This article defines amicable pairs in the complex numbers and finds that some amicable pairs in the natural numbers are also amicable in the complex numbers. Unlike the case in the natural numbers, it is proved that no $(2,1)$ pairs made up of natural numbers where the common factor is a power of $2$ exist as Gaussian amicable pairs. Many pairs are found with complex parts using the DivisorSigma function in Mathematica. The factorizations into primes is given so that the type of pair might be determined.

In 1986, the contestants of the $27$th International Mathematical Olympiad were given a game of numbers played on a pentagon. In 1987, Mozes generalized this game to an arbitrary undirected, weighted, connected graph. The convergence properties and total number of moves of any convergent game have been resolved by Mozes using Weyl groups. Eriksson provided an alternate proof using matrix theory and graph theory. In this paper, we briefly discuss the results of Mozes and Eriksson on undirected graphs. Then we generalize this game to arbitrary directed, strongly connected graphs and investigate the convergence properties of the game of numbers.

In this note, we extend the notion of standard backgammon to a more general setting and call it hyper dice backgammon (or HD-gammon for short) of size $n \geq 6$ (a positive even integer) by extending the regular die to a hyper die (i.e., hyper cube) with $n$ faces and the board from 24 pips to $4n$ pips, where $n = 2k \geq 6$ and there are $4k+3$ checkers for each player. The rules of the game are similar to the rules of standard backgammon when $n = 2k = 6$ and the number of the $n$-sided dice depends on $n$. Finally, we include a list of references related to some theoretical studies on standard backgammon.

A translational surface is a rational surface generated from two rational space curves by translating one curve along the other curve. In this paper, we utilize matrices to represent translational surfaces, and give necessary and sufficient conditions for a real rational surface to be a translational surface.

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted volume with minimum weighted perimeter. According to Chambers' recent proof of the log-convex density conjecture, for many densities on $\mathbb{R}^n$, the answer is a sphere about the origin. We seek to generalize his results to some other spaces of revolution or to two different densities for volume and perimeter. We provide general results on existence and boundedness and a new approach to proving circles about the origin isoperimetric.

We extend results of Bongiovanni et al. [1] on double bubbles on the line with log-convex density to the case where the derivative of the log of the density is bounded. We show that the tie function between the double interval and the triple interval still exists, but may blow up to infinity in finite time. For the first time, a density is presented for which the blowup time is positive and finite.

Let $p$, $q$ be two distinct primes. We consider pointed Hopf algebras of dimension $pq$ over an algebraically closed field of characteristic $p$. We compute higher Frobenius-Schur indicators of these Hopf algebras through the associated graded Hopf algebras with respect to their coradical filtrations. The resulting indicators are gauge invariants for the monoidal representation categories of these algebras.

Let $T$ be a bounded linear operator on a complex Hilbert space $\mathcal{H}$. In this paper we introduce two new classes of operators: class $Q(N)$ and class $Q^*(N)$. An operator $T\in \mathcal{L}(\mathcal{H})$ is of class $Q(N)$ for a fixed real number $N\geq 1$, if $T$ satisfies $N\|Tx\|^{2} \leq \| T^2 x\|^{2}+ \| x\|^{2}$ for all $x\in \mathcal{H}$. And an operator $T\in \mathcal{L}(\mathcal{H})$ is of class $Q^*(N)$ for a fixed real number $N\geq 1$, if $T$ satisfies $N\|T^*x\|^{2} \leq \| T^2 x\|^{2}+ \| x\|^{2}$ for all $x\in \mathcal{H}$. We prove the basic properties of these classes of operators.

Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable $x$. As we follow this link, we find that these polynomials are ready to spill two of their secrets: (i) There exists a unique “base-$x$” representation of such polynomials that makes the ring $\mathbb{Z}[x]$ into an ordered domain; and (ii) There is a 1-1 correspondence between positive rational primes $p$ and certain infinite sets of irreducible polynomials $f(x)$ that attain the value $p$ at sufficiently large $x$, each generated in finitely many steps from the $p$th cyclotomic polynomial. The base-$x$ representation provides practical conversion methods among numeric bases (not to mention a polynomial factorization algorithm), while the prime-irreducible correspondence puts a new angle on the Bouniakowsky Conjecture, a generalization of Dirichlet's Theorem on Primes in Arithmetic Progressions.