A type-B associahedron

The (type-A) associahedron is a polytope related to polygon dissections which arises in several mathematical subjects. We propose a B-analogue of the associahedron. Our original motivation was to extend the analogies between type-A and type-B noncrossing partitions, by exhibiting a simplicial polytope whose h-vector is given by the rank-sizes of the type-B noncrossing partition lattice, just as the h-vector of the (simplicial type-A) associahedron is given by the Narayana numbers. The desired polytope Q^B_n is constructed via stellar subdivisions of a simplex, similarly to Lee's construction of the associahedron. As in the case of the (type-A) associahedron, the faces of Q^B_n can be described in terms of dissections of a convex polygon, and the f-vector can be computed from lattice path enumeration. Properties of the simple dual $\text{Q}{}^{\mathrm{<mtext>B</mtext><mtext>1</mtext>}}{}_{\mathrm{n}}$ are also discussed and the construction of a space tessellated by $\text{Q}{}^{\mathrm{<mtext>B</mtext><mtext>1</mtext>}}{}_{\mathrm{n}}$ is given. Additional analogies and relations with type A and further questions are also discussed.     

Convex hull realizations of the multiplihedra

We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. We use this realization to unite the approach to An-maps of Iwase and Mimura to that of Boardman and Vogt. We include a review of the appearance of the nth multiplihedron for various n in the studies of higher homotopy commutativity, (weak) n-categories, A_∞-categories, deformation theory, and moduli spaces. We also include suggestions for the use of our realizations in some of these areas as well as in related studies, including enriched category theory and the graph-associahedra.     

On the number of non-equivalent differentiable structures on 4-manifolds

By using results from [7], [8] we show that for any positive integer k there exist k simply-conected algebraic surfaces of general type which are pairwise homeomorphic but not diffeomorphic.     

On the asymptotic number of non-equivalent q-ary linear codes

Let be the group of monomial matrices, i.e., the group generated by all permutation matrices and diagonal matrices in . The group acts on the set of all subspaces of . The number of orbits of this action, denoted by N_n,q, is the number of non-equivalent linear codes in . It was conjectured by Lax that as n→∞. We confirm this conjecture in this paper.     

Crystalline realizations of 1-motives

We consider the crystalline realization of Delignes 1-motives in positive characteristics and prove a comparison theorem with the De Rham realization of (formal) liftings to zero characteristic. We then show that one dimensional crystalline cohomology of an algebraic variety, defined by forcing universal cohomological descent via de Jongs alterations, coincides with the crystalline realization of the Picard 1-motive, over perfect fields of characteristic >2.     

Topological realizations of ortho-projection graphs

We discuss the production of ortho-projection graphs from alternating knot diagrams, and introduce a more general construction of such graphs from “splittings” of closed, non-orientable surfaces. As our main result, we prove that this new topological construction generates all ortho-projection graphs. We present a minimal example of an ortho-projection graph that does not arise from a knot diagram, and provide a surface-splitting that realizes this graph.     

Minimal realizations of nonlinear systems

We generalize the notion of the minimal realization from the linear theory to the nonlinear case. We justify a relationship of this notion with the notion of first integral of a system. We suggest algorithms for the computation of first integrals and minimal realizations.     

A finite-dimensional normed space with two non-equivalent symmetric bases

A sequence of finite-dimensional normed spaces is constructed, each with two symmetric bases, such that the sequence of equivalence constants between these bases is unbounded. An essential tool in the proof is the edge-isoperimetric inequality in the discrete cube.     

Weighting operator patterns of Pritchard-Salamon realizations

In this paper a complete characterization is given of those operator kernels which appear as weighting patterns of Pritchard-Salamon realizations. The result is based on an extension of the standard shift realization to weightedL ₂-spaces of vector-valued functions.     

On realizations related to Weyl operators

The paper deals with the minimal and the maximal realizations (L ^w )^~ and (L ^w ):L ₂L ₂ of linear operators of Weyl type