Some remarks on Liouville type results for quasilinear elliptic equations

For a wide class of nonlinearities satisfying

0$\space in $(0,a)$\space and $f(u)<0$\space in $(a,\infty)$ ,}\end{displaymath}">

we show that any nonnegative solution of the quasilinear equation over the entire must be a constant. Our results improve or complement some recently obtained Liouville type theorems. In particular, we completely answer a question left open by Du and Guo.

     

Hyperbolic -spheres with conical singularities, accessory parameters and Kähler metrics on

We show that the real-valued function on the moduli space of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic -sphere with conical singularities of arbitrary orders , generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kähler metrics on parameterized by the set of orders , explicitly relate accessory parameters to these metrics, and prove that the functions are their Kähler potentials.

     

Quantum Monodromy and Bohr–Sommerfeld Rules

This article is based on a talk given in Dijon for the 2000 summer school Topological and Geometrical Methods: Applications to Dynamical Systems. The standard definitions of the monodromy invariant for completely integrable classical systems are reviewed, and the link to the quantum monodromy observed in the joint spectrum of commuting operators is explained. The mathematical treatment relies on a modern and attractive version of the Bohr–Sommerfeld quantisation rules.     

Minimum Distance Bounds for s-Regular Codes

A code C F ⁿ is s-regular provided, forevery vertex x F ⁿ, if x is atdistance at most s from C then thenumber of codewords y C at distance ifrom x depends only on i and the distancefrom x to C. If denotesthe covering radius of C and C is -regular,then C is said to be completely regular. SupposeC is a code with minimum distance d,strength t as an orthogonal array, and dual degrees ^*. We prove that d 2t + 1 whenC is completely regular (with the exception of binaryrepetition codes). The same bound holds when C is(t + 1)-regular. For unrestricted codes, we show thatd s ^* + t unless C is a binary repetitioncode.     

Entire solutions of semilinear elliptic equations in and a conjecture of De Giorgi

In 1978 De Giorgi formulated the following conjecture. Let be a solution of in all of such that and 0$"> in . Is it true that all level sets of are hyperplanes, at least if ? Equivalently, does depend only on one variable? When , this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for . The question, however, remains open for . The results for and 3 apply also to the equation for a large class of nonlinearities .

     

On the Eigenstates of the Elliptic Calogero–Moser Model

It is known that the trigonometric Calogero–Sutherland model is obtained by the trigonometric limit (–1) of the elliptic Calogero–Moser model, where (1, ) is a basic period of the elliptic function. We show that for all square-integrable eigenstates and eigenvalues of the Hamiltonian of the Calogero–Sutherland model, if exp(2–1) is small enough then there exist square-integrable eigenstates and eigenvalues of the Hamiltonian of the elliptic Calogero–Moser model which converge to the ones of the Calogero–Sutherland model for the 2-particle and the coupling constant l is positive integer cases and the 3-particle and l=1 case. In other words, we justify the regular perturbation with respect to the parameter exp(2–1). With some assumptions, we show analogous results for N-particle and l is positive integer cases.     

可积开边界条件下的q形变玻色子模型

利用代数Bethe Ansatz方法在可积开边界条件下推广了q形变玻色子模型，得到可积开边界条件下此模型的哈密顿量及其本征方程.该工作可为在更小尺度下研究具有相互作用的玻色子系统提供有效的理论基础. 关键词： 代数Bethe Ansatz q形变玻色子模型')" href="#">q形变玻色子模型 开边界 可积系统     

Noninvariant zeta-function regularization in quantum Liouville theory

We consider two possible zeta-function regularization schemes of quantum Liouville theory. One refers to the Laplace–Beltrami operator covariant under conformal transformations, the other to the naive noninvariant operator. The first produces an invariant regularization which however does not give rise to a theory invariant under the full conformal group. The other is equivalent to the regularization proposed by A.B. Zamolodchikov and Al.B. Zamolodchikov and gives rise to a theory invariant under the full conformal group.     

Fractional diffusion models of option prices in markets with jumps

Most of the recent literature dealing with the modeling of financial assets assumes that the underlying dynamics of equity prices follow a jump process or a Lévy process. This is done to incorporate rare or extreme events not captured by Gaussian models. Of those financial models proposed, the most interesting include the CGMY, KoBoL and FMLS. All of these capture some of the most important characteristics of the dynamics of stock prices. In this article we show that for these particular Lévy processes, the prices of financial derivatives, such as European-style options, satisfy a fractional partial differential equation (FPDE). As an application, we use numerical techniques to price exotic options, in particular barrier options, by solving the corresponding FPDEs derived.     

A HIERARCHY OF INTEGRABLE LATTICE SOLITON EQUATIONS AND ITS INTEGRABLE SYMPLECTIC MAP

1TheLatticeSolitonHierarchyDenoteEf(n)-f(n 1),nEZ,andwritef(n)=f,f(n k)=E*f.Considerthediscrete3x3matrixspectralproblem:whereajb,carethreepotentials,Aisaconstantspectralparameter.Theadjointrepresentationequationof(1)iswhereeachentryVj=Kj(B(A),C(A),D(A))0fthe3x3matrixVisaLaurentexpansionofA:Then(2)isequivaentt0'therecursi0nrelations:\,Theaboverecursionequationscanbesolvedsuccessivelyt0deducethefollowingresults:Wedefine{Dj}bythefollowingrelation:From(3)we0btainKGj-1=JGj,Gj=(Aj,Bj,Cj…