order to calculate the relaxation modulus in the Andrade model. Here, we generalize this result, calculating the inverse Laplace transform of a given function Fα,βs by using two different approaches: the Bromwich integral and the decomposition of Fα,βs in simple fractions. From both calculations, we obtain a set of novel Laplace and Stieltjes transforms.

Performance Analysis of Feeding System in Sugar Mill with Consideration of Bagasse Jamming

Objectives: The sugar industry comprises various units, including feeding, crushing, evaporation, refining, and crystallization. The feeding system is the most crucial aspect of the sugar mill as the sugar-making process starts from here. So faults which occur here are divided into four categories minor, major, cease faults and walkout faults. This paper sets out to showcase a comprehensive analysis of the system's performance, availability, and profit. The study takes into account how minor, major, and cease faults can potentially impact the system's overall effectiveness. The findings of this research hold significant importance for sugar mills that prioritize optimal performance and profit. Methods: Primary data regarding various failures is collected from Ch. Devilal Co-Operative Sugar Mills Limited, Ahulana, Gohana. To find MTSF, Reliability, Availability, Profit etc., a mathematical model has been created. This model is based on the Semi-Markov process and Regenerative Point Technique and equations are drawn using exponential distribution and solved with Cremer’s rule and Laplace ­- Stieltjes transformation. Findings: A fault, Bagasse Jamming is considered a cease fault which occurs very frequently in the feeding system. Therefore, it is found that cease faults and major faults have more adverse impacts on the system's performance and availability than minor faults. MTSF, Availability and Profit are inversely proportional to these faults. When a major fault is 0.0035 and the cease fault is 0.0042, MTSF is nearly 100. So, to gain more profit we have to pay more attention to these faults. Novelty: The paper's results will aid in fault removal, increase availability, and optimize maintenance tactics in the sugar industry. Keywords: MTSF, Performance Analysis, Availability, Reliability, Minor, Major, Cease Faults

A study on the spectral properties of covariance type matrices with isotropic log-concave column vectors.

The limiting spectral distribution of matrix [Formula: see text] is considered in this paper. Existing results always focus on the condition of modifying Tn, but for Xn, it is usually assumed to be a matrix composed of n × N independent identically distributed elements. Here we specify the joint distribution of column vectors of Xn. In particular, entries on the same column of Xn are correlated, in contrast with more common independence assumptions. Assuming that the columns of Xn are random vectors following the isotropic log-concave distribution, and under some additional regularity conditions, we prove that the empirical spectral distribution [Formula: see text] of matrix Bn converges to a deterministic probability distribution F almost surely. Moreover, the Stieltjes transformation m = m(z) of F satisfies a deterministic form of equation, and for any [Formula: see text], it is the unique solution of the equation.

Stieltjes Transforms and R-Transforms Associated with Two-Parameter Lambert-Tsallis Functions.

In this paper, we study a two-parameter family of Stieltjes transformations related to holomorphic Lambert-Tsallis functions, which are a two-parameter generalization of the Lambert function. Such Stieltjes transformations appear in the study of eigenvalue distributions of random matrices associated with some growing statistically sparse models. A necessary and sufficient condition on the parameters is given for the corresponding functions being Stieltjes transformations of probabilistic measures. We also give an explicit formula of the corresponding R-transformations.

Some Relations between Stieltjes Transform and Hankel Transform with Applications

In the present paper four theorems connecting Stieltjes transform and Hankel transform are established. The theorems are general in nature. Four integral formulae involving special functions are obtained with the help of these theorems. Otherwise it is very difficult to evaluate such type of integrals. Other several integrals may be evaluated with the help of these theorems.

Some New Inequalities on Laplace–Stieltjes Transforms Involving Logarithmic Growth

This article is devoted to exploring the properties on the logarithmic growth of entire functions represented by Laplace–Stieltjes transforms of zero order. In order to describe the growth of Laplace–Stieltjes transforms more finely, we introduce some concepts of the logarithmic indexes of the maximum term and the center index of the maximum term of Laplace–Stieltjes transforms, and establish some new inequalities focusing on the above logarithmic indexes, the logarithmic order, the (lower) logarithmic type and the coefficients of Laplace–Stieltjes transforms. Moreover, we obtain two estimation forms on the (lower) logarithmic type of entire functions represented by Laplace–Stieltjes transform by applying these inequalities. One estimation is mainly by the center indexes of the maximum term, the other is by the logarithmic order, exponent and coefficients. Finally, we obtain the equivalence condition of entire functions with the perfectly logarithmic linear growth. This result shows that the two estimation forms can be equivalent to some extent.

