On UC-multipliers for multiple trigonometric systems

Investigates the problem of asymptotic stability for a class of linear time-invariant multidimensional discrete systems. We establish the equivalence of the present problem and the problem of robust stability for a class of linear time-varying one-dimensional discrete systems with the matrix uncertainty set defined by the coefficient matrices of the original multidimensional system. On the basis of this equivalence, by using the variational method and the Lyapunov second method, necessary and sufficient conditions for asymptotic stability are obtained in different forms for the class or multidimensional systems considered in the present paper. The parametric classes of Lyapunov functions which define the necessary and sufficient conditions of asymptotic stability are determined. We use the piecewise linear polyhedral Lyapunov functions of the infinity vector norm type to derive an algebraic criterion for asymptotic stability of the given class of multidimensional systems in the form of solvability conditions of a set of matrix equations.

It is shown that the double sequence {Xm"} with X"" = 1 if n < m and 0 otherwise is an Lp multiplier for the Walsh system in two dimensions only if p = 2.This result is then used to show that the one-dimensional trigonometric system and the Walsh system are nonequivalent bases of the Banach space Lp[0, 1], and hence have different Lp multipliers, 1 < p < oo, p # 2.1. Let {Xmn}_x<min<x be the double sequence defined by Xmn = 1 if n < m and 0 otherwise.This is a consequence of the one-dimensional result of M. Riesz for the conjugate function [6,1, p. 253].The Lp boundedness of Tx was used, for example, in C. Fefferman's proof of the almost everywhere convergence of double Fourier series [1].We now turn our attention to the Walsh system {wn).For 00 /~ 2 amnwmix)wn(y)GL"i[0,l]x[0,l]), m,n = 0 consider the corresponding operator T2 defined by 00 Tlf~ 2 amnKnWmiX)Wn(y)m,n = 0Because of the great similarity between the Walsh system and the trigonometric system, one would expect T2 to be bounded on Z/([0, 1] X [0, 1]), 1 < p < oo.However, this is not the case.In 2 we will show that T2 is not bounded on Lp except for p = 2.This result is then used in 3 to give a negative answer to a question of P. Enflo: Are the trigonometric system and the Walsh system equivalent bases of the Banach space Lp[0, I], 1 < p < oo, p ^ 21 Finally in 4 we will deduce from the nonequivalence that the

Multidimensional (MD) systems describe relations between signals depending on two or more independent variables. They are also called distributed parameter systems, if the independent variables are time and space. The only conventional models for their description are partial differential equations. This is in contrast to onedimensional (lumped parameter) systems, where a variety of different models including transfer functions is used. This paper extends the concept of transfer function models to multidimensional systems with bounded spatial domains, i.e., systems which can be described as initial-boundary-value problems. These transfer function models are useful for system analysis and as a starting point for the derivation of numerically efficient discrete simulation models.

We prove that the condition is necessary for an increasing sequence of numbers to be an almost everywhere unconditional convergence Weyl multiplier for the trigonometric system. This property was known long ago for Haar, Walsh, Franklin and some other classical orthogonal systems. The proof of this result is based on a new sharp logarithmic lower bound on for the majorant operator related to the rearranged trigonometric system. Bibliography: 32 titles.

In connection with the problems of development and implementation of perfect automatic and automated control systems for complex technological processes based on modern information technology tools, the development of research methods for multidimensional nonlinear discrete control systems is of particular relevance. The variety of these systems is due to the types of modulation, non-linearities, modes of operation of impulse elements, characteristics of continuous parts. The article deals with the problems of modeling and research of one-dimensional and multidimensional pulse-frequency systems with delay based on the use of dynamic graphs. One of the key points in solving the problem is an approach based on the maximum consideration of the physical features of these systems. Given the structural complexity of systems, it is necessary first of all to establish the possibility of decomposing the system into simpler subsystems. Then the study of the system can be reduced to the study of individual subsystems and the nature of their interaction, that is, to the determination of the properties of the system as a whole by the properties of its elements. Systems with frequency-pulse modulation are essentially non-linear systems. The topological method based on dynamic graph models allows solving the fundamental difficulties of studying nonlinear discrete systems with delay, allows clearly and strictly formalizing their description, and makes it possible to obtain effective algorithms for analyzing one-dimensional and multidimensional systems of this class. The article presents an example of using the obtained algorithm.

