The compact support property for solutions to the heat equation with noise

Summary We consider all solutions of a martingale problem associated with the stochastic pde and show thatu(t,·) has compact support for allt0 ifu(0,·) does and if <1. This extends a result of T. Shiga who derived this compact support property for 1/2 and complements a result of C. Mueller who proved this property fails if 1.The author's research was supported by an NSF grant and an NSERC operating grantThe author's research was supported by an NSERC operating grant     

Rigidity and the lower bound theorem 1

Summary For an arbitrary triangulated (d-1)-manifold without boundaryC withf ₀ vertices andf ₁ edges, define . Barnette proved that (C)0. We use the rigidity theory of frameworks and, in particular, results related to Cauchy's rigidity theorem for polytopes, to give another proof for this result. We prove that ford4, if (C)=0 thenC is a triangulated sphere and is isomorphic to the boundary complex of a stacked polytope. Other results: (a) We prove a lower bound, conjectured by Björner, for the number ofk-faces of a triangulated (d-1)-manifold with specified numbers of interior vertices and boundary vertices. (b) IfC is a simply connected triangulatedd-manifold,d4, and (lk(v, C))=0 for every vertexv ofC, then (C)=0. (lk(v,C) is the link ofv inC.) (c) LetC be a triangulatedd-manifold,d3. Then Ske1₁( _d+2) can be embedded in skel₁ (C) iff (C)>0. ( _d is thed-dimensional simplex.) (d) IfP is a 2-simpliciald-polytope then . Related problems concerning pseudomanifolds, manifolds with boundary and polyhedral manifolds are discussed.     

Minimal Surfaces with an Elastic Boundary

LetD:= { C ³ (

On the computation of modules of long quadrilaterals

We study quadrilateralsQ which are given by two intervals on {:Im = 0} and {:Im = 1}, and two Jordan arcs ₁, ₂, in {:0 Im 1} connecting these two intervals. Many practical problems require the determination of the modulem(Q) ofQ, but ifQ is long, i.e., if

The Orders of Related Elements of a Finite Field

All finite fields q (q 2, 3, 4, 5, 7, 9, 13, 25, 121) contain a primitive element for which + 1/ is also primitive. All fields of square order q ² (q 3, 5) contain an element of order q + 1 for which + 1/ is a primitive element of the subfield q. These are unconditional versions of general asymptotic results.     

Some asymptotic results related to the law of iterated logarithm for Brownian motion

For 0<<1, let . The questions addressed in this paper are motivated by a result due to Strassen: almost surely, lim sup _t U ((t))=1–exp{–4(^–1)–1}. We show that Strassen's result is closely related to a large deviations principle for the family of random variablesU (t), t>0. Also, when =1,U (t)0 almost surely and we obtain some bounds on the rate of convergence. Finally, we prove an analogous limit theorem for discounted averages of the form as 0, whereD is a suitable discount function. These results also hold for symmetric random walks.     

Impossibility of a Certain Cyclotomic Equation with Applicationsto Difference Sets

Under certain conditions, we show the nonexistence ofan element in the p-th cyclotomicfield over , that satisfies . As applications, we establish the nonexistence ofsome difference sets and affine difference sets.     

Remark on the Helly number for strongly convex sets on Riemannian manifolds

Replacing convex by strongly convex we show that Helly's famous intersection theorem holds on every Riemannian n-manifold in the following form: The intersection of k relatively compact, strongly convex subsets of M (kn+i2) is nonvoid as soon as any n+i of these sets have a nonvoid intersection, where i=2 if M is homeomorphic to the standard n-sphere and i=1 otherwise.     

An Erdős-Révész type law of the iterated logarithm for stationary Gaussian processes

Summary Let {X(t),t 0} be a stationary Gaussian process withEX(t)=0,EX ²(t)=1 and covariance function satisfying (i)r(t) = 1 2212;C |t | + o (|t|)ast0 for someC>0, 0<2; (ii)r(t)=0(t ^–2) as t for some >0 and (iii) sup_ts|r(t)|<1 for eachs>0. Put (t)= sup {s:0 s t,X(s) (2logs)^1/2}. The law of the iterated logarithm implies a.s. This paper gives the lower bound of (t) and obtains an Erds-Rèvèsz type LIL, i.e., a.s. if 0<<2 and . Applications to infinite series of independent Ornstein-Uhlenbeck processes and to fractional Wiener processes are also given.Research supported by the Fok Yingtung Education Foundation of China and by Charles Phelps Taft Postdoctoral Fellowship of the University of Cincinnati     

ИНтЕгРАльНыЕ пРЕДст АВлЕНИь В тРУБЧАтых ОБлАстьх НАД пОлИЁДРАМИ И ВОпРОсы пРИБлИжЕНИ ь ФУНкцИИ

Let B ⁿ be a domain and (y), y B and arbitrary positive, continuous function. If p, s (0, +), denote byH _s, ^p (T _B ) the class of the functionsf(z)f(x+iy), holomorphic in the tube domain

The critical value for some non-attractive long range nearest particle systems

We consider the nearest particle system which gives birth rate to each vacant interval, concentrated on the interval's midpoint(s). We prove that a critical value for exists and equals one. The proof extends to a large class of nearest particle systems. This paper solves a problem suggested by Liggett (1985).In the following we deal with nearest particle systems { _t :t0}. These can be described as particle systems with the following flip rates:

