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.

     

Pointwise van der Corput lemma for functions of several variables

A multidimensional version of the well-known van der Corput lemma is presented. A class of phase functions is described for which the corresponding oscillatory integrals satisfy a multidimensional decay estimate. The obtained estimates are uniform with respect to parameters on which the phases and amplitudes may depend.     

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.

     

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 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     

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.     

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).     

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.     

Fibonacci, Van der Corput and Riesz-Nágy

What do the three names in the title have in common? The purpose of this paper is to relate them in a new and, hopefully, interesting way. Starting with the Fibonacci numeration system — also known as Zeckendorff's system — we will pose ourselves the problem of extending it in a natural way to represent all real numbers in (0,1). We will see that this natural extension leads to what is known as the ?-system restricted to the unit interval. The resulting complete system of numeration replicates the uniqueness of the binary system which, in our opinion, is responsible for the possibility of defining the Van der Corput sequence in (0,1), a very special sequence which besides being uniformly distributed has one of the lowest discrepancy, a measure of the goodness of the uniformity.Lastly, combining the Fibonacci system and the binary in a very special way we will obtain a singular function, more specifically, the inverse of one of the family of Riesz-Nágy.     

Bounds on some van der Waerden numbers

For positive integers s and k₁,k₂,…,ks, the van der Waerden number w(k₁,k₂,…,ks;s) is the minimum integer n such that for every s-coloring of set {1,2,…,n}, with colors 1,2,…,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this lower bound for m=3. We also give a lower bound for w(k,k,…,k;s) that slightly improves previously-known bounds. Upper bounds for w(k,4;2) and w(4,4,…,4;s) are also provided.     

Asymptotic stability of periodic solution for compressible viscous van der Waals fluids

This paper is concerned with the asymptotic stability of the periodic solution to a one-dimensional model system for the compressible viscous van der Waals fluid in Eulerian coordinates. If the initial density and initial momentum are suitably close to the average density and average momentum, then the solution is proved to tend toward a stationary solution as t -→∞.     

A simple solution of the van der Waerden permanent problem

The van der Waerden permanent problem was solved using mainly algebraic methods. A much simpler analytic proof is given using a new concept in optimization theory which may be of importance in the general theory of mathematical programming.     

Stability and bifurcation analysis in the delay-coupled van der Pol oscillators

In this paper, the dynamics of a system of two van der Pol equations with a finite delay are investigated. We show that there exist the stability switches and a sequence of Hopf bifurcations occur at the zero equilibrium when the delay varies. Using the theory of normal form and the center manifold theorem, the explicit expression for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.     

Models and numerical schemes for generalized van der Pol equations

This paper presents three generalizations of the van der Pol equation (VDPE) using newly proposed three new generalized K-, A- and B-operators. These operators allow kernel to be arbitrary. As a result, these operators provide a greater generalization of the VDPE than the fractional integral and differential operators do. Like the original VDPE, the generalized van der Pol equations (GVDPEs) are also nonlinear equations, and in most cases, they can not be solved analytically. Numerical algorithms are presented and used to solve the GVDPEs. Results for several examples are presented to demonstrate the effectiveness of the numerical algorithms, and to examine the behavior of the GVDPEs and the limit cycles associated with them. Although the numerical algorithms have been used to solve the GVDPEs only, they can also be used to solve many other generalized oscillators and generalized differential equations. This will be considered in the future.     

Horizontal fast excitation in delayed van der Pol oscillator

This paper investigates the interaction effect of horizontal fast harmonic parametric excitation and time delay on self-excited vibration in van der Pol oscillator. We apply the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic of the oscillator. The method of averaging is then performed on the slow dynamic to obtain a slow flow which is analyzed for equilibria and periodic motion. This analysis provides analytical approximations of regions in parameter space where periodic self-excited vibrations can be eliminated. A numerical study is performed on the original oscillator and compared to analytical approximations. It was shown that in the delayed case, horizontal fast harmonic excitation can eliminate undesirable self-excited vibrations for moderate values of the excitation frequency. In contrast, the case without delay requires large excitation frequency to eliminate such motions. This work has application to regenerative behavior in high-speed milling.     

Bifurcation analysis in van der Pol's oscillator with delayed feedback

In this paper, a classical van der Pol's equation with generally delayed feedback is considered. It is shown that there are Bogdanov–Takens bifurcation, triple zero and Hopf-zero singularities by analyzing the distribution of the roots of the associated characteristic equation. In the situation that the zero is as a simple eigenvalue, the normal forms of the reduced equations are obtained by the center manifold theory and normal form method for functional differential equation, and hence the stability of the fixed point is determined, and transcritical and pitchfork bifurcations are found.     

Multiple Hopf bifurcations of three coupled van der Pol oscillators with delay

A system of three coupled van der Pol oscillators with delay is considered. Hopf bifurcations at the zero equilibrium as the delay increases are exhibited. The existence and stability of multiple periodic solutions are established using a symmetric Hopf bifurcation result of Wu (Trans. Amer. Math. Soc. 350 (1998) 4799-4838).     

A brief remark on van der Waerden spaces

We demonstrate that Martin's axiom for -centered notions of forcing implies the existence of a van der Waerden space that is not a Hindman space. Our proof is an adaptation of the one given by M. Kojman and S. Shelah that such a space exists if one assumes the continuum hypothesis to be true.