Spectra inhabiting the left half-plane that are universally realizable

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.

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

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.

Singular Values of Two Parameter Families $\lambda\dfrac{e^{az}-1}{z}$ and $\lambda\dfrac{z}{e^{az}-1}$

<p>The singular values of two parameter families of entire functions $f_{\lambda,a}(z)=\lambda\frac{e^{az}-1}{z}$, $f_{\lambda,a}(0)=\lambda a$ and meromorphic functions $g_{\lambda,a}(z)=\lambda\frac{z}{e^{az}-1}$, $g_{\lambda,a}(0)=\frac{\lambda}{a}$, $\lambda, a \in \mathbb{R} \backslash \{0\}$, $z \in \mathbb{C}$, are investigated. It is shown that all the critical values of $f_{\lambda,a}(z)$ and $g_{\lambda,a}(z)$ lie in the right half plane for $a&lt;0$ and lie in the left half plane for $a&gt;0$. It is described that the functions $f_{\lambda,a}(z)$ and $g_{\lambda,a}(z)$ have infinitely many singular values. It is also found that all the singular values $f_{\lambda,a}(z)$ are bounded and lie inside the open disk centered at origin and having radius $|\lambda a|$ and all the critical values of $g_{\lambda,a}(z)$ belong to the exterior of the disk centered at origin and having radius $|\frac{\lambda}{a}|$.</p>

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

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 $|\lambda|$.