The purpose of this paper is to study the higher order asymptotic distributions of the eigenvalues associated with a class of Sturm-Liouville problem with equation of the form w??=(?2f(x)?R(x)) (1), on [a,b, where ? is a real parameter and f(x) is a real valued function in C2(a,b which has a single zero (so called turning point) at point 0x=x and R(x) is a continuously differentiable function. ...

Let W [n] be an independent random sequence with constant mean µ W = 0 and variance σ 2 W. Define a new random sequence X[n] as follows: X[0] = 0 X[n] = ρX[n − 1] + W [n] for n ≥ 1. (a) Find the mean value of X[n] for n ≥ 0. (b) Find the autocovariance of X[n], denoted as K XX [m, n]. (c) For what values of ρ does K XX [m, n] tend to G[m − n], for some finite-valued function G, as m and n becom...

In this paper, we define a new concept of factorization for a bounded bilinear mapping $f:Xtimes Yto Z$, depended on  a natural number $n$ and a cardinal number $kappa$; which is called $n$-factorization property of level $kappa$. Then we study the relation between $n$-factorization property of  level $kappa$ for $X^*$ with respect to $f$ and automatically boundedness and $w^*$-$w^*$-continuity...

Abstract Z.-W. Sun [‘Refining Lagrange’s four-square theorem’, J. Number Theory 175 (2017), 169–190] conjectured that every positive integer n can be written as $ x^2+y^2+z^2+w^2\ (x,y,z,w\in \mathbb {N}=\{0,1,\ldots \})$ with $x+3y$ a square and also $n=x^2+y^2+z^2+w^2\ (x,y,z,w \in {Z})$ $x+3y\in \{4^k:k\in {N}\}$ . In this paper, we confirm these conjectures via the arithmetic theory of tern...

Let $X = (X_1, X_2)$ be a 2-dimensional random variable and $X(n), n \in \mathbb{N}$ sequence of i.i.d. copies $X$. The associated walk is $S(n)= X(1) + \cdots +X(n)$. corresponding absorbed-reflected $W(n), in the first quadrant given by $W(0) x \mathbb{R}_+^2$ $W(n) \max \{ 0, W(n-1) - X(n) \}$, where maximum taken coordinate-wise. This often called Lindley process models waiting times two-se...

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