In this paper, we characterize the finite commutative rings such that their comaximal graph (or zero-divisor graph) are ring graphs, and we also study the case where they are outerplanar.

In this article, we define the index of maximality $\mathfrak{m}(y)$ of a positive integer $y$, associated with the vanishing of certain power sums over $\mathbb{F}_q[T]$, related to the set $V_m(y)$ of ``valid'' decompositions of $y=X_1+\cdots+X_m$ of length~$m$. The index of maximality determines the maximum positive integer $m$ for which the sets $V_m(y)$ are not empty. An algorithm is provided to find $V_i(y)$, $1\leq i\leq \mathfrak{m}(y)$ explicitly.

The invariance, under some action, of the index of maximality $\mathfrak{m}(y)$ and of the property of divisibility by $q-1$ of $\ell_q (y)$, the sum of the $q$-adic digits of $y$, implies the invariance of the degree of Goss zeta function; it is illustrated here for two cases.

Finally, we generalize, to all $q$, the properties of an equivalence relation on $\mathbb{Z}_p$, which depends on the Newton polygon of the Goss zeta function.

We introduce an exact functor defined on multigraded modules which we call the expansion functor and study its homological properties. The expansion functor applied to a monomial ideal amounts to substitute the variables by monomial prime ideals and to apply this substitution to the generators of the ideal. This operation naturally occurs in various combinatorial contexts.

Let $k=\mathbb{F}_q(t)$ be the rational function field over $\mathbb{F}_q$ and $f(\mathbf{x}) \in k[x_1, \ldots, x_s]$ be a form of degree~$d$. For $l \in \mathbb{N}$, we establish that whenever \[ s > l + \sum_{w=1}^{d} w^2 \binom{d-w+l-1}{l-1}, \] the projective hypersurface $f(\mathbf{x})=0$ contains a $k$-rational linear space of projective dimension~$l$. We also show that if $s> 1+ d(d+1)(2d+1)/6$, then for any $k$-rational zero $\mathbf{a}$ of $f(\mathbf{x})$ there are infinitely many $s$-tuples $(\varpi_1, \ldots, \varpi_s)$ of monic irreducible polynomials over $k$, with the $\varpi_i$ not all equal, and $f(a_1\varpi_1, \ldots, a_s \varpi_s) =0$. We establish in fact more general results of the above type for systems of forms over $C_i$-fields.

We compute the monoid $\mathcal{V} [ L_{K} (E) ]$ of isomorphism classes of finitely generated projective modules of a Leavitt path algebra over an arbitrary directed graph. Our result generalizes the result of Ara, Moreno and Pardo in which they computed the monoid $\mathcal{V}[ L_{K} (E) ]$ of a Leavitt path algebra over a countable row-finite directed graph.

We give new characterizations of minimal cusco maps in the class of all set-valued maps extending results from \cite{BZ1, GM}. Let $X$ be a topological space and $Y$ a Hausdorff locally convex linear topological space. Let $F: X \to Y$ be a set-valued map. The following are equivalent: (1)~$F$ is minimal cusco; (2)~$F$ has nonempty compact values, there is a quasicontinuous, subcontinuous selection $f$ of $F$ such that $F(x) = \overline{co\,}\overline f(x)$ for every $x \in X$; (3)~$F$ has nonempty compact values, there is a densely defined subcontinuous, quasicontinuous selection $f$ of $F$ such that $F(x) = \overline{co}\,\overline f(x)$ for every $x \in X$; (4)~$F$ has nonempty compact convex values, $F$ has a closed graph, every extreme function of $F$ is quasicontinuous, subcontinuous and any two extreme functions of $F$ have the same closures of their graphs. Some applications to known results are given.

Consider the elliptic curves given by \[ \ent : y^2=x^3+2s n x^2-(r^2-s^2) n^2 x \] where $0 \lt \ta\lt \pi$, $\cos(\ta)=s/r$ is rational with $0\leq |s| \lt r$ and $\gcd (r,s)=1$. These elliptic curves are related to the $\ta$-congruent number problem as a generalization of the congruent number problem. For fixed $\ta$, this family corresponds to the quadratic twist by $n$ of the curve $\ttt: y^2=x^3+2s x^2-(r^2-s^2) x$. We study two special cases: $\ta=\pi/3$ and $\ta=2\pi/3$. We have found a subfamily of $n=n(w)$ having rank at least $3$ over $\Q(w)$ and a subfamily with rank~$4$ parametrized by points of an elliptic curve with positive rank. We also found examples of $n$ such that $E_{n, \ta}$ has rank up to $7$ over $\Q$ in both cases.

