Let A be a commutative nilpotent finitely-dimensional algebra over a field $F$ of characteristic $p \gt 0$. A conjecture of Eggert says that $p^. \operatorname{dim} A^{(p)} \operatorname{dim} A$, where $A^{(p)}$ is the subalgebra of $A$ generated by elements $a^p , a ∈ A$. We show that the conjecture holds if $A^{(p)}$ is at most 2-generated.

Groups defined by presentations of the form $\langle s_1,\ldots,s_n | s_i^2 = 1, (s_is_j)^{m_{ij}} = 1(i,j=1,\ldots,n)\rangle$ are called Coxeter groups. The exponents $m_{i,j} ∈ N ∪ {∞}$ form the Coxeter matrix, which characterizes the group up to isomorphism. The Coxeter groups that are most important for applications are the Weyl groups and affine Weyl groups. For example, the symmetric group $S_n$ is isomorphic to the Coxeter group with presentation $\langle s_1,\ldots,s_n | s_i^2 = 1 (i=1,\ldots,n),(s_is_{i+1})^3=1(i=1,\ldots,n-1)\rangle$, and is also known as the Weyl group of type $A_{n-1}$.

We described the structure of jordan $δ$-derivations and jordan $δ$-prederivations of unital associative algebras. We gave examples of nonzero jordan 1/2 -derivations, but not 1/2 -derivations.

Tail-biting-trellis representations of codes allow for iterative decoding algorithms, which are limited in effectiveness by the presence of pseudocodewords. We introduce a multivariate weight enumerator that keeps track of these pseudocodewords. This enumerator is invariant under many linear transformations, often enabling us to compute it exactly. The extended binary Golay code has a particularly nice tail-biting-trellis and a famous unsolved question is to determine its minimal AWGN pseudodistance. The new enumerator provides an inroad to this problem.

We consider the holomorphic unramified mapping of two arbitrary finite bordered Riemann surfaces. Extending the map to the doubles $X_1$ and $X_2$ of Riemann surfaces we define the vector bundle on the second double as a direct image of the vector bundle on first double. We choose line bundles of half-order differentials $Δ_1$ and $Δ_2$ so that the vector bundle $V_{\chi_2}^{X_2}⊗Δ_2$ on $X_2$ would be the direct image of the vector bundle $V_{\chi_1}^{X_1}⊗Δ_2$. We then show that the Hardy spaces $H_{2,J_1(p)}(S_1,V_{χ_1}⊗Δ_1$ and $H_{2,J_2(p)}(S_2,V_{χ_2}⊗Δ_2$ are isometrically isomorphic. Proving that we construct an explicit isometric isomorphism and a matrix representation $χ_2$ of the fundamental group $π_1(X_2, p_0)$ given a matrix representation $χ_1$ of the fundamental group $π_1(X_1, p'_0)$. On the basis of the results of Alpay et al. and Theorem 3.1 proven in the present work we then conjecture that there exists a covariant functor from the category $\mathcal{RH}$ of finite bordered surfaces with vector bundle and signature matrices to the category of Kreĭn spaces and isomorphisms which are ramified covering of Riemann surfaces.

It is well known that given a differential module $E$ with a differential d we can measure the non-exactness of this differential module by its homologies which are based on the key relation $d^2 =0$. This relation is a basis for several important structures in modern mathematics and theoretical physics to point out only two of them which are the theory of de Rham cohomologies on smooth manifolds and the apparatus of BRST-quantization in gauge field theories.

The orthogonal group acts on the space of several $n × n$ matrices by simultaneous conjugation. For an infinite field of characteristic different from two, relations between generators for the algebra of invariants are described. As an application, the maximal degree of elements of a minimal system of generators is described with deviation 3.

In this paper, we study weak bialgebras and weak Hopf algebras. These algebras form a class wider than bialgebras respectively Hopf algebras. The main results of this paper are Kaplansky’s type constructions which lead to weak bialgebras or weak Hopf algebras starting from a regular algebra or a bialgebra. Also we provide a classification of 2-dimensional and 3-dimensional weak bialgebras and weak Hopf algebras. We determine then the stabilizer group and the representative of these classes, the action being that of the linear group.

Chaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. In this survey paper, we treat questions of characterizing which of the spaces from a given class support a semigroup of prescribed shape satisfying a given topological ergodic property.

In particular, we characterize countable inductive limits of separable Banach spaces that admit a hypercyclic operator, show that there is a non-mixing hypercyclic operator on a separable infinite dimensional complex Fréchet space $X$ if and only if $X$ is non-isomorphic to the space $ω$ of all sequences with coordinatewise convergence topology. It is also shown for any $k ∈ \mathbb{N}$, any separable infinite dimensional Fréchet space $X$ non-isomorphic to $ω$ admits a mixing uniformly continuous group $\{T_t\}_{t∈C^n}$ of continuous linear operators and that there is no supercyclic strongly continuous operator semigroup $\{T_t\}_{t≥0}$ on $ω$. We specify a wide class of Fréchet spaces $X$, including all infinite dimensional Banach spaces with separable dual, such that there is a hypercyclic operator $T$ on $X$ for which the dual operator $T′$ is also hypercyclic. An extension of the Salas theorem on hypercyclicity of a perturbation of the identity by adding a backward weighted shift is presented and its various applications are outlined.