Localized Solutions of a Piecewise Linear Model of the Stationary Swift–Hohenberg Equation on the Line and on the Plane

In this paper we study a simplified model of the stationary Swift–Hohenberg equation, where the cubic nonlinearity is replaced by a piecewise linear function with similar properties. The main goal is to prove the existence of so-called localized solutions of this equation, i.e., solutions decaying to a homogeneous zero state with unbounded increase of the space variable. The following two cases of the space variable are considered: one-dimensional (on the whole line) and two-dimensional; in the latter case, radially symmetric solutions are studied. The existence of such solutions and increase of their number with change in the equation parameters are shown.     

Group methodology in production management: A solution based on a hierarchical classification method

We consider the problem submitted by Garcia and Proth and we propose for it a solution obtained by simply applying our very general method of hierarchical classification based on the likelihood of the links, in the case of an incidence table of data. This application is done on each of the two sides of the incidence data table. In our approach, the common number p of classes has not to be fixed arbitrarily, but is an observed possible result—provided by the notions of ‘significant’ levels and nodes of the classification tree—for the ‘natural’ association between two partitions. We compare our solution with the previous one.     

Light transmission through a circular metallic grating under broadband radial and azimuthal polarization illumination

We studied the characteristics of a circular metallic grating illuminated by broadband radial and azimuthal polarizations. We demonstrated that this scenario is the cylindrical analogue of a one-dimensional Cartesian grating illuminated by TM and TE polarizations. We measured the transmission spectra of this structure and observed strong polarization selectivity and, specifically, a resonance for radial polarization excitation, indicating a strong coupling to surface plasmons. The structure may be attractive for applications where pure radial polarization is needed, such as tight focusing, material processing, and particle trapping.     

Solution of Dynamic Problems for Thin-Walled Structural Elements with Nonuniform Dynamic Boundary Conditions

The characteristics of the stress–strain state of thin-walled structural elements are determined in the case where dynamic boundary loads or displacements described by pulse functions are specified. A general scheme for realization of the method of natural-mode expansion is stated as applied to differential equations with unknown functions of one spatial coordinate and time. Theoretical relations for rods, plates, and shells are given. The potential of the approach developed is illustrated by solving specific problems     

Hamiltonian torus actions on symplectic orbifolds and toric varieties

In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly, we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each open facet and that all such orbifolds are algebraic toric varieties.

     

A comparison of the solutions of the problems of the theory of shells and the three-dimensional theory of elasticity

By comparing with the results obtained by numerical solution of a three-dimensional problem of the theory of elasticity we evaluate the different versions of the theory of elastic orthotropic cylindrical shells.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 96–101.