Band 2010, Heft 646

We study the average order of the divisor function, as it ranges over the values of binary quartic forms that are reducible over ℚ.

Associated to any uniform finite layered graph Γ there is a noncommutative graded quadratic algebra A (Γ) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW-complexes, X . Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups H X ( n, k ), generalizing the usual cohomology groups H n ( X ). Along with several other results, our methods give a new and primarily topological proof of the main result of [Serconek and Wilson, J. Algebra 278: 473–493, 2004] and [Piontkovski, J. Alg. Comput. 15, 643–648, 2005].

Given a couple of smooth positive measures of same total mass on a compact connected Riemannian manifold M , we look for a smooth optimal transportation map G , pushing one measure to the other at a least total squared distance cost, directly by using the continuity method to produce a classical solution of the elliptic equation of Monge–Ampère type satisfied by the potential function u , such that G = exp(grad u ). This approach boils down to proving an a priori upper bound on the Hessian of u , which was done on the flat torus by the first author. The recent local C 2 estimate of Ma–Trudinger–Wang enabled Loeper to treat the standard sphere case by overcoming two difficulties, namely: in collaboration with the first author, he kept the image G ( m ) of a generic point m ∈ M , uniformly away from the cut-locus of m ; he checked a fourth-order inequality satisfied by the squared distance cost function, proving its so-called (strict) regularity. In the present paper, we treat along the same lines the case of manifolds with curvature sufficiently close to 1 in C 2 norm—specifying and proving a conjecture stated by Trudinger.

We prove that the mapping stack Map(𝒴, 𝒳) of topological stacks 𝒳 and 𝒴 is again a topological stack if 𝒴 admits a groupoid presentation [ Y 1 ⇉ Y 0 ] such that Y 0 and Y 1 are compact topological spaces. If Y 0 and Y 1 are only locally compact, we show that Map(𝒴, 𝒳) is a paratopological stack. In particular, it has a classifying space (hence, a natural weak homotopy type). We also show that the weak homotopy type of the mapping stack Map( Y , 𝒳) does not change if we replace 𝒳 by its classifying space, provided that Y is a paracompact topological space. As an example, we describe the loop stack of the classifying stack ℬ G of a topological group G in terms of twisted loop groups of G .

We prove that any rational linear combination of Pontryagin numbers that is not a multiple of the signature is unbounded on connected closed oriented manifolds of nonnegative sectional curvature. Combining our result with Gromov's boundedness result for the signature yields a new characterization of the L -genus.

We introduce a class of variational principles on measure spaces which are causal in the sense that they generate a relation on pairs of points, giving rise to a distinction between spacelike and timelike separation. General existence results are proved. It is shown in examples that minimizers need not be unique. Counter examples to compactness are discussed. The existence results are applied to variational principles formulated in indefinite inner product spaces.

We study the asymptotic property for the heat kernel on Heisenberg type groups. Also, we give sharp estimates for derivatives of the heat kernel. In particular, we obtain sharp upper and lower bounds for the heat kernel on Heisenberg type groups, and an estimate of type for all h > 0 and g, g ′.