Picard-Vessiot theory for real fields

The existence of a Picard-Vessiot extension for a homogeneous linear differential equation has been established when the differential field over which the equation is defined has an algebraically closed field of constants. In this paper, we prove the existence of a Picard-Vessiot extension for a homogeneous linear differential equation defined over a real differential field K with real closed field of constants. We give an adequate definition of the differential Galois group of a Picard-Vessiot extension of a real differential field with real closed field of constants and we prove a Galois correspondence theorem for such a Picard-Vessiot extension.     

Linearized gravity in the Randall-Sundrum model with stabilized distance between branes

We consider linearized gravity in the Randall-Sundrum model in which the distance between branes is stabilized by introducing the scalar Goldberger-Wise field. We construct the second variation Lagrangian for fluctuations of gravitational and scalar fields over the background solution and investigate its gauge invariance. We obtain, separate, and solve the corresponding equations of motion. For physical degrees of freedom, we obtain the effective four-dimensional Lagrangian describing the massless graviton, massive gravitons, and the set of massive scalar fields. We also find masses and coupling constants of these fields to the matter on the negative-tension brane. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 339–353, December, 2006.     

Remark about scalar curvature and Riemannian submersions

We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space and fibers. We give an application concerning scalar curvatures of smooth limit spaces arising in bounded curvature collapses.

     

Linear mappings preserving ρ-orthogonality

We define a ρ-orthogonality in a real normed space and we consider the class of linear mappings preserving this relation. We show that a linear mapping preserving ρ-orthogonality has to be a similarity, i.e., a scalar multiple of an isometry. As a result, we give a characterization of smooth spaces in terms of this orthogonality.     

Partial data inverse problems for Maxwell equations via Carleman estimates

In this article we consider an inverse boundary value problem for the time-harmonic Maxwell equations. We show that the electromagnetic material parameters are determined by boundary measurements where part of the boundary data is measured on a possibly very small set. This is an extension of earlier scalar results of Bukhgeim–Uhlmann and Kenig–Sjöstrand–Uhlmann to the Maxwell system. The main contribution is to show that the Carleman estimate approach to scalar partial data inverse problems introduced in those works can be carried over to the Maxwell system.     

Comparison of perturbative and non-perturbative models for description of nucleon properties

We compare the static nucleon properties in the perturbative (chiral perturbation theory) and the non-perturbative model. We consider a chiral model for the nucleon based on the linear sigma model with scalar–isoscalar scalar–isovector mesons coupled to quarks, and solved it using the coherent pair approximation.     

Effective fermion models near a tricritical point

We consider the class of fermion models with a momentum cutoff Λ and an interaction containing derivatives. For the quasilocal fermion model with two coupling constants near a tricritical point, we classify the solutions of mass-gap equations for different values of coupling constants. The mass spectrum of the scalar excitations in the vicinity of the tricritical point is derived and it is found that two scalar exitations always arise when at least one of the constants exceeds its critical value. The mass formation for scalar bound states induced by static gluon fields (gluon condensates) is investigated. Bibliography 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 5–19. Translated by V. A. Andrianov.     

Vacuum and Thermal Energies for Two Oscillators Interacting Through A Field

We consider a simple (1+1)-dimensional model for the Casimir–Polder interaction consisting of two oscillators coupled to a scalar field. We include dissipation in a first-principles approach by allowing the oscillators to interact with heat baths. For this system, we derive an expression for the free energy in terms of real frequencies. From this representation, we derive the Matsubara representation for the case with dissipation. We consider the case of vanishing intrinsic frequencies of the oscillators and show that the contribution from the zeroth Matsubara frequency is modified in this case and no problem with the laws of thermodynamics appears.     

On Romanovski–Jacobi polynomials and their related approximation results

The aim of this article is to present the essential properties of a finite class of orthogonal polynomials related to the probability density function of the F -distribution over the positive real line. We introduce some basic properties of the Romanovski–Jacobi polynomials, the Romanovski–Jacobi–Gauss type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kinds of finite orthogonal polynomials and other classes of infinite orthogonal polynomials. Moreover, we derive spectral Galerkin schemes based on a Romanovski–Jacobi expansion in space and time to solve the Cauchy problem for a scalar linear hyperbolic equation in one and two space dimensions posed in the positive real line. Two numerical examples demonstrate the robustness and accuracy of the schemes.     

Linear programming approaches for multicategory support vector machines

We consider linear programming approaches for support vector machines (SVM). The linear programming problems are introduced as an approximation of the quadratic programming problems commonly used in SVM. When we consider the kernel based nonlinear discriminators, the approximation can be viewed as kernel principle component analysis which generates an important subspace from the feature space characterized the kernel function. We show that any data points nonlinearly, and implicitly, projected into the feature space by kernel functions can be approximately expressed as points lying a low dimensional Euclidean space explicitly, which enables us to develop linear programming formulations for nonlinear discriminators. We also introduce linear programming formulations for multicategory classification problems. We show that the same maximal margin principle exploited in SVM can be involved into the linear programming formulations. Moreover, considering the low dimensional feature subspace extraction, we can generate nonlinear multicategory discriminators by solving linear programming problems.Numerical experiments on real world datasets are presented. We show that the fairly low dimensional feature subspace can achieve a reasonable accuracy, and that the linear programming formulations calculate discriminators efficiently. We also discuss a sampling strategy which might be crucial for huge datasets.     

