Given a family $$\varphi = (\varphi_1, \ldots, \varphi_d)\in \mathbb{Z}[T]^d$$ of d distinct nonconstant polynomials, positive integer $$k\le d$$ and real parameter $$\rho$$ , we consider the mean value $$M_{k, \rho} (\varphi, N) \int_{{\rm x} \in [0,1]^k} \sup_{{\rm y} [0,1]^{d-k}} | S_{\varphi}({\rm x}, {\rm y}; |^\rho \,d{\rm $$ of exponential sums $$S_{\varphi}({\rm \sum_{n=1}^{N} \...

Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denoted by $Gamma(R)$, is a graph with vertex set $RZ(R)$ and two vertices $a$ and $b$ are adjacent iff $ab=ba$. In this paper, we consider the commuting graph of non-commutative rings of order pq and $p^2q$ with Z(R) = 0 and non-commutative rings with unity of order $p^3q$. It is proved that $C_R(a)$ is a commutative ring...

Abstract Kreisel’s conjecture is the statement: if, for all $n\in \mathbb {N}$ , $\mathop {\text {PA}} \nolimits \vdash _{k \text { steps}} \varphi (\overline {n})$ then \forall x.\varphi (x)$ . For a theory of arithmetic T given recursive function h $T _{\leq h} $ holds if there proof $\varphi in whose code at most $h(\#\varphi )$ This notion depends on underlying coding. ${P}^h_T(x)$ predicat...

Let f be an entire function, f(0) = 1, F(z) zf′ (z)/f(z), and Γm $$\bigcup_{j=1}^{m}\left\{z:\mathrm{arg}z={\psi }_{j}\right\},$$ 0 ≤ ψ1 < ψ2 ...< ψm 2π. An function is called a of improved regular growth if for some ρ ∈ (0;+∞) ρ2 (0; ρ), 2π-periodic ρ-trigonometrically convex h(φ) ≢ −∞, there exists set U ⊂ $${\mathbb{C}}$$ contained in the union disks with finite sum radii such that $$\begin{...

‎The purpose of this paper is to introduce a new mapping for a finite‎ ‎family of accretive operators and introduce an iterative algorithm‎ ‎for finding a common zero of a finite family of accretive operators‎ ‎in a real reflexive strictly convex Banach space which has a‎ ‎uniformly G^ateaux differentiable norm and admits the duality‎ ‎mapping $j_{varphi}$‎, ‎where $varphi$ is a gauge function ...

We consider the perturbed Stark operator $$H_q\varphi = -\varphi '' + x\varphi q(x)\varphi $$ , $$\varphi (0)=0$$ in $$L^2({\mathbb R}_+)$$ where q is a real function that belongs to $${\mathfrak {A}}_r =\left\{ q\in \mathcal {A}_r\cap \text {AC}[0,\infty ): q'\in {A}_r\right\} $$\mathcal {A}_r L^2_{\mathbb R}({\mathbb R}_+,(1+x)^r dx)$$ and $$r>1$$ arbitrary but fixed. Let $$\left\{ \lambda _n...

The flat torus $${{\mathbb T}}=\mathbb S^1\left( \frac{1}{2} \right) \times \mathbb $$ admits a proper biharmonic isometric immersion into the unit 4-dimensional sphere $$\mathbb S^4$$ given by $$\Phi =i \circ \varphi , where $$\varphi :{{\mathbb T}}\rightarrow S^3(\frac{1}{\sqrt{2}})$$ is minimal Clifford and $$i:\mathbb S^3(\frac{1}{\sqrt{2}}) \rightarrow small hypersphere. first goal of this...

Abstract Let D D be a digraph with vertex set V V and arc E E . For u u , the out-degree in-degree of are denoted by d + {d}_{u}^{+} − {d}_{u}^{-} respectively. A vertex-degree-based (VDB) topologic...