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We will show in this paper that if λ is very close to 1, then can be attained, where M is a compact–manifold with boundary. This result gives a counter–example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" ( Comm. Pure. Appl. Math., 55, 135–152, 2002). [ABSTRACT FROM AUTHOR]

In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer. [ABSTRACT FROM AUTHOR]

So far the study of exponential bounds of an empirical process has been restricted to a bounded index class of functions. The case of an unbounded index class of functions is now studied on the basis of a new symmetrization idea and a new method of truncating the original probability space; the exponential bounds of the tail probabilities for the supremum of the empirical process over an unbounded class of functions are obtained. The exponential bounds can be used to establish laws of the logarithm for the empirical processes over unbounded classes of functions. [ABSTRACT FROM AUTHOR]

Let $${\fancyscript D}$$ be an increasing sequence of positive integers, and consider the divisor functions: where [ d, δ] = l.c.m.( d, δ). A probabilistic argument is introduced to evaluate the series $$ {\sum\nolimits_{n = 1}^\infty {\alpha _{n} d{\left( {n,{\fancyscript D}} \right)}} } $$ and $$ {\sum\nolimits_{n = 1}^\infty {\alpha _{n} d_{2} {\left( {n,{\fancyscript D}} \right)}} } $$ . [ABSTRACT FROM AUTHOR]