Means on equivalence relations

Suppose that X is a Polish space and E is a countable Borel equivalence relation on X. We show that if there is a Borel assignment of means to the equivalence classes of E, then E is smooth. We also show that if there is a Baire measurable assignment of means to the equivalence classes of E, then E is generically smooth. Research partially supported by NSF Grants DMS-9987437 and DMS-0455285. Research partially supported by NSF VIGRE Grant DMS-0502315.     

The rotation number and the herpolhode angle in Euler’s top

We describe the relation between the rotation number and the herpolhode angle in the rotation of the force-free rigid body.Received: September 23, 2002Research partially supported by NSERC.Research partially supported by European Community funding for the Research and Training Network MASIE (HPRN-CT-2000-00113).     

Solving combinatorial optimization problems using Karmarkar's algorithm

We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.Computational facilities provided by the Cornell Computational Optimization Project supported by NSF Grant DMS-8706133 and by the Cornell National Supercomputer Facility. The Cornell National Supercomputer Facility is a resource of the Center for Theory and Simulation in Science and Engineering at Cornell Unversity, which is funded in part by the National Science Foundation, New York State, and the IBM Corporation. The research of both authors was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by ONR Grant N00014-90-J-1714.Research partially supported by NSF Grant ECS-8602534 and by ONR Contract N00014-87-K-0212.     

The Lévy-Hinčin representation for random compact convex subsets which are infinitely divisible under Minkowski addition

Summary This paper establishes the Lévy-Hinčin representation for all random compact convex subsets of ℝ which are infinitely divisible for Minkowski addition. Research partially supported by NSF grants No. MCS 8100728 and DMS-8318610 Research partially supported by NSF grant No. MCS 8301326     

Operator means and networks

The operator means of two variables are known to be weighted arithmetic averages of weighted harmonic means. The passage from 2 to 3 or more variables offers conceptual difficulties, and besides the symmetric means of Anderson, Morley, and Trapp next to nothing is known.In this paper we construct and study a new family of operator means in any number of variables. Our construction is motivated by network considerations and it generalizes the above mentioned symmetric means.Research partially supported by NSF Grant No. DMS 8702371.     

The dimension of cycle-free orders

The existence of a four-dimensional cycle-free order is proved. This answers a question of Ma and Spinrad. Two similar problems are also discussed.Research partially supported by Office of Naval Research grant N00014-90-J-1206Research partially supported by the National Science Foundation under grant DMS     

Super efficiency in convex vector optimization

We establish a Lagrange Multiplier Theorem for super efficiency in convex vector optimization and express super efficient solutions as saddle points of appropriate Lagrangian functions. An example is given to show that the boundedness of the base of the ordering cone is essential for the existence of super efficient points.Research is supported partially by NSERC.Research is supported partially by NSERC and Mount St. Vincent University grant.     

The Manin—Drinfeld theorem and Ramanujan sums

The Manin—Drinfeld theorem asserts the finiteness of the cuspidal divisor class group of a modular curve corresponding to a congruence subgroup. The purpose of the note is to draw attention to the connection between this theorem and Ramanujan sums, and to the question of what happens for non-congruence subgroups. Research partially supported by an NSERC grant. Alfred P Sloan Fellow. Research partially supported by the NSF.     

The horospherical geometry of surfaces in hyperbolic 4-space

We study some geometrical properties associated to the contacts of surfaces with hyperhorospheres inH ₊ ⁴ (−1). We introduce the concepts of osculating hyperhorospheres, horobinormals, horoasymptotic directions and horospherical points and provide conditions ensuring their existence. We show that totally semiumbilical surfaces have orthogonal horoasymptotic directions. Work partially supported by Grant-in-Aid for Scientific Research, JSPS, No. 12874007. Work partially supported by Grant-in-Aid for Scientific Research, JSPS, No. 12000266. Work partially supported by DGCYT grant no. BFM2003-02037.     

Local and parallel finite element algorithms for the stokes problem

Based on two-grid discretizations, some local and parallel finite element algorithms for the Stokes problem are proposed and analyzed in this paper. These algorithms are motivated by the observation that for a solution to the Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shape-regular grids. Y. He was partially subsidized by the NSF of China 10671154 and the National Basic Research Program under the grant 2005CB321703; A. Zhou was partially supported by the National Science Foundation of China under the grant 10425105 and the National Basic Research Program under the grant 2005CB321704; J. Li was partially supported by the NSF of China under the grant 10701001. J. Xu was partially supported by Alexander von Humboldt Research Award for Senior US Scientists, NSF DMS-0609727 and NSFC-10528102.     

