On the representation of solutions of some classes of two-linear dimensional difference equations

Herein, some classes of linear two-dimensional difference equations of Volterra type are considered. Representations of solutions using analogs of the resolvent and the Riemann matrix are obtained.

Representation of a solution for a system of linear inhomogeneous two-dimensional difference equations of fractional order

One linear inhomogeneous two-parameter discrete fractional system is considered, and the boundary condition is a solution of an analogue of the Cauchy problem for a linear ordinary difference equation. Equation coefficients are given by discrete matrix functions. By introducing an analogue of the Riemann matrix, representations of solutions of the considered boundary value problem are obtained. Note that the result obtained plays an essential role in the linear case for establishing a necessary and sufficient optimality condition in the form of the Pontryagin maximum principle, and also in the general case for studying special control in discrete optimal control problems for systems of 2D fractional orders.

Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential Equations

The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.

ABOUT RIEMANN MATRIX OPERATOR IN THE SPACE OF SMOOTH VECTOR FUNCTIONS

В пространстве гладких на единичной окружности векторфункций рассматривается матричный оператор линейного сопряжения, порождаемый краевой задачей Римана. Предполагается, что коэффициенты краевой задачи являются гладкими матрицамифункциями. Вводится и изучается понятие гладкой вырожденной факторизации типов плюс и минус гладкой матрицыфункции. В терминах вырожденных факторизаций даются необходимые и достаточные условия нетеровости рассматриваемого матричного оператора Римана в пространстве гладких векторфункций. Для гладкой на окружности функции, имеющей не более чем конечное число нулей конечных порядков, вводится и изучается понятие сингулярного индекса, обобщающее понятие индекса невырожденной непрерывной функции. Для нетерового матричного оператора Римана получена формула для вычисления индекса этого оператора, совпадающая с общеизвестной аналогичной формулой в случае, когда коэффициенты оператора Римана невырождены.

Peak-to-average power ratio reduction in wavelet packet modulation Using Binary Phase Sequence.

<p class="1">A novel method of Peak–to-Average–Power-Ratio reduction is presented in this paper using WPT for multi carrier modulationand Riemann matrix for phase sequencing in SLM. Wavelet Packet Multicarrier Modulation(WPM) is a betteralternative to the existing multicarrier transmission system,OFDM system which suffers from high PAPR.Further reduction in Peak to Average Power Ratio (PAPR)is achieved by using Wavelet Packet Modulation(WPM) with SLM technique. Classical SLM uses random phases for phase rotations which suffers from the additional load of transmitting SI(side Information) to receiver. So to overcome a novel method is proposed here which uses the principle of varying the phases of the subcarrier data block to be transmitted ,using rows of the normalised Reimann matrix as phase sequence vectors for the Selected Mapping (SLM)Technique. The generated WPM frame with minimum PAPR is selected and transmitted .The proposed system achieves a significant improvement in PAPR reduction over the OFDM-classical SLM and WPM-classical SLM.</p>

A note on proper curvature symmetry in general cylindrically symmetric four-dimensional Lorentzian manifolds

We considered the most general form of non-static cylindrically symmetric space-times for studying proper curvature symmetry by using the rank of the [Formula: see text] Riemann matrix and direct integration techniques. Studying proper curvature symmetry in each case of the above space-times, we show that when the above space-times admit proper curvature symmetry, they form an infinite dimensional vector space. It is important to note that here we also find the case when the rank of the [Formula: see text] Riemann matrix is one and no covariantly constant vector fields exist.

EXACT ONE-PERIODIC AND TWO-PERIODIC WAVE SOLUTIONS TO HIROTA BILINEAR EQUATIONS IN (2+1) DIMENSIONS

Riemann theta functions are used to construct one-periodic and two-periodic wave solutions to a class of (2+1)-dimensional Hirota bilinear equations. The basis for the involved solution analysis is the Hirota bilinear formulation, and the particular dependence of the equations on independent variables guarantees the existence of one-periodic and two-periodic wave solutions involving an arbitrary purely imaginary Riemann matrix. The resulting theory is applied to two nonlinear equations possessing Hirota bilinear forms: ut + uxxy - 3uuy - 3uxv = 0 and ut + uxxxxy - (5uxxv + 10uxyu - 15u2v)x = 0 where vx = uy, thereby yielding their one-periodic and two-periodic wave solutions describing one-dimensional propagation of waves.