Fatgraphs are multigraphs enriched with a cyclic order of the edges incident to a vertex. This paper presents algorithms to: (1) generate the set Rg,n of fatgraphs, given the genus g and the number of boundary cycles n; (2) compute automorphisms of any given fatgraph; (3) compute the homology of the fatgraph complex Rg,n. The algorithms are suitable for effective computer implementation. In par...

The authors T.Harima, J.C.Migliore, U.Nagel and J.Watanabe characterize in [8] the Hilbert function of algbebras with the Lefschetz property. We extend this characterization to algebras with the Lefschetz property m times. We also give upper bounds for the Betti numbers of Artinian algebras with a given Hilbert function and with the Lefschetz property m times and describe the cases in which the...

The characteristic cycle of a complex of sheaves on a complex analytic space provides weak information about the complex; essentially, it yields the Euler characteristics of the hypercohomology of normal data to strata. We show how perverse cohomology actually allows one to extract the individual Betti numbers of the hypercohomology of normal data to strata, not merely the Euler characteristics...

We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semialgebraic sets. Aside from describing these results, we discuss briefly the background as well as the importance of these problems, and also describe the main tools from algorithmic semi-al...

This paper considers the generalized Stirling numbers of the first and second kinds. First, we show that the sequences of the above generalized Stirling numbers are both log-concave under some mild conditions. Then, we show that some polynomials related to the above generalized Stirling numbers are q-log-concave or q-log-convex under suitable conditions. We further discuss the log-convexity of ...

This article grew out of my talk in ‘The Legacy of Srinivasa Ramanujan’ conference where I spoke about some techniques to prove algebraicity results for the special values of symmetric cube L-functions attached to the Ramanujan ∆-function. If one wishes to compare these different techniques, then one needs to compare various automorphic periods attached to the symmetric cube transfer of ∆. Moti...

We show that when certain cohomological finiteness conditions hold, then the dimensions of the cohomology groups of modules over a finite dimensional algebra are given by polynomials. This applies in particular to quantum complete intersections, and among these all exterior algebras.

In this paper we show that every ideal with linear quotients is componentwise linear. We also generalize the Eliahou-Kervaire formula for graded Betti numbers of stable ideals to homogeneous ideals with linear quotients.

Motivated by some earlier Diophantine works on triangular numbers by Ljunggren and Cassels, we consider similar problems for general polygonal numbers.