Separation of Variables in the Kovalevskaya–Goryachev–Chaplygin Gyrostat

We construct separation variables for the Kovalevskaya–Goryachev–Chaplygin gyrostat for arbitrary values of the parameters. We show that different separation variables can be constructed for the same integrable system if different integrals of motion are chosen.     

Functional Equations and Quantum Separation of Variables for 3d Spin Models

The recently proposed invariant formulation of the auxiliary linear problem for 3d integrable models provides several new ideas for solving the spectral problem of 3d spin models, e.g., the Zamolodchikov–Bazhanov–Baxter model in its vertex formulation. This paper announces results following from the invariant formulation. We formulate the class of 3d spin models that are essentially appropriately parameterized inhomogeneous Zamolodchikov–Bazhanov–Baxter models, present an expression for the generating function of the complete set of matrices commuting with the transfer matrix of this model (integrals of motion), give the functional equations defining the eigenvalues of the integrals of motion and the transfer matrices, explicitly describe the groupoid of isospectral transformations of the initial system of integrals of motion, and finally give an explicit parameterization of a projection operator onto the separated states in the sense of the quantum separation of variables (functional Bethe ansatz).     

Local Points of Motives in Semistable Reduction

In this paper we study – for a semistable scheme – a comparison map between its log-syntomic cohomology and the p-adic étale cohomology of its generic fiber. The image can be described in terms of what Bloch and Kato call the local points of the underlying motive. The results extend a proven conjecture of Schneider which treats the good reduction case. The proof uses the theory of logarithmic schemes, some crystalline cohomology theories defined on them and various techniques in p-adic Hodge theory, in particular the Fontaine–Jannsen conjecture proven by Kato and Tsuji as well as Fontaine's rings of p-adic periods and their properties. The comparison result may become useful with respect to cycle class maps.     

Toda Chains in the Jacobi Method

We use the Jacobi method to construct various integrable systems, such as the Stäckel systems and Toda chains, related to various root systems. We find canonical transformations that relate integrals of motion for the generalized open Toda chains of types B _n, C _n, and D _n.     

A Sato–Tate Law for Drinfeld Modules

We formulate and prove a Sato–Tate equidistribution law for Drinfeld modules.     

本征值问题的统一推导方法及讨论

针对一般情形的本征方程X″(x)-2bX′(x)+λX(x)=0结合第一、二、三类齐次边界条件的统一形式,给出有关本征值问题的统一结果,从而可直接利用分离变量法求解2U/t2=a22U/x2+a1u/x+a2t/u+a3u型等含有ux项的泛定方程的定解问题.     

Motives with Galois Group of type G2>: an Exceptional Theta-Correspondence

In this paper, we study an exceptional theta correspondence, obtained by restricting the minimal automorphic representation of the adjoint group of type E₇ and rank 3 over Q to the dual pair GxPGSp₆. Here G is the anisotropic form of G₂ over Q; using the correspondence, we lift certain automorphic forms on G to holomorphic cusp forms on PGSp₆. This lifting provides the first step in a project to construct motives of rank 7 and weight O over Q with Galois group of type G₂.     

On the orthogonality of the MacDonald's functions

A proof of an orthogonality relation for the MacDonald's functions with identical arguments but unequal complex lower indices is presented. The orthogonality is derived first via a heuristic approach based on the Mehler–Fock integral transform of the MacDonald's functions, and then proved rigorously using a polynomial approximation procedure.     

A supplement to the Davis–Gut law

Let {X,X_n;n1} be a sequence of i.i.d. real-valued random variables and set , n1. Let h() be a positive nondecreasing function such that . Define Lt=log_emax{e,t} for t0. In this note we prove that

if and only if E(X)=0 and E(X²)=1, where , t1. When h(t)≡1, this result yields what is called the Davis–Gut law. Specializing our result to h(t)=(Lt)^r, 0<r1, we obtain an analog of the Davis–Gut law.     

An inversion formula for the Segal–Bargmann transform on a symmetric space of non-compact type

We prove analogs of the heat kernel transform inversion formulae of Golse, Leichtnam and the author [E. Leichtnam, F. Golse, M. Stenzel, Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds, Ann. Sci. École Norm. Sup. (4) 29 (6) (1996) 669–736. MR1422988 (97h:58153), Theorems 0.3, 0.4] in the setting of a Riemannian symmetric space of Helgason's non-compact type.     

