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We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…
Can we do arithmetic in a completely different way, with a radically different data structure? Could this approach provide practical benefits, like operations on giant numbers while having an average performance similar to traditional…
Numerous exact relations exist that relate the effective elastic properties of composites to the elastic properties of their components. These relations can not only be used to determine the properties of certain composites, but also…
A relatively new topic in computability theory is the study of notions of computation that are robust against mistakes on some kind of small set. However, despite the recent popularity of this topic relatively foundational questions about…
Let $q_1, \ldots , q_t$ be distinct prime numbers. Let $a_1, \ldots , a_t$ be nonnegative integers. We establish effective lower bounds for $|z^d - q_1^{a_1} \ldots q_t^{a_t}|$ and for its greatest prime factor, which tend to infinity with…
In this paper, we focus on the associated primes of powers of monomial ideals and asymptotic behavior properties such as normally torsion-freeness, normality, the strong persistence property, and the persistence property. In particular, we…
We consider the geometry relaxation of an isolated point defect embedded in a homogeneous crystalline solid, within an atomistic description. We prove a sharp convergence rate for a periodic supercell approximation with respect to uniform…
To compute the local cohomology of powers of edge ideals one needs to know their saturations. The saturation of the second and third powers has been described in terms of the graph in [13] and [10]. In this article, we give a combinatorial…
The celebrated Trotter approximation theorem provides a sufficient condition for the convergence of a sequence of operator semigroups in terms of the corresponding sequence of infinitesimal generators. There exist a few results on the rate…
In the paper, the author studies properties of three functions relating to the exponential function and the existence of partitions of unity, including accurate and explicit computation of their derivatives, analyticity, complete…
We study recurrence, and multiple recurrence, properties along the $k$-th powers of a given set of integers. We show that the property of recurrence for some given values of $k$ does not give any constraint on the recurrence for the other…
The classical Fibonacci sequence is known to exhibit many fascinating properties. In this paper, we explore the Fibonacci sequence and integer sequences generated by second order linear recurrence relations with positive integer…
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…
Computing $\log\det(A)$ for large symmetric positive definite matrices arises in Gaussian process inference and Bayesian model comparison. Standard methods combine matrix-vector products with polynomial approximations. We study a different…
This paper concerns optimal gradient estimates of solutions for the perfect conductivity problem with closely spaced interfacial boundaries. The problem arises from composite material. Our estimates exhibit different blow up rates of the…
We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$…
Fast matching of regular expressions with bounded repetition, aka counting, such as (ab){50,100}, i.e., matching linear in the length of the text and independent of the repetition bounds, has been an open problem for at least two decades.…
The paper discusses the problem of the Lorentz contraction in accelerated systems, in the context of the special theory of relativity. Equal proper accelerations along different world lines are considered, showing the differences arising…
We give optimal convergence rates in the central limit theorem for a large class of martingale difference sequences with bounded third moments. The rates depend on the behaviour of the conditional variances and for stationary sequences the…
An integer sequence that is defined by initial values and a linear recurrence with constant integer coefficients, can be represented by the difference of two arithmetic terms containing exponentiation. All constants occuring in the term are…