A Semi-Lagrangian Scheme with Radial Basis Approximation for Surface Reconstruction

We propose a Semi-Lagrangian scheme coupled with Radial Basis Function interpolation for approximating a curvature-related level set model, which has been proposed by Zhao et al. in \cite{ZOMK} to reconstruct unknown surfaces from sparse, possibly noisy data sets. The main advantages of the proposed scheme are the possibility to solve the level set method on unstructured grids, as well as to concentrate the reconstruction points in the neighbourhood of the data set, with a consequent reduction of the computational effort. Moreover, the scheme is explicit. Numerical tests show the accuracy and robustness of our approach to reconstruct curves and surfaces from relatively sparse data sets.Comment: 14 pages, 26 figure

Flux form Semi-Lagrangian methods for parabolic problems

A semi-Lagrangian method for parabolic problems is proposed, that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations. A basic consistency and convergence analysis are proposed. Numerical examples validate the proposed method and display its potential for consistent semi-Lagrangian discretization of advection--diffusion and nonlinear parabolic problems

An estimate for the multiplicity of binary recurrences

In this paper we improve drastically the estimate for the multiplicity of a binary recurrence. The main contribution comes from an effective version of the Faltings' Product Theorem

A generalization of the Subspace Theorem with polynomials of higher degree

Recently, Corvaja and Zannier obtained an extension of the Subspace Theorem with arbitrary homogeneous polynomials of arbitrary degreee instead of linear forms. Their result states that the set of solutions in P^n(K) (K number field) of the inequality being considered is not Zariski dense. In our paper we prove by a different method a generalization of their result, in which the solutions are taken from an arbitrary projective variety X instead of P^n. Further, we give a quantitative version which states in a precise form that the solutions with large height lie ina finite number of proper subvarieties of X, with explicit upper bounds for the number and for the degrees of these subvarieties.Comment: 31 page

A further improvement of the quantitative Subspace Theorem

In 2002, Evertse and Schlickewei obtained a quantitative version of the so-called Absolute Parametric Subspace Theorem. This result deals with a parametrized class of twisted heights. One of the consequences of this result is a quantitative version of the Absolute Subspace Theorem, giving an explicit upper bound for the number of subspaces containing the solutions of the Diophantine inequality under consideration. In the present paper, we further improve Evertse's and Schlickewei's quantitative version of the Absolute Parametric Subspace Theorem, and deduce an improved quantitative version of the Absolute Subspace Theorem. We combine ideas from the proof of Evertse and Schlickewei (which is basically a substantial refinement of Schmidt's proof of his Subspace Theorem from 1972, with ideas from Faltings' and Wuestholz' proof of the Subspace Theorem.Comment: 93 page

Blended numerical schemes for the advection equation and conservation laws

In this paper we propose a method to couple two or more explicit numerical schemes approximating the same time-dependent PDE, aiming at creating new schemes which inherit advantages of the original ones. We consider both advection equations and nonlinear conservation laws. By coupling a macroscopic (Eulerian) scheme with a microscopic (Lagrangian) scheme, we get a new kind of multiscale numerical method

A fully semi-Lagrangian discretization for the 2D Navier--Stokes equations in the vorticity--streamfunction formulation

A numerical method for the two-dimensional, incompressible Navier--Stokes equations in vorticity--streamfunction form is proposed, which employs semi-Lagrangian discretizations for both the advection and diffusion terms, thus achieving unconditional stability without the need to solve linear systems beyond that required by the Poisson solver for the reconstruction of the streamfunction. A description of the discretization of Dirichlet boundary conditions for the semi-Lagrangian approach to diffusion terms is also presented. Numerical experiments on classical benchmarks for incompressible flow in simple geometries validate the proposed method

Supergravity and matrix theory do not disagree on multi-graviton scattering

We compare the amplitudes for the long-distance scattering of three gravitons in eleven dimensional supergravity and matrix theory at finite N. We show that the leading supergravity term arises from loop contributions to the matrix theory effective action that are not required to vanish by supersymmetry. We evaluate in detail one type of diagram---the setting sun with only massive propagators---reproducing the supergravity behavior.Comment: 10 pages, 1 eps figure, it requires JHEP.cl