Simple choreographies of the planar Newtonian NN-body Problem

In the $N$-body problem, a simple choreography is a periodic solution, where all masses chase each other on a single loop. In this paper we prove that for the planar Newtonian $N$-body problem with equal masses, $N \ge 3$, there are at least $2^{N-3} + 2^{[(N-3)/2]}$ different main simple choreographies. This confirms a conjecture given by Chenciner and etc. in \cite{CGMS02}.Comment: 31pages, 6 figures. Refinements in notations and proof

Simulation-based reachability analysis for nonlinear systems using componentwise contraction properties

A shortcoming of existing reachability approaches for nonlinear systems is the poor scalability with the number of continuous state variables. To mitigate this problem we present a simulation-based approach where we first sample a number of trajectories of the system and next establish bounds on the convergence or divergence between the samples and neighboring trajectories. We compute these bounds using contraction theory and reduce the conservatism by partitioning the state vector into several components and analyzing contraction properties separately in each direction. Among other benefits this allows us to analyze the effect of constant but uncertain parameters by treating them as state variables and partitioning them into a separate direction. We next present a numerical procedure to search for weighted norms that yield a prescribed contraction rate, which can be incorporated in the reachability algorithm to adjust the weights to minimize the growth of the reachable set

Lagrangian Reachabililty

We introduce LRT, a new Lagrangian-based ReachTube computation algorithm that conservatively approximates the set of reachable states of a nonlinear dynamical system. LRT makes use of the Cauchy-Green stretching factor (SF), which is derived from an over-approximation of the gradient of the solution flows. The SF measures the discrepancy between two states propagated by the system solution from two initial states lying in a well-defined region, thereby allowing LRT to compute a reachtube with a ball-overestimate in a metric where the computed enclosure is as tight as possible. To evaluate its performance, we implemented a prototype of LRT in C++/Matlab, and ran it on a set of well-established benchmarks. Our results show that LRT compares very favorably with respect to the CAPD and Flow* tools.Comment: Accepted to CAV 201

Guaranteed optimal reachability control of reaction-diffusion equations using one-sided Lipschitz constants and model reduction

We show that, for any spatially discretized system of reaction-diffusion, the approximate solution given by the explicit Euler time-discretization scheme converges to the exact time-continuous solution, provided that diffusion coefficient be sufficiently large. By "sufficiently large", we mean that the diffusion coefficient value makes the one-sided Lipschitz constant of the reaction-diffusion system negative. We apply this result to solve a finite horizon control problem for a 1D reaction-diffusion example. We also explain how to perform model reduction in order to improve the efficiency of the method

Wykorzystanie technologii rolnictwa precyzyjnego na terenie województwa podlaskiego

The objective of the paper was to evaluate the precision agriculture technology on the territory of Podlaskie Voivodeship. Research material consisted of data from the survey carried out in 100 randomly selected farms located in Podlaskie Voivodeship. The concept of "precision agriculture" is known among the surveyed farmers. Based on the research which was carried out, a relation between the surface area of the farm, education and age of respondents was reported. Precision agriculture technology is more popular among farmers who are less than 40 years old, who have higher education and big agricultural farms. Majority of farmers participating in the survey carries out regular research on the soil richness. Moreover, 46% uses in their farms modern technologies for plant production, only 10% uses the system of positioning and 8% of the investigated farms uses the system for guiding agricultural machines, 14% of the investigated farmers uses the system for parallel guiding of a tractor, less than 40% applies the system of variable dosing of fertilizers and plant protection substances. Majority of the investigated farmers claims that new technologies are necessary and they should be applied; however, they indicate too high costs and distribution of farms as the main reason for limitations.Celem badań była ocena stopnia wykorzystania technologii rolnictwa precyzyjnego na terenie województwa podlaskiego. Materiał badawczy stanowiły dane pochodzące z badań ankietowych przeprowadzonych wśród 100 losowo wybranych gospodarstw rolnych, położonych na terenie województwa podlaskiego. Spośród badanych rolników większości znane jest pojęcie „rolnictwa precyzyjnego”. Na podstawie przeprowadzonych badań zauważono związek między powierzchnią gospodarstwa, wykształceniem i wiekiem respondentów. Technologia rolnictwa precyzyjnego bardziej znana jest wśród rolników w wieku do 40 lat, którzy posiadają wyższe wykształcenie oraz duże gospodarstwa rolne. Większość rolników biorących udział w badaniu przeprowadza regularnie badania zasobności gleb. Ponadto: 46% wykorzystuje w swoim gospodarstwie nowoczesne technologie w produkcji roślinnej, systemu pozycjonowania używa zaledwie 10% a naprowadzania maszyn rolniczych 8% badanych gospodarstw, system prowadzenia równoległego ciągnika wykorzystuje 14% badanych rolników, niecałe 40% stosuje system zmiennego dawkowania nawozów oraz środków ochrony roślin. Większość badanych rolników uważa, że nowe technologie są potrzebne i należy z nich korzystać, jednak jako główny powód ograniczenia wskazują zbyt wysokie koszty oraz rozdrobnienie gospodarstw