There is a question attributed to Irving Kaplansky concerning the solvability of the quadratic equation $x^2-p{y}^2=a$ in the case that the prime $p=a^2+{\left(2b\right)}^2$. This question was answered in the affirmative by Mollin [1], although according to [3], this result is implicit in the work of Gauss and Legendre. The proof appearing in [1] was later simplified in [4], and it was also shown therein that Kaplansky’s question was a special case of a more general result. Using the method of proof in [4], Mollin [2] has recently extended the results of [4], but upon further consideration, it appears that there is a more general phenomenon occurring, and also, that one of the assumptions in the main theorem of [2] is unnecessary. In this paper we prove this generalization, and eliminate one of the assumptions stated in the main result of [2]. The proof is again based on the method described in [4].

The aim of the present paper is to establish a new result, which guarantees the instability of zero solution to a certain class of non-autonomous ordinary differential equations of sixth order. Our result improves some known results in the literature for non- autonomous case (see, [20,Theorem 4], [31, Theorem 1]).

In this paper, we introduce and consider a new class of Wiener-Hopf equations involving the nonlinear operator and nonexpansive operators, which is called the implicit Wiener-Hopf equations. Essentially using the projection technique, we establish the equivalence between the implicit Wiener-Hopf equations and quasi variational inequalities. Using this alternative equivalent formulation, we suggest and analyze an iterative method for finding the common element of the set of fixed points of nonexpansive mappings and the set of solutions of the quasi variational inequalities. We also study the convergence criteria of iterative methods under some mild conditions. Our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results.

We obtain expressions for sums of the form $\sum _{j=0}^m{(-1)}^j\frac{j^d\left(\begin{array}{c}m\\ j\end{array}\right)}{\left(\begin{array}{c}n+j\\ j\end{array}\right)}$ and deduce, for an even integer $d\ge 0$ and , that this sum is $0$ or $\frac{1}{2}$ according as to whether or not. Further, we prove for even that $\sum _{l=1}^d{c}_{l-1}\frac{{(-1)}^l\left(\begin{array}{c}n\\ l\end{array}\right)l!}{(l+1)\left(\begin{array}{c}2n\\ l+1\end{array}\right)}=0$ where $c_r=\frac{1}{r!}\sum _{s=0}^r{(-1)}^s\left(\begin{array}{c}r\\ s\end{array}\right){(r-s+1)}^{d-1}$. Similarly, we show when is even that $\sum _{r=0}^d{a}_r\frac{r!\left(\begin{array}{c}n\\ r+1\end{array}\right)}{\left(\begin{array}{c}2n\\ r+1\end{array}\right)}=0$, where $a_r=\frac{{(-1)}^{d+r}}{r!}\sum _{s=0}^r{(-1)}^s\left(\begin{array}{c}r\\ s\end{array}\right){(r-s+1)}^d$.

Let $\{X_{n,}n\ge 1\}$ be a sequence of independent random variables with common continuous distribution function $F$. In this article we discuss distributional and asymptotical properties of the point process $N_{n,m}(·)=\sum _{i=1}^{n}1(X_{n-m+1:n}-X_i\in ·)$ driven by the $m$th upper order statistic $X_{n-m+1:n}$. Further we derive some limiting results for related sums, which are of some interest in insurance applications.