Simple homotopy types of Hom-complexes, neighborhood complexes, Lovász complexes, and atom crosscut complexes

In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes.To start with, we give a sequence of elementary collapses leading from the barycentric subdivision of the neighborhood complex to the Lovász complex of a graph. Then, for an arbitrary lattice L we describe a formal deformation of the barycentric subdivision of the atom crosscut complex Γ(L) to its order complex . We proceed by proving that the complex of sets bounded from below J(L) can also be collapsed to .Finally, as a pinnacle of our project, we apply all these results to certain graph complexes. Namely, by describing an explicit formal deformation, we prove that, for any graph G, the neighborhood complex N(G) and the polyhedral complex Hom(K₂,G) have the same simple homotopy type in the sense of Whitehead.     

Quadratic addition rules for quantum integers

For every positive integer n, the quantum integer [n]_q is the polynomial [n]_q=1+q+q²+?+q^n-1. A quadratic addition rule for quantum integers consists of sequences of polynomials , , and such that for all m and n. This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials that satisfy the associated functional equation .     

The heat equation with time-independent multiplicative stable Lévy noise

We study the heat equation with a random potential term. The potential is a one-sided stable noise, with positive jumps, which does not depend on time. To avoid singularities, we define the equation in terms of a construction similar to the Skorokhod integral or Wick product. We give a criterion for existence based on the dimension of the space variable, and the parameter p$p$ of the stable noise. Our arguments are different for p<1$p<1$ and p?1$p?1$.     

Adsorption of biopolyester depolymerase on silicon wafer and poly[(R)-3-hydroxybutyric acid] single crystal revealed by real-time AFM

The adsorption behavior of PHB depolymerase from R. pickettii T1 on a silicon wafer and on P(3HB) single crystals has been studied by real-time and AFM in air and a buffer solution. First, the morphology of PHB depolymerase adsorbed on a silicon wafer was characterized to show that one molecule of PHB depolymerase has dimensions of 2.2 +/- 0.7 nm height and 16 +/- 5 nm width. The observation of PHB depolymerase adsorbed on a P(3HB) single crystal indicated that the dimensions of enzyme on the crystalline surface in air were 1.2 +/- 0.5 nm high and 28 +/- 7 nm wide, while enzyme molecules with dimensions of 2.1 +/- 0.6 nm height and 16 +/- 7 nm width were detected in a buffer solution. Comparison of the dimensions of PHB depolymerase in air with those in a buffer solution showed that the enzyme was squashed in air, but not in a buffer solution. In addition, the influence of enzymatic adsorption on the molecular state of the P(3HB) crystalline surface was investigated. The AFM images of P(3HB) single crystals after enzymatic adsorption and washing with ethanol indicated that the adhesion of PHB depolymerase changed the molecular state and generated holes on the crystalline surface.     

Coleman maps for modular forms at supersingular primes over Lubin-Tate extensions

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Given an elliptic curve with supersingular reduction at an odd prime p, Iovita and Pollack have generalised results of Kobayashi to define even and odd Coleman maps at p over Lubin-Tate extensions given by a formal group of height 1. We generalise this construction to modular forms of higher weights.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=KQpsht0JaME.     

A characterization of semisimple plane polynomial automorphisms

It is well-known that an element of the linear group is semisimple if and only if its conjugacy class is Zariski closed. The aim of this paper is to show that the same result holds for the group of complex plane polynomial automorphisms.     

Generalizations of the image conjecture and the Mathieu conjecture

We first propose a generalization of the image conjecture Zhao (submitted for publication) [31] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture Mathieu (1997) [21] from integrals of G-finite functions over reductive Lie groups G to integrals of polynomials over open subsets of Rⁿ with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of the Duistermaat and van der Kallen theorem Duistermaat and van der Kallen (1998) [14] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly introduced notion.     

Equivariant simplicial cohomology with local coefficients and its classification

We introduce equivariant twisted cohomology of a simplicial set equipped with simplicial action of a discrete group and prove that for suitable twisting function induced from a given equivariant local coefficients, the simplicial version of Bredon-Illman cohomology with local coefficients is isomorphic to equivariant twisted cohomology. The main aim of this paper is to prove a classification theorem for equivariant simplicial cohomology with local coefficients.