Numerical solutions of the m-membranes problem

In this paper numerical approximation for the m-membrane problem is considered. We make a change of variables that leads to a different expression of the quadratic functional that allows after discretizing the problem to reformulate it as finite dimensional bound constrained quadratic problem. To our knowledge this is the first paper on numerical approximation of the m-membrane problem. We reformulate the m-membrane problem as a bound constraint quadratic minimization problem. The bound constraint quadratic form is solved with the gradient projection method.     

Increasing smoothness of solutions to a hyperbolic system on the plane with delay in the boundary conditions

Under consideration is a mixed problem in the half-strip Π = {(x, t): 0 < x < 1, t > 0} for a first order homogeneous linear hyperbolic system with delay in t in the boundary conditions. We study the behavior of the Laplace transform of a solution to this problem for the large values of the complex parameter. The boundary conditions are found under which the smoothness of a solution to the corresponding mixed problem increases with t.     

An iterative rounding 2-approximation algorithm for the k-partial vertex cover problem

We study a generalization of the vertex cover problem. For a given graph with weights on the vertices and an integer k, we aim to find a subset of the vertices with minimum total weight, so that at least k edges in the graph are covered. The problem is called the k-partial vertex cover problem. There are some 2-approximation algorithms for the problem. In the paper we do not improve on the approximation ratios of the previous algorithms, but we derive an iterative rounding algorithm. We present our technique in two algorithms. The first is an iterative rounding algorithm and gives a （2 ＋ Q/OPT ）-approximation for the k-partial vertex cover problem where Q is the largest finite weight in the problem definition and OPT is the optimal value for the instance. The second algorithm uses the first as a subroutine and achieves an approximation ratio of 2.     

On the Dirichlet problem for the prescribed mean curvature equation over general domains

We study and solve the Dirichlet problem for graphs of prescribed mean curvature in Rn+1 over general domains Ω without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result by Schulz and Williams we can then also solve the Dirichlet problem for boundary values satisfying a Lipschitz condition.     

A numerical method for solving the Dirichlet problem for the wave equation

In this paper we present a numerical method for solving the Dirichlet problem for a two-dimensional wave equation. We analyze the ill-posedness of the problem and construct a regularization algorithm. Using the Fourier series expansion with respect to one variable, we reduce the problem to a sequence of Dirichlet problems for one-dimensional wave equations. The first stage of regularization consists in selecting a finite number of problems from this sequence. Each of the selected Dirichlet problems is formulated as an inverse problem Aq = f with respect to a direct (well-posed) problem. We derive formulas for singular values of the operator A in the case of constant coefficients and analyze their behavior to judge the degree of ill-posedness of the corresponding problem. The problem Aq = f on a uniform grid is reduced to a system of linear algebraic equations A _ll q = F. Using the singular value decomposition, we find singular values of the matrix A _ll and develop a numerical algorithm for constructing the r-solution of the original problem. This algorithm was tested on a discrete problem with relatively small number of grid nodes. To improve the calculated r-solution, we applied optimization but observed no noticeable changes. The results of computational experiments are illustrated.     

Integrability of the Problem of the Motion of a Cylinder and a Vortex in an Ideal Fluid

In this paper, we obtain a nonlinear Poisson structure and two first integrals in the problem of the plane motion of a circular cylinder and n point vortices in an ideal fluid. This problem is a priori not Hamiltonian; specifically, in the case n= 1 (i.e., in the problem of the interaction of a cylinder with a vortex) it is integrable.     

Optimal design of the damping set for the stabilization of the wave equation

We consider the problem of optimizing the shape and position of the damping set for the internal stabilization of the linear wave equation in RN, N=1,2. In a first theoretical part, we reformulate the problem into an equivalent non-convex vector variational one using a characterization of divergence-free vector fields. Then, by means of gradient Young measures, we obtain a relaxed formulation of the problem in which the original cost density is replaced by its constrained quasi-convexification. This implies that the new relaxed problem is well-posed in the sense that there exists a minimizer and, in addition, the infimum of the original problem coincides with the minimum of the relaxed one. In a second numerical part, we address the resolution of the relaxed problem using a first-order gradient descent method. We present some numerical experiments which highlight the influence of the over-damping phenomena and show that for large values of the damping potential the original problem has no minimizer. We then propose a penalization technique to recover the minimizing sequences of the original problem from the optimal solution of the relaxed one.     

