Related papers: The mean value problem of Smale's problems

In this paper, we will continue to estmate g_1(n|m) for general n and m.

In this paper we provide a proof of the Riemann Hypothesis by relating the non-trivial zeros of the zeta function to a certain Sturm-Liouville eigenvalue problem on a finite interval.

In this paper some open problems for Painlev\'e equations are discussed. In particular the following open problems are described: (i) the Painlev\'e equivalence problem; (ii) notation for solutions of the Painlev\'e equations; (iii)…

In this article we give, for the fist time the solution of the general difference equation of 2-degree. We also give as application the expansion of a continued fraction into series, which was first proved, found in the past by the author.

This article gives explicit solutions to the Yang-Mills equations. The solutions have positive energy that can be made arbitrarily small by selection of a parameter showing that Yang-Mills field theories do not have a mass gap.

We consider the initial-value problem for the Sasa-Satsuma equation on the line with decaying initial data. Using a Riemann-Hilbert formulation and steepest descent arguments, we compute the long-time asymptotics of the solution in the…

The main purpose of this paper is to study the hybrid mean value problem involving generalized Dedekind sums, generalized Hardy sums and Kloosterman sums, and give some exact computational formulae for them by using the properties of Gauss…

We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is non-strictly hyperbolic and does not admit a fully conservative form, and we establish the…

In this note we obtain sharp bounds for the Seiffert mean in terms of a two parameter family of means. Our results generalize and extend the recent bounds presented in the Journal of Inequalities and Applications (2012) and Abstract and…

In this note, we solve the Gauss image problem given two Borel measures on the unit sphere, one of which is absolutely continuous with respect to the uniform measure.

This note presents a new equivalence to the Riemann Hypothesis by means of the Salem integral equation.

In the present article for the generalized bi-axially symmetric multivariable Helmholtz equation four fundamental solutions are constructed in explicit form. Furthermore, some properties of these solutions are shown, which will be used for…

This paper deals with the computation of the eigenvalues of two-parameter Sturm- Liouville (SL) problems using the Regularized Sampling Method, a method which has been effective in computing the eigenvalues of broad classes of SL problems…

In this paper, we prove a mean value formula for bounded subharmonic Hermitian matrix valued function on a complete Riemannian manifold with nonnegative Ricci curvature. As its application, we obtain a Liouville type theorem for the complex…

A counter-example to lower bounds for the singular values of the sum of two matrices in [1] and [2] is given. Correct forms of the bounds are pointed out.

We prove Dual Smale's mean value conjecture for all odd polynomials with nonzero linear term. Precisely, if $P$ is an odd polynomial of degree $d\ge3$ with $P(0)=0$ and $P'(0)=1$, then there exists a critical point $\zeta$ of $P$ such that…

In this paper we study Appell polynomials by connecting them to random variables. This probabilistic approach yields, e.g., the mean value property which is fundamental in the sense that many other properties can be derived from it. We also…

Various forms of Mean Value Theorems are available in the literature. If we use Flett's Mean Value Theorem in Extended Generalized Mean Value Theorem then what would the new theorem look like. A sincere effort is done to develop this…

We obtain upper bounds on the number of finite sets $\mathcal S$ of primes below a given bound for which various $2$ variable $\mathcal S$-unit equations have a solution.

In this note we present solutions to two problems which appeared in the American Mathematical Monthly. Although the problems seem to be of different nature when it comes to the hypothesis we show that they can be proved using essentially…