Cauchy Matrix Research Articles

Asymptotically almost periodic solutions for certain differential equations with piecewise constant arguments

It is well known that differential equations with piecewise constant arguments is a class of functional differential equations, which has fascinated many scholars in recent years. These delay differential equations have been successfully applied to diverse models in real life, especially in biology, physics, economics, etc. In this work, we are interested in the existence and uniqueness of asymptotically almost periodic solution for certain differential equation with piecewise constant arguments. Due to the particularity of the equations, we cannot use the traditional method to convert it into the difference equation with exponential dichotomy. Through constructing Cauchy matrix of the investigated system to find the corresponding Green matrix of the difference equation, we need the concept of exponential dichotomy and the Banach contraction fixed point theorem of the corresponding system. Then we give some sufficient conditions to obtain the existence and uniqueness of asymptotically almost periodic solutions for these systems.

Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

Structural Stress Determination at a Hot-Spot

Abstract Stress analysis by a finite element analysis program sometimes causes singularities in hot-spots. In a failure assessment, the structural stress, and membrane and bending stress should be determined in a highly stressed spot (hot-spot). When the structural stress at a hot-spot is disturbed by singularities, the stress result not only diverges, by reducing the element size, but also shows a decreased value compared to nominal stress because the stress is substituted into the von Mises equivalent stress equation. In the three-dimensional (3D) finite element model, it is hard to avoid the singularity problem at hot-spots particularly when it is subjected to loads in three axial directions. For the alternative, this study converts the 3D model into two-dimensional (2D) plane models to remove the singularity and to obtain reasonable structural stress values excluding the peak stress. The structural stress-estimating approaches applied in the 2D model were examined for whether they could avoid the structural stress reduction near the hot-spot with mesh insensitivity. The implemented approaches are stress linearization, single point away method, stress equilibrium, stress extrapolation, and the nodal force method. The results computed by each approach are compared and reconstructed to 3D stress by the Cauchy stress matrix. This study found that the difference in structural stress in 2D was eliminated after the 3D stress reconstruction.

К вопросу об устойчивости системы двух линейных гибридных функционально-дифференциальных систем с последействием

We consider a system of two hybrid vector equations containing linear difference (defined on a discrete set) and functional differential (defined on a half-axis) parts. To study it, a model system of two vector equations is chosen, one of which is linear difference with aftereffect (LDEA), and the other is a linear functional differential with aftereffect (LFDEA). Two equivalent representations of this system are shown: the first representation in the form of LFDEA, the second — in the form of LDEA. This allows us to study the stability issues of the system under consideration using the well-known results on the stability of LFDEA and LDEA. Using the results of the article [Gusarenko S. A. On the stability of a system of two linear differential equations with delayed argument // Boundary value problems. Interuniversity collection of scientific papers. Perm: PPI, 1989. P. 3–9], two examples are shown when a joint system of four equations will be stable with respect to the right side. In the first example, we use the LFDEA for which sufficient conditions for the sign-definiteness of the elements of the 2 Ч 2 Cauchy matrix function are known (in terms of the LFDEA coefficients). In the second example, LFDEA is given such that LFDEA is a system of linear ordinary differential equations (LODE) of the second order. In both cases, estimates of the components of the Cauchy matrix function are known. An exponential estimate with a negative exponent is given for the components of the Cauchy matrix function of LDEA.

К оценке значений линейных функционалов на решениях систем с последействием

For a wide class of linear functional differential systems with Volterra operators, a constructive technique is proposed to obtain estimates of linear functionals values over solutions in conditions of uncertainty of external perturbations. It can be applied to solutions of boundary value problems with arbitrary number of boundary conditions as well as to description of attainability sets in control problems with respect to given on-target functionals. External perturbations are constrained by a given linear inequalities system on the main time segment. The technique is based on the results of general theory of functional differential equations about the solvability of boundary value problems with general linear boundary conditions and the representation of solutions. The problem under consideration is reduced to the generalized moment problem. Therewith the results on the properties of the Cauchy matrix to systems with aftereffect are of essential importance. The general form of functionals allows one to cover many cases being topical in applications such as multipoint, integral ones, as well as hybrids of those.

A lattice CKP equation expressed by the [formula omitted] function

Various integrable systems, both continuous and discrete, having soliton solutions have been studied through the generalized Cauchy matrix method. Here we employ this approach to investigate a lattice CKP equation which is expressed by a τ function. We start from introducing a Sylvester equation and then show how to solve it. The variables in the Sylvester equation help us construct the τ function of the lattice CKP equation, which yields different types of exact solutions to the lattice CKP equation as well.

