Classifying magnetic and superfluid equilibrium states in magnets with the spin <Emphasis Type="Italic">s</Emphasis> = 1

Based on the method of quasiaverages, we classify magnetic and superfluid equilibrium states in magnets with the spin s = 1. Under certain simplifications, assumptions about the residual symmetry of degenerate states and the transformation properties of order parameter operators under transformations generated by additive integrals of motions lead to linear algebraic equations for a classification of the equilibrium means of the order parameters. We consider different cases of the magnetic SO(3) or SU(3) symmetry breaking and obtain solutions for the vector and tensor order parameters for particular forms of the parameters of the residual symmetry generators. We study the equilibriums of magnets with simultaneously broken phase and magnetic symmetries. We find solutions of the classification equations for superfluid equilibrium states and establish relations between the parameters of the residual symmetry generator that allow the thermodynamic coexistence of nonzero equilibrium means of the order parameters.     

Quasi-averages in the solution of the classification problem for equilibriums of condensed media with a spontaneously broken symmetry

We develop a statistical approach for solving the classification problem for equilibriums of degenerate condensed media. We introduce generators of unbroken and spatial symmetries of an equilibrium and use them to derive the classification equations for the order parameter. We elucidate the mechanism of the appearance of additional thermodynamic parameters characterizing both homogeneous and inhomogeneous equilibriums. We solve the classification problem for equilibriums of various liquid crystals analytically.     

Fundamental solutions, transition densities and the integration of Lie symmetries

In this paper we present some new applications of Lie symmetry analysis to problems in stochastic calculus. The major focus is on using Lie symmetries of parabolic PDEs to obtain fundamental solutions and transition densities. The method we use relies upon the fact that Lie symmetries can be integrated with respect to the group parameter. We obtain new results which show that for PDEs with nontrivial Lie symmetry algebras, the Lie symmetries naturally yield Fourier and Laplace transforms of fundamental solutions, and we derive explicit formulas for such transforms in terms of the coefficients of the PDE.     

Effective interactions in the periodic Anderson model in the regime of mixed valency with strong correlations

For the periodic Anderson model in the strong correlation regime, we construct the effective Hamiltonian H _eff up to terms of the fourth order in the parameter V/U, where V is the hybridization interaction intensity and U is the intra-atom Coulomb repulsion strength. This Hamiltonian contains interactions inducing both magnetic ordering and Cooper instability under conditions of a mixed valency of rare-earth ions. Based on numerical calculations, we obtain information about the dependences of the effective interaction parameters on the distance between crystal lattice sites. We demonstrate that realizing exchange interactions corresponds to a strongly frustrated system of localized spin moments and facilitates the suppression of the antiferromagnetic order parameter with a possible transition to the state of a quantum spin liquid. It is essential that among the terms in H _eff inducing the transition to the superconductivity phase, there are terms resulting in the d-type symmetry of the superconductivity order parameter; such a symmetry is realized in many heavy-fermion compounds. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 2, pp. 235–249, November, 2008.     

Normal form construction for nearly-integrable systems with dissipation

We consider a dissipative vector field which is represented by a nearly-integrable Hamiltonian flow to which a dissipative contribution is added. The vector field depends upon two parameters, namely the perturbing and dissipative parameters, and by a drift term. We study an ?-dimensional, time-dependent vector field, which is motivated by mathematical models in Celestial Mechanics. Assuming to start with non-resonant initial conditions, we provide the construction of the normal form up to an arbitrary order. To construct the normal form, a suitable choice of the drift parameter must be performed. The normal form allows also to provide an explicit expression of the frequency associated to the normalized coordinates. We also give an example in which we construct explicitly the normal form, we make a comparison with a numerical integration, and we determine the parameter values and the time interval of validity of the normal form.     

Statistical field theory of a nonadditive system

Based on quantum field methods, we develop a statistical theory of complex systems with nonadditive potentials. Using the Martin-Siggia-Rose method, we find the effective system Lagrangian, from which we obtain evolution equations for the most probable values of the order parameter and its fluctuation amplitudes. We show that these equations are unchanged under deformations of the statistical distribution while the probabilities of realizing different phase trajectories depend essentially on the nonadditivity parameter. We find the generating functional of a nonadditive system and establish its relation to correlation functions; we introduce a pair of additive generating functionals whose expansion terms determine the set of multipoint Green’s functions and their self-energy parts. We find equations for the generating functional of a system having an internal symmetry and constraints. In the harmonic approximation framework, we determine the partition function and moments of the order parameter depending on the nonadditivity parameter. We develop a perturbation theory that allows calculating corrections of an arbitrary order to the indicated quantities.     

New algebraic approaches to classical boundary layer problems

Classical non-steady boundary layer equations are fundamental nonlinear partial differential equations in the boundary layer theory of fluid dynamics. In this paper, we introduce various schemes with multiple parameter functions to solve these equations and obtain many families of new explicit exact solutions with multiple parameter functions. Moreover, symmetry transformations are used to simplify our arguments. The technique of moving frame is applied in the three-dimensional case in order to capture the rotational properties of the fluid. In particular, we obtain a family of solutions singular on any moving surface, which may be used to study turbulence. Many other solutions are analytic related to trigonometric and hyperbolic functions, which reflect various wave characteristics of the fluid. Our solutions may also help engineers to develop more effective algorithms to find physical numeric solutions to practical models.     

A modified quasi-boundary value method for ill-posed problems

In this paper, we study a final value problem for first order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.     

Lebesgue functions corresponding to a family of Lagrange interpolation polynomials

In this paper we obtain various explicit forms of the Lebesgue function corresponding to a family of Lagrange interpolation polynomials defined at an even number of nodes. We study these forms by using the derivatives up to the second order inclusive. We estimate exact values of Lebesgue constants for this family from below and above in terms of known parameters. In a particular case we obtain new convenient formulas for calculating these estimates.     

Positivity and time behavior of a linear reaction–diffusion system,non-local in space and time

We consider a general linear reaction–diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle and positivity of the solution and investigate its asymptotic behavior. Moreover, we give an explicit expression of the limit of the solution for large times. In order to obtain these results, we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution to a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction–diffusion system. Using this and the facts that the diffusion equation on manifolds satisfies the maximum principle and its solution converges to a easily calculated constant, we can obtain analogous properties for the original system. Copyright © 2008 John Wiley & Sons, Ltd.     

A probability measure which has Markov property

We define a probability measure which has Markov property on the unit interval, compute a higher order partial derivative of its distribution function about the parameter. As application, we obtain explicit formulas of a digital sum of the block of the binary number.     

Classification of Integrable Divergent N-Component Evolution Systems

We use a symmetry approach to solve the classification problem for integrable N-component evolution systems having the form of conservation laws. We obtain complete lists of both isotropic and anisotropic systems of this type and find auto-Bäcklund transformations with a spectral parameter for all systems.     

The Riemann problem for an isentropic ideal dusty gas flow with a magnetic field

This paper deals with Riemann problem for one-dimensional inviscid, isentropic, and perfectly conducting ideal dusty gas flow with a transverse magnetic field. The explicit expressions of elementary waves are derived in terms of the density, velocity, and transverse magnetic induction of an ideal dusty gas flow. The analytical properties of elementary wave curves and the influence of parameter on the elementary waves are discussed. A new approach is proposed to resolve the Riemann problem. By applying this approach, we obtain 10 kinds of exact solutions and their corresponding criteria.     

Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions

A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in n>1 dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors.     

An exact solution of a quasilinear Fisher equation in cylindrical coordinates

Lie symmetry method is applied to analyse Fisher equation in cylindrical coordinates. Symmetry algebra is found and symmetry invariance is used to reduce the equation to a first-order ODE. The first-order ODE is further analysed to obtain exact solution of Fisher equation in explicit form.     

An exact solution of a quasilinear Fisher equation in cylindrical coordinates

Lie symmetry method is applied to analyse Fisher equation in cylindrical coordinates. Symmetry algebra is found and symmetry invariance is used to reduce the equation to a first-order ODE. The first-order ODE is further analysed to obtain exact solution of Fisher equation in explicit form.     

Properties and numerical testing of a parallel global optimization algorithm

In the present paper, we have considered three methods with which to control the error in the homotopy analysis of elliptic differential equations and related boundary value problems, namely, control of residual errors, minimization of error functionals, and optimal homotopy selection through appropriate choice of auxiliary function H(x). After outlining the methods in general, we consider three applications. First, we apply the method of minimized residual error in order to determine optimal values of the convergence control parameter to obtain solutions exhibiting central symmetry for the Yamabe equation in three or more spatial dimensions. Secondly, we apply the method of minimizing error functionals in order to obtain optimal values of the convergnce control parameter for the homotopy analysis solutions to the Brinkman?CForchheimer equation. Finally, we carefully selected the auxiliary function H(x) in order to obtain an optimal homotopy solution for Liouville??s equation.     

An efficient computational method for moments of order statistics under progressive censoring

Thomas and Wilson (Technometrics 14 (1972) 679) developed a computational method for calculating the single and product moments of order statistics from progressively censored samples by making use of the corresponding moments of the usual order statistics. The absence of an explicit representation for the marginal and joint density function of order statistics under progressive censoring makes their method extremely tedious. By deriving the required marginal and joint density functions in explicit form, we obtain an alternative, highly efficient, method for computing the desired moments.     

An integrable nonlinear diffusion hierarchy with a two-dimensional arbitrary function

By introducing the hereditary condition to a general first order differential strong symmetry operator, we obtain general 1+1 and 2+1 dimensional integrable nonlinear diffusion hierarchies with infinitely many symmetries and Lax pairs. For a special example the infinitely many nonlocal conservation laws and some explicit and implicit exact solutions are also given.