The classification of nonsingular multidimensional Dubrovin-Novikov brackets

In this paper, the well-known Dubrovin-Novikov problem posed as long ago as in 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type, is solved. In contrast to the one-dimensional case, in the general case, a nondegenerate multidimensional Poisson bracket of hydrodynamic type cannot be reduced to a constant form by a local change of coordinates. Generally speaking, such Poisson brackets are generated by nontrivial canonical special infinite-dimensional Lie algebras. In this paper, we obtain a classification of all nonsingular nondegenerate multidimensional Poisson brackets of hydrodynamic type for any number N of components and for any dimension n by differential-geometric methods. A key role in the solution of this problem is played by the theory of compatible metrics earlier constructed by the present author.     

The Arab Spring: A simple compartmental model for the dynamics of a revolution

We introduce a simple compartmental model for the dynamics of a revolution in dictatorial regimes that employ censorship and police repression. A defining property of the model is the use of visibility and policing terms that feature rapid transitions as a function of the size of the revolution, for which we provide conceptual and network-based mathematical justifications. The complete mathematical classification of the dynamical behaviour of the model leads to a division in parameter space that is interpreted naturally in terms of stability of the regime (stable police state, meta-stable police state, unstable police state, and failed state). We show that these dynamical properties of the model are generic for a broad class of visibility and policing functions that feature rapid transitions. We investigate how the model can be applied to the peaceful revolutions of the Arab Spring in Tunisia and Egypt, taking into account the influence of the Internet and new media on the visibility of the revolution and the ensuing reduced effectivity of censorship. Within the model this leads to significant, discontinuous changes in regime stability, which greatly increase the probability of realized revolutions. These properties of the model inform possible answers to questions on causes and timing of the Arab Spring revolutions, and the role of the Internet and new media. The broader relevance of the model classification is also investigated by applying it to the current political situation in some other countries with regimes that employ censorship and police repression.     

A memetic algorithm approach for solving the multidimensional multi-way number partitioning problem

In this paper, we describe a generalization of the multidimensional two-way number partitioning problem (MDTWNPP) where a set of vectors has to be partitioned into p sets (parts) such that the sums per every coordinate should be exactly or approximately equal. We will call this generalization the multidimensional multi-way number partitioning problem (MDMWNPP). Also, an efficient memetic algorithm (MA) heuristic is developed to solve the multidimensional multi-way number partitioning problem obtained by combining a genetic algorithm (GA) with a powerful local search (LS) procedure. The performances of our memetic algorithm have been compared with the existing numerical results obtained by CPLEX based on an integer linear programming formulation of the problem. The solution reveals that our proposed methodology performs very well in terms of both quality of the solutions obtained and the computational time compared with the previous method of solving the multidimensional two-way number partitioning problem.     

Adaptive FEM–BEM coupling with a Schur complement error indicator

In this paper an a-posteriori error estimate for non-linear coupled FEM–BEM equations is derived by using the Steklov–Poincaré operator and hierarchical basis techniques. We obtain “local” error indicators which are based on two-level subspace decompositions with the additive Schwarz operator. We present an algorithm for adaptive error control which is driven by these error indicators and numerical results are included.     

An information global minimization algorithm using the local improvement technique

In this paper, the global optimization problem with a multiextremal objective function satisfying the Lipschitz condition over a hypercube is considered. An algorithm that belongs to the class of information methods introduced by R.G. Strongin is proposed. The knowledge of the Lipschitz constant is not supposed. The local tuning on the behavior of the objective function and a new technique, named the local improvement, are used in order to accelerate the search. Two methods are presented: the first one deals with the one-dimensional problems and the second with the multidimensional ones (by using Peano-type space-filling curves for reduction of the dimension of the problem). Convergence conditions for both algorithms are given. Numerical experiments executed on more than 600 functions show quite a promising performance of the new techniques.     

Generalized k-d-trees and local reorganizations

This work is devoted to the problem of organization of data structure for the realization of a multidimensional dictionary. In this paper, a generalization of k-d-trees is suggested, which admits local reorganizations in contrast to the classical version. Also, in addition to the multidimensional analog of rotations, a new type of local reorganizations is introduced. Unfortunately, in contrast to the classical one-dimensional case, reorganizations of this kind are not always applicable to a particular node; this work suggests necessary and sufficient conditions for all kinds of local reorganizations under study to be applicable to a particular node of the tree. By elementary means we prove a theorem that generalizes a well known result on binary search trees, namely, any tree of this kind may be transformed into any other tree with the same set of keys by a sequence of local reorganizations each involving only one node and its children. This kind of trees may be employed in the organization of storage of multi-dimensional data. Here, perfect analogy with the one-dimensional case is hardly feasible, because the application of the reorganizations described above to a particular node requires special processing of both of its children. However, in systems which involve a separate process for tree structure optimization concurrently with the use of the tree, the feasibility of performing an arbitrary reorganization of the tree structure with the use of a sequence of only local reorganizations seems to be a significant advantage.     

