The algebraic structure of discrete zero curvature equations associated with integrable couplings and application to enlarged Volterra systems

An algebraic structure of discrete zero curvature equations is established for integrable coupling systems associated with semi-direct sums of Lie algebras. As an application example of this algebraic structure, a τ-symmetry algebra for the Volterra lattice integrable couplings is engendered from this theory.     

Explicit examples of arbitrarily large analytic ergodic potentials with zero Lyapunov exponent

We give explicit examples of arbitrarily large analytic ergodic potentials for which the Schr?dinger equation has zero Lyapunov exponent for certain energies. For one of these energies there is an explicit solution. In the quasi-periodic case we prove that one can have positive Lyapunov exponent on certain regions of the spectrum and zero on other regions. We also show the existence of 1-dependent random potentials with zero Lyapunov exponent. Research partially supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), Institutional Grant 2002-2052. Received: February 2005; Accepted: May 2005     

An improved GPS method with a new pseudo-peripheral nodes finder in finite element analysis

Finding pseudo-peripheral nodes with the largest eccentricity is important in matrix bandwidth and profile reduction algorithms in finite element analysis. A heuristic parameter, called the “width-depth ratio” and denoted by κ, is presented for finding the pseudo-peripheral nodes with larger pseudo-diameter compared with the GPS (Gibbs-Poole-Stockmeyer) pseudo-peripheral nodes finder. A novel nodes renumbering algorithm is thus developed by using our nodes finder based on GPS method. Simulations show that proposed nodes finder is reliable and effective in locating the proper pseudo-peripheral nodes with larger pseudo-diameters. A shielded microstrip line is given as an example to testify the ability of the proposed algorithm in application. The results, including time, pseudo-diameter, bandwidths and profiles, all indicate that our method is more competitive than GPS algorithm to be used as the nodes renumbering algorithm.     

Convergence of Bieberbach polynomials in domains with interior cusps

We extend the results on the uniform convergence of Bieberbach polynomials for domains with certain interior zero angles (outward pointing cusps) and show that they play a special role in the problem. Namely, we construct a Keldysh-type example on the divergence of Bieberbach polynomials at an outward pointing cusp and discuss thecritical order of tangency at this interior zero angle, separating the convergent behavior of Bieberbach polynomials from the divergent one for sufficiently thin cusps. Research of both authors was supported in part by the National Science Foundation grant DMS-9707359. Research of the second author was also supported in part by the National Science Foundation grant DMS-9970659.     

Large deviations for diffusion processes in duals of nuclear spaces

Motivated by applications to neurophysiological problems, various authors have studied diffusion processes in duals of countably Hilbertian nuclear spaces governed by stochastic differential equations. In these models the diffusion coefficients describe the random stimuli received by spatially extended neurons. In this paper we present a large deviation principle for such processes when the diffusion terms tend to zero in terms of a small parameter. The lower bounds are established by making use of the Girsanov formula in abstract Wiener space. The upper bounds are obtained by Gaussian approximation of the diffusion processes and by taking advantage of the nuclear structure of the state space to pass from compact sets to closed sets.This research was partially supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620-92-J-0154 and the Army Research Office Grant No. DAAL03-92-G-0008.     

Switching between modified Newton and fix-point iteration for implicit ode-solvers

In this paper we discuss the automatic switch between modified Newton iteration with the Jacobian not equal zero and fix-point iteration, i.e. the Jacobian equal zero. The switching strategy is based on the ratio between the norm of the displacement and the norm of the residual. Several examples are discussed both from non-stiff and stiff systems. These examples show that our strategy is working as predicted.This work was supported by NATO Research Grant Programme, project RG 096.80 and by National Sciences and Engineering Research Council of Canada, Grants: A1244, A3597, A4076, A8111, A8239, A8639.     

Harnack inequalities and difference estimates for random walks with infinite range

Difference estimates and Harnack inequalities for mean zero, finite variance random walks with infinite range are considered. An example is given to show that such estimates and inequalities do not hold for all mean zero, finite variance random walks. Conditions are then given under which such results can be proved.Research supported by the National Science Foundation.     

An efficient simplicial algorithm for computing a zero of a convex union of smooth functions

We present an efficient simplicial algorithm for computing a zero of a point-to-set mapping that is formed by piecing together smooth functions. Such mappings arise in nonlinear programming and economic equilibrium problems. Our algorithm, under suitable regularity conditions on the problem, generates a sequence converging at least Q-superlinearly to a zero of the mapping. Asymptotically, it operates in a space of reduced dimension, analogous to an active set strategy in the optimization setting, but it switches active sets automatically. Results of computational experiments are given. Research of this author was supported by a Fellowship from the Rockeffeller Foundation. Research of this author was partially supported by a fellowhip from the John Simon Guggenheim Memorial Foundation and by National Science Foundation Grant ECS-7921279.     

Convergence of finite volume schemes for triangular systems of conservation laws

We consider non-strictly hyperbolic systems of conservation laws in triangular form, which arise in applications like three-phase flows in porous media. We device simple and efficient finite volume schemes of Godunov type for these systems that exploit the triangular structure. We prove that the finite volume schemes converge to weak solutions as the discretization parameters tend to zero. Some numerical examples are presented, one of which is related to flows in porous media. The research of K. H. Karlsen was supported by an Outstanding Young Investigators Award from the Research Council of Norway.     

