On a Laplace Transform Based Metric for Probabilities on a Hilbert Space

Let D be a closed subset of a real separable Hilbert space H. Let (D) denote the set of all Borel probability measures on D and (D) the set of all probabilities with integrable Laplace transform. A metric d, based on the Laplace transform, is defined on (D). Topological properties, viz., separability, connectedness, completeness, compactness and local compactness, of (D, d are investigated, and the d-topology is compared with the topology of weak convergence.     

Hilbert spaces of holomorphic functions on bounded domains

In [6] the notion of Hilbert-Stein complex spaces was introduced. The purpose of this note is to illustrate the conjecture that any Stein space whose points can be separated by bounded holomorphic functions, is Hilbert-Stein. We prove this fact for the following kinds of bounded domains of holomorphyD in C ⁿ : 1)n=1 and ₁(D) is finitely generated, 2) the group of automorphisms ofD in the compactopen topology is compact, 3)D is strongly pseudoconvex with smooth boundary andn2, 4)D is symmetric of type I.     

On Duality in Spaces of Solutions to Elliptic Systems

Suppose that D is a bounded domain in (n2) with connected real-analytic boundary, A is an elliptic system with real-analytic coefficients in a neighborhood of the closure of D, and sol(A,D) is the space of solutions to the system Au=0 in D furnished with the standard Frechet–Schwartz topology. Then the dual of sol(A,D) represents the space sol(A, ) of solutions to the system Au=0 in a neighborhood of furnished with the standard inductive limit topology over some decreasing net of neighborhoods of . The corresponding pairing is generated by the inner product in the Lebesgue space L ²(D).     

On the Volume of a Domain Obtained by a Holomorphic Motion Along a Complex Curve

Let D be a domain obtained by a holomorphic motion of a domain D _p M _p ^n–1 along a complex curve P in a complex space form M ⁿ . We prove that, if n= 2, the volume of D depends only on the geometry of D _p and the intrinsic geometry of P, but not on the extrinsic geometry of P. When M is closed (compact without boundary), then the dependence on P is only through its topology. When n > 2, and for arbitrary domains D _p, a similar result holds only for Frenet motions, but when D _p has certain integral symmetries (and only in this case) this result is still true for any motion .     

The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Summary The Skorohod oblique reflection problem for (D, , w) (D a general domain in ^d , (x),xD, a convex cone of directions of reflection,w a function inD(⁺, ^d )) is considered. It is first proved, under a condition on (D, ), corresponding to (x) not being simultaneously too large and too much skewed with respect to D, that given a sequence {w ⁿ} of functions converging in the Skorohod topology tow, any sequence {(x ⁿ, ⁿ)} of solutions to the Skorohod problem for (D, , w ⁿ) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, , w). Next it is shown that if (D, ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, , w) exist for everywD(⁺, ^d ) with small enough jump size. The requirement is met in the case when D is piecewiseC _b ¹ , is generated by continuous vector fields on the faces ofD and (x) makes and angle (in a suitable sense) of less than /2 with the cone of inward normals atD, for everyxD. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.     

Lower estimate of the isoperimetric deficit of convex domains inR n in terms of asymmetry

For convex bodiesD inR ⁿ it is shown that the isoperimetric deficit ofD is minorized by a constant times the square of thebarycentric asymmetry (D) ofD. Here (D) is defined as the volume ofDB _D divided by the volume ofD, whereB _D denotes the ball centred at the barycentre ofD and having the same volume asD.Dedicated to the memory of Børge Jessen     

Classification of Distributions by the Arithmetic Means of Their Fourier Series

We consider the class C ^m of functions that are m times differentiable on the one-dimensional torus group G = R/2Z with respect to addition mod 2; and the class of D ^m of distributions of order at most m. Clearly, D ^m can be identified as the dual space of C ^m . One of our main results says that a formal trigonometric series is the Fourier series of a distribution in D ^m if and only if the sequence of its arithmetic means _N (u) as distributions is bounded for all u C ^m; or equivalently, if sup _N ¦ _{D m} < . Another result says that the arithmetic mean _N F of a distribution converges to F in the strong topology of D ^m if F D ^m–1, which is not true in general if F D ^m .     

A sufficient condition for the existence of an anti-directed 2-factor in a directed graph

Let D be a directed graph with vertex set V, arc set A, and order n. The graph underlyingD is the graph obtained from D by replacing each arc (u,v)∈A by an undirected edge {u,v} and then replacing each double edge by a single edge. An anti-directed (hamiltonian) cycleH in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. An anti-directed 2-factor in D is a vertex-disjoint collection of anti-directed cycles in D that span V. It was proved in Busch et al. (submitted for publication) [3] that if the indegree and the outdegree of each vertex of D is greater than then D contains an anti-directed Hamilton cycle. In this paper we prove that given a directed graph D, the problem of determining whether D has an anti-directed 2-factor is NP-complete, and we use a proof technique similar to the one used in Busch et al. (submitted for publication) [3] to prove that if the indegree and the outdegree of each vertex of D is greater than then D contains an anti-directed 2-factor.     

On the domain monotonicity of the first eigenvalue of a boundary value problem

Summary This paper is concerned with the monotonicity of the first eigenvalue ₁ (D) of (1) as a functional of the domainD.For 0<k< we prove that ₁(D) is increasing withD in several cases, but a counterexample is given showing that this is not true in general. In the same cases we prove that, for –<k<0, ₁(D) decreases whenD increases.

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Zusammenfassung Diese Arbeit behandelt die Monotonie des ersten Eigenwertes ₁(D) von (1) als Funktional des GebietesD. Für 0<k< beweisen wir, dass ₁(D) in mehreren Fällen mitD wächst; aber ein Gegenbeispiel zeigt, dass dies nicht allgemein gilt. In denselben Fällen beweisen wir, dass ₁(D) für –<k<0 bei wachsendemD abnimmt.

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Technical Report TR-75-13.     

The lifetime of conditioned Brownian motion in planar domains of infinite area

Summary In this paper, it is shown that all expected lifetimes ofh-processes inD are finite if and only if the area ofD is finite ifD={(x,y):ø_(x)>y<ø₊(x), – <x<}, where ø_(x)<ø₊ are two Lipschitz functions. We show that if is a bounded convex region in the plane, there is anh-process in with expected lifetime at leastc area (). We also give an example of a planar domainD of infinite area such that the expected lifetime of eachh-process inD is finite.     

A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations

LetDR ^N be a region with smooth boundaryD. Letp·q>1,p, q1. We consider the system:u _t=u+v ^p,v _t=u+u ^q inD×[0, ) withu=v=0 inD×[0, ) andu ₀,v ₀ nonnegative. Let=max(p, q). We show that ifD isR ^N, a cone or the exterior of a bounded domain, then there is a numberpc(D) such that (a) if (+1)/(pq–1)>pc(D) no nontrivial global positive solutions of the system exist while (b) if (+1)/(pq–1)<pc(D) both nontrivial global and nonglobal solutions exist. In caseD is a cone orD=R ^N, (a) holds with equality. An explicit formula forpc(D) is given.This research was supported in part by NSF Grant DMS-8822788 and in part by the Air Force Office of Scientific Research.     

Green Integrals on Manifolds with Cracks

We prove the existence of a limit in H ^m (D)of iterations of a double layer potential constructed from the Hodge parametrix on a smooth compact manifold with boundary, X, and a crack S D, D being a domain in X. Using this result we obtain formulas for Sobolev solutions to the Cauchyproblem in D with data on S, for an elliptic operator A of order m 1, whenever these solutions exist.This representation involves the sum of a series whose terms are iterationsof the double layer potential. A similar regularisation is constructed also for a mixed problem in D.     

<Emphasis Type="Italic">C</Emphasis>*-Independence,Product States and Commutation

Let D be a unital C*-algebra generated by C*-subalgebras A and B possessing the unit of D. Motivated by the commutation problem of C*-independent algebras arising in quantum field theory, the interplay between commutation phenomena, product type extensions of pairs of states and tensor product structure is studied. Rooss theorem [11] is generalized in showing that the following conditions are equivalent: (i) every pair of states on A and B extends to an uncoupled product state on D; (ii) there is a representation of D such that (A) and (B) commute and is faithful on both A and B; (iii) is canonically isomorphic to a quotient of D.The main results involve unique common extensions of pairs of states. One consequence of a general theorem proved is that, in conjunction with the unique product state extension property, the existence of a faithful family of product states forces commutation. Another is that if D is simple and has the unique product extension property across A and B then the latter C*-algebras must commute and D be their minimal tensor product.Communicated by Klaus Fredenhagensubmitted 03/12/03, accepted 26/04/04     

Geometric Properties of Universally Measurable Mappings

We study conditions under which universally measurable mappings from a separable topological space S into a metric space R belong to the class D of mappings f : SR: such that for any compact subset KS and number > 0 there exists an open (in the induced topology) set VK such that the oscillation (f;V) of an R-valued function f on V is less than . Bibliography: 7 titles.     

A survey of partial difference sets

LetG be a finite group of order . Ak-element subsetD ofG is called a (,k, , )-partial difference set if the expressionsgh ^–1, forg andh inD withgh, represent each nonidentity element inD exactly times and each nonidentity element not inD exactly times. IfeD andgD iffg ^–1D, thenD is essentially the same as a strongly regular Cayley graph. In this survey, we try to list all important existence and nonexistence results concerning partial difference sets. In particular, various construction methods are studied, e.g., constructions using partial congruence partitions, quadratic forms, cyclotomic classes and finite local rings. Also, the relations with Schur rings, two-weight codes, projective sets, difference sets, divisible difference sets and partial geometries are discussed in detail.     

Paradoxical subsets of hyperbolic and spherical discs

Given a disc D of radius r in H ² (resp. S ²) with <r (resp. <r), we determine the pairs (m,n) for which there is an (m,n)-paradoxical subset of D but not an (m–1, n)-paradoxical subset of D or an (m, n–1)-paradoxical subset of D.     

A constrained matching problem

We show that certain manpower scheduling problems can be modeled as the following constrained matching problem. Given an undirected graphG = (V,E) with edge weights and a digraphD = (V,A). AMaster/Slave-matching (MS-matching) ofG with respect toD is a matching ofG such that for each arc (u, v) A for which the nodeu is matched, the nodev is matched, too. TheMS-Matching Problem is the problem of finding a maximum-weight MS-matching. Letk(D) be the maximum size of a (weakly) connected component ofD. We prove that MS-matching is an NP-hard problem even ifG is bipartite andk(D) 3. Moreover, we show that in the relevant special case wherek(D) 2, the MS-Matching Problem can be transformed to the ordinary Matching Problem.This research was supported by Grant 03-KL7PAS-6 of the German Federal Ministry of Research and Technology.