A positive integer $n$ is called a square-free number if it is not divisible by a perfect square except $1$. Let $p$ be an odd prime. For $n$ with $(n,p)=1$, the smallest positive integer $f$ such that $n^f\equiv 1(modp)$ is called the exponent of $n$ modulo $p$. If the exponent of $n$ modulo $p$ is $p-1$, then $n$ is called a primitive root mod $p$. Let $A\left(n\right)$ be the characteristic function of the square-free primitive roots modulo $p$. In this paper we study the distribution $$\sum _{n\le x}A\left(n\right)A(n+1),$$ and give an asymptotic formula by using properties of character...

A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let $𝒮$ be a nontrivial finite regular linear space and $G\le \mathrm{Aut}\left(𝒮\right).$ Suppose that $G$ is extremely primitive on points and let rank$\left(G\right)$ be the rank of $G$ on points. We prove that rank$\left(G\right)\ge 4$ with few exceptions. Moreover, we show that $\mathrm{Soc}\left(G\right)$ is neither a sporadic group nor an alternating group, and $G=\mathrm{PSL}(2,q)$ with $q+1$ a Fermat prime if $\mathrm{Soc}\left(G\right)$ is a finite classical simple group. ...

We compare the growth of the least common multiple of the numbers $u_{a_1},...,u_{a_n}$ and $|u_{a_1}\cdots u_{a_n}|$, where ${\left(u_n\right)}_{n\ge 0}$ is a Lucas sequence and ${\left(a_n\right)}_{n\ge 0}$ is some sequence of positive integers.

In this paper we solve the problem of finding integrals of equations determining the Killing tensors on an $n$-dimensional differentiable manifold $M$ endowed with an equiaffine $SL(n,)$-structure and discuss possible applications of obtained results in Riemannian geometry.

If $(M,g)$ is a Riemannian manifold, we have the well-known base preserving   vector bundle isomorphism $TM\widetilde{=}{T}^{*}M$ given by $v\to g(v,-)$ between the tangent $TM$ and the cotangent $T^{*}M$ bundles of $M$. In the present note, we generalize this isomorphism to the one $T^{\left(r\right)}M\widetilde{=}{T}^{r*}M$ between the $r$-th order vector tangent $T^{\left(r\right)}M={\left(J^r{(M,R)}_0\right)}^{*}$ and the $r$-th order cotangent $T^{r*}M=J^r{(M,R)}_0$ bundles of $M$. Next, we describe all base preserving  vector bundle maps $C_M\left(g\right):T^{\left(r\right)}M\to T^{r*}M$ depending on a Riemannian metric $g$ in terms of natural (in $g$) tensor fields on $M$.

Let $G$ be a simple algebraic group over an algebraically closed field $𝕜$ of characteristic 0, and $𝔤=LieG$. Let $(e,h,f)$ be an $𝔰{𝔩}_2$-triple in $𝔤$ with $e$ being a long root vector in $𝔤$. Let $(·,·)$ be the $G$-invariant bilinear form on $𝔤$ with $(e,f)=1$ and let $\chi \in {𝔤}^{*}$ be such that $\chi \left(x\right)=(e,x)$ for all $x\in 𝔤$. Let $𝒮$ be the Slodowy slice at $e$ through the adjoint orbit of $e$ and let $H$ be the enveloping algebra of $𝒮$; see [31]. In this article we give an explicit presentation of $H$ by generators and relations. As a consequence we deduce that $H$ contains...

Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$. We consider the image ${𝒩}_1(D,X)$ of ${𝒩}_1\left(D\right)$ in ${𝒩}_1\left(X\right)$ under the natural push-forward of $1$-cycles. We show that ${\rho }_X-{\rho }_D\le codim{𝒩}_1(D,X)\le 8$. Moreover if $codim{𝒩}_1(D,X)\ge 3$, then either $X\cong S\times T$ where $S$ is a Del Pezzo surface, or $codim{𝒩}_1(D,X)=3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that ${\rho }_X-{\rho }_T=4$.

Let $X$ be a Stein manifold of complex dimension $n\ge 2$ and $\Omega ⋐X$ be a relatively compact domain with $C^2$ smooth boundary in $X$. Assume that $\Omega$ is a weakly $q$-pseudoconvex domain in $X$. The purpose of this paper is to establish sufficient conditions for the closed range of $\overline{\partial }$ on $\Omega$. Moreover, we study the $\overline{\partial }$-problem on $\Omega$. Specifically, we use the modified weight function method to study the weighted $\overline{\partial }$-problem with exact support in $\Omega$. Our method relies on the $L^2$-estimates by Hörmander (1965) and by Kohn (1973). ...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

For $n=2m\ge 4$, let $\Omega \in {ℝ}^n$ be a bounded smooth domain and $𝒩\subset {ℝ}^L$ a compact smooth Riemannian manifold without boundary. Suppose that $\left\{u_k\right\}\in W^{m,2}(\Omega ,𝒩)$ is a sequence of weak solutions in the critical dimension to the perturbed $m$-polyharmonic maps $$\frac{\mathrm{d}}{\mathrm{d}t}{|}_{t=0}{E}_m\left(\Pi (u+t\xi )\right)=0$$ with ${\Phi }_k\to 0$ in ${\left(W^{m,2}(\Omega ,𝒩)\right)}^{*}$ and $u_k\rightharpoonup u$ weakly in $W^{m,2}(\Omega ,𝒩)$. Then $u$ is an $m$-polyharmonic map. In particular, the space of $m$-polyharmonic maps is sequentially compact for the weak-$W^{m,2}$ topology.

We prove some results in the context of isoperimetric inequalities with quantitative terms. In the $2$-dimensional case, our main contribution is a method for determining the optimal coefficients $c_1,...,c_m$ in the inequality $\delta P\left(E\right)\ge {\sum }_{k=1}^m{c}_k\alpha {\left(E\right)}^k+o\left(\alpha {\left(E\right)}^m\right)$, valid for each Borel set $E$ with positive and finite area, with $\delta P\left(E\right)$ and $\alpha \left(E\right)$ being, respectively, the $\mathrm{𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡}$ and the $\mathrm{𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦}$ of $E$. In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of $\mathrm{𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠}$ including the lower semicontinuous extension of $\frac{\delta P\left(E\right)}{\alpha {\left(E\right)}^2}$, we...

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle {𝒰}_n:n\in \omega \rangle$ of open covers of $X$, there exists $U_n\in {𝒰}_n$ for each $n\in \omega$ such that $X={\bigcup }_{n\in \omega }{U}_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p{(\Psi \left(𝒜\right),2)}^n$ to have the Rothberger property when $𝒜$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $𝒜$ for which the space $C_p{(\Psi \left(𝒜\right),2)}^n$ is Rothberger for all $n\in \omega$.

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...