Duality of Anderson tt-motives having N≠0N\ne0

This paper extends the main result of the paper "Duality of Anderson $t$-motives", that the lattice of the dual of a t-motive $M$ is the dual lattice of $M$, to the case when the nilpotent operator $N$ of $M$ is non-zero.Comment: 21 pages; minor correction

Lattice map for Anderson t-motives : first approach

There exists a lattice map from the set of pure uniformizable Anderson t-motives to the set of lattices. It is not known what is the image and the fibers of this map. We prove a local result that sheds the first light to this problem and suggests that maybe this map is close to 1 -- 1. Namely, let $M(0)$ be a t-motive of dimension $n$ and rank $r=2n$ \ --- \ the $n$-th power of the Carlitz module of rank 2, and let $M$ be a t-motive which is in some sense "close" to $M(0)$. We consider the lattice map $M \mapsto L(M)$, where $L(M)$ is a lattice in $C^n$. We show that the lattice map is an isomorphism in a "neighborhood" of $M(0)$. Namely, we compare the action of monodromy groups: (a) from the set of equations defining t-motives to the set of t-motives themselves, and (b) from the set of Siegel matrices to the set of lattices. The result of the present paper gives that the size of a neighborhood, where we have an isomorphism, depends on an element of the monodromy group. We do not know whether there exists a universal neighborhood. Method of the proof: explicit solution of an equation describing an isomorphism between two t-motives by a method of successive approximations using a version of the Hensel lemma.Comment: 26 pages. Minor improvement

Kimmerle conjecture for the Held and O'Nan sporadic simple groups

Using the Luthar--Passi method, we investigate the Zassenhaus and Kimmerle conjectures for normalized unit groups of integral group rings of the Held and O'Nan sporadic simple groups. We confirm the Kimmerle conjecture for the Held simple group and also derive for both groups some extra information relevant to the classical Zassenhaus conjecture.Comment: 9 page

Kimmerle conjecture for the Held and O'Nan sporadic simple groups

Using the Luthar--Passi method, we investigate the Zassenhaus and Kimmerle conjectures for normalized unit groups of integral group rings of the Held and O'Nan sporadic simple groups. We confirm the Kimmerle conjecture for the Held simple group and also derive for both groups some extra information relevant to the classical Zassenhaus conjecture

On a construction of self-dual gauge fields in seven dimensions

We consider gauge fields associated with a semisimple Malcev algebra. We construct a gauge-invariant Lagrangian and found a solution of modified Yang-Mills equations in seven dimensions.Comment: 10 pages, LaTeX, no figure

On filtered multiplicative bases of some associative algebras

We deal with the existing problem of filtered multiplicative bases of finite-dimensional associative algebras. For an associative algebra A over a field, we investigate when the property of having a filtered multiplicative basis is hereditated by homomorphic images or by the associated graded algebra of $A$. These results are then applied to some classes of group algebras and restricted enveloping algebras.Comment: 10 page