On the Calculation of Functional Safety Parameters of Technical Systems

Now the scientific methodology is created, the theory and practice of the analysis and synthesis of functional safety of responsible electronic programmable devices and systems at all stages of their life cycle are developed. The basics of the methodology are fixed by standards. Methods of analysis and synthesis of functional safety are strictly formalized. They are based on the calculations of functional safety indicators with respect to failures of constituent elements and, especially, dangerous and protective failures of the system. Known methods of calculation are focused on determining the intensity and probability of dangerous failures. The objective of the proposed method lies in the fact that, in graph form, without resorting to the solution of the system of equations in the operator transformations to establish the distribution function of time until the threat or security failure, or any unhealthy condition of the system. These distribution functions determine all the necessary indicators of mean time (and, if necessary, the variance of this time) to a dangerous or protective failure. The proposed semi-Markov (Markov) operator method allows to solve a number of problems of calculation and prediction of functional safety of critical (responsible) systems. The method is formalized and suitable for subsequent computer implementation. This fact testifies to the expediency of further development of graph methods, convenient for the study of the safety of complex critical systems, devoid of the shortcomings of the proposed method in terms of the complexity of the preparatory work to determine the analytical expressions of transition probabilities in the Laplace - Stieltjes transformations. The given example of using the method has an independent value – it allows you to assess the advantages and disadvantages of ensuring functional safety by building a two-channel system without restarting the channels

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.

Renewal Redundant Systems Under the Marshall–Olkin Failure Model. A Probability Analysis

In this paper a two component redundant renewable system operating under the Marshall–Olkin failure model is considered. The purpose of the study is to find analytical expressions for the time dependent and the steady state characteristics of the system. The system cycle process characteristics are analyzed by the use of probability interpretation of the Laplace–Stieltjes transformations (LSTs), and of probability generating functions (PGFs). In this way the long mathematical analytic derivations are avoid. As results of the investigations, the main reliability characteristics of the system—the reliability function and the steady state probabilities—have been found in analytical form. Our approach can be used in the studies of various applications of systems with dependent failures between their elements.

Analysis of an unreliable single server batch arrival queue with two types of services under Bernoulli vacation policy

This article considers a single server batch arrival queue providing two types of general heterogeneous parallel services under Bernoulli Vacation Policy. The server assumed here is unreliable and incorporates the concept of delayed repair period. An extensive study of this model is carried out and the queue size distribution at random as well as departure epoch is obtained. Further, Laplace Stieltjes Transform of a busy period and waiting time distribution is also being derived. Finally, some important performance measures along with the reliability indices are accomplished in the study.

Markov – Stieltjes transformation of measures and some of its applications

The paper is devoted to the study of the properties of the Markov – Stieltjes transformation of measures. In the works of J. Anderson, A. A. Pekarsky, N. S. Vyacheslavov, E. P. Mochalina et al., the functions of Markov – Stieltjes type were studied from the point of view of the approximation theory. In the works of A.R. Mirotin and the author, the Markov – Stieltjes transform of functions was studied as an operator in Hardy and Lebesgue spaces. In this paper, the general properties of the Markov – Stieltjes transform of measures are studied, the theorem of analyticity and the uniqueness theorem are proved, the Markov – Stieltjes transformations of positive and complex measures are described, the inversion formula and the continuity theorem are established, the boundary behavior of the given transformation is investigated. In particular, the analogues of the Sokhotsky – Plemelya formulas are established. Applications to the theory of self-conjugate operators are given. In addition, the results obtained can find use in the theory of functions and integral operators, as well as in the theory of information transfer, in particular, in the theory of signal processing.

Doubly infinite Jacobi matrices revisited: Resolvent and spectral measure

We study the resolvent and spectral measure of certain doubly infinite Jacobi matrices via asymptotic solutions of two-sided difference equations. By finding the minimal (or subdominant) solutions or calculating the continued fractions for the difference equations, we derive explicit formulas for the matrix entries of resolvent of doubly infinite Jacobi matrices corresponding to Lommel polynomials, associated ultraspherical polynomials, and Al-Salam–Ismail polynomials. The spectral measures are then obtained by inverting Stieltjes transformations.

