On the two-point boundary-value problem for the Riccati matrix differential equation

Constructive sufficient conditions for univalent resolvability of a two-point boundary value problem for nonlinear Riccati equation are obtained. An illustrative example is given.     

On quasi-solution to infeasible linear complementarity problem obtained by Lemke’s method

For a linear complementarity problem with inconsistent system of constraints a notion of quasi-solution of Tschebyshev type is introduced. It’s shown that this solution can be obtained automatically by Lemke’s method if the constraint matrix of the original problem is copositive plus or belongs to the intersection of matrix classes P ₀ and Q ₀.     

Transverse compression of PPTA fibers

Results of single transverse compression testing of PPTA and PIPD fibers, using a novel test device, are presented and discussed. In the tests, short lengths of single fibers are compressed between two parallel, stiff platens. The fiber elastic deformation is analyzed as a Hertzian contact problem. The inelastic deformation is analyzed by elastic-plastic FE simulation and by laser-scanning confocal microscopy of the compressed fibers ex post facto. The results obtained are compared to those in the literature and to the theoretical predictions of PPTA fiber transverse elasticity based on PPTA crystal elasticity.Presented at the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000).3TEX, Inc. 109, MacKenan Drive, Cary, North Carolina 27511, USA. Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 533–544, March–April, 2000.     

Quasilinearization for the periodic boundary value problem for hybrid differential equation

The method of quasilinearization for a periodic boundary value problem for nonlinear hybrid differential equations is studied. It is shown that the convergence is quadratic.     

The Dirichlet problem for Baire-one functions

Let X be a compact convex set and let ext X stand for the set of all extreme points of X. We characterize those bounded function defined on ext X which can be extended to an affine Baire-one function on the whole set X.     

Tchebotaröv’s extremal problem

Central European Journal of Mathematics - We give the complete solution of the extremal problem posed by N.G. Tchebotaröv in 20th of the last century, and we establish explicit parametric...     

Higher-Order Optimality Conditions for Set-Valued Optimization

This paper deals with higher-order optimality conditions of set-valued optimization problems. By virtue of the higher-order derivatives introduced in (Aubin and Frankowska, Set-Valued Analysis, Birkhäuser, Boston, [1990]) higher-order necessary and sufficient optimality conditions are obtained for a set-valued optimization problem whose constraint condition is determined by a fixed set. Higher-order Fritz John type necessary and sufficient optimality conditions are also obtained for a set-valued optimization problem whose constraint condition is determined by a set-valued map.     

Two Interior-Point Methods for Nonlinear P *(τ)-Complementarity Problems

Two interior-point algorithms using a wide neighborhood of the central path are proposed to solve nonlinear P _*-complementarity problems. The proof of the polynomial complexity of the first method requires the problem to satisfy a scaled Lipschitz condition. When specialized to monotone complementarity problems, the results of the first method are similar to those in Ref. 1. The second method is quite different from the first in that the global convergence proof does not require the scaled Lipschitz assumption. However, at each step of this algorithm, one has to compute an approximate solution of a nonlinear system such that a certain accuracy requirement is satisfied.     

The flatness properties of <Emphasis Type="Italic">S</Emphasis>-poset <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">I</Emphasis>) and Rees factor <Emphasis Type="Italic">S</Emphasis>-posets

Let S be a pomonoid and I a proper right ideal of S. In a previous paper, using the amalgamated coproduct A(I) of two copies of _S S over I, we were able to solve one of the problems posed in S. Bulman-Fleming et al. (Commun. Algebra 34:1291–1317, 2006). In the present paper, we investigate further flatness properties of A(I). We also solve another problem stated in the paper cited above. Namely, we determine the condition under which Rees factor S-posets have property (P _w ). Research supported by nwnu-kjcxgc-03-18.     

On solvability of an initial-boundary-value problem for equations of magnetohydrodynamics

In this paper, we consider a nonstationary problem of magnetohydrodynamics (MHD) for viscous incompressible fluid under the condition that the medium is poorly conducting. The problem is analyzed in a bounded one-connected domain Ω ⊂ ℝⁿ, n = 2,3, for t > 0 under the condition of ideal conductivity on the boundary. We prove a theorem on the unique solvability of the problem “in the small,” on a small time interval, and on a given time interval ]0, T[ (including T = +∞) when the given data of the problem are sufficiently small (precise formulations are given in Sect. 2). To investigate the nonlinear problem, several auxiliary linear problems are preliminarily considered. The results of this paper were announced by the author in the Trakai Conference on Mathematical Modeling and Analysis in spring of 2005. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 234–279, April–June, 2007.     

Lagrangean decomposition: A model yielding stronger lagrangean bounds

Given a mixed-integer programming problem with two matrix constraints, it is possible to define a Lagrangean relaxation such that the relaxed problem decomposes additively into two subproblems, each having one of the two matrices of the original problem as its constraints. There is one Lagrangean multiplier per variable. We prove that the optimal value of this new Lagrangean dual dominates the optimal value of the Lagrangean dual obtained by relaxing one set of constraints and give a necessary condition for a strict improvement. We show on an example that the resulting bound improvement can be substantial. We show on a complex practical problem how Lagrangean decomposition may help uncover hidden special structures and thus yield better solution methodology. Research supported by the National Science Foundation under grant ECS-8508142.     

