Cauchy problem for the Mathieu equation away from parametric resonance

Four solutions of the Cauchy problem for Mathieu’s equation away from parametric resonance domains are analytically constructed using an asymptotic averaging method in the fourth approximation. Three solutions occur near fractional parameter values at which slow combination phases exist. The fourth solution occurs in the absence of slow phases away from parametric resonance domains and the fractional parameter values.     

Spatial inhomogenity due to turing bifurcation in a system of Gierer-Meinhardt type

This paper treats the conditions for the existence and stability properties of stationary solutions of reaction-diffusion equations of Gierer-Meinhardt type, subject to Neumann boundary data. The domains in which diffusion takes place are of three types: a regular hexagon, a rectangle and an isosceles rectangular triangle. Considering one of the relevant features of the domains as a bifurcation parameter it will be shown that at a certain critical value a diffusion driven instability occurs and Turing bifurcation takes place: a pattern emerges.     

Some Boundary — Contact Problems of the Elasticity Theory with Mixed Boundary Conditions Outside the Contact Surface

n — Dimensional (n ≥ 2) boundary-contact problems of statics of the elasticity theory for homogeneous anisotropic media are investigated when the contact of two bounded domains occurs from the outside on some part of boundaries with mixed boundary conditions. Theorems on the existence and uniqueness of solutions of boundary-contact problems in Besov and Bessel potential spaces are obtained. The smoothness of solutions is studied in closed domains occupied by elastic media.     

Phase separation of multi-component Bose–Einstein condensates in optical lattices

In this paper, we analyze phase separation of multi-component Bose–Einstein condensates (BECs) in the presence of strong optical lattices. This paper is in threefold. We first prove that when the inter-component scattering lengths go to infinity, phase separation of a multi-component BEC occurs. Furthermore, particles repel each other and form segregated nodal domains. Secondly, we show that the union of these segregated nodal domains equal to the entire domain. Thirdly, we show that if the intra-component scattering lengths are bounded by some finite number, each nodal domain is connected. For large intra-component scattering lengths, however, the third result is not true and a counter example of non-connected nodal domains is given.     

Unimodular quadratic forms over global function fields

The isometry problem is studied for unimodular quadratic forms over the Hasse domains of global function fields. Over the polynomial ring k[x] the problem reduces to classification of forms over k; but examples are provided showing that in general no such reduction occurs, even when the underlying ring is Euclidean. Connections with the structure of the ideal class group are given, and a complete invariant for the isometry class is found in the ternary isotropic case.     

The Liouville Foliation of Nonconvex Topological Billiards

Together with the classical plane billiards, topological billiards can be considered, where the motion occurs on a locally flat surface obtained by isometrically gluing together several plane domains along their boundaries, which are arcs of confocal quadrics. A point moves inside each of the domains along straight line segments; when it reaches the boundary of a domain, it passes to another domain. Previously, the author gave a Liouville classification of all topological billiards obtained by gluing along convex boundaries. In the present paper, all topological integrable billiards obtained by gluing along convex or nonconvex boundaries from elementary billiards bounded by arcs of confocal quadrics are classified. For some of such nonconvex topological billiards, the Fomenko–Zieschang invariants (marked molecules W*) for Liouville equivalence are calculated.     

Asymptotic analysis for blow-up solutions in parabolic equations involving variable exponents

In this article, we consider non-negative solutions of the homogeneous Dirichlet problems of parabolic equations with local or nonlocal nonlinearities, involving variable exponents. We firstly obtain the necessary and sufficient conditions on the existence of blow-up solutions, and also obtain some Fujita-type conditions in bounded domains. Secondly, the blow-up rates are determined, which are described completely by the maximums of the variable exponents. Thirdly, we show that the blow-up occurs only at a single point for the equations with local nonlinearities, and in the whole domain for nonlocal nonlinearities.     

Turing bifurcation in a system with cross diffusion

This paper treats the conditions for the existence and stability properties of stationary solutions of reaction–diffusion equations subject to Neumann boundary data. Hence, we assume that there are two substances in a two-dimensional bounded spatial domain where they are diffusing according to Fick's law: the velocity of the flow of diffusing substance is directed opposite to the (spatial) gradient of the density and is proportional to its modulus, but the spatial flow of each substance is influenced not only by its own but also by the other one's density (cross diffusion). The domains in which the substances are diffusing are of three type: a regular hexagon, a rectangle and an isosceles rectangular triangle. It will be assumed that there is no migration across the boundary of these domains. Considering one of the relevant features of the domains as a bifurcation parameter it will be shown that at a certain critical value a diffusion driven (Turing type) instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion) but becomes unstable with respect to the system with diffusion and that Turing bifurcation takes place: a spatially non-homogenous (non-constant) solution (structure or pattern) arises.     

