Universally coupled massive gravity

We derive Einstein’s equations from a linear theory in flat space-time using free-field gauge invariance and universal coupling. The gravitational potential can be either covariant or contravariant and of almost any density weight. We adapt these results to yield universally coupled massive variants of Einstein’s equations, yielding two one-parameter families of distinct theories with spin 2 and spin 0. The Freund-Maheshwari-Schonberg theory is therefore not the unique universally coupled massive generalization of Einstein’s theory, although it is privileged in some respects. The theories we derive are a subset of those found by Ogievetsky and Polubarinov by other means. The question of positive energy, which continues to be discussed, might be addressed numerically in spherical symmetry. We briefly comment on the issue of causality with two observable metrics and the need for gauge freedom and address some criticisms by Padmanabhan of field derivations of Einstein-like equations along the way. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 311–336, May, 2007.     

Derivation of equations of seismic and acoustic wave propagation and equations of filtration via homogenization of periodic structures

A linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve nonsmooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As the results for different ratios between physical parameters, we derive Biot’s equations of poroelasticity, a system consisting of nonisotropic Lamé’s equations for the solid component and acoustic equations for the liquid component, nonisotropic Lamé’s equations or equations of viscoelasticity for one-velocity continuum, decoupled system consisting of Darcy’s system of filtration or acoustic equations for the liquid component (first approximation) and nonisotropic Lamé’s equations for the solid component (second approximation), a system consisting of nonisotropic Stokes equations for the liquid component and acoustic equations for the solid component, nonisotropic Stokes equations for one-velocity continuum, or, finally a different type of acoustic equations for one- or two-velocity continuum. The proofs are based on Nguetseng’s two-scale convergence method of homogenization in periodic structures.     

High-order accurate monotone difference schemes for solving gasdynamic problems by Godunov’s method with antidiffusion

An approach to the construction of high-order accurate monotone difference schemes for solving gasdynamic problems by Godunov’s method with antidiffusion is proposed. Godunov’s theorem on monotone schemes is used to construct a new antidiffusion flux limiter in high-order accurate difference schemes as applied to linear advection equations with constant coefficients. The efficiency of the approach is demonstrated by solving linear advection equations with constant coefficients and one-dimensional gasdynamic equations.     

Positivity conditions and bounds for Green’s functions for higher order two-point BVP

We consider a linear nonautonomous higher order ordinary differential equation and establish the positivity conditions and two-sided bounds for Green’s function for the two-point boundary value problem. Applications of the obtained results to nonlinear equations are also discussed.     

Generalized McCormick relaxations

Convex and concave relaxations are used extensively in global optimization algorithms. Among the various techniques available for generating relaxations of a given function, McCormick’s relaxations are attractive due to the recursive nature of their definition, which affords wide applicability and easy implementation computationally. Furthermore, these relaxations are typically stronger than those resulting from convexification or linearization procedures. This article leverages the recursive nature of McCormick’s relaxations to define a generalized form which both affords a new framework within which to analyze the properties of McCormick’s relaxations, and extends the applicability of McCormick’s technique to challenging open problems in global optimization. Specifically, relaxations of the parametric solutions of ordinary differential equations are considered in detail, and prospects for relaxations of the parametric solutions of nonlinear algebraic equations are discussed. For the case of ODEs, a complete computational procedure for evaluating convex and concave relaxations of the parametric solutions is described. Through McCormick’s composition rule, these relaxations may be used to construct relaxations for very general optimal control problems.     

Algebraic independence of the values ofE-functions at singular points and Siegel’s conjecture

We prove a generalization of Shidlovskii’s theorem on the algebraic independence of the values ofE-functions satisfying a system of linear differential equations that is well known in the theory of transcendental numbers. We consider the case in which the values ofE-functions are taken at singular points of these systems. Using the obtained results, we prove Siegel’s conjecture that, for the case of first-order differential equations, anyE-function satisfying a linear differential equation is representable as a polynomial in hypergeometricE-functions. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 174–190, February, 2000.     

