Distoriton of area and conditioned Brownian motion

Summary In a simply connected planar domainD the expected lifetime of conditioned Brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain. We show that each hyperbolic geodesic induces a decomposition ofD into disjoint subregions and that the subregions are obtained in a natural way using Euclidean geometric quantities relating toD. The lifetime associated with on each _j is then shown to be bounded by the product of the diameter of the smallest ball containing _j and the diameter of the largest ball in _j . Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in _j it leads to finite lifetime estimates in a variety of domains of infinite area.Research of the first author was supported in part by NSF Grant DMS-9100811Research of the second author was supported in part by NSF Grant DMS-9105407     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

On the γ-Interior and γ-Closure of a Set

For a set X, let : exp X exp X satisfy A B whenever A B X. In [4], -open subsets of X, -interior iA and -closure cA of A X have been defined. The purpose of the present paper is to show that, under suitable conditions on , explicit formulas furnish iA and cA.     

Long paths and cycles in tough graphs

For a graphG, letp(G) andc(G) denote the length of a longest path and cycle, respectively. Let (t,n) be the minimum ofp(G), whereG ranges over allt-tough connected graphs onn vertices. Similarly, let (t,n) be the minimum ofc(G), whereG ranges over allt-tough 2-connected graphs onn vertices. It is shown that for fixedt>0 there exist constantsA, B such that (t,n)A·log(n) and (t,n)·log((t,n))B·log(n). Examples are presented showing that fort1 there exist constantsA, B such that (t,n)A·log(n) and (t,n)B· log(n). It is conjectured that (t,n) B·log(n) for some constantB. This conjecture is shown to be valid within the class of 3-connected graphs and, as conjectured in Bondy [1] forl=3, within the class of 2-connectedK _1.l-free graphs, wherel is fixed.     

A cuspidal edge of first kind with a point of asymptotic type

We shall say that an analytic surface supports a line with a singularity of order 2 if is a singular line of the surface (the first fundamental form of anyC -parametrization of is degenerate on ) and there exists aC -parametrization(s, v) of the surface such that the second derivative of in the direction orthogonal to is nonzero. We shall assume that has no singular points or points of straightening or flattening, that the Gaussian curvature of is nonpositive, and that the Gaussian curvature of has a finite limita(s) on (s is arc length on ), and the osculating plane of is tangent to in the sense that this plane contains as a half-plane the contingency of at the corresponding point of . Finally supposeb(0)=0,b(s)0 fors0, andb _ss (0)0, whereb(s)=a(s)+ ²(s) and is the torsion of . It is shown that under these hypotheses envelopes the asymptotes of one of the families of. All the asymptotes of the other family are tangent to except the asymptote passing through the points=0 of the singular line.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 66–76.     

Global inequalities for curves and surfaces in three-space

Summary The aim of the paper is to give upper bounds for the total curvature of smooth curves and surfaces embedded in euclidean space, in terms of other global geometric characters; in particular, for a plane curve , we prove the inequality K() < (2 + f()d()/2), where d() is the geometric degree of and f() is the number of its inflection points. In the case of a surface S, a bound is given in terms of the genus g(S), the number of components of the parabolic points on S and the geometry of its apparent contour.     

Reflected Brownian motion in a cone: Semimartingale property

Summary In this paper, the object of study is reflected Brownian motion in a cone ind-dimensions (d3) with nonconstant oblique reflection on each radial line emanating from the vertex of the cone. The basic question considered here is When is this process a semimartingale?. Conditions for the existence and uniqueness of the process for which the vertex is an instantaneous state were given by Kwon, which is resolved in terms of a real parameter depending on the cone and the direction of reflection. It is shown that starting from any point of the cone, the process is a semimartingale if < 1, + 0 and not a semimartingale if < < 2.This research is supported by KOSEF grant 941-0100-011-1     

On strong solutions of the Stokes equations in exterior domains

We construct strong solutionsu, p/of the general nonhomogeneous Stokes equations -u + p=f inG, ·u=g inG, u= on in an exterior domainG ⁿ (n3) with boundary of class C². Our approach uses a localization technique: With the help of suitable cut-off functions and the solution of the divergence equation ·=g inG, = 0 on , the exterior domain problem is reduced to the entire space problem and an interior problem.     

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     

Mittelwert der Eulerschen φ-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern

LetK be an algebraic number field, and for every integer K let () andd(), respectively, denote the number of relatively prime residue classes and the number of divisors of the principal ideal (). Asymptotic equalities are proved for the sums () and d ²(), where runs through certain finite sets of integers ofK.     

On the semimartingale representation of reflecting Brownian motion in a cusp

Summary LetC be the symmetric cusp {(x, y)²:–x yx ,x0} where >1. In this paper we decide whether or not reflecting Brownian motion inC has a semimartingale representation. Here the reflecting Brownian motion has directions of reflection that make constant angles with the unit inward normals to the boundary. Our results carry through for a wide class of asymmetric cusps too.     

