Метод многопараметрической интерполяции и теоремы вложения пространств БесоваB_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)

In this paper we determine the method of multi-parameter interpolation and the scales of Lebesgue spaces $B_{\vec p} \left[ {0,2\pi } \right)$ and Besov spaces $B_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)$ , which are generalizations of the Lorentz spacesL _pq [0, 2π) and Besov spacesB _pq ^α [0, 2π). We also prove imbedding theorems.     

The boundedness of Bochner-Riesz means B R δ (f) on the spaces\dot F_p^{\alpha,q}

Let 0<p≤1. The authors give out the boundedness of δ-order Bouchner-Riesz means B _R ^δ (f) on the Triebel-Lizorkin spaces \(\dot F_p^{\alpha ,q} \) for \(\delta > J - \tfrac{{n + 1}}{2} - \min \{ \alpha ,0\} \) , where J=n/min{p,q}. The above results cann't be improved, if α≥0 and p≤q.     

Asymptotic behavior of sums of the type\sum\limits_{\kappa = m}^{n - 1} {\exp } (i\omega \sqrt {n\kappa } ) as n,m → ∞

The asymptotic behavior asn, m → ∞ of the sum $$\sum\limits_{\kappa ,\ell = m}^{n - 1} {\exp \left[ {i\omega \sqrt n \left( {\sqrt \kappa + \sqrt \ell } \right)} \right]} \Phi \left( {1 - \frac{{\left| {\sqrt \kappa - \sqrt \ell } \right|}}{\Delta }} \right)$$ is studied where π(t)=0 for t?0 and φ(t)=t for t > 0.     

Unlösbarkeit der Gleichung $\alpha x^n + \beta y^n = z^n$ und Gleichverteilung von $\root n \of{\alpha x^n + \beta y^n}$

We show that the unsolvability of the Diophantine equation is equivalent to a good uniform distribution of the set . The proof depends on the asymptotic evaluation of the Gauss sum .     

The elliptic functions and integrals of the “{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}” problem

We describe the functions needed in the determination of the rate of convergence of best $L^\infty $ rational approximation to $\exp ( - x)$ on [0,∞) when the degree n of the approximation tends to ∞ (“1/9” problem).     

The behavior of \sum\nolimits_{n = 1}^\infty {{{\zeta ^{ \llcorner n\theta \lrcorner } } \mathord{\left/ {\vphantom {{\zeta ^{ \llcorner n\theta \lrcorner } } n}} \right. \kern-\nulldelimiterspace} n}} for particular values of θ

Let ζ be a primitive q′-root of unity. We prove that the series $ \sum\nolimits_{n = 1}^\infty {{{\zeta ^{ \llcorner n\theta \lrcorner } } \mathord{\left/ {\vphantom {{\zeta ^{ \llcorner n\theta \lrcorner } } n}} \right. \kern-0em} n}} $ for θ ∈ Q converges if and only if θ = p/q with (p,q) = 1 and q′ ? p, and that there exists an uncountable set S of Liouville’s numbers such that the series does not converge when θ ∈ S.     

The Diederich–Fornaess index and the global regularity of the \bar{\partial }-Neumann problem

We describe along the guidelines of Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999), the constant ${\mathcal {E}}_s$ which is needed to control the commutator of a totally real vector field $T_{{\mathcal {E}}}$ with $\bar{\partial }^*$ in order to have $H^s$ a-priori estimates for the Bergman projection $B_k, k\ge q-1$ , on a smooth $q$ -convex domain $D\subset \subset {\mathbb {C}}^{n}$ . This statement, not explicit in Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999), yields regularity of $B_k$ in specific Sobolev degree $s$ . Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function $r$ of $D$ , the operators $(T^+)^{-\frac{\delta }{2}}$ and $(-r)^{\frac{\delta }{2}}$ . We are thus able to extend to general degree $k\ge 0$ of $B_k$ , the conclusion of (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999) which only holds for $q=1$ and $k=0$ : if for the Diederich–Fornaess index $\delta $ of $D$ , we have $(1-\delta )^{\frac{1}{2}}\le {\mathcal {E}}_s$ , then $B_k$ is $H^s$ -regular.     

从$A^{p}_{\alpha}$到$A^{\infty}(\varphi)$的加权复合算子

设$u \in H(D), \ \phi$为$D$上的解析自映射,定义$H(D)$上的加权复合算子为$u C_{\phi}(f)=$$uf\circ\phi$, \ $f\in H(D)$.本文得到了从$A^{p}_{\alpha}$到$A^{\infty}(\varphi)\ (A_{0}^{\infty}(\varphi))$的加权复合算子$u C_{\phi}$的有界性和紧性的充要条件.     

