Möbius parametrizations of curves in {\mathbb{R}}^n

We use Ahlfors’ definition of Schwarzian derivative for curves in euclidean spaces to present new results about M?bius or projective parametrizations. The class of such parametrizations is invariant under compositions with M?bius transformations, and the resulting curves are simple. The analysis is based on the oscillatory behavior of the associated linear equation , where k = k(s) is the curvature as a function of arclength. Received: 24 November 2008     

On Confining Potentials and Essential Self-Adjointness for Schrödinger Operators on Bounded Domains in {\mathbb{R}}^n

Let Ω be a bounded domain in with C²-smooth boundary, , of co-dimension 1, and let be a Schr?dinger operator on Ω with potential . We seek the weakest conditions we can find on the rate of growth of the potential V close to the boundary which guarantee essential self-adjointness of H on . As a special case of an abstract condition, we add optimal logarithmic type corrections to the known condition where . More precisely, we show that if, as x approaches ,

Beurling’s theorem for SL(2,$${\mathbb{R}}$$)

We prove Beurling’s theorem for the full group SL(2,). This is the master theorem in the quantitative uncertainty principle as all the other theorems of this genre follow from it.     

A Generalization Of Reifenberg’s Theorem In {\mathbb{R}}^3

In 1960 Reifenberg proved the topological disc property. He showed that a subset of which is well approximated by m-dimensional affine spaces at each point and at each (small) scale is locally a bi-H?lder image of the unit ball in . In this paper we prove that a subset of which is well approximated in the Hausdorff distance sense by one of the three standard area-minimizing cones at each point and at each (small) scale is locally a bi-H?lder deformation of a minimal cone. We also prove an analogous result for more general cones in . Received: July 2006, Revised: August 2007, Accepted: January 2008     

On Nikol’skii Inequalities for Domains in $${\mathbb {R}}^d$$

Nikol’skii inequalities for various sets of functions, domains, and weights will be discussed. Much of the work is dedicated to the class of algebraic polynomials of total degree n on a bounded convex domain D. That is, we study \(\sigma := \sigma (D,d)\) for which

$$\begin{aligned} \Vert P\Vert _{L_q(D)}\le c n^{\sigma (\frac{1}{p}-\frac{1}{q})}\Vert P\Vert _{L_p(D)},\quad 0<p\le q\le \infty , \end{aligned}$$

where P is a polynomial of total degree n. We use geometric properties of the boundary of D to determine \(\sigma (D,d)\) with the aid of comparison between domains. Computing the asymptotics of the Christoffel function of various domains is crucial in our investigation. The methods will be illustrated by the numerous examples in which the optimal \(\sigma (D,d)\) will be computed explicitly.

     

一类$L^2(\mathbb{R}^n)$的紧支撑不可分正交小波

本篇文章给出一类$L^{2}(\mathbb{R}^{n})$, $n\geq2$的紧支撑不可分正交小波基的具体构造算法,其中正交小波的伸缩矩阵为$\alpha I_{n}~(\alpha\geq2,\ \alpha \in \mathbb{Z})$, $I_{n}$是$n$阶单位矩阵.最后给出两个不可分正交小波基的构造算例.     

Time-periodic solutions of the compressible Navier–Stokes equations in {\mathbb{R}^{4}}

This paper deals with the existence of time-periodic solutions to the compressible Navier–Stokes equations effected by general form external force in \({\mathbb{R}^{N}}\) with \({N = 4}\). Using a fixed point method, we establish the existence and uniqueness of time-periodic solutions. This paper extends Ma, UKai, Yang’s result [5], in which, the existence is obtained when the space dimension \({N \ge 5}\).     

