Analytical sequences of upper and lower solutions for a class of elliptic equations

Comparison arguments are applied to derive decreasing sequences of upper solutions and increasing sequences of lower solutions for a class of nonlinear elliptic equations. The monotonicity of the two sequences is proven. These polynomial sequences are obtained by applying new algorithms and solving linear differential equations. The obtained upper and lower solutions are analytic and have closed forms. Different examples are presented to explore the effectiveness of the new algorithms. The presented ideas and algorithms can be extended to deal with different classes of equations.     

非线性二元算子方程组的迭代求解方法

Using the cone and partial ordering theory and mixed monotone operator theory, the existence and uniqueness of solutions for some classes of systems of nonlinear two binary operator equations in a Banach space with a partial ordering are discussed. And the error estimates that the iterative sequences converge to solutions are also given. Some relevant results of solvability of two binary operator equations and systems of operator equations are improved and generalized.     

Construction of binary and ternary self-orthogonal linear codes

We construct new binary and ternary self-orthogonal linear codes. In order to do this we use an equivalence between the existence of a self-orthogonal linear code with a prescribed minimum distance and the existence of a solution of a certain system of Diophantine linear equations. To reduce the size of the system of equations we restrict the search for solutions to solutions with special symmetry given by matrix groups. Using this method we found at least six new distance-optimal codes, which are all self-orthogonal.     

The General Theory of Linear Difference Equations over the Max-Plus Semi-Ring

We present the mathematical theory underlying systems of linear difference equations over the max-plus semi-ring. The result provides an analog of isomonodromy theory for ultradiscrete Painlevé equations, which are extended cellular automata, and provide evidence for their integrability. Our theory is analogous to that developed by Birkhoff and his school for linear q -difference equations, but stands independently of the latter. As an example, we derive linear problems in this algebra for ultradiscrete versions of the symmetric P_IV equation and show how it is a necessary condition for isomonodromic deformation of a linear system.     

Numerical solutions for some coupled systems of nonlinear boundary value problems

Summary In the well-known Volterra-Lotka model concerning two competing species with diffusion, the densities of the species are governed by a coupled system of reaction diffusion equations. The aim of this paper is to present an iterative scheme for the steady state solutions of a finite difference system which corresponds to the coupled nonlinear boundary value problems. This iterative scheme is based on the method of upper-lower solutions which leads to two monotone sequences from some uncoupled linear systems. It is shown that each of the two sequences converges to a nontrivial solution of the discrete equations. The model under consideration may have one, two or three nonzero solutions and each of these solutions can be computed by a suitable choice of initial iteration. Numerical results are given for these solutions under both the Dirichlet boundary condition and the mixed type boundary condition.     

Modelling nonlinear sequence generators in terms of linear cellular automata

In this work, a wide family of LFSR-based sequence generators, the so-called clock-controlled shrinking generators (CCSGs), has been analyzed and identified with a subset of linear cellular automata (CA). In fact, a pair of linear models describing the behavior of the CCSGs can be derived. The algorithm that converts a given CCSG into a CA-based linear model is very simple and can be applied to CCSGs in a range of practical interest. The linearity of these cellular models can be advantageously used in two different ways: (a) for the analysis and/or cryptanalysis of the CCSGs and (b) for the reconstruction of the output sequence obtained from this kind of generators.     

非线性非单调二元算子方程组的解及其应用

利用非线性泛函分析中的锥理论和单调迭代的方法,研究了一类非线性非单调二元算子方程组的解的存在性,并给出了收敛于解的迭代序列,然后作为应用,得到了B anach空间中的一类非线性V olterra型积分方程组的解,改进了最近的许多结果.     

Fundamental sequences of functions in the approximation of solutions to exterior problems in the bending of thin micropolar plates

In this paper we show how to construct fundamental sequences for approximate solutions to exterior Dirichlet and Neumann-type problems in the bending of micropolar plates. The sequences are based on singular solutions of the governing equilibrium equations. These singular solutions are, however, unbounded at infinity. This leads to difficulties when applying the usual methods for proving linear independence and completeness. By decomposing the sequences into divergent and convergent parts we show that they can be accommodated in a more general framework developed in previous work. This allows us to overcome the difficulties mentioned above. Kupradze's method of generalised Fourier series is then modified and used to construct approximations which converge uniformly to the corresponding exact solutions     

Reversibility of linear cellular automata

In this paper the reversibility problem for linear cellular automata defined by a characteristic matrix of the form of a pentadiagonal matrix is tackled. Specifically, a criterion for the reversibility in terms of the number of cells is stated and, in these cases, the inverse cellular automata are explicitly computed.     

基于矩阵模型表示的(线性)有限自动机的同步序列判定

在有限自动机矩阵模型表示方法的基础上,采用矩阵理论和布尔代数为工具,分别给出了判定输入序列是否是(线性)有限自动机的同步序列的新充要条件和求解线性有限自动机的最短同步序列的新算法.     

