It is shown that $f_{t}(x)=x^{5}+(t^{2}-3125)x-4(t^{2}-3125)$ ($t \in \mathbb{Q}$) is reducible in $\mathbb{Q} [x]$ if and only if $t=0$. When $t \neq 0$ it is shown that $\text{Gal} (f_{t}) \simeq D_{5}$ or $A_{5}$, and necessary and sufficient conditions are given for each possibility.

In this note we provide a geometrical interpretation for the basic assumptions made in the method of variation of parameters applied to second order ordinary differential equations. We also discuss a physical motivation drawn from celestial mechanics.

Every real polynomial with all real, distinct zeros has a corresponding ratio vector that measures the distribution of critical numbers relative to the zeros. We prove several results on the ratio vectors of Chebyshev polynomials, as well as polynomials with equally spaced zeros, and contrast the behaviors of these families of polynomials.

Next to a power series, the classical Legendre series offers the simplest method of representing a function using polynomial expansion means. In 1862, Neumann established results for complex Legendre expansions that are analogous to Taylor's Theorem and the Cauchy-Hadamard Formula for power series, the primary difference being that results are stated in terms of ellipses, as opposed to discs, of convergence. After a simple change of variable, the foci of these ellipses may vary, each leading to a modified Legendre expansion of the original function. Our main result is that as the foci of these ellipses tend to one another, the limit of the corresponding Legendre expansions is the Taylor series representation.

A strong law of large numbers for arrays of rowwise pairwise negatively quadrant dependent (NQD) random variables is obtained which relaxes the usual assumption of rowwise negatively dependent (ND). The moment conditions of the main result are weaker than the previous results. The result obtained generalizes and improves the result of Taylor (2002).

In this note we provide a new method of constructing the Nagel Point.

Let $G$ be a finite metabelian $p$-group whose non-linear irreducible character degrees lie between $p^a$ and $p^b$, where $1 < a \le b$. In this paper it is shown that the nilpotence class of $G$ is bounded by a function of $p$ and $b-a$.

The objective of this paper is to obtain properties of strongly compact spaces by using nets, filterbases, pre-complete accumulation points and so on.

In this paper we introduce $T_{ac}$ spaces as a generalization of $T_{D}$ spaces and we present some properties of these spaces related to subspaces, quotients, continuous functions, and homeomorphisms. Also, we show that the property $T_{ac}$ does not imply $T_{0}$ and $T_{0}$ does not imply $T_{ac}$.