Kleene’s theorem on recognizable languages in free monoids is considered to be of eminent importance in theoretical computer science. It has been generalized into various directions, including trace and rational monoids. Here, we investigate divisibility monoids which are defined by and capture algebraic properties sufficient to obtain a characterization of the recognizable languages by certain...

A class of Laplace transforms is examined to show that particular cases of this class are associated with production-destruction and reaction-diffusion problems in physics, study of differences of independently distributed random variables and the concept of Laplacianness in statistics, α-Laplace and Mittag-Leffler stochastic processes, the concepts of infinite divisibility and geometric infini...

A sequence a = Han L is B−automatic if it can be recognized by a finite state machine in the sense that, on input n, the machine outputs an . Here n is usually written in standard base B notation, though other numeration systems are also considered. A typical example is the Morse−Thue sequence Htn L which is usually defined in terms of iterated morphisms. However, there is an alternative charac...

We study the the polytopes of binary n-strings that encode (positive) integers that are not divisible by a particular positive integer p | the indivisibility polytopes, as well as the more general \clipped cubes". Also, we discuss a potential application to factoring. Finally, we present some results concerning divisibility polytopes.

In this paper we determine the solvability of families of systems of two polynomial equations over finite fields by computing the exact divisibility of the exponential sums associated to the systems. This generalizes a theorem of Carlitz to systems of two equations. Our result gives an upper bound for the Waring number of systems of diagonal equations. Also, as a by-product, we also obtain info...

Wilf stated that the Lagrange inversion formula (LIF) is a remarkable tool for solving certain kinds of functional equations, and at its best it can give explicit formulas where other approaches run into stone walls. Here we present the LIF combinatorially in the form of lattice paths, and apply it to the divisibility property of the coefficients of a formal power series expansion. For the LIF,...

For all positive integers n, we prove the following divisibility properties: (2n + 3) ( 2n n ) ∣∣∣∣3(6n 3n )( 3n n ) and (10n + 3) ( 3n n ) ∣∣∣∣21(15n 5n )( 5n n ) . This confirms two recent conjectures of Z.-W. Sun. Some similar divisibility properties are given. Moreover, we show that, for all positive integers m and n, the product am ( am+bm−1 am )( an+bn an ) is divisible by m + n. In fact,...

In this paper, we prove a result linking the square and the rectangular R-transforms, which consequence is a surprising relation between the square and rectangular free convolutions, involving the Marchenko-Pastur law. Consequences on infinite divisibility and on the arithmetics of Voiculescu’s free additive and multiplicative convolutions are given.

We give an estimate for the Kolmogorov distance between an infinitely divisible distribution (with mean zero and variance one) and the standard Gaussian distribution in terms of the difference between the fourth moment and 3. In a similar fashion we give an estimate for the Kolmogorov distance between a freely infinitely divisible distribution and the Semicircle distribution in terms of the dif...

Stable laws can be tempered bymodifying the Lévymeasure to cool the probability of large jumps. Tempered stable laws retain their signature power law behavior at infinity, and infinite divisibility. This paper develops randomwalk models that converge to a tempered stable law under a triangular array scheme. Since tempered stable laws and processes are useful in statistical physics, these random...