Efficient application of nonlinear stationary operators in adaptive wavelet methods—the isotropic case

This paper deals with the efficient application of nonlinear operators in wavelet coordinates using a representation based on local polynomials. In the framework of adaptive wavelet methods for solving, e.g., PDEs or eigenvalue problems, one has to apply the operator to a vector on a target wavelet index set. The central task is to apply the operator as fast as possible in order to obtain an efficient overall scheme. This work presents a new approach of dealing with this problem. The basic ideas together with an implementation for a specific PDE on an L-shaped domain were presented firstly in [38]. Considering the approximation of a function based on wavelets consisting of piecewise polynomials, e.g., spline wavelets, one can represent each wavelet using local polynomials on cells of the underlying domain. Because of the multilevel structure of the wavelet spaces, the generated polynomial usually consists of many overlapping pieces living on different spatial levels. Since nonlinear operators, by definition, cannot generally be applied to a linear decomposition exactly, a locally unique representation is sought. The application of the operator to these polynomials now has a simple structure due to the locality of the polynomials and many operators can be applied exactly to the local polynomials. From these results, the values of the target wavelet index set can be reconstructed. It is shown that all these steps can be applied in optimal linear complexity. The purpose of the presented paper is to provide a self-consistent development of this operator application independent of the particular PDE, operator, underlying domain, types of wavelets, or space dimension, thereby extending and modifying the previous ideas from [38].     

Combined phase I—phase II methods of feasible directions

This paper presents several new algorithms, generalizing feasible directions algorithms, for the nonlinear programming problem, min{f ⁰ (z) f ^j (z) 0,j = 1, 2, ,m}. These new algorithms do not require an initial feasible point. They automatically combine the operations of initialization (phase I) and optimization (phase II).Research sponsored by the National Science Foundation (RANN) Grant ENV76-04264 and the National Science Foundation Grant ENG73-08214-A01.     

On four-sheeted polynomial mappings of ℂ2. I. The case of an irreducible ramification curve

The paper is devoted to the Jacobian Conjecture: a polynomial mappingf²² with a constant nonzero Jacobian is polynomially invertible. The main result of the paper is as follows. There is no four-sheeted polynomial mapping whose Jacobian is a nonzero constant such that after the resolution of the indeterminacy points at infinity there is only one added curve whose image is not a point and does not belong to infinity.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 847–862, December, 1998.The authors are grateful to A. G. Vitushkin and P. Cassou-Nogues for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01218. The work of the second author was done under the financial support of DGICYT (Spain), grant No. SAB95-0502.     

The Darling-Erdős theorem for sums of I.I.D. random variables

Summary We obtain a general Darling-Erds type theorem for the maximum of appropriately normalized sums of i.i.d. mean zero r.v.'s with finite variances. We infer that the Darling-Erds theorem holds in its classical formulation if and only ifE[X ² 1 {|X|t}]=o((loglogt)^-1) ast. Our method is based on an extension of the truncation techniques of Feller (1946) to non-symmetric r.v.'s. As a by-product we are able to reprove fundamental results of Feller (1946) dealing with lower and upper classes in the Hartman-Wintner LIL.     

Numerical aspects of the XFEM — The XFEM p–Version

The XFEM is known to approximate the displacements and stresses around a crack tip in a very efficient way. But as we will present in this paper we have to deal with a phenomenon coming along with this method that compels us to use higher order shape functions for those elements that are enriched by the crack tip functions. For the computation of the stress–intensity–factors we are using a J–integral over a circular domain Ω. The accuracy of the results depend on • the radius of Ω • the number of elements used in the XFEM computation • the number of nodes which were enriched by the crack tip functions (number of layers) and • the shape functions which were used for the standard FE term For more information about the XFEM we refer to [1]. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – The stationary case

The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schrödinger equations, mainly the compactness of the support and its spatial localization. This question touches the very foundations underlying the derivation of the Schrödinger equation, since it is well-known a solution of a linear Schrödinger equation perturbed by a regular potential never vanishes on a set of positive measure. A fact, which reflects the impossibility of locating the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution is a compact set, and so any estimate on its spatial localization implies very rich information on places not accessible by the particle. Our results are obtained by the application of certain energy methods which connect the compactness of the support with the local vanishing of a suitable “energy function” which satisfies a nonlinear differential inequality with an exponent less than one. The results improve and extend a previous short presentation by the authors published in 2006.     

