Modified Shephard’s problem on projections of convex bodies

We disprove a conjecture of A. Koldobsky asking whether it is enough to compare (n − 2)-derivatives of the projection functions of two symmetric convex bodies in the Shephard problem in order to get a positive answer in all dimensions. The author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953     

Carbon Nanostructured Screen‐printed Electrodes Modified with CuO/Glucose Oxidase/Maltase/SiO2 Composite Film for Maltose Determination

A sensitive voltammetric method was developed to determine maltose in beverage products using a carbon nanostructured screen‐printed electrode modified with CuO/glucose oxidase/maltase/SiO₂ biocomposite film. Adding CuO particles was done to possess catalytic activity toward hydrogen peroxide. Electrode modified by glucose oxidase and maltase shows a good response to maltose. A well‐defined reduction peak was registered at the potential of ?0.55 V (vs. Ag/AgCl) which intensity increases linearly with the concentration of maltose ranging from 0.01 to 0.1 mmol L^?1. The calculated limit of detection was 0.005 mmol L^?1. Tested on the beer samples, the developed CuO/glucose oxidase/maltase/SiO₂ biocomposite film covered carbon nanostructured screen‐printed electrode is showed to be a prospective sensitive element of the third generation biosensor for maltose.     

Kolmogorov's problem on the class of multiply monotone functions

In this paper we give necessary and sufficient conditions for the system of positive numbers _Mk1,_Mk2,…,_Mkd$M_{k_1},M_{k_2},\dots ,M_{k_d}$, 0≤_k1<…<_kd≤r$0\le k_1<\dots <k_d\le r$, to guarantee the existence of an r  -monotone function defined on the negative half-line _R−${\mathbb{R}}_{-}$ and such that _{‖x(ki)‖∞}=_Mki${\Vert x^{(k_i)}\Vert }_{\infty }=M_{k_i}$, i=1,2,…,d$i=1,2,\dots ,d$. We also discuss some applications of the obtained results and connections with other problems.     

Vibration localization in one-dimensional linear and nonlinear lattices: discrete and continuum models

The phenomenon of vibration localization plays an important role in the dynamics of inhomogeneous and nonlinear materials and structures. The vibration localization can occur in the case of inhomogeneity under the following conditions: (i) the frequency spectrum of the periodic structure includes stopbands, (ii) a perturbation of periodicity is present, and (iii) the eigenfrequency of the perturbed element falls into a stopband. Under these conditions, the energy can be spatially localized in the vicinity of the defect with an exponential decay in the infinity. The influence of nonlinearity can shift frequency into the stopband zone. In the present paper, the localization of vibrations in one-dimensional linear and nonlinear lattices is investigated. The localization frequencies are determined and the attenuation factors are calculated. Discrete and continuum models are developed and compared. The limits of the applicability of the continuum models are established. Analysis of the linear problem has allowed a better understanding of specifics of the nonlinear problem and has led to developing a new approach for the analysis of nonlinear lattices alternative to the method of continualization.     

Analytical study of the load transfer in fibre-reinforced 2D composite materials

The phenomenon of the load diffusion from a fibre to a surrounding matrix is analysed for the 2D case. We use an approximate analytical approach based on the asymptotic reduction of the governing biharmonic problem into two harmonic problems. The comparison of the obtained solutions with known results of other authors shows an acceptable accuracy of the proposed asymptotic simplifications. All solutions are obtained in closed analytical form.The case of perfect bonding between fibre and matrix for a single fibre and for a periodic system of fibres is firstly considered. Then we study the influence of the interface stiffness. In the case when only a single fibre is loaded, the influence of all other fibres is predicted by means of a three-phase model. The proposed approach gives a possibility to solve the problems for a broken fibre and for a broken matrix, as well as for partly debonded fibres. The important problem of infinite matrix cracks is also solved in the present paper.The obtained results can be used for the calculation of pull-out and push-out tests, as well as for the investigation of the fracture of composite materials.     

Simple estimation on effective transport properties of a random composite material with cylindrical fibres

Improved bounds for effective transport properties of a random non-percolated composite with cylindrical fibres are developed by means of the security-spheres approach. The key point of the method is to obtain a solution for a regular composite that can be valid for all values of the volume fractions and properties of the components. For this aim we use the asymptotic homogenization method; a cell problem is solved by a modified version of the boundary shape perturbation technique.      

Study of wave dispersion in periodic composites by higher-order asymptotic homogenization method

We present [1] an application of the higher-order asymptotic homogenization method (AHM) to the study of wave dispersion in periodic composite materials. When the wavelength of a travelling signal becomes comparable to the size of heterogeneities, successive reflections and refractions of the waves at the components interfaces lead to the formation of a complicated sequence of pass and stop frequency bands. The AHM provides a long-wave approximation valid in the low frequency range. Solution for the high frequencies is obtained on the basis of the Floquet–Bloch theorem by the plane-wave (PW) expansions method. Anti-plane shear waves in a fibre-reinforced composite with a square lattice of cylindrical fibres are considered. The dispersion curves are obtained, the pass and stop bands are identified. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)