在${\mathbb{R}}^m$中重对数广义律的精确速率

令\{$X$, $X_n$, $n\ge 1$\}是期望为${\mathbb{E}}X=(0,\ldots,0)_{m\times 1}$和协方差阵为${\rm Cov}(X,X)=\sigma^2I_m$的独立同分布的随机向量列, 记$S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. 对任意$d>0$和$a_n=o((\log\log n)^{-d})$, 本文研究了${{\mathbb{P}}(|S_n|\ge (\varepsilon+a_n)\sigma \sqrt{n}(\log\log n)^d)$的一类加权无穷级数的重对数广义律的精确速率.     

Leray’s inequality in general multi-connected domains in {\mathbb{R}^n}

Consider the stationary Navier–Stokes equations in a bounded domain whose boundary consists of L + 1 smooth (n − 1)-dimensional closed hypersurfaces Γ₀, Γ₁, . . . , Γ _L , where Γ₁, . . . , Γ _L lie inside of Γ₀ and outside of one another. The Leray inequality of the given boundary data β on plays an important role for the existence of solutions. It is known that if the flux on Γ _i (ν: the unit outer normal to Γ _i ) is zero for each i = 0, 1, . . . , L, then the Leray inequality holds. We prove that if there exists a sphere S in Ω separating in such a way that Γ₁, . . . , Γ _k (1 ≦ k ≦ L) are contained inside of S and that the others Γ _k+1, . . . , Γ _L are outside of S, then the Leray inequality necessarily implies that γ ₁ + · · · +  γ _k =  0. In particular, suppose that there are L spheres S ₁, . . . , S _L in Ω lying outside of one another such that Γ _i lies inside of S _i for all i = 1, . . . , L. Then the Leray inequality holds if and only if γ ₀ = γ ₁ = · · · = γ _L = 0.     

Fredholm and regularity theory of Douglis–Nirenberg elliptic systems on {\mathbb{R}^{N}}

We give a fairly complete exposition of the Fredholm properties of the Douglis–Nirenberg elliptic systems on ${\mathbb{R}^{N}}$ in the classical (unweighted) L ^p Sobolev spaces and under “minimal” assumptions about the coefficients. These assumptions rule out the use of classical pseudodifferential operator theory, although it is indirectly of assistance in places. After generalizing a necessary and sufficient condition for Fredholmness, already known in special cases, various invariance properties are established (index, null space, etc.), with respect to p and the Douglis–Nirenberg numbers. Among other things, this requires getting around the problem that the L ^p spaces are not ordered by inclusion. In turn, with some work, invariance leads to a regularity theory more general than what can be obtained by the method of differential quotients.     

The Helmholtz Operator on Higher Dimensional M?bius Strips Embedded in {\mathbb{R}^{4}}

In this paper we present explicit formulas for the fundamental solution to the Helmholtz operator on a higher-dimensional analogue of the M?bius strip in three real variables (embedded in ${\mathbb{R}^{4}}$ ) with values in distinct pinor bundles. Herefore we use an approach that uses classical harmonic analysis methods combined with some Clifford analysis tools and adapt it to this special geometry. The fundamental solution is described in terms of generalizations of the Weierstrass ${\wp}$ -function that are adapted to the context of these geometries. As our main result we present an analytic integral representation formula to express the solutions of the inhomogeneous time-independent Klein-Gordon problem on M?bius strips.     

Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\). We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in \(\mathbb {S}^3 \times \mathbb {S}^3\). We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\) can only be \(\frac{2}{\sqrt{3}}\) or \(\frac{4}{\sqrt{3}}\).     

${\mathbb{Z}}/3{\mathbb{Z}}$-量子群的Gr\"{o}bner-Shirshov 基

通过计算合成, 我们证明了Yamane 给出的关系是 ${\mathbb{Z}}/3{\mathbb{Z}}$-量子群的一个Gr\"{o}bner-Shirshov 基.     

On the Fučik point spectrum for Schrödinger operators on {\mathbb{R}}^{N}

We investigate the Fučik point spectrum of the Schr?dinger operator when the potential V_λ has a steep potential well for sufficiently large parameter λ > 0. It is allowed that S_λ has essential spectrum with finitely many eigenvalues below the infimum of . We construct the first nontrivial curve in the Fučik point spectrum by minimax methods and show some qualitative properties of the curve and the corresponding eigenfunctions. As applications we establish some results on existence of multiple solutions for nonlinear Schr?dinger equations with jumping nonlinearity.      

Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in {\mathbb {R}^{3}}

We consider a class of semilinear elliptic equations of the form $$ \label{eq:abs}-\Delta u(x,y,z)+a(x)W'(u(x,y,z))=0,\quad (x,y,z)\in\mathbb {R}^{3},$$ where ${a:\mathbb {R} \to \mathbb {R}}$ is a periodic, positive, even function and, in the simplest case, ${W : \mathbb {R} \to \mathbb {R}}$ is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the one dimensional heteroclinic problem $$-\ddot q(x)+a(x)W^{\prime}(q(x))=0,\ x\in\mathbb {R},\quad q(x)\to\pm1\,{\rm as}\, x\to \pm\infty,$$ we show, via variational methods the existence of infinitely many geometrically distinct solutions u of (0.1) verifying u(x, y, z) → ± 1 as x → ± ∞ uniformly with respect to ${(y, z) \in \mathbb {R}^{2}}$ and such that ${\partial_{y}u \not \equiv0, \partial_{z}u \not\equiv 0}$ in ${\mathbb {R}^{3}}$ .     

Lp estimates for non-smooth bilinear Littlewood–Paley square functions on $${\mathbb{R}}$$

In this work, we study some non-smooth bilinear analogues of linear Littlewood–Paley square functions on the real line. We prove boundedness-properties in Lebesgue spaces for them. Let us consider the functions \({\phi_{n}}\) satisfying \({\widehat{\phi_n}(\xi)={\bf 1}_{[n,n+1]}(\xi)}\) and define the bilinear operator \({S_n(f,g)(x):=\int f(x+y)g(x-y) \phi_n(y) dy}\) . These bilinear operators are closely related to the bilinear Hilbert transforms. Then for exponents \({p,q,r'\in[2,\infty)}\) satisfying \({\frac{1}{p}+\frac{1}{q}=\frac{1}{r}}\) , we prove that

$\left\| \left( \sum_{n\in \mathbb{Z}}\left|S_n(f,g) \right|^2 \right)^{1/2}\right\|_{L^{r}(\mathbb{R})}\lesssim \|f\|_{L^p(\mathbb{R})}\|g\|_{L^q(\mathbb{R})}.$

     

Classification of solutions to the higher order Liouville’s equation on {\mathbb{R}^{2m}}

We classify the solutions to the equation (−Δ) ^m u = (2m − 1)!e ^2mu on giving rise to a metric with finite total Q-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Δu at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric at infinity, and we observe that the pull-back of this metric to S ^2m via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.     

Pitt’s Inequality and Logarithmic Uncertainty Principle for the Dunkl Transform on {\mathbb{R}}

We establish Pitt’s inequality and deduce Beckner’s logarithmic uncertainty principle for the Dunkl transform on \({\mathbb{R}}\) . Also, we prove Stein–Weiss inequality for the Dunkl–Bessel potentials.     

\mathbb{R}�϶Գ�Cauchy�����񼯵�ȷ��Hausdorff���

??This paper establishes limsup type law of the iterated logarithm of the occupation measure, using the asymptotic equivalence relation between the occupation measure and the number of excursion process of a symmetric Cauchy process. Furthermore, by using the density theorem and the economic coverage method, it derives the exact Hausdorff measure for the range of a symmetric Cauchy process in \mathbb{R}.     

���ܶȹ��ƵĶԳƼ�����~$L_1(\mathbb{R}^d)$~�µ���ƫ��

Let f_n be a non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in \mathbb{R}^d. In this paper we prove two moderate deviation theorems in L_1(\mathbb{R}^d) for \{f_n(x)-f_n(-x),\,n\ge1\}.