Solution of the hyperbolic mild‐slope equation using the finite volume method

A finite volume solver for the 2D depth‐integrated harmonic hyperbolic formulation of the mild‐slope equation for wave propagation is presented and discussed. The solver is implemented on unstructured triangular meshes and the solution methodology is based upon a Godunov‐type second‐order finite volume scheme, whereby the numerical fluxes are computed using Roe's flux function. The eigensystem of the mild‐slope equations is derived and used for the construction of Roe's matrix. A formulation that updates the unknown variables in time implicitly is presented, which produces a more accurate and reliable scheme than hitherto available. Boundary conditions for different types of boundaries are also derived. The agreement of the computed results with analytical results for a range of wave propagation/transformation problems is very good, and the model is found to be virtually paraxiality‐free. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical analysis for stochastic age-dependent population equations with Poisson jumps

In this paper, stochastic age-dependent population equations with Poisson jumps are considered. In general, most of stochastic age-dependent population equations with jumps do not have explicit solutions, thus numerical approximation schemes are invaluable tools for exploring their properties. The main purpose of this paper is to develop a numerical Euler scheme and show the convergence of the numerical approximation solution to the true solution.     

Quasi-Hamiltonian structure and Hojman construction

Given a smooth vector field Γ and assuming the knowledge of an infinitesimal symmetry X, Hojman [S. Hojman, The construction of a Poisson structure out of a symmetry and a conservation law of a dynamical system, J. Phys. A Math. Gen. 29 (1996) 667-674] proposed a method for finding both a Poisson tensor and a function H such that Γ is the corresponding Hamiltonian system. In this paper, we approach the problem from geometrical point of view. The geometrization leads to the clarification of several concepts and methods used in Hojman's paper. In particular, the relationship between the nonstandard Hamiltonian structure proposed by Hojman and the degenerate quasi-Hamiltonian structures introduced by Crampin and Sarlet [M. Crampin, W. Sarlet, Bi-quasi-Hamiltonian systems, J. Math. Phys. 43 (2002) 2505-2517] is unveiled in this paper. We also provide some applications of our construction.     

Estimates from Below for the Spectral Function and for the Remainder in Local Weyl’s Law

We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl’s law on manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Key ingredients of the proof include the wave equation parametrix, a pretrace formula and the Dirichlet box principle. Our results develop and extend the unpublished thesis of A. Karnaukh [Ka]. The first author was supported by NSERC, FQRNT, Alfred P. Sloan Foundation Fellowship and Dawson Fellowship. The second author was supported by NSERC and FQRNT. Received: April 2006 Revision: October 2006 Accepted: October 2006     

On solving energy-dependent partitioned eigenvalue problem by genetic algorithm: The case of real symmetric Hamiltonian matrices

An energy-dependent partitioning scheme is explored for extracting a small number of eigenvalues of a real symmetric matrix with the help of genetic algorithm. The proposed method is tested with matrices of different sizes (30 × 30 to 1000 × 1000). Comparison is made with Löwdin’s strategy for solving the problem. The relative advantages and disadvantages of the GA-based method are analyzed     

Dielectric properties of KDP-type ferroelectric crystals in the presence of external electric field

Considering external electric field as well as third- and fourth-order phonon anharmonic interaction terms in the pseudospin-lattice coupled mode (PLCM) model Hamiltonian for KDP-type ferroelectrics, expressions for field-dependent shift, width, renormalized soft mode frequency, Curie temperature, dielectric constant and dielectric loss are evaluated. For the calculation, method of statistical double-time temperature-dependent Green’s function has been used. By fitting model values of physical quantities, temperature and electric field dependences of soft mode frequency, dielectric constant and loss have been calculated which compare well with experimental results of Baumgartner [8] and Choi and Lockwood [9]. Both dielectric constant and loss decrease with electric field.     

Extension of Isometries Between the Unit Spheres of Normed Space E and C（Ω）

In this paper, we study the extension of isometries between the unit spheres of normed space E and C（Ω）. We obtain that any surjective isometry between the unit spheres of normed space E and C（Ω） can be extended to be a linear isometry on the whole space E and give an affirmative answer to the corresponding Tingley＇s problem （where Ω be a compact metric space）.     

On a generalization of Pólya’s theorem

We consider generalizations of Pólya’s theorem to the case of entire functions taking algebraic values at natural or integer points.     

Fourier Analysis of Semistable Distributions

We consider the generalized convolution powers G _α ^*u (x) of an arbitrary semistable distribution function G _α (x) of exponent α∈(0,2), and prove that for all j, k∈{0,1,2,…} and u>0 the derivatives G _α ^(k,j)(x;u)=∂ ^k+j G _α ^*u (x)/∂ x ^k ∂ u ^j , x∈ℝ, are of bounded variation on the whole real line ℝ. The proof, along with an integral recursion in j, is new even in the special case of stable laws, and the result provides a framework for possible asymptotic expansions in merge theorems from the domain of geometric partial attraction of semistable laws. An erratum to this article can be found at