Average Number of Real Roots of Random Harmonic Equations

Let {g_k}be a sequence of normally distributed independent random variables with mathematical expectation zero and variance unity. Let _k (t ) (k = 0, 1, 2,...) be the normalized Jacobi polynomials orthogonal with respect to the interval [ – 1, 1 ]. Then it is proved that the average number of real roots of the random equations, _k=0 ⁿ g_kk(1)=C where Cis a constant, is asymptotically equal to n/in the same interval when nis large and even for C as long as C=O (n ²).     

Real Zeros of Algebraic Polynomials with Dependent Random Coefficients

The expected number of real zeros of polynomials a ₀ + a ₁ x + a ₂ x ² +…+a _n?1 x ^n?1 with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π)log n. For the dependent cases studied so far it is shown that this asymptotic value remains O(log n). In this article, we show that when cov(a _i , a _j ) = 1 ? |i ? j|/n, for i = 0,…, n ? 1 and j = 0,…, n ? 1, the above expected number of real zeros reduces significantly to O(log n)^1/2.     

Trigonometric polynomials with many real zeros and a Littlewood-type problem

We examine the size of a real trigonometric polynomial of degree at most having at least zeros in (counting multiplicities). This result is then used to give a new proof of a theorem of Littlewood concerning flatness of unimodular trigonometric polynomials. Our proof is shorter and simpler than Littlewood's. Moreover our constant is explicit in contrast to Littlewood's approach, which is indirect.

     

Complex Zeros of Algebraic Polynomial with Non-Zero Mean Random Coefficients

In this paper we obtain a formula for the average density of the distribution of complex zeros of an algebraic polynomial with random coefficients. The coefficients are assumed independent identical normally distributed random variables with mean and variance ². The value of the average density for the case of =0 and ²=1 was obtained previously. Some limits of the distribution of the complex zeros are provided using the presented formula.     

Number Variance of Random Zeros on Complex Manifolds

We show that the variance of the number of simultaneous zeros of m i.i.d. Gaussian random polynomials of degree N in an open set with smooth boundary is asymptotic to , where is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on . Our results hold more generally for the simultaneous zeros of random holomorphic sections of the N-th power of any positive line bundle over any m-dimensional compact K?hler manifold. Received: August 2006 Revision: March 2007 Accepted: April 2007     

Zeros of a random algebraic polynomial with coefficient means in geometric progression

This paper provides the mathematical expectation for the number of real zeros of an algebraic polynomial with non-identical random coefficients. We assume that the coefficients {a_j}ⁿ⁻¹_j=0 of the polynomial T(x)=a₀+a₁x+a₂x²+?+a_n−1xⁿ⁻¹ are normally distributed, with mean E(a_j)=μ^j+1, where μ≠0, and constant non-zero variance. It is shown that the behaviour of the random polynomial is independent of the variance on the interval (−1,1); it differs, however, for the cases of |μ|<1 and |μ|>1. On the intervals (−∞,−1) and (1,∞) we find the expected number of real zeros is governed by an interesting relationship between the means of the coefficients and their common variance. Our result is consistent with those of previous works for identically distributed coefficients, in that the expected number of real zeros for μ≠0 is half of that for μ=0.     

三角多项式类中的Birkhoff型三角插值

本文在三角多项式类中讨论了2π周期函数的一类Birkhoff型等距结点的三角插值问题,给出了此问题有解的充要条件,并构造出插值基.     

The Expected Number of Local Maxima of a Random Algebraic Polynomial

In this work we obtain an asymptotic estimate for the expected number of maxima of the random algebraic polynomial , where a _j (j=0, 1,...,n–1) are independent, normally distributed random variables with mean and variance one. It is shown that for nonzero , the expected number of maxima is asymptotic to log n, when n is large.     

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of Obrechkoff and certain explicit connection coefficients for these polynomials. As a consequence, sharp bounds for the zeros of the Hermite polynomials are obtained.

     

多项式零点保持线性映射

设H是维数大于2的复Hilbert空间,β(H)代表H上所有有界线性算子全体．假定Φ是从β(H)到其自身的弱连续线性双射．我们证明了映射Φ满足对所有的A,B∈β(H),AB=BA~*蕴涵Φ(A)Φ(B)=Φ(B)Φ(A)~*当且仅当存在非零实数c和酉算子U∈(?)(H),使得Φ(A)=cUAU~*对所有的A∈β(H)成立．     

Preserving Zeros of a Polynomial

We study nonlinear surjective mappings on ? _n () and its subsets, which preserve the zeros of some fixed polynomials in noncommuting variables.     

一般随机缺项三角级数表示断片的Bouligand维数

该文对［１］中的Ｂｏｕｌｉｇａｎｄ维数计算公式进行了改进，用对称原理和简化原理，得到了一般随机缺项三角级数所表示断片的Ｂｏｕｌｉｇａｎｄ维数的一些计算公式．     

An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

Let ${\{{\phi }_i\}}_{i=0}^{\infty }$ be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure μ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say ${\mathbb{E}}_n(\mu )$, of random polynomials

$P_n(z):=\sum _{i=0}^n{\eta }_i{\phi }_i(z),$

where ${\eta }_0,\dots ,{\eta }_n$ are i.i.d. standard Gaussian random variables. When μ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that ${\mathbb{E}}_n(|\mathrm{d}\xi |)$ admits an asymptotic expansion of the form

${\mathbb{E}}_n(|\mathrm{d}\xi |)～\frac{2}{\pi }\log ?(n+1)+\sum _{p=0}^{\infty }{A}_p{(n+1)}^{?p}$

(Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where μ is absolutely continuous with respect to arclength measure and its Radon–Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case ${\mathbb{E}}_n(\mu )$ admits an analogous expansion with the coefficients $A_p$ depending on the measure μ for $p\ge 1$ (the leading order term and $A_0$ remain the same).     

Zeros of real polynomials

We study vector subspaces of the set of zeros of real valued symmetric odd polynomials and of real homogeneous polynomials of low degree.     

Zeros of real polynomials

We study vector subspaces of the set of zeros of real valued symmetric odd polynomials and of real homogeneous polynomials of low degree.