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Long Wang - One of the best experts on this subject based on the ideXlab platform.

  • stability analysis for continuous time positive systems with time varying delays
    IEEE Transactions on Automatic Control, 2010
    Co-Authors: Xingwen Liu, Long Wang
    Abstract:

    This note addresses the stability problem of continuous-time positive systems with time-varying delays. It is shown that such a system is asymptotically stable for any continuous and bounded delay if and only if the sum of all the system matrices is a Hurwitz Matrix. The result is a time-varying version of the widely-known asymptotic stability criterion for constant-delay positive systems. A numerical example illustrates the correctness of our result.

Mohammad Adm - One of the best experts on this subject based on the ideXlab platform.

Olga Holtz - One of the best experts on this subject based on the ideXlab platform.

  • generalized Hurwitz matrices generalized euclidean algorithm and forbidden sectors of the complex plane
    Computational Methods and Function Theory, 2016
    Co-Authors: Olga Holtz, Sergey Khrushchev, Olga Y Kushel
    Abstract:

    Given a polynomial $$\begin{aligned} f(x)=a_0x^n+a_1x^{n-1}+\cdots +a_n \end{aligned}$$ with positive coefficients \(a_k\), and a positive integer \(M\le n\), we define an infinite generalized Hurwitz Matrix \(H_M(f):= (a_{Mj-i})_{i,j}\). We prove that the polynomial f(z) does not vanish in the sector $$\begin{aligned} \left\{ z\in \mathbb {C}: |\arg (z)| < \frac{\pi }{M}\right\} \end{aligned}$$ whenever the Matrix \(H_M\) is totally non-negative. This result generalizes the classical Hurwitz’ Theorem on stable polynomials (\(M=2\)), the Aissen–Edrei–Schoenberg–Whitney theorem on polynomials with negative real roots (\(M=1\)), and the Cowling–Thron theorem (\(M=n\)). In this connection, we also develop a generalization of the classical Euclidean algorithm, of independent interest per se.

  • generalized Hurwitz matrices generalized euclidean algorithm and forbidden sectors of the complex plane
    arXiv: Classical Analysis and ODEs, 2015
    Co-Authors: Olga Holtz, Sergey Khrushchev, Olga Y Kushel
    Abstract:

    Given a polynomial \[ f(x)=a_0x^n+a_1x^{n-1}+\cdots +a_n \] with positive coefficients $a_k$, and a positive integer $M\leq n$, we define a(n infinite) generalized Hurwitz Matrix $H_M(f):=(a_{Mj-i})_{i,j}$. We prove that the polynomial $f(z)$ does not vanish in the sector $$ \left\{z\in\mathbb{C}: |\arg (z)| < \frac{\pi}{M}\right\} $$ whenever the Matrix $H_M$ is totally nonnegative. This result generalizes the classical Hurwitz' Theorem on stable polynomials ($M=2$), the Aissen-Edrei-Schoenberg-Whitney theorem on polynomials with negative real roots ($M=1$), and the Cowling-Thron theorem ($M=n$). In this connection, we also develop a generalization of the classical Euclidean algorithm, of independent interest per se.

Mikhail Tyaglov - One of the best experts on this subject based on the ideXlab platform.

Müller-hermes Alexander - One of the best experts on this subject based on the ideXlab platform.

  • Annihilating Entanglement Between Cones
    2021
    Co-Authors: Aubrun Guillaume, Müller-hermes Alexander
    Abstract:

    Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The entanglement annihilating maps transform the $k$-fold maximal tensor product of a cone $C_1$ into the $k$-fold minimal tensor product of a cone $C_2$, and the pair $(C_1,C_2)$ is called resilient if all entanglement annihilating maps are entanglement breaking. Our main result is that $(C_1,C_2)$ is resilient if either $C_1$ or $C_2$ is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz Matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.Comment: 38 pages, no figure

  • Annihilating Entanglement Between Cones
    HAL CCSD, 2021
    Co-Authors: Aubrun Guillaume, Müller-hermes Alexander
    Abstract:

    Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The entanglement annihilating maps transform the $k$-fold maximal tensor product of a cone $C_1$ into the $k$-fold minimal tensor product of a cone $C_2$, and the pair $(C_1,C_2)$ is called resilient if all entanglement annihilating maps are entanglement breaking. Our main result is that $(C_1,C_2)$ is resilient if either $C_1$ or $C_2$ is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz Matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied