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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]

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]

A right R–module E over a ring R is said to be maximally injective in case for any maximal right ideal m of R, every R–homomorphism f : m → E can be extended to an R–homomorphism f' : R → E. In this paper, we first construct an example to show that maximal injectivity is a proper generalization of injectivity. Then we prove that any right R–module over a left perfect ring R is maximally injective if and only if it is injective. We also give a partial affirmative answer to Faith's conjecture by further investigating the property of maximally injective rings. Finally, we get an approximation to Faith's conjecture, which asserts that every injective right R–module over any left perfect right self–injective ring R is the injective hull of a projective submodule. [ABSTRACT FROM AUTHOR]

The article provides information on how to determine the eigenvalue problem and solution for Sturm-Liouville mathematical equation. It discusses the procedures in determining the equation's boundary conditions by using the arbitrary complex number and complex-valued function. It presents the equation's fundamental system of solutions, along with initial conditions. It highlights the tips in calculating the equation, where a certain given problem can be reduced into simple mathematical form. Furthermore, the theorems as well as the functions of the equation are also highlighted.

The article analyzes the boundary asymptotic behavior of solutions for weighted p(x)-Laplacian equations that take infinite value on a bounded domain. It was found that the boundary asymptotic of solutions of the Laplacian function is continuous and positive. The p(x)-Laplacian possesses more complicated inhomogeneous nonlinearities, thus, some special techniques are needed. Its main difficulty arises from the lack of compactness. The uniqueness and asymptotic behavior of solutions for problem is a nonnegative function which is allowed to vanish on the boundary.

The equations under consideration have the following structure: where 0 < x n < ∞, ( x 1, ..., x n−1) ∈ Ω, Ω is a bounded Lipschitz domain, $$f(0,x_n ) \equiv 0,\tfrac{{\partial f}}{{\partial u}}(0,x_n ) \equiv 0$$ is a function that is continuous and monotonic with respect to u, and all coefficients are bounded measurable functions. Asymptotic formulas are established for solutions of such equations as x n → + ∞; the solutions are assumed to satisfy zero Dirichlet or Neumann boundary conditions on ∂Ω. Previously, such formulas were obtained in the case of a ij, ai depending only on ( x 1, ..., x n−1). [ABSTRACT FROM AUTHOR]

The classical proofs of the De Giorgi–Nash–Moser Theorem are based on the iteration of some inequality through countably many concentric balls. In this note, we present a new approach to the Hölder continuity of solutions to elliptic equations in divergence form, which avoids any form of discrete iteration. In particular, we prove that a suitable energy function satisfies a differential inequality, whose integration yields a new proof of the crucial step in the regularity result. [ABSTRACT FROM AUTHOR]

The article analyzes enough conditions on the lower order coefficients of the parabolic equations in the second-order. It discusses the solution of the Cauchy problem in classes of rising initial functions stables x-uniformly to zero on each compact set K ∈ RN. It presents the reality of coefficients αik = αki (i, k = 1, 2, ..., N) and the satisfaction of the uniform parabolicity conditions; Ǝ there exist constants ɻ0 0 and ɻ1.