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]
In this paper, a class of global optimization problems is considered. Corresponding to each local minimizer obtained, we introduced a new modified function and construct a corresponding optimization subproblem with one constraint. Then, by applying a local search method to the one-constraint optimization subproblem and using the local minimizer as the starting point, we obtain a better local optimal solution. This process is continued iteratively. A termination rule is obtained which can serve as stopping criterion for the iterating process. To demonstrate the efficiency of the proposed approach, numerical examples are solved. [ABSTRACT FROM AUTHOR]