We show that the Jacobian of a certain Artin-Schreier curve over the field Fp has a a 1-dimensional formal summand of height (p− 1)f for any positive integer f . We give two proofs, the classical one which was known to Manin in 1963 and which requires knowledge of the zeta function of the curve, and a new simpler one using methods of Honda. This is the first step toward contructing the cohomolo...

Concave compositions are compositions (i.e. ordered partitions) of a number in which the parts decrease up to the middle summand(s) and increase thereafter. Perhaps the most surprising result is for even length, concave compositions where the generating function turns out to be the quotient of two instances of the pentagonal number theorem with variations of sign. The false theta function disco...

In this short note we study the entropy for algebraic actions of certain amenable groups. The possible values for this entropy are studied. Various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant. We provide a natural decomposition of the entropy into summands contributed by individual primes and a summand correspond...

Weyl modules play a central role in the representation theory of the general and special linear groups. There has been great interest recently in the modular case of these groups over fields of prime characteristic p. In particular, Glover [l] has proved some remarkable and comprehensive results about these modules for SL(2,p). We shall extend one of these theorems to the case of SL(2, q) where...

Let G be a group, F a field of prime characteristic p and V a finite-dimensional FGmodule. Let L(V ) denote the free Lie algebra on V regarded as an FG-submodule of the free associative algebra (or tensor algebra) T (V ). For each positive integer r, let Lr(V ) and T r(V ) be the rth homogeneous components of L(V ) and T (V ), respectively. Here Lr(V ) is called the rth Lie power of V . Our mai...

We describe explicitly the algebras of degree zero operations in connective and periodic p-local complex K-theory. Operations are written uniquely in terms of certain infinite linear combinations of Adams operations, and we give formulas for the product and coproduct structure maps. It is shown that these rings of operations are not Noetherian. Versions of the results are provided for the Adams...

When a positive integer is expressed as a sum of squares, with each successive summand as large as possible, the summands decrease rapidly in size until the very end, where one may find two 4’s, or several 1’s. We find that the set of integers for which the summands are distinct does not have a natural density but that the counting function oscillates in a predictable way.

A method for analysing a class of divergent series is developed from the Euler– Maclaurin summation formula. The conditions that the summand must satisfy are explored, and a significant simplification is obtained for cases where the summation ranges over all integers. As an example, we consider the Ewald representation for Schlömilch series, and show that this includes Twersky’s dual series for...

Triangular subnorms are associative commutative non-decreasing operations on the unit interval, upper bounded by the minimum. Continuous triangular subnorms are shown to be ordinal sum of Archimedean continuous t-subnorms with at most one proper t-subnorm summand. Special attention is paid to generate continuous t-subnorms. An application of continuous t-subnorms to the construction of left-con...

We show that for several variations of partially observable Markov decision processes, polynomial-time algorithms for nding control policies are unlikely to or simply don't have guarantees of nding policies within a constant factor or a constant summand of optimal. Here \unlikely" means \unless some complexity classes collapse," where the collapses considered are P = NP, P = PSPACE, or P = EXP....