Let $R$ be a finite commutative ring with identity. We form the zero divisor graph of $R$ by taking the nonzero zero divisors as the vertices and connecting two vertices, $x$ and $y$, by an edge if and only if $xy=0$. We establish that if the zero divisor graph of a finite commutative ring with identity is planar, then the graph has a planar supergraph with a Hamiltonian cycle. We also determine the book thickness of all planar zero divisor graphs.

We find three new ways to construct Smith numbers from primes that contain small digits and a known digit sum. The resulting Smith numbers may have a fairly large number of digits.

Let $f(x)$ be an irreducible polynomial of degree $n$ defined over a field $F$, and let $G$ be the Galois group of $f$, identified as a transitive subgroup of $S_n$. Let $K/F$ be the stem field of $f$. We show the automorphism group of $K/F$ is isomorphic to the centralizer of $G$ in $S_n$. We include two applications to computing Galois groups; one in the case $F$ is the rational numbers, the other when $F$ is the 5-adic numbers.

Let $R$ be a commutative ring with unity and $M_R$ be a nonzero unital right $R$-module. We say that $M$ is an OI $R$-module if for each $x \in R$, $Mx = M$ implies $x$ is invertible in $R$. We give a characterization of OI torsion abelian groups in terms of their direct summands.

The excedent function, $e(n) := \sigma(n) - 2n$, measures the amount by which the sum of the divisors of an integer exceeds that integer. Despite having been in the mathematical consciousness for more than $2000$ years, there are many unanswered questions concerning the function. Of particular importance to us is the question of explaining and classifying values in the image of $e(n)$ — especially in understanding the ``small'' values. We look at extensive calculated data, and use them as inspiration for new results, generalizing theorems in the literature, to better understand a family of values in this image.

A number of landmark existence theorems of nonlinear functional analysis follow in a simple and direct way from the basic separation of convex closed sets in finite dimension via elementary versions of the Knaster-Kuratowski-Mazurkiewicz principle - which we extend to arbitrary topological vector spaces - and a coincidence property for so-called von Neumann relations. The method avoids the use of deeper results of topological essence such as the Brouwer Fixed Point Theorem or the Sperner's Lemma and underlines the crucial role played by convexity. It turns out that the convex KKM Principle is equivalent to the Hahn-Banach Theorem, the Markov-Kakutani Fixed Point Theorem, and the Sion-von Neumann Minimax Principle.

In a recent paper G. A. Cannon and K. M. Neuerburg point out that if $A=\mathbb{Z}$ and $B$ is an arbitrary ring with unity, then $\mathbb{Z}\star{B}$, the Dorroh extension of $B$, is isomorphic to the direct product $\mathbb{Z}\times{B}$. Thus, the ideal structure of $\mathbb{Z}\star{B}$ can be completely described. The aim of this note is to point out that this result may be extended to any pair $(A,B)$ in which $B$ is an $A$-algebra with unity, and to study the construction of extensions of algebras without zero divisors and their behavior with respect to algebra maps.

In the paper, the authors establish some Hermite-Hadamard type inequalities for the product of two $(\alpha, m)$-convex functions.

For $r \in [0,1]$, the Bernoulli measure $\mu_r$ on the Cantor space $\{0,1\}^{\mathbb{N}}$ assigns measure $r$ to the set of sequences with $1$ at a fixed position. In [5] it is shown that for $r,s \in [0,1]$, $\mu_s$ is continuously reducible to $\mu_r$ if any only if $r$ and $s$ satisfy certain purely number theoretic conditions (binomial reduciblility). We bring these results into the context of computability theory and Martin-Löf randomness and show that the continuous maps arising in [5] are truth-table functionals (tt-functionals) on $\{0,1\}^{\mathbb{N}}$. This allows us extend the characterization of continuous reductions between Bernoulli measures to include tt-functionals. It then follows from the conservation of randomness under tt-functionals that if $s$ is binomially reducible to $r$, then there is a tt-functional that maps every Martin-Löf random sequence for $\mu_s$ to a Martin-Löf random sequences for $\mu_r$. We are also able to show using results in [2] that the converse of this statement is not true.

Two new forms of contra-somewhat continuity are introduced. Characterizations and the basic properties of both forms investigated.

In this short note we give an exceedingly short proof that the harmonic series diverges. The proof is virtually a one-liner.