Article history: Received 12 May 2009 Available online 6 October 2009 Submitted by B.S. Thomson

The Laplace-Beltrami operator for graphs has been been widely used in many machine learning issues, such as spectral clustering and transductive inference. Functions on the nodes of a graph with vanishing Laplacian are called harmonic functions. In differential geometry, the Laplace-de Rham operator generalizes the Laplace-Beltrami operator. It is a differential operator on the exterior algebra...

A method is suggested for the calculation of the DeWitt-Seeley-Gilkey (DWSG) coefficients for the operator √ −∇2 + V (x) basing on a generalization of the pseudodifferential operator technique. The lowest DWSG coefficients for the operator √ −∇2 + V (x) are calculated by using the method proposed. It is shown that the method admits a generalization to the case of operators of the type (−∇2 + V ...

The notion of a Laplace ladder for a discrete analogue of the Laplace equation is presented. The adjoint of the discrete Moutard equation and a discrete counterpart of the nonlinear form of Goursat equation are introduced. 1 Notation Value of functions of continuous variables we denote by f(u, v) i.e. f : R ∋ (u, v) 7→ f(u, v) ∈ R while functions of discrete variables by f(m1, m2) i.e. f : Z 2 ...

A'transputer-based multiprocessor system for visual communication applications is presented. H.261, the encoding/decoding standard for video telephones recommended by CCITT, is implemented in this system for demonstration. Fundamental video processing operators, such as 3 x 3 discrete Laplacian, are also implemented. Up to 20.4 frames per second (fps) video replaying rate and about 7.2 fps proc...

The migration section is the reflectivity distribution modulated by the discrete Green’s function for migration. For a poorly sampled trace distribution, the Green’s function introduces noise into the migration image that can obscure its interpretation. Using this migration Green’s function and assuming shift invariance, we propose an economical method to deconvolve the Green’s function from th...

We consider traveling wave solutions to a class of diierential-diierence equations. Our interest is in understanding propagation failure, directional dependence due to the discrete Laplacian, and the relationship between traveling wave solutions of the spatially continuous and spatially discrete limits of this equation. The diierential-diierence equations that we study include damped and undamp...

We study the properties of a discrete Schrödinger operator. We prove that if its spectrum is discrete in the complement of [−2d, 2d], then it contains [−2d, 2d].

Many asset classes, such as interest rates, exchange rates, commodities, and equities, often exhibit a strong relationship between asset prices and asset volatilities. This paper examines an analytical model that takes into account this level dependence of volatility. We demonstrate how prices of European options under stochastic volatility can be calculated analytically via inverse Laplace tra...