We prove that the number of fractions $$h_1/h_2$$ integers $$h_1,h_2$$ a subset $$\mathcal {A}\subset \mathcal {H}\cap [1,X]$$ is at least $$ \alpha X/(\log X)^{3/2}$$ , where {H}$$ set $$p-1$$ p being prime such $$p+1$$ sum two coprime squares. So, this $$\gg _\varepsilon ^{1+\varepsilon }|\mathcal {A}|^2$$ $$\varepsilon any positive real number. take opportunity to describe geometrical view s...

In this paper, we introduce $(mathcal{C},mathcal{C}')$-controlled continuous $g$-Bessel families and their multipliers in Hilbert spaces and investigate some of their properties. We show that under some conditions sum of two $(mathcal{C},mathcal{C}')$-controlled continuous $g$-frames is a $(mathcal{C},mathcal{C}')$-controlled continuous $g$-frame. Also, we investigate when a $(mathcal{C},mathca...

Abstract In this paper we prove a result on the effective generation of pluri-canonical linear systems foliated surfaces general type. Fix function {P:\mathbb{Z}_{\geq 0}\to\mathbb{Z}} , then there exists an integer {N>0} such that if {(X,{\mathcal{F}})} is canonical or nef model foliation type with Hilbert polynomial {\chi(X,{\mathcal{O}}_{X}(mK_{\mathcal{F}}))=P(m)} for all {m\in\mathbb{Z}...

let $d$ be a digraph with skew-adjacency matrix $s(d)$. then the skew energyof $d$ is defined to be the sum of the norms of all eigenvalues of $s(d)$. denote by$mathcal{o}_n$ the class of digraphs on order $n$ with no even cycles, and by$mathcal{o}_{n,m}$ the class of digraphs in $mathcal{o}_n$ with $m$ arcs.in this paper, we first give the minimal skew energy digraphs in$mathcal{o}_n$ and $mat...

This paper aims to establish the existence of a weak solution for non-local problem: \begin{equation*} \left\{\begin{array}{ll} -a\left(\int_{\Omega}\mathcal{H}(x,|\nabla u|)dx \right) \Delta_{\mathcal{H}}u &=f(x,u) \ \hbox{in} \Omega, \\ \hspace{3.3cm} u &= 0 \hbox{on} \partial \end{array}\right. \end{equation*} where $\Omega\subseteq \mathbb{R}^{N},\, N\geq 2$ is bounded and smooth domain con...

In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vall\'ee Poussin's form: $$ \pi(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) Instead of performing asymptotic expansion on Chebyshev functions as conventional methods, new approach uses contour-integration analyze Riemann's counting function $J(x)$, which only differs from $\pi(x)$ by $\math...

Let K be a discretely valued field with ring of integers $$\mathcal {O}_K$$ perfect residue field. K(x) the rational function in one variable. $${\mathbb {P}}^1_{\mathcal {O}_K}$$ standard smooth model {P}}^1_K$$ coordinate x. $$f(x) \in \mathcal {O}_K[x]$$ squarefree polynomial corresponding divisor zeroes $${{\,\mathrm{div}\,}}_0(f)$$ on . We give an explicit description minimal embedded reso...

Abstract Let $\mathbb{K}$ be a cyclic extension of degree 3 $\mathbb{Q}$. Take $G={\rm Gal}(\mathbb{K}/ \mathbb{Q})$ and χ the character non trivial representation G. In this case, is non-principal Dirichlet quantity $r_3(n)$ defined by $$r_3(n):=\big(1*\chi*\chi^2\big)(n)$$ counts number ideals $O_{\mathbb{K}}$ norm n. paper, using new result on Hooley’s Delta function from [11], we prove an a...

Consider a system of n polynomial equations and r polynomial inequations in n indeterminates of degree bounded by d with coefficients in a polynomial ring of s parameters with rational coefficients of bit-size at most $\sigma$. From the real viewpoint, solving such a system often means describing some semi-algebraic sets in the parameter space over which the number of real solutions of the cons...