For some text algorithms, the real measure for the complexity analysis is not the string itself but its structure stored in its prefix table or equivalently border table. In this paper, we define the combinatorial class of prefix lists, namely a sequence of integers together with their size, and an injection ψ from the class of prefix tables to the class of prefix lists. We call a valid prefix list the image by ψ of a prefix table. In particular, we describe algorithms converting a prefix/border table to a prefix list and inverse linear algorithms from computing from a prefix list L = ψ(P) two words respectively in a minimal size alphabet and on a maximal size alphabet with P as prefix table. We then give a new upper bound on the number of prefix tables for strings of length n (on any alphabet) which is of order (1 + ϕ)n (with $\varphi=\frac{1+\sqrt{5}}{2}$ the golden mean) and also present a corresponding lower bound.

The objective of the Apollon project is the generation of 10 PW peak power pulses of 15 fs at 1 shot/minute. In this paper the Apollon facility design, the technological challenges and the current progress of the project will be presented.

Let $\mathcal{V}$ be a variety of associative algebras generated by an algebra with 1 over a field of characteristic zero. This paper is devoted to the classification of the varieties $\mathcal{V}$ that are minimal of polynomial growth (i.e., their sequence of codimensions grows like ${{n}^{k}}$, but any proper subvariety grows like ${{n}^{t}}$ with $t\,<\,k$). These varieties are the building blocks of general varieties of polynomial growth.

It turns out that for $k\,\le \,4$ there are only a finite number of varieties of polynomial growth ${{n}^{k}}$, but for each $k\,>\,4$, the number of minimal varieties is at least $\left| F \right|$, the cardinality of the base field, and we give a recipe for their construction.

In a previous paper [L. Giambruno and S. Mantaci, Theoret. Comput. Sci. 411 (2010) 1785–1792] a bideterministic transducer is defined for the bidirectional deciphering of words by the method introduced by Girod [IEEE Commun. Lett. 3 (1999) 245–247]. Such a method is defined using prefix codes. Moreover a coding method, inspired by the Girod’s one, is introduced, and a transducer that allows both right-to-left and left-to-right decoding by this method is defined. It is proved also that this transducer is minimal. Here we consider the number of states of such a transducer, related to some features of the considered prefix code X. We find some bounds of such a number of states in relation with different notions of “size” of X. In particular, we give an exact formula for the number of states of transducers associated to maximal prefix codes. We moreover consider two special cases of codes: maximal uniform codes and a class of codes, that we name string-codes. We show that they represent, for maximal codes, the extreme cases with regard to the number of states in terms of different sizes. Moreover we prove that prefix codes corresponding to isomorphic trees have transducers that are isomorphic as unlabeled graphs.

We investigate the intersection of two finitely generated submonoids of the free monoid on a finite alphabet. To this purpose, we consider automata that recognize such submonoids and we study the product automata recognizing their intersection. By using automata methods we obtain a new proof of a result of Karhumäki on the characterization of the intersection of two submonoids of rank two, in the case of prefix (or suffix) generators. In a more general setting, for an arbitrary number of generators, we prove that if H and K are two finitely generated submonoids generated by prefix sets such that the product automaton associated to $H \cap K$ has a given special property then $\widetilde{rk}(H \cap K) \leq \widetilde{rk}(H) \widetilde{rk}(K)$ where $\widetilde{rk}(L)=\max(0,rk(L)-1)$ for any submonoid L.

In this paper we describe the group gradings by a finite abelian group $G$ of the matrix algebra ${{M}_{n}}(F)$ over an algebraically closed field $F$ of characteristic different from 2, which respect an involution (involution gradings). We also describe, under somewhat heavier restrictions on the base field, all $G$-gradings on all finite-dimensional involution simple algebras.

Let $F$ be an algebraically closed field of characteristic $0$ , and let $A$ be a $G$ -graded algebra over $F$ for some finite abelian group $G$ . Through $G$ being regarded as a group of automorphisms of $A$ , the duality between graded identities and $G$ -identities of $A$ is exploited. In this framework, the space of multilinear $G$ -polynomials is introduced, and the asymptotic behavior of the sequence of $G$ -codimensions of $A$ is studied.

Two characterizations are given of the ideal of $G$ -graded identities of such algebra in the case in which the sequence of $G$ -codimensions is polynomially bounded. While the first gives a list of $G$ -identities satisfied by $A$ , the second is expressed in the language of the representation theory of the wreath product $G \wr S_n$ , where $S_n$ is the symmetric group of degree $n$ .

As a consequence, it is proved that the sequence of $G$ -codimensions of an algebra satisfying a polynomial identity either is polynomially bounded or grows exponentially.

Let RG denote the group ring of a group G over a semiprime ring R. We prove that, if the center of G is of finite index and some natural restrictions hold, then every R-derivation of RG is inner. We also give an example of a group G which is both locally finite and nilpotent and such that, for every field F, there exists an F-derivation of FG which is not inner.

Let M2(K, *) be the algebra of 2 × 2 matrices with involution over a field K of characteristic 0. We obtain the exact values of the cocharacters, codimensions and Hilbert series of the *-T-ideal of the polynomial identities for M2(K, *).

The hypercenter theorem [6] asserts that in a ring with no non-zero nil ideals an element commuting with a suitable power of each element of the ring must be central. In this paper we shall be concerned with a similar problem in the setting of rings with involution. Let R be a ring with involution *, let Z denote the center of R and let S = {x ∈ R|x = x*} be the set of symmetric elements in R. We define the symmetric hypercenter of R to be

What can one hope to say about H? That H need not equal Z is clear. For instance, in the ring R = F2 of 2 X 2 matrices over a field, if * is the symplectic involution, all symmetric elements are central, hence H = R but Z ≠ R. Furthermore if R is a noncommutative ring in which every symmetric element is nilpotent then even in this case H = R and Z ≠ R follows.

Let R be an associative ring and f(x1,…, xd) a polynomial in noncommuting variables. We say that f is periodic or nil in R if for all r1,…, rd ∈ R we have that f(r1,…, rd) is periodic, respectively nilpotent (recall that a ∈ R is periodic if for some integer ).