Narrow Results

Let $K$ be any field. In this paper, we give a complete list of the indecomposable left injective module over the Jacobson algebra $K〈X,Y|XY=1〉$, i.e. the free associative $K$-algebra on two (non-commuting) generators, modulo the single relation $XY=1$. This is the natural continuation of the paper of the second two authors with Gene Abrams on the characterization of the injective envelope of the simple modules over $K〈X,Y|XY=1〉$.

Let S denote a subset of the positive integers, and let p_S(n) be the associated partition function, that is, p_S(n) denotes the number of partitions of the positive integer n into parts taken from S. Thus, if S is the set of positive integers, then p_S(n) is the ordinary partition function p(n). In this paper, working in the ring of formal power series in one variable over the field of two elements Z/2Z, we develop new methods for deriving lower bounds for both the number of even values and the number of odd values taken by p_S(n), for n ≤ N. New very general theorems are obtained, and applications are made to several partition functions.

We investigate Büchi automata with weights for the transitions. Assuming that the weights are taken in a suitable ordered semiring, we show how to define the behaviors of these automata on infinite words. Our main result shows that the formal power series arising in this way are precisely the ones which can be constructed using ω-rational operations. This extends the classical Kleene–Schützenberger result for weighted finite automata to the case of infinite words and generalizes Büchi's theorem on languages of infinite words. We also derive versions of our main result for non-complete semirings and for other automata models.

We show a reduction of Hilbert's tenth problem to the solvability of the matrix equation over non-commuting integral matrices, where Z is the zero matrix, thus proving that the solvability of the equation is undecidable. This is in contrast to the case whereby the matrix semigroup is commutative in which the solvability of the same equation was shown to be decidable in general.

The restricted problem where k = 2 for commutative matrices is known as the "A-B-C Problem" and we show that this problem is decidable even for a pair of non-commutative matrices over an algebraic number field.

The aim of this paper is to characterize the formal power series which have purely periodic β-expansions in Pisot or Salem unit base under some condition. Furthermore, we will prove that if β is a quadratic Pisot unit base, then every rational f in the unit disk has a purely periodic β-expansion and discuss their periods.

Conway hemirings are Conway semirings without a multiplicative unit. We also define iteration hemirings as Conway hemirings satisfying certain identities associated with the finite groups. Iteration hemirings are iteration semirings without a multiplicative unit. We provide an analysis of the relationship between Conway hemirings and (partial) Conway semirings and describe several free constructions. In the second part of the paper we define and study hemimodules of Conway and iteration hemirings, and show their applicability in the analysis of quantitative aspects of the infinitary behavior of weighted transition systems. These include discounted and average computations of weights investigated recently.

A ring R with Jacobson radical J(R) is a homogeneous semilocal ring if R/J(R) is simple artinian. In this paper, we study the transfer of the property of being homogeneous semilocal from a ring R to the formal power series ring R[[x]], the skew formal power series ring R[[x, α]] and the Hurwitz series ring HR. The results of the paper generalize those proved for commutative local rings. We also consider finite centralizing extensions proving that if the ring of matrices M_n(R) is a homogeneous semilocal ring, then so is R. More generally, if e is an idempotent of a homogeneous semilocal ring S, then eSe is homogeneous semilocal.

Let $𝜀={({𝜀}_n)}_{n\in \mathbb{N}}$ be an integer sequence and $f(x)={\sum }_{n=0}^{\infty }{𝜀}_n{x}^n$ be its ordinary generating function. In this paper, we study the behavior of 2-adic valuations of the sequence ${(c_m(n))}_{n\in \mathbb{N}}$, where $m\in \mathbb{Z}$ is fixed and