Positivity, monotonicity, and consensus on lie groups

Dynamical systems whose linearizations along trajectories are positive in the sense that they infinitesimally contract a smooth cone field are called differentially positive. The property can be thought of as a generalization of monotonicity, which is differential positivity in a linear space with respect to a constant cone field. Differential positivity places significant constraints on the asymptotic behavior of trajectories under mild technical conditions. This paper studies differentially positive systems defined on Lie groups. The geometry of a Lie group allows for the generation of invariant cone fields over the tangent bundle given a single cone in the Lie algebra. We outline the mathematical framework for studying differential positivity of discrete and continuous-time dynamics on a Lie group with respect to an invariant cone field and motivate the use of this analysis framework in nonlinear control, and, in particular in nonlinear consensus theory. We also introduce a generalized notion of differential positivity of a dynamical system with respect to an extended notion of cone fields generated by cones of rank k. This new property provides the basis for a generalization of differential Perron-Frobenius theory, whereby the Perron-Frobenius vector field which shapes the one-dimensional attractors of a differentially positive system is replaced by a distribution of rank k that results in k-dimensional integral submanifold attractors instead

Invariant Differential Positivity and Consensus on Lie Groups

Differential positivity of a dynamical system refers to the property that its linearization along trajectories is positive, that is, infinitesimally contracts a smooth cone field defined in the tangent bundle. The property can be thought of as a generalization of monotonicity, which is differential positivity in a linear space with respect to a constant cone field. Differential positivity induces a conal order which places significant constraints on the asymptotic behavior of solutions. This paper studies differentially positive systems defined on Lie groups, which constitute an important and basic class of manifolds with the structure of a homogeneous space. The geometry of a Lie group allows for the generation of invariant cone fields over the tangent bundle given a single cone in the Lie algebra. We outline the mathematical framework for studying differential positivity of a nonlinear flow on a Lie group with respect to an invariant cone field and motivate the use of this analysis framework in nonlinear control, and, in particular in nonlinear consensus theory

Target formation on the circle by monotone system design

Positivity and Perron-Frobenius theory provide an elegant framework for the convergence analysis of linear consensus algorithms. Here we consider a generalization of these ideas to the analysis of nonlinear consensus algorithms on the circle and establish tools for the design of consensus protocols that monotonically converge to target formations on the circle

Physics of the Non-Abelian Coulomb Phase: Insights from Pad\'e Approximants

We consider a vectorial, asymptotically free SU($N_c$) gauge theory with $N_f$ fermions in a representation $R$ having an infrared (IR) fixed point. We calculate and analyze Pad\'e approximants to scheme-independent series expansions for physical quantities at this IR fixed point, including the anomalous dimension, $\gamma_{\bar\psi\psi,IR}$, to $O(\Delta_f^4)$, and the derivative of the beta function, $\beta'_{IR}$, to $O(\Delta_f^5)$, where $\Delta_f$ is an $N_f$-dependent expansion variable. We consider the fundamental, adjoint, and rank-2 symmetric tensor representations. The results are applied to obtain further estimates of $\gamma_{\bar\psi\psi,IR}$ and $\beta'_{IR}$ for several SU($N_c$) groups and representations $R$, and comparisons are made with lattice measurements. We apply our results to obtain new estimates of the extent of the respective non-Abelian Coulomb phases in several theories. For $R=F$, the limit $N_c \to \infty$ and $N_f \to \infty$ with $N_f/N_c$ fixed is considered. We assess the accuracy of the scheme-independent series expansion of $\gamma_{\bar\psi\psi,IR}$ in comparison with the exactly known expression in an ${\cal N}=1$ supersymmetric gauge theory. It is shown that an expansion of $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ is quite accurate throughout the entire non-Abelian Coulomb phase of this supersymmetric theory.Comment: 28 pages, 3 figure

Differential positivity with respect to cones of rank k ≥ 2

We consider a generalized notion of differential positivity of a dynamical system with respect to cone fields generated by cones of rank k. The property refers to the contraction of such cone fields by the linearization of the flow along trajectories. It provides the basis for a generalization of differential Perron-Frobenius theory, whereby the Perron-Frobenius vector field which shapes the one-dimensional attractors of a differentially positive system is replaced by a distribution of rank $k$ that results in $k$-dimensional integral submanifold attractors instead. We further develop the theory in the context of invariant cone fields and invariant differential positivity on Lie groups and illustrate the key ideas with an extended example involving consensus on the space of rotation matrices SO(3).C. Mostajeran is supported by the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom. The research leading to these results has also received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n.67064

What can we learn about SARS-CoV-2 prevalence from testing and hospital data?

Measuring the prevalence of active SARS-CoV-2 infections is difficult because tests are conducted on a small and non-random segment of the population. But people admitted to the hospital for non-COVID reasons are tested at very high rates, even though they do not appear to be at elevated risk of infection. This sub-population may provide valuable evidence on prevalence in the general population. We estimate upper and lower bounds on the prevalence of the virus in the general population and the population of non-COVID hospital patients under weak assumptions on who gets tested, using Indiana data on hospital inpatient records linked to SARS-CoV-2 virological tests. The non-COVID hospital population is tested fifty times as often as the general population. By mid-June, we estimate that prevalence was between 0.01 and 4.1 percent in the general population and between 0.6 to 2.6 percent in the non-COVID hospital population. We provide and test conditions under which this non-COVID hospitalization bound is valid for the general population. The combination of clinical testing data and hospital records may contain much more information about the state of the epidemic than has been previously appreciated. The bounds we calculate for Indiana could be constructed at relatively low cost in many other states

Contraction Analysis of Monotone Systems via Separable Functions

In this paper, we study incremental stability of monotone nonlinear systems through contraction analysis. We provide sufficient conditions for incremental asymptotic stability in terms of the Lie derivatives of differential one-forms or Lie brackets of vector fields. These conditions can be viewed as sum- or max-separable conditions, respectively. For incremental exponential stability, we show that the existence of such separable functions is both necessary and sufficient under standard assumptions for the converse Lyapunov theorem of exponential stability. As a by-product, we also provide necessary and sufficient conditions for exponential stability of positive linear time-varying systems. The results are illustrated through examples