Holomorphic functions which are highly nonintegrable at the boundary

Given a bounded convex domainD in ℂ^N with smooth boundary and a positive continuous function ϕ ond, it is proved that there is a holomorphic functionf onD such that |f|ϕ is nonintegrable onM∩D wheneverM is a real submanifold of a neighbourhood of a point ofbD which intersectsbD transversely.     

A geometric characterization of inner product spaces

A normed linear spaceE is an inner product space if (and only if) there is a nontrivial closed bounded setD inE such that each translate ofD which misses the origin is taken by the inversion ofE to a convex set. Research supported in part by NSF Grant No. MCS 79-02734.     

The Dirichlet problem of a discontinuous Markov process

Given a Markov process satisfying certain general type conditions, whose paths are not assumed to be continuous. LetD be an open subset of the state spaceE. Any bounded function defined on the complement ofD extends to be a function onE such that it is harmonic inD and satisfies the Dirichlet boundary condition at any regular boundary point ofD. The relation between harmonic functions and the characteristic operator of the given process is discussed.     

Chaotic Motion of the N-Vortex Problem on a Sphere: I. Saddle-Centers in Two-Degree-of-Freedom Hamiltonians

We study the motion of N point vortices with N∈ℕ on a sphere in the presence of fixed pole vortices, which are governed by a Hamiltonian dynamical system with N degrees of freedom. Special attention is paid to the evolution of their polygonal ring configuration called the N -ring, in which they are equally spaced along a line of latitude of the sphere. When the number of the point vortices is N=5n or 6n with n∈ℕ, the system is reduced to a two-degree-of-freedom Hamiltonian with some saddle-center equilibria, one of which corresponds to the unstable N-ring. Using a Melnikov-type method applicable to two-degree-of-freedom Hamiltonian systems with saddle-center equilibria and a numerical method to compute stable and unstable manifolds, we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle-centers and hence chaotic motions occur. Especially, the evolution of the unstable N-ring is shown to be chaotic.      

Steiner symmetrals and their distance from a ball

It is known that given any convex bodyK ⊂ ℝ ⁿ there is a sequence of suitable iterated Steiner symmetrizations ofK that converges, in the Hausdorff metric, to a ball of the same volume. Hadwiger and, more recently, Bourgain, Lindenstrauss and Milman have given estimates from above of the numberN of symmetrizations necessary to transformK into a body whose distance from the equivalent ball is less than an arbitrary positive constant. In this paper we will exhibit some examples of convex bodies which are “hard to make spherical”. For instance, for any choice of positive integersn≥2 andm, we construct ann-dimensional convex body with the property that any sequence ofm symmetrizations does not decrease its distance from the ball. A consequence of these constructions are some lower bounds on the numberN.     

Konstruktion regulärer Hüllen konstanter Breite

LetM be a compact, convex set of diameter 2 inE ^d. There exists a bodyK of constant width 2 containingM such that every symmetry ofM is one ofK and every singular boundary point ofK is a boundary point ofM, for which the set of antipodes inK is the convex hull of the antipodes, which are already inM.

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Mit 1 Abbildung     

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     

On Functional Separately Convex Hulls

Let D be a set of vectors in R ^d . A function f: R ^d → R is called D-convex if its restriction to each line parallel to a nonzero vector of D is a convex function. For a set A⊆ R ^d , the functional D-convex hull of A, denoted by co ^D (A) , is the intersection of the zero sets of all nonnegative D -convex functions that are 0 on A . We prove some results concerning the structure of functional D -convex hulls, e.g., a Krein—Milman-type theorem and a result on separation of connected components. We give a polynomial-time algorithm for computing co ^D (A) for a finite point set A (in any fixed dimension) in the case of D being a basis of R ^d (the case of separate convexity). This research is primarily motivated by questions concerning the so-called rank-one convexity, which is a particular case of D -convexity and is important in the theory of systems of nonlinear partial differential equations and in mathematical modeling of microstructures in solids. As a direct contribution to the study of rank-one convexity, we construct a configuration of 20 symmetric 2 x 2 matrices in a general (stable) position with a nontrivial functionally rank-one convex hull (answering a question of K. Zhang on the existence of higher-dimensional nontrivial configurations of points and matrices). Received October 3, 1995, and in revised form June 24, 1996.     

A graph theoretic approach to matrix inversion by partitioning

LetM be a square matrix whose entries are in some field. Our object is to find a permutation matrixP such thatPM P ^–1 is completely reduced, i.e., is partitioned in block triangular form, so that all submatrices below its diagonal are 0 and all diagonal submatrices are square and irreducible. LetA be the binary (0, 1) matrix obtained fromM by preserving the 0's ofM and replacing the nonzero entries ofM by 1's. ThenA may be regarded as the adjacency matrix of a directed graphD. CallD strongly connected orstrong if any two points ofD are mutually reachable by directed paths. Astrong component ofD is a maximal strong subgraph. Thecondensation D ^* ofD is that digraph whose points are the strong components ofD and whose lines are induced by those ofD. By known methods, we constructD ^* from the digraph,D whose adjacency matrixA was obtained from the original matrixM. LetA ^* be the adjacency matrix ofD ^*. It is easy to show that there exists a permutation matrixQ such thatQA ^* Q ^–1 is an upper triangular matrix. The determination of an appropriate permutation matrixP from this matrixQ is straightforward.This was an informal talk at the International Symposium on Matrix Computation sponsored by SIAM and held in Gatlinburg, Tennessee, April 24–28, 1961 and was an invited address at the SIAM meeting in Stillwater, Oklahoma on August 31, 1961     