Numerical investigation of a liquid displacing a gas in thin porous layers

This article deals with a liquid displacing a gas in a thin heterogeneous porous material, which occurs e.g. during the filling process of a lithium-ion battery with an electrolyte. The investigation is based upon the local volume-averaged Navier-Stokes equations, using a Volume-of-Fluid method to treat the interface. For the flow the wall effect and capillary forces have to be considered. Capillary rise experiments are used to determine the permeability. Since the layers are thin and the characteristic size of the particles is comparatively large, friction with the electrode is taken into account with respect to the mobility of the contact line. The implemented models are validated against analytical results, showing a good agreement. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive solutions in semilinear critical problems for polyharmonic operators

In this work, we study the existence of positive solutions in semilinear critical problems for polyharmonic operators. Minimizing on some infinite-dimensional Finsler manifold, we prove the existence result in some general domain under the appropriate assumptions. Alternatively, the concentration phenomenon occurs if minimizing method does not work. This permits us to search for the instable solutions in higher level set by topological arguments in domains perforated with the small holes.     

Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains

Consider the nonstationary Stokes equations in exterior domains with the compact boundary . We show first that the solution decays like for all as . This decay rate is optimal in the sense that for some as occurs if and only if the net force exerted by the fluid on is zero. Received: 15 June 2000 / Published online: 18 June 2001     

Non-steady interaction of fluids with free surfaces and rigid bodies

Interaction between fluids and rigid bodies is a phenomenon, which occurs for example in floating bearings or turbines. Mostly the focus is on domains with rigid boundaries on every side or defined influx or effusion of fluid over the boundaries. The interaction between fluids with free surfaces and a rigid body were mostly studied under the aspect of stability for steady-state conditions, e. g. for fluid-filled centrifuges. A characteristic property is the instability over non-empty intervalls of angular velocities, so the analysis of non-steady behaviour is essential to investigate the stability of the drive through these instable domains. A first approach to this topic is the qualitative investigation of a fluid domain with the shallow water theory. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singular Perturbations of Elliptic Problems on Domains with Small Holes

Singular perturbation techniques are used to study the solutions of nonlinear second order elliptic boundary value problems defined on arbitrary plane domains from which a finite number of small holes of radius ρ_i(ε) have been removed, in the limit ε → 0. Asymptotic outer and inner expansions are constructed to describe the behavior of solutions at simple bifurcation and limit points. Since bifurcation usually occurs a eigenvalues of a linearized problem, we study in detail the dependence of the eigenvalues and eigenfunctions on ε, for ε → 0. These results are applied to the vibration of a rectangular membrane with one or two circular holes. The asymptotic analysis predicts a remarkably large sensitivity of eigenvalues and limit points to the ε-domain perturbation considered in this paper.     

The problem of highly constrained tasks in group decision support systems

Most experimental uses of group decision support systems (GDSS) are associated with relatively unrestricted domains, for example, idea generation and preference specification, where few restrictions on potential solutions exist. However, an important GDSS task is that of resource allocation across functional areas of the organization, including supply chain applications. These types of tasks, such as budget planning and production planning, are typically highly constrained and difficult to solve optimally, necessitating the use of decision aids, such as those found in GDSS.We use a model based on adaptive search of a genetic algorithm as the analogy for the group decision making process. We apply this model to experimental data gathered from GDSS groups solving a production planning task. The results indicate very low estimated crossover rates in the experimental data. We also run computational experiments based on adaptive search to mimic the GDSS data and find that the low estimated crossover rate might be due to the highly constrained search space explored by the decision making groups. The results suggest further investigation into the presumed beneficial effects of group interaction in such highly constrained task domains, as it appears very little true information exchange occurs between group members in such an environment. Furthermore, the simulation technique can be used to help predict certain GDSS behaviors, thus improving the entire GDSS process.     