Spectral methods based on Hermite functions for linear hyperbolic equations

We consider the approximation by spectral and pseudo‐spectral methods of the solution of the Cauchy problem for a scalar linear hyperbolic equation in one space dimension posed in the whole real line. To deal with the fact that the domain of the equation is unbounded, we use Hermite functions as orthogonal basis. Under certain conditions on the coefficients of the equation, we prove the spectral convergence rate of the approximate solutions for regular initial data in a weighted space related to the Hermite functions. Numerical evidence of this convergence is also presented. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

An aperiodic set of 13 Wang tiles

A new aperiodic tile set containing only 13 tiles over 5 colors is presented. Its construction is based on a recent technique developed by Kari. The tilings simulate the behavior of sequential machines that multiply real numbers in balanced representations by real constants.     

Multigrid preconditioned Krylov subspace methods for three-dimensional numerical solutions of the incompressible Navier–Stokes equations

We consider numerical methods for the incompressible Reynolds averaged Navier–Stokes equations discretized by finite difference techniques on non-staggered grids in body-fitted coordinates. A segregated approach is used to solve the pressure–velocity coupling problem. Several iterative pressure linear solvers including Krylov subspace and multigrid methods and their combination have been developed to compare the efficiency of each method and to design a robust solver. Three-dimensional numerical experiments carried out on scalar and vector machines and performed on different fluid flow problems show that a combination of multigrid and Krylov subspace methods is a robust and efficient pressure solver. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic Behavior of Solutions of Dynamic Equations

We consider linear dynamic systems on time scales, which contain as special cases linear differential systems, difference systems, or other dynamic systems. We give an asymptotic representation for a fundamental solution matrix that reduces the study of systems in the sense of asymptotic behavior to the study of scalar dynamic equations. In order to understand the asymptotic behavior of solutions of scalar linear dynamic equations on time scales, we also investigate the behavior of solutions of the simplest types of such scalar equations, which are natural generalizations of the usual exponential function.     

Krylov-Bogolyubov averaging of asymptotically autonomous differential equations

We apply the Krylov and Bogolyubov asymptotic integration procedure to asymptotically autonomous systems. First, we consider linear systems with quasi-periodic coefficient matrix multiplied by a scalar factor vanishing at infinity. Next, we study the asymptotically autonomous Van-der-Pol oscillator.

     

Ibragimov-type invariants for a system of two linear parabolic equations

We obtain new semi-invariants for a system of two linear parabolic type partial differential equations (PDEs) in two independent variables under equivalence transformations of the dependent variables only. This is achieved for a class of systems of two linear parabolic type PDEs that correspond to a scalar complex linear (1 + 1) parabolic equation. The complex transformations of the dependent variables which map the complex scalar linear parabolic PDE to itself provide us with real transformations that map the corresponding system of linear parabolic type PDEs to itself with different coefficients in general. The semi-invariants deduced for this class of systems of two linear parabolic type equations correspond to the complex Ibragimov invariants of the complex scalar linear parabolic equation. We also look at particular cases of the system of parabolic type equations when they are uncoupled or coupled in a special manner. Moreover, we address the inverse problem of when systems of linear parabolic type equations arise from analytic continuation of a scalar linear parabolic PDE. Examples are given to illustrate the method implemented.     

A gap theorem for hypersurfaces of the sphere with constant scalar curvature one

We consider closed hypersurfaces of the sphere with scalar curvature one, prove a gap theorem for a modified second fundamental form and determine the hypersurfaces that are at the end points of the gap. As an application we characterize the closed, two-sided index one hypersurfaces with scalar curvature one in the real projective space. Received: October 12, 2001     

Quantum mechanical model in gravity theory

We consider a model of a real massive scalar field defined as homogeneous on a d-dimensional sphere such that the sphere radius, time scale, and scalar field are related by the equations of the general theory of relativity. We quantize this system with three degrees of freedom, define the observables, and find dynamical mean values of observables in the regime where the scalar field mass is much less than the Planck mass.     

Capacity planning for phased implementation of flexible manufacturing systems under budget restrictions

We consider a problem of gradually replacing conventional dedicated machines with flexible manufacturing modules (FMMs) under budget restrictions over a finite planning horizon assuming that dedicated machines cannot be purchased during the planning horizon and acquired FMMs are kept until the end of the horizon. In the problem, a replacement schedule is to be determined and operations are to be assigned to the FMMs or the dedicated machines with the objective of minimizing the sum of discounted costs of acquisition and operation of FMMs and operation costs of conventional dedicated machines. In this research, the problem is formulated as a mixed integer linear program and solved by a Lagrangean relaxation approach. A subgradient optimization method is employed to obtain lower bounds of solutions and a multiplier adjustment method is devised to improve the lower bounds. We develop a linear programming-based Lagrangean heuristic algorithm to find a good feasible solution of the original problem in a reasonable amount of computation time. The algorithm is tested on randomly generated test problems and the results are reported.     

Real Liouville Extensions

We characterize linear differential equations defined over a real differential field with a real closed field of constants C, which are solvable by real Liouville functions, as those having a differential Galois group whose identity component is solvable and C-split.