Diffusion approximation of branching migration processes

We study the behavior of a Galton-Watson process with homogeneous migration component stopped at zero (i.e., the state zero is absorbing). Assuming that the process is initiated at time zero by a large number of particles, we find a diffusion approximation for this process in the case where the average number of offspring per individual is close to one. Supported by the Russian Foundation for Fundamental Research (grant Nos. 96-01-00338 and 96-15-96092) and INTAS-RFBR (grant No. 95-0099). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

On the divergence of Walsh-Fejér means of bounded functions on sets of measure zero

The aim of this paper is to prove that for an arbitrary set of measure zero there exists a bounded function for which the Fejér means of the Walsh-Fourier series of the function diverge. Research supported by the Georgian National Fundation for Scientific Research, Grant no. 07_225_3-100.     

Torsion sections of elliptic surfaces

Given a torsion section of a semistable elliptic surface we prove equidistribution results for the components of singular fibers which are hit by the section and for the root of unity (identifying the zero component withC) which is hit by the section in case the section hits the zero component Research supported in part by the NSF under grant DMS 9104058 and the NSA under grant MDA 904-92 H 3022     

Zero energy asymptotics of the resolvent for a class of slowly decaying potentials

We prove a limiting absorption principle at zero energy for two-body Schrödinger operators with long-range potentials having a positive virial at infinity. More precisely, we establish a complete asymptotic expansion of the resolvent in weighted spaces when the spectral parameter tends to zero in cones which are adjacent to the positive real axis. The principal tools are absence of eigenvalue at zero, singular Mourre theory and microlocal estimates.in final form: 14 November 2003S. Fournais was supported by a grant from the Carlsberg Foundation (before 31.12.02) and by a Marie Curie Fellowship of the European Community Programme Improving the Human Research Potential and the Socio-Economic Knowledge Base under contract number HPMF-CT-2002-01822 (from 01.01.03).E. Skibsted is (partially) supported by MaPhySto – A Network in Mathematical Physics and Stochastics funded by The Danish National Research Foundation.     

Finiteness conditions for finitely generated monoids

In this paper, we show that every finitely generated monoid, with a disjunctive zero 0 such that abc=0 implies cba=0, is finite. This research has been supported by Grant A7877 of the National Research Council of Canada.     

Nonconservation of the fermion number and the limiting density of fermionic matter (two-dimensional gauge model)

Conclusions Thus, at the present time there are two possible ways for instability to develop in gauge theories at high fermion density. In the four-dimensional Abelian model considered in [1] there is ultimately formed an anomalous state characterized by zero density of the real fermions, zero scalar condensate, and large gauge field condensate. In the two-dimensional model considered in the present paper, the effects of the complicated vacuum structure have the consequence that the system undergoes a transition to a normal state with low fermion density above a topologically nontrivial vacuum, this transition being accompanied by nonconservation of the fermion number. It is of undoubted interest to clarify which of these possibilities is realized in realistic non-Abelian four-dimensional theories (for example, in the standard model of electroweak interactions), i. e., to consider the existence of stable anomalous states in such theories. This question will be considered in later papers.Institute of Nuclear Research, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 68, No. 1, pp. 3–17, July, 1986.     

Bases for rings of coinvariants

We study the multiplicative structure of rings of coinvariants for finite groups. We develop methods that give rise to natural monomial bases for such rings over their ground fields and explicitly determine precisely which monomials are zero in the ring of coinvariants. We apply our methods to the Dickson, upper triangular and symmetric coinvariants. Along the way, we recover theorems of Steinberg [17] and E. Artin [1]. Using these monomial bases we prove that the image of the transfer for a general linear group over a finite field is a principal ideal in the ring of invariants. This research is supported in part by the Natural Sciences and Engineering Research Council of Canada     

The behavior of super-Brownian motion near extinction

We start from a super-Brownian motion with the branching mechanism presented by Dawson and Vinogradov. Its behaviour near extinction is studied in this paper, and the main result is that the diameter of the support tends to zero almost surely at the time of extinction. Research supported by Tianyuan Foundation and Postdoctoral Foundation     

Tubes and eigenvalues for negatively curved manifolds

We investigate the structure of the spectrum near zero for the Laplace operator on a complete negatively curved Riemannian manifoldM. If the manifold is compact and its sectional curvaturesK satisfy 1 ≤K < 0, we show that the smallest positive eigenvalue of the Laplacian is bounded below by a constant depending only on the volume ofM. Our result for a complete manifold of finite volume with sectional curvatures pinched between −a² and −1 asserts that the number of eigenvalues of the Laplacian between 0 and (n− 1)²/4 is bounded by a constant multiple of the volume of the manifold with the constant depending ona and the dimension only. Research supported in part by the Swiss National Science Foundation, the US National Science Foundation, and the PSC-CUNY Research Award Program.     

Intersection of ultrafilters may have measure zero

We show that it is consistent with ZFC that the intersection of some family of less than ultrafilters have measure zero. This answers a question of D. Fremlin.The author thanks the Lady Davis Fellowship Trust for full supportPartially supported by Basic Research Fund, Israel Academy of Sciences, publication 436