The structure of the Mitchell order—I

We isolate here a wide class of well-founded orders called tame orders, and show that each such order of cardinality at most κ can be realized as the Mitchell order on a measurable cardinal κ, from a consistency assumption weaker than o(κ) = κ⁺.     

The effect of suspended particles on Rayleigh-Bénard convection I. A nonlinear stability analysis of a thermal equilibrium model

The weakly nonlinear stability of the pure conduction solution for an appropriate aerosol one-layer Rayleigh-Bénard model of a Boussinesq particle-gas system in thermal equilibrium which retains both the particle and collision pressures is investigated. The main result of this analysis is in qualitative accord with the dominant but heretofore anomalous characteristic of columnar instabilities observed in smoke-air mixtures: namely, that lowering the threshold temperature gradient associated with the occurrence of the supercritically equilibrated rolls predicted for a clean gas leads to reduction increasing with decreasing layer depth which becomes quite severe in the case of very thin layers.     

Long—time behaviour of strong solutions of retarded nonlinear P.D.E.s

We prove the existence of strong solutions for a class of retarded partial differential equations of second order with respect to the time variable, and study the long-time behaviour of these solutions. We prove the existence of a global finite-dimensional attractor when the parameters of the system range over a “large” domain and investigate the dependence of the attractor on these parameters. MOS subject classification: 58F39, 58F12, 35B40, 73K70.     

Numerical method for finding 3D solitons of the nonlinear Schrödinger equation in the axially symmetric case

A system of two nonlinear Schrödinger equations is considered that governs the frequency doubling of femtosecond pulses propagating in an axially symmetric medium with quadratic and cubic nonlinearity. A numerical method is proposed to find soliton solutions of the problem, which is previously reformulated as an eigenvalue problem. The practically important special case of a single Schrödinger equation is discussed. Since three-dimensional solitons in the case of cubic nonlinearity are unstable with respect to small perturbations in their shape, a stabilization method is proposed based on weak modulations of the cubic nonlinearity coefficient and variations in the length of the focalizing layers. It should be emphasized that, according to the literature, stabilization was previously achieved by alternating layers with oppositely signed nonlinearities or by using nonlinear layers with strongly varying nonlinearities (of the same sign). In the case under study, it is shown that weak modulation leads to an increase in the length of the medium by more than 4 times without light wave collapse. To find the eigenfunctions and eigenvalues of the nonlinear problem, an efficient iterative process is constructed that produces three-dimensional solitons on large grids.     

The genuinely nonlinear Graetz—Nusselt ultraparabolic equation

We study a second-order quasilinear ultraparabolic equation whose matrix of the coefficients of the second derivatives is nonnegative, depends on the time and spatial variables, and can change rank in the case when it is diagonal and the coefficients of the first derivatives can be discontinuous. We prove that if the equation is a priori known to enjoy the maximum principle and satisfies the additional “genuine nonlinearity” condition then the Cauchy problem with arbitrary bounded initial data has at least one entropy solution and every uniformly bounded set of entropy solutions is relatively compact in L _loc ¹ . The proofs are based on introduction and systematic study of the kinetic formulation of the equation in question and application of the modification of the Tartar H-measures proposed by E. Yu. Panov.     

Rigidity of measures—The high entropy case and non-commuting foliations

We consider invariant measures for partially hyperbolic, semisimple, higher rank actions on homogeneous spaces defined by products of real andp-adic Lie groups. In this paper we generalize our earlier work to establish measure rigidity in the high entropy case in that setting. We avoid any additional ergodicity-type assumptions but rely on, and extend the theory of conditional measures. To Hillel Furstenberg with friendship and admiration Manfred Einsiedler is partially supported by the NSF Grant DMS 0400587. Anatole Katok is partially supported by the NSF Grant DMS 0071339.     

The use of integral methods in solving partial differential equations — I. drying of very moist soil

An integral method technique is used in solving the governing partial differential equation which models the processes of drying very moist soil. The solution is in the form of a series of convenient closed-form equations.     

Numerical approximation of solution for the coupled nonlinear Schrödinger equations

In this article, a compact finite difference scheme for the coupled nonlinear Schrödinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O(τ² + h⁴) are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis.