In this paper, we find the genus field of finite abelian extensions of the global rational function field. We introduce the term conductor of constants for these extensions and determine it in terms of other invariants. We study the particular case of finite abelian $p$-extensions and give an explicit description of their genus field.

Let $B$ be a smooth projective surface, and let $\mathcal{L} $ be an ample line bundle on $B$. The aim of this paper is to study the families of elliptic Calabi-Yau threefolds sitting in the bundle $\mathbb{P} (\mathcal{L}^a \oplus \mathcal{L}^b \oplus \mathcal{O}_B)$ as anticanonical divisors. We will show that the number of such families is finite.

Let $E$ be an elliptic curve over $\mathbb {Q}$ described by $y^2= x^3+ Kx+ L$, where $K, L \in \mathbb {Q}$. A set of rational points $(x_i,y_i) \in E(\mathbb {Q})$ for $i=1, 2, \ldots , k$, is said to be a sequence of consecutive cubes on $E$ if the $x$-coordinates of the points $x_i$'s for $i=1, 2,\ldots $, form consecutive cubes. In this note, we show the existence of an infinite family of elliptic curves containing a length-$5$-term sequence of consecutive cubes. Moreover, these five rational points in $E (\mathbb {Q})$ are linearly independent, and the rank $r$ of $E(\mathbb {Q})$ is at least $5$.

Let $D$ be an integral domain, $X$ an indeterminate over $D$, $d \in D$, and $R = D[X, {d}/{X}]$ a subring of $D[X,{1}/{X}]$. In this paper, we show that $R$ is a weakly factorial domain if and only if $D$ is a weakly factorial GCD-domain and $d=0$, $d$ is a unit of $D$ or $d$ is a prime element of $D$. We also show that, if $D$ is a weakly factorial GCD-domain, $p$ is a prime element of $D$, and $n \geq 2$ is an integer, then $D[X, {p^n}/{X}]$ is an almost weakly factorial domain with $Cl(D[X, {p^n}/{X}]) = \mathbb {Z}_n$.

This paper is concerned with the following systems of Kirchhoff-type equations: \begin{equation} \begin{cases} -\left (a+b\int _{\mathbb {R}^{N}}|\nabla u|^{2}\mathrm {d}x\right )\Delta u\\ \quad +V(x)u=F_{u}(x, u, v)\quad & x\in \mathbb {R}^{N},\\ -\left (c+d\int _{\mathbb {R}^{N}}|\nabla v|^{2}\mathrm {d}x\right )\Delta v\\ \quad +V(x)v=F_{v}(x, u, v) & x\in \mathbb {R}^{N},\\ u(x)\rightarrow 0,\quad v(x)\rightarrow 0 &\mbox {as } |x|\rightarrow \infty . \end{cases} \end{equation} Under some more relaxed assumptions on $V(x)$ and $F(x, u, v)$, we prove the existence of infinitely many negative-energy solutions for the above system via the genus properties in critical point theory. Some recent results from the literature are greatly improved and extended.

In this note, we investigate Liu's work on the truncated Gaussian square exponent theorem and obtain more truncations. We also discuss some possible multiple summation extensions of Liu's results.

We introduce and study a combinatorially defined notion of the root basis of a (real) root system of a possibly infinite Coxeter group. Known results on conjugacy up to sign of root bases of certain irreducible finite rank real root systems are extended to abstract root bases, to a larger class of real root systems and, with a short list of (genuine) exceptions, to infinite rank irreducible Coxeter systems.

We give some new refinements and reverses of Young inequalities in both additive and multiplicative-type for two positive numbers/operators. We show our advantages by comparing with known results. A few applications are also given. Some results relevant to the Heron mean are also considered.

Let $G$ be a group. The intersection graph $\Gamma (G)$ of $G$ is an undirected graph without loops and multiple edges, defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$, where $1$ denotes the trivial subgroup of $G$. In this paper, we study the dominating sets in intersection graphs of finite groups. We classify abelian groups by their domination number and find upper bounds for some specific classes of groups. Subgroup intersection is related to Burnside rings. We introduce the notion of an intersection graph of a $G$-set (somewhat generalizing the ordinary definition of an intersection graph of a group) and establish a general upper bound for the domination number of $\Gamma (G)$ in terms of subgroups satisfying a certain property in the Burnside ring. The intersection graph of $G$ is the $1$-skeleton of the simplicial complex. We name this simplicial complex intersection complex of $G$ and show that it shares the same homotopy type with the order complex of proper non-trivial subgroups of $G$. We also prove that, if the domination number of $\Gamma (G)$ is 1, then the intersection complex of $G$ is contractible.

