A nonconforming finite element method for a two-dimensional curl–curl and grad-div problem

A numerical method for a two-dimensional curl–curl and grad-div problem is studied in this paper. It is based on a discretization using weakly continuous P ₁ vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive ) in both the energy norm and the L ₂ norm are established on graded meshes. The theoretical results are confirmed by numerical experiments. The work of the first author was supported in part by the National Science Foundation under Grant No. DMS-03-11790 and by the Humboldt Foundation through her Humboldt Research Award. The work of the third author was supported in part by the National Science Foundation under Grant No. DMS-06-52481.     

Stair Laguerre pseudospectral method for differential equations on the half line

A stair Laguerre pseudospectral method is proposed for numerical solutions of differential equations on the half line. Some approximation results are established. A stair Laguerre pseudospcetral scheme is constructed for a model problem. The convergence is proved. The numerical results show that this new method provides much more accurate numerical results than the standard Laguerre spectral method. Dedicated to Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 65N35, 41A10. Li-lian Wang: The work of this author is partially supported by The Shanghai Natural Science Foundation N. 00JC14057, The Shanghai Natural Science Foundation for Youth N. 01QN85 and The Special Funds for Major Specialities of Shanghai Education Committee. Ben-yu Guo: The work of this author is partially supported by The Special Funds for Major State Basic Research Projects of China G1999032804, The Shanghai Natural Science Foundation N. 00JC14057 and The Special Funds for Major Specialities of Shanghai Education Committee.     

Mean convergence of hermite interpolation revisited

The first author is supported by the Hungarian National Foundation for Scientific Research Grant No. 1910 and No. T7570, and the second author is supported by the National Science Foundation Grant No. 9302721. The work was done during the first author's visit in Eugene, Oregon in 1993, and was completed during the second author's visit to the Mathematisches Institut, University of Erlange-Nürnberg, supported by the Alexander von Humboldt Foundation.     

Asymptotic expansion and extrapolation for the Eigenvalue approximation of the biharmonic eigenvalue problem by Ciarlet–Raviart scheme

We propose and analyze the Ciarlet–Raviart mixed scheme for solving the biharmonic eigenvalue problem with bilinear finite elements. We derive a higher order convergence rate for eigenvalue and eigenfunction approximations. Furthermore, we give an asymptotic expansion of the eigenvalue error, from which an efficient extrapolation and an a posteriori error estimate for the eigenvalue are given. Finally, numerical experiments illustrating the theoretical results are reported. This author was supported by China Postdoctoral Sciences Foundation.     

On the method of complementary functions for nonlinear boundary-value problems

We show that, like the method of adjoints, the method of complementary functions can be effectively used to solve nonlinear boundary-value problems.This work was supported by the Alexander von Humboldt Foundation. The author is thankful to Prof. G. Hämmerlin for providing the facilities and to Miss J. Gumberger for performing numerical tests. The author is also indebted to Dr. S. M. Roberts for his suggestions on the first draft of this paper.     

On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces

In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.     

Remarks on the Regularity to 3-D Ideal Magnetohydrodynamic Equations

In this paper we are interested in the sufficient conditions which guarantee the regularity of solutions of 3-D ideal magnetohydrodynamic equations in the arbitrary time interval [0,T]. Five sufficient conditions are given. Our results are motivated by two main ideas: one is to control the accumulation of vorticity alone; the other is to generalize the corresponding geometric conditions of 3-D Euler equations to 3-D ideal magnetohydrodynamic equations.     

Strong Normalizability of Typed Lambda-Calculi for Substructural Logics

The strong normalization theorem is uniformly proved for typed λ-calculi for a wide range of substructural logics with or without strong negation. We would like to thank the referees for their valuable comments and suggestions. This research was supported by the Alexander von Humboldt Foundation. The second author is grateful to the Foundation for providing excellent working conditions and generous support of this research. This work was also supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Young Scientists (B) 20700015, 2008.     

Discrete weighted residual methods: A sensitive nonlinear boundary-value problem

The foundations, applications, and convergence properties of discrete weighted residual methods (DWRM's) are presented in Refs. 1–3. This paper serves to illustrate DWRM's for solving a sensitive nonlinear discrete boundary-value problem. The results indicate that DWRM's can be applied to provide models of increasing complexity which can then be utilized for the analysis and design of physical systems.The work of the first author was supported in part by the National Science Foundation, Grant No. GJ-1075. The work of the second author was supported by a Hertz Foundation Fellowship.     

On the distribution of imaginary parts of zeros of the Riemann zeta function,II

We continue our investigation of the distribution of the fractional parts of αγ, where α is a fixed non-zero real number and γ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to Montgomery’s pair correlation function and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function. The first author is supported by National Science Foundation Grant DMS-0555367. The second author is partially supported by the National Science Foundation and the American Institute of Mathematics (AIM). The third author is supported by National Science Foundation Grant DMS-0456615.     

Estimation of the effect of numerical integration in finite element eigenvalue approximation

Summary Finite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration. The effect of this numerical integration on the eigenvalue and eigenfunction error is estimated. Specifically, for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well-known estimates for approximations without quadrature, provided the quadrature rules have appropriate degrees of precision.The work of this author was partially supported by the National Science Foundation under Grant DMS-84-10324     

On critical orientations in the kedem-sharir motion planning algorithm

We discuss a technical problem arising in the motion planning algorithm of Kedem and Sharir [KS], and propose a way to overcome it without increasing the asymptotic complexity of the algorithm The first author was supported by the Eshkol Grant 04601-90 from the Israeli Ministry of Science and Technology. The second author was partly supported by the Fund for Basic Research administered by the Israeli Academy of Sciences, by National Science Foundation Grants CCR-91-22103 and CCR-93-11127, and by grants from the U.S.-Israeli Binational Science Foundation, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. The third author was partly supported by the Interdisciplinary Program at Tel-Aviv University. The third author’s current address is: Department of Computer Science, MIT, Boston, MA, USA.     

