Estimates and structure of α-harmonic functions

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on D ^c ∪ {∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test. K. Bogdan was supported by KBN grant 1 P03A 026 29 and RTN contract HPRN-CT-2001-00273-HARP. T. Kulczycki was supported by KBN grant 1 P03A 020 28 and RTN contract HPRN-CT-2001-00273-HARP. M. Kwaśnicki was supported by KBN grant 1 P03A 020 28 and RTN contractHPRN-CT-2001-00273-HARP.     

Boundary Harnack Principle for <Emphasis Type="Italic">p</Emphasis>-harmonic Functions in Smooth Euclidean Domains

We establish a scale-invariant version of the boundary Harnack principle for p-harmonic functions in Euclidean C ^1,1-domains and obtain estimates for the decay rates of positive p-harmonic functions vanishing on a segment of the boundary in terms of the distance to the boundary. We use these estimates to study the behavior of conformal Martin kernel functions and positive p-superharmonic functions near the boundary of the domain. H. A. was partially supported by Grant-in-Aid for (B) (2) (No. 15340046) Japan Society for the Promotion of Science. N. S. was partially supported by NSF grant DMS-0355027. X. Z. was partially supported by the Taft foundation.     

On boundaries and influences

We prove an inequality relating the size of the boundary of a monotone subset of {0,1} ⁿ and the influences of the coordinates upon this set. It shows that if the boundary (resp. the influences) are small, the influences (resp. the boundary) are large.Work partially supported by an NSF grant.     

An approximate Dirac-type theorem for k-uniform hypergraphs

A k-uniform hypergraph is hamiltonian if for some cyclic ordering of its vertex set, every k consecutive vertices form an edge. In 1952 Dirac proved that if the minimum degree in an n-vertex graph is at least n/2 then the graph is hamiltonian. We prove an approximate version of an analogous result for uniform hypergraphs: For every K ≥ 3 and γ > 0, and for all n large enough, a sufficient condition for an n-vertex k-uniform hypergraph to be hamiltonian is that each (k − 1)-element set of vertices is contained in at least (1/2 + γ)n edges. Research supported by NSF grant DMS-0300529. Research supported by KBN grant 2P03A 015 23 and N201036 32/2546. Part of research performed at Emory University, Atlanta. Research supported by NSF grant DMS-0100784.     

The numerical solution of Fredholm integral equations of the second kind with singular kernels

A numerical method is given for integral equations with singular kernels. The method modifies the ideas of product integration contained in [3], and it is analyzed using the general schema of [1]. The emphasis is on equations which were not amenable to the method in [3]; in addition, the method tries to keep computer running time to a minimum, while maintaining an adequate order of convergence. The method is illustrated extensively with an integral equation reformulation of boundary value problems for u–P(r ²)u=0; see [9].This research was supported in part by NSF grant GP-8554.     

Asymptotic coarsening of proper metric spaces and index problems

 We introduce an asymptotic coarse structure on proper metric spaces and study the associated C ^* -algebras and assembly maps. We establish an asymptotic Lipschitz homotopy invariance theorem for the K-theory of these C ^* -algebras and the K-homology of the metric space, and show that the assembly map is an isomorphism over an asymptotically scaleable space. Received: 12 July 2001 / Revised version: 29 October 2002 Published online: 3 March 2003 The first author is supported in part by the National Basic Research Project (973), NSF and the Educational Ministry of P. R. China. The second author is supported by the NSF grant 10201007, P. R. China, and a grant from Shanghai Science and Technology Commission, No. 01ZA14003. Mathematics Subject Classification (2000): Primary 46L80     

Hearing pseudoconvexity with the Kohn Laplacian

A bounded domain in with connected Lipschitz boundary is pseudoconvex if the bottom of the essential spectrum of the Kohn Laplacian on the space of (0,q)-forms, 1qn–1, with L²-coefficients is positive.The author was supported in part by NSF grant DMS 0070697 and by an AMS centennial fellowship.Revised version: 9 July 2004     

Non-trivial involutions on Floer Homology

 We construct the first examples of irreducible 3-manifolds with the homology of S ¹×S ² admitting an involution acting non-trivially on their Floer Homology. The examples are obtained by 0-surgery along certain composite amphicheiral knots whose SU(2)-representation variety satisfies a non-degeneracy condition. Received: 13 September 1999 / Revised version: 23 March 2000 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 57R58, 57M25 When this article was submitted and revised, the author was supported by an NSERC post-doctoral fellowship and Harvard University     

Shortest watchman routes in simple polygons

In this paper we present an O(n ⁴, log logn) algorithm to find a shortest watchman route in a simple polygon through a point,s, in its boundary. A watchman route is a route such that each point in the interior of the polygon is visible from at least one point along the route. S. Ntafos was supported in part by a grant from Texas Instruments, Inc.     