Structure of A*-Fibrations over One-Dimensional Seminormal Semilocal Domains

In this paper we give a structure theorem for an A*-fibration over a one-dimensional noetherian seminormal semilocal domain and show that, in this situation, any A*-fibration whose spectrum occurs as an affine open subscheme of the spectrum of an A¹-fibration (equivalently, an affine line A¹) is actually A*. The structure theorem provides examples of A*-fibrations over one-dimensional noetherian seminormal semilocal domains whose spectra are not affine open subschemes of any affine line A¹ over the base ring. We also construct examples of nontrivial A*-fibrations over one-dimensional noetherian non-seminormal local domains whose spectra are open subschemes of A¹-fibrations over the base ring.     

Optimization of neural network for ionic conductivity of nanocomposite solid polymer electrolyte system (PEO-LiPF6-EC-CNT)

In this study, the ionic conductivity of a nanocomposite polymer electrolyte system (PEO-LiPF₆-EC-CNT), which has been produced using solution cast technique, is obtained using artificial neural networks approach. Several results have been recorded from experiments in preparation for the training and testing of the network. In the experiments, polyethylene oxide (PEO), lithium hexafluorophosphate (LiPF₆), ethylene carbonate (EC) and carbon nanotubes (CNT) are mixed at various ratios to obtain the highest ionic conductivity. The effects of chemical composition and temperature on the ionic conductivity of the polymer electrolyte system are investigated. Electrical tests reveal that the ionic conductivity of the polymer electrolyte system varies with different chemical compositions and temperatures. In neural networks training, different chemical compositions and temperatures are used as inputs and the ionic conductivities of the resultant polymer electrolytes are used as outputs. The experimental data is used to check the system’s accuracy following the training process. The neural network is found to be successful for the prediction of ionic conductivity of nanocomposite polymer electrolyte system.     

An accurate semi-analytic finite difference scheme for two-dimensional elliptic problems with singularities

A high-order semi-analytic finite difference scheme is presented to overcome degradation of numerical performance when applied to two-dimensional elliptic problems containing singular points. The scheme, called Least-Square Singular Finite Difference Scheme (L-S SFDS), applies an explicit functional representation of the exact solution in the vicinity of the singularities, and a conventional finite difference scheme on the remaining domain. It is shown that the L-S SFDS is “pollution” free, i.e., no degradation in the convergence rate occurs because of the singularities, and the coefficients of the asymptotic solution in the vicinity of the singularities are computed as a by-product with a very high accuracy. Numerical examples for the Laplace and Poisson equations over domains containing re-entrant corners or abrupt changes in the boundary conditions are presented. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 281–296, 1998     

Mathematical modeling of industrial aluminum electrolysis

The problem of aluminum electrolysis is discussed. The mathematical model of an industrial electrolyzer presented in the paper is written under the assumption that the electrolyte and metal media are immiscible. At the basis of the mathematical statement is a three-dimensional, nonstationary, and nonlinear system of magnetic hydrodynamics equations which is written separately in the aluminum medium and in the electrolyte medium with a geometric account for wall accretion, skull, and arrangement of anodes. The proposed system allows one to model various forms of anodes, the number of anodes in a bath, and their sizes. Interfaces of media are connected by a viscous friction. Initial values of speeds and electromagnetic fields in the media and the medium interface are considered as set. On the skull, bottom, and anodes the attachment conditions are set. The speed of change of a magnetic field in the metal and electrolyte on the interface is considered zero. On the basis of a numerical method of solution of the system of equations, there is a well-proved method of division over physical processes. The analysis of results of the numerical experiment has shown that it is actually possible to allocate a ??middle?? layer for modeling of the electrolysis process. The proposed model allows one to investigate media behavior upon the occurrence of a long anode effect due to sharp reduction of the electric conductivity of the electrolyte and subsequent sharp growth of the electric-field strength.     

Generalizations of the Wirtinger-Northcott Inequality

Wirtinger and Northcott introduced an inequality bounding thenorm of a 2 periodic function by the norm of its derivatives.It is shown here that this type of result occurs for other operators,defined on various domains and spaces. 2000 Mathematics SubjectClassification 26D10 (primary); 41A17 (secondary).     

Quadrature Domains with Infinite Volume

We discuss quadrature domains for subharmonic functions and prove the existence of core quadrature domains for certain positive measures. The core quadrature domains are the smallest quadrature domains as measures and inherit good properties from quadrature domains with finite volume. We next discuss new balayage for the class of harmonic functions integrable in a neighborhood of ∞. We give several estimates of balayage measures. The new balayage is introduced to construct quadrature domains for harmonic functions. Submitted: June 26, 2008. Accepted: July 24, 2008.