Each sphere in $\ell^{2}$ contains uncountably many mutually disjoint, simply connected, Frechet differentiable and contractible subsets, each of which is dense in the sphere.

Spiral knots are a generalization of torus knots we define by a certain periodic closed braid representation. For spiral knots with prime power period, we calculate their genus, bound their crossing number, and bound their $m$-alternating excess.

This paper shows that the nullity and rank of $aP+bQ-cQAP$ is a constant, where $P$ and $Q$ are outer inverses of a given matrix $A$, $c=a+b$ ($a,b\neq 0$) or $c\neq a+b$, $a, b, c \in \mathbb{C}$. In addition, the rank of $aP+bQ-cQAP$ is equal to the rank of $P-Q$ if $c=a+b$ and to $P+Q$ if $c\neq a+b$.

We present several new bounds for certain sums of deviation probabilities involving sums of nonnegative random variables. These are based upon upper bounds for the moment generating functions of the sums. We compare these new bounds to those of Maurer [2], Bernstein [4], Pinelis [16], and Bentkus [3]. We also briefly discuss the infinitely divisible distributions case.

Falcón Santana and Díaz-Barrero [Missouri Journal of Mathematical Sciences, 18.1, pp. 33-40, 2006] proved that the sum of the first $4n+1$ Pell numbers is a perfect square for all $n \ge 0$. They also established two divisibility properties for sums of Pell numbers with odd index. In this paper, the sum of the first $n$ Pell numbers is characterized in terms of squares of Pell numbers for any $n \ge 0$. Additional divisibility properties for sums of Pell numbers with odd index are also presented, and divisibility properties for sums of Pell numbers with even index are derived.

In this note, we define adherent and accumulation points of a set in an intuitionistic fuzzy metric space, and prove the Sierpinski and Hine Theorem for intuitionistic fuzzy metric spaces.