О росте вдоль прямой ц елых функций экспоне нциального типа с заданными нулями

The following result is proved. Let=_n} be a sequence of complex numbers with ¦Re _n¦¦ _n ¦, >0, and letg be an entire function of exponential type with a sequence of zeros which satisfies the same condition. There exists an entire function of exponential typef0 such thatf()=0 and ¦f(iy)¦¦g(iy)¦,yR, if and only if there exists a constantM such that for all numbersr andR, 0rR<>, we have

Interfaces and typical Gibbs configurations for one-dimensional Kac potentials

Summary We consider a one dimensional Ising spin system with a ferromagnetic Kac potential J(|r|),J having compact support. We study the system in the limit, »0, below the Lebowitz-Penrose critical temperature, where there are two distinct thermodynamic phases with different magnetizations. We prove that the empirical spin average in blocks of size ^–1 (for any positive ) converges, as »0, to one of the two thermodynamic magnetizations, uniformly in the intervals of size ^–p , for any given positivep1. We then show that the intervals where the magnetization is approximately constant have lengths of the order of exp(c ^–1),c>0, and that, when normalized, they converge to independent variables with exponential distribution. We show this by proving large deviation estimates and applying the Ventsel and Friedlin methods to Gibbs random fields. Finally, if the temperature is low enough, we characterize the interface, namely the typical magnetization pattern in the region connecting the two phases.The research has been partially supported by CNR, GNFM, GNSM and by grant SC1CT91-0695 of the Commission of European Communities     

On convergence of types and processes in Euclidean space

Let be an Euclidean space; Y _n , Z, U random vectors in ; h _n , g _n affine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that g_n and where Z is nonsingular. The behaviour of _n = h _n g _n ^–1 as n is discussed first. The results are used then to prove that if for all t(0, ), where h _n þ and Z ₁ is nonsingular and nonsymmetric with respect to þ then H, for all t(0,) and is a continuous homomorphism of the multiplicative group of (0, ) into þ. The explicit forms of the possible are shown.     

On the theory of imploding shocks

This paper is concerned with the non-monotonic distribution of the static pressure behind self-similar imploding shocks as revealed by the numerical solutions of Landau and Stanyukovich. It is found that in terms of the ratio of the reduced gas velocity to the similarity variable, the exponent equation derived from the condition of zero pressure gradient and that derived from the theory of singularity are of identical form. The identicalness is shown to be formal only. It is shown that both equations can be obtained from a unified equation. Using these results, it is shown that within the framework of inviscid self-similar solutions, the zero pressure gradient cannot exist immediately behind the shock for2+3. Applying the unified equation, it is analytically shown that as approaches unity the similarity exponents also approach unity independent of the dimensionality, a result previously obtained by Lazarus and Richtmyer.

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Zusammenfassung Die nicht-monotone statische Druckverteilung hinter selbst-ähnlichen implodierenden Stößen, dargestellt in den numerischen Lösungen von Landau uad Stanyukovich, ist Gegenstand dieser Arbeit. Es wird gezeigt, daß in bezug auf das Verhältnis zwischen reduzierter Gasgeschwindigkeit und der Ähnlichkeits-Veränderlichen, die unter der Null-Druckgradient-Bedingung und der Singularitätstheorie abgeleiteten Exponentengleichungen identisch sind. Des weiteren wird gezeigt, daß die Identität nur formal ist, daß beide Gleichungen von einer Einheitsgleichung abgeleitet werden können und daß im reibungsfreien Bereich der Null-Druckgradient unmittelbar hinter dem Stoß für2+3 nicht bestehen kann. Bei Anwendung der Einheitsgleichung läßt sich zeigen, daß unabhängig von ihrer Dimensionalität, sowohl als auch die Ähnlichkeitsexponenten sich Eins annähern, wie bereits von Lazarus und Richtmyer angegeben wurde.

     

Integration of Primal Lower Nice Functions in Hilbert Spaces

In this paper, we obtain some integration results from subdifferential inclusions for primal lower nice functions by using the Moreau envelopes. A general result concerns an enlarged subdifferential inclusion. It says that, for g primal lower nice at x, the inclusion around x entails that, for any ]0; [, f – g is - Lipschitz continuous on an appropriate neighborhood of x.     

Long time existence for the heat equation with a noise term

Summary We consider the equationu _t =u _xx +u W forx on a finite interval, with Dirichlet boundary conditions. W is spacetime white noise. The initial condition is continuous and nonnegative. We show existence and uniqueness for all time, provided 1 <3/2.Supported by an NSF grant     

On the Mean Value of the Index of Composition of an Integer

For each integer n 2, let be the index of composition of n, where . For convenience, we write (1)=(1)=1. We obtain sharp estimates for and , as well as for and . Finally we study the sum of running over shifted primes.Research supported in part by a grant from NSERC.Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA.     

Some notes about affine diameters of convex figures

Given a pointx in a convex figureM, let(x) denote the number of all affine diameters ofM passing throughx. It is shown that, for a convex figureM, the following conditions are equivalent.

Infinitesimal,first order bendings of smooth,convex surfaces of revolution subject to conic,sleeve-like constraints along the boundary

We prove that on a closed, smooth, convex surface of revolution , whose poles are not flattening points, there exists only a countable set of parallels _n. Each of these parallels cuts surface into two parts so that one of the parts, , admits nontrivial, infinitesimal bendings in the process of which all the points of its boundary _n are displaced on a preassigned, conic sleeve K that is coaxial with the surface. The sequence of such parallels _n converges to parallel ^*, which has the following properties: 1) the tangent cone to surface along ^* is orthogonal to sleeve K; 2) surface , cut off from surface by parallel ^*, has rigidity of first order in the indicated class of bendings.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 3–8, 1990.