Relative Equilibria of the (1+N)-Vortex Problem

We examine existence and stability of relative equilibria of the n-vortex problem specialized to the case where N vortices have small and equal circulation and one vortex has large circulation. As the small circulation tends to zero, the weak vortices tend to a circle centered on the strong vortex. A special potential function of this limiting problem can be used to characterize orbits and stability. Whenever a critical point of this function is nondegenerate, we prove that the orbit can be continued via the Implicit Function Theorem, and its linear stability is determined by the eigenvalues of the Hessian matrix of the potential. For N≥3 there are at least three distinct families of critical points associated to the limiting problem. Assuming nondegeneracy, one of these families continues to a linearly stable class of relative equilibria with small and large circulation of the same sign. This class becomes unstable as the small circulation passes through zero and changes sign. Another family of critical points which is always nondegenerate continues to a configuration with small vortices arranged in an N-gon about the strong central vortex. This class of relative equilibria is linearly unstable regardless of the sign of the small circulation when N≥4. Numerical results suggest that the third family of critical points of the limiting problem also continues to a linearly unstable class of solutions of the full problem independent of the sign of the small circulation. Thus there is evidence that linearly stable relative equilibria exist when the large and small circulation strengths are of the same sign, but that no such solutions exist when they have opposite signs. The results of this paper are in contrast to those of the analogous celestial mechanics problem, for which the N-gon is the only relative equilibrium for N sufficiently large, and is linearly stable if and only if N≥7.     

Donovan’s conjecture,crossed products and algebraic group actions

Donovan’s conjecture, on blocks of finite group algebras over an algebraically closed field of prime characteristicp, asserts that for any finitep-groupD, there are only finitely many Morita equivalence classes of blocks with defect groupD. The main result of this paper is a reduction theorem: It suffices to prove the conjecture for groups generated by conjugates ofD. A number of other finiteness results are proved along the way. The main tool is a result on actions of algebraic groups.     

Asymmetric prime ends

Summary Each simply connected domain in the plane has at most countably many prime ends whose right and left wings do not coincide. On the other hand, to each countable setE on the unit circleC there corresponds a function which is holomorphic and univalent in the unit diskD and which has the property that it carries each point ofE and no point ofC\E onto a prime end with unequal wings.     

Stationary Configurations of Point Vortices on a Cylinder

In this paper we study the problem of constructing and classifying stationary equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex positions, we derive an ordinary differential equation satisfied by the polynomials. We prove that this equation can be used to find any stationary configuration. The multivortex systems containing point vortices with circulation Γ₁ and Γ₂ (Γ₂ = ?μΓ₁) are considered in detail. All stationary configurations with the number of point vortices less than five are constructed. Several theorems on existence of polynomial solutions of the ordinary differential equation under consideration are proved. The values of the parameters of the mathematical model for which there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface are found. New point vortex configurations are obtained.     

Unions of increasing and intersections of decreasing sequences of convex sets

We prove that a convex setC is a polytope if and only ifC is not the union of any strictly increasing sequence of convex sets. In addition, we attempt (with partial success) to characterize, in intrinsic geometric terms, those convex subsetsC of a convex setX such thatC is not the intersection of any strictly decreasing sequence of convex subsets ofX.     

Covering the plane with convex polygons

It is proved that for any centrally symmetric convex polygonal domainP and for any natural numberr, there exists a constantk=k(P, r) such that anyk-fold covering of the plane with translates ofP can be split intor simple coverings.     

Projections of 3-polytopes

Given any 3-dimensional convex polytopeP, and any simple circuitC in the 1-skeleton ofP, there is a convex polytopeP′ combinatorially equivalent toP, and a direction such that ifP′ is projected orthogonally in this direction, then the inverse image of the boundary of the projection is the circuit inP′ corresponding to the circuitC inP. Research supported by NSF Grant GP-8470.     

Moduli of non-dentability and the radon-nikodým property in banach spaces

We show that for a separable Banach spaceX failing the Radon-Nikodym property (RNP), andε > 0, there is a symmetric closed convex subsetC of the unit ball ofX such that every extreme point of the weak-star closure ofC in the bidualX** has distance fromX bigger than 1 −ε. An example is given showing that the full strength of this theorem does not carry over to the non-separable case. However, admitting a renorming, we get an analogous result for this theorem in the non-separable case too. We also show that in a Banach space failing RNP there is, forε > 0, a convex setC of diameter equal to 1 such that each slice ofC has diameter bigger than 1 −ε. Some more related results about the geometry of Banach spaces failing RNP are given.     

Generalized convex kernels

The notion of the convex kernel of a setD is generalized to that of then-th order kernel ofD. Such kernels are studied for compact, simply connected subsets of the Euclidean plane. In particular, it is shown that under certain circumstances [see Theorem 4 and also section 5], these kernels have rather simple structures. One of the authors was supported by NSF grant GP-1592.     

Renormages de Quelquesℒ(K)

LetD([0, 1]) be the space of left continuous real valued functions on [0, 1] which have a right limit at each point. We show thatD([0, 1]) has no equivalent norm which is Gateau differentiable. Hence the class of spaces which can be renormed by a Gateau differentiable norm fails the three spaces property. We show that there is no norm onℒ([0, Ω]) such that its dual is strictly convex. However, there is an equivalent Fréchet differentiable norm on this space.