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Given a positive integern, letand. The generator matrixof thekth-order Reed–Muller code RMis an important tool in the study of Boolean functions' algebraic immunity. In this paper, choosing the lastscolumn vectors inas a basis of the vector space, we study the values of the coefficients in the linear expressions of's column vectors over this basis. As an application, we present a new construction of balanced even-variable Boolean functions onwith optimal algebraic immunity by modifying the outputs of majority function. The nonlinearities of these constructed functions are also determined. [ABSTRACT FROM AUTHOR]

This paper considers a repairablek-out-of-n:Gsystem with repairman's multiple delayed vacations where the operating times and the repair times of components are governed by exponential distributions and general distributions, respectively. After completion of repair of all broken components, the repairman remains idle for a period of time called changeover time, and then takes a vacation following an arbitrary distribution if there is no operating component breaks down during the changeover time. Applying the supplementary variable method, several reliability measures of the system including the steady-state availability, the steady-state rate of occurrence of failures and the mean time to the first failure are obtained. Meanwhile, some numerical illustrations are presented to demonstrate how the various parameters influence the behaviour of the system. Finally, a special case-out-of-n:Gsystem is discussed to validate the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with the optimal control (OC) problem that arise when a hydraulic system with fixed-head hydroplants is considered. In the frame of a deregulated electricity market, the resulting Hamiltonian for such OC problems is linear in the control variable and results in an optimal singular/bang-bang control policy. To avoid difficulties associated with the computation of optimal singular/bang-bang controls, an efficient and simple optimization algorithm is proposed. The computational technique is illustrated on one example. [ABSTRACT FROM AUTHOR]

For the global optimization problems with continuous variables, evolutionary algorithms (EAs) are often used to find the approximate solutions. The number of generations for an EA to find the approximate solutions, called the first hitting time, is an important index to measure the performance of the EA. However, calculating the first hitting time is still difficult in theory. This paper proposes some new drift conditions that are used to estimate the upper bound of the first hitting times of EAs for finding the approximate solutions. Two case studies are given to show how to apply these conditions to estimate the first hitting times. [ABSTRACT FROM AUTHOR]

This paper is a matrix iterative method presented to compute the solutions of the matrix equation, AXB=C, with unknown matrix X∈S, where S is the constrained matrices set like symmetric, symmetric-R-symmetric and (R, S)-symmetric. By this iterative method, for any initial matrix X0∈S, a solution X* can be obtained within finite iteration steps if exact arithmetics were used, and the solution X* with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. The solution [image omitted] , which is nearest to a given matrix ~X in Frobenius norm, can be obtained by first finding the minimum Frobenius norm solution of a new compatible matrix equation. The numerical examples given here show that the iterative method proposed in this paper has faster convergence and higher accuracy than the iterative methods proposed in [G.-X. Huang, F. Yin, and K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C, J. Comput. Appl. Math. 212 (2008), pp. 231-244; Y. Lei and A.-P. Liao, A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetric matrices, Appl. Math. Comput. 188 (2007), pp. 499-513; Z.-Y. Peng, An iterative method for the least squares symmetric solution of the linear matrix equation AXB=C, Appl. Math. Comput. 170 (2005), pp. 711-723; Y.-X. Peng, X.-Y. Hu, and L. Zhang, An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C, Appl. Math. Comput. 160 (2005), pp. 763-777]. [ABSTRACT FROM AUTHOR]

