A refinement of Selberg’s asymptotic equation

Asymptotic distributions of Neumann problem for Sturm-Liouville equation

In this paper we apply the Homotopy perturbation method to derive the higher-order asymptotic distribution of the eigenvalues and eigenfunctions associated with the linear real second order equation of Sturm-liouville type on $[0,pi]$ with Neumann conditions $(y'(0)=y'(pi)=0)$ where $q$ is a real-valued Sign-indefinite number of $C^{1}[0,pi]$ and $lambda$ is a real parameter.

On a Refinement Type Equation

Let (Ω,A, P ) be a complete probability space. We show that the trivial function is the unique L-solution of the following refinement type equation f(x) = Z Ω |φx(x, ω)|f(φ(x, ω))dP (ω) for a wide class of the given functions φ. This class contains functions of the form φ(x, ω) = α(ω)x−β(ω) with −∞ < R Ω log |α(ω)|dP (ω) < 0.

The Umbral Refinement Equation

Asymptotic States of a Smoluchowski Equation

∗ Department of Mathematics, The University of Chicago, Chicago, Il 60637. †Chemical Engineering, PACM and Mathematics, Princeton University, Princeton, NJ 08544. ‡Department of Mathematics, and Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697-3875, Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100,...

Asymptotic Refinement of the Berry-Esseen Constant

For sum of independent and identically distributed (i.i.d.) random variables {Xi}i=1, the Berry-Esseen theorem states that sup y∈< ∣∣∣∣Pr { 1 sn (X1 + X2 + · · ·+ Xn) ≤ y } − Φ(y) ∣∣∣∣ ≤ C ρ σ3 √ n , where σ and ρ are respectively the variance and the absolute third moment of the parent distribution, Φ(·) is the unit normal cumulative distribution function, and C is an absolute constant. In thi...