On positive solutions for a nonlinear boundary value problem with impulse

In this paper we study nonlinear second order differential equations subject to separated linear boundary conditions and to linear impulse conditions. Sign properties of an associated Green’s function are investigated and existence results for positive solutions of the nonlinear boundary value problem with impulse are established. Upper and lower bounds for positive solutions are also given.     

Nonholonomic Mechanical Systems and Kaluza–Klein Theory

In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection and adapted frame fields, we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affine constraints. In the second part of the paper, we show that the Kaluza–Klein theory is best presented and explained by using the framework of nonholonomic mechanical systems. We show that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projections on the spacetime are the geodesics from general relativity. Any other geodesic of the Kaluza–Klein space that is not tangent to the electromagnetic distribution is also a solution of Lagrange–d’Alembert equations, but for affine constraints. In particular, some of these geodesics project exactly on the solutions of the Lorentz force equations of the spacetime.     

The Hermite–Krichever Ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite-gap potentials

Several results including integral representation of solutions and Hermite– Krichever Ansatz on Heun’s equation are generalized to a certain class of Fuchsian differential equations, and they are applied to equations which are related with physics. We investigate linear differential equations that produce Painlevé equation by monodromy preserving deformation and obtain solutions of the sixth Painlevé equation which include Hitchin’s solution. The relationship with finite-gap potential is also discussed. We find new finite-gap potentials. Namely, we show that the potential which is written as the sum of the Treibich–Verdier potential and additional apparent singularities of exponents − 1 and 2 is finite-gap, which extends the result obtained previously by Treibich. We also investigate the eigenfunctions and their monodromy of the Schr?dinger operator on our potential.     

On estimates for integral solutions of linear inequalities

Recently, Bombieri and Vaaler obtained an interesting adelic formulation of the first and the second theorems of Minkowski in the Geometry of Numbers and derived an effective formulation of the well-known “Siegel’s lemma” on the size of integral solutions of linear equations. In a similar context involving linearinequalities, this paper is concerned with an analogue of a theorem of Khintchine on integral solutions for inequalities arising from systems of linear forms and also with an analogue of a Kronecker-type theorem with regard to euclidean frames of integral vectors. The proof of the former theorem invokes Bombieri-Vaaler’s adelic formulation of Minkowski’s theorem.     

An efficient algorithm for solving rank one perturbed linear Diophantine systems using Rosser��s approach

Recently, we described a generalization of Rosser’s algorithm for a single linear Diophantine equation to an algorithm for solving systems of linear Diophantine equations. Here, we make use of the new formulation to present a new algorithm for solving rank one perturbed linear Diophantine systems, based on using Rosser’s approach. Finally, we compare the efficiency and effectiveness of our proposed algorithm with the algorithm proposed by Amini and Mahdavi-Amiri (Optim Methods Softw 21:819–831, 2006).     

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

We show that the Belavin-Polyakov-Zamolodchikov equation of the minimal model of conformal field theory with the central charge c = 1 for the Virasoro algebra is contained in a system of linear equations that generates the Schlesinger system with 2×2 tmatrices. This generalizes Suleimanov’s result on the Painlevé equations. We consider the properties of the solutions, which are expressible in terms of the Riemann theta function.     

A new integral representation for quasi-periodic scattering problems in two dimensions

Boundary integral equations are an important class of methods for acoustic and electromagnetic scattering from periodic arrays of obstacles. For piecewise homogeneous materials, they discretize the interface alone and can achieve high order accuracy in complicated geometries. They also satisfy the radiation condition for the scattered field, avoiding the need for artificial boundary conditions on a truncated computational domain. By using the quasi-periodic Green’s function, appropriate boundary conditions are automatically satisfied on the boundary of the unit cell. There are two drawbacks to this approach: (i) the quasi-periodic Green’s function diverges for parameter families known as Wood’s anomalies, even though the scattering problem remains well-posed, and (ii) the lattice sum representation of the quasi-periodic Green’s function converges in a disc, becoming unwieldy when obstacles have high aspect ratio.     

