Elektrodynamik mit Spin

Atomistic equations of the electromagnetic field for a particle with spin are derived from a Lagrangian. These equations are consistent with the equations of motion for such a particle. The resulting phenomenological equations are the well-known equations of Maxwell for the electromagnetic field in matter. The atomistic field equations for a particle with spin and magnetic moment give a dipole field. This result and the corresponding quantum mechanics for a particle with spin are applied to compute the hyperfine structure of the hydrogen atom by perturbation theory.     

On mixing sets arising in chance-constrained programming

The mixing set with a knapsack constraint arises in deterministic equivalent of chance-constrained programming problems with finite discrete distributions. We first consider the case that the chance-constrained program has equal probabilities for each scenario. We study the resulting mixing set with a cardinality constraint and propose facet-defining inequalities that subsume known explicit inequalities for this set. We extend these inequalities to obtain valid inequalities for the mixing set with a knapsack constraint. In addition, we propose a compact extended reformulation (with polynomial number of variables and constraints) that characterizes a linear programming equivalent of a single chance constraint with equal scenario probabilities. We introduce a blending procedure to find valid inequalities for intersection of multiple mixing sets. We propose a polynomial-size extended formulation for the intersection of multiple mixing sets with a knapsack constraint that is stronger than the original mixing formulation. We also give a compact extended linear program for the intersection of multiple mixing sets and a cardinality constraint for a special case. We illustrate the effectiveness of the proposed inequalities in our computational experiments with probabilistic lot-sizing problems.     

Implicit construction of multiple eigenvalues for trees

We are generally concerned with the possible lists of multiplicities for the eigenvalues of a real symmetric matrix with a given graph. Many restrictions are known, but it is often problematic to construct a matrix with desired multiplicities, even if a matrix with such multiplicities exists. Here, we develop a technique for construction using the implicit function theorem in a certain way. We show that the technique works for a large variety of trees, give examples and determine all possible multiplicities for a large class of trees for which this was not previously known.     

Optimality conditions for extremal controls in bilinear and quadratic problems

We consider the optimization problem for a bilinear functional with respect to a linear phase system with a modularly constrained control. On the base of exact formulas for the functional increment we establish sufficient optimality conditions for extremal controls. These conditions are stated as inequalities for one-dimensional functions on a time interval. They supplement the maximum principle, keeping the implementation complexity at the same level. The optimization problem for a quadratic functional is reduced to the bilinear case with the help of the matrix conjugate function.     

On the properties of orthorecursive expansions with respect to subspaces

In a Hilbert space, for orthorecursive expansions with respect to closed subspaces, we establish a criterion for expansions of elements of a certain finite-dimensional subspace with respect to a finite sequence of subspaces to coincide with the expanded elements. This implies a criterion for an element to be equal to its orthorecursive expansion with respect to a finite sequence of subspaces. We also obtain a number of results related to the best approximations of elements by partial sums of their orthorecursive expansions with respect to a sequence of finite-dimensional subspaces.     

Euler-Poisson-Darboux differential-operator equation with variable domains of smooth operators

We prove the strong well-posed solvability of the Cauchy problem for a second-order singular hyperbolic differential equation with variable domain of variable unbounded operator coefficients and for the mixed problem for a complete equation of string vibrations with a strong singularity in time and with a time-dependent boundary condition.     

Progressive Enlargement of Filtrations and Backward Stochastic Differential Equations with Jumps

This work deals with backward stochastic differential equations (BSDEs for short) with random marked jumps, and their applications to default risk. We show that these BSDEs are linked with Brownian BSDEs through the decomposition of processes with respect to the progressive enlargement of filtrations. We prove that the equations have solutions if the associated Brownian BSDEs have solutions. We also provide a uniqueness theorem for BSDEs with jumps by giving a comparison theorem based on the comparison for Brownian BSDEs. We give in particular some results for quadratic BSDEs. As applications, we study the pricing and the hedging of a European option in a market with a single jump, and the utility maximization problem in an incomplete market with a finite number of jumps.     

A numerical method for some initial-value problems of one scalar hyperbolic conservation law

We examine the existence and regularity results for a scalar conservation law with a convexity condition and solve its weak solution with shocks by using a special method of characterization combined with a representation formula for the weak solution.     

Coexistence in host–pathogen systems

Lanchier and Neuhauser have initiated the study of host–symbiont systems but have concentrated on the case in which the birth rates for unassociated hosts are equal. Here we allow the birth rates to be different and identify cases in which a host with a specialist pathogen can coexist with a second species. Our calculations suggest that it is possible for two hosts with specialist pathogens to coexist but it is not possible for a host with a specialist mutualist to coexist with a second species.     

