A general theorem concerning many absolute summability methods is proved. 2000 Mathematics Subject Classification. 40F05, 40G05, 40D15, 40D25.

Some generalized Hölder’s inequalities for positive as well as negative exponents are obtained. 2000 Mathematics Subject Classification. 26D07, 26D15.

A relation between Weyl connections and Gaussian thermostats is exposed and exploited. 2000 Mathematics Subject Classification: 37D, 53C, 70K, 82C05.

Some comments and clarifications are made to the author’s paper [1]. 2000 Mathematics Subject Classification 03F50, 03C62, 03E70, 03E40 (primary)

In this study, we establish weighted Čebyšev-Grüss type inequalities on time scales. Mathematics subject classification (2000): 26D15.

We show that for any fixed base a, a positive proportion of primes become composite after any one of their digits in the base a expansion is altered; the case where a = 2 has already been established by Cohen and Selfridge [‘Not every number is the sum or difference of two prime powers’, Math. Comput. 29 (1975), 79–81] and Sun [‘On integers not of the form ±pa ± qb’, Proc. Amer. Math. Soc. 128 ...

Given positive integers a, b, c and d such that c and d are coprime we show that the primes p ≡ c(mod d) dividing ak + bk for some k ≥ 1 have a natural density and explicitly compute this density. We demonstrate our results by considering some claims of Fermat that he made in a 1641 letter to Mersenne. Mathematics Subject Classification (2001). 11N37, 11R45.

The Fishburn numbers, ξ(n), are defined by a formal power series expansion ∞ ∑ n=0 ξ(n)q = 1 + ∞ ∑ n=1 n ∏ j=1 (1− (1− q)). For half of the primes p, there is a non–empty set of numbers T (p) lying in [0, p− 1] such that if j ∈ T (p), then for all n ≥ 0, ξ(pn+ j) ≡ 0 (mod p). 2010 Mathematics Subject Classification: 05A19, 11F20, 11P83

The explicit construction of infinite families of d-regular graphs which are Ramanujan is known only in the case d 1 is a prime power. In this paper, we consider the case when d 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of ‘‘almost’’ Ramanujan graphs for...

We consider generalized gradients in the general context of G-structures. They are natural first order differential operators acting on sections of vector bundles associated to irreducible G-representations. We study their geometric properties and show in particular their conformal invariance. 2000 Mathematics Subject Classification: Primary 53C10, 53A30, 58J60.