On the complexity of preemptive open-shop scheduling problems

The problem of preemptively scheduling a set of n independent jobs on an m-machine open shop is studied, and two results are obtained. The first indicates that constructing optimal flow-time schedules is NP-hard for m larger than two. The second result shows that the problem remains NP-hard for the two-processor case when all jobs must be completed by their respective deadlines.     

Systems of convolution equations of the first and second kind on a finite interval and factorization of matrix-functions

Systems of n convolution equations of the first and second kind on a finite interval are reduced to a Riemann boundary value problem for a vector function of length 2n. We prove a theorem about the equivalence of the Riemann problem and the initial system. Sufficient conditions are obtained for the well-posedness of a system of the second kind. Also under study is the case of the periodic kernel of the integral operator of a system of the first and second kind.     

On the Cauchy problem for a semilinear fractional elliptic equation

We study, for the first time in the literature on the subject, the Cauchy problem for a semilinear fractional elliptic equation. Under an a priori assumption on the solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic type is established.     

An initial-boundary value problem for the Maxwell equations

In this paper, by using methods from complex analysis and quaternionic analysis, we investigate an initial-boundary value problem for the Maxwell equations and obtain the general solutions and solvable conditions of the problem respectively in different cases. In addition, by using a similar method, we also discuss an initial-boundary value problem for a hyperbolic complex system of first order equations in R³.     

Modified algorithms for the minimum volume enclosing axis-aligned ellipsoid problem

Consider the problem of computing a (1+?)-approximation to the minimum volume axis-aligned ellipsoid (MVAE) enclosing a set of m points in Rⁿ. We first provide an extension and improvement to algorithm proposed in Kumar and Y?ld?r?m (2008) [5] (the KY algorithm) for the MVAE problem. The main challenge of the MVAE problem is that there is no closed form solution in the line search step (beta). Therefore, the KY algorithm proposed a certain choice of beta that leads to their complexity and core set results in solving the MVAE problem. We further analyze the line search step to derive a new beta, relying on an analysis of up to the fourth order derivative. This choice of beta leads to the improved complexity and core set results. The second modification is given by incorporating “away steps” into the first one at each iteration, which obtains the same complexity and core set results as the first one. In addition, since the second modification uses the idea of “dropping points”, it has the potential to compute smaller core sets in practice. Some numerical results are given to show the efficiency of the modified algorithms.     

Optimal control of the layer size in the problem of equilibrium of elastic bodies with overlapping domains

We consider the problem of equilibrium of a two-layer elastic body. The first of the layers contains a crack,while the second is a circle centered at one of the crack tips. The round layer is glued by its boundary to the first layer. The unique solvability is proved for this problem in the nonlinear formulation. An optimal control problem is also considered. The radius a of the second layer is chosen as a varying parameter under assumption that a takes positive values from a closed interval. It is shown that there are a value of a minimizing the functional that characterizes how potential energy depends on the crack length and a value of a minimizing the functional that characterizes the opening of the crack.     

Problem F2∥Cmax with forbidden jobs in the first or last position is easy

Saadani et al. [N.E.H. Saadani, P. Baptiste, M. Moalla, The simple F2∥C_max with forbidden tasks in first or last position: A problem more complex that it seems, European Journal of Operational Research 161 (2005) 21–31] studied the classical n-job flow shop scheduling problem F2∥C_max with an additional constraint that some jobs cannot be placed in the first or last position. There exists an optimal job sequence for this problem, in which at most one job in the first or last position is deferred from its position in Johnson’s [S.M. Johnson, Optimal two- and three-stage production schedules with setup times included, Naval Research Logistics Quarterly 1 (1954) 61–68] permutation. The problem was solved in O(n²) time by enumerating all candidate job sequences. We suggest a simple O(n) algorithm for this problem provided that Johnson’s permutation is given. Since Johnson’s permutation can be obtained in O(n log n) time, the problem in Saadani et al. (2005) can be solved in O(n log n) time as well.     