Applications of Darboux transformations to the self-dual Yang-Mills equations

The linear problem associated with the self-dual Yang-Mills equations is covariant with respect to Darboux and binary Darboux transformations of almost classical type. This technique is used to construct solutions of the problem in the form of Wronskian-like and Gramm-like determinants. The self-dual conditions can be properly realized for only the latter type of solutions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 284–293, February, 2000.     

Characterization of 2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-periodic binary sequences with fixed 2-error or 3-error linear complexity

The linear complexity of sequences is an important measure of the cryptographic strength of key streams used in stream ciphers. The instability of linear complexity caused by changing a few symbols of sequences can be measured using k-error linear complexity. In their SETA 2006 paper, Fu et al. (SETA, pp. 88–103, 2006) studied the linear complexity and the 1-error linear complexity of 2 ⁿ -periodic binary sequences to characterize such sequences with fixed 1-error linear complexity. In this paper we study the linear complexity and the k-error linear complexity of 2 ⁿ -periodic binary sequences in a more general setting using a combination of algebraic, combinatorial, and algorithmic methods. This approach allows us to characterize 2 ⁿ -periodic binary sequences with fixed 2- or 3-error linear complexity. Using this characterization we obtain the counting function for the number of 2 ⁿ -periodic binary sequences with fixed k-error linear complexity for k = 2 and 3.     

The solution of fuzzy linear systems by non-linear programming: a financial application

Fuzzy linear systems of equations play a major role in various financial applications. In this paper we analyse a particular fuzzy linear system: the derivation of the risk neutral probabilities in a fuzzy binary tree. This system has previously been investigated and different solutions to different forms of the same system have been proposed.     

Hodograph Transformations and Cauchy Problem to Systems of Nonlinear Parabolic Equations

In this paper, we provide a method to solve the Cauchy problem of systems of quasi‐linear parabolic equations, such systems can be transformed to the systems of linear parabolic equations with variable coefficients via the hodograph transformations. Our approach to solve the linear systems with variable coefficients is to use their fundamental solutions, which are constructed by using the Lie's symmetry method. In consequence, we can derive explicit solutions to the Cauchy problem of the quasi‐linear systems in terms of the solutions of the linear systems and the hodograph transformations relating to the quasi‐linear and the linear systems.     

Existence, upper and lower solutions and quasilinearization for singular differential equations

In this paper, we discuss existence theorems in the presenceof upper and lower solutions as well as the method of quasilinearization(QSL) for general non-linear second-order singular ordinarydifferential equations. We show the existence of solutions underthe assumption of weak continuity of the non-linear part. Ifthe non-linear part is monotone decreasing, a solution may beobtained by the QSL method as the strong limit of a quadraticallyconvergent sequence of approximate solutions. Under strongerassumptions on the linear and the non-linear parts, a solutionis quadratically bracketed between two monotone sequences ofapproximate solutions of certain related linear equations.     

On the linear complexity of binary lattices

Linear complexity is an important and frequently used measure of unpredictability and pseudorandomness of binary sequences. In this paper our goal is to extend this notion to two dimensions. We will define and study the linear complexity of binary lattices. The linear complexity of a truly random binary lattice will be estimated. Finally, we will analyze the connection between linear complexity and correlation measures, and we will utilize the inequalities obtained in this way for estimating the linear complexity of an important special binary lattice. Finally, we will study the connection between the linear complexity of binary lattices and of the associated binary sequences.     

Absolute stability of the solutions of linear differential equations with time lags in Banach space

The connection between the exponential stability of the solutions of linear differential equations in space of multidimensional bounded vector sequences and the absolute asymptotic stability of the solutions of differential equations with several time lags is investigated.     

On periodic solutions of countable systems of linear and quasilinear difference equations with periodic coefficients

We present conditions for the existence of periodic solutions of linear difference equations with periodic coefficients in spaces of bounded number sequences. In the case where the generating linear equation has a unique periodic solution, we indicate sufficient conditions for the existence of a periodic solution of a quasilinear difference equation.     

一类非单调二元算子方程解的Mann迭代序列的收敛性

利用Mann迭代技巧,讨论了不具有连续性和紧性条件的非单调二元算子方程解的存在唯一性,并给出了迭代序列收敛于解的误差估计,所得结果是某些已知结果本质改进和推广.     

On the Use of Cellular Automata in Symmetric Cryptography

In this work, pseudorandom sequence generators based on finite fields have been analyzed from the point of view of their cryptographic application. In fact, a class of nonlinear sequence generators has been modelled in terms of linear cellular automata. The algorithm that converts the given generator into a linear model based on automata is very simple and is based on the concatenation of a basic structure. Once the generator has been linearized, a cryptanalytic attack that exploits the weaknesses of such a model has been developed. Linear cellular structures easily model sequence generators with application in stream cipher cryptography.Work supported by Ministerio de Educación y Ciencia (Spain), Projects SEG2004-02418 and SEG2004-04352-C04-03.