A Borsuk-Ulam type theorem for sphere bundles over stunted projective spaces

A CW complex B is described as I-trivial if there does not exist a Z₂-map from Si−1 to S(α) for any vector bundle α over B and any integer i with i>dimα. For n>1, we determine all positive integers m for which the stunted projective space is I-trivial, where F=R,C or H.     

Tame parts of free summands in coproducts of Priestley spaces

It is well known that a sum (coproduct) of a family of Priestley spaces is a compactification of their disjoint union, and that this compactification in turn can be organized into a union of pairwise disjoint order independent closed subspaces Xu, indexed by the ultrafilters u on the index set I. The nature of those subspaces Xu indexed by the free ultrafilters u is not yet fully understood.In this article we study a certain dense subset satisfying exactly those sentences in the first-order theory of partial orders which are satisfied by almost all of the Xi's. As an application we present a complete analysis of the coproduct of an increasing family of finite chains, in a sense the first non-trivial case which is not a ?ech-Stone compactification of the disjoint union I_?Xi. In this case, all the Xu's with u free turn out to be isomorphic under the Continuum Hypothesis.     

The connected Vietoris powerlocale

The connected Vietoris powerlocale is defined as a strong monad Vc on the category of locales. VcX is a sublocale of Johnstone's Vietoris powerlocale VX, a localic analogue of the Vietoris hyperspace, and its points correspond to the weakly semifitted sublocales of X that are “strongly connected”. A product map ×:VcX×VcY→Vc(X×Y) shows that the product of two strongly connected sublocales is strongly connected. If X is locally connected then VcX is overt. For the localic completion of a generalized metric space Y, the points of are certain Cauchy filters of formal balls for the finite power set FY with respect to a Vietoris metric.Application to the point-free real line R gives a choice-free constructive version of the Intermediate Value Theorem and Rolle's Theorem.The work is topos-valid (assuming natural numbers object). Vc is a geometric construction.     

Real dicompact textures

A notion of real compactness for completely biregular bi-T₂ ditopological texture spaces is defined and studied under the name real dicompactness. In particular it is shown that real dicompact spaces are nearly plain ∗-spaces, and an important characterization is presented. Finally the connection of this work with topological and bitopological real compactness is discussed in a categorical setting.     

A refinement of the Conway-Gordon theorems

In 1983, Conway-Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this article, we give integral lifts of the Conway-Gordon theorems above in terms of the square of the linking number and the second coefficient of the Conway polynomial. As applications, we give alternative topological proofs of theorems of Brown-Ramírez Alfonsín and Huh-Jeon for rectilinear spatial complete graphs which were proved by computational and combinatorial methods.     

Monotone generalized contractions in partially ordered probabilistic metric spaces

In this paper, a concept of monotone generalized contraction in partially ordered probabilistic metric spaces is introduced and some fixed and common fixed point theorems are proved. Presented theorems extend the results in partially ordered metric spaces of Nieto and Rodriguez-Lopez [Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239; Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205-2212], Ran and Reurings [A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443] to a more general class of contractive type mappings in partially ordered probabilistic metric spaces and include several recent developments.     

Non-existence of a secondary bifurcation point for a semilinear elliptic problem in the presence of symmetry

We give two sufficient conditions for a branch consisting of non-trivial solutions of an abstract equation in a Banach space not to have a (secondary) bifurcation point when the equation has a certain symmetry. When the nonlinearity f is of Allen-Cahn type (for instance f(u)=u−u³), we apply these results to an unbounded branch consisting of non-radially symmetric solutions of the Neumann problem on a disk D⊂R²

     

On certain exponential sums over primes

Let f(x) be a real valued polynomial in x of degree k?4 with leading coefficient α. In this paper, we prove a non-trivial upper bound for the quantity

     

Finite Weil representations and associated Hecke algebras

An algebra H(Gm) of double cosets is constructed for every finite Weil representation Gm. For the Clifford-Weil groups Gm=Cm(ρ) associated to some classical Type ρ of selfdual codes over a finite field, this algebra is shown to be commutative. Then the eigenspace decomposition of H(Cm(ρ)) acting on the space of degree N invariants of Cm(ρ) may be obtained from the kernels of powers of the coding theory analogue of the Siegel Φ-operator.     

One-parameter toric deformations of cyclic quotient singularities

In the case of two-dimensional cyclic quotient singularities, we classify all one-parameter toric deformations in terms of certain Minkowski decompositions introduced by Altmann [Minkowski sums and homogeneous deformations of toric varieties, Tohoku Math. J. (2) 47 (2) (1995) 151-184.]. In particular, we show how to induce each deformation from a versal family, describe exactly to which reduced versal base space components each such deformation maps, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions.