Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation

In this paper we study the equation

in a bounded domain of , , with homogeneous boundary conditions of the Neumann type, as a model of aggregating population with a migration rate determined by , and total birth and mortality rates characterized by . We will show that the aggregating mechanism induced by allows the survival of a species in danger of extinction. Numerical simulations suggest that the solutions stabilize asymptotically in time to a not necessarily homogeneous stationary solution. This is shown to be the case for a particular version of the function .

     

Non-linear Semigroup of a Class of Abstract Semilinear Functional Differential Equations with a Non-Dense Domain

In this work, we are concerned with a general class of abstract semilinear autonomous functional differential equations with a non-dense domain on a Banach space. Our objective is to study, using the Crandall-Liggett approach, the solutions as a semigroup of non-linear operators.     

A Property of <Emphasis Type="Italic">g</Emphasis>-Expectation

This paper proves that, under the hypothesis g(t, 0, 0) ≡ 0 and some natural assumptions, the generator g of a backward stochastic differential equation can be uniquely determined by the corresponding g-expectations with all terminal conditions. The main result of this paper also confirms and extends Peng Shige‘s conjecture.     

Schrödinger Operators with Random Sparse Potentials. Existence of Wave Operators

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.The research of the author is supported in part by a Grant-in-Aid for Scientific Research (B) (No. 15340005) from the Ministry of Education, Culture, Sports, Science and Technology.Mathematics Subject classifications (2000). primary, 34M35, secondary, 33E20.This revised version was published online in March 2005 with corrections to the cover date.     

A Class of Orthogonal Polynomials Suggested by a Trigonometric Hamiltonian: Symmetric States

A new subclass of the Jacobi polynomials arising in the exact analytical solution of the one-dimensional Schrödinger equation with a trigonometric potential has been introduced. The polynomials which consist of a free parameter are not ultraspherical polynomials and have been simply named the -polynomials since they are generated by a trigonometric Hamiltonian. In certain sense, it is shown that the -polynomials can be regarded as a generalisation of the airfoil polynomials or the Chebyshev polynomials of the third kind. This paper is intended to discuss the basic properties of the polynomials so defined.     

Constructing irregular surfaces to enclose macromolecular complexes for mesoscale modeling using the discrete surface charge optimization (DISCO) algorithm

Salt-mediated electrostatics interactions play an essential role in biomolecular structures and dynamics. Because macromolecular systems modeled at atomic resolution contain thousands of solute atoms, the electrostatic computations constitute an expensive part of the force and energy calculations. Implicit solvent models are one way to simplify the model and associated calculations, but they are generally used in combination with standard atomic models for the solute. To approximate electrostatics interactions in models on the polymer level (e.g., supercoiled DNA) that are simulated over long times (e.g., milliseconds) using Brownian dynamics, Beard and Schlick have developed the DiSCO (Discrete Surface Charge Optimization) algorithm. DiSCO represents a macromolecular complex by a few hundred discrete charges on a surface enclosing the system modeled by the Debye-Hückel (screened Coulombic) approximation to the Poisson-Boltzmann equation, and treats the salt solution as continuum solvation. DiSCO can represent the nucleosome core particle (>12,000 atoms), for example, by 353 discrete surface charges distributed on the surfaces of a large disk for the nucleosome core particle and a slender cylinder for the histone tail; the charges are optimized with respect to the Poisson-Boltzmann solution for the electric field, yielding a approximately 5.5% residual. Because regular surfaces enclosing macromolecules are not sufficiently general and may be suboptimal for certain systems, we develop a general method to construct irregular models tailored to the geometry of macromolecules. We also compare charge optimization based on both the electric field and electrostatic potential refinement. Results indicate that irregular surfaces can lead to a more accurate approximation (lower residuals), and the refinement in terms of the electric field is more robust. We also show that surface smoothing for irregular models is important, that the charge optimization (by the TNPACK minimizer) is efficient and does not depend on the initial assigned values, and that the residual is acceptable when the distance to the model surface is close to, or larger than, the Debye length. We illustrate applications of DiSCO's model-building procedure to chromatin folding and supercoiled DNA bound to Hin and Fis proteins. DiSCO is generally applicable to other interesting macromolecular systems for which mesoscale models are appropriate, to yield a resolution between the all-atom representative and the polymer level.     

On Approximations of First Integrals for Strongly Nonlinear Oscillators

In this paper strongly nonlinear oscillator equations will be studied.It will be shown that the recently developed perturbation method based onintegrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how in a rather efficient way the existence and stability oftime-periodic solutions can be obtained from these approximations. In particularthe generalized Rayleigh oscillator equation will be studied in detail, and it will beshown that at least five limit cycles can occur.     

Fundamental frequency of a doubly connected membrane: a modified perturbation method

The fundamental frequency of a membrane is the square root ofthe lowest eigenvalue of the negative Laplace operator withDirichlet boundary conditions. A doubly connected membrane withthe inner region of vanishing maximal dimension 2c is consideredin this paper. A modified perturbation method is developed toprovide an asymptotic expansion (c 0) for the fundamental frequencyof the membrane. The first three order terms of the asymptoticexpansion for the fundamental frequency of a doubly connectedmembrane with the circular inner region are derived explicitly.The results are compared with the exact solutions and the approximationsdetermined by other investigators. The error of the perturbationcalculations compared with the exact values is less than 1%as c is less than or equal to 0·25 and is less than 4%as c is less than or equal to 0·35.     

FRACTIONAL DIFFUSION EQUATIONS WITH INTERNAL DEGREES OF FREEDOM

We present a generalization of the linear one-dimensional diffusion equation by com-bining the fractional derivatives and the internal degrees of freedom. The solutions areconstructed from those of the scalar fractional diffusion equation. We analyze the in-terpolation between the standard diffusion and wave equations defined by the fractionalderivatives. Our main result is that we can define a diffusion process depending on theinternal degrees of freedom associated to the system.     

On Integral Graphs Which Belong to the Class

Let G be a simple graph and let ¯G denote its complement. A graph G is said to be integral if its spectrum contains integral values. If is integral with >1 and a>b, where mG denotes the m-fold union of the graph G, we show that it belongs to the class of integral graphs where (i) (kn ² s ²+km nst+ms,km ² t ²+kmnst+ms)= such that (m,)=1 and (s,)=1; (ii) k,m,n,s,t,z such that (k,m)=1, (k,s)=1, (m,n)=1, (m,s)=1, (n,t)=1 and (s,t)=1; and (iii) ns>m t. Acknowledgments.The author is very grateful to the referees for their valuable comments and suggestions concerning this paper.1991 Mathematics Subject Classification. 05 C 50