Note on roots location of a symmetric polynomial with respect to the imaginary axis

2012 ◽  
Vol 60 (4) ◽  
pp. 499-510
Author(s):  
Hui-Feng Hao ◽  
Yong-Jian Hu ◽  
Gong-Ning Chen
Keyword(s):  
Imaginary Axis ◽  
Symmetric Polynomial

On a Subclass of Univalent Harmonic Mappings Convex in the Imaginary Direction

2020 ◽  
pp. 445-454
Author(s):  
Deepali Khurana ◽  
Sushma Gupta ◽  
Sukhjit Singh
Keyword(s):  
Present Article ◽  
Unit Disk ◽  
Imaginary Axis ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Distortion Bounds ◽  
Univalent Harmonic Mappings

In the present article, we consider a class of univalent harmonic mappings, $\mathcal{C}_{T} = \left\{ T_{c}[f] =\frac{f+czf'}{1+c}+\overline{\frac{f-czf'}{1+c}}; \; c>0\;\right\}$ and $f$ is convex univalent in $\mathbb{D}$, whose functions map the open unit disk $\mathbb{D}$ onto a domain convex in the direction of the imaginary axis. We estimate coefficient, growth and distortion bounds for the functions of the same class.

An Algorithm for Factoring Hermitian Matrix Polynomials Relative to the Imaginary Axis

1996 ◽  
Vol 29 (1) ◽  
pp. 1263-1268
Author(s):  
Chyi Hwang ◽  
Bo-Win Lin ◽  
Tong-Yi Guo
Keyword(s):  
Imaginary Axis ◽  
Hermitian Matrix ◽  
Matrix Polynomials

scholarly journals Large amplitude oscillations for a class of symmetric polynomial differential systems in R³

2007 ◽  
Vol 79 (4) ◽  
pp. 563-575 ◽  
Author(s):  
Jaume Llibre ◽  
Marcelo Messias
Keyword(s):  
Large Amplitude ◽  
Symmetric Polynomial ◽  
The Other ◽  
Poincaré Compactification ◽  
Heteroclinic Loop ◽  
Differential Systems ◽  
Heteroclinic Cycles ◽  
Polynomial Differential Systems ◽  
Global Study ◽  
Straight Lines

In this paper we study a class of symmetric polynomial differential systems in R³, which has a set of parallel invariant straight lines, forming degenerate heteroclinic cycles, which have their two singular endpoints at infinity. The global study near infinity is performed using the Poincaré compactification. We prove that for all n <FONT FACE=Symbol>Î</FONT> N there is epsilonn > 0 such that for 0 < epsilon < epsilonn the system has at least n large amplitude periodic orbits bifurcating from the heteroclinic loop formed by the two invariant straight lines closest to the x-axis, one contained in the half-space y > 0 and the other in y < 0.

scholarly journals Computation of Green’s Function of the Bounded Solutions Problem

2018 ◽  
Vol 18 (4) ◽  
pp. 673-685 ◽  
Author(s):  
Vitalii G. Kurbatov ◽  
Irina V. Kurbatova
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Numerical Experiments ◽  
Imaginary Axis ◽  
Approximate Calculation ◽  
Special Function ◽  
Bounded Solution ◽  
Free Term ◽  
Bounded Solutions ◽  
Square Matrix

AbstractIt is well known that the equation {x^{\prime}(t)=Ax(t)+f(t)}, where A is a square matrix, has a unique bounded solution x for any bounded continuous free term f, provided the coefficient A has no eigenvalues on the imaginary axis. This solution can be represented in the formx(t)=\int_{-\infty}^{\infty}\mathcal{G}(t-s)f(s)\,ds.The kernel {\mathcal{G}} is called Green’s function. In this paper, for approximate calculation of {\mathcal{G}}, the Newton interpolating polynomial of a special function {g_{t}} is used. An estimate of the sensitivity of the problem is given. The results of numerical experiments are presented.

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