A Diophantine approximation problem with two primes and one k-power of a prime

We refine a result of the last two Authors on a Diophantine approximation problem with two primes and a k-th power of a prime which was only proved to hold for 1<k<4/3. We improve the k-range to 1<k 643 by combining Harman's technique on the minor arc with a suitable estimate for the L4-norm of the relevant exponential sum over primes Sk. In the common range we also give a stronger bound for the approximation

Long-Lasting Bank Relationships and Growth of Firms

A puzzling but consistent result in the empirical literature on banking is that firms with close bank ties do not grow faster than bank-independent firms. In this paper, we reconsider the link between relationship lending and firms’ growth, distinguishing firms by size and “health”. The idea is that the beneficial effects of relationship lending on information asymmetries can be compensated by other negative capture, risk and externality effects which make relational banks reluctant to support long-term growth projects of client firms, and that the strength of these compensating effects varies with firm size and health status. We explore the influence of long-lasting bank relationships on employment and asset growth of a large sample of Italian firms. The main finding is that relationship lending hampers the efforts of small firms to increase their size, while it mitigates the negative growth of troubled, medium-large enterprises.relationship lending, capture effects, firms’ growth

Analysis of Solving a Cauchy Problem Using an Eye-Tracker

In this paper we present a qualitative analysis based on data collected by means of an eye-tracker tool, concerning the outcome of a mathematical analysis question administered to a group of candidates of university level or higher. One of the research aims is to highlight similarities and differences in the visual observation of the question itself between candidates considered experts and non-experts. This analysis should provide useful information and clues about the way a mathematical problem is tackled (and the underlying cognitive processes used to tackle it) for eventual research use in Mathematics education, also at university level

On an average ternary problem with prime powers

We continue our work on averages for ternary additive problems with powers of prime numbers in short intervals by computing the average number of representations of a positive integer $n$ as $p_1^{k_1} + p_2^{k_2} + p_3^{k_3}$, where $p_1$, $p_2$ and $p_3$ are prime numbers and $2 le k_1 le k_2 le k_3$ are natural numbers