Lower bounds for boundary roughness for droplets in Bernoulli percolation

We consider boundary roughness for the ``droplet' created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least l², for large l. The maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull; we show that for large l this maximum is at least of order l^1/3(logl)^–2/3. This complements the upper bound of order l^1/3(logl)^–2/3 proved in [Al3] for the average local roughness. The exponent 1/3 on l here is in keeping with predictions from the physics literature for interfaces in two dimensions. The research of the first author was supported by NSF grant DMS-9802368. The research of the second author was supported by NSF grants DMS-9802368 and DMS-0103790.Mathematics Subject Classification (2000): Primary 60K35; Secondary 82B20, 82B43     

A volume-preserving counterexample to the Seifert conjecture

We prove that every 3-manifold possesses aC ¹, volume-preserving flow with no fixed points and no closed trajectories. The main construction is a volume-preserving version of the Schweitzer plug. We also prove that every 3-manifold possesses a volume-preserving,C ^∞ flow with discrete closed trajectories and no fixed points (as well as a PL flow with the same geometry), which is needed for the first result. The proof uses a Dehn-twisted Wilson-type plug which also preserves volume. The author was supported by an NSF Postdoctoral Fellowship, grant #DMS-9107908.     

Solutions for the prescribing mean curvature equation

By variational methods, for a kind of Yamabe problem whose scalar curvature vanishes in the unit ball BN and on the boundary S^N-1 the mean curvature is prescribed, we construct multi-peak solutions whose maxima are located on the boundary as the parameter tends to 0^＋ under certain assumptions. We also obtain the asymptotic behaviors of the solutions.     

AnO(h 6) quintic spline collocation method for fourth order two-point boundary value problems

AnO(h ⁶) collocation method based on quintic splines is developed and analyzed for general fourth-order linear two-point boundary value problems. The method determines a quintic spline approximation to the solution by forcing it to satisfy a high order perturbation of the original boundary value problem at the nodal points of the spline. A variation of this method is formulated as a deferred correction method. The error analysis of the new method and its numerical behavior is presented.This research was supported by AFOSR grant 84-0385.     

Braids and their monotone clones

In this paper we investigate properties of the monotone clones of certain ordered sets known asbraids. This class of ordered sets arose naturally in the study of how the clone of monotone functions on an ordered set could satisfy, or fail to satisfy, Mal'cev conditions. One version of the main result can be stated as follows. IfB is a finite braid with reachr(B)>2 (defined in the text), then the only idempotent order-preserving functionsfBⁿB are then projections. It then follows, for example, that no algebra of monotone functions on a finite braidB withr(B)>2 generates a congruence-modular variety.Dedicated to Bjarni Jónsson on the occasion of his 70th birthday.Presented by G. McNulty.This research was supported by ARC grant A68831070 (Davey) and in part by NSF grant 87-03540 (Nation) and 89-04014 (McKenzie).     

On the Circuit-cocircuit Intersection Conjecture

Oxley has conjectured that for k≥4, if a matroid M has a k-element set that is the intersection of a circuit and a cocircuit, then M has a (k−2)-element set that is the intersection of a circuit and a cocircuit. In this paper we prove a stronger version of this conjecture for regular matroids. We also show that the stronger result does not hold for binary matroids. The second author was partially supported by CNPq (grant no 302195/02-5) and the ProNEx/CNPq (grant no 664107/97-4).     

On the computation of the projective surgery obstruction groups

The computation of the projective surgery obstruction groupsL _n ^p (ZG), forG a hyperelementary finite group, is reduced to standard calculations in number theory, mostly involving class groups. Both the exponent of the torsion subgroup and the precise divisibility of the signatures are determined. ForG a 2-hyperelementary group, theL _n ^p (ZG) are detected by restriction to certain subquotients ofG, and a complete set of invariants is given for oriented surgery obstructions.Partially supported by NSERC grant A4000.Partially supported by NSF grant DMS-8610730(1) and the Danish Research Council.     

A characterization of domains in <Emphasis Type="Bold">C</Emphasis><Superscript>2</Superscript> with noncompact automorphism group

Let D be a bounded domain in C ² with a non-compact group of holomorphic automorphisms. Model domains for D are obtained under the hypotheses that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type. The author was supported by DST (India) Grant No.: SR/S4/MS-283/05 and in part by a grant from UGC under DSA-SAP, Phase IV.     

Is there always an extremal Teichmüller mapping?

Given a quasisymmetric self-homeomorphismh of the unit circleS ¹, letQ(h) be the set of all quasiconformal mappings with the boundary correspondenceh. In [1], it was shown that there existsh for which no extremal extension inQ(h) as a Teichmüller mapping is possible. This disproved some conjectures of long standing. In the example constructed there, the boundary correspondence has a single extremal quasiconformal extension. We show that even when there are infinitely many extremal extensions of the boundary values, it may still happen that none of the extensions is a Teichmüller mapping. An infinitesimal version of this result is also obtained. The research was supported by the China Postdoctoral Science Foundation and by the National Natural Science Foundation of China (Grant No. 10401036).     

On the countable definability of sets

LetQ be a connected set in ^p . Denote byD[Q] the set of all domains containingQ, and letW(Q) be the set of all convex domains fromD[Q]. We present tests for classesD[Q] andW(Q) (in the case whenQ is convex for the last one) to have a countable basis. The results are expressed in terms of properties of the boundary FrQ of the setQ.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 382–395, March, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-011-242.     

On regular semiovals in PG(2,q)

In this paper we prove that a point set in PG(2,q) meeting every line in 0, 1 or r points and having a unique tangent at each of its points is either an oval or a unital. This answers a question of Blokhuis and Szőnyi [1]. Research was partially supported by OTKA Grants T 043758, F 043772; the preparation of the final version was supported by OTKA Grant T 049662 and TéT grant E-16/04.