Fixed point theorems in homotopically trivial spaces

We introduce a new concept of generalized Knaster–Kuratowski–Mazurkiewicz (KKM) family of sets and related to this we obtain fixed point theorems and sections results in homotopically trivial spaces.     

On continua with homotopically fixed boundary

The paper presents two subcontinua of Rn, one Peano-continuum, and one cellular continuum with trivial fundamental group. Both of them have the remarkable property that neither the entire spaces nor (roughly speaking) any part of them is homotopy equivalent to a lower-dimensional space. This extends work of the last three authors and of Karimov from the planar case to the higher-dimensional case, but it also contains in the cellular case the first example with all these properties in dimension two.     

A tutorial proof that extreme points are basic feasible solutions

Given a set of m linear equations in n unknowns with the requirement that the solution space be nonnegative, a simple, heuristic proof is offered which shows that the extreme points of the set of feasible solutions are also basic feasible solutions. This proof can be used in many text treatments of Linear Programming which omit the proof on the grounds that it is too difficult to prove.     

Rationally trivial hermitian spaces are locally trivial

Let R be a regular local ring, K its field of fractions and A an Azumaya algebra with involution over R. Let h be an -hermitian space over A. We show that if is hyperbolic over , then h is hyperbolic over A. Received September 28, 1998; in final form December 6, 1999 / Published online March 12, 2001     

An effective proof that open sets are Ramsey

Solovay has shown that if is an open subset of with code and no infinite set avoids , then there is an infinite set hyperarithmetic in that lands in . We provide a direct proof of this theorem that is easily formalizable in . Received January 22, 1996     

Infinite families of links with trivial Jones polynomial

For each k?2, we exhibit infinite families of prime k-component links with Jones polynomial equal to that of the k-component unlink.     

An algebraic proof that pseudovarieties are defined by pseudoidentities

An algebraic proof is given which shows that any pseudovariety of finite algebras is determined by a set of pseudoidentities.Presented by Boris M. Schein.     

A geometric non-existence proof of an extremal additive code

We use a geometric approach to solve an extremal problem in coding theory. Expressed in geometric language we show the non-existence of a system of 12 lines in PG(8,2) with the property that no hyperplane contains more than 5 of the lines. In coding-theoretic terms this is equivalent with the non-existence of an additive quaternary code of length 12, binary dimension 9 and minimum distance 7.     

A simple proof of the geometric fractional monodromy theorem

A simple proof of the “geometric fractional monodromy theorem” (Broer-Efstathiou-Lukina 2010) is presented. The fractional monodromy of a Liouville integrable Hamiltonian system over a loop γ ? ?² is a generalization of the classic monodromy to the case when the Liouville foliation has singularities over γ. The “geometric fractional monodromy theorem” finds, up to an integral parameter, the fractional monodromy of systems similar to the 1: (?2) resonance system. A handy equivalent definition of fractional monodromy is presented in terms of homology groups for our proof.     

A geometric proof of Asplund’s differentiability theorem

A geometric consequence inB of local uniform rotundity inB ^* is used to prove Asplund’s theorem on Fréchet differentiability of convex functionals.     

A direct proof that vwb processes are closed in the\bar d-metric

As defined in the literature, a process is very weak Bernoulli if a certain propertyP(ε) is satisfied for everyε>0. By means of an easy proof, it is shown that givenε>0, there existsδ>0 such that given any two stationary processes whose -distance is less thanδ, if one of the processes is very weak Bernoulli then the other process is “almost” very weak Bernoulli in the sense that the propertyP(ε) is satisfied. Using this result a direct proof can be given that the very weak Bernoulli processes are closed under the -distance, and also that a finitely determined process is very weak Bernoulli. Relativized versions of these results are also considered. Supported by NSF Grant MCS-7821335.     

On geometries that are connected with general geometric lattices

In this paper, we consider general geometries connected with geometric lattices. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 22, Algebra and Geometry, 2004.     

A geometric proof of the definability of Hausdorff limits

We give a geometric proof of the following well-established theorem for o-minimal expansions of the real field: the Hausdorff limits of a compact, definable family of sets are definable. While previous proofs of this fact relied on the model-theoretic compactness theorem, our proof explicitly describes the family of all Hausdorff limits in terms of the original family.     

An elementary proof that rationally isometric quadratic forms are isometric

Let R be a valuation ring with fraction field K and 2 ∈ R ^×. We give an elementary proof of the following known result: two unimodular quadratic forms over R are isometric over K if and only if they are isometric over R. Our proof does not use cancelation of quadratic forms and yields an explicit algorithm to construct an isometry over R from a given isometry over K. The statement actually holds for hermitian forms over valuated involutary division rings, provided mild assumptions.