Cauchy matrix approach to integrable equations with self-consistent sources and the Yajima–Oikawa system

A new approach by means of the Cauchy matrix is developed to obtain integrable equations with self-consistent sources. We derive a lower order Kadomtsev–Petviashvili equation with self-consistent sources which is then reduced to the multi-component Yajima–Oikawa system. This approach allows explicit multiple-pole solutions. New solutions of the Yajima–Oikawa system are obtained, of which there is more freedom in dispersion relations and amplitudes.

Stability of functional differential systems applied to the model of testosterone regulation

In this paper we propose a method for stability studies of functional differential systems. The idea of our method is to reduce the analysis of an n-dimensional system to one for an (n+m)-dimensional system, where m is a natural number, to obtain stability and then to come back and make conclusions on the stability of the given n-dimensional system. As an example, a model describing testosterone by distributed inputs feedback control is considered. The aim of the regulation is to hold testosterone concentration above an appropriate level. The feedback control with integral term is proposed. We have to increase the testosterone level to the normal one. The control we proposed could destroy the stability of the model. That is why we have to choose the parameters of our distributed control, namely a dosage or intensity of assimilation of a medicine in a human body in such a form that the stability of our system is preserved. Thus the problem of regulation of testosterone level leads us to the stability analysis of the functional differential system describing a connection between the concentrations of hormones (GnRH), (LH), and testosterone (Te). Constructing the system, we discard the connections which seem nonessential. To estimate the effect of these connections is an important problem. We construct the Cauchy matrix of integro-differential system to estimate this influence.

Lamb Waves in Anisotropic Functionally Graded Plates: A Closed Form Dispersion Solution

ABSTRACTPropagation of harmonic Lamb waves in plates made of functionally graded materials (FGM) with transverse inhomogeneity is studied by combination of the Cauchy six-dimensional formalism and matrix exponential mapping. For arbitrary transverse inhomogeneity a closed form implicit solution for dispersion equation is derived and analyzed. Both the dispersion equation and the corresponding solution resemble ones obtained for stratified media. The dispersion equation and the corresponding solution are applicable to media with arbitrary elastic (monoclinic) anisotropy.

Around the Model of Infection Disease: The Cauchy Matrix and Its Properties

In this paper the model of infection diseases by Marchuk is considered. Mathematical questions which are important in its study are discussed. Among them there are stability of stationary points, construction of the Cauchy matrices of linearized models, estimates of solutions. The novelty we propose is in a distributed feedback control which affects the antibody concentration. We use this control in the form of an integral term and come to the analysis of nonlinear integro-differential systems. New methods for the study of stability of linearized integro–differential systems describing the model of infection diseases are proposed. Explicit conditions of the exponential stability of the stationary points characterizing the state of the healthy body are obtained. The method of the paper is based on the symmetry properties of the Cauchy matrices which allow us their construction.

Discrete potential Ablowitz–Kaup–Newell–Segur equation

We establish a discrete model for the potential Ablowitz–Kaup–Newell–Segur equation via a generalized Cauchy matrix approach. Soliton solutions and Jordan block solutions of this equation are presented by solving the determining equation set. By applying appropriate continuum limits, we obtain two semi-discrete potential Ablowitz–Kaup–Newell–Segur equations. The reductions to real and complex discrete and semi-discrete potential modified Korteweg-de Vries equations are also discussed.

Cauchy Matrix Type Solutions for the Nonlocal Nonlinear Schrödinger Equation

In this paper, we study some Cauchy matrix type solutions for the nonlocal nonlinear Schrodinger equations, including Cauchy matrix type multisoliton solutions and Cauchy matrix type Jordan block solutions. These solutions are obtained from the Cauchy matrix type solutions of Ablowitz–Kaup–Newell–Segur equation through nonlocal reduction. The reduction procedure developed in this paper can apply to other systems which have Cauchy matrix type solutions and admit local and nonlocal reductions.

On a Class of Optimal Control Problems for Functional Differential Systems

A linear functional differential control system of general form with aftereffect is considered. An optimal control problem with linear constraints on the state and control variables is studied. The control is realized by a linear operator of general form. The cases of distributed and lumped delay in the control loop, as well as the case of impulsive control, are covered. The Cauchy matrix is used to reduce the problem under consideration to a problem formulated only in terms of control variables with the use of some auxiliary variables linked with the defining relations for the Cauchy matrix of the system. In the case where the control is chosen from a finite-dimensional subspace of the control space, a problem effectively solvable by standard software tools is written explicitly. An example of an applied optimal control problem that arises in economic dynamics is presented. A class of hybrid systems (systems with continuous and discrete times) reducible to the system under consideration is described.

