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

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

Fully automated countrywide monitoring of fuel break maintenance operations

PTDC/CCI-COM/30344/2017 PCIF/SSI/0102/2017 UIDB/00239/2020 UIDB/00066/2020Fuel break (FB) networks are strategic locations for fire control and suppression. In order to be effective for wildfire control, they need to be maintained through regular interventions to reduce fuel loads. In this paper, we describe a monitoring system relying on Earth observations to detect fuel reduction inside the FB network being implemented in Portugal. Two fast automated pixel-based methodologies for monthly monitoring of fuel removals in FB are developed and compared. The first method (M1) is a classical supervised classification using the difference and postdisturbance image of monthly image composites. To take into account the impact of different land cover and phenology in the detection of fuel treatments, a second method (M2) based on an innovative statistical change detection approach was developed. M2 explores time series of vegetation indices and does not require training data or user-defined thresholds. The two algorithms were applied to Sentinel-2 10 m bands and fully processed in the cloud-based platform Google Earth Engine. Overall, the unsupervised M2, which is based on a Welch t-test of two moving window averages, gives better results than the supervised M1 and is suitable for an automated countrywide fuel treatment detection. For both methods, two vegetation indices, the Modified Excess of Green and the Normalized Difference Vegetation Index, were compared and exhibited similar performances.publishersversionpublishe

Revisiting niche fundamentals with Tukey depth

1. The first attempts to describe species ecological niches were simple geometric procedures that depict the niche boundaries directly from environmental data. The convex hull was one of such procedures, popular for its simplicity, clear ecological rational and precise definition of the niche. However, it lacked the ability to differentiate areas of the niche with different probabilities of occurrence according to environmental suitability. 2. We incorporate the Tukey depth, a mathematical tool to measure the centrality of a point within a cloud of points on a multidimensional space, in the convex hull approach to (i) propose a new procedure (CH-Tukey) to estimate species’ environmental suitability, and (ii) estimate niche overlap coherently. In addition to a clear ecological rational and simplicity the CHTukey procedure has a number of attractive features: use of presence-only data; independence from background data; invariance to scale; robustness to outliers; and the decomposition of the niche into a finite number of isosuitability levels, permitting the computation of consistent overlap indices. We illustrate the use of CH-Tukey, using occurrence data of the main Quercus species and subspecies from Western Mediterranean Europe, comparing its outputs with BIOCLIM and MAXENT. 3. Results showed distinct niche geometries among the different approaches. BIOCLIM produced rectilinear niches reflecting the assumption that ecological variables are independent in their action on the species. CHTukey, relaxing this assumption, adjusts niche outer boundary and the inner suitability levels to the known occurrences. MAXENT produced unbounded niche geometries, showing abrupt shifts in the species response to the environmental variables. 4. The niche predictions obtained with geometric approaches, BIOCLIM and CH Tukey, are simpler but better aligned with Hutchinson’s niche concept than those obtained with MAXENT, this latter showing ecologically implausible relationships with the environmental variables. CH-Tukey and the related overlap measures provide an adequate tool to explore niche properties and species-environment relationships

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

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

Ecological implications of fine-scale fire patchiness and severity in tropical savannas of northern Australia

Research ArticleUnderstanding fine-scale fire patchiness has significant implications for ecological processes and biodiversity conservation. It can affect local extinction of and recolonisation by relatively immobile fauna and poorly seed-dispersed flora in fire-affected areas. This study assesses fine-scale fire patchiness and severity, and associated implications for biodiversity, in north Australian tropical savanna systems. We used line transects to sample burning patterns of ground layer vegetation in different seasons and vegetation structure types, within the perimeter of 35 fires that occurred between 2009 and 2011. We evaluated two main fire characteristics: patchiness (patch density and mean patch length) and severity (inferred from char and scorch heights, and char and ash proportions). The mean burned area of ground vegetation was 83 % in the early dry season (EDS: May to July) and 93 % in the late dry season (LDS: August to November). LDS fires were less patchy (smaller and fewer unburned patches), and had higher fire severity (higher mean char and scorch heights, and twice the proportion of ash) than EDS fires. Fire patchiness varied among vegetation types, declining under more open canopy structure. The relationship between burned area and fire severity depended on season, being strongly correlated in the EDS and uncorrelated in the LDS. Simulations performed to understand the implications of patchiness on the population dynamics of fire-interval sensitive plant species showed that small amounts of patchiness substantially enhance survival. Our results indicate that the ecological impacts of high frequency fires on firesensitive regional biodiversity elements are likely to be lower than has been predicted from remotely sensed studies that are based on assumptions of homogeneous burninginfo:eu-repo/semantics/publishedVersio

Characterizing Computable Analysis with Differential Equations

The functions of Computable Analysis are defined by enhancing the capacities of normal Turing Machines to deal with real number inputs. We consider characterizations of these functions using function algebras, known as Real Recursive Functions. Bournez and Hainry 2006 [5] used a function algebra to characterize the twice continuously differentiable functions of Computable Analysis, restricted to certain compact domains. In a similar model, Shannon’s General Purpose Analog Computer, Bournez et. al. 2007 [3] also characterize the functions of Computable Analysis. We combine the results of [5] and Graça et. al. [13], to show that a different function algebra also yields Computable Analysis. We believe that our function algebra is an improvement due to its simple definition and because the operations in our algebra are less obviously designed to mimic the operations in the usual definition of the recursive functions using the primitive recursion and minimization operators.

The elementary computable functions over the real numbers: Applying two new techniques

The basic motivation behind this work is to tie together various computational complexity classes, whether over different domains such as the naturals or the reals, or whether defined in different manners, via function algebras (Real Recursive Functions) or via Turing Machines (Computable Analysis). We provide general tools for investigating these issues, using two techniques we call approximation and lifting. We use these methods to obtain two main theorems. First we provide an alternative proof of the result from Campagnolo, Moore and Costa [3], which precisely relates the Kalmar elementary computable functions to a function algebra over the reals. Secondly, we build on that result to extend a result of Bournez and Hainry [1], which provided a function algebra for the C 2 real elementary computable functions; our result does not require the restriction to C 2 functions. In addition to the extension, we provide an alternative approach to the proof. Their proof involves simulating the operation of a Turing Machine using a function algebra. We avoid this simulation, using a technique we call lifting, which allows us to lift the classic result regarding the elementary computable functions to a result on the reals. The two new techniques bring a different perspective to these problems, and furthermore appear more easily applicable to other problems of this sort