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} \...

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 ...

Let $X$ be a finite connected poset, $F$ field and $I(X,F)$ the incidence algebra of over $F$. We describe bijective linear idempotent preservers $\varphi:I(X,F)\to I(X,F)$. Namely, we prove that, whenever $\mathrm{char}(F)\ne 2$, $\varphi$ is either an automorphism or anti-automorphism $I(X,F)$. If $\mathrm{char}(F)=2$ $|F|>2$, then (in general, non-proper) Lie Finally, if $F=\mathbb{Z}_2$, co...

 In this paper, we prove some coupled coincidence point theorems for mappings satisfying generalized contractive conditions under a new invariant set in ordered cone metric spaces. In fact, we obtain sufficient conditions for existence of coupled coincidence points in the setting of cone metric spaces. Some examples are provided to verify the effectiveness and applicability of our results.

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...