On hyperbolic polynomial-like functions and their derivatives

2007 ◽  
Vol 137 (04) ◽  
pp. 819-845 ◽  
Author(s):  
Vladimir Petrov Kostov
Keyword(s):  
Hyperbolic Polynomial

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 A Geometric Modeling Method Based on TH-Type Uniform B-Splines

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jin Xie
Keyword(s):  
Geometric Modeling ◽  
Modeling Method ◽  
Control Points ◽  
Hyperbolic Functions ◽  
Hyperbolic Polynomial ◽  
B Splines ◽  
Control Polygon ◽  
The Given

A geometric modeling method based on TH-type uniform B-splines which are composed of trigonometric and hyperbolic polynomial with parameters is introduced in this paper. The new splines possess many important properties of quadratic and cubic B-splines. Taking different values of the parameters, one can not only locally adjust the shape of the curves, but also change the type of some segments of a curve between trigonometric and hyperbolic functions as well. The given curves can also interpolate directly control polygon locally by selecting special parameters. Moreover, the introduced splines can represent some quadratic curves and transcendental curves with selecting proper control points and parameters.

scholarly journals Level crossings and turning points of random hyperbolic polynomials

1999 ◽  
Vol 22 (3) ◽  
pp. 579-586
Author(s):  
K. Farahmand ◽  
P. Hannigan
Keyword(s):  
Asymptotic Estimate ◽  
Turning Points ◽  
Random Variables ◽  
Random Polynomial ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Level Crossings ◽  
Hyperbolic Polynomials ◽  
Normally Distributed

In this paper, we show that the asymptotic estimate for the expected number ofK-level crossings of a random hyperbolic polynomiala1sinhx+a2sinh2x+⋯+ansinhnx, whereaj(j=1,2,…,n)are independent normally distributed random variables with mean zero and variance one, is(1/π)logn. This result is true for allKindependent ofx, providedK≡Kn=O(n). It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomiala1coshx+a2cosh2x+⋯+ancoshnx, withaj(j=1,2,…,n)as before, is also(1/π)logn.

scholarly journals On arrangements of roots for a real hyperbolic polynomial and its derivatives

2002 ◽  
Vol 126 (1) ◽  
pp. 45-60 ◽  
Author(s):  
Vladimir Petrov Kostov ◽  
Boris Zalmanovich Shapiro
Keyword(s):  
Hyperbolic Polynomial

scholarly journals Symbolic computation in hyperbolic programming

2018 ◽  
Vol 17 (10) ◽  
pp. 1850192
Author(s):  
Simone Naldi ◽  
Daniel Plaumann
Keyword(s):  
Semidefinite Programming ◽  
Linear Function ◽  
Symbolic Computation ◽  
Exact Algorithms ◽  
Hyperbolic Polynomial ◽  
Hyperbolicity Cone ◽  
Optimal Value ◽  
Hyperbolic Programming

Hyperbolic programming is the problem of computing the infimum of a linear function when restricted to the hyperbolicity cone of a hyperbolic polynomial, a generalization of semidefinite programming (SDP). We propose an approach based on symbolic computation, relying on the multiplicity structure of the algebraic boundary of the cone, without the assumption of determinantal representability. This allows us to design exact algorithms able to certify the multiplicity of the solution and the optimal value of the linear function.

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