Each Cantor measure $\mu$ with scaling factor $\frac{1}{2n}$ has at least one associated orthonormal basis of exponential functions (ONB) for $L^2(\mu)$. In the particular case where the scaling constant for the Cantor measure is $\frac14$ and two specific ONBs are selected for $L^2(\mu_{1/4})$, there is a unitary operator $U$ defined by mapping one ONB to the other. This paper focuses on the case in which one ONB $\Gamma$ is the original Jorgensen-Pedersen ONB for the Cantor measure $\mu_{{1}/{4}}$ and the other ONB is $5\Gamma$. The main theorem of the paper states that the corresponding operator $U$ is ergodic in the sense that only the constant functions are fixed by $U$.

A group $G$ with subgroups $S$ and $T$ satisfying $G = ST$ and $S\cap T = \{e\}$ gives rise to functions $[t,s] \in S$ and $\\lt t,s\> \in T$ such that $(st)(s't') = (s[t,s'])(\\lt t,s'\>t')$. This notion may be extended to arbitrary semigroups $S$, $T$ with identities, producing the Zappa product of $S$ and $T$, a generalization of direct and semidirect product. Necessary and sufficient conditions are given for a semigroup compactification of a Zappa product $G$ of topological semigroups $S$ and $T$ to be canonically isomorphic to a Zappa product of compactifications of $S$ and $T$. The result is applied to various types of compactifications of $G$, including the weakly almost periodic and almost periodic compactifications.

Let $M$ be a real hypersurface in a complex space form $\mn$, $c \neq 0$. In this paper we prove that if $\rx\li = \li\rx$ holds on $M$, then $M$ is a Hopf hypersurface, where $\rx$ and $\li$ denote the structure Jacobi operator and the induced operator from the Lie derivative with respect to the structure vector field $\xi$, respectively. We characterize such Hopf hypersurfaces of $\mn$.

The objective of this paper is to obtain an upper bound to the second Hankel determinant $|a_{p+1}a_{p+3}-a_{p+2}^{2}|$ for certain $p$-valent analytic functions, using Toeplitz determinants.

Let $\mathcal{A}$ be a unital separable approximately divisible C$^*$-algebra. We show that $\mathcal{A}$ is generated by two self-adjoint elements and the topological free entropy dimension of any finite generating set of $\mathcal{A}$ is less than or equal to 1.

In this paper, by the extension of beta functions containing three extra parameters, we generalize Appell's and Lauricella's hypergeometric functions. Some integral representations, transformation formulae, differentiation formulae and recurrence relations are obtained for these new generalizations.

We compute a lower bound of the canonical height on quadratic twists of elliptic curves over $\Q$. Also, we show a simple method for constructing families of quadratic twists with an explicit rational point. Using the above lower bound, we show that the explicit rational point is primitive as an element of the Mordell-Weil group.

For $n\geq 3$, we shall show that the group $\mbox{Aut\,}(NS(M))$ of simplicial automorphisms of the complex $NS(M)$ of non-separating embedded spheres in the manifold $M=\sharp_{n} S^{2}\times S^{1}$ is isomorphic to the group $\mbox{Out\,}(\mathbf F_n)$ of outer automorphisms of the free group $\mathbf F_n$, where $\mathbf F_n$ is identified with the fundamental group of the manifold $M$ up to conjugacy of the base point in $M$.

In this paper we consider the multiparameter system introduced in \cite{sch}, which corresponds to an extension of the classic minimal Daphnia-algae model. It is shown that there is a neighborhood in the parameter space where the system in the realistic quadrant has a unique equilibrium point which is a repelling weak focus of order four enclosed by a global attractor hyperbolic limit cycle. For a small enough change of the parameters in this neighborhood, bifurcation occurs from the weak focus four infinitesimal Hopf limit cycles (alternating the type of stability) such that the last bifurcated limit cycle is an attractor. Moreover, for certain values of parameters, we concluded that this applied model has five concentric limit cycles, three of them being stable hyperbolic limit cycles. This gives a positive answer to a question raised in \cite{col, Llo}.

Suppose $m(\alpha)$ denotes the Mahler measure of the non-zero algebraic number $\alpha$. For each positive real number $t$, the author studied a version $m_t(\alpha)$ of the Mahler measure that has the triangle inequality. The construction of $m_t$ is generic and may be applied to a broader class of functions defined on any Abelian group $G$. We prove analogs of known results with an abstract function on $G$ in place of the Mahler measure. In the process, we resolve an earlier open problem stated by the author regarding $m_t(\alpha)$.

We prove that Kalton twisting of Schatten classes is strictly singular as $B(H)$-modules. We also identify the dual construction.