Modeling of spraying with time-dependent material feed rate

This paper presents a mathematical model for predicting the thickness of coatings deposited by means of spraying onto rotating parts with circular symmetry, for the case of time-dependent material feed rate. A procedure for calculating the material feed rate control law providing production of coatings with uniform thickness or coatings with a predefined law of thickness variation is developed. The proposed procedure was used to analyze the process of spraying onto the surface of a rotating disc. A material feed rate variation law providing production of uniform coatings and coatings with linear thickness variation is calculated. It is demonstrated that for the first case the optimal law of material feed rate variation is described by a linear function and for the second case it can be successfully approximated by a quadratic function. The proposed calculation technique can be easily used for the case of coatings with a more complex law of thickness variation.     

Reduced order observer design for nonlinear systems

This work is a geometric study of reduced order observer design for nonlinear systems. Our reduced order observer design is applicable for Lyapunov stable nonlinear systems with a linear output equation and is a generalization of Luenberger’s reduced order observer design for linear systems. We establish the error convergence for the reduced order estimator for nonlinear systems using the center manifold theory for flows. We illustrate our reduced order observer construction for nonlinear systems with a physical example, namely a nonlinear pendulum without friction.     

Minimum dimension of a functional observer with specifiable dynamical properties

The article considers the construction of a functional observer with a given rate of convergence for the most general case: a vector state functional of a linear dynamical system with a vector output. An upper bound is derived on the minimum dimension of such an observer, which holds for almost all functionals. An algorithm is proposed for constructing an observer that achieves this bound. The algorithm can be used to construct a functional observer for almost all specified spectra (i.e., with the exception of a set of measure zero). The scalar observer method previously developed by the authors and proposed in the present article is based on canonical form Luenberger observability for systems with vector output.     

Global solutions to a degenerate solutal phase-field model for the solidification of a binary alloy

We consider a degenerate solutal phase-field model for the solidification of a binary alloy. This model is devoted to the evolution of the phase-field variable together with the relative concentration of the alloy for which the equation may degenerate. The existence of global weak solutions is proved for the degenerate case with a loss of regularity for the concentration in comparison with the non-degenerate case.     

A cylindrical plate with a weakly fixed curvilinear edge made of a transversally isotropic material

Free oscillations and stability under an axial compression of a thin cylindrical plate with a weakly fixed rectilinear edge made of a transversally isotropic material with low stiffness with respect to transverse displacements are considered. The curvilinear edges of the plate are assumed to be hingedly supported. The oscillation frequencies and the critical load for a plate with a free or weakly fixed edge are smaller than those for a shell closed in the circumferential direction. The shapes of oscillations and the forms of stability loss localized near the weakly fixed edge and damped at a distance from it are considered. The Timoshenko-Reissner model is used. Localized forms are analyzed by using a system of equations for Timoshenko-Reissner shallow shells, which is derived for this purpose. The main special feature of this system is that it contains a separate equation describing a solution with large variability. For the example of the stability problem under consideration, the error involved in the system of equations for Timoshenko-Reissner shallow shells is studied. The critical load values obtained with the use of the Kirchhoff-Love and Timoshenko-Reissner models are compared.     

A model for lime consolidation of porous solids

We propose a mathematical model describing the process of filling the pores of a building material with lime water solution with the goal to improve the consistency of the porous solid. Chemical reactions produce calcium carbonate which glues the solid particles together at some distance from the boundary and strengthens the whole structure. The model consists of a 3D convection–diffusion system with a nonlinear boundary condition for the liquid and for calcium hydroxide, coupled with the mass balance equations for the chemical reaction. The main result consists in proving that the system has a solution for each initial data from a physically relevant class. A 1D numerical test shows a qualitative agreement with experimental observations.     

Inversion of vector delay systems

We consider stationary linear vector systems with commensurable delays. We obtain sufficient conditions for the reducibility of such systems to canonical form with the extraction of null dynamics. A constructive algorithm for the reduction of a system to that form is presented. We suggest a method for estimating the unknown input for vector delay systems with given accuracy.     

Asymptotic observers for <Emphasis Type="Italic">n</Emphasis>-dimensional bilinear systems

We consider the construction of exponential observers for n-dimensional bilinear dynamical systems with arbitrary n. An algorithm is proposed to construct observers with any prespecified degree of stability for a system with a multidimensional output. Observers are constructed for systems with a scalar output and a singular bilinearity matrix. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 3, pp. 19-26, 2003.     

A type of new posteriori error estimators for stokes problems

1. IntroductionIn the numerical approximation of PDE, it is often very importals to detect regionswhere the accuracy of the numerical solution is degraded by local singularities of the solutionof the continuous problem such as the singularity near the re-entrant corller. An obviousremedy is to refine the discretization in the critical regions, i.e., to place more gridpointswhere the solution is less regular. The question is how to identify these regions automdticallyand how to determine a goo…     

Blow-up of solutions of some nonlinear wave equations

We analyze the problem of blow-up of global solutions of a semilinear wave equation with a potential and with a possible degeneration at infinity for nonnegative initial data with compact support. By using the nonlinear capacity method, we prove the theorem on the nonexistence of such a solution for a subcritical and the critical nonlinearity exponent.