Variational inequalities over Euclidean balls

This paper investigates solution stability of parametric variational inequalities over Euclidean balls in finite dimensional spaces. We provide exact formulas for computing required coderivatives of the normal cone mappings to Euclidean balls via the initial data. On the basis of these formulas, we establish necessary and sufficient conditions for Lipschitzian stability of the solution maps of the aforementioned variational inequalities.     

Stochastic matrices over cones

The notion of a stochastic operator in an ordered Banach space is specialized to a finite dimensional ordered real vector space. The classical limit theorems are obtained, and an application is made to non-homogeneous Markov chains. Finally, groups of nonnegative matrices are discussed.     

Coverings of 3-manifolds by open balls and two open solid tori

Hempel and McMillan showed that a closed 3-manifold that can be covered by three open balls is a connected sum of S³- and S²-bundles over S¹. In this paper we obtain a classification of all closed 3-manifolds that can be covered by two open balls and one open solid torus or by one open ball and two open solid tori.     

Stochastic matrices over cones

The notion of a stochastic operator in an ordered Banach space is specialized to a finite dimensional ordered real vector space. The classical limit theorems are obtained, and an application is made to non-homogeneous Markov chains. Finally, groups of nonnegative matrices are discussed.     

Cohomology of tori over p-adic curves

 We establish a duality in the cohomology of arbitrary tori over smooth but not necessarily projective curves over a p-adic field. This generalises Lichtenbaum–Tate duality between the Picard group and the Brauer group of a smooth projective curve. Received: 28 January 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 14G20, 14F22, 14L15, 11S25     

Multiple criteria problems over Minkowski balls

Under study are some vector optimization problems over the space of Minkowski balls, i.e., symmetric convex compact subsets in Euclidean space. A typical problem requires to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a symmetric convex body $\mathfrak{x}$ , we try to maximize the volume of $\mathfrak{x}$ and minimize the width of $\mathfrak{x}$ simultaneously.     

Unsmoothing over balls via plane wave decomposition

Let

Lyapunov-type least-squares problems over symmetric cones

The Lyapunov-type least-squares problem over symmetric cone is to find the least-squares solution of the Lyapunov equation with a constraint of symmetric cone in the Euclidean Jordan algebra, and it contains the Lyapunov-type least-squares problem over cone of semidefinite matrices as a special case. In this paper, we first give a detailed analysis for the image of Lyapunov operator in the Euclidean Jordan algebra. Relying on these properties together with some characterizations of symmetric cone, we then establish some necessary and?or sufficient conditions for solution existence of the Lyapunov-type least-squares problem. Finally, we study uniqueness of the least-squares solution.     

Counting cones over reducible cubic scrolls

Here we construct an explicit parameter space for cones in ℙ ⁿ for 4 ≤ n ≤ 6 with vertices of codimension 4 and 5, respectively (when this makes sense), over a reducible cubic scroll of codimension 2 in a hyperplane (in ℙ ⁿ ) and use it to compute the numbers of them that are incident to the appropriate number of linear spaces.     

Enumeration of cones over cubic scrolls

We enumerate the cones over cubic scrolls satisfying appropriate incidence conditions to lines. The first and second authors were partially supported by CNPq. The third author was partially supported by PADCT/CNPq.     

Generalized nonsmooth invexity over cones in vector optimization

In this paper K-nonsmooth quasi-invex and (strictly or strongly) K-nonsmooth pseudo-invex functions are defined. By utilizing these new concepts, the Fritz–John type and Kuhn–Tucker type necessary optimality conditions and number of sufficient optimality conditions are established for a nonsmooth vector optimization problem wherein Clarke’s generalized gradient is used. Further a Mond Weir type dual is associated and weak and strong duality results are obtained.     

Higher order duality in vector optimization over cones

In this paper higher order cone convex, pseudo convex, strongly pseudo convex, and quasiconvex functions are introduced. Higher order sufficient optimality conditions are given for a weak minimum, minimum, strong minimum and Benson proper minimum solution of a vector optimization problem. A higher order dual is associated and weak and strong duality results are established under these new generalized convexity assumptions.     

Majorants of functions with vanishing integrals over balls

We prove the existence of nontrivial functions in ℝⁿ, n ≥ 2, with vanishing integrals over balls of fixed radius and given majorant of growth. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 857–861, June, 2008.     

Nondifferentiable multiobjective symmetric dual programs over cones

Wolfe and Mond-Weir type nondifferentiable multiobjective symmetric dual programs are formulated over arbitrary cones and appropriate duality theorems are established under K-preinvexity/K-convexity/pseudoinvexity assumptions.     

Lie bialgebra structures on derivation Lie algebra over quantum tori

We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H ¹(W, W ? W) is trivial.     

Smoothing algorithms for complementarity problems over symmetric cones

There recently has been much interest in studying optimization problems over symmetric cones. In this paper, we first investigate a smoothing function in the context of symmetric cones and show that it is coercive under suitable assumptions. We then extend two generic frameworks of smoothing algorithms to solve the complementarity problems over symmetric cones, and prove the proposed algorithms are globally convergent under suitable assumptions. We also give a specific smoothing Newton algorithm which is globally and locally quadratically convergent under suitable assumptions. The theory of Euclidean Jordan algebras is a basic tool which is extensively used in our analysis. Preliminary numerical results for second-order cone complementarity problems are reported.     

Isoperimetric inequalities for the fundamental groups of torus bundles over the circle

We give upper bounds for isoperimetric functions of semi-direct products in terms of the asymptotic behaviour of ||A ^k || ask . In the caseA Sp(n, ) we show that these bounds are sharp. This enables us to describe infinite families of nilpotent groups whose Dehn functions are bounded above and below by polynomials of degree the nilpotency class plus 1. We also recover the isoperimetric inequalities of cocompact lattices inNil andSol.Partially supported by NSF grant DMS-9203500.     

Modified objective function method in nonsmooth vector optimization over cones

In this paper optimality for a nonsmooth vector optimization problem having generalized cone-invex objective and constraint functions is considered. An equivalent $\eta $ -approximated vector optimization problem is constructed by a modification of the objective function. The relationships between weakly efficient solutions and saddle points of the two problems are studied.