The Peaceman-Rachford splitting method (PRSM) is well studied for solving the two-block separable convex minimization problems with linear constraints recently. In this paper, we consider the separable convex minimization problem where its objective function is the sum of more than two functions without coupled variables, when applying the PRSM to this case directly, it is not necessarily convergent. To remedy this difficulty, we propose a proximal Peaceman-Rachford splitting method for solving this multi-block separable convex minimization problems, which updates the Lagrangian multiplier two times at each iteration and solves some subproblems parallelly. Under some mild conditions, we prove global convergence of the new method and analyse the worst-case convergence rate in both ergodic and nonergodic senses. In addition, we apply the new method to solve the robust principal component analysis problem and report some preliminary numerical results to indicate the feasibility and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper treats a bulk arrival queue with randomized working vacation policy. Whenever the system becomes empty, the server takes a vacation. During the vacation period, customers are to be served at a lower rate. Once the vacation ends, the server will return to the normal working state and begin to serve the customers in the system if any. Otherwise, the server either remains idle with probabilitypor leaves for another vacation with probability 1−p. This pattern continues until the number of vacations taken reachesJ. If the system is empty at the end of theJth vacation, the server will wait idly for a new arrival. By using supplementary variable technique, we derive the system size distribution at arbitrary epoch, at departure epoch and at busy period initial epoch, as well as some important system characteristics. Numerical examples are provided to illustrate the influence of system parameters on several performance measures. [ABSTRACT FROM AUTHOR]

In this paper, a class of variable-order fractional convection diffusion equations have been solved with assistance of the second kind Chebyshev wavelets operational matrix. The operational matrix of variable-order fractional derivative is derived for the second kind Chebyshev wavelets. By implementing the second kind Chebyshev wavelets functions and also the associated operational matrix, the considered equations will be reduced to the corresponding Sylvester equation, which can be solved by some appropriate iterative solvers. Also, the convergence analysis of the proposed numerical method to the exact solutions and error estimation are given. A variety of numerical examples are considered to show the efficiency and accuracy of the presented technique. [ABSTRACT FROM AUTHOR]

In this paper, we propose a technique for constructing balanced Boolean functions on even numbers of variables. The main technique is to utilize a set of disjoint spectra functions and a special Boolean permutation to derive a balanced Boolean function with high nonlinearity and optimal algebraic degree. It is shown that the functions we construct are different from both Maiorana-McFarland's (M-M) super-class functions introduced by Carlet and modified M-M super-class functions presented by Zeng and Hu. Furthermore, we show that they have no nonzero linear structures. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a numerical method for the solution of an inverse problem of recovering lacking data on some part of the boundary of a domain from the Cauchy data on other part for a variable coefficient elliptic Cauchy problem. In the process, the Cauchy problem is transformed into the problem of solving a compact linear operator equation. As a remedy to the ill-posedness of the problem, we use a projection method which allows regularization solely by discretization. The discretization level plays the role of regularization parameter in the case of projection method. The balancing principle is used for the choice of an appropriate discretization level. Several numerical examples show that the method produces a stable good approximate solution. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to build up a set of semi-analytical integrators based on the use of a class of anomalies defined by means of a generalized Sundman transformation. The integrators are based on the developments of the mean anomaly and the vector radius according to the new anomaly. To manipulate these developments, a Poisson series processor called poison.h has been used. This processor has been written as a C++class and it contains a set of methods to manage the most common arithmetic and functional operations with these objects. [ABSTRACT FROM PUBLISHER]

In this paper, a numerical method is developed to solve an N-carrier system with Neumann boundary conditions. First, we apply the compact finite difference scheme of fourth order for discretizing spatial derivatives at the interior points. Then, we develop a new combined compact finite difference scheme for the boundary, which also has fourth-order accuracy. Lastly, by using a Padé approximation method for the resulting linear system of ordinary differential equations, a new compact finite difference scheme is obtained. The present scheme has second-order accuracy in time direction and fourth-order accuracy in space direction. It is shown that the scheme is unconditionally stable. The present scheme is tested by two numerical examples, which show that the convergence rate with respect to the spatial variable from the new scheme is higher and the solution is much more accurate when compared with those obtained by using other previous methods. [ABSTRACT FROM PUBLISHER]

This paper presents a method for finding the optimal control of linear systems with delays in state and input and the quadratic cost functional using an orthonormal basis for square integrable functions. The state and control variables and their delays are expanded in an orthonormal basis with unknown coefficients. Using operational matrices of integration, delay and product, the relation between coefficients of variables is provided. Then, necessary condition of optimality is driven as a linear system of algebraic equations in terms of the unknown coefficients of state and control variables. As an application of this method, the linear Legendre multiwavelets as an orthonormal basis for L2[0, 1] are used. Two time-delayed optimal control problems are approximated to establish the usefulness of the method. [ABSTRACT FROM AUTHOR]

