On the smoothness of the convex hull in negatively curved manifolds

In Hadamard manifolds the existence of suitable large convex sets is important for solving the Dirichlet problem at infinity. In this note we proveC ¹ boundary regularity of the convex hull of any compact setK away from points which lie on geodesics connecting points inK.The research was supported by the Hungarian Science Foundation Grant 4232 and by the Kuwait University Research Grant SM 080.     

Geodesic convexity in nonlinear optimization

The properties of geodesic convex functions defined on a connected RiemannianC ² k-manifold are investigated in order to extend some results of convex optimization problems to nonlinear ones, whose feasible region is given by equalities and by inequalities and is a subset of a nonlinear space.This research was supported in part by the Hungarian National Research Foundation, Grant No. OTKA-1044.     

Groups with super-exponential subgroup growth

We show that, if the subgroup growth of a finitely generated (abstract or profinite) groupG is super-exponential, then every finite group occurs as a quotient of a finite index subgroup ofG. The proof involves techniques from finite permutation groups, and depends on the Classification of Finite Simple Groups.The first author was partially supported by the Hungarian National Foundation for Scientific Research, Grant No. T7441. The second author was partially supported by the Israeli National Science Foundation.     

Some Geometrical Aspects of Efficient Points in Vector Optimization

We study efficient point sets in terms of extreme points, positive support points and strongly positive exposed points. In the case when the ordering cone has a bounded base, we prove that the efficient point set of a weakly compact convex set is contained in the closed convex hull of its strongly positive exposed points, thereby extending the Phelps theorem. We study also the density of positive proper efficient point sets. This research was supported by a Central Research Grant of Hong Kong Polytechnic University, Grant G-T 507. Research of the first author was also supported by the National Natural Science Foundation of P.R. China, Grant 10361008, and the Natural Science Foundation of Yunnan Province, China, Grant 2003A002M. Research of the second author was also supported by the Natural Science Foundation of Chongqing. Research of the third author was supported by a research grant from Australian Research Counsil.     

The limit shape of convex lattice polygons

It is proved here that, asn→∞, almost all convex (1/n)ℤ²-lattice polygons lying in the square [−1, 1]² are very close to a fixed convex set. This research was partially supported by Hungarian Science Foundation Grants 1907 and 1909.     

Finite and uniform stability of sphere coverings

By attaching cables to the centers of the balls and certain intersections of the boundaries of the balls of a ball covering ofE ^d with unit balls, we can associate to any ball covering a collection of cabled frameworks. It turns out that a finite subset of balls can be moved, maintaining the covering property, if and only if the corresponding finite subframework in one of the cabled frameworks is not rigid. As an application of this cabling technique we show that the thinnest cubic lattice sphere covering ofE ^d is not finitely stable. The first two authors were partially supported by the Hungarian National Science Foundation under Grant No. 326-0413.     

An optimization problem on subsets of the symmetric positive-semidefinite matrices

Motivated by the metricsstress problem in multidimensional scaling, the authors consider the more general problem of minimizing a strictly convex function on a particular subset ofR ^{n × n} . The subset in question is the intersection of a linear subspace with the symmetric positive-semidefinite matrices of rank p. Because of the rank restriction, this subset is not convex. Several equivalent formulations of this problem are derived, and the advantages and disadvantages of each formulation are discussed.Part of this research was conducted while the authors were visitors at the Center for Research on Parallel Computation, Rice University, Houston, Texas. The first author was partially supported by National Science Foundation Grant RII-89-05080.     

Positivity aspects of the Fantappiè transform

A Hilbert space approach to the classical Fantappiè transform, based on the concept of Gel'fand triples of locally convex spaces, leads to a novel proof of Martineau-Aizenberg duality theorem. A study of Fantappiè transforms of positive measures on the unit ball inC ⁿ relates ideas of realization theory of multivariate linear systems, locally convex duality and pluripotential theory. This is applied to obtain von Neumann type estimates on the joint numerical range of tuples of Hilbert space operators. Partially supported by National Science Foundation Grants DMS 0070639 and DMS 0322255. Partially supported by National Science Foundation Grant DMS 0100367.     

Vertical decomposition of arrangements of hyperplanes in four dimensions

We show that, for any collection ℋ ofn hyperplanes in ℜ⁴, the combinatorial complexity of thevertical decomposition of the arrangementA(ℋ) of ℋ isO(n ⁴ logn). The proof relies on properties of superimposed convex subdivisions of 3-space, and we also derive some other results concerning them. Work on this paper by Leonidas Guibas and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Leonidas Guibas was also supported by National Science Foundation Grant CCR-9215219. Work by Micha Sharir was also supported by National Science Foundation Grant CCR-91-22103, and by grants from the G.I.F.—the German Isreali Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences. Work by Jiří Matouŝek was done while he was visiting Tel Aviv University, and its was partially supported by a Humboldt Research Fellowship. Work on this paper by Dan Halperin was carried out while he was at Tel Aviv University.     

Conditions for a function to be a centralizer on an H*-algebra

Research partially supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA 1652.     

