The Frobenius number F (a) of an integer vector a with positive coprime coefficients is defined as the largest number that does not have a representation as a positive integer linear combination of the coefficients of a. We show that if a is taken to be random in an expanding d-dimensional domain, then F (a) has a limit distribution, which is given by the probability distribution for the coveri...

Abstract The aim of this paper is to represent any polynomial in terms degenerate Frobenius–Euler polynomials and, more generally, higher-order polynomials. Explicit formulas with the help umbral calculus are derived and obtained results illustrated by some examples.

We establish a stochastic nonlinear analogue of the PerronFrobenius theorem on eigenvalues and eigenvectors of positive matrices. The result is formulated in terms of an automorphism T of a probability space (Ω,F , P ) and a random mapping D(ω, ·) : R+ → R+. Under assumptions of monotonicity and homogeneity of D(ω, ·), we prove the existence of scalar and vector measurable functions α(ω) > 0 an...

We define the Frobenius morphism of certain class of noncommutative blowups in positive characteristic. Thanks to a nice property of the class, the defined morphism is always flat. Therefore we say that the noncommutative blowups in this class are Kunz regular. One of such blowups is the one associated to a regular Galois alteration. From de Jong’s theorem, we see that for every variety over an...