In this paper, for an integer n≥10, two classes of n-variable Boolean functions with optimum algebraic immunity (AI) are constructed, and their nonlinearities are also determined. Based on non-degenerate linear transforms to the proposed functions, we can obtain 1-resilient n-variable Boolean functions with optimum AI and high nonlinearity if n-1 is never equal to any power of 2. [ABSTRACT FROM AUTHOR]

In the paper, we show some new convergence conditions on waveform relaxation (WR) for general differential-algebraic equations (DAEs). The main conclusion is that the convergence conditions on index r+1 can be derived from that of index r, in which the corresponding system is composed by ordinary differential equations if r=0. The approach of analysing relaxation process is novel for WR solutions of DAEs. It is also the first time to give the convergence conclusions for general index systems of DAEs in the WR field. [ABSTRACT FROM AUTHOR]

We first introduce trivariate multiresolution analysis and trivariate orthogonal wavlets. We then give a method for constructing a class of trivariate nonseparable compactly supported orthogonal scaling functions and the related wavelets. Finally, we study the regularity of these wavelets. We can get arbitrarily smooth trivariate orthogonal wavelets. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to compute two unified fractional integrals involving the product of twoH-functions, a general class of polynomials and Appell function. These integrals are further applied in proving two theorems on Saigo–Maeda fractional integral operators. Some consequent results and special cases are also pointed out in the concluding section. [ABSTRACT FROM AUTHOR]

This paper addresses all possible equivalence classes of 1-variable Boolean functions and from these classes using recursion and Cartesian product of sets, 15 different ways of classifications ofn-variable Boolean functions are obtained. The properties with regard to the size and the number of classes for these 15 different ways are also elaborated. [ABSTRACT FROM AUTHOR]

In this paper, we study a repairableK-out-of-(M+W) retrial system withMidentical primary components,Wstandby components and one repair facility. The time-to-failure and time-to-repair of the primary and standby components are assumed to be exponential and general distributions, respectively. The failed components are immediately for repair if the server is idle, otherwise the failed machines would enter an orbit. It is assumed that the retrial times are exponentially distributed. We present a recursive method using the supplementary variable technique and treating the supplementary variable as the remaining repair time to obtain the steady-state probabilities of down components at arbitrary epoch. Then, a unified and efficient algorithm is developed to compute the steady-state availability. The method is illustrated analytically for the exponential repair time distribution. Sensitivity analysis of the steady-state availability with respect to system parameters for a variety of repair time distributions is also investigated. [ABSTRACT FROM AUTHOR]

Kumaret al.[Generalized bent functions and their properties, J. Comb. Theory Ser. A 40 (1985), pp. 90–107] have extended the notion of classical bent Boolean functions in the generalized setup on. They have provided an analogue of classical Maiorana-McFarland type bent functions. In this paper, we study the cross-correlation of a subclass of such generalized Maiorana-McFarland type bent functions. We provide a characterization of quaternary (q=4) bent functions onn+1 variables in terms of their subfunctions onn-variables. Analogues of sum-of-squares’ indicator and absolute indicator of cross-correlation of Boolean functions are defined in the generalized setup. Further,q-ary functions are studied in terms of these indicators and some upper bounds of these indicators are obtained. Finally, we provide some constructions of balanced quaternary functions with high nonlinearity under Lee metric. [ABSTRACT FROM AUTHOR]

This paper presents a new trust-region procedure for solving symmetric nonlinear systems of equations having several variables. The proposed approach takes advantage of the combination of both an effective adaptive trust-region radius and a non-monotone strategy. It is believed that the selection of an appropriate adaptive radius and the application of a suitable non-monotone strategy can improve the efficiency and robustness of the trust-region framework as well as decrease the computational costs of the algorithm by decreasing the required number of subproblems to be solved. The global convergence and the quadratic convergence of the proposed approach are proved without the non-degeneracy assumption of the exact Jacobian. The preliminary numerical results of the proposed algorithm indicating the promising behaviour of the new procedure for solving nonlinear systems are also reported. [ABSTRACT FROM PUBLISHER]

