Algebraic Polynomials with Non-identical Random Coefficients

The asymptotic estimate for the expected number of real zeros of a random algebraic polynomial is known. The identical random coefficients a_j(ω) are normally distributed defined on a probability space , ω ∈Ω. The estimate for the expected number of zeros of the derivative of the above polynomial with respect to x is also known, which gives the expected number of maxima and minima of Q_n(x, ω). In this paper we provide the asymptotic value for the expected number of zeros of the integration of Q_n(x,ω) with respect to x. We give the geometric interpretation of our results and discuss the difficulties which arise when we consider a similar problem for the case of .     

On algebraic polynomials with random coefficients

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form

where , are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the zero crossings of the curve representing the polynomial are perpendicular to the axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form which was previously the most studied.

     

Algebraic polynomials with non-identical random coefficients

There are many known asymptotic estimates for the expected number of real zeros of a random algebraic polynomial The coefficients are mostly assumed to be independent identical normal random variables with mean zero and variance unity. In this case, for all sufficiently large, the above expected value is shown to be . Also, it is known that if the have non-identical variance , then the expected number of real zeros increases to . It is, therefore, natural to assume that for other classes of distributions of the coefficients in which the variance of the coefficients is picked at the middle term, we would also expect a greater number of zeros than . In this work for two different choices of variance for the coefficients we show that this is not the case. Although we show asymptotically that there is some increase in the number of real zeros, they still remain . In fact, so far the case of is the only case that can significantly increase the expected number of real zeros.

     

Local maxima of a random trigonometric polynomial

In this paper we obtain a formula for the expected number of maxima of a normal process (t) which occur below a levelu. The main condition assumed in the derivation is that (t) and its first and second derivative have, with probability one, continuous one-dimensional distributions. The expected number of such maxima of a trigonometric polynomial with random coefficients follow from this result. It is shown that, wich probability one, all the maxima of this type of polynomial occur below a levelu if , and a sizeable number of maxima exist below zero level.     

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.     

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.     

On random algebraic polynomials

This paper provides asymptotic estimates for the expected number of real zeros and -level crossings of a random algebraic polynomial of the form , where are independent standard normal random variables and is a constant independent of . It is shown that these asymptotic estimates are much greater than those for algebraic polynomials of the form .

     

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).     

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.     

n次代数多项式有m个不同根的充要条件

本文利用方阵的迹及顺序主子式 ,给出了 n次代数多项式有 m( m n)个不同根的充要条件 .     

Exceedence Measure of Classes of Algebraic Polynomials

There is both mathematical and physical interest in the behaviour of the polynomial of the form . The coefficients a _j , j = 0,...,n are assumed to be independent normally distributed random variables with mean zero and variance ². In this paper by using the motion of exceedence measure for stochastic processes, for n large, we derive an asymptotic estimate for the expected area of the curve representing the above polynomial cut off by the x-axis. We show that our method can be used to obtain results for similar random polynomials.     

Local Limit Theorems for Sums of Power Series Distributed Random Variables and for the Number of Components in Labelled Relational Structures

A Local limit theorem for the distribution of the number of components in random labelled relational structures of size n (e.g., a type of random graphs on n vertices, random permutations of n elements, etc.) is proved as n→∞. The case when the corresponding exponential generating functions diverge at their radii of convergence is considered.     

Improved approximations to distributions of the largest and the smallest latent roots of a wishart matrix

Summary Normalizing transformations of the largest and the smallest latent roots of a sample covariance matrix in a normal sample are obtained, when the corresponding population roots are simple. Using our results, confidence intervals for population roots may easily be constructed. Some numerical comparisons of the resulting approximations are made in a bivariate case, based on exact values of the probability integral of latent roots.     

有限域上单变量代数函数域中的素数定理

设q是素数的幂次，Fq为一有限域；F为Fq上的单变量代数函数域．在这篇文章中我们证明了下面的素数定理，πF（x）=1/（q-1）.x/logqx＋O（x/log^2qx）.x=q^n→∞其中logqx以q为底的对数，这一结果改进了M．Kruse，H．Stichtenoth的结果．