Solution of two problems on wavelets

We solve two problems on wavelets. The first is the nonexistence of a regular wavelet that generates a wavelet basis for the Hardy space ?²(?). The second is the existence, given any regular wavelet basis for $\mathbb{H}^2 (\mathbb{R})$ , of aMulti-Resolution Analysis generating the wavelet. Moreover, we construct a regular scaling function for this Multi-Resolution Analysis. The needed regularity conditions are very mild and our proofs apply to both the orthonormal and biorthogonal situations. Extensions to more general cases in dimension 1 and higher are given. In particular, we show in dimension larger than 2 that a regular wavelet basis for $\mathbb{L}^2 (\mathbb{R}^n )$ arises from a Multi-Resolution Analysis that is regular modulo the action of a unitary operator, which is whenn = 2 a Calderón-Zygmund operator of convolution type.     

Solution of two fractional packing problems of lovász

Lovász asked whether the following is true for each hypergraph H and natural number k:(1) if $\text{v}{}_{\mathrm{k}}\text{(H′) = k · v}{}^1\text{(H′)}$ holds for each hypergraph H′ arising from H by multiplication of points, then v_k(H) = τ_k(H);(7) if $\text{τ}{}_{\mathrm{k}}\text{(H′) = k · τ}{}^1\text{(H′)}$ holds for each hypergraph H′ arising by removing edges, then τ_k (H) = v_k (H).We prove and generalize assertion (1) and give a counterexample to (7).     

Solution of boundary-value problems for xk-analytical functions in a halfplane with two cuts

We consider two boundary-value problems for x^k-analytical fonctions (k=2m+1, m=1,2,3,...) of the variable z=x+iy in a halfplane with cuts along the intervals (0, 1) and (b, ) of the real line (b > a), when the imaginary or the real part of the x-analytical function is given at the ends of the finite cut and resp. the real or the imaginary part of the function is given at the ends of the semi-infinite cut. Closed-form solutions are obtained for this problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 3–10, 1986     

Consistency and monotonicity in sequencing problems

We explore the implications of consistency and monotonicity in sequencing problems. We first identify all rules satisfying Pareto indifference, individual rationality from random arrival, and consistency. Next, we ask whether there is a rule satisfying the three axioms together with either one of two monotonicity requirements, time monotonicity and cost monotonicity. As it turns out, the minimal transfer rule is the only rule satisfying Pareto indifference, individual rationality from random arrival, consistency together with either time monotonicity or cost monotonicity. We also investigate how the maximal transfer rule responds to changes in the service time and waiting cost.     

Solution of real-world postman problems

The paper presents the results of a study performed by the Deutsche post endowed chair of optimization of distribution networks in collaboration with Deutsche Post World Net with the aim of improving the planning of letter mail delivery. Modelling and solution methods for real-world postman problems are presented which extend one of the most general postman problems studied in the literature, the windy rural postman problem, with regard to several aspects. The discussed extensions include turn and street crossing restrictions, cluster constraints, the option to have alternative service modes (including ‘zigzag deliveries’), and the use of public transport to reach the postal district. The solution method is based on a transformation to the asymmetric TSP and uses non-standard neighbourhood search techniques. Extensive computational experiments show that the solution method clearly and consistently outperforms standard TSP heuristics on real-world instances.     

Solution of boundary value problems in cylinders separated by a three-layer film into two semicylinders

Boundary value problems are considered for the class of equations ? _x ² u + L[u] = 0 in cylinders D = (x ?? R, y ?? Q ? R ^m ) with an infinitely thin film at x = 0 consisting of three sublayers with alternating high and low permeability (L-linear differential operator with respect to y _i ). The solutions of the problems are expressed in terms of those of the corresponding classical boundary value problems in homogeneous cylinders D with no film. The resulting formulas have the form of simple quadrature rules, which are amenable to numerical computations.     

On some sequencing problems in finite groups

A finite group is called Z-sequenceable if its non-identity elements can be listed x₁, x₂, …, x_n so that x_ix_i+1 for i = 1, 2, …, n ? 1. Various necessary and sufficient conditions are determined for such sequencings to exist. In particular, it is proved that if n ? 3, then the symmetric group S_n is not Z-sequenceable.     