In this paper, a collocation method is presented for the solutions of the system of the Riccati-type differential equations with variable coefficients. The proposed approach consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions in terms of the Bessel polynomials with unknown coefficients. The unknown coefficients of the Bessel polynomials are found by using the matrix operations of derivatives together with the collocation method. The proposed method gives the analytic solutions when the exact solutions are polynomials. Also, an error analysis technique based on the residual function is introduced for the suggested method. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. Comparing the methodology with some known techniques shows that the presented approach is relatively easy and highly accurate. All of the numerical calculations have been done by using a program written in Maple. [ABSTRACT FROM AUTHOR]

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, a numerical implementation of a direct united boundary-domain integral equation (BDIE) related to the Neumann boundary value problem for a scalar elliptic partial differential equation with a variable coefficient is discussed. The BDIE is reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretization of the BDIEs with quadrilateral domain elements leads to a system of linear algebraic equations (discretized BDIE). Then, the system is solved by LU decomposition and Neumann iterations. Convergence of the iterative method is discussed in relation to the distribution of eigenvalues of the corresponding discrete operators calculated numerically. [ABSTRACT FROM AUTHOR]

In this paper, starting with a basis of , we define some sets in that are the supports of bent functions of 2k variables. We also establish some results in order to count the number of bent functions we can construct and we provide a complete classification of all bases of (for k=2) providing the same supports of bent functions. [ABSTRACT FROM AUTHOR]

In this paper, a split least-squares characteristic mixed finite element method is proposed for solving nonlinear nonstationary convection–diffusion problem. By selecting the least-squares functional property, the resulting least-squares procedure can be split into two independent symmetric positive definite sub-schemes. The first sub-scheme is for the unknown variable u, which is the same as the standard characteristic Galerkin finite element approximation. The second sub-scheme is for the unknown flux σ. Theoretical analysis shows that the method yields the approximate solutions with optimal accuracy in L 2(Ω) norm for the primal unknown and in H(div; Ω) norm for the unknown flux, respectively. Some numerical examples are given to confirm our theory results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the numerical simulation of the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation. After introducing a new variable, the integro-differential equation is transformed into an equivalent coupled system of first-order differential equations. A second-order accurate difference scheme is constructed for the new system of equations, which is proved to be local uncoupled by separation of variables. It is also proved that the scheme is uniquely solvable and second-order convergent in both time and space in L 2-norm. A numerical example is given to demonstrate the theoretical results. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a characteristic centred finite difference method on non-uniform grids to solve the problem of the air pollution model. Numerical solutions and error estimates of the air pollution concentration and its first-order derivatives for space variables are obtained. The computational cost of the method is the same as that of the characteristic difference method based on a linear interpolation. The error order of the numerical solutions is the same as that of the characteristic difference method based on a quadratic interpolation. At last, we give numerical examples to illustrate feasibility and efficiency of this method. [ABSTRACT FROM AUTHOR]

In this paper we study numerical solutions for a hyperbolic system of equations using finite differences. In this setting, we propose the method of lines, with high precision in space. A class of some explicit, implicit and also semi-implicit schemes, with code variable methods, are presented. Finally, the analysis of some qualitative and quantitative proprieties of these methods is included. [ABSTRACT FROM AUTHOR]

In this paper, we present a new three-step predictor corrector type iterative method for finding simple and real roots of nonlinear equations in a single variable. Experiments show that the new method is more efficient than the well known methods available in the literature. Comparison tables are given demonstrating the importance of the proposed approach. [ABSTRACT FROM AUTHOR]

