Let $X$ be a finite set of positive integers. The divisibility graph $\mathscr {D}\,(X)$ is a directed graph with vertex set $X\backslash \{1\}$ and an arc from $a$ to $b$ whenever $a$ divides $b$. Since the divisibility graph and its underlying graph have the same number of connected components, we consider the underlying graph of $\mathscr {D}\,(X)$, and we denote it by $\rm D (X)$. In this paper, we will find some graph theoretical parameters of $\rm D (X)$, some relations between the structure of $\rm D (X)$, and the structure of known graphs $\Gamma (X)$, $\Delta (X)$ and $B(X)$ will be considered. In addition, we investigate some properties of $\rm D (XY)$ for the product of two non-empty subsets $X$ and $Y$ of positive integers.

We give a characterization of the existence of a nonuniform exponential dichotomy for a linear impulsive differential equation. The characterization uses quadratic functions and the symmetric matrices defining them. As an application, we give a simple proof of the robustness property of a nonuniform exponential dichotomy under sufficiently small linear perturbations.

We develop an abstract framework for the investigation of quantization and dequantization procedures based on orthogonality relations that do not necessarily involve group representations. To illustrate the usefulness of our abstract method, we show that it behaves well with respect to infinite tensor products. This construction subsumes examples from the study of magnetic Weyl calculus, magnetic pseudo-differential Weyl calculus, metaplectic representation on locally compact abelian groups, irreducible representations associated with finite-dimensional coadjoint orbits of some special infinite-dimensional Lie groups, and square-integrability properties shared by arbitrary irreducible representations of nilpotent Lie groups.

Let $E/\mathbb{Q} $ be an elliptic curve. Silverman and Stange define primes $p$ and $q$ to be an elliptic, amicable pair if $\#E(\mathbb{F} _p) = q$ and $\#E(\mathbb{F} _q) = p$. More generally, they define the notion of aliquot cycles for elliptic curves. Here, we study the same notion in the case that the elliptic curve is defined over a number field~$K$. We focus on proving the existence of an elliptic curve~$E/K$ with aliquot cycle $(\mathfrak{p} _1, \ldots , \mathfrak{p} _{n})$ where the $\mathfrak{p} _{i}$ are primes of~$K$ satisfying mild conditions.

We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. Speci\-fically, we consider a Yang-Baxter state model for the {\rm sl($n$)} polynomial of classical links and extend it to oriented singular links and balanced oriented 4-valent knotted graphs with rigid vertices. We also define a representation of the singular braid monoid into a matrix algebra and seek conditions for further extending the invariant to contain topological knotted graphs. In addition, we show that the resulting Yang-Baxter-type invariant for singular links yields a version of the Murakami-Ohtsuki-Yamada state model for the {\rm sl($n$)} polynomial for classical links.

Let $E$ be an elliptic curve defined over $\mathbb{Q} $. We study the relationship between the torsion subgroup $E(mathbb{Q} )_{tors}$ and the torsion subgroup $E(K)_{tors}$, where $K$ is a cubic number field. In particular, we study the number of cubic number fields $K$ such that $E(\mathbb{Q} )_{tors}\neq E(K)_{tors}$.

In this paper, the authors establish an explicit formula for computing Bernoulli polynomials at nonnegative integer points in terms of $r$-Stirling numbers of the second kind.

In this paper, we will prove the existence and H\"{o}lder continuity of smooth center-unstable and center-stable manifolds for random dynamical systems based on their Lyapunov exponents. Furthermore, we obtain the existence and H\"{o}lder continuity of smooth center manifolds.

Substitution Markov chains have been introduced \cite {KoslickiThesis2012} as a new model to describe molecular evolution. In this note, we study the associated Martin boundaries from a probabilistic and topological viewpoint. An example is given that, although having a boundary homeomorphic to the well-known coin tossing process, has a metric description that differs significantly.

A functional equation between the $\zeta $ distributions can be obtained from the theory of prehomogeneous vector spaces. We show that the functional equation can be extended from the Schwartz space to certain degenerate principal series.

In this paper, we extend the notions of strongly copure projective, injective and flat modules to that of complexes and characterize these complexes. We show that the strongly copure projective precover of any finitely presented complex exists over $n$-FC rings, and a strongly copure injective envelope exists over left Noetherian rings. We prove that strongly copure flat covers exist over arbitrary rings and that $(\mathcal {SCF},\mathcal {SCF}^\bot )$ is a perfect hereditary cotorsion theory where $\mathcal {SCF}$ is the class of strongly copure flat complexes.

Let $\mathcal{A} $ be a finite subset of $L^2(\mathbb{R} )$ and $p,q\in \mathbb{N} $. We characterize the Schauder basis properties in $L^2(\mathbb{R} )$ of the Gabor system \[ G(1,p/q,\mathcal{A} )=\{e^{2\pi i m x}g(x-np/q) : m,n\in \mathbb{Z} , g\in \mathcal{A} \}, \] with a specific ordering on $\mathbb{Z} \times \mathbb{Z} \times \mathcal{A} $. The characterization is given in terms of a Muckenhoupt matrix $A_2$ condition on an associated Zibulski-Zeevi type matrix.

Let $\mathscr {E}$ be a rational elliptic surface over a number field~$k$. We study the interplay between a geometric property, the configuration of its singular fibers, and arithmetic features such as its Mordell-Weil rank over the base field and its possible minimal models over~$k$.

Let $R$ be an integral domain with quotient field $K$, and let $X$ be an indeterminate over $R$. In this paper, we consider content formulae for power series in terms of $*$-operations for PVMDs, Krull domains and Dedekind domains, where $*$ is the star-operation, $d$, $w$, $t$, or $v$. We prove that $R$ is a Krull domain if and only if $c(f/g)_w=(c(f)c(g)^{-1})_w$ for all $f,g\in R[[X]]^*$ with $c(f/g)$ a fractional ideal if and only if $c(f/g)_t=(c(f)c(g)^{-1})_t$ for all $f,g\in R[[X]]^*$ with $c(f/g)$ a fractional ideal, and $R$ is a Dedekind domain if and only if for all $f,g\in R[[X]]^*$ with $c(f/g)$ a fractional ideal, $c(f/g)=c(f)c(g)^{-1}$.