We announce a new type of " Jacobi identity " for vertex operator algebras, incorporating values of the Riemann zeta function at negative integers. Using this we " explain " and generalize some recent work of S. Bloch's relating values of the zeta function with the commutators of certain operators and Lie algebras of differential operators.

We prove that for any one-dimensional Schrr odinger operator with potential V (x) satisfying decay condition jV (x)j C x ?3=4? ; the absolutely continuous spectrum lls the whole positive semi-axis. The description of the set in R + on which the singular part of the spectral measure might be supported is also given. Analogous results hold for Jacobi matrices.

We study modular differential equations for the basic weak Jacobi forms in one abelian variable with applications to elliptic genus of Calabi--Yau varieties. show that any $CY_3$ satisfies a equation degree respect heat operator. For $K3$ surface or $CY_5$ is $3$. prove general $CY_4$ its $5$. give examples two operator similar Kaneko--Zagier variable. find type $2$ $3$ second, third and fourth...

This paper is concern with the boundary-value problem in the Hilbert space ) ( 2 N w l generated by an infinite Jacobi matrix with a spectral parameter in the boundary condition. We proved theorems on the completeness of the system of eigenvalues and eigenvectors of operator generated by boundary value problem.

We prove that the hierarchy of higher antibrackets (aka higher Koszul brackets, aka Koszul braces) of a linear operator ∆ on a commutative superalgebra can be defined by some universal formulas involving iterated Nijenhuis–Richardson brackets having as arguments ∆ and the multiplication operators. As a byproduct, we can immediately extend higher antibrackets to noncommutative algebras in a way ...

Various infinite-dimensional versions of the Calogero–Moser operator are discussed. The related class of Jack–Laurent symmetric functions is studied. In the special case when parameter k = −1 the analogue of Jacobi–Trudy formula is given and the relation with representation theory of Lie superlagebra gl(m, n) is discussed.

Convergence to a single steady state is shown for non-negative and radially symmetric solutions to a diffusive Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions, the diffusion being the p-Laplacian operator, p ≥ 2, and the source term a power of the norm of the gradient of u. As a first step, the radially symmetric and non-increasing stationary solutions are characterized.

In this work we solve the problem of the minimization of the spectral norm of the SOR operator associated with a block two-cyclic consistently ordered matrix A ∈ C, assuming that the corresponding Jacobi matrix has eigenvalues μ ∈ [−β, β] ∪ [−ıα, ıα], with β ∈ [0, 1), α ∈ [0,+∞) and ı = √ −1. Previous results obtained by other researchers are extended.

We consider the 1D periodic Jacobi matrices. The spectrum of this operator is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) in terms of vertical slits on the quasimomentum domain . Furthermore, we obtain a priori two-sided estimates for vertical slits in terms of Jacoby matrices.

Curvature collineations of a spray manifold induced by the Lie symmetries of the underlying spray are studied. The basic observation is that the Jacobi endomorphism and the Berwald curvature are invariant under these symmetries; this implies the invariance of the further curvature data. Our main technical tool is an appropriate Lie derivative operator along the tangent bundle projection. M.S.C....