Circles are one of the basic drawing primitives for computers and while the naive way of setting up an equation for drawing circles is simple, implementing it in an efficient way using integer arithmetic has resulted in quite a few different algorithms. We present a short chronological overview of the most important publications of such digital circle generation algorithms. Bresenham is often assumed to have invented the first all integer circle algorithm. However, there were other algorithms published before his first official publication, which did not use floating point operations. Furthermore, we present both a 4- and an 8-connected all integer algorithm. Both of them proceed without any multiplication, using just one addition per iteration to compute the decision variable, which makes them more efficient than previously published algorithms. [ABSTRACT FROM AUTHOR]

Two direct pseudospectral methods based on nonclassical orthogonal polynomials are proposed for solving finite-horizon and infinite-horizon variational problems. In the proposed finite-horizon and infinite-horizon methods, the rate variables are approximated by theNth degree weighted interpolant, using nonclassical Gauss-Lobatto and Gauss points, respectively. Exponential Freud type weights are introduced for both of nonclassical orthogonal polynomials and weighted interpolation. It is shown that the absolute error in weighted interpolation is dependent on the selected weight, and the weight function can be tuned to improve the quality of the approximation. In the finite-horizon scheme, the functional is approximated based on Gauss-Lobatto quadrature rule, thereby reducing the problem to a nonlinear programming one. For infinite-horizon problems, an strictly monotonic transformation is used to map the infinite domain onto a finite interval. We transcribe the transformed problem to a nonlinear programming using Gauss quadrature rule. Numerical examples demonstrate the accuracy of the proposed methods. [ABSTRACT FROM PUBLISHER]

The construction of an approximate solution to an initial boundary value problem for the Rakib–Sivashinsky equation is of concern. The Fourier method is combined with the Adomian decomposition method in order to provide the approximate solution. The variables are separated by the Fourier method and the approximate solution to the nonlinear system of ordinary differential equations is obtained by the Adomian decomposition method. One example of application is presented. [ABSTRACT FROM AUTHOR]

Under a jump-diffusion process, the option pricing function satisfies a partial integro-differential equation. A fourth-order compact scheme is used to discretize the spatial variable of this equation. The boundary value method is then utilized for temporal integration because of its unconditional stability and high-order accuracy. Two approaches, the local mesh refinement and the start-up procedure with refined step size, are raised to avoid the numerical malfunction brought by the nonsmooth payoff function. The GMRES method with a preconditioner which comes from the Crank-Nicolson formula is employed to solve the resulting large-scale linear system. Numerical experiments demonstrate the efficiency of the proposed method when pricing European and double barrier call options in the jump-diffusion model. [ABSTRACT FROM AUTHOR]

In this work, we successfully extended two-dimensional differential transform method and their reduced form, by presenting and proving some theorems, to obtain the solution of partial differential equations (PDEs) with proportional delay in t and shrinking in x. Theorems are presented in the most general form to cover a wide range of PDEs, being linear or nonlinear and constant or variable coefficient. In order to show the power and robustness of the present methods and to illustrate the pertinent features of related theorems, some examples are presented. [ABSTRACT FROM AUTHOR]

We present second-order difference schemes for a class of parabolic problems with variable coefficients and mixed derivatives. The solvability, stability and convergence of the schemes are rigorously analysed by the discrete energy method. Using the Richardson extrapolation technique, the fourth-order accurate numerical approximations both in time and space are obtained. It is noted that the Richardson extrapolation algorithms can preserve stability of the original difference scheme. Finally, numerical examples are carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The initial value problem for fractional Bagley–Torvik equations is investigated by considering variable coefficients and the fractional order as. Making use of the integration method, a Volterra integral equation of the second kind is obtained. Then the contraction operator theorem in Banach spaces is further used to address the uniqueness of the solution for the obtained Volterra integral equation. A novel numerical method is proposed to find the approximate solution of the given Volterra integral equation. Moreover, the convergence and error estimate of the approximate solution are analysed. Finally, some numerical examples are carried out to show the effectiveness of the proposed method by comparing with the existing ones. The developed method will be helpful for finding a good approximation solution of fractional differential equations in practical applications. [ABSTRACT FROM AUTHOR]