On some properties of the positive m-Hessian operators in C 2(Ω)

We consider the p-Hessian operator T _p [u] := tr _p u _xx on the set ${\{u \in C^2(\bar{\Omega}), T_p[u] \geq \nu > 0, 1 \leq p \geq 0\}}$ and extend the common approach to the uniqueness and existence theorems. For instance, we prove the uniqueness of solutions to the Dirichlet problem for m-Hessian equations in ${C^2(\bar{\Omega})}$ for constant boundary data. We also give nonexistence theorems and, in addition, we establish the extremal properties of the functionals I _p [u; f] in ${C^2(\bar{\Omega})}$ and we derive some applications.     

New fixed point theorems for $${\alpha}$$-$${H\Theta}$$HΘ-contractions in ordered metric spaces

Recently, Jleli and Samet [J. Inequal. Appl. (2014), 2014:38] introduced and studied a new contraction to prove a generalization of the Banach contraction principle. In this paper, we introduce the concept of \({\alpha}\)-\({H\Theta}\)-contraction with respect to a general family of functions H and we establish Jleli–Samet-type fixed point results in metric and ordered metric spaces. As an application of our results we deduce Suzuki-type fixed point results for \({H\Theta}\)-contractions. We also derive certain fixed and periodic point results for orbitally continuous generalized \({\Theta}\)-contractions. Moreover, we present an illustrative example to highlight the obtained improvements.     

On the prescribing �� 2 curvature equation on {\mathbb S^4}

Prescribing ?? _k curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. For a given a positive function K to be prescribed on the 4-dimensional round sphere, we obtain asymptotic profile analysis for potentially blowing up solutions to the ?? ₂ curvature equation with the given K; and rule out the possibility of blowing up solutions when K satisfies a non-degeneracy condition. Under the same non-degeneracy condition on K, we also prove uniform a priori estimates for solutions to a family of ?? ₂ curvature equations deforming K to a positive constant; and under an additional, natural degree condition on a finite dimensional map associated with K, we prove the existence of a solution to the ?? ₂ curvature equation with the given K using a degree argument involving fully nonlinear elliptic operators to the above deformation.     

图$(\bigcup\limits_{i{\in}A}P_i){\bigcup}(\bigcup\limits_{j{\in}B}U_j)$伴随唯一的充要条件

设$h(G; x) =h(G)$和$[G]_h$分别表示图$G$的伴随多项式和伴随等价类. 文中给出了$[G]_h$的一个新应用. 利用$[G]_h$, 给出了图$H{\;}(H \cong G)$伴随唯一的充要条件, 其中$H=(\bigcup_{i{\in}A}P_i){\bigcup}(\bigcup_{j{\in}B}U_j)$, $A \subseteq A^{'}=\{1,2,3,5\} \bigcup \{2n|n \in N, n \geq 3\}$, $B \subseteq B^{'}     

On the monodromy of the moduli space of Calabi–Yau threefolds coming from eight planes in {\mathbb{P}^3}

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi–Yau threefolds coming from eight planes in ${\mathbb{P}^3}$ does not have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is naturally isomorphic to wedge product of a variation of Hodge structures of weight one. It turns out the hyperelliptic locus does not extend to a Shimura subvariety of type III (Siegel space) within the moduli space. Besides general Hodge theory, representation theory and computational commutative algebra, one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of Hodge structures.     

Über die Summe\mathop \Sigma \limits_{p \leqq N} e(\alpha p)

Ohne Zusammenfassung     

广义P$\grave{\bf o}$lya 组合计

文中引入了P-置换图的概念．作为置换群的指标多项式和函数等价类配置多项式的推广形式分别定义了P-置换图的容量指标多项式与色权多项式，并给出了递归公式和相关定理，由此建立了计算P-置换图的色权多项式的一般方法和P-置换图的色轨道多项式的表达公式．Pblya计数定理是这一公式当约束图是空图时的特例．最后给出了P-置换图的色权多项式的一些基本性质和两个计算实例．     

A proof of the Anderson–Badawi $$ \mathbf{rad}\varvec{(I)}^{\varvec{n}} \varvec{\subseteq } {\varvec{I}}$$ formula for $$\varvec{n}$$n-absorbing ideals

In [1], Anderson and Badawi conjectured that \(\mathrm{rad}(I)^n \subseteq I\) for every n-absorbing ideal I of a commutative ring. In this article, we prove their conjecture. We also prove related conjectures for radical ideals.