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Abstract

To study the behavior of Italian researchers living in Italy with a view to creating appropriate policies to tackle the brain drain and discourage academics from weight in driving emigrating, we constructed a survey based on a sample of 4,700 Italian researchers (assistant professors) in several universities in Italy. The outlook is far from rosy: Italian researchers are generally dissatisfied with the economic and social situation of the country. Strong family ties represent the element keeping them at home in Italy. In this regard, no particular differences were noted between the North and South of the country. In analyzing the Italian academic system we identified factors that have greater weight in driving Italian intellectual talent to emigrate: the country's higher education system leaves all dissatisfied. Furthermore, we discovered other factors that, albeit weak, keep Italian researchers in Italy. However, one wonders how much longer family and national ties will be able to keep Italian skilled agents in Italy, and whether such dissatisfaction may jeopardize the country's future economic development.

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Abstract  

Let (n k)k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying n k+1/n k > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝0 1 f(x) dx = 0. Then the probabilistic behavior of the system (f(n k x))k≧1 is very similar to the behavior of sequences of i.i.d. random variables. For example, Erdős and Gál proved in 1955 the following law of the iterated logarithm (LIL) for f(x) = cos 2πx and lacunary

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(n_k )_{k \geqq 1}$$ \end{document}
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\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim \sup }\limits_{N \to \infty } (2N\log \log N)^{1/2} \sum\limits_{k = 1}^N {f(n_k x)} = \left\| f \right\|_2$$ \end{document}
((1))
for almost all x ∈ (0, 1), where ‖f2 = (∝0 1 f(x)2 dx)1/2 is the standard deviation of the random variables f(n k x). If (n k)k≧1 has certain number-theoretic properties (e.g. n k+1/n k → ∞), a similar LIL holds for a large class of functions f, and the constant on the right-hand side is always ‖f2. For general lacunary (n k)k≧1 this is not necessarily true: Erdős and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence (n k)k≧1, such that the lim sup in the LIL (1) is not equal to ‖f2 and not even a constant a.e. In this paper we show that the class of possible functions on the right-hand side of (1) can be very large: we give an example of a trigonometric polynomial f such that for any function g(x) with sufficiently small Fourier coefficients there exists a lacunary sequence (n k)k≧1 such that (1) holds with √‖f2 2 + g(x) instead of ‖f2 on the right-hand side.

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