Sample path large deviations for super-Brownian motion

Summary We derive two large deviation principles of Freidlin-Wentzell type for rescaled super-Brownian motion. For one of the appearing rate functions an integral representation is given and interpreted as Kakutani-Hellinger energy. As a tool we develop estimates for the Laplace functionals of (historical) super-Brownian motion and certain maximal inequalities. Also it is shown that the Hölder norm of index <1/2 of the processtf, X _t possesses some finite exponential moments provided the functionf is smooth.This work was supported in part by the Graduiertenkolleg Algebraische, analytische und geometrische Methoden und ihre Wechselwirkung in der modernen Mathematik, Bonn     

Angular limits of holomorphic functions which satisfy an integrability condition

Let (–1,1), let 2/(1–)p<, letp denote the Hölder conjugate ofp, and let be an open arc of the unit circle. It is shown that, iff is a holomorphic function on the unit disc such that: (i) (1–|z|)log⁺|f(z)| isL ^p -integrable on the sector {r:0f has an infinite asymptotic value has -finite (2–(1+)p)-dimensional Hausdorff, measure, thenf has finite angular limits on a subset of of positive linear measure. In fact, a stronger conclusion will be established.     

An extension of the fenchel theorem

We give an extension of the Fenchel-Borsuk result by introducing the total absolute torsion-curvature KT() for regular curves whose tangent indicatrix is a piecewise regular curve (-closed curves). We prove that KT() is low bounded by 2 and we give a geometric characterization for the -closed curves whose KT() is minimal.     

Quelques propriétés des solutions de l'équation de Ginzburg-Landau sur un ouvert de ∝2

For a solution u of –u=u(1–|u|²) on the whole plane, |u|<1 holds everywhere unless u=eⁱ for some ; the derivatives of order k have moduli a constant M _kdepending only on k. For a solution u on an open set ², the moduli of u and its derivatives have upper bounds depending only on the distance to ²\ therefore the set of solutions on a given is compact in C() for the topology of uniform convergence on compact subsets of . For a solution u such that |u|<1, 1–|u| satisfies an estimation similar to the classical Harnack inequality for positive harmonic functions.Finally, if is bounded and |u| has a lim supm at each boundary point, the |u|m in if m1, but if m<1 then |u| admits only a majorant S _m with values in ]m, 1[ and sufficient conditions are given for lim S _m =0 or S _m =O(m) as m0.

     

A certain periodic control problem for differential equations with impulses in the space of bounded numerical sequences

One considers the differential equation dx/dt=f(t, x) with the impulse action ¦_t=t_i=H_i(t_i,x) in the space of bounded numerical sequences, where f(t, x), H_i(t, x) are T-periodic, countable-dimensional vector-valued functions, is a positive parameter. One gives conditions for the existence of a control (₁,₂) such that the solution of the equation dx/dt=f(t, x)–₁ with impulse action x¦_t=t_i=H_i(t_i,x)–₂ assuming for t= the value x=x₀, be T-periodic.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 271–275, February, 1990.     

On the asymptotic and invariantσ-algebras of random walks on locally compact groups

Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , ^a ,Q) andL (G , ⁱ ,Q) where ^a and ⁱ stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , ^a ,Q) L (G , ⁱ ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers ⁿ.     

A result concerning the existence of certain finite Minkowski - 2 - Structures

Under a special assumption, a theorem concerning the order of Minkowski - 2 - Structures is proved. In particular it is shown that if is a sharply 4 - transitive set of permutations onn elements (n7, integer), such that contains the identical permutation and implies ^–1, then isn=11.     

Global solutions describing the collapse of a spherical or cylindrical cavity

The collapse of a spherical (cylindrical) cavity in air is studied analytically. The global solution for the entire domain between the sound front, separating the undisturbed and the disturbed gas, and the vacuum front is constructed in the form of infinite series in time with coefficients depending on an appropriate similarity variable. At timet=0⁺, the exact planar solution for a uniformly moving cavity is assumed to hold. The global analytic solution of this initial boundary value problem is found until the collapse time (=(–1)/2) for 1+(2/(1+v)), wherev=1 for cylindrical geometry, andv=2 for spherical geometry. For higher values of , the solution series diverge at timet — 2(–1)/ (v(1+)+(1–)2) where =2/(–1). A close agreement is found in the prediction of qualitative features of analytic solution and numerical results of Thomaset al. [1].     

γ-Subdifferential and γ-convexity of functions on a normed space

In this paper, the -subdifferential is introduced for investigating the global behavior of real-valued functions on a normed spaceX. Iff: DX attains its global minimum onD atx ^*, then 0 f(x ^*). This necessary condition always holds, even iff is not continuous orx ^* is at the boundary of its domain. Nevertheless, it is useful because, by choosing a suitable ₊, many local minima cannot satisfy this necessary condition. For the sufficient conditions, the so-called -convex functions are defined. The class of these functions is rather large. For example, every periodic function on the real line is a -convex function. There are -convex functions which are not continuous everywhere. Every function of bounded variation can be represented as the difference of two -convex functions. For all that, -convex functions still have properties similar to those of convex functions. For instance, each -local minimizer off is at the same time a global one. Iff attains its global minimum onD, then it does so at least at one point of its -boundary.This research was supported by the Alexander von Humboldt Foundation. The author thanks Professors R. Bulirsch, K. H. Hoffmann, and H. G. Bock for inviting him to Munich and Augsburg where this research was done.