The heteroclinic cycle in the model of competition betweenn species and its stability

This paper gives a sufficient condition for the existence of heteroclinic cycle in the model of competition betweenn species and a criterion for determining the stability of the heteroclinic cycle. The results given in this paper extend the results obtained by May and Leonard in and by Hofbauer and Sigmund in. A conjecture on the permanence of the model and a open problem on the stability of the heteroclinic cycle for the critical case are given at the end of this paper. This research is supported by the National Natural Science Foundation of China     

Finite and p-adic Polylogarithms

The finite nth polylogarithm li _n (z) /p(z) is defined as _k=1 ^p–1 z ^k /k ⁿ . We state and prove the following theorem. Let Li _k : _{p p} be the p-adic polylogarithms defined by Coleman. Then a certain linear combination F _n of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p ^1–n DF _n (z) reduces modulo p>n+1 to li _n–1((z)), where D is the Cathelineau operator z(1–z)d/dz and is the inverse of the p-power map. A slightly modified version of this theorem was conjectured by Kontsevich. This theorem is used by Elbaz-Vincent and Gangl to deduce functional equations of finite polylogarithms from those of complex polylogarithms.     

Some Results about Dissipativity of Kolmogorov Operators

Given a Hilbert space H with a Borel probability measure , we prove the m-dissipativity in L ¹(H, ) of a Kolmogorov operator K that is a perturbation, not necessarily of gradient type, of an Ornstein-Uhlenbeck operator.     

Nonlinear Dynamics Equation in p-Adic String Theory

We investigate nonlinear pseudodifferential equations with infinitely many derivatives. These are equations of a new class, and they originally appeared in p-adic string theory. Their investigation is of interest in mathematical physics and its applications, in particular, in string theory and cosmology. We undertake a systematic mathematical investigation of the properties of these equations and prove the main uniqueness theorem for the solution in an algebra of generalized functions. We discuss boundary problems for bounded solutions and prove the existence theorem for spatially homogeneous solutions for odd p. For even p, we prove the absence of a continuous nonnegative solution interpolating between two vacuums and indicate the possible existence of discontinuous solutions. We also consider the multidimensional equation and discuss soliton and q-brane solutions.     

<Emphasis Type="Italic">BV</Emphasis> functions and parabolic initial boundary value problems on domains

Given a uniformly elliptic second order operator on a possibly unbounded domain , let (T(t)) _t≥0 be the semigroup generated by in L ¹(Ω), under homogeneous co-normal boundary conditions on ∂Ω. We show that the limit as t → 0 of the L ¹-norm of the spatial gradient D _x T(t)u ₀ tends to the total variation of the initial datum u ₀, and in particular is finite if and only if u ₀ belongs to BV(Ω). This result is true also for weighted BV spaces. A further characterization of BV functions in terms of the short-time behaviour of (T(t)) _t≥0 is also given.      

Stability of the analytic and numerical solutions for impulsive differential equations

This paper deals with the stability analysis of the analytic and numerical solutions of impulsive differential equations. In particular, the linear equation with variable coefficients and the nonlinear equation are considered. The stability conditions of the analytic solutions of these impulsive differential equations and the numerical solutions of the θ-methods are obtained. Finally, some numerical experiments are given.     

费马q-差分微分方程整函数解的增长性研究

本文研究了费马q-差分微分方程的整函数解的相关问题.利用经典和差分的Nevanlinna理论和函数方程理论的研究方法,获得了q-差分微分方程整函数解增长性的几个结果.     

A Note on Fuzzy Relation Programming Problems with Max-Strict-t-Norm Composition

The fuzzy relation programming problem is a minimization problem with a linear objective function subject to fuzzy relation equations using certain algebraic compositions. Previously, Guu and Wu considered a fuzzy relation programming problem with max-product composition and provided a necessary condition for an optimal solution in terms of the maximum solution derived from the fuzzy relation equations. To be more precise, for an optimal solution, each of its components is either 0 or the corresponding component's value of the maximum solution. In this paper, we extend this useful property for fuzzy relation programming problem with max-strict-t-norm composition and present it as a supplemental note of our previous work.     

