In this paper, using module endomorphisms, we extend the concept of the zero-divisor graph of a ring to a module over an arbitrary commutative ring. The main aim of this article is studying the interplay of module-theoretic properties of a module with graph properties of its zero-divisor graph.

In this article, we deal with complete linear Weingarten spacelike hypersurfaces (that is, complete spacelike hypersurfaces whose mean and scalar curvatures are linearly related) immersed in a Lorentz space form. By assuming that the mean curvature attains its maximum and supposing appropriated restrictions on the norm of the traceless part of the second fundamental form, we apply Hopf's strong maximum principle in order to prove that such a spacelike hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of the ambient space.

In the topological classification of phase portraits no distinctions are made between a focus and a node and neither are they made between a strong and a weak focus or between foci of different orders. These distinctions are however important in the production of limit cycles close to the foci in perturbations of the systems. The distinction between the one direction node and the two directions node, which plays a role in understanding the behavior of solution curves around the singularities at infinity, is also missing in the topological classification.

In this work we introduce the notion of \textit{geometric equivalence relation of singularities} which incorporates these important purely algebraic features. The \textit{geometric} equivalence relation is finer than the \textit{topological} one and also finer than the \textit{qualitative equivalence relation} introduced in \cite{J_L}. We also list all possibilities we have for finite and infinite singularities, taking into consideration these finer distinctions, and introduce notation for each one of them. %Our %long term goal is to use this finer and deeper equivalence relation to classify %the quadratic family according to their different \textit{geometric %configurations of singularities}, finite and infinite.

In this work we give the classification theorem and bifurcation diagram in the 12-dimensional parameter space, using the \textit{geometric equivalence relation}, of the class of quadratic systems according to the configuration of singularities at infinity of the systems. Our classification theorem, stated in terms of invariant polynomials, is an algorithm for computing the \textit{geometric configurations} of infinite singularities for any family of quadratic systems, in any normal form. %The theorem we give also %contains a bifurcation diagram, done in the 12-parameter space, of %the \textit{geometric configurations} of singularities at infinity, %and this bifurcation set is algebraic in the parameter space. To %determine the bifurcation diagram of configurations of singularities %at infinity for any family of quadratic systems, given in any normal %form, becomes thus a simple task using computer algebra %calculations.

In previous works, Jones and Roberts and Pauli and Roblot have studied finite extensions of the $p$-adic numbers $\mathbb{Q}_p$. This paper focuses on results for local fields of characteristic~$p$. In particular, we are able to produce analogous results to Jones and Roberts in the case that the characteristic does not divide the degree of the field extension. Also, in this case, following from the work of Pauli and Roblot, we prove that the defining polynomials of these extensions can be written in a simple form amenable to computation. Finally, if $p$ is the degree of the extension, we show there are infinitely many extensions of this degree and thus these cannot be classified in the same manner.

In this paper, we study the asymptotic behavior of the solutions of a neutral type difference equation of the form% \[ \Delta \bigg[ x(n)+\sum_{j=1}^{w}c_{j}x(\tau _{j}(n))\bigg] +p(n)x(\sigma (n))=0,\quad n\geq 0 \] where $\tau _{j}(n)$, $j=1,\ldots,w$, are general retarded arguments, $\sigma (n) $ is a general deviated argument (retarded or advanced), $c_{j}\in \mathbb{R}$, $j=1,\ldots,w$, $(p(n))_{n\geq 0}$ is a sequence of positive real numbers such that $p(n)\geq p$, $p\in\mathbb{R}_{+}$, and $\Delta $ denotes the forward difference operator $\Delta x(n)=x(n+1)-x(n)$.

We also examine the convergence of the solutions when these are continuous and differentiable with respect to $c_{j}$, $j=1,\ldots,w$.

Let $\varphi(z)(\not\equiv0)$ be a function holomorphic in a domain $D$, $k\in\mathbb{N}$, and let $\mathcal{F}$ be a family of meromorphic functions defined in $D$, all of whose zeros have multiplicity at least $k+2$ such that, for every $f\in\mathcal{F}$, $f^{(k)}(z)\neq\varphi(z)$. The non-normal sequences in $\mathcal{F}$ are characterized.

This paper is meant to be a gentle introduction to Carleson's theorem on pointwise convergence of Fourier series.

Given a metric continuum $X$, we consider the following hyperspaces of $X$: $2^{X}$, $C_{n}(X)$ and $F_{n}(X)$ ($n\in \mathbb{N}$). Let $F_{1}(X)=\{\{x\}:x\in X\}$. A hyperspace $K(X)$ of $X$ is said to be rigid, provided that for every homeomorphism $h:K(X)\rightarrow K(X)$, we have $h(F_{1}(X))=F_{1}(X)$. In this paper, we study conditions under which a continuum $X$ has a rigid hyperspace $C_{n}(X)$. Among others, we consider families of continua, such as dendroids, Peano continua, hereditarily indecomposable continua and smooth fans.