We consider a predator–prey population model with prey gathering together for defence purposes. A transmissible unrecoverable disease affects the prey. We characterize the system behaviour, establishing that ultimately either only the susceptible prey survive, or the disease becomes endemic, but the predators are wiped out. Another alternative is that the disease is eradicated, with sound prey and predators thriving at an equilibrium or through persistent population oscillations. Finally, the populations can thrive together, with the endemic disease. The only impossible alternative in these circumstances is predators thriving just with infected prey. But this follows from the model assumptions, in that infected prey are too weak to sustain themselves. A mathematical peculiarity of the model is the singularity-free reformulation, which leads to three entirely new dependent variables to describe the system. The model is then extended to encompass the situation in which ingestion of diseased prey is fatal for the predators and to the cases where the predators find the infected prey less palatable. [ABSTRACT FROM AUTHOR]

In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, Ran and Yuan [On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems, Appl. Math. Comput. 217 (2010), pp. 3050–3068] presented the block symmetric successive over-relaxation (BSSOR) and the modified BSSOR iteration methods based on the special structures of the coefficient matrices. In this study, we present the modified alternating direction-implicit (MADI) iteration method for solving the linear systems. Under suitable conditions, we establish convergence theorems for the MADI iteration method. In addition, the optimal parameter involved in the MADI iteration method is estimated in detail. Numerical experiments show that the MADI iteration method is a feasible and effective iterative solver. [ABSTRACT FROM AUTHOR]

A new, improved split-step backward Euler method is introduced and analysed for stochastic differential delay equations (SDDEs) with generic variable delay. The method is proved to be convergent in the mean-square sense under conditions (Assumption 3.1) that the diffusion coefficient g(x, y) is globally Lipschitz in both x and y, but the drift coefficient f(x, y) satisfies the one-sided Lipschitz condition in x and globally Lipschitz in y. Further, the exponential mean-square stability of the proposed method is investigated for SDDEs that have a negative one-sided Lipschitz constant. Our results show that the method has the unconditional stability property, in the sense, that it can well reproduce stability of the underlying system, without any restrictions on stepsize h. Numerical experiments and comparisons with existing methods for SDDEs illustrate the computational efficiency of our method. [ABSTRACT FROM AUTHOR]

Two posterior distributions for the mean of the normal distribution are obtained by deriving the distributions of the product XY and the ratio X/Y when X and Y are normal and Student's t random variables distributed independently of each other. Estimation of the associated credible intervals is considered and computer programs are provided for generating them. [ABSTRACT FROM AUTHOR]

We study the geometric structure of the set of solutions of a random ε-1-in-k SAT problem [D. Achlioptas, A. Chtcherba, G. Istrate, and C. Moore, The phase transition in 1-in-K SAT and NAE3SAT, in Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms, 2001, pp. 721-722; A. Percus, G. Istrate, and C. Moore (eds.), Computational Complexity and Statistical Physics, Oxford University Press, Oxford, UK, 2006]. For l≥1, two satisfying assignments A and B are l-connected if there exists a sequence of satisfying assignments connecting them by changing at most l bits at a time. We first identify a subregion of the satisfiable phase where the set of solutions provably forms one cluster. Next, we provide a range of parameters (c, ε) such that w.h.p. (with high probability) two assignments of a random ε-1-in-k SAT instance with n variables and cn clauses are O(log n)-connected, conditional on being satisfying assignments. Also, for random instances of 1-in-k SAT in the satisfiable phase we show that there exists νk∈(0, 1/(k-2)] such that w.h.p. no two satisfying assignments at distance at least νk·n form a 'hole'. We believe that this is true for all νk>0, and in fact solutions of a random 1-in-k SAT instance in the satisfiable phase form one cluster. [ABSTRACT FROM AUTHOR]

A study of different higher-order time-varying singular systems of initial value problems (IVPs) using the generalized single-term Walsh series (STWS technique) is presented. The existing STWS technique is used to solve IVPs of first- and second-order systems and is extended to determine discrete solutions for the systems of IVPs of any higher order n with p variables. The effectiveness of the technique is demonstrated by using it to find discrete solutions for time-varying singular systems of different orders. [ABSTRACT FROM AUTHOR]