A Mathematica package for q-holonomic sequences and power series

We describe a Mathematica package for dealing with q-holonomic sequences and power series. The package is intended as a q-analogue of the Maple package gfun and the Mathematica package GeneratingFunctions. It provides commands for addition, multiplication, and substitution of these objects, for converting between various representations (q-differential equations, q-recurrence equations, q-shift equations), for computing sequence terms and power series coefficients, and for guessing recurrence equations given initial terms of a sequence. C. Koutschan partially supported by the Austrian Science Foundation (FWF) grants SFB F1305.     

Asymptotic behavior of the solutions of the p -Laplacian equation

The asymptotic behavior of the solutions for p-Laplacian equations as p → ∞ is studied.     

Integral formulas for closed spacelike hypersurfaces in anti-de Sitter space <Emphasis Type="Italic">H</Emphasis><Stack><Subscript>1</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript></Stack> (−1)

In this paper, we study closed k-maximal spacelike hypersurfaces M ⁿ in anti-de Sitter space H ₁ ⁿ⁺¹ (−1) with two distinct principal curvatures and give some integral formulas about these hypersurfaces. The first author was supported by Japan Society for Promotion of Science. The third author was supported by grant Proj. No. R17-2008-001-01000-0 from Korea Science & Engineering Foundation.     

B-Theory of Runge-Kutta methods for stiff volterra functional differential equations

B-stability andB-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra functional differential equations (VFDEs) are established which provide unified theoretical foundation for the study of Runge-Kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type which appear in practice.     

Rank-<Emphasis Type="Italic">k</Emphasis> solutions of hydrodynamic-type systems

We present a variant of the conditional symmetry method for obtaining rank-k solutions in terms of Riemann invariants for first-order quasilinear hyperbolic systems of PDEs in many dimensions and discuss examples of applying the proposed approach to fluid dynamics equations in n+1 dimensions in detail. We obtain several new types of algebraic, rational, and soliton-like solutions (including kinks, bumps, and multiple-wave solutions). __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 83–100, July, 2007.     

A pathwise solution for nonlinear parabolic equations with stochastic perturbations

We analyse here a semilinear stochastic partial differential equation of parabolic type where the diffusion vector fields are depending on both the unknown function and its gradient ∂ _xu with respect to the state variable, ∈ ℝⁿ. A local solution is constructed by reducing the original equation to a nonlinear parabolic one without stochastic perturbations and it is based on a finite dimensional Lie algebra generated by the given diffusion vector fields.     

Arithmetic properties of the Ramanujan function

We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisorP (τ(n)) and the number of distinct prime divisors ω (τ (n)) of τ(n) for various sequences ofn. In particular, we show thatP(τ(n)) ≥ (logn)^33/31+o(1) for infinitely many n, and

Duality for Knizhnik–Zamolodchikov and Dynamical Equations

We consider the Knizhnik–Zamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of (gl _k ,gl _n ) duality. We show that the KZ and dynamical equations naturally exchange under the duality.     

On S-Shaped Bifurcation Curves for a Class of Perturbed Semilinear Equations

By making use of bifurcation analysis and continuation method, the authors discuss the exact number of positive solutions for a class of perturbed equations. The nonlinearities concerned are the so-called convex-concave functions and their behaviors may be asymptotic sublinear or asymptotic linear. Moreover, precise global bifurcation diagrams are obtained.     

<Emphasis Type="Italic">m</Emphasis> Components-admissible solutions of systems of higher-order partial differential equations on <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Using value distribution theory and techniques in several complex variables,we investigate the problem of existence of m components-admissible solutions of a class of systems of higher-order partial differential equations in several complex variables and estimate the number of admissible components of solutions.Some related results will also be obtained.