On priority queues with impatient customers

In this paper, we study three different problems where one class of customers is given priority over the other class. In the first problem, a single server receives two classes of customers with general service time requirements and follows a preemptive-resume policy between them. Both classes are impatient and abandon the system if their wait time is longer than their exponentially distributed patience limits. In the second model, the low-priority class is assumed to be patient and the single server chooses the next customer to serve according to a non-preemptive priority policy in favor of the impatient customers. The third problem involves a multi-server system that can be used to analyze a call center offering a call-back option to its impatient customers. Here, customers requesting to be called back are considered to be the low-priority class. We obtain the steady-state performance measures of each class in the first two problems and those of the high-priority class in the third problem by exploiting the level crossing method. We furthermore adapt an algorithm from the literature to obtain the factorial moments of the low-priority queue length of the multi-server system exactly.      

Realization of primitive branched coverings over closed surfaces following the hurwitz approach

Let V be a closed surface, H⊑π₁(V) a subgroup of finite index l and D=[A ₁,...,A _m ] a collection of partitions of a given number d≥2 with positive defect v(D). When does there exist a connected branched covering f:W→V of order d with branch data D and f∶W→V It has been shown by geometric arguments [4] that, for l=1 and a surface V different from the sphere and the projective plane, the corresponding branched covering exists (the data D is realizable) if and only if the data D fulfills the Hurwitz congruence v(D)э0 mod 2. In the case l>1, the corresponding branched covering exists if and only if v(D)э0 mod 2, the number d/l is an integer, and each partition A _i ∈D splits into the union of l partitions of the number d/l. Here we give a purely algebraic proof of this result following the approach of Hurwitz [11]. The realization problem for the projective plane and l=1 has been solved in [7,8]. The case of the sphere is treated in [1, 2, 12, 7].     

Equivalence and existence of weak Pareto optima for multiobjective optimization problems with cone constraints

In this work, a differentiable multiobjective optimization problem with generalized cone constraints is considered, and the equivalence of weak Pareto solutions for the problem and for its η-approximated problem is established under suitable conditions. Two existence theorems for weak Pareto solutions for this kind of multiobjective optimization problem are proved by using a Karush–Kuhn–Tucker type optimality condition and the F-KKM theorem.     

A new geometric condition for Fenchel's duality in infinite dimensional spaces

In 1951, Fenchel discovered a special duality, which relates the minimization of a sum of two convex functions with the maximization of the sum of concave functions, using conjugates. Fenchel's duality is central to the study of constrained optimization. It requires an existence of an interior point of a convex set which often has empty interior in optimization applications. The well known relaxations of this requirement in the literature are again weaker forms of the interior point condition. Avoiding an interior point condition in duality has so far been a difficult problem. However, a non-interior point type condition is essential for the application of Fenchel's duality to optimization. In this paper we solve this problem by presenting a simple geometric condition in terms of the sum of the epigraphs of conjugate functions. We also establish a necessary and sufficient condition for the ε-subdifferential sum formula in terms of the sum of the epigraphs of conjugate functions. Our results offer further insight into Fenchel's duality. Dedicated to Terry Rockafellar on his 70th birthday     

On the computation of scaling coefficients of Daubechies' wavelets

In the present paper, Daubechies' wavelets and the computation of their scaling coefficients are briefly reviewed. Then a new method of computation is proposed. This method is based on the work [7] concerning a new orthonormality condition and relations among scaling moments, respectively. For filter lengths up to 16, the arising system can be explicitly solved with algebraic methods like Gröbner bases. Its simple structure allows one to find quickly all possible solutions.     

A necessary and sufficient condition for the existence of an exponential attractor

We give a necessary and sufficient condition for the existence of an exponential attractor. The condition is formulated in the context of metric spaces. It also captures the quantitative properties of the attractor, i.e., the dimension and the rate of attraction. As an application, we show that the evolution operator for the wave equation with nonlinear damping has an exponential attractor.     

Mean–variance hedging under transaction costs

The paper proposes a new approach to the mean–variance-hedging problem under transaction costs. This approach is based on the idea of dividing the gain functional into two parts. One part representing the gains resulting from a pure buying strategy, and the other part representing the gains resulting from a pure selling strategy. The problem will be studied in a general incomplete market in discrete time. Some technical assumptions such as the RAS condition are excluded.      

Exact and stable least squares solution to the linear programming problem

A linear programming problem is transformed to the finding an element of polyhedron with the minimal norm. According to A. Cline [6], the problem is equivalent to the least squares problem on positive ortant. An orthogonal method for solving the problem is used. This method was presented earlier by the author and it is based on the highly developed least squares technique. First of all, the method is meant for solving unstable and degenerate problems. A new version of the artifical basis method (M-method) is presented. Also, the solving of linear inequality systems is considered.     

A Characterization of Graphs without Even Factors

In this paper, we first reduce the problem of finding a minimum parity (g,f)-factor of a graph G into the problem of finding a minimum perfect matching in a weighted simple graph G*. Using the structure of G*, a necessary and sufficient condition for the existence of an even factor is derived. This paper was accomplished while the second author was visiting the Center for Combinatorics, Nankai University. The research is supported by NSFC