On the instability of the Millionshchikov linear systems depending on a real parameter

The present article considers one-parameter families of second-order linear differential systems with a coefficient matrix depending on the real parameter, which is a diagonal matrix at each odd time interval of unit length. The Cauchy matrix is the rotation matrix at each odd time interval, whereas the angle is the sum of a parameter value and some real number. Earlier, it has been has proved that the upper Lyapunov exponent of each such a system, which is considered to be the function of parameter, is positive on the set of the positive Lebesque measure if the diagonal part of the coefficient matrix is independent on a parameter and separated from zero. The proof of this result essentially uses a complex matrix of special type. In recent article, the author has given another way to prove this theorem based on implementing the Parseval equality for trygonometric sums. Besides, the author considers the special case of the above systems. Now the diagonal part of the coefficient matrix is time-independent and is sufficiently big, whereas the rotation angle is defined by a maximum degree of two that divides the number of the corresponding time interval. For such a system, in the case of a continious coefficient dependence on a parameter it is proved that such a value exists, at which the corresponding system is unstable.

A Reliable Workload Management Based on Predictive Analysis and Characterization of Workload Resources in HPC

Execution of Big Data workloads upon High Performance Computing (HPC) infrastructures has become an attractive way to improve their performances. In resource management, a large volume of multi-structured log data of Cloud Data Center (CDC) is generated regarding job arrival patterns, CPU memory consumption, task duration and many others. The system also provides the mechanism for CACO Cauchy matrix method for automatic data recovery from disk failure; it can also remove the lengthy process like replica management. In this paper we proposed a system for dynamic load rebalancing and resource allocation technique using machine learning algorithm. Q-Learning based on ML algorithm has used for validating the system. Experimental analysis illustrates that how proposed system eliminates the present approaches drawbacks. Our key is to identify the various workload patterns generation in heterogeneous storage environments using machine learning algorithm.

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.

Discrete analogues for two nonlinear Schrödinger type equations

Discrete versions for nonlinear Schrödinger equation and modified nonlinear Schrödinger equation are presented, using the method of generalized Cauchy matrix approach based on a Sylvester equation. By solving the determining equation set, some exact solutions including soliton solutions, Jordan block solutions and mixed solutions for the resulting discrete equations are generated. The associated semi-discrete equations as well as continuous equations are also discussed, applying appropriate continuum limits.

Solitons of the (2+2)-dimensional Toda lattice

We use the generalized Cauchy matrix approach to derive the N-soliton solutions for the -dimensional Toda lattice.

Cauchy Matrix Factorization for Tag-Based Social Image Retrieval

User-provided tags associated with social images are essential information for social image retrieval. Unfortunately, these tags are often imperfect to describe the visual contents, which severely degrades the performance of image retrieval. Tag relevance learning models are proposed to improve the descriptive powers of tags mostly based on the Gaussian noise assumption. However, the intrinsic probability distribution of the noise is unknown and other probability distributions may be much better. Towards this end, this paper investigates the applicable probability distributions of tag noise and proposes a novel Cauchy Matrix Factorization (CMF) method for tag-based image retrieval. The Cauchy probability distribution is robust to all kinds of noise and more suitable to model the tagging noise of social images. Therefore, we utilize Cauchy distribution to model noise under the matrix factorization framework. Besides, other five probability density functions, i.e., Gaussian, Laplacian, Poisson, Student-t and Logistic, are investigated to model noise of social tags. To evaluate the performance of different probability distributions, extensive experiments on two widely-used datasets are conducted and results show the robustness of CMF to noisy tags of social images.

Matrix WKB solution for electromagnetic waves in an inhomogeneous gyrotropic layer

A matrix solution 4x4 was obtained using the Wenzel-Kramers-Brillouin method for inclined incidence of a plane electromagnetic wave on an inhomogeneous gyrotropic torsion layer. The medium is affected by an external nonhomogeneous magnetic field reverses direction within the layer. Solutions for fields in the form of a fundamental matrix of the solution and the Cauchy matrix are obtained. The absolute values of the reflectance matrix coefficients and the reflection energy coefficients for the s and p polarized waves are calculated. The dependence of the cross-polarization coefficients of the reflection matrix of the total rotation angle of the external magnetic field vector within the layer is shown.