scholarly journals On a Number of Zeros of a Polynomial inside a Given Circle in the Left Half Plane

2001 ◽  
Vol 37 (6) ◽  
pp. 587-589
Author(s):  
Fumitake FUJII ◽  
Kenzo WADA
Keyword(s):  
Half Plane ◽  
Left Half ◽  
Number Of Zeros ◽  
Left Half Plane

scholarly journals Conditions for the physical realizability of a typical component z(y)-matrix of a matching quadrupole of a general form in a concentrated elemental basis

2020 ◽  
pp. 13-20
Author(s):  
Gennady Devyatkov ◽  
Keyword(s):  
Half Plane ◽  
Single Point ◽  
Sufficient Conditions ◽  
Point Of View ◽  
Signal Source ◽  
Left Half ◽  
Left Half Plane ◽  
Broadband Matching ◽  
Typical Component ◽  
Physical Realizability

When solving problems of broadband matching, very often there is a need for a certain form of the amplitude-frequency characteristic. In connection with this, the problem comes up of synthesizing broadband matching devices that simultaneously have correcting properties, i.e. having a given frequency dependence of the power conversion coefficient in the operating frequency band. The use of broadband reactive matching - correcting circuits in most practical cases is difficult because of the reflected power. This leads to the problem of the synthesis of broadband matching-correcting circuits with arbitrary immittances of the signal source and load in an elemental basis of a general form, containing along with reactive and active elements, which has not been adequately solved. Therefore, it becomes necessary to find the conditions for the physical realizability of a typical component of the immitance matrix of a two-port network of general form containing poles in the left half-plane of complex frequencies. In this paper the necessary and sufficient conditions are defined for the physical realizability of the immitance matrix of a typical component of a subclass of two-terminal networks of general form in a lumped elemental electric basis, when the poles of the Eigen functions in the Foster representation can be in the left half-plane of complex frequencies, excluding the imaginary and real axes. This allows to synthesis of broadband dissipative matching, matching-correcting circuits and matched attenuators in an elemental basis of a general form with arbitrary immitances of the signal source and load from a single point of view.

scholarly journals Spectra inhabiting the left half-plane that are universally realizable

2021 ◽  
Vol 10 (1) ◽  
pp. 180-192
Author(s):  
Ricardo L. Soto
Keyword(s):  
Canonical Form ◽  
Half Plane ◽  
New Left ◽  
Nonnegative Matrix ◽  
Complex Numbers ◽  
Jordan Canonical Form ◽  
Positivity Condition ◽  
Left Half ◽  
Left Half Plane ◽  
New Criteria

Abstract Let Λ = {λ1, λ2, . . ., λ n } be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably positively realizable, then Λ is universally realizable. The positivity condition is essential for the proof of Minc, and the question whether the result holds for nonnegative realizations has been open for almost forty years. Recently, two extensions of the Minc’s result have been proved in ([5], 2018) and ([12], 2020). In this work we characterize new left half-plane lists (λ1 > 0, Re λ i ≤ 0, i = 2, . . ., n) no positively realizable, which are universally realizable. We also show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results.

scholarly journals Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane

2012 ◽  
Vol 218 (11) ◽  
pp. 6557-6565 ◽  
Author(s):  
Rosihan M. Ali ◽  
Naveen K. Jain ◽  
V. Ravichandran
Keyword(s):  
Half Plane ◽  
Left Half ◽  
Lemniscate Of Bernoulli ◽  
Left Half Plane

The Number of Zeros of a Polynomial in a Half-Plane

1960 ◽  
Vol 56 (2) ◽  
pp. 132-147 ◽  
Author(s):  
A. Talbot
Keyword(s):  
Half Plane ◽  
Continued Fraction ◽  
Complex Polynomial ◽  
Continued Fraction Expansion ◽  
Left Half ◽  
Number Of Zeros ◽  
Full Discussion ◽  
The Subject ◽  
The Right

The determination of the number of zeros of a complex polynomial in a half-plane, in particular in the upper and lower, or right and left, half-planes, has been the subject of numerous papers, and a full discussion, with many references, is given in Marden (l) and Wall (2), where the basis for the determination is a continued-fraction expansion, or H.C.F. algorithm, in terms of which the number of zeros in one of the half-planes can be written down at once. In addition, determinantal formulae for the relevant elements of the algorithm can be obtained, and these lead to determinantal criteria for the number of zeros, including that of Hurwitz (3) for the right and left half-planes.

scholarly journals Singular Values of One Parameter Family \(\lambda ((e^{z}-1)/z)^{m}\)

2015 ◽  
Vol 4 (2) ◽  
pp. 295 ◽  
Author(s):  
Mohammad Sajid
Keyword(s):  
Half Plane ◽  
Entire Functions ◽  
Singular Values ◽  
Critical Values ◽  
Parameter Family ◽  
Open Disk ◽  
Left Half ◽  
Left Half Plane

In the present paper, the singular values of one parameter family of entire functions $f_{\lambda}(z)=\lambda\bigg(\dfrac{e^{z}-1}{z}\bigg)^{m}$ and $f_{\lambda}(0)=\lambda$, $m\in \mathbb{N}\backslash \{0\}$, $\lambda\in \mathbb{R} \backslash \{0\}$, $z \in \mathbb{C}$ is investigated. It is shown that all the critical values of $f_{\lambda}(z)$ lie in the left half plane. It is also found that the function $f_{\lambda}(z)$ has infinitely many bounded singular values and lie inside the open disk centered at origin and having radius $|\lambda|$.

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