A simple proof of a theorem unifying Bolzano's Theorem [8], the Intermediate Value Theorem, Rouché's Theorem [3] and its extensions to differentiable maps to $\mathbb{R}^n$ [2, 6, 9] is obtained. This unifying theorem in particular shows that in Professor Baker's [1] examples where the number of solutions of $f(x) = y$ for a continuous map $f \colon B^2 \to \mathbb{R}^2$, $y \not \in f ( \partial B^2 )$, from the unit ball $B^2$ in the plane $\mathbb{R}^2$ is not exactly the absolute value of the winding number of the curve $f ( \partial B^2)$ about $y$, the number of the connected components of the solution set counted with multiplicity coincides with the winding number.

In this note we prove that for all positive integers $n$, the sum $S_{4n+1}$ of the first $4n+1$ Pell numbers is a perfect square. As a consequence, an identity involving binomial coefficients and Pell numbers is given. Also, sums of an even and odd number of terms of odd order are evaluated and some divisibility properties are obtained.

In the present paper the real ideals of relatively compact operators of $W^*$-algebras are considered. Similar to the complex case, a description (up to isomorphism) of the real two-sided ideal of relatively compact operators of the complex $W^*$-factors is given.

We present a simple algorithm that reduces the time complexity of solving the linear system $Gx=b$, where $G$ is a centrosymmetric/skew-centrosymmetric matrix. We also reduce the time complexity of solving some complex linear systems. We propose efficient methods for multiplying centrosymmetric/skew-centrosymmetric matrices.

This paper deals with some new properties of $T_{D}$-spaces. These properties are used in order to give an intrinsic topological characterization of the Goldman spectrum of a commutative ring.

In this paper we introduce the concepts of $\omega$-paracompactness and countable $\omega$-paracompactness as generalizations of paracompactness. We characterize each of them. The study deals with subspaces, products and mappings of each. We conclude this paper with various counterexamples relevant to the relations among the concepts of this paper.