A splitting method for stochastic programs

This paper derives a new splitting-based decomposition algorithm for convex stochastic programs. It combines certain attractive features of the progressive hedging algorithm of Rockafellar and Wets, the dynamic splitting algorithm of Salinger and Rockafellar and an algorithm of Korf. We give two derivations of our algorithm. The first one is very simple, and the second one yields a preconditioner that resulted in a considerable speed-up in our numerical tests. The first author was supported by the Finnish Foundation for Economic Education under grants 20728 and 21599, and by Jenny ja Antti Wihuri Foundation.     

A likelihood-MPEC approach to target classification

In this paper we develop a method for classifying an unknown data vector as belonging to one of several classes. This method is based on the statistical methods of maximum likehood and borrowed strength estimation. We develop an MPEC procedure (for Mathematical Program with Equilibrium Constraints) for the classification of a multi-dimensional observation, using a finite set of observed training data as the inputs to a bilevel optimization problem. We present a penalty interior point method for solving the resulting MPEC and report numerical results for a multispectral minefield classification application. Related approaches based on conventional maximum likehood estimation and a bivariate normal mixture model, as well as alternative surrogate classification objective functions, are described. Received: October 26, 1998 / Accepted: June 11, 2001?Published online March 24, 2003 RID="***" ID="***"The authors of this work were all partially supported by the Wright Patterson Air Force Base via Veda Contract F33615-94-D-1400. The first and third author were also supported by the National Science Foundation under grant DMS-9705220. RID="*" ID="*"The work of this author was based on research supported by the U.S. National Science Foundation under grant CCR-9624018. RID="**" ID="**"The work of this author was supported by the Office of Naval Research under grant N00014-95-1-0777.     

Variational inequalities with cone constraints

The present paper considers variational inequalities with cone and operator constraints. The characterization of solutions is given. An algorithm for numerical solution of the problem is proposed, and the convergence of the generated sequence of points is proved.This research was supported in part by the Science Foundation of the Republic of Serbia.The author thanks two anonymous referees for their helpful remarks.     

Variational approach to serial multilevel production/inventory systems

We discuss stochastic, continuous-time models of serial multilevel production/inventory systems using quasi-variational inequalities. The value functions are characterized as maximum solutions of suitable equivalent problems. Then, we show how to construct optimal centralized policies and, if possible, decentralized policies. Finally, we present some numerical examples for the deterministic case.The work of the second author was supported by the National Science Foundation under Grant No. DMS-87-02236.     

Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps

We show equivalence of several standard conditions for non-uniform hyperbolicity of complex rational functions, including the Topological Collet-Eckmann condition (TCE), Uniform Hyperbolicity on Periodic orbits, Exponential Shrinking of components of pre-images of small discs, backward Collet-Eckmann condition at one point, positivity of the infimum of Lyapunov exponents of finite invariant measures on the Julia set. The condition TCE is stated in purely topological terms, so we conclude that all these conditions are invariant under topological conjugacy.?For rational maps with one critical point in Julia set all the conditions above are equivalent to the usual Collet-Eckmann and backward Collet-Eckmann conditions. Thus the latter ones are invariant by topological conjugacy in the unicritical setting. We also prove that neither part of this stronger statement is valid in the multicritical case. Oblatum 2-IV-2002 & 2-V-2002?Published online: 6 August 2002 RID="*" ID="*"All authors are supported by the European Science Foundation program PRODYN. The first author is also supported by the Foundation for Polish Sciences and Polish KBN grant 2P03A 00917. The second author is grateful to IMPAN and KTH for hospitality and is also supported by a Polish-French governmental agreement, Fundacion Andes and a “Beca Presidente de la Republica,” Chile. The third author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.     

Hybrid Steepest Descent Methods for Zeros of Nonlinear Operators with Applications to Variational Inequalities

In this paper, the hybrid steepest descent methods are extended to develop new iterative schemes for finding the zeros of bounded, demicontinuous and φ-strongly accretive mappings in uniformly smooth Banach spaces. Two iterative schemes are proposed. Strong convergence results are established and applications to variational inequalities are given. In this research, the first author was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality (075105118). The third author was partially supported by Grant NSC 96-2628-E-110-014-MY3.     

Non-constant stationary solutions to a prey-predator model with diffusion

In this paper, a nonlinear predator reproduction and prey competition model with diffusion is discussed. Some existence and non-existence results concerning non-constant positive steady-states are presented using topological degree argument and the energy method, respectively. The first author was supported by the National Natural Science Foundation of China (No. 10771032) and the Natural Science Foundation of Jiangsu province BK2006088, and the second author was supported by National Natural Science Foundation of China (No. 10601011).     

An invariant for certain smooth manifolds

Summary Starting with Hirzebruch's A-genus we define a numerical invariantμ for certain (4k-1)-manifolds. We produce examples to show howμ can distinguish differentiable structures on certain topological manifolds. To Enrico Bompiani on his scientific Jubilee Research partially supported by the Office of Naval Rescarch under Contract 266 (57). This author has a research grant from the National Science Foundation NSF-G-13989.