LOCAL CLASSICAL SOLUTION OF FREE BOUNDARY PROBLEM FOR A COUPLED SYSTEM

This paper considers a two-phase free boundary problem for coupled system including one parabolic equation and two elliptic equations. The problem comes from the discussion of a growth model of self-lnaintaining protocell in multidimensional case. The local classical solution of the problem with free boundary F : y = g(x,t) between two domains is being seeked. The local existence and uniqueness of the problem will be proved in multidimensional case.     

Jump type stochastic differential equations with non-Lipschitz coefficients: Non-confluence,Feller and strong Feller properties,and exponential ergodicity

This paper considers multidimensional jump type stochastic differential equations with super linear and non-Lipschitz coefficients. After establishing a sufficient condition for nonexplosion, this paper presents sufficient local non-Lipschitz conditions for pathwise uniqueness. The non-confluence property for solutions is investigated. Feller and strong Feller properties under local non-Lipschitz conditions are investigated via the coupling method. Sufficient conditions for irreducibility and exponential ergodicity are derived. As applications, this paper also studies multidimensional stochastic differential equations driven by Lévy processes and presents a Feynman–Kac formula for Lévy type operators.     

Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations

The multidimensional piston problem is a special initial-boundary value problem. The boundary conditions are given in two conical surfaces: one is the boundary of the piston, and the other is the shock whose location is to be determined later. In this paper, we are concerned with spherically symmetric piston problem for the relativistic Euler equations. A local shock front solution with the state equation p = a ² ρ, a is a constant and has been established by the Newton iteration. To overcome the difficulty caused by the free boundary, we introduce a coordinate transformation to fix it and employ the linear iteration scheme to establish a sequence of approximate solutions to the auxiliary problems by iteration. In each step, the value of the solution of the previous problem is taken as the data to determine the solution of the next problem. We obtain the existence of the original problem by establishing the convergence of these sequences. Meanwhile, we establish the convergence of the local solution as c → ∞ to the corresponding solution of the classical non-relativistic Euler equations.     

3D dynamic responses of a multi-layered transversely isotropic saturated half-space under concentrated forces and pore pressure

Dynamic Green's function plays an important role in the study of various wave radiation, scattering and soil-structure interaction problems. However, little research has been done on the response of transversely isotropic saturated layered media. In this paper, the 3D dynamic responses of a multi-layered transversely isotropic saturated half-space subjected to concentrated forces and pore pressure are investigated. First, utilizing Fourier expansion in circumferential direction accompanied by Hankel integral transform in radial direction, the wave equations for transversely isotropic saturated medium in cylindrical coordinate system are solved. Next, with the aid of the exact dynamic stiffness matrix for in-plane and out-of-plane motions, the solutions for multi-layered transversely isotropic saturated half-space under concentrated forces and pore pressure are obtained by direct stiffness method. A FORTRAN computer code is developed to achieve numerical evaluation of the proposed method, and its accuracy is validated through comparison with existing solutions that are special cases of the more general problems addressed. In addition, selected numerical results for a homogeneous and a layered material model are performed to illustrate the effects of material anisotropy, load frequency, drainage condition and layering on the dynamic responses. The presented solutions form a complete set of Green's functions for concentrated forces (including horizontal load in x(y)-direction, vertical load in z-direction) as well as pore pressure, which lays the foundation for further exploring wave propagation of complex local site in a layered transversely isotropic saturated half-space by using the BEMs.     

Minimax pointwise estimation of an anisotropic regression function with unknown density of the design

Our aim in this paper is to estimate with best possible accuracy an unknown multidimensional regression function at a given point where the design density is also unknown. To reach this goal, we will follow the minimax approach: it will be assumed that the regression function belongs to a known anisotropic Hölder space. In contrast to the parameters defining the Hölder space, the density of the observations is assumed to be unknown and will be treated as a nuisance parameter. New minimax rates are exhibited as well as local polynomial estimators which achieve these rates. As these estimators depend on a tuning parameter, the problem of its selection is also discussed.     

Homogenization and simulation for compositional flow in naturally fractured reservoirs

A dual porosity model of multidimensional, multicomponent, multiphase flow in naturally fractured reservoirs is derived by the mathematical theory of homogenization. A fully compositional model is considered where there are N chemical components, each of which may exist in any or all of the three phases: gas, oil, and water. Special attention is paid to developing a general approach to incorporating gravitational forces, pressure gradient effects, and effects of mass transfer between phases. In particular, general equations for the interactions between matrix and fracture systems are obtained under homogenization by a careful scaling of these effects. Using this dual porosity compositional model, numerical experiments are reported for the benchmark problems of the sixth comparative solution project organized by the society of petroleum engineers.     

Global existence and asymptotic behavior for a multidimensional nonisentropic hydrodynamic semiconductor model with the heat source

In this paper, a multidimensional nonisentropic hydrodynamic model for semiconductors with the nonconstant lattice temperature is studied. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. Global existence to the Cauchy problem for the multidimensional nonisentropic hydrodynamic semiconductor model with the small perturbed initial data is established, and the asymptotic behavior of these smooth solutions is investigated, namely, that the solutions converge to the general steady-state solution exponentially fast as t→+∞ is obtained. Moreover, the existence and uniqueness of the stationary solutions are investigated.     

The splitting of separatrices,the branching of solutions and non-integrability in the problem of the motion of a spherical pendulum with an oscillating suspension point

The effectiveness of the results obtained previously in [Dovbysh SA. Transversal intersection of separatrices and non-existence of an analytical integral in multidimensional systems. In: Ambrosetti A, Dell Antonio GF, editors. Variational and Local Methods in the Study of Hamiltonian Systems. Singapore, etc: World Scientific; 1995. p. 156–65; Dovbysh SA. Transversal intersection of separatrices, the structure of a set of quasi-random motions and the non-existence of an analytic integral in multidimensional systems. Uspekhi Mat Nauk 1996; 51(4): 153–54; Dovbysh SA. Transversal intersection of separatrices and branching of solutions as obstructions to the existence of an analytic integral in many-dimensional systems. I. Basic result: Separatrices of hyperbolic periodic points. Collect Math 1999; 50(2): 119–97; Dovbysh SA. Branching of the solutions in the complex domain from the point of view of symbolic dynamics and the non-integrability of multidimensional systems. Dokl Ross Akad Nauk 1998; 361(3): 303–6] on the non-integrability of multidimensional systems is illustrated using the example of the problem of the motion of a spherical pendulum with a suspension point performing small periodic oscillations. With this aim, the splitting of the separatrices of the unstable equilibrium position and the branching of the solutions are investigated. It is shown that the separatrices are split for any law of motion of the suspension point, and a simple criterion of the presence of their transversal intersection is obtained. The validity of the non-integrability result, based on a combination of the conditions related to the splitting of multidimensional separatrices and to the branching of the solutions, is also pointed out.     

Demilitarized zone planning: A multiobjective approach

The demilitarized zone has been used frequently as a means to separate conflicting nations. Imposition of a buffer zone free from military activities is considered to be an effective first step to restore peace between hostile nations. The demilitarized zone is usually patrolled by third-party forces to discourage military hostilities. This paper deals with the problem of allocating the third-party supervisory forces in order to assure effective policing of demilitarized zones where there are multiple and conflicting objectives.     

Local a-posteriori error indicators for the Galerkin discretization of boundary integral equations

In this paper we present local a-posteriori error indicators for the Galerkin discretization of boundary integral equations. These error indicators are introduced and investigated by Babuška-Rheinboldt [3] for finite element methods. We transfer them from finite element methods onto boundary element methods and show that they are reliable and efficient for a wide class of integral operators under relatively weak assumptions. These local error indicators are based on the computable residual and can be used for controlling the adaptive mesh refinement. Received March 4, 1996 / Revised version received September 25, 1996     

On the algebra of multidimensional integral operators with homogeneous SO(n)-invariant kernels and weakly radially oscillating coefficients

In this paper, we study the Banach algebra B generated by multidimensional integral operators whose kernels are homogeneous functions of degree (?n) invariant with respect to the rotation group SO(n) and by the operators of multiplication by radial weakly oscillating functions. A symbolic calculus is developed for the algebra 25. The Fredholm property and the formula for calculating the index are described in terms of this calculus.     

Improving police control rooms using simulation

Police command and control centres are the main point of contact for the public who require help. Like other areas of UK public services, police forces are set targets for their performance. Some of these targets relate to the speed at which they respond to calls for assistance from the public. In this paper we share our experience in improving the performance of command and control centres of a UK Police Force; a project which started as a classical simulation exercise and ended up with a significant reorganization in a UK Police Force.     

Admissible Functions for the Positive Octant

In this paper, we give examples of admissible functions for the positive octant, which are multidimensional generalizations of regularly varying functions of a single variable introduced by J. Karamata in 1930. For an arbitrary closed convex acute solid n-dimensional cone, admissible functions were introduced by Yu. N. Drozhzhinov and B. I. Zav'yalov in 1984 in connection with applications to Tauberian theory and mathematical physics. Results in the asymptotics of multidimensional infinitely divisible distribution laws at infinity were obtained by the author in 2003 by applying admissible functions of the cone.