A randomized algorithm for the min-max selecting items problem with uncertain weights

This paper deals with the min-max version of the problem of selecting p items of the minimum total weight out of a set of n items, where the item weights are uncertain. The discrete scenario representation of uncertainty is considered. The computational complexity of the problem is explored. A randomized algorithm for the problem is then proposed, which returns an O(ln K)-approximate solution with a high probability, where K is the number of scenarios. This is the first approximation algorithm with better than K worst case ratio for the class of min-max combinatorial optimization problems with unbounded scenario set.     

(Cyclopentadienyl){(N,N-dimethylaminoethyl)cyclopentadienyl} complexes of zirconium: Crystal structure of [(η-C5H5)(η-C5H4CH2CH2NHMe2)ZrCl2]2[ZrCl6]

[(η⁵-C₅H₅)ZrCl₃] reacts with [C₅H₄CH₂CH₂NMe₂]Li yielding the coordination polymer [(C₅H₅)(C₅H₄CH₂CH₂NMe₂)ZrCl₂]_n (1) as a brown solid which is very sensitive to moisture. The reaction of 1 with 1.35 equivalent of HCl (methanolic solution) yields pale yellow green crystals of [(η⁵-C₅H₅)(η⁵-C₅H₄CH₂CH₂NHMe₂)ZrCl₂]₂[ZrCl₆] (2). Compound 2 was fully characterized on the basis of NMR data and X-ray crystal structure analysis. The formation of this product indicates the elimination of C₅H₄CH₂CH₂NMe₂ as well as C₅H₅ ligands from the Zr(IV) metal centre.     

Polynomial approximation of rational Bézier curves with constraints

We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bézier curve by the polynomial Bézier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, and exploits their recursive properties. Examples are given, showing the effectiveness of the algorithm.     

The Love's problem for hyperbolic thermoelasticity

In our paper we investigated the initial-boundary value problem for elastic layer situated on half space of another elastic medium. In this medium the thermomechanical interactions were taken into consideration. The system of equations with initial-boundary conditions describes the phenomenon of wave propagation with finite speed. In our problem there are two surfaces ie. free surface and contact surface between layer and half space. On the free surface are setting boundary conditions for normal and tangent surface force. We consider two types of contact between layer and half-space: rigid contact and slip contact. The initial-boundary value problem was solved by using integral transformations and Cagniard-de Hoope methods. From the solution of this problem follows that in layer and half space exist some kind of thermoelastic waves. We investigated moreover the conditions which should be fullfiled for propagation of Rayleigh and Love's type waves on the contact surface between layers and half space. The results obtained in our investigation were used in technical applications especially engineering design and diagnostics of roads and airfields. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Response of the mechanical system with the hysteresis in a high frequency limit

Vibrations of a single-degree-of-freedom oscillator with a hysteretic damper subjected to kinematic excitations in a high frequency limit have been explored. By using of two time scales expansion we derived an effective damping coefficient. It has been shown that in the limit of the high excitation frequency the hysteretic loop could be eliminated for the slow time scale dynamics. For simplicity, to examine the effect of hysteresis we used the model of a single hysteron. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)