A regularized Newton method for the efficient approximation of tensors represented in the canonical tensor format

In the present survey, we consider a rank approximation algorithm for tensors represented in the canonical format in arbitrary pre-Hilbert tensor product spaces. It is shown that the original approximation problem is equivalent to a finite dimensional ? ₂ minimization problem. The ? ₂ minimization problem is solved by a regularized Newton method which requires the computation and evaluation of the first and second derivative of the objective function. A systematic choice of the initial guess for the iterative scheme is introduced. The effectiveness of the approach is demonstrated in numerical experiments.     

Simultaneous stabilization: Construction of a universal stabilizer for linear plants with delay with the use of spectral reducibility

We consider the problem of the construction of a common stabilizing controller for a family of k linear stationary nth-order plants with commensurable delays in the controls and the state variables. Our approach to stabilization is based on the construction of a common two-loop stabilizer. The controller of the first (internal) loop solves the problem of reducing each plant of the family to a finite spectrum, and the controller of the second (external) contour solves the stabilization problem.     

Series solution of the differential equation determining the nth-shell one-electron density of a bare Coulomb problem in quantum physics

The differential equation determining the nth-shell one-electron density of a bare Coulomb problem is analytically investigated by using the differential function method and extended homogeneous balance idea. New series solution of the differential equation has been obtained in quantum physics for the first time. The explicit series analytical expression for the mathematical characteristic of the bare Coulomb problem in the density-functional theory can further describe the dynamical behaviors of the nth-shell one-electron density.     

A generalized insertion algorithm for the seriation problem

Given a binary matrix E, the Seriation Problem consists of determining a permutation of the rows of E minimizing the sum over all columns of the differences between the last and first nonzero element. This problem is derived from an archaeological context, but has applications in several other fields. We present a new generalized insertion heuristic for this problem. The proposed heuristic outperforms other methods on a series of benchmark problems.     

Landscape analysis and efficient metaheuristics for solving the n-queens problem

The n-queens problem is a classical combinatorial optimization problem which has been proved to be NP-hard. The goal is to place n non-attacking queens on an n×n chessboard. In this paper, two single-solution-based (Local Search (LS) and Tuned Simulated Annealing (SA)) and two population-based metaheuristics (two versions of Scatter Search (SS)) are presented for solving the problem. Since parameters of heuristic and metaheuristic algorithms have great influence on their performance, a TOPSIS-Taguchi based parameter tuning method is proposed, which not only considers the number of fitness function evaluations, but also aims to minimize the runtime of the presented metaheuristics. The performance of the suggested approaches was investigated through computational analyses, which showed that the Local Search method outperformed the other two algorithms in terms of average runtimes and average number of fitness function evaluations. The LS was also compared to the Cooperative PSO (CPSO) and SA algorithms, which are currently the best algorithms in the literature for finding the first solution to the n-queens problem, and the results showed that the average fitness function evaluation of the LS is approximately 21 and 8 times less than that of SA and CPSO, respectively. Also, a fitness analysis of landscape for the n-queens problem was conducted which indicated that the distribution of local optima is uniformly random and scattered over the search space. The landscape is rugged and there is no significant correlation between fitness and distance of solutions, and so a local search heuristic can search a rugged plain landscape effectively and find a solution quickly. As a result, it was statistically and analytically proved that single-solution-based metaheuristics outperform population-based metaheuristics in finding the first solution of the n-queens problem.     

Particular features of implementation of an unsaturated numerical method for the exterior axisymmetric Neumann problem

Using Babenko’s profound ideas, we construct a fundamentally new unsaturated numerical method for solving the spectral problem for the operator of the exterior axisymmetric Neumann problem for Laplace’s equation. We estimate the deviation of the first eigenvalue of the discretized problem from the eigenvalue of the Neumann operator. More exactly, the unsaturated discretization of the spectral Neumann problem yields an algebraic problem with a good matrix, i.e., a matrix inheriting the spectral properties of the Neumann operator. Thus, its spectral portrait lacks “parasitic” eigenvalues provided that the discretization error is sufficiently small. The error estimate for the first eigenvalue involves efficiently computable parameters, which in the case of C ^∞-smooth data provides a foundation for a guaranteed success.