Solution of linear-viscoelasticity boundary value problems

The basic elements of the algebra of exponential-fractional operators [1,9] are generalized to include any arbitrary resolvent operator. As a consequence, it is found that solutions of viscoelasticity boundary-value problems within the framework of the Volterra principle can always be realized in time without specifying the starting operators. Analytic functions of resolvent operators are discussed. The results are demonstrated with reference to some common integral operators of viscoelasticity.Mekhanika Polimerov, Vol. 3, No. 6, pp. 999–1011, 1967     

Solution of projection problems over polytopes

Summary In this paper, we present variants of a convergent projection and contraction algorithm [25] for solving projection problems over polytope. By using the special struture of the projection problems, an iterative algorithm with constant step-size is given, which is globally linearly convergent. These algorithms are simple to implement and each step of the method requires only a few matrix-vector multiplications. Especially, for minimums norm problems over transportation or general network polytopes onlyO(n) additions andO(n) multiplications are needed at each iteration. Numerical results for randomly generated test problems over network polytopes, up to 10000 variables, indicate that the presented algorithms are simple and efficient even for large problems.     

Solution of some two-dimensional problems of elasticity

Using the method of boundary elements, we obtain numerical solutions of two-dimensional (plane deformation) boundary-value problems on the elastic equilibrium of infinite and finite homogeneous isotropic bodies having elliptic holes with cracks and cuts of finite length. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.     

Solution techniques for some allocation problems

This paper presents methods for solving allocation problems that can be stated as convex knapsack problems with generalized upper bounds. Such bounds may express upper limits on the total amount allocated to each of several subsets of activities. In addition our model arises as a subproblem in more complex mathematical programs. We therefore emphasize efficient procedures to recover optimality when minor changes in the parameters occur from one problem instance to the next. These considerations lead us to propose novel data structures for such problems. Also, we introduce an approximation method to solve certain equations, which arise during the procedures.     

Solution of large-scale symmetric travelling salesman problems

In this paper we report on a cutting plane procedure with which we solved symmetric travelling salesman problems of up to 1000 cities to optimality. Our implementation is based on a fast LP-solver (IBM's MPSX) and makes effective use of polyhedral results on the symmetric travelling salesman polytope. We describe the important ingredients of our code and give an extensive documentation of its computational performance.Supported by DFG-Schwerpunkt Anwendungsbezogene Optimierung und Steuerung Universität Augsburg, Germany.Supported by SFB 303 (DFG), Forschungsinstitut für Diskrete Mathematik, Institut für Operations Research, Universität Bonn, Germany.     

Solution of spectral problems for polynomial matrices

For polynomial matrices of full rank, including matrices of the form A - I and A - B, numerical methods for solving the following problems are suggested: find the divisors of a polynomial matrix whose spectra coincide with the zeros of known divisors of its characteristic polynomial; compute the greatest common divisor of a sequence of polynomial matrices; solve the inverse eigenvalue problem for a polynomial matrix. The methods proposed are based on the W and V factorizations of polynomial matrices. Applications of these methods to the solution of certain algebraic problems are considered. Bibliography: 3 titles._________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 122–138.     

Solving real car sequencing problems with ant colony optimization

An automobile assembly line is usually configured as three successive shops in which the body is constructed, painted, and then assembled together with all component parts into a finished vehicle. However, many published production sequencing models ignore the first two shops and base their results only on the requirements and constraints of the assembly shop. In this article, we propose to more closely follow the actual industrial structure. We therefore first propose a single objective mathematical model for scheduling the paint and assembly shops. We then propose an ACO metaheuristic for solving a multiple-objective formulation. Data provided by Groupe Renault show that the proposed approach offers better solutions than those of current practice.     

Solution behavior for parametric implicit complementarity problems

In this paper we study the solution behavior for a special class of quasi-variational inequalities, namely implicit complementarity problems. We derive conditions under which the perturbed solution of a parametric implicit complementarity problem is locally unique, continuous and Fréchet differentiable.     

Solution procedures for three-dimensional eddy current problems

The problem under consideration is that of the scattering of time periodic electromagnetic fields by metallic obstacles. A common approximation here is that in which the metal is assumed to have infinite conductivity. The resulting problem, called the perfect conductor problem, involves solving Maxwell's equations in the region exterior to the obstacle with the tangential component of the electric field zero on the obstacle surface. In the interface problem different sets of Maxwell equations must be solved in the obstacle and outside while the tangential components of both electric and magnetic fields are continuous across the obstacle surface. Solution procedures for this problem are given. There is an exact integral equation procedure for the interface problem and an asymptotic procedure for large conductivity. Both are based on a new integral equation procedure for the perfect conductor problem. The asymptotic procedure gives an approximate solution by solving a sequence of problems analogous to the one for perfect conductors.     

Solution semicontinuity of parametric generalized vector equilibrium problems

In this paper, the lower semicontinuity and continuity of the solution mapping to a parametric generalized vector equilibrium problem involving set-valued mappings are established by using a new proof method which is different from the ones used in the literature.