Efficient representations of Green’s functions for some elliptic equations with piecewise-constant coefficients

Convenient for immediate computer implementation equivalents of Green’s functions are obtained for boundary-contact value problems posed for two-dimensional Laplace and Klein-Gordon equations on some regions filled in with piecewise homogeneous isotropic conductive materials. Dirichlet, Neumann and Robin conditions are allowed on the outer boundary of a simply-connected region, while conditions of ideal contact are assumed on interface lines. The objective in this study is to widen the range of effective applicability for the Green’s function version of the boundary integral equation method making the latter usable for equations with piecewise-constant coefficients.     

NEW PERIODIC SOLUTIONS OF ITO＇S 5th-ORDER mKdV EQUATION AND ITO＇S 7th-ORDER mKdV EQUATION

Based on the modified Jocobi elliptic function expansion method and the modified extended tanh-function method, a new algebraic method is presented to obtain multiple travelling wave solutions for nonlinear wave equations. By using the method ,Ito‘s 5th-order and 7th-order mKdV equations are studied in detail and more new exact Jocobi elliptic function periodic solutions are found. With modulus m→1 or m→0, these solutions degenerate into corresponding solitary wave solutions, shock wave solutions and trigonometric function solutions.     

New variants of Jarratt’s method with sixth-order convergence

In this paper, by using the two-variable Taylor expansion formula, we introduce some new variants of Jarratt’s method with sixth-order convergence for solving univariate nonlinear equations. The proposed methods contain some recent improvements of Jarratt’s method. Furthermore, a new variant of Jarratt’s method with sixth-order convergence for solving systems of nonlinear equations is proposed only with an additional evaluation for the involved function, and not requiring the computation of new inverse. Numerical comparisons are made to show the performance of the presented methods.     

A generalization of Clausen’s identity

The paper gives an extension of Clausen’s identity to the square of any Gauss hypergeometric function. Accordingly, solutions of the related third-order linear differential equation are found in terms of certain bivariate series that can reduce to ₃F₂ series similar to those in Clausen’s identity. The general contiguous variation of Clausen’s identity is found as well. The related Chaundy’s identity is generalized without any restriction on the parameters of the Gauss hypergeometric function. The special case of dihedral Gauss hypergeometric functions is underscored.     

On the Finite Termination of an Entropy Function Based Non-Interior Continuation Method for Vertical Linear Complementarity Problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that under some milder than usual assumptions the proposed algorithm finds an exact solution of VLCP in a finite number of iterations. Some computational results are included to illustrate the potential of this approach. This author’s work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10271002 and 10401038). This author’s work was partially supported by the Scientific Research Foundation of Tianjin University for the Returned Overseas Chinese Scholars and the Scientific Research Foundation of Liu Hui Center for Applied Mathematics, Nankai University-Tianjin University.     

Taylor series method for dynamical systems with control: Convergence and error estimates

To optimize a complicated function constructed from a solution of a system of ordinary differential equations (ODEs), it is very important to be able to approximate a solution of a system of ODEs very precisely. The precision delivered by the standard Runge-Kutta methods often is insufficient, resulting in a “noisy function” to optimize. We consider an initial-value problem for a system of ordinary differential equations having polynomial right-hand sides with respect to all dependent variables. First we show how to reduce a wide class of ODEs to such polynomial systems. Using the estimates for the Taylor series method, we construct a new “aggregative” Taylor series method and derive guaranteed a priori step-size and error estimates for Runge-Kutta methods of order r. Then we compare the 8,13-Prince-Dormand’s, Taylor series, and aggregative Taylor series methods using seven benchmark systems of equations, including van der Pol’s equations, the “brusselator,” equations of Jacobi’s elliptic functions, and linear and nonlinear stiff systems of equations. The numerical experiments show that the Taylor series method achieves the best precision, while the aggregative Taylor series method achieves the best computational time. The final section of this paper is devoted to a comparative study of the above numerical integration methods for systems of ODEs describing the optimal flight of a spacecraft from the Earth to the Moon. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.