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The coefficients and delays in models describing various processes are usually obtained as a results of measurements and can be obtained only approximately. We deal with the question of how to estimate the influence of ‘mistakes’ in coefficients and delays on solutions’ behavior of the delay differential neutral system x i ′ ( t ) − q i ( t ) x i ′ ( t − θ i ( t ) ) + ∑ j = 1 n ( p i j ( t ) − Δ p i j ( t ) ) x j ( t − τ i j ( t ) − Δ τ i j ( t ) ) = f i ( t ) , i = 1 , … , n , t ∈ [ 0 , ∞ ) . This topic is known in the literature as uncertain systems or systems with interval defined coefficients. The goal of this paper is to obtain stability of uncertain systems and to estimate the difference between solutions of a ‘real’ system with uncertain coefficients and/or delays and corresponding ‘model’ system. We develop the so-called Azbelev W -transform, which is a sort of the right regularization allowing researchers to reduce analysis of boundary value problems to study of systems of functional equations in the space of measurable essentially bounded functions. In corresponding cases estimates of norms of auxiliary linear operators (obtained as a result of W -transform) lead researchers to conclusions about existence, uniqueness, positivity and stability of solutions of given boundary value problems. This method works efficiently in the case when a ‘model’ used in W -transform is ‘close’ to a given ‘real’ system. In this paper we choose, as the ‘models’, systems for which we know estimates of the resolvent Cauchy operators. We demonstrate that systems with positive Cauchy matrices present a class of convenient ‘models’. We use the W -transform and other methods of the general theory of functional differential equations. Positivity of the Cauchy operators is studied and then used in the analysis of stability and estimates of solutions. Results: We propose results about exponential stability of the given system and obtain estimates of difference between the solution of this uncertain system and the ‘model’ system x i ′ ( t ) − q i ( t ) x i ′ ( t − θ i ( t ) ) + ∑ j = 1 n p i j ( t ) x j ( t − τ i j ( t ) ) = f i ( t ) , i = 1 , … , n , t ∈ [ 0 , ∞ ) . New tests of stability and in the future of existence and uniqueness of boundary value problems for neutral delay systems can be obtained on the basis of this technique.

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

Let p 1 , p 2 , … , p n , ( n ≥ 2 ) , be distinct positive numbers and r > 0 . We propose to study a comparison of the positivity properties of two families of matrices, K r + 1 = p i r + 1 + p j r + 1 p i + p j and B r = | p i - p j | r in full. Indeed, Bhatia and Jain (Spectr. Theory 5(1):71–87, 2015) studied about B r and carried out rigorous analysis on the study of K r . They conjectured therein that inertia of K r + 1 and B r are same for all r > 0 . We settle a congruence relation between these two families in this paper.

Based on the higher-order restricted flows, the first type of integrable deformed fourth-order matrix NLS equations, that is, the fourth-order matrix NLS equations with self-consistent sources (FMNLSSCS), is derived. By virtue of the ∂ ¯ -dressing method, the second type of integrable deformed fourth-order matrix NLS equations called the fourth-order matrix NLS–Maxwell–Bloch system (FMNLS-MB) is presented. We prove the equivalence of the FMNLSSCS and the FMNLS-MB successfully. Furthermore, N-soliton solutions are explicitly obtained by means of the Cauchy matrix method starting from corresponding Sylvester equation.

In this paper, we seek connections between the Sylvester equation and Kadomtsev–Petviashvili system. By introducing Sylvester equation LM are bold, please chekc if bold neceaasry, if not, please remove all bold of equation −MK = rsT together with an evolution equation set of r and s, master function S(i,j)=sTKjC(I + MC)−1Lir is used to construct the Kadomtsev–Petviashvili system, including the Kadomtsev–Petviashvili equation, modified Kadomtsev–Petviashvili equation and Schwarzian Kadomtsev–Petviashvili equation. The matrix M provides τ-function by τ = |I + MC|. With the help of some recurrence relations, the reductions to the Korteweg–de Vries and Boussinesq systems are discussed.

An extended -dimensional Toda lattice equation is investigated by means of the Cauchy matrix approach. We introduce a direction parameter in the extension and represented the equation as a coupled system in a -dimensional space. The equation can also be considered as a negative-order member in one direction of the discrete Kadomtsev–Petviashvili equation. By introducing the -function and an auxiliary direction, the equation can be bilinearized in a -dimensional space with a single -function.

Applying the maximum separable distance (MDS) matrices is one of the most common approaches to meet diffusion layer in modern block ciphers. Using Cauchy and extensions of Cauchy matrices are classical methods to generate MDS matrices. In this paper, using generalized Cauchy matrices, an approach to construct MDS matrices is proposed so that if A is an MDS matrix constructed with the proposed approach and B is a 3 × 3 sub-matrix of A , then interesting connections between the entries of B are introduced. More precisely, every element of the matrix B can be uniquely determined by eight other entries of B . Moreover, using generalized Cauchy matrices and also applying the proposed approach, some common forms of MDS matrices such as Hadamard, circulant and MDS matrices with maximum entries equal to the unit element, have been investigated.

Two novel extended semidiscrete KP-type systems, namely, partial differential–difference systems with one continuous and two discrete variables, are investigated. Introducing an arbitrary function into the Cauchy matrix function or the plane wave factor allows implementing extended integrable systems within the Cauchy matrix approach. We introduce the bilinear KP system, the extended pKP, pmKP, and SKP systems, all of which are based on the Cauchy matrix approach. This results in a diversity of solutions for these extended systems as contrasted to the usual multiple soliton solutions.

The Cauchy matrix approach is developed for solving nonisospectral Kadomtsev–Petviashvili equation and the nonisospectral modified Kadomtsev–Petviashvili equation. By means of a Sylvester equation , a set of scalar master functions are defined. We derive the evolution of scalar functions using the nonisospectral dispersion relations. Some explicit solutions are illustrated together with the analysis of their dynamics.