Convergence theorems for set-valued amarts and uniform amarts

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Newton polyhedra and the Bergman kernel

The purpose of this paper is to study singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities. Under some assumptions on a domain in ⁿ⁺¹ , the Bergman kernel B(z) of takes the form near a boundary point p: where (w,) is some polar coordinates on a nontangential cone with apex at p and means the distance from the boundary. Here admits some asymptotic expansion with respect to the variables ^{1/ m} and log(1/) as 0 on . The values of d _F >0, m _F ₊ and m are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of as 0 on is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szegö kernel. Mathematics Subject Classification (2000):32A25, 32A36, 32T25, 14M25.     

Quantum Integrable and Nonintegrable Nonlinear Schrödinger Models for Realizable Bose–Einstein Condensation in d+1 Dimensions (d = 1, 2, 3)

We evaluate finite-temperature equilibrium correlators for thermal time ordered Bose fields to good approximations by new methods of functional integration in d=1,2,3 dimensions and with the trap potentials V(r)0. As in the translationally invariant cases, asymptotic behaviors fall as to longer-range condensate values for and only for d=3 in agreement with experimental observations; but there are generally significant corrections also depending on due to the presence of the traps. For d=1, we regain the exact translationally invariant results as the trap frequencies 0. In analyzing the attractive cases, we investigate the time-dependent c-number Gross–Pitaevskii (GP) equation with the trap potential for a generalized nonlinearity –2c||²ⁿ and c<0. For n=1, the stationary form of the GP equation appears in the steepest-descent approximation of the functional integrals. We show that collapse in the sense of Zakharov can occur for c=0 and nd2 and a functional E _NLS[]0 even when V(r)0. The singularities typically arise as -functions centered on the trap origin r=0.     

Invariance of closed convex sets and domination criteria for semigroups

Let a and b be two positive continuous and closed sesquilinear forms on the Hilbert space H=L ²(, ). Denote by T=T(t) _t0and S=S(t) _t0the semigroups generated by a and b on H. We give criteria in terms of a and b guaranteeing that the semigroup T is dominated by S, i.e. |T(t)f|S(t)|f| for all t0 and fH. The method proposed uses ideas on invariance of closed convex sets of H under semigroups. Applications to elliptic operators and concrete examples are given.     

Foliated control theory, II

The main result is a control theorem for the structure space of E with control near the leaves F in M, where : E M is a fiber bundle over the Riemannian manifold M having a compact closed manifold for fiber and F is a smooth foliation of M, each leaf of which inherits a flat Riemannian geometry from M. A similar result has been proved by the authors under the assumption that each leaf of F is one-dimensional and the fiber of : E M is homotopy stable.Both authors were supported in part by the National Science Foundations.     

The Darboux-Type Theorems for ∂-Symplectic and ∂-Contact Structures

A -symplectic structure on a complex manifold M of complex dimension2n is given by a smooth -closed (2, 0)-form such that ⁿ is nonvanishing. We prove that a version of the Darboux theorem isvalid for such a structure: locally can be represented as _i=1 ⁿ f _i f _{n + i} for appropriate smooth complex valuedfunctions f ₁, ..., f _2n . We also present a contact version of this theorem.     

A conjecture on semi-field planes. II

Conclusion The results in the previous sections lend strong support to the conjecture made in the Introduction. Furthermore, if the long-standing conjecture concerning the solvability of autotopism groups for semi-field planes is correct then the probability of our conjecture being true is greatly increased. In any case the existence of a semi-field plane for which u() = 2, 3, or 4 would provide a counterexample to the earlier conjecture.There are examples of semi-field planes with u() = %. As mentioned in Example 2 of Section 3, one of the semi-field planes of order 16 has u()-5. For that plane, the five orbits of the autotopism group G in (G) have lengths 27, 36, 54, 54, 54. The union of the orbit having length 36 and one of those having 54 is the union of the points in (G) on 6 lines through a vertex U and the union of the remaining three orbits consists of the 135 points on the remaining 9 lines through U. There are also non-Desarguesian A-planes in which u() = 5; the semi-field plane of order 3⁴ coordinatized by the twisted field of Albert has u() = 5.Supported in part by NSF Grants No. MPS 75-05260 and MPS 76-06661     

Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements

Summary A generalized Stokes problem is addressed in the framework of a domain decomposition method, in which the physical computational domain is partitioned into two subdomains ₁ and ₂.Three different situations are covered. In the former, the viscous terms are kept in both subdomains. Then we consider the case in which viscosity is dropped out everywhere in . Finally, a hybrid situation in which viscosity is dropped out only in ₁ is addressed. The latter is motivated by physical applications.In all cases, correct transmission conditions across the interface between ₁ and ₂ are devised, and an iterative procedure involving the successive resolution of two subproblems is proposed.The numerical discretization is based upon appropriate finite elements, and stability and convergence analysis is carried out.We also prove that the iteration-by-subdomain algorithms which are associated with the various domain decomposition approaches converge with a rate independent of the finite element mesh size.This work was partially supported by CIRA S.p.A. under the contract Coupling of Euler and Navier-Stokes equations in hypersonic flowsDeceased     

Representing workloads in GI/G/1 queues through the preemptive-resume LIFO queue discipline

We give in this paper a detailed sample-average analysis of GI/G/1 queues with the preemptive-resume LIFO (last-in-first-out) queue discipline: we study the long-run state behavior of the system by averaging over arrival epochs, departure epochs, as well as time, and obtain relations that express the resulting averages in terms of basic characteristics within busy cycles. These relations, together with the fact that the preemptive-resume LIFO queue discipline is work-conserving, imply new representations for both actual and virtual delays in standard GI/G/1 queues with the FIFO (first-in-first-out) queue discipline. The arguments by which our results are obtained unveil the underlying structural explanations for many classical and somewhat mysterious results relating to queue lengths and/or delays in standard GI/G/1 queues, including the well-known Bene's formula for the delay distribution in M/G/l. We also discuss how to extend our results to settings more general than GI/G/1.     

