105.04 Polynomials with no real zeros

2021 ◽  
Vol 105 (562) ◽  
pp. 117-120
Author(s):  
Aaron Melman
Keyword(s):  
Real Zeros

SOME EXTENSIONS OF A LEMMA OF KOTLARSKI

2012 ◽  
Vol 28 (4) ◽  
pp. 925-932 ◽  
Author(s):  
Kirill Evdokimov ◽  
Halbert White
Keyword(s):  
Characteristic Function ◽  
Characteristic Functions ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Weaker Conditions

This note demonstrates that the conditions of Kotlarski’s (1967, Pacific Journal of Mathematics 20(1), 69–76) lemma can be substantially relaxed. In particular, the condition that the characteristic functions of M, U1, and U2 are nonvanishing can be replaced with much weaker conditions: The characteristic function of U1 can be allowed to have real zeros, as long as the derivative of its characteristic function at those points is not also zero; that of U2 can have an isolated number of zeros; and that of M need satisfy no restrictions on its zeros. We also show that Kotlarski’s lemma holds when the tails of U1 are no thicker than exponential, regardless of the zeros of the characteristic functions of U1, U2, or M.

scholarly journals Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mina Ketan Mahanti ◽  
Amandeep Singh ◽  
Lokanath Sahoo
Keyword(s):  
Random Variables ◽  
Random Coefficients ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Gaussian Random Variables ◽  
Hyperbolic Polynomials ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the formy=Pnt=n1a1cosh⁡t+n2a2cosh⁡2t+⋯+nnancosh⁡nt, wherea1,…,anis a sequence of standard Gaussian random variables, isn/2+op(1). It is shown that the asymptotic value of expected number of times the polynomial crosses the levely=Kis alson/2as long asKdoes not exceed2neμ(n), whereμ(n)=o(n). The number of oscillations ofPn(t)abouty=Kwill be less thann/2asymptotically only ifK=2neμ(n), whereμ(n)=O(n)orn-1μ(n)→∞. In the former case the number of oscillations continues to be a fraction ofnand decreases with the increase in value ofμ(n). In the latter case, the number of oscillations reduces toop(n)and almost no trace of the curve is expected to be present above the levely=Kifμ(n)/(nlogn)→∞.

scholarly journals DEsingle for detecting three types of differential expression in single-cell RNA-seq data

2017 ◽  
Author(s):  
Zhun Miao ◽  
Ke Deng ◽  
Xiaowo Wang ◽  
Xuegong Zhang
Keyword(s):  
Single Cell ◽  
Differential Expression ◽  
Negative Binomial ◽  
Single Cells ◽  
R Package ◽  
Supplementary Information ◽  
Binomial Model ◽  
Supplementary Data ◽  
Rna Seq ◽  
Real Zeros

AbstractSummaryThe excessive amount of zeros in single-cell RNA-seq data include “real” zeros due to the on-off nature of gene transcription in single cells and “dropout” zeros due to technical reasons. Existing differential expression (DE) analysis methods cannot distinguish these two types of zeros. We developed an R package DEsingle which employed Zero-Inflated Negative Binomial model to estimate the proportion of real and dropout zeros and to define and detect 3 types of DE genes in single-cell RNA-seq data with higher accuracy.Availability and ImplementationThe R package DEsingle is freely available at https://github.com/miaozhun/DEsingle and is under Bioconductor’s consideration [email protected] informationSupplementary data are available at bioRxiv online.

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