Bifurcations in a predator-prey model with memory and diffusion II: Turing bifurcation

Research partially supported by the Hungarian National Foundation for Scientific Research, grant numbers 1186, 1994.     

Some universal graphs

It is shown that various classes of graphs have universal elements. In particular, for eachn the class of graphs omitting all paths of lengthn and the class of graphs omitting all circuits of length at leastn possess universal elements in all infinite powers. Research partially supported by Hungarian Science Research Fund No. 1805. Research partially supported by NSERC of Canada Grant #A8948.     

Bifurcations in a predator-prey model with memory and diffusion. I: Andronov-Hopf bifurcation

Research partially supported by the Hungarian Foundation for Scientific Research, grant no. 1186, 1994.     

Ideals and Clots in Pointed Regular Categories

We clarify the relationship between ideals, clots, and normal subobjects in a pointed regular category with finite coproducts. Partially supported by South African NRF Research partially supported by the Hungarian National Foundation for Scientific Research grant no. K61007.     

Lower order eigenvalues of Dirichlet Laplacian

In this paper, we investigate an eigenvalue problem for the Dirichlet Laplacian on a domain in an n-dimensional compact Riemannian manifold. First we give a general inequality for eigenvalues. As one of its applications, we study eigenvalues of the Laplacian on a domain in an n-dimensional complex projective space, on a compact complex submanifold in complex projective space and on the unit sphere. By making use of the orthogonalization of Gram–Schmidt (QR-factorization theorem), we construct trial functions. By means of these trial functions, estimates for lower order eigenvalues are obtained. Qing-Ming Cheng research was partially supported by a Grant-in-Aid for Scientific Research from JSPS. Hejun Sun and Hongcang Yang research were partially supported by NSF of China.     

Complete solution of a family of quartic Thue equations

This paper evolved from a visit of the first author to the University of Debrecen, Hungary, which was supported by the Austrian-Hungarian Science Cooperation project Nr. 10-U-3. Research of the second author was partially supported by Hungarian National Foundation for Scientific Research Grant 1641/90.     

An axiomatization of the consistent non-transferable utility value

The Maschler–Owen consistent value for non-transferable utility games is axiomatized, by means of a marginality axiom.Previous versions: October 2003 (Center for the Study of Rationality DP-337), December 2004. Research partially supported by a grant of the Israel Academy of Sciences and Humanities. The author thanks Andreu Mas-Colell, Bezalel Peleg, Peter Sudhölter, the referees, and the editor for their comments and suggestions.     

Duality for almost convex optimization problems via the perturbation approach

We deal with duality for almost convex finite dimensional optimization problems by means of the classical perturbation approach. To this aim some standard results from the convex analysis are extended to the case of almost convex sets and functions. The duality for some classes of primal-dual problems is derived as a special case of the general approach. The sufficient regularity conditions we need for guaranteeing strong duality are proved to be similar to the ones in the convex case. The research of the first and third authors was partially supported by DFG (German Research Foundation), project WA 922/1. The research of the second author was supported by the grant PN II, ID 523/2007.     

A version of Rolle's theorem and applications

This paper deals with global injectivity of vector fields defined on euclidean spaces. Our main result establishes a version of Rolle's Theorem under generalized Palais-Smale conditions. As a consequence of this, we prove global injectivity for a class of vector fields defined on n-dimensional spaces.Research partially supported by CNPq/Brazil under grant # 307014/89-4Research partially supported by CNPq/Brazil under grant # 301251/78-9     

Bootstrapped confidence bands for percentile lifetime

Summary We propose bootstrapped confidence bands for the percentile lifetime function. Our method is based on a joint approximation of the empirical and quantile processes and their bootstrapped counterparts. Modest simulations support the results. Confidence bands are applied to quantile and median residual lifetimes of tractor rear brakes. Research done while at Carleton University Norman Paterson Centre: also partially supported by NSERC Canada grants of M. Cs?rgő. Research supported in part by a NSERC Canada operating grant. Research done while at Carleton University; also partially supported by NSERC Canada grants of m. Cs?rgő and D. A. Dawson and by an EMR Canada grant of M. Cs?gő. Research supported in part by a United States Department of Agriculture CSRS grant and a United States NSF grant of D. M. Bates, G. Wahba and B. S. Yandell.