Related papers: Factorization length distribution for affine semig…
An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F(T)\in \mathbb{Z}[T]$ with $f(p)=F(p)$ for all primes $p$. We study the distribution in coprime residue…
Nonunique factorization in commutative monoids is often studied using factorization invariants, which assign to each monoid element a quantity determined by the factorization structure. For numerical monoids (co-finite, additive submonoids…
We consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power $q$. For fixed $d$, we restrict to moduli $q$ so that there is a unique subgroup of invertible classes modulo $q$ of order $d$. We…
We describe quantizations on monoidal categories of modules over finite groups. They are given by quantizers which are elements of a group algebra. Over the complex numbers we find these explicitly. For modules over S3 and A4 we are given…
Nonunique factorization in cancellative commutative semigroups is often studied using combinatorial factorization invariants, which assign to each semigroup element a quantity determined by the factorization structure. For numerical…
A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequality $f_1x_1+\cdots +f_nx_n \mod b \le g_1x_1+\cdots +g_nx_n$ where $g_1,\dots,g_n,$ $f_1,\ldots ,f_n\in \mathbb{Z}$ and…
We give an intrinsic criterion to tell whether a reflection factorization in the general linear group is reduced, and give a formula for computing reflection length in the general affine group.
Exponential Puiseux semirings are additive submonoids of $\qq_{\geq 0}$ generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we…
We prove a new equidistribution estimate for the divisor function in arithmetic progression to moduli that have two small factors. We give two applications. First, we show an asymptotic formula for the divisor function over arithmetic…
A quasi-automatic semigroup is defined by a finite set of generators, a rational (regular) set of representatives, such that if a is a generator or neutral, then the graph of right multiplication by a on the set of representatives is a…
We prove an asymptotic equidistribution result for word values for words with constants in the symmetric group. We also speculate about simultaneous asymptotic equidistribution results for values of $d$-tuples of elements of $\mathbb F_d$.
A numerical semigroup is a submonoid of ${\mathbb Z}_{\ge 0}$ whose complement in ${\mathbb Z}_{\ge 0}$ is finite. For any set of positive integers $a,b,c$, the numerical semigroup $S(a,b,c)$ formed by the set of solutions of the inequality…
We study processes with unstable particles in intermediate time-like states. It is shown that the amplitudes squared of such processes factor exactly in the framework of the model of unstable particles with continuous masses. Decay widths…
We establish asymptotic expansions for factorial moments of following distributions: number of cycles in a random permutation, number of inversions in a random permutation, and number of comparisons used by the randomized quick sort…
Oftentimes the elements of a ring or semigroup $H$ can be written as finite products of irreducible elements, say $a=u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_{\ell}$, where the number of irreducible factors is distinct. The set…
Each restriction semigroup is proved to be embeddable in a factorisable restriction monoid, or, equivalently, in an almost factorisable restriction semigroup. It is also established that each restriction semigroup has a proper cover which…
A numerical semigroup is an additive subsemigroup of the natural numbers that contains zero and has finite complement. A numerical semigroup is irreducible if it cannot be written as an intersection of numerical semigroups properly…
In this paper we present the set of saturated numerical semigroups with prime multiplicity. We also characterize the catenary degree of these semigroups that we acquire. The catenary degree of a numerical semigroup is the variant which…
A finite group $G$ is called $k$-factorizable if for every ordered factorization $|G|=a_1\cdots a_k$ into integers each greater than $1$ there exist subsets $A_1,\dots,A_k\subseteq G$ such that $|A_i|=a_i$ for each $i$ and $G=A_1\cdots…
We consider actions of the free semigroup with two generators on the real line, where the generators act as affine maps, one contracting and one expanding, with distinct fixed points. Then every orbit is dense in a half-line, which leads to…