一类二阶非线性常微分方程零解的全局稳定性

本文用变量分离法研究了二阶非线性微分方程ｘ＋ φ（ｘ）ｐ（ｘ ）＋ ｇ（ｘ）ｆ （ｘ）＝ ０ 零解的全局稳定性     

Precise Asymptotics in Spitzer's Law of Large Numbers

Let X, X ₁, X ₂,... be a sequence of i.i.d. random variables such that EX=0, assume the distribution of X is attracted to a stable distribution with exponent <1, and set S _n=X ₁+ ··· +X _n. We prove that

The Potts model representation and a Robinson–Schensted correspondence for the partition algebra

We construct a correspondence between the set of partitions of a finite set M and the set of pairs of walks to the same vertex on a graph giving the Bratteli diagram of the partition algebra on M. This is the precise analogue of the correspondence between the set of permutations of a finite set and the set of pairs of Young tableaux of the same shape, called the Robinson–Schensted correspondence.     

Finite-Dimensional Motives and the Conjectures of Beilinson and Murre

We relate the notion of finite dimensionality of the Chow motive M(X) of a smooth projective variety X (as defined by S. Kimura) with the conjectures of Beilinson, Bloch and Murre on the existence of a filtration on the Chow ring CH*(X). We show (Theorem 3) that finite dimensionality of M(X) implies uniqueness, up to isomorphism, of Murre's decomposition of M(X). Conversely (Theorem 4), Murre's conjecture for X ^m ×X ^m (for a suitable m) implies finite-dimensionality of M(X). We also show (Theorem 7) that, for a surface X with p _g = 0, the motive M(X) is finite-dimensional if and only if the Chow group of 0-cycles of X is finite-dimensional in the sense of Mumford, i.e. iff the Bloch conjecture holds for X.The second named author is a member of GNSAGA of CNR.     

An extension of Bar-Hen's preliminary test procedure

An asymptotic test procedure, proposed by Bar-Hen (J. Multivariate Anal. 57 (1996) 266), for deciding if a given set of data represents a new population or one of k a priori known populations, is extended to the case when the new population is described by more than one parameter.     

General Solution for Hamiltonians with Extended Cubic and Quartic Potentials

In terms of hyperelliptic functions, we integrate a two-particle Hamiltonian with quartic potential and additional linear and nonpolynomial terms in the Liouville integrable cases 1:6:1 and 1:6:8.     

On the least values of -norms for the Kontorovich–Lebedev transform and its convolution

We establish analogs of the Hausdorff–Young and Riesz–Kolmogorov inequalities and the norm estimates for the Kontorovich–Lebedev transformation and the corresponding convolution. These classical inequalities are related to the norms of the Fourier convolution and the Hilbert transform in L_p spaces, 1p∞. Boundedness properties of the Kontorovich–Lebedev transform and its convolution operator are investigated. In certain cases the least values of the norm constants are evaluated. Finally, it is conjectured that the norm of the Kontorovich–Lebedev operator is equal to . It confirms, for instance, by the known Plancherel-type theorem for this transform when p=2.     

Bi-Hamiltonian systems of natural form

We propose a new method for constructing integrable systems of natural form. In this method, integrals of motion are solutions of an overdetermined system of algebraic and partial differential equations obtained from the compatibility condition for Poisson tensors polynomial in the momenta and from the condition that the bi-Lagrangian distribution corresponding to the integrals of motion is invariant under the action of the recursion operator. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 161–182, November, 2006.     

Exact and numerical solution of Black–Scholes matrix equation

In this paper a new method for solving Black–Scholes equation is proposed. The approach is based on the Mellin transform. A numerical procedure for the approximation of the solution is given.     

Nonstationary Initial Boundary Value Contact Problems of Generalized Elastothermodiffusion

The initial boundary value problems with mixed boundary conditions are considered for a system of partial differential equations of generalized electrothermodiffusion. Approximate solutions are constructed and a mathematical substantiation of the method is given.