Downward Sets and their separation and approximation properties

We develop a theory of downward subsets of the space ^I, where I is a finite index set. Downward sets arise as the set of all solutions of a system of inequalities x^I,f_t(x)0 (tT), where T is an arbitrary index set and each f _t (tT) is an increasing function defined on ^I. These sets play an important role in some parts of mathematical economics and game theory. We examine some functions related to a downward set (the distance to this set and the plus-Minkowski gauge of this set, which we introduce here) and study lattices of closed downward sets and of corresponding distance functions. We discuss two kinds of duality for downward sets, based on multiplicative and additive min-type functions, respectively, and corresponding separation properties, and we give some characterizations of best approximations by downward sets. Some links between the multiplicative and additive cases are established.     

Approximation of the efficient point set by perturbation of the ordering cone

A vector optimization problem is given by a feasible setZ ⁿ , a vector-valued objective functionf: ^{n l} , and an ordering coneC ^l . We perturb the ordering cone in such a way that the weakly efficient points of the perturbed vector optimization problem given byZ, f, and the perturbed cone are efficient points of the original problem. Especially this means that scalarization methods, which compute in general only weakly efficient points, determine efficient points of the original problem, when they were applied to the perturbed problem.It turns out that the efficient points are the limits of weakly efficient points of the perturbed problems, letting the perturbation tend to zero. On the basis of this, a reference point algorithm is formulated. Finally, we apply this algorithm to a structural optimization problem.     

Reducible Dupin submanifolds

Pinkall's standard constructions for obtaining a Dupin hypersurface W in ^N from a Dupin hypersurface M in ⁿ , N>n, are studied in the context of Lie sphere geometry. It is shown that a compact Dupin hypersurface W in ^N with g distinct principal curvatures at each point is reducible to a compact Dupin hypersurface M in ⁿ if and only if g=2.This research was supported by NSF Grant No. DMS 87-06015.     

Polyedrische 2-Mannigfaltigkeiten mit wenigen nicht-konvexen Ecken

LetP denote a polyhedral 2-manifold in ³, i.e. a 2-dimensional cell-complex in ³ whose underlying point-set is a closed connected 2-manifold. A vertexv ofP is called convex if at least one of the two components into whichP divides a sufficiently small ball centered atv is convex. It is shown that every polyhedral 2-manifold in ³ of genusg>–1 contains at least five non-convex vertices and that for every positive integerg this bound is attained, i.e. there exists a polyhedral 2-manifold in ³ of genusg with precisely five non-convex vertices.

---

---

Mit 2 Abbildungen     

Complete controllability

A linear autonomous control system in ⁿ is said to be completely controllable iff there existsT>0 such that eachx ⁿ can be steered to anyy ⁿ in timeT. This paper presents a geometric characterization of this property in the case in which there are constraints on the values which the control maps can assume. A necessary and sufficient condition to get instant controllability (i.e., complete controllability for anyT>0) is also derived. This condition generalizes the well-known Kalman condition to the constrained case.     

On Non-Negative Elliptic-Type Differential Inequalities in ℝ n , Vanishing Almost Everywhere on Some Open Subset in ℝ n

It is proved in this article that any generalized solution of a sufficiently general class of elliptic-type differential inequalities in  ⁿ that is non-negative almost everywhere in  ⁿ and vanishes almost everywhere on an open set ⁿ is trivial in  ⁿ .     

On Two-Point Boundary Conditions in Optimal Control Problems

Let x=g(t,x(t),u(t)) be the governing equation of an optimal control problem with two-point boundary conditions h ₀(x(a))+h ₁(x(b)) = 0, where x: [a,b] ⁿ is continuous, u: [a,b] ^k-n is piecewise continuous and left continuous, h₀,h₁: ^{n q} are continuously differentiable, and g:[a,b]× ^{k n} is continuous. The paper finds functions _i C¹([a,b]× ⁿ ) such that (x(t),u(t)) is a solution of the governing equation if and only if

Norming operators for generalized domains of attraction

A sequence of independent and identically distributed random vectorsX _n on ^k is said to belong to the generalized domain of attraction of a nondegenerate random vectorY on ^k provided that there exist linear operatorsA _n on ^k and nonrandom constantsb _n ^k such that the centered and normalized partial sumsA _n (X ₁++X _n –b _n converge in distribution toY. In this paper we show that the sequence of norming operatorsA _n can always be chosen to vary regularly.Partially supported by NSF Grant DMS-91-03131 at Albion College.     

A partial solution of the Pompeiu problem

A nonempty bounded open subset D of ⁿ is said to have the Pompeiu property if and only if for every continuous complex-valued function f on ⁿ which does not vanish identically there is a rigid motion of ⁿ onto itself — taking D onto (D) — such that the integral of f over (D) is not zero. This article gives a partial solution of the Pompeiu problem, the problem of finding all sets D with the Pompeiu property.In the special case that D is the interior of a homeomorphic image of an(n–1)-dimensional sphere, the main result states that if D has a portion of an(n–1)-dimensional real analytic surface on its boundary, then either D has the Pompeiu property or any connected real analytic extension of the surface also lies on the boundary of D. Thus, for example, any such region D having a portion of a hyperplane as part of its boundary must have the Pompeiu property, since the entire hyperplane cannot lie in the boundary of the bounded set D.The research for this paper was done in part while on sabbatical at the Courant Institute of Mathematical Sciences, New York University.     