Hydrodynamic Limit of Multiple SLE

Recently del Monaco and Schlei{\ss}inger addressed an interesting problem whether one can take the limit of multiple Schramm--Loewner evolution (SLE) as the number of slits $N$ goes to infinity. When the $N$ slits grow from points on the real line ${\mathbb{R}}$ in a simultaneous way and go to infinity within the upper half plane ${\mathbb{H}}$, an ordinary differential equation describing time evolution of the conformal map $g_t(z)$ was derived in the $N \to \infty$ limit, which is coupled with a complex Burgers equation in the inviscid limit. It is well known that the complex Burgers equation governs the hydrodynamic limit of the Dyson model defined on ${\mathbb{R}}$ studied in random matrix theory, and when all particles start from the origin, the solution of this Burgers equation is given by the Stieltjes transformation of the measure which follows a time-dependent version of Wigner's semicircle law. In the present paper, first we study the hydrodynamic limit of the multiple SLE in the case that all slits start from the origin. We show that the time-dependent version of Wigner's semicircle law determines the time evolution of the SLE hull, $K_t \subset {\mathbb{H}} \cup {\mathbb{R}}$ , in this hydrodynamic limit. Next we consider the situation such that a half number of the slits start from $a>0$ and another half of slits start from $-a < 0$, and determine the multiple SLE in the hydrodynamic limit. After reporting these exact solutions, we will discuss the universal long-term behavior of the multiple SLE and its hull $K_t$ in the hydrodynamic limit.

Self-Similar Traffic in G/M/1 Queue Defined by the Weibull Distribution

The work considers a queuing system of the G/M/1 type that simulates service of self-similar traffic in a NodeB (e-NodeB) base station of a mobile operator. The feature of quality of service (QoS) characteristics estimation process for the self-similar traffic defined by the Weibull distribution is the solution based on the Laplace–Stieltjes transformation. The Laplace transformation for an infinite number of items under the Weibull distribution condition was found. It was shown that this series was equiconvergent to some convergence domain. The following QoS characteristics were obtained for the self-similar traffic: the average amount of time that a request spends in the system; the average number of requests waiting in the queue and the average queue length. The obtained results allowed to consider the real values of traffic serviced by a NodeB (e-NodeB) for their optimal deployment over a covered territory at the stage of frequency planning and operation of the 3G/UMTS and 4G/LTE networks.

Local Semicircle Law under Moment Conditions: The Stieltjes Transform, Rigidity, and Delocalization

We consider a random symmetric matrix ${X} = [X_{jk}]_{j,k=1}^n$ where the upper triangular entries are independent identically distributed random variables with zero mean and unit variance. We add...

The approximation of Laplace–Stieltjes transformations with finite order on the left half plane

In this paper, we study the error in approximating the analytic function defined by a Laplace–Stieltjes transformation of finite order, which converges on the left half plane, and obtain the relation theorems between the error, the coefficients, and the proximate order of the Laplace–Stieltjes transformation.

On the growth and approximation of entire functions represented by Laplace–Stieltjes’ transformation

In the present paper, we obtain the characterization of various growth parameters of an entire function F(s) represented by Laplace–Stieltjes transformation in terms of the rate of decrease of \(E_n ( {F,\beta } ),\)where \(E_n ( {F,\beta } )\) represents the error in approximating the function F(s) by exponential polynomials.

Stieltjes transformation as the iterated Laplace transformation

Stieltjes transformation as the iterated Laplace transformation

Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples

For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular, we retrieve the examples given by the author in a previous paper and relating generalized Stieltjes transforms of special beta distributions to powers of (ordinary) Stieltjes ones. We also provide further examples of similar relations which are motivated by the representation theory of symmetric groups. Remarkably, the power of the Stieltjes transform of the symmetric Bernoulli distribution is a generalized Stietljes transform of a probability distribution if and only if the power is greater than one. As to the free Poisson distribution, it corresponds to the product of two independent Beta distributions in $[0,1]$ while another example of Beta distributions in $[-1,1]$ is found and is related with the Shrinkage process. We close the exposition by considering the generalized Stieltjes transform of a linear functional related with Humbert polynomials and generalizing the symmetric Beta distribution.