Be careful with the Exp-function method

An application of the Exp-function method to search for exact solutions of nonlinear differential equations is analyzed. Typical mistakes of application of the Exp-function method are demonstrated. We show it is often required to simplify the exact solutions obtained. Possibilities of the Exp-function method and other approaches in mathematical physics are discussed. The application of the singular manifold method for finding exact solutions of the Fitzhugh–Nagumo equation is illustrated. The modified simplest equation method is introduced. This approach is used to look for exact solutions of the generalized Korteweg–de Vries equation.     

On “new travelling wave solutions” of the KdV and the KdV–Burgers equations

The Korteweg-de Vries and the Korteweg-de Vries–Burgers equations are considered. Using the travelling wave the general solutions of these equations are presented. “New travelling wave solutions” of the KdV and the KdV–Burgers equations by Wazzan [Wazzan L. Commun Nonlinear Sci Numer Simulat 2009;14:443–50] are analyzed. We demonstrate that all his solutions are not new and are transformed to known solutions.     

A tensorial framework for strain induced damage of plastically deformed metals

A tensorial framework for strain induced ductile damage of plastically deformed metals is developed in terms of both plastic flow theory and continuum damage mechanics. A symmetric second order damage rate tensor is used in order to study various processes with large finite deformations in combination with damage analysis. The definition of this tensor is physically meaningful since its volumetric and deviatoric parts describe the damage increments caused by an increase in the void volume and by a change in the shape of the void, respectively. Such a view on damage kinetics leads to the introduction of two measures for damage assessment which allow predicting not only a risk of macroscopic failure but also the onset of void coalescence. Material functions appearing in the constitutive equations for damage are determined both by own experiments and by known results from literature. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Collapse criteria of foam cells under various loading

Recently porous materials are widely used in civil and mechanical engineering. In particular, such porous materials as metal and polymer foams have applications in lightweight structures. From mechanics point of view foams can demonstrate unusual behavior such as strain localization related to foam cells buckling under certain loads. The aim of this work is the elaboration of the model of foam material taking into account the cell collapse. We consider the cell collapse initiation during the elastic instability and its further evolution under loading. The geometrical structure of foam is generated with the use of the Voronoi algorithm. Based on stochastic distributions of cells we create various geometrical models of foams. The influence of the cell volume, wall thicknesses and material properties of the foam material on critical loads is obtained. The calculations are performed with the use of Abaqus CAD/CAE system. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multi-agent model describing self-renewal of differentiation effects on the blood cell population

In this work, a new multi-agent model is used to describe blood cell population dynamics. More particularly, we focus our simulations here on differentiation and self-renewal process based on cell communication. We consider the different cases where progenitor cells are able to self-renew or not in the bone marrow. As a consequence of this study, we give some possible explanations of the mechanism for recovery of the system under important blood loss or blood diseases such as anemia.     

Alkylation of schleyer's lithiocarbon C4Li4 by methyl iodide

Alkylation of lithiocarbon C₄Li₄ gives the hydrocarbon C₈H₁₂ to which the structure of tetramethyltetrahedrane has been ascribed.     

Epitaxial composite layers of electron donors and acceptors from very large polycyclic aromatic hydrocarbons

Large polycyclic aromatic hydrocarbons (PAHs) can be considered as nanographenes, whose electron donating or accepting properties are controlled by their size and shape as well as functionalities in their periphery. Epitaxial thin films of them are targets for optoelectronic applications; however, large PAHs are increasingly difficult to process. Here we show that epitaxial layers of very large unsubstituted PAHs (C(42)H(18) and C(132)H(34)), as well as a mixed layer of C(42)H(18) with an electron acceptor, can be obtained by self-assembly from solution. The C(132)H(34) is by far the largest nanographene that up to now has been processed into ordered thin films; due to its size it cannot be sublimed in a vacuum. Scanning tunneling microscopy (STM) studies reveal that the interaction with the substrate induces a strong perturbation of the electronic structure of the pure donor in the first epitaxial monolayer. In a second epitaxial layer with a donor acceptor stoichiometry of 2:1 the molecules are unperturbed.     

Linear resolvent growth of rank one perturbation of a unitary operator does not imply its similarity to a normal operator

The main result of this paper is that for any unitary (selfadjoint) operatorU with non-trivial absolutely continuous part of the spectrum, there exists a rank-one perturbationK = ba* = (a)b such that the operatorT = U + K satisfies the Linear Resolvent Growth condition (LRG),

Transformation of Quadratic Forms to Perfect Squares for Broken Extremals

In this paper we study a quadratic form which corresponds to an extremal with piecewise continuous control in variational problems. This form, compared with the classical one, has some new terms connected with the set of all points of discontinuity of the control. Its positive definiteness is a sufficient optimality condition for broken extremals. We show that if there exists a solution to corresponding Riccati equation satisfying some jump condition at each point of the set , then the quadratic form can be transformed to a perfect square, just as in the classical case. As a result we obtain sufficient conditions for positive definiteness of the quadratic form in terms of the Riccati equation and hence, sufficient optimality conditions for broken extremals.     

Behavior of the Bergman kernel and metric near convex boundary points

The boundary behavior of the Bergman metric near a convex boundary point of a pseudoconvex domain is studied. It turns out that the Bergman metric at points in the direction of a fixed vector tends to infinity, when is approaching , if and only if the boundary of does not contain any analytic disc through in the direction of .