Equal-order interpolations: a unified approach to stabilize the incompressible and advective effects

ABSTRACTMultidimensional systems are becoming increasingly important as they provide a promising tool for estimation, simulation and control, while going beyond the traditional setting of one-dimensional systems. The analysis of multidimensional systems is linked to multivariate polynomials, and is therefore more difficult than the well-known analysis of one-dimensional systems, which is linked to univariate polynomials. In the current paper, we relate the realisation theory for overdetermined autonomous multidimensional systems to the problem of solving a system of polynomial equations. We show that basic notions of linear algebra suffice to analyse and solve the problem. The difference equations are associated with a Macaulay matrix formulation, and it is shown that the null space of the Macaulay matrix is a multidimensional observability matrix. Application of the classical shift trick from realisation theory allows for the computation of the corresponding system matrices in a multidimensional state-space setting. This reduces the task of solving a system of polynomial equations to computing an eigenvalue decomposition. We study the occurrence of multiple solutions, as well as the existence and analysis of solutions at infinity, which allow for an interpretation in terms of multidimensional descriptor systems.

A generalization of the notions of minimality, reachability, and observability for parameter dependent and multidimensional systems modeled by linear fractional transformations over noncommuting operators is presented. The multidimensional system results developed herein are formally similar to and directly specialize to the standard results when one-dimensional systems are considered.

Complex Semiclassical Description of Scattering Problem in Systems with 1.5 Degrees of Freedom

This paper discusses miscible gas injection in fractured reservoirs, based on compositional reservoir modeling. Developed miscibility in conventional one-dimensional systems is dependent on fluid properties alone, and can be estimated by well known experimental and numerical procedures. This is not the case for heterogeneous and multi-dimensional systems such as fractured reservoirs. Based on a multi-cell algorithm a method is proposed to determine conditions of developed miscibility in fractured reservoirs. The proposed method makes it possible to evaluate gas injection processes, and developed miscibility in particular, in specific regions of the reservoir. A systematic step-by-step extension from a single one-dimensional matrix block to more complex matrix-fracture systems shows that the minimum miscibility pressure/enrichment (MMP/MME) level in a fractured reservoir is significantly higher than for a conventional one-dimensional single-porosity system. This is mainly due to multi-dimensional flow and molecular diffusion. On the other hand, even at pressures significantly below the fracture system MMP, substantial enhancements in the recovery rate and ultimate oil recovery can be expected by non-equilibrium gas injection. It is shown that reduction of the gas-oil interfacial tension and internal Darcy flow induced by interfacial tension gradients play a key role. This capillary driven process may lead to accelerated production and very high ultimate recoveries.

The requirement for two or more retention parameters to describe the final position or emergence point of the component is consistent with the concept of multiple dimensions used throughout science and mathematics. In coupled column multidimensional separation, the effluent from one column is divided into sequential portions and shunted individually to one or more subsequent columns where they are subjected to new separative conditions. Multidimensional separation procedures become especially important for complex samples consisting of numerous components. The planar 2D system is highly important in its own right, but it also serves as a simple reference system to which all other multidimensional systems can be compared. The situation is entirely analogous to the overlap of peaks in one dimension as they expand from infinitely thin lines to realistically broadened peaks. The multiplicative form assumed by peak capacity in higher dimensional systems provides a mechanism for amplifying peak capacities far beyond that possible in one-dimensional systems.

