On a filtered multiplicative basis of group algebras

Let K be a field of characteristic p and G a nonabelian metacyclic finite p-group. We give an explicit list of all metacyclic p-groups G, such that the group algebra KG over a field of characteristic p has a filtered multiplicative K-basis. We also present an example of a non-metacyclic 2-group G, such that the group algebra KG over any field of characteristic 2 has a filtered multiplicative K-basis.     

On the existence of $\lambda $-projective splitters

Let D be an R-module over an arbitrary ring R of projective dimension at most 1. We construct an R-module G containing D such that Ext(D, G) = 0 = Ext(G, G). Moreover, we show that if D is l\lambda -projective over a hereditary ring R, for some infinite cardinal l\lambda , then G is also l\lambda -projective.     

Periodic structure of $ \sigma $-permutations maps on In

Summary. We present a result concerning the periodic structure of certain maps on Iⁿ = [0,1]ⁿ which are called s \sigma -permutation maps. This structure resembles that given by the xarkovskii's ordering for continuous interval maps.     

Chaos in the dynamics generated by sequences of maps, with applications to chaotic advection in flows with aperiodic time dependence

In this paper we develop analytical techniques for proving the existence of chaotic dynamics in systems where the dynamics is generated by infinite sequences of maps. These are generalizations of the Conley-Moser conditions that are used to show that a (single) map has an invariant Cantor set on which it is topologically conjugate to a subshift on the space of symbol sequences. The motivation for developing these methods is to apply them to the study of chaotic advection in fluid flows arising from velocity fields with aperiodic time dependence, and we show how dynamics generated by infinite sequences of maps arises naturally in that setting. Our methods do not require the existence of a homoclinic orbit in order to conclude the existence of chaotic dynamics. This is important for the class of fluid mechanical examples considered since one cannot readily identify a homoclinic orbit from the structure of the equations.¶We study three specific fluid mechanical examples: the Aref blinking vortex flow, Samelson's tidal advection model, and Min's rollup-merge map that models kinematics in the mixing layer. Each of these flows is modelled as a type of "blinking flow", which mathematically has the form of a linked twist map, or an infinite sequence of linked twist maps. We show that the nature of these blinking flows is such that it is possible to have a variety of "patches" of chaos in the flow corresponding to different length and time scales.     

Sur les tours localement cyclotomiques

For a given number field K and any prime l\ell we construct an increasing sequence of l\ell -extensions K_n of K which are locally cyclotomic over K. We give various criterious of finiteness or non-finiteness of this l\ell -tower and we characterise the number fields which have such a finite tower in terms of Iwasawa theory.     

How Tight Is the Bollobás-Komlós Conjecture?

The bipartite case of the Bollobás and Komlós conjecture states that for every j₀, %>0 there is an !=!(j₀, %) >0 such that the following statement holds: If G is any graph with minimum degree at least n$\displaystyle {n \over 2}+%n then G contains as subgraphs all n vertex bipartite graphs, H, satisfying¶H)hj₀ \quad {\rm and} \quad b(H)h!n.$j (H)hj₀ \quad {\rm and} \quad b(H)h!n.¶Here b(H), the bandwidth of H, is the smallest b such that the vertices of H can be ordered as v₁, …, v_n such that v_i~_Hv_j implies |imj|hb.¶ This conjecture has been proved in [1]. Answering a question of E. Szemerédi [6] we show that this conjecture is tight in the sense that as %̂ then !̂. More precisely, we show that for any 0 such that that !(j₀, %)Д %.     

Abelian gradings on upper-triangular matrices

Let G be an arbitrary finite abelian group. We describe all possible G-gradings on an upper-triangular matrix algebra over an algebraically closed field of characteristic zero.     

Lavrentiev''s phenomenon for totally unconstrained variational problems in one dimension

We give examples of one-dimensional variational problems with free ends that exhibit Lavrentiev's phenomenon, i.e. the infimum of the functional over one class X of admissible functions is strictly greater than the infimum over another class Y of admissible functions - even though X is dense in Y.     

Mal'cev nilpotent algebras

We show that the Mal'cev semigroup identity x_n = y_n holds in the circle semigroup of an associative algebra over an infinite field precisely when the algebra is Lie nilpotent of class at most n. The Mal'cev semigroup law x_n = y_n holds in a group if and only if the group is nilpotent of class at most n.     

Power residues on Abelian varieties

Given a prime l and an elliptic curve E defined over a number field k, we show that a non-zero point P] E(k) lies in lE(k) if and only if P lies in lE(k)(mod ) for almost all finite primes  of k. We give conditions on l under which analogous results hold for Abelian varieties and with one point replaced by a finite number of points. We also construct examples to show that these conditions are essential.