A new algorithm for recognizing the unknot

The topological underpinnings are presented for a new algorithm which answers the question: `Is a given knot the unknot?' The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and only if it is the boundary of a disc with a combinatorial foliation. The main problems which are solved in this paper are: how to systematically enumerate combinatorial braid foliations of a disc; how to verify whether a combinatorial foliation can be realized by an embedded disc; how to find a word in the the braid group whose conjugacy class represents the boundary of the embedded disc; how to check whether the given knot is isotopic to one of the enumerated examples; and finally, how to know when we can stop checking and be sure that our example is not the unknot.Comment: 46 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper9.abs.htm

Boundedness of the domain of definition is undecidable for polynomial odes

Consider the initial-value problem with computable parameters dx dt = p(t, x) x(t0) = x0, where p : Rn+1 ! Rn is a vector of polynomials and (t0, x0) 2 Rn+1. We show that the problem of determining whether the maximal interval of definition of this initial-value problem is bounded or not is in general undecidable

Computational bounds on polynomial differential equations

In this paper we study from a computational perspective some prop-erties of the solutions of polynomial ordinary di erential equations. We consider elementary (in the sense of Analysis) discrete-time dynam-ical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary diferential equations with coe cients in Q[ ]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines. We also apply the previous methods to show that the problem of de-termining whether the maximal interval of defnition of an initial-value problem defned with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56. Combined with earlier results on the computability of solutions of poly-nomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines

A systems approach to evaluate One Health initiatives

Challenges calling for integrated approaches to health, such as the One Health (OH) approach, typically arise from the intertwined spheres of humans, animals, and ecosystems constituting their environment. Initiatives addressing such wicked problems commonly consist of complex structures and dynamics. As a result of the EU COST Action (TD 1404) “Network for Evaluation of One Health” (NEOH), we propose an evaluation framework anchored in systems theory to address the intrinsic complexity of OH initiatives and regard them as subsystems of the context within which they operate. Typically, they intend to influence a system with a view to improve human, animal, and environmental health. The NEOH evaluation framework consists of four overarching elements, namely: (1) the definition of the initiative and its context, (2) the description of the theory of change with an assessment of expected and unexpected outcomes, (3) the process evaluation of operational and supporting infrastructures (the “OH-ness”), and (4) an assessment of the association(s) between the process evaluation and the outcomes produced. It relies on a mixed methods approach by combining a descriptive and qualitative assessment with a semi-quantitative scoring for the evaluation of the degree and structural balance of “OH-ness” (summarised in an OH-index and OH-ratio, respectively) and conventional metrics for different outcomes in a multi-criteria-decision-analysis. Here, we focus on the methodology for Elements (1) and (3) including ready-to-use Microsoft Excel spreadsheets for the assessment of the “OH-ness”. We also provide an overview of Element (2), and refer to the NEOH handbook for further details, also regarding Element (4) (http://neoh.onehealthglobal.net). The presented approach helps researchers, practitioners, and evaluators to conceptualise and conduct evaluations of integrated approaches to health and facilitates comparison and learning across different OH activities thereby facilitating decisions on resource allocation. The application of the framework has been described in eight case studies in the same Frontiers research topic and provides first data on OH-index and OH-ratio, which is an important step towards their validation and the creation of a dataset for future benchmarking, and to demonstrate under which circumstances OH initiatives provide added value compared to disciplinary or conventional health initiatives

Computing domains of attraction for planar dynamics

In this note we investigate the problem of computing the domain of attraction of a ow on R2 for a given attractor. We consider an operator that takes two inputs, the description of the ow and a cover of the attractors, and outputs the domain of attraction for the given attractor. We show that: (i) if we consider only (structurally) stable systems, the operator is (strictly semi-)computable; (ii) if we allow all systems de ned by C1-functions, the operator is not (semi-)computable. We also address the problem of computing limit cycles on these systems

Fractional smoothness and applications in finance

This overview article concerns the notion of fractional smoothness of random variables of the form $g(X_T)$, where $X=(X_t)_{t\in [0,T]}$ is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance mainly concern the analysis of discrete time hedging errors. We close the review by indicating some further developments.Comment: Chapter of AMAMEF book. 20 pages

Computability with polynomial differential equations

In this paper, we show that there are Initial Value Problems de ned with polynomial ordinary di erential equations that can simulate univer- sal Turing machines in the presence of bounded noise. The polynomial ODE de ning the IVP is explicitly obtained and the simulation is per- formed in real time

Computation with perturbed dynamical systems

This paper analyzes the computational power of dynamical systems robust to infinitesimal perturbations. Previous work on the subject has delved on very specific types of systems. Here we obtain results for broader classes of dynamical systems (including those systems defined by Lipschitz/analytic functions). In particular we show that systems robust to infinitesimal perturbations only recognize recursive languages. We also show the converse direction: every recursive language can be robustly recognized by a computable system. By other words we show that robustness is equivalent to decidability. (C) 2013 Elsevier Inc. All rights reserved.INRIA program "Equipe Associee" ComputR; Fundacao para a Ciencia e a Tecnologia; EU FEDER POCTI/POCI via SQIG - Instituto de Telecomunicacoes through the FCT project [PEst-OE/EEI/LA0008/2011]info:eu-repo/semantics/publishedVersio

Ground and excited states Gamow-Teller strength distributions of iron isotopes and associated capture rates for core-collapse simulations

This paper reports on the microscopic calculation of ground and excited states Gamow-Teller (GT) strength distributions, both in the electron capture and electron decay direction, for $^{54,55,56}$Fe. The associated electron and positron capture rates for these isotopes of iron are also calculated in stellar matter. These calculations were recently introduced and this paper is a follow-up which discusses in detail the GT strength distributions and stellar capture rates of key iron isotopes. The calculations are performed within the framework of the proton-neutron quasiparticle random phase approximation (pn-QRPA) theory. The pn-QRPA theory allows a microscopic \textit{state-by-state} calculation of GT strength functions and stellar capture rates which greatly increases the reliability of the results. For the first time experimental deformation of nuclei are taken into account. In the core of massive stars isotopes of iron, $^{54,55,56}$Fe, are considered to be key players in decreasing the electron-to-baryon ratio ($Y_{e}$) mainly via electron capture on these nuclide. The structure of the presupernova star is altered both by the changes in $Y_{e}$ and the entropy of the core material. Results are encouraging and are compared against measurements (where possible) and other calculations. The calculated electron capture rates are in overall good agreement with the shell model results. During the presupernova evolution of massive stars, from oxygen shell burning stages till around end of convective core silicon burning, the calculated electron capture rates on $^{54}$Fe are around three times bigger than the corresponding shell model rates. The calculated positron capture rates, however, are suppressed by two to five orders of magnitude.Comment: 18 pages, 12 figures, 10 table