Rough singular integrals associated with surfaces of Van der Corput type简

In this paper,the authors establish the Lp-mapping properties for a class of singular integrals along surfaces in Rn of the form {φ(|u|)u : u ∈ Rn} as well as the related maximal operators provided that the function φ satisfies certain oscillatory integral estimates of Van der Corput type,and the integral kernels are given by the radial function h ∈Δγ(R+) for γ 1 and the sphere function Ω∈ Fβ(Sn.1) for some β 0,which is distinct from H1(Sn.1).     

Van der Corput Sequences and Linear Permutations

This extended abstract is concerned with the irregularities of distribution of one-dimensional permuted van der Corput sequences that are generated from linear permutations. We show how to obtain upper bounds for the discrepancy and diaphony of these sequences, by relating them to Kronecker sequences and applying earlier results of Faure and Niederreiter.     

Sharp van der Corput estimates and minimal divided differences

We find the sharp constant in a sublevel set estimate which arises in connection with van der Corput's lemma. In order to do this, we find the nodes that minimise divided differences. We go on to find the sharp constant in the first instance of the van der Corput lemma. With these bounds we improve the constant in the general van der Corput lemma, so that it is asymptotically sharp.

     

Lp- and Sp,qrB-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases

We study the local discrepancy of a symmetrized version of the well-known van der Corput sequence and of modified two-dimensional Hammersley point sets in arbitrary base $b$. We give upper bounds on the norm of the local discrepancy in Besov spaces of dominating mixed smoothness $S_{p,q}^{r}B\left({[0,1)}^s\right)$, which will also give us bounds on the $L_p$-discrepancy. Our sequence and point sets will achieve the known optimal order for the $L_p$- and $S_{p,q}^{r}B$-discrepancy. The results in this paper generalize several previous results on $L_p$- and $S_{p,q}^{r}B$-discrepancy estimates and provide a sharp upper bound on the $S_{p,q}^{r}B$-discrepancy of one-dimensional sequences for $r>0$. We will use the $b$-adic Haar function system in the proofs.     

Distribution functions for subsequences of the van der Corput sequence

For an integer b>1$b>1$ let _(?b(n))n≥0${\left({?}_b\left(n\right)\right)}_{n\ge 0}$ denote the base b$b$ van der Corput sequence in [0,1)$[0,1)$. Answering a question of O. Strauch, C. Aisleitner and M. Hofer showed that the distribution function of _{(?b(n),?b(n+1),…,?b(n+s−1))n≥0}${({?}_b\left(n\right),{?}_b(n+1),\dots ,{?}_b(n+s-1))}_{n\ge 0}$ on [0,1)^s${[0,1)}^s$ exists and is a copula. In this note we show that this phenomenon extends to a broad class of subsequences of the van der Corput sequences.     

A van der Corput lemma for the -adic numbers

We prove a version of van der Corput's lemma for polynomials over the -adic numbers.

     

Multidimensional van der Corput and sublevel set estimates

Van der Corput's lemma gives an upper bound for one-dimensional oscillatory integrals that depends only on a lower bound for some derivative of the phase, not on any upper bound of any sort. We establish generalizations to higher dimensions, in which the only hypothesis is that a partial derivative of the phase is assumed bounded below by a positive constant. Analogous upper bounds for measures of sublevel sets are also obtained. The analysis, particularly for the sublevel set estimates, has a more combinatorial flavour than in the one-dimensional case.

     

A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets

We study the star discrepancy of Hammersley nets and van der Corput sequences which are important examples of low-dimensional quasi-Monte Carlo point sets. By a so-called digital shift, the quality of distribution of these point sets can be improved. In this paper, we advance and extend existing bounds on digitally shifted Hammersley and van der Corput point sets and establish criteria for the choice of digital shifts leading to optimal results. Our investigations are partly based on a close analysis of certain sums of distances to the nearest integer. Mathematics Subject Classi cation (2000) 11K38; 11K09     

Coupled Van der Pol oscillator with non-integer order connection

In this paper the dynamics of a system of two oscillators with strong nonlinear connection is considered. The two mass system connected with a spring with pure nonlinear force of any positive rational order (integer or noninteger), on which some additional small nonlinear forces act, is analyzed. The mathematical model of the system contains two coupled second order differential equations of oscillatory type with strong pure nonlinearity and small additional terms. In the paper an analytical solving procedure which introduces the periodical Ateb function is developed. The averaging solution method is adopted to this special function and gives the new type of averaged differential equations.The special attention is given to the steady-state motion of a two-degree-of-freedom Van der Pol oscillator system of positive rational order of nonlinearity. The influence of the order of nonlinearity on the motion of the system is analyzed. using the suggested approximate method three numerical examples are solved. The obtained results are much more accurate than those obtained by the already published methods based on the trigonometric functions.     

