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.

Note to the behavior of the maximal term of Dirichlet series absolutely convergent in half-plane

By $S_0(\Lambda)$ denote a class of Dirichlet series $F(s)=\sum_{n=0}^{\infty}a_n\exp\{s\lambda_n\} (s=\sigma+it)$ withan increasing to $+\infty$ sequence $\Lambda=(\lambda_n)$ of exponents ($\lambda_0=0$) and the abscissa of absolute convergence $\sigma_a=0$.We say that $F\in S_0^*(\Lambda)$ if $F\in S_0(\Lambda)$ and $\ln \lambda_n=o(\ln |a_n|)$ $(n\to\infty)$. Let$\mu(\sigma,F)=\max\{|a_n|\exp{(\sigma\lambda_n)}\colon n\ge 0\}$ be the maximal term of Dirichlet series. It is proved that in order that $\ln (1/|\sigma|)=o(\ln \mu(\sigma))$ $(\sigma\uparrow 0)$ for every function $F\in S_0^*(\Lambda)$ it is necessary and sufficient that $\displaystyle \varlimsup\limits_{n\to\infty}\frac{\ln \lambda_{n+1}}{\ln \lambda_n}<+\infty. $For an analytic in the disk $\{z\colon |z|<1\}$ function $f(z)=\sum_{n=0}^{\infty}a_n z^n$ and $r\in (0, 1)$ we put $M_f(r)=\max\{|f(z)|\colon |z|=r<1\}$ and $\mu_f(r)=\max\{|a_n|r^n\colon n\ge 0\}$. Then from hence we get the following statement: {\sl if there exists a sequence $(n_j)$ such that $\ln n_{j+1}=O(\ln n_{j})$ and $\ln n_{j}=o(\ln |a_{n_{j}}|)$ as $j\to\infty$, then the functions $\ln \mu_f(r)$ and $\ln M_f(r)$ are or not are slowly increasing simultaneously.

Study of the strengthened state near the forces for the semi-plan

Many of the engineering applications have faced the delicate contact problem in the area close to the forces where it is very difficult to experimentally carry out various measurements and draw important conclusions on the condition of the contact points. In this paper the forced state in the vicinity of the forces for the half-plane will be studied. Furthermore, the qualities displayed by the half-plane under the action of normal forces, tangential forces and the moment caused by a pair of forces will be analyzed, as well as changes in the elastic characteristics for the forced plane state and the deformed plane state.

Harmonic mappings and its directional convexity

For any $\mu _{j}\ (\mu _{j}\in \mathbb{C},\left\vert \mu _{j}\right\vert =1,j=1,2)$, we consider the rotations $f_{\mu _{1}}$ and $F_{\mu _{2}}$ of right half-plane harmonic mappings $f,F\in S_{\mathcal{H}}$ which are CHD with the prescribed dilatations $\omega _{f}(z)=\left( a-z\right) /\left(1-az\right) $ for some $a$ $\left( -1<a<1\right) $ and $\omega _{F}(z)=$ $e^{i\theta }z^{n}$ $\left( n\in \mathbb{N},\theta \in \mathbb{R}\right) $, $\omega _{F}(z)=$ $\left( b-z\right) /\left( 1-bz\right) $, $\omega_{F}(z)=\left( b-ze^{i\phi }\right) /\left( 1-bze^{i\phi }\right) $ $(-1<b<1,\phi \in \mathbb{R})$, respectively. It is proved that the convolution $f_{\mu _{1}}\ast F_{\mu _{2}}\in S_{\mathcal{H}}$ and is convex in the direction of $\overline{\mu _{1}\mu _{2}}$ under certain conditions on the parameters involved.