Automated computation of robust normal forms of planar analytic vector fields

We construct an auto-validated algorithm that calculates a close to identity change of variables which brings a general saddle point into a normal form. The transformation is robust in the underlying vector field, and is analytic on a computable neighborhood of the saddle point. The normal form is suitable for computations aimed at enclosing the flow close to the saddle, and the time it takes a trajectory to pass it. Several examples illustrate the usefulness of this method

Fixed points of arbitrarily deep 1-dimensional neural networks

In this paper, we establish a sharp upper bound on the the number of fixed points a certain class of neural networks can have. The networks under study (autoencoders) can be viewed as discrete dynamical systems whose nonlinearities are given by the choice of activation functions. To this end, we introduce a new class $\mathcal{F}$ of $C^1$ activation functions that is closed under composition, and contains e.g. the logistic sigmoid function. We use this class to show that any 1-dimensional neural network of arbitrary depth with activation functions in $\mathcal{F}$ has at most three fixed points. Due to the simple nature of such networks, we are able to completely understand their fixed points, providing a foundation to the much needed connection between application and theory of deep neural networks.Comment: 9 pages, 3 figure

Combination of exhaustive search and continuation method for the study of sinks in the Hénon map

Abstract-The problem of existence of stable periodic orbits (sinks) for the Hénon map in a neighborhood of classical parameter values is studied numerically. Several parameter values which sustain a sink are found. It is shown rigorously that the sinks exist. Regions of existence in the parameter space of the sinks are located using the continuation method

A note on the convergence of parametrised non-resonant invariant manifolds

Truncated Taylor series representations of invariant manifolds are abundant in numerical computations. We present an aposteriori method to compute the convergence radii and error estimates of analytic parametrisations of non-resonant local invariant manifolds of a saddle of an analytic vector field, from such a truncated series. This enables us to obtain local enclosures, as well as existence results, for the invariant manifolds

Computer-assisted techniques for the verification of the Chebyshev property of Abelian integrals

We develop techniques for the verification of the Chebyshev property of Abelian integrals. These techniques are a combination of theoretical results, analysis of asymptotic behavior of Wronskians, and rigorous computations based on interval arithmetic. We apply this approach to tackle a conjecture formulated by Dumortier and Roussarie in [Birth of canard cycles, Discrete Contin. Dyn. Syst. 2 (2009), 723-781], which we are able to prove for q < 2

On a computer-aided approach to the computation of Abelian integrals

An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four

Electrical stimulation devices for the prevention of venous thromboembolism: Preliminary studies of physiological efficacy and user satisfaction.

Introduction: Electrical stimulation could provide an alternative method for preventing venous thromboembolism in stroke patients. The purpose of this preliminary study was to explore the effects of electrical stimulation and intermittent pneumatic compression on enhancing lower limb venous return in healthy and chronic stroke patients and also to evaluate patient and nurse satisfaction. Methods: We investigated the effectiveness of two electrical stimulation devices: Geko (Firstkind Ltd, High Wycombe, UK) and Orthopaedic Microstim 2V2 (Odstock Medical Ltd, Salisbury, UK); and one intermittent pneumatic compression device: Huntleigh Flowstron Universal (Huntleigh Healthcare Ltd, Cardiff, UK). We recruited 12 healthy and 5 chronic stroke participants. The devices were fitted sequentially, and Doppler ultrasound measurements were taken. Eight patients and nurses were also recruited for a separate usability evaluation. Results: The electrical stimulation devices emulated the blood flow characteristics of intermittent pneumatic compression in both healthy and stroke participants provided that the intensity of electrical stimulation was sufficient. Patients and nurses also felt that the electrical stimulation devices were acceptable. Conclusions: Electrical stimulation may offer benefit as an alternative method for venous thromboembolism prevention in stroke patients. The apparent benefit is sufficient to warrant further investigation in a full powered randomised controlled trial

Impressions of action and critical action learning:exploring the leadership development of senior doctors in an English healthcare organization

This paper aims to explore the influence of one cycle of a learning set experience in a postgraduate medical leadership development programme. It does so from two perspectives: first, from the self-reports of nine senior doctors working in leadership roles in England in the National Health Service; and second from a researcher perspective as we present our research process, findings and perceptions on the use of action learning (AL) and critical action learning (CAL) for leadership development in the complex and unpredictable context of that service. The paper affirms other study findings that CAL in the development of participants’ collective reflexivity has the potential to deal with emotions and power relations in organizational life. An original contribution lies in advancing the idea that CAL can help build resilience in doctor leaders and groups in uncertain conditions such that they are able to challenge current care delivery and effect change in organizational performance