On an Extension and a Refinement of Van der Corput's Inequality

By introducing a parameterα,we give an extension of Van der Corput's inequal- ity.Also a new strengthened version of it is considered.     

关于Ven der Curput不等式的一个推广及改进

By introducing a parameter α, we give an extension of Van der Corput's inequality. Also a new strengthened version of it is considered.     

The Asymptotic Expansion and Numerical Verification of Van der Pol's Equation

A technique is presented in this paper to verify the order of accuracy of asymptotic expansion of Van der Pol's equation. The technique is focused on using numerical solutions as an independent means of verifying the validity of asymptotic expansions.     

On an Extension and a Refinement of Van der Corput＇s Inequality

By introducing a parameterα,we give an extension of Van der Corput's inequal- ity.Also a new strengthened version of it is considered.     

The degree sequence of Fibonacci and Lucas cubes

The Fibonacci cube Γ_n is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1’s. The Lucas cube Λ_n is obtained from Γ_n by removing vertices that start and end with 1. It is proved that the number of vertices of degree k in Γ_n and Λ_n is and , respectively. Both results are obtained in two ways, since each of the approaches yields additional results on the degree sequences of these cubes. In particular, the number of vertices of high resp. low degree in Γ_n is expressed as a sum of few terms, and the generating functions are given from which the moments of the degree sequences of Γ_n and Λ_n are easily computed.     

Horseshoe in a modified Van der Pol–Duffing circuit

In this paper, we study a modified Van der Pol–Duffing circuit, and present a rigorous verification for existence of chaos in this system. The arguments are given in the manner of computer-assisted proof by using topological horseshoe theory.     

谐和随机噪声联合作用下Van der PolDuffing振子的参数主共振

研究了Van der Pol－Duffing振子在谐和与随机噪声联合激励下的参数主共振响应和稳定性问题。用多尺度法分离了系统的快变项，并求出了系统的最大Liapunov指数和稳态概率密度函数，还分析了失稳、分 叉和跳跃现象，讨论了系统的阻尼项、非线性项、随机项和确定性参激强度等参数对系统响应的影响。数值模拟表明所提出的方法是有效的。     

Two-dimensional Riemann problem of the Euler equations to the Van der Waals gas around a sharp corner

In this paper, we study the Riemann problem of the two-dimensional (2D) pseudo-steady supersonic flow with Van der Waals gas around a sharp corner expanding into vacuum. The essence of this problem is the interaction of the centered simple wave with the planar rarefaction wave, which can be solved by a Goursat problem or a mixed characteristic boundary value and slip boundary value problem for the 2D self-similar Euler equations. We establish the hyperbolicity and a priori C¹ estimates of the solution through the methods of characteristic decompositions and invariant regions. Moreover, we construct the pentagon invariant region in order to obtain the global solution. In addition, based on the generality of the Van der Waals gas, we construct the subinvariant regions and get the hyperbolicity of the solution according to the continuity of the subinvariant region. At last, the global existence of solution to the gas expansion problem is obtained constructively.     

Van der Waals气体Euler方程的Riemann问题及基本波的相互作用

研究了修正的等熵Van der Waals气体动力学Euler方程Riemann问题及其基本波的相互作用.利用Maxwell提出的等面积法则,将Van der Waals气体状态方程修正为与实际相符,从而守恒律方程组从混合型转化为双曲型.利用广义特征线分析法,构造性地得到了Riemann问题的解是存在的.进一步,得到了基本波相互作用.     

对数螺线、黄金分割与斐波那契数列的完美统一的

本文对对数螺线、黄金分割与斐波那契数列之间的关系进行研究,把线段上黄金分割点的定义推广到射线上的黄金分割点列,发现过极轴上任意一点有且仅有一条特殊的对数螺线与极轴的交点所成的点列为黄金分割点列,并把这个点列所对应的坐标定义为黄金分割数列,我们发现首项为1/√15黄金分割数列无限逼近于斐波那契数列。