Consider a quartic $q$-Weil polynomial $f$. Motivated by equidistribution considerations, we define, for each prime $\ell$, a local factor that measures the relative frequency with which $f$$ \bmod \,\ell $ occurs as the characteristic polynomial of a symplectic similitude over ${{\mathbb{F}}_{\ell }}$. For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over ${{\mathbb{F}}_{q}}$ with Weil polynomial $f$.

Let $\Gamma$ be a connection on a smooth manifold $M$. In this paper we give some properties of $\Gamma$ by studying the corresponding Lie algebras. In particular, we compute the first Chevalley–Eilenberg cohomology space of the horizontal vector fields Lie algebra on the tangent bundle of $M$, whose the corresponding Lie derivative of $\Gamma$ is null, and of the horizontal nullity curvature space.

The Chowla conjecture states that if $t$ is any given positive integer, there are infinitely many prime positive integers $N$ such that $\text{Per}\left( \sqrt{N} \right)\,=\,t$, where $\text{Per}\left( \sqrt{N} \right)$ is the period length of the continued fraction expansion for $\sqrt{N}$. C. Friesen proved that, for any $k\,\in \,\mathbb{N}$, there are infinitely many square-free integers $N$, where the continued fraction expansion of $\sqrt{N}$ has a fixed period. In this paper, we describe all polynomials $Q\,\in \,{{\mathbb{F}}_{q}}\left[ X \right]$ for which the continued fraction expansion of $\sqrt{Q}$ has a fixed period. We also give a lower bound of the number of monic, non-squares polynomials $Q$ such that $\deg \,Q=\,2d$ and $Per\sqrt{Q}\,=\,t$.

Motivated by Almgren’s work on the isoperimetric inequality, we prove a sharp inequality relating the length and maximum curvature of a closed curve in a complete, simply connected manifold of sectional curvature at most −1. Moreover, if equality holds, then the norm of the geodesic curvature is constant and the torsion vanishes. The proof involves an application of the maximum principle to a function defined on pairs of points.

We show that a class of semilinear Laplace–Beltrami equations on the unit sphere in ${{\mathbb{R}}^{n}}$ has infinitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as ${{\left| s \right|}^{p-1}}s$ for $\left| s \right|$ large with $1\,<\,p\,<\,\left( n+5 \right)/\left( n-3 \right)$.

Let $p$ be a prime number and $F$ a field containing a root of unity of order $p$. We relate recent results on vanishing of triple Massey products in the $\bmod-p $ Galois cohomology of $F$, due to Hopkins, Wickelgren, Mináč, and Tân, to classical results in the theory of central simple algebras. We prove a stronger form of the vanishing property for global fields.

Let $\mathcal{E}$ be an injectively resolving subcategory of left $R$-modules. A left $R$-module $M$ (resp. right $R$-module $N$) is called $\mathcal{E}$-injective (resp. $\mathcal{E}$-flat) if $\text{Ext}_{R}^{1}\left( G,\,M \right)\,=\,0$ (resp. $\text{Tor}_{1}^{R}\left( N,\,G \right)\,=\,0$) for any $G\,\in \,\mathcal{E}$. Let $\mathcal{E}$ be a covering subcategory. We prove that a left $R$-module $M$ is $\mathcal{E}$-injective if and only if $M$ is a direct sum of an injective left $R$-module and a reduced $\mathcal{E}$-injective left $R$-module. Suppose $\mathcal{F}$ is a preenveloping subcategory of right $R$-modules such that ${{\mathcal{E}}^{+}}\,\subseteq \,\mathcal{F}$ and ${{\mathcal{F}}^{+}}\,\subseteq \,\mathcal{E}$. It is shown that a finitely presented right $R$-module $M$ is $\mathcal{E}$-flat if and only if $M$ is a cokernel of an $\mathcal{F}$-preenvelope of a right $R$-module. In addition, we introduce and investigate the $\mathcal{E}$-injective and $\mathcal{E}$-flat dimensions of modules and rings. We also introduce $\mathcal{E}$-(semi)hereditary rings and $\mathcal{E}$-von Neumann regular rings and characterize them in terms of $\mathcal{E}$-injective and $\mathcal{E}$-flat modules.

In this article we prove an embedding theorem for inhomogeneous Besov and Triebel–Lizorkin spaces on $\text{RD}$-spaces. The crucial idea is to use the geometric density condition on the measure.

We prove character sum estimates for additive Bohr subsets modulo a prime. These estimates are analogous to the classical character sum bounds of Pólya–Vinogradov and Burgess. These estimates are applied to obtain results on recurrence $\bmod \,p$ by special elements.

In this paper, we generalize the finite generation result of Sormani to non-branching $RCD\left( 0,\,N \right)$ geodesic spaces (and in particular, Alexandrov spaces) with full supportmeasures. This is a special case of the Milnor’s Conjecture for complete non-compact $RCD\left( 0,\,N \right)$ spaces. One of the key tools we use is the Abresch–Gromoll type excess estimates for non-smooth spaces obtained by Gigli–Mosconi.

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is said to be $s$-semipermutable in $G$ if $H{{G}_{p}}\,=\,{{G}_{p}}H$ for any Sylow $p$-subgroup ${{G}_{p}}$ of $G$ with $\left( p,\,\left| H \right| \right)\,=\,1$; $H$ is said to be $s$-quasinormally embedded in $G$ if for each prime $p$ dividing the order of $H$, a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-quasinormal subgroup of $G$. In every non-cyclic Sylow subgroup $P$ of $G$ we fix some subgroup $D$ satisfying $1\,<\,\left| D \right|\,<\,\left| P \right|$ and study the structure of $G$ under the assumption that every subgroup $H$ of $P$ with $\left| H \right|\,=\,\left| D \right|$ is either $s$-semipermutable or $s$-quasinormally embedded in $G$. Some recent results are generalized and unified.

This paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained.

We consider the class of polynomial differential systems of the form $\dot{x} =\,\lambda x\,-\,y\,+\,{{P}_{n}}\left( x,\,y \right)$ , $\dot{y} =\,x\,+\,\lambda y\,+\,{{Q}_{n}}\left( x,\,y \right)$ where ${{P}_{n}}$ and ${{Q}_{n}}$ are homogeneous polynomials of degree $n$ . For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence.

Let $Q$ be a finite acyclic quiver, let $J$ be an ideal of $kQ$ generated by all arrows in $Q$, and let $A$ be a finite-dimensional $k$ -algebra. The category of all finite-dimensional representations of $\left( Q,\,{{J}^{2}} \right)$ over $A$ is denoted by $\text{rep}\left( Q,\,{{J}^{2}},\,A \right)$ . In this paper, we introduce the category $\text{exa}\left( Q,{{J}^{2}},A \right),$ which is a subcategory of $\text{rep}\left( Q,\,{{J}^{2}},\,A \right)$ of all exact representations. The main result of this paper explicitly describes the Gorenstein-projective representations in $\text{rep}\left( Q,\,{{J}^{2}},\,A \right)$, via the exact representations plus an extra condition. As a corollary, $A$ is a self-injective algebra if and only if the Gorenstein-projective representations are exactly the exact representations of $\left( Q,\,{{J}^{2}} \right)$ over $A$.

We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians with harmonic curvature with respect to the generalized Tanaka–Webster connection if they satisfy some further conditions.

Let $A\,\in \,{{M}_{n}}\left( \mathbb{R} \right)$ be an invertible matrix. Consider the semi-direct product ${{\mathbb{R}}^{n}}\rtimes \,\mathbb{Z}$ where the action of $\mathbb{Z}$ on ${{\mathbb{R}}^{n}}$ is induced by the left multiplication by $A$. Let $\left( \alpha ,\,\tau \right)$ be a strongly continuous action of ${{\mathbb{R}}^{n}}\rtimes \,\mathbb{Z}$ on a ${{C}^{*}}$-algebra $B$ where $\alpha$ is a strongly continuous action of ${{\mathbb{R}}^{n}}$ and $\tau$ is an automorphism. The map $\tau$ induces a map $\widetilde{\tau }\,\text{on}\,\text{B}\,{{\rtimes }_{\alpha }}\,{{\mathbb{R}}^{n}}$. We show that, at the $K$-theory level, $\tau$ commutes with the Connes–Thom map if $\det \left( A \right)\,>\,0$ and anticommutes if $\det \left( A \right)\,>\,0$. As an application, we recompute the $K$-groups of the Cuntz–Li algebra associated with an integer dilation matrix.

Let $\eta \left( z \right)\,\left( z\,\in \,\mathbb{C},\,\operatorname{Im}\left( z \right)\,>\,0 \right)$ denote the Dedekind eta function. We use a recent product-to-sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotients

$$f\left( z \right)\,:=\,{{\eta }^{a\left( {{m}_{1}} \right)}}\,\left( {{m}_{1}}z \right)\,.\,.\,.\,{{\eta }^{a\left( {{m}_{r}} \right)}}\,\left( {{m}_{r}}z \right)\,=\,\sum\limits_{n=1}^{\infty }{c\left( n \right){{e}^{2\pi inz}},\,\,\,z\,\in \,\mathbb{C},\,\operatorname{Im}\left( z \right)\,>\,0,}$$

such that the Fourier coefficients $c\left( n \right)$ vanish for all positive integers $n$ in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if $f\left( z \right)\,=\,{{\eta }^{4}}\left( z \right){{\eta }^{9}}\left( 4z \right){{\eta }^{-2}}\left( 8z \right)$ we have $c\left( n \right)\,=\,0$ for all $n$ in each of the arithmetic progressions ${{\{16k\,+\,14\}}_{k\ge 0}},\,{{\{64k\,+\,56\}}_{k\ge 0}},\,{{\{256k,\,224\}}_{k\ge 0}},\,{{\{1024k\,+\,869\}}_{k\ge 0}},\,.\,.\,.$

Under sufficiently strong assumptions about the first term in an arithmetic progression, we prove that for any integer $a$, there are infinitely many $n\,\in \,\mathbb{N}$ such that for each prime factor $p\,\text{ }\!\!|\!\!\text{ }\,n$, we have $p\,-\,a\,\text{ }\!\!|\!\!\text{ }\,n\,-\,a$. This can be seen as a generalization of Carmichael numbers, which are integers $n$ such that $p\,-\,1\,\text{ }\!\!|\!\!\text{ }\,n\,-\,1$ for every $p\,\text{ }\!\!|\!\!\text{ }\,n$.

We show that if $v$ is a regular semi-classical form (linear functional), then the symmetric form $u$ defined by the relation ${{x}^{2}}\sigma u\,=\,-\lambda v$, where $\left( \sigma f \right)\left( x \right)\,=\,f\left( {{x}^{2}} \right)$ and the odd moments of $u$ are 0, is also regular and semi-classical form for every complex $\lambda $ except for a discrete set of numbers depending on $v$. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with $u$ and the class of the form $u$ knowing that of $v$. We conclude with an illustrative example.