In this work the following problem in the hyperbolic plane is investigated. Find the perimeter-minimizing regions of prescribed area between two parallel horocycles. An explicit and detailed description of all such regions is given through isoperimetric inequalities.

In this paper, we examine computationally the results of combining two well-known, simple, and imperfect tests for primality: the Fermat base-2 test, and the Fibonacci test. Although considerable attention has been paid to various properties of composite integers which pass the base-2 test (base-2 pseudoprimes), no comparable study of Fibonacci and base-2 Fibonacci tests exists in the literature. Our study tabulates various empirical properties of these numbers. Among other things, we conclude that there are no base-2 Fibonacci pseudoprimes less than $10^{15}$ which are congruent to 2 or 3 (modulo 5).

If $(S, \ast)$ is an arbitrary mathematical structure on a set $S$, three universal problems are to find all groups $(S, \cdot)$ on the same set that left-distribute or right-distribute or both left-distribute and right-distribute over $(S, \ast)$, if such a group exists. These concepts are defined in this paper. Also, we give a solution to the first two of these three problems for a naturally occurring example that involves what we call an $n$-star (which is structurally the same as $n$ lines in the plane intersecting in ${n \choose 2}$ district points).

The primary purpose of this paper is to present the definition of the a-pedal hypersurface with respect to a point in the interior of a closed, convex and smooth hypersurface $M$. The secondary purpose of this paper is to give some new characteristic properties of the a-pedal hypersurfaces related to the support function, Gauss curvature, mean curvature, the first and second fundamental forms and their coefficients of $M$ (Section 3). Using the classical methods of the hypersurfaces in differential geometry we have established that the support function $h_{a}$ of the a-pedal hypersurface $M_{a}$ is equal to $\frac{h^{a+1}}{P_{a}}$ where $P_{a}^{2}=h^{2}+a^{2}\overset{III}{\nabla }(h,h)$.

The planar soap bubble problem seeks the least perimeter way to enclose and separate $m$ regions of $m$ given areas. We discuss a useful approach, especially for $m\le 8$.

The Rellich-Kondrachov Theorem is a fundamental result in the theory of Sobolev spaces. We prove an analogue of this theorem in the context of sequence spaces, using elementary methods.

We study the lattice structure of the set $ \Omega (X)$ of all $T_1$-$L$ topologies on a given set $X$. It is proved that $\Omega ( X ) $ has dual atoms (anti atoms) if and only if membership lattice $L$ has dual atoms (anti atoms). Some other properties of this lattice are also discussed.

Suppose $P(x)$ and $Q(x)$ are two arbitrary polynomials. In this paper we use the theory of resultants of two polynomials including Sylvester's Matrix to specify a large number of polynomial relations involving $P(x)$, $Q(x)$ and their derivatives. We also include a research problem for the reader to consider.