Hybrid Proximal-Point Methods for Common Solutions of Equilibrium Problems and Zeros of Maximal Monotone Operators

The purpose of this paper is to introduce and study two hybrid proximal-point algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of solutions to the equation 0∈Tx for a maximal monotone operator T in a uniformly smooth and uniformly convex Banach space X. Strong and weak convergence results of these two hybrid proximal-point algorithms are established, respectively. The research of L.C. Ceng was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Shanghai Leading Academic Discipline Project (S30405). The research of J.C. Yao was partially supported by Grant NSC 97-2115-M-110-001. Research was carried on within the agreement between National Sun Yat-Sen University of Kaohsiung, Taiwan and Pisa University, Pisa, Italy, 2008.     

Some geometric applications of Dilworth’s theorem

A geometric graph is a graph drawn in the plane such that its edges are closed line segments and no three vertices are collinear. We settle an old question of Avital, Hanani, Erdős, Kupitz, and Perles by showing that every geometric graph withn vertices andm>k ⁴ n edges containsk+1 pairwise disjoint edges. We also prove that, given a set of pointsV and a set of axis-parallel rectangles in the plane, then either there arek+1 rectangles such that no point ofV belongs to more than one of them, or we can find an at most 2·10⁵ k ⁸ element subset ofV meeting all rectangles. This improves a result of Ding, Seymour, and Winkler. Both proofs are based on Dilworth’s theorem on partially ordered sets. The research by János Pach was supported by Hungarian National Foundation for Scientific Research Grant OTKA-4269 and NSF Grant CCR-91-22103.     

Variable metric gradient projection processes in convex feasible sets defined by nonlinear inequalities

A capture-convergence rate theorem is proved for variable metric gradient projection processes near nonsingular local minimizers in convex feasible sets defined by nonlinear inequalities in a real Hilbert spaceX. The minimizers in question satisfy standard Kuhn-Tucker second-order sufficient conditions for local optimality.Investigation partially supported by National Science Foundation Research Grant No. DMS-85-03746.     

On a convex body with odd hadwiger number

We construct a convex body K ⊃ ℝ³ such that the maximum number of mutually nonoverlapping translates of K which touch K is 15. The work is in part supported by grant no. T043556 of the Hungarian National Science Foundation (OTKA).     

On the distribution of imaginary parts of zeros of the Riemann zeta function,II

We continue our investigation of the distribution of the fractional parts of αγ, where α is a fixed non-zero real number and γ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to Montgomery’s pair correlation function and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function. The first author is supported by National Science Foundation Grant DMS-0555367. The second author is partially supported by the National Science Foundation and the American Institute of Mathematics (AIM). The third author is supported by National Science Foundation Grant DMS-0456615.     

Convex hull theorem for multiply connected domains in the plane with an estimate of the quasiconformal constant

For any multiply connected domain Ω in R2, let S be the boundary of the convex hull in H3 of R2\Ω which faces Ω. Suppose in addition that there exists a lower bound l > 0 of the hyperbolic lengths of closed geodesics in Ω. Then there is always a K-quasiconformal mapping from S to Ω, which extends continuously to the identity on S = Ω, where K depends only on l. We also give a numerical estimate of K by using the parameter l.     

Chebyshevian multistep methods for ordinary differential equations

Summary In this paper some theory of linear multistep methods fory ^(r) (x)=f(x,y) is extended to include smooth, stepsize-dependent coefficients. Treated in particular is the case where exact integration of a given set of functions is desired.Work on this paper was supported in part by U.S. Army Research Office (Durham) Grant DA-ARO(D)-31-124-G1050 and National Science Foundation Grant GP-23655 with The University of Texas at Austin.     

Existence theory for generalized nonlinear complementarity problems

The nonlinear complementarity problem is generalized by replacing the usual nonnegative ordering ofR ⁿ by an ordering generated by a convex cone. Two new classes of operators are introduced, each of which is used to guarantee the existence of a solution to the generalized problem. The new classes can be seen to be broader than previously studied classes. In addition, conditions are presented under which the solution set of the generalized linear complementarity problem is shown to have at most a finite number of solutions.This research was partially supported by National Science Foundation, Grant No. GP-16293, and constitutes part of the junior author's doctoral thesis. The authors are indebted to Dr. Carlton E. Lemke for many helpful discussions.     

A generalization of the Frank—Wolfe theorem

The Frank—Wolfe theorem states that a quadratic function, bounded below on a nonempty polyhedral convex set, attains its infimum there. This paper gives sufficient conditions under which a function either attains its infimum on a nonempty polyhedral convex set or is unbounded below on some halfline of that set. Quadratic functions are shown to satisfy these sufficient conditions.Research and reproduction of this report were partially supported by the National Science Foundation Grant MCS76-81259; and the Office of Naval Research Contract N00014-75-C-0267.     

Uniqueness in the dirichlet problem for some elliptic operators with discontinuous coefficients

Uniqueness is proved for the Dirichlet problem for second order nondivergence form elliptic operators with coefficients continuous except at a countable set of points having at most one accumulation point. Moreover, gradient estimates are proved.The authors are partially supported by the National Science Foundation Grant no. NSF/DMS 8421377-04.