Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length

We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\operatorname{PTIME}$ of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous (time and space) elegant and simple characterization of $\operatorname{PTIME}$. This is the first time such classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of computable analysis. This extends to deterministic complexity classes above polynomial time. This may provide a new perspective on classical complexity, by giving a way to define complexity classes, like $\operatorname{PTIME}$, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations, i.e.~by using the framework of analysis

Computing the exact number of periodic orbits for planar flows

In this paper, we consider the problem of determining the \emph{exact} number of periodic orbits for polynomial planar flows. This problem is a variant of Hilbert's 16th problem. Using a natural definition of computability, we show that the problem is noncomputable on the one hand and, on the other hand, computable uniformly on the set of all structurally stable systems defined on the unit disk. We also prove that there is a family of polynomial planar systems which does not have a computable sharp upper bound on the number of its periodic orbits

Polynomial differential equations compute all real computable functions on computable compact intervals

In the last decade, the eld of analog computation has experienced renewed interest. In particular, there have been several attempts to un- derstand which relations exist between the many models of analog com- putation. Unfortunately, most models are not equivalent. It is known that Euler's Gamma function is computable according to computable analysis, while it cannot be generated by Shannon's General Purpose Analog Computer (GPAC). This example has often been used to argue that the GPAC is less powerful than digital computation. However, as we will demonstrate, when computability with GPACs is not restricted to real-time generation of functions, we obtain two equiva- lent models of analog computation. Using this approach, it has been shown recently that the Gamma func- tion becomes computable by a GPAC [1]. Here we extend this result by showing that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals. Since GPACs are equivalent to systems of polynomial di erential equations then we show that all real computable functions over compact intervals can be de ned by such models

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

Computability of ordinary differential equations

In this paper we provide a brief review of several results about the computability of initial-value problems (IVPs) defined with ordinary differential equations (ODEs). We will consider a variety of settings and analyze how the computability of the IVP will be affected. Computational complexity results will also be presented, as well as computable versions of some classical theorems about the asymptotic behavior of ODEs.info:eu-repo/semantics/publishedVersio

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

Analiza pedagoškog znanja budućih nastavnika tjelesne i zdravstvene kulture

This study examined how pedagogical knowledge emerges in the teaching practice preservice teaching and supports the construction of pedagogical content knowledge of prospective physical education (PE) teachers. Four pairs of prospective PE teachers were purposefully selected to represent each of the four stages of a PE licensure. Data were collected during three academic semesters by means of interviews and structured reflective logs. The transcribed material was first divided into meaningful units, and then classified into three categories of analytical scope of pedagogical concern (macro, meso, and micro levels). Themes and patterns were sought by looking for similarities and differences among the data from analytical levels and prospective teacher pairs. Findings demonstrated that prospective PE teachers at the beginning of the program confined their primary educational concerns at the micro level, while those about to finish can also discern pedagogical issues at the meso level, but not yet at the macro level. Nevertheless, excessive concerns with a single lesson, with surviving, self-centered objectives and with a strict adherence to the planned strategies suggest a superficial share of pedagogical knowledge in the teaching practice preservice teaching and in the construction of the pedagogical content knowledge of the prospective teachers investigated.Ova studija je istraživala kako se pedagoško znanje očituje tijekom stručne prakse te kako podržava konstrukciju sadržaja pedagoškog znanja u budućih nastavnika tjelesne i zdravstvene kulture. Četiri para budućih profesora tjelesne i zdravstvene kulture bila su ciljano odabrana kao reprezentanti svakoga od četiri stupnja u procesu stjecanja kompetencija za provođenje nastave tjelesne i zdravstvene kulture. Podaci su se prikupljali tijekom tri semestra sveučilišne nastave intervjuima i strukturiranim dnevnicima rada. Prepisani dokumenti su najprije raspodijeljeni u smislene jedinice, a potom klasificirani u tri kategorije analitičkog raspona pedagoškog promišljanja (na makro, mezo i mikro razini). Teme i struktura znanja su definirane na temelju sličnosti i razlika između podataka koji su prikupljeni s različitih analitičkih razina od parova budućih nastavnika. Rezultati su pokazali da budući profesori tjelesne i zdravstvene kulture na početku svojeg školovanja ograničeno usmjeravaju pozornost na edukacijske vještine na mikro razini, dok oni koji su pri kraju svoje edukacije mogu prepoznavati i pedagoške probleme na mezo razini, ali ne i na makro razini. Ipak, pretjerana zaokupljenost pojedinačnim nastavnim satom, preživljavanjem, osobnim ciljevima te strogom provedbom planiranih strategija sugerira površan doprinos pedagoškog znanja procesu obrazovanja i konstrukciji sadržaja pedagoškog znanja budućih nastavnika tjelesne i zdravstvene kulture koji su bili sudionici ovoga kvalitativnog istraživanja

The connection between computability of a nonlinear problem and its linearization: the Hartman-Grobman theorem revisited

As one of the seven open problems in the addendum to their 1989 book Computability in Analysis and Physics Pour-El and Richards (1989)[17], Pour-El and Richards asked, "What is the connection between the computability of the original nonlinear operator and the linear operator which results from it?" Yet at present, systematic studies of the issues raised by this question seem to be missing from the literature. In this paper, we study one problem in this direction: the Hartman-Grobman linearization theorem for ordinary differential equations (ODEs). We prove, roughly speaking, that near a hyperbolic equilibrium point x(0) of a nonlinear ODE (x) over dot = f(x), there is a computable homeomorphism H such that H circle phi = L circle H, where phi is the solution to the ODE and L is the solution to its linearization (x) over dot = Df (x(0)) x. (C) 2012 Elsevier B.V. All rights reserved.Fundacao para a Ciencia e a Tecnologia; EU FEDER POCTI/POCI via SQIG - Instituto de Telecomunicacoes through the FCT [PEst-OE/EEI/LA0008/2011