In this paper the problem of locating eigenvalues of negative Krein signature is considered for operators of the form JL, where J is skew-symmetric with bounded inverse and L is self-adjoint. A finite-dimensional matrix, hereafter referred to as the Krein matrix, associated with the eigenvalue problem JLu = λu is constructed with the property that if the Krein matrix has a nontrivial kernel for...

A game on a convex geometry is a real-valued function de0ned on the family L of the closed sets of a closure operator which satis0es the 0nite Minkowski–Krein–Milman property. If L is the boolean algebra 2 then we obtain an n-person cooperative game. Faigle and Kern investigated games where L is the distributive lattice of the order ideals of the poset of players. We obtain two classes of axiom...

Known properties of ‘‘canonical connections’’ from database theory and of ‘‘closed sets’’ from statistics implicitly define a hypergraph convexity, here called canonical convexity (cconvexity), and provide an efficient algorithm to compute c-convex hulls. We characterize the class of hypergraphs in which c-convexity enjoys the Minkowski–Krein–Milman property. Moreover, we compare c-convexity wi...

We extend recent work on nonlinear optimal control problems with integer restrictions on some of the control functions (mixed-integer optimal control problems, MIOCP). We improve a theorem [25] that states that the solution of a relaxed and convexified problem can be approximated with arbitrary precision by a solution fulfilling the integer requirements. Unlike in previous publications the new ...

Violator Spaces were introduced by J. Matoušek et al. in 2008 as generalization of Linear Programming problems. Convex geometries were invented by Edelman and Jamison in 1985 as proper combinatorial abstractions of convexity. Convex geometries are defined by antiexchange closure operators. We investigate an interrelations between violator spaces and closure spaces and show that violator spaces ...

There are two basic angles associated with a pair of linear subspaces: the Diximier angle and Friedrichs angle. These classical notions, which date back to first half 20th century, have been thoroughly studied due their utility in description convergence rates for various projection-based algorithms solving feasibility problems. The Dixmier orthogonal complements is same as original provided ...

Krein-de Branges spectral theory establishes a correspondence between the class of differential operators called canonical Hamiltonian systems and measures on real line with finite Poisson integral. We further develop this area by giving description whose have logarithmic integral converging over line. This result can be viewed as version classical Szego theorem in polynomials orthogonal unit c...