Sufficient conditions for local equivalence of mappings

Let $$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$ be $$C^{r+1}$$ mappings and let $$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$ , $$0\in Z$$ , $$m\le n$$ . We will show that if there exist a neighbourhood U of $$0\in {\mathbb {R}}^n$$ and constants $$C,C'>0$$ and $$k>1$$ such that for $$x\in U$$ $$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$ for any $$i\in \{1,\dots , m\}$$ and for any $$s \in \mathbf {\mathbb {N}}^n_0$$ such that $$|s|\le r$$ , then there exists a $$C^r$$ diffeomorphism $$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$ such that $$f=g\circ \varphi $$ in a neighbourhood of $$0\in {\mathbb {R}}^n$$ . By $$\nu (df)$$ we denote the Rabier function.     

THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2     

A Class of Integral Operators on the Unit Ball of \mathbb{C}^{n}

For real parameters a, b, c, and t, where c is not a nonpositive integer, we determine exactly when the integral operator

Stability of the Derivative of a Canonical Product

With each sequence \(\alpha =(\alpha _n)_{n\in \mathbb{N }}\) of pairwise distinct and non-zero points which are such that the canonical product $$\begin{aligned} P_\alpha (z) := \lim _{r\rightarrow \infty }\prod _{|\alpha _n|\le r}\left( 1-\frac{z}{\alpha _n}\right) \end{aligned}$$ converges, the sequence $$\begin{aligned} \alpha ^{\prime } := \bigl (P_\alpha ^{\prime }(\alpha _n)\bigr )_{n\in \mathbb{N }} \end{aligned}$$ is associated. We give conditions on the difference \(\beta -\alpha \) of two sequences which ensure that \(\beta ^{\prime }\) and \(\alpha ^{\prime }\) are comparable in the sense that $$\begin{aligned} \exists \,c,C>0:\quad c|\alpha ^{\prime }_n| \le |\beta ^{\prime }_n| \le C|\alpha ^{\prime }_n|, \quad n\in \mathbb{N }. \end{aligned}$$ The values \(\alpha ^{\prime }_n\) play an important role in various contexts. As a selection of applications we present: an inverse spectral problem, a class of entire functions and a continuation problem.     

Multiple positive solutions for a quasilinear system of Schrödinger equations

We consider the quasilinear system

Growth Estimates for Cauchy Integrals of Measures and Rectifiability

We show that if μ is a finite Borel measure on the complex plane such that

对带混合非线性项的基尔霍夫方程规范解问题的补充

在本文中,我们研究了带有质量约束条件的基尔霍夫方程规范解的存在性问题, $$-\Big(a+b \int_{\mathbb{R}^{3}}|\nabla u|^{2} \text{d} x\Big)\Delta u=\lambda u+\mu|u|^{q-2} u+|u|^{p-2} u,~~x\in \mathbb{R}^{3},$$ 其中质量约束条件为$$S_{c}:=\Big\{u \in H^{1}(\mathbb{R}^{3}):\int_{\mathbb{R}^{3}}|u|^{2} \text{d} x=c\Big\},$$ 这里$a$, $b$, $c>0$, $\mu\in \mathbb{R}$, $20$, 当$(p, q)$ 属于$\mathbb{R}^{2}$中的某个域时. 我们通过使用约束最小化、集中紧性原理和Minimax方法证明了规范解的存在性.我们部分地扩展了已经被研究的结果.     

Hardy-Sobolev类上的Bohr型不等式

Let β 〉 0 and Sβ ：= {z ∈ C ： ｜Imz｜ 〈β} be a strip in the complex plane. For an integer r ≥ 0, let H∞^Г,β denote those real-valued functions f on R, which are analytic in Sβ and satisfy the restriction ｜f^（r）（z）｜ ≤ 1, z ∈ Sβ. For σ 〉 0, denote by Bσ the class of functions f which have spectra in （-2πσ, 2πσ）. And let Bσ^⊥ be the class of functions f which have no spectrum in （-2πσ, 2πσ）. We prove an inequality of Bohr type
‖f‖∞≤π/√λ∧σ^r∑k=0^∞（-1）^k（r＋1）/（2k＋1）^rsinh（（2k＋1）2σβ）,f∈H∞^r,β∩B1/σ,
where λ∈（0,1）,∧and ∧′are the complete elliptic integrals of the first kind for the moduli λ and λ′=√1- λ^2,respectively,and λ satisfies
4∧β/π∧′=1/σ.
The constant in the above inequality is exact.     

About entire functions with specialL 2-properties on one-dimensional subspaces of ℂ n

In this paper we consider special elements of the Fock space #x2131; _n . That is the space of entire functionsf:ℂ: ⁿ →ℂ, such that the followingL ²- condition is satisfied: . Here we show that there exists an entire functiong:ℂ ⁿ →ℂ such that for every one-dimensional subspace Π⊂ℂ ⁿ and for all 0<∈<2 we have , but in the limit case ∈=0 we have . This result is analogue to a result from [1]. There holomorphic functions on the unit-ball are investigated. Furthermore the proof — as the one in [1] — uses a theorem from [2]. Therefore we give another application of the results from [2] — namely for spaces of entire functions.     