Numerical studies concerning the efficiency of various techniques in time-independent surface hopping calculations

The systems of rational functions {Φn(z)}, n ∈ ℤ; that are orthonormalized on the real axis ℝ and are defined by the fixed set of points a := {ak}k = 0∞, (Im ak > 0) and b := {bk}k = 1∞, (Im bk 1; and the pointwise convergence of Fourier series on the systems {Φn(t)}, n ∈ ℤ; are studied under the certain restrictons on the sequences of poles of these systems. Some analogs of the classical Jordan–Dirichlet and Dini–Lipschitz criteria of convergence of Fourier series in a trigonometric system are constructed.

Difference analog equations of high-order accuracy describing stable miscible displacement are presented. The high-order-difference scheme eliminates almost all the numerical smearing, which is the result of the normally used approximations, and leaves only the effect of the physical dispersion in the solution. In a one-dimensional system, the technique involves an addition of a negative dispersion term to the continuity equation. The negative dispersion term, which is called the numerical dispersion coefficient, depends upon the flow velocity, the time-step size and the block size. The procedure is extended to multidimensional systems. As a check, comparisons of the computed results with analytical solutions in one- and two-dimensional systems are made. The error-function solution in a one-dimensional miscible system and a five-spot fractional flow curve computed from the potentiometric model are considered as the analytical potentiometric model are considered as the analytical solutions in the two cases. Based on the computed results, it has been concluded that the effect of the term with the negative numerical dispersion coefficient accounts for almost all the truncation error within the practical limits of oilfield problems. Introduction Miscible flooding is one of the most efficient hydrocarbon recovery processes known at the present time. Its application in the field, however, present time. Its application in the field, however, remains limited because of economic requirements. Recently, the development of vertical miscible floods has made the process economically attractive. The result is that many oil companies have proposed and started such floods in suitable reservoirs. The amount of solvent required to flood an entire reservoir miscibly can be roughly calculated by analytical methods. Since these methods are derived for a one-dimensional system, some important questions remain unanswered. Some of these questions are:How will the solvent bank develop under the proposed injection scheme?What will be the condition of the solvent bank at breakthrough? These questions can only be answered by two- or three-dimensional studies. Known numerical techniques simulating miscible displacement give considerable smearing of concentration in the liquid-liquid transition zone when applied to a Held case. The effect of numerical smearing in this case is significantly larger than the effect of physical dispersion. Therefore, in the past, immiscible simulators with controlled numerical past, immiscible simulators with controlled numerical smearing were used to simulate miscible floods. For example, in one case the two-phase saturation profile was displaced as near piston-like as possible, profile was displaced as near piston-like as possible, neglecting mixing due to molecular diffusion and convective dispersion. In another case, capillary pressure and relative permeability functions were pressure and relative permeability functions were modified to simulate the dispersion effect. It was stated that the second method is applicable to a system where numerical smearing is considerably less than the effect of physical dispersion. This limits the use of the second method to a system such as gas-gas displacement. In this article a high-order difference scheme for miscible equations is given. This scheme eliminates most of the numerical smearing, leaving only the effect of physical dispersion in the transition zone. The procedure is developed for one- and two-dimensional systems. Calculated results are compared with analytical solutions. It is shown that the procedure can be extended to a three-dimensional system. FORMULATION OF THE PROBLEM The differential equations governing a miscible displacement process for an incompressible system are the continuity equations and the flow equations. Numerical solutions of this set of equations have been described by Peaceman and Rachford and Warren et al. Their methods are adequate to simulate miscible displacement in laboratory systems. If one applies them to a field-size system, significant truncation errors are introduced in the results. Usually these errors are large compared with the effect of physical dispersion on the results. SPEJ P. 277

The main distinction between one-dimensional and multi-dimensional hyperbolic systems is that in the case of the latter information propagates in an infinite number of directions. In this paper we present a new class of bicharacteristic methods which take all the infinitely many directions of propagation into account. The principal issue involved in constructing bicharacteristic methods reduces to evolving the system of equations approximately along the bicharacteristics. For the spatial approximation we mainly concentrate on a discontinuous Galerkin finite element discretisation. We exemplify the technique on Maxwell’s equations with constant coefficients in two space dimensions.