On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323     

A remark on fixed points of functors in topological categories

For a topological category over Set we prove that if a functor T: has a fixed cardinal (i.e. for each object K with card (UK)= we have card (UTK)), then T has a least fixed point, and if T has a successive pair of fixed cardinals and ⁺, then T has a greatest fixed point. This extends results of Adámek and Koubek.Partial financial support of the Grant Agency of the Czech Republic under Grant No. 201/93/0950 is gratefully acknowledged.     

Stability analysis of numerical methods for delay differential equations

Summary This paper deals with the stability analysis of step-by-step methods for the numerical solution of delay differential equations. We focus on the behaviour of such methods when they are applied to the linear testproblemU(t)=U(t)+U(t–) with >0 and , complex. A general theorem is presented which can be used to obtain complete characterizations of the stability regions of these methods.     

Classical integrable lattice models through quantum group related formalism

We effectively translate our earlier quantum constructions to the classical language and, using Yang-Baxterisation of the Faddeev-Reshetikhin-Takhtajan algebra, are able to construct the Lax operators and associated -matrices of classical integrable models. Thus, new as well as known lattice systems of different classes are generated, including new types of collective integrable models and canonical models with nonstandard matrices.Fachbereich 17-Mathematik/Informatik, GH-Universitän Kassel, Holländische Str. 36, 34109 Kassel, Germany. (Permanent address: Saha Institute of Nuclear Physics, AF/1 Bidhan Nagar, Calcutta 700 064, India.) Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 428–434, June, 1994.     

On Conservative Confidence Intervals

The subject of the paper – (conservative) confidence intervals – originates in applications to auditing. Auditors are interested in upper confidence bounds for an unknown mean for all sample sizes n. The samples are drawn from populations such that often only a few observations are nonzero. The conditional distribution of an observation given that it is nonzero usually has a very irregular shape. However, it can be assumed that observations are bounded. We propose a way to reduce the problem to inequalities for tail probabilities of certain relevant statistics. Note that a traditional approach involving limit theorems forces to impose additional conditions on regularity of samples and leads to approximate or asymptotic bounds. In the case of , as a statistic we can use sample mean, say , and we have to use Hoeffding [7] inequalities, since currently they are the best available. This leads to upper confidence bounds for which are of (asymptotic) size at most in the case of risk =0.05, where is the unknown standard deviation. We have , where is the bound in a model with normally distributed observations. It seems that the bound is very robust and can be improved replacing Hoeffding's inequalities by more refined ones. The commonly used Stringer bound (it is still not known whether it is an upper confidence bound) is of asymptotic size c with equality only for Bernoulli distributions, and the ratio c / can be arbitrary large already for rather simple distributions. Our bounds can involve a priori information (professional judgment of an auditor) of type ₀ or/and ₀, which leads to improvements. Most of the results also hold for sampling without replacement from finite populations. The i.i.d. condition can be replaced by a martingale-type dependence assumption. Finally, the results can be extended to the noni.i.d. case and for settings with several samples.     

On the integrability of “finite energy” solutions for <Emphasis Type="Italic">p</Emphasis>-harmonic equations

We prove a higher integrability result for the gradient of solutions to some degenerate elliptic PDEs, whose model arises in the study of mappings with finite distortion.The nonnegative function which measures the degree of degeneracy of ellipticity bounds lies in the exponential class, i.e. is integrable for some > 0.Our result states that if is sufficiently large, then the gradient of a finite energy solution actually belongs to the Zygmund space L^plog^{L, 1}.     

Superstable implicit Runge-Kutta methods for second order initial value problems

Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy=f(x, y, y),y(x _o)=y _o,y(x _o)=y _o. These methods when applied to the test equationy+2y+ ² y=0, ,0, +>0, are superstable with the exception of a finite number of isolated values ofh. These methods can be successfully used for solving singular perturbation problems for which f/y and/or f/y are negative and large. Numerical results demonstrate the efficiency of these methods.     

The general Neville-Aitken-algorithm and some applications

Summary In this note we will present the most general linear form of a Neville-Aitken-algorithm for interpolation of functions by linear combinations of functions forming a ebyev-system. Some applications are given. Expecially we will give simple new proofs of the recurrence formula for generalized divided differences [5] and of the author's generalization of the classical Neville-Aitkena-algorithm[8]applying to complete ebyev-systems. Another application of the general Neville-Aitken-algorithm deals with systems of linear equations. Also a numerical example is given.     

The asymptotic stability of one-parameter methods for neutral differential equations

This paper deals with the asymptotic stability of theoretical solutions and numerical methods for systems of neutral differential equationsx=Ax(t–)+Bx(t)+Cx(t–), whereA, B, andC are constant complexN ×N matrices, and >0. A necessary and sufficient condition such that the differential equations are asymptotically stable is derived. We also focus on the numerical stability properties of adaptations of one-parameter methods. Further, we investigate carefully the characterization of the stability region.     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

On the approximation of singular integral equations by equations with smooth kernels

Singular integral equations with Cauchy kernel and piecewise-continuous matrix coefficients on open and closed smooth curves are replaced by integral equations with smooth kernels of the form(t–)[(t–) ²–n ² (t) ²]^–1,0, wheren(t), t , is a continuous field of unit vectors non-tangential to . we give necessary and sufficient conditions under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the spaceL ₂() these conditions coincide with the strong ellipticity of the given equation.This work was fulfilled during the first author's visit to the Weierstrass Institute for Applied Analysis and Stochastics, Berlin in October 1993.