In this study, we deal with stability for the linearized neural field model for two neuron populations. We determine the asymptotic stability region by using the D-subdivision method for different delay terms including time delays. Also, we find the number of unstable characteristic exponents for the unstable regions. We observe that the stability region for the model becomes smaller as the delay term $\tau $ increases.

Let $G$ be a finite abelian group which acts symplectically on a K3 surface. The Néron-Severi lattice of the projective K3 surfaces admitting $G$ symplectic action and with minimal Picard number was computed by Garbagnati and Sarti. We consider a four-dimensional family of projective K3 surfaces with $\mathbb {Z}_2^2$ symplectic action which do not fall into the above cases. If $X$ is one of these K3 surfaces, then it arises as the minimal resolution of a specific $\mathbb {Z}_2^3$-cover of $\mathbb {P}^2$ branched along six general lines. We show that the Néron-Severi lattice of $X$ with minimal Picard number is generated by $24$ smooth rational curves and that $X$ specializes to the Kummer surface $\textrm{Km}(E_i\times E_i)$. We relate $X$ to the K3 surfaces given by the minimal resolution of the $\mathbb {Z}_2$-cover of $\mathbb {P}^2$, branched along six general lines, and the corresponding Hirzebruch-Kummer covering of exponent $2$ of $\mathbb {P}^2$.

Let $\{f_j\}_{j=0}^n$ be a sequence of orthonormal polynomials where the orthogonality relation is satisfied on either the real line or on the unit circle. We study zero distribution of random linear combinations of the form $$P_n(z)=\sum _{j=0}^n\eta _jf_j(z),$$ where $\eta _0,\ldots ,\eta _n$ are complex-valued iid standard Gaussian random variables. Using the Christoffel-Darboux formula, the density function for the expected number of zeros of $P_n$ in these cases takes a very simple shape. From these expressions, under the mere assumption that the orthogonal polynomials are from the Nevai class, we give the limiting value of the density function away from their respective sets where the orthogonality holds. In the case when $\{f_j\}$ are orthogonal polynomials on the unit circle, the density function shows that the expected number of zeros of $P_n$ are clustering near the unit circle. To quantify this phenomenon, we give a result that estimates the expected number of complex zeros of $P_n$ in shrinking neighborhoods of compact subsets of the unit circle.

In this article, we present a general method that can be used to deduce weighted Hardy-type inequalities from a particular non-linear partial differential inequality in a relatively simple and unified way on the sub-Riemannian manifold $\mathbb {R}^{2n+1}=\mathbb {R}^{n}\times \mathbb {R}^{n}\times \mathbb {R}$, defined by the Greiner vector fields \[X_{j}=\frac {\partial }{\partial x_{j}}+2ky_{j} \vert z\vert ^{2k-2}\frac {\partial }{\partial l},\] \[Y_{j}=\frac {\partial }{\partial y_{j}}-2kx_{j}|z|^{2k-2}\frac {\partial }{\partial l},\] $j=1,\ldots ,n$, where $z=x+iy\in \mathbb {C}^{n},$ $l\in \mathbb {R}$, $k\geq 1$. Our method allows us to improve, extend, and unify many previously obtained sharp weighted Hardy-type inequalities as well as to yield new ones. These cases are illustrated by giving many concrete examples, including radial, logarithmic, hyperbolic and non-radial weights. Furthermore, we introduce a new technique for constructing two-weight $L^{p}$ Hardy-type inequalities with remainder terms on smooth bounded domains $\Omega $ in $\mathbb {R}^{2n+1}$. We also give several applications leading to various weighted Hardy inequalities with remainder terms.

This paper presents a local Morrey regularity with the optimal exponents for linear parabolic equations in divergence form under the assumption that the leading coefficient is independent of $t$ and not necessarily symmetric based on a rather different approach. Here, we achieve it by applying natural growth properties of Green's functions through the use of parabolic operators and the hole-filling technique.