A search of the literature reveals only a few studies on the necessary as well as sufficient conditions for the existence of eventually positive and/or monotone solutions of higher order impulsive differential equations that also allow delays. To fill this gap, we study a general class of higher order impulsive delay differential equations and establish necessary and/or sufficient conditions for the existence of eventually positive and monotone solutions. Our results are sharp in the sense that, in special cases, they are necessary and sufficient. Illustrative examples are included.

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The singular curve $C\subset \mathbb{P}^2$ defined over $\mathbb{F}_p$ for a prime $p$ by the equation $x^2t^2+y^2t^2+x^2y^2-t^4=0$ is known as Gauss's curve. For $p\equiv 3$ mod $4$, we give a proof that the zeta function of $C$ is \[ Z_{C}(u)=\frac{(1+pu^2)(1+u)^2}{(1-pu)(1-u)}. \] We define the (Hasse-Weil) global zeta function for any geometric genus~1 singular curve and, in particular, find that the global zeta function of $C$ is \[ \zeta_C(s)=\frac{\zeta(s)\zeta(s-1)}{L_E(s)L(s,\chi^{\prime})^{2}}, \] where $E$ is a projective nonsingular model for $C$, $L_E(s)$ is its $L$-function, and $L(s, \chi^{\prime})$ is a Dirichlet $L$-series for a character $\chi^{\prime}$ that we specify. We then consider more generally the ratio $R_{X}(s)$ of the Hasse-Weil global zeta function of a singular curve $X$ and that of its normalization $\widetilde{X}$. We finish with questions about the analytic properties of $R_{X}(s)$.

We introduce three concepts, called $I$-vector, $IP$-vector, and $P$-vector, which are related to isosceles orthogonality and Pythagorean orthogonality in normed linear spaces. Having the Banach-Mazur rotation problem in mind, we prove that an almost transitive real Banach space, whose dimension is at least three and which contains an $I$-vector (an $IP$-vector, a $P$-vector, or a unit vector whose pointwise James constant is $\sqrt2$, respectively) is a Hilbert space.

We consider the exponential Diophantine equation $a^{x}+b^{y}=c^{z}$ in positive integers $x$, $y$ and $z$, where $a$, $b$ and $c$ are fixed pair-wise relatively prime positive integers greater than one. In this paper, we obtain several upper bounds for solutions $x$, $y$ and $z$ for which two of $x$, $y$ and $z$ are even. As their applications, we solve exponential Diophantine equations in which $a$, $b$ and $c$ are expressed as terms of linearly recurrence sequences.

Let $\mathcal{L}$ be a $\mathcal{J}$-subspace lattice on a complex Banach space $X$ and $\mbox{\rm Alg\,}{\mathcal L}$ the associated $\mathcal{J}$-subspace lattice algebra. We say that an operator $Z\in {\rm Alg\,}{\mathcal L}$ is a full-derivable point of $\mbox{\rm Alg\,}{\mathcal L}$ if every linear map $\delta$ from $\mbox{\rm Alg\,}{\mathcal L}$ into itself derivable at $Z$ (i.e., $\delta(A)B+A\delta(B)=\delta(Z)$ for any $A,B \in {\rm Alg\,}{\mathcal L}$ with $AB=Z$) is a derivation and is a full-generalized-derivable point of $\mbox{\rm Alg\,}{\mathcal L}$ if every linear map $\delta$ from $\mbox{\rm Alg\,}{\mathcal L}$ into itself generalized derivable at $Z$ (i.e., $\delta(A)B+A\delta(B)-A\delta(I)B=\delta(Z)$ for any $A,B \in {\rm Alg\,}{\mathcal L}$ with $AB=Z$) is a generalized derivation. In this paper, we prove that if $Z\in\mbox{\rm Alg\,}{\mathcal L}$ is an injective operator or an operator with dense range, then $Z$ is a full-derivable point as well as a full-generalized-derivable point of ${\rm Alg\,}{\mathcal L}$.

In \cite{Cab}, Cabras, Ianus and Pitis proved that in a cosymplectic manifold there does not exist any extrinsic sphere tangent to the structure vector field $\xi$. We consider the structure vector field $\xi$ normal to the submanifold in the sense of Papaghiuc \cite{Pap} and derive that a totally umbilical CR-submanifold of a cosymplectic manifold is either (i)~totally geodesic, (ii)~anti-invariant or (iii)~an extrinsic sphere.