Random continued fractions and inverse Gaussian distribution on a symmetric cone

In this paper we introduce the inverse Gaussian and Wishart distributions on the cone of real (n, n) symmetric positive definite matricesH _n ⁺ () and more generally on an irreducible symmetric coneC. Then we study the convergence of random continued fractions onH _n ⁺ () andC by means of real Lagrangians forH _n ⁺ () and by new algebraic identities on symmetric cones forC. Finally we get a characterization of the inverse Gaussian distribution onH _n ⁺ () andC.     

The (n + 1)2 m -ray algorithm: a new simplicial algorithm for the variational inequality problem on ℝ + m ×S n

In this paper we propose a variable dimension simplicial algorithm for solving the variational inequality problem on the cross product of the nonnegative orthant ₊ ^m of them-dimensional Euclidean space ^m and then-dimensional unit simplexS ⁿ of ⁿ⁺¹. Starting from an arbitrary point (u, v) є ₊ ^m ×S ⁿ, the algorithm generates a piecewise linear path in ₊ ^m ×S ⁿ. The path is traced by making alternately linear programming pivot operations and replacement steps in an appropriate simplicial subdivision of ₊ ^m ×S ⁿ. The algorithm differs from the thus far known algorithm in the number of directions in which it may leave the starting point. More precisely, the algorithm has (n+1)2 ^m rays to leave the starting point whereas the existing algorithm hasn+m+1 rays. A convergence condition is presented and the accuracy estimation of an approximate solution generated is also given.     

Isospectral sets for akns systems on the unit interval with generalized periodic boundary conditions

We analyze isospectral sets of potentials associated with generalized periodic boundary conditionsB inSL (2, ) for the 2 ×2AKNS systems on the unit interval. WhenB is a rotation we get the usual periodic case. WhenB is not a rotation isospectral sets are cylindrical real analytic submanifolds ofL ² ([0, 1])² ×SL(2, ) and their sections for fixed boundary conditions are real analytic submanifolds ofL ² ([0, 1])².     

Rigid Sets of Dimension n – 1 in ℝ n

We give conditions allowing an intrinsic isometry on a dense subset to be extended to an isometry of the whole set. This enables us to find examples of (n-1)-dimensional sets rigid in ⁿ .     

On homogenization of periodic parabolic systems

We study homogenization in the small period limit for a periodic parabolic Cauchy problem in ^d and prove that the solutions converge in L ₂(^d) to the solution of the homogenized problem for each t > 0. For the L₂(^d)-norm of the difference, we obtain an order-sharp estimate uniform with respect to the L ₂(^d)-norm of the initial value.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 86–90, 2004Original Russian Text Copyright © by T. A. SuslinaSupported by RFBR grant No. 02-01-00798.     

Die konvexen Polytope im ℝ4, bei denen alle Facetten reguläre Tetraeder sind

All Convex Polytopes in ⁴ the Facets of Which Are Regular Tetrahedra. Continuing in ⁿ (n4) the study of convex polytopes the facets of which are regular, it is proved: Regular polytopes and the two bipyramids over the tetrahedron and the icosahedron are the only convex polytopes in ⁴ the facets of which are regular tetrahedra.     

On the connectivity of unit distance graphs

For a number fieldK , consider the graphG(K^d), whose vertices are elements ofK ^d, with an edge between any two points at (Euclidean) distance 1. We show thatG(K²) is not connected whileG(K^d) is connected ford 5. We also give necessary and sufficient conditions for the connectedness ofG(K³) andG(K⁴).     

Submanifolds of restricted type

A submanifoldM ⁿ of the Euclidean space ^m is said to be of restricted type if the shape operator of the mean curvature vector is the tangent part of a fixed linear transformation of ^m. We show that a hypersurface of restricted type is either minimal, a part of the product of a sphere and a linear subspace or a cylinder on a plane curve of restricted type. Finally we classify plane curves of restricted type.Supported by a research fellowship of the Research Council of the Katholieke Universiteit LeuvenResearch Assistant of the National Fund for Scientific Research (Belgium)     

Generalized solutions to semilinear hyperbolic systems

In this article semilinear hyperbolic first order systems in two variables are considered, whose nonlinearity satisfies a global Lipschitz condition. It is shown that these systems admit unique global solutions in the Colombeau algebraG(²). In particular, this provides unique generalized solutions for arbitrary distributions as initial data. The solution inG(²) is shown to be consistent with the locally integrable or the distributional solutions, when they exist.     

Bochner-Riesz means of functions in weak-L p

The Bochner-Riesz means of order 0 for suitable test functions on ^N are defined via the Fourier transform by . We show that the means of the critical index , do not mapL ^p,( ^N ) intoL ^p,( ^N ), but they map radial functions ofL ^p,( ^N ) intoL ^p,( ^N ). Moreover, iff is radial and in theL ^p,( ^N ) closure of test functions,S _R f(x) converges, asR+, tof(x) in norm and for almost everyx in ^N . We also observe that the means of the function|x| ^–N/p, which belongs toL ^p,( ^N ) but not to the closure of test functions, converge for nox.     

On the semiboundedness ofδ-perturbations of the laplacian supported by curves with angle points

Elementary self-adjoint perturbations of the Laplacian supported by curves with singular angle points in ³ and ⁴ are studied. The perturbations are shown to be semibounded in ³ and not semibounded in ⁴. In the latter case semiboundedness may take place in subspaces with a given symmetry, as simple examples illustrate.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 1, pp. 3–17, October, 1995.