Elliptic Systems with a Partially Sublinear Local Term

Let $1 0.$ This is in sharp contrast to D'Aprile and Mugnai's non-existence results.     

增益图与其底图的正惯性指数之间的关系

设$\mathbb{T}$是模为1的复数乘法子群.图$G=(V,E)$,这里$V,E$分别表示图的点和边.增益图是将底图中的每条边赋于$\mathbb{T}$中的某个数值$\varphi(v_iv_j)$,且满足$\varphi(v_iv_j) =\overline{\varphi(v_jv_i)}$.将赋值以后的增益图表示为$(G,\varphi)$.设$i_+(G,\varphi)$和$i_+(G)$分别表示增益图与底图的正惯性指数,本文证明了如下结论: $$ - c( G ) \le {i_ + } ( {G,\varphi } ) - {i_ + }( G ) \le c( G ), $$ 这里$c(G)$表示圈空间维数,并且刻画了等号成立时候的所有极图.     

ESTIMATE OF d_0/d~* FOR STARLIKE FUNCTIONS

Let S~* be the class of functionsf(z)analytic,univalent in the unit disk|z|<1 andmap|z|<1 onto a region which is starlike with respect to w=0 and is denoted as D_f.Letr_0=r_0(f)be the radius of convexity of f(2).In this note,the author proves the following result:(d_0/d~*)≥0.4101492,where d_0= f(z),d~*=|β|.     

The Krzyz Problem and Polynomials with Zeros on the Unit Circle

Let $ \Pi_{n,M} $ be the class of all polynomials $ p(z) = \sum _{0}^{n} a_{k}z^{k} $ of degree n which have all their zeros on the unit circle $ |z| = 1$ , and satisfy $ M = \max _{|z| = 1}|\,p(z)| $ . Let $ \mu _{k,n} = \sup _{p\in \Pi _{n,M}} |a_{k}| $ . Saff and Sheil-Small asked for the value of $\overline {\lim }_{n\rightarrow \infty }\mu _{k,n} $ . We find an equivalence between this problem and the Krzyz problem on the coefficients of bounded non-vanishing functions. As a result we compute $$ \overline {\lim }_{n\rightarrow \infty }\mu _{k,n} = {{M} \over {e}}\quad {\rm for}\ k = 1,2,3,4,5.$$ We also obtain some bounds for polynomials with zeros on the unit circle. These are related to a problem of Hayman.     

Variation of Potentials in the Unit Ball of $ {\shadC}^n$

In this paper, we study the variation of invariant Green potentials G w in the unit ball B of $ {\shadC}^n$ , which for suitable measures w are defined by $$ G_{\mu}(z) = \int_{B}G(z,w)\, d\mu(w), $$ where G is the invariant Green function for the Laplace-Beltrami operator ¨ j on B . The main result of the paper is as follows. Let w be a non-negative regular Borel measure on B satisfying $$ \int_{B}(1-|w|^2)^n\log {1 \over (1-|w|^2)}\, \d\mu(w) ] B , { z denotes the holomorphic automorphism of B satisfying { z (0) = z , { z ( z ) = 0 and ( { z { z )( w ) = w for every w ] B .     

On the Existence of Trace for Elements of \mathbb{C}_p

Let T be a transcendental element of and the orbit of T. On we have a Haar measure . The goal of this paper is to characterize all the elements of for which the integral , called the trace of T, is well defined.Presented by A. Verschoren     

Semiclassical limits of ground state solutions to Schrödinger systems

This paper is concerned with the existence and concentration properties of the ground state solutions to the following coupled Schrödinger systems $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)G_{v}(z)~\hbox { in }\ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)G_{u}(z)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ and $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)(G_{v}(z)+|z|^{2^*-2}v)~\hbox {in } \ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)(G_{u}(z)+|z|^{2^*-2}u)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where \(z=(u,v)\in {\mathbb {R}}^2\) , \(G\) is a power type nonlinearity, having superquadratic growth at both \(0\) and infinity but subcritical, \(V\) can be sign-changing and \(\inf W>0\) . We prove the existence, exponential decay, \(H^2\) -convergence and concentration phenomena of the ground state solutions for small \(\varepsilon >0\) .     

一类振荡积分算子在Wiener 共合空间上的有界性

假设a,b0并且K_(a,b)(x)=(e~(i|x|~(-b)))/(|x|~(n+a))定义强奇异卷积算子T如下:Tf(x)=(K_(a,b)*f)(x),本文主要考虑了如上定义的算子T在Wiener共合空间W(FL~p,L~q)(R~n)上的有界性.另一方面,设α,β0并且γ(t)=|t|~k或γ(t)=sgn(t)|t|~k.利用振荡积分估计,本文还研究了算子T_(α,β)f(x,y)=p.v∫_(-1)~1f(x-t,y-γ(t))(e~(2πi|t|~(-β)))/(t|t|~α)dt及其推广形式∧_(α,β)f(x,y,z)=∫_(Q~2)f(x-t,y-s,z-t~ks~j)e~(-2πit)~(-β_1_s-β_2)t~(-α_1-1)s~(-α_2-1)dtds在Wiener共合空间W(FL~p,L~q)上的映射性质.本文的结论足以表明,Wiener共合空间是Lebesgue空间的一个很好的替代.