Bounds for algorithms in differential algebra

We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials F, let M(F) be the sum of maximal orders of differential indeterminates occurring in F. We propose a modification of the Rosenfeld-Groebner algorithm, in which for every intermediate polynomial system F, the bound M(F) is less than or equal to (n-1)!M(G), where G is the initial set of generators of the radical ideal. In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in a characteristic decomposition of a radical differential ideal. We also give an algorithm for converting a characteristic decomposition of a radical differential ideal from one ranking into another. This algorithm performs all differentiations in the beginning and then uses a purely algebraic decomposition algorithm.Comment: 40 page

Canonical Characteristic Sets of Characterizable Differential Ideals

We study the concept of canonical characteristic set of a characterizable differential ideal. We propose an efficient algorithm that transforms any characteristic set into the canonical one. We prove the basic properties of canonical characteristic sets. In particular, we show that in the ordinary case for any ranking the order of each element of the canonical characteristic set of a characterizable differential ideal is bounded by the order of the ideal. Finally, we propose a factorization-free algorithm for computing the canonical characteristic set of a characterizable differential ideal represented as a radical ideal by a set of generators. The algorithm is not restricted to the ordinary case and is applicable for an arbitrary ranking.Comment: 26 page

Efficient computation of regular differential systems by change of rankings using Kähler differentials

We present two algorithms to compute a regular differential system for some ranking, given an equivalent regular differential system for another ranking. Both make use of Kähler differentials. One of them is a lifting for differential algebra of the FGLM algorithm and relies on normal forms computations of differential polynomials and of Kähler differentials modulo differential relations. Both are implemented in MAPLE V. A straightforward adaptation of FGLM for systems of linear PDE is presented too. Examples are treated

Shape recognition and classification in electro-sensing

This paper aims at advancing the field of electro-sensing. It exhibits the physical mechanism underlying shape perception for weakly electric fish. These fish orient themselves at night in complete darkness by employing their active electrolocation system. They generate a stable, high-frequency, weak electric field and perceive the transdermal potential modulations caused by a nearby target with different admittivity than the surrounding water. In this paper, we explain how weakly electric fish might identify and classify a target, knowing by advance that the latter belongs to a certain collection of shapes. Our model of the weakly electric fish relies on differential imaging, i.e., by forming an image from the perturbations of the field due to targets, and physics-based classification. The electric fish would first locate the target using a specific location search algorithm. Then it could extract, from the perturbations of the electric field, generalized (or high-order) polarization tensors of the target. Computing, from the extracted features, invariants under rigid motions and scaling yields shape descriptors. The weakly electric fish might classify a target by comparing its invariants with those of a set of learned shapes. On the other hand, when measurements are taken at multiple frequencies, the fish might exploit the shifts and use the spectral content of the generalized polarization tensors to dramatically improve the stability with respect to measurement noise of the classification procedure in electro-sensing. Surprisingly, it turns out that the first-order polarization tensor at multiple frequencies could be enough for the purpose of classification. A procedure to eliminate the background field in the case where the permittivity of the surrounding medium can be neglected, and hence improve further the stability of the classification process, is also discussed.Comment: 10 pages, 15 figure

Efficient computation of regular differential systems by change of rankings using Kähler differentials

We present two algorithms to compute a regular differential system for some ranking, given an equivalent regular differential system for another ranking. Both make use of Kähler differentials. One of them is a lifting for differential algebra of the FGLM algorithm and relies on normal forms computations of differential polynomials and of Kähler differentials modulo differential relations. Both are implemented in MAPLE V. A straightforward adaptation of FGLM for systems of linear PDE is presented too. Examples are treated

Differential Elimination and Biological Modelling

International audienceThis paper describes applications of a computer algebra method, differential elimination, to applied mathematics problems mostly borrowed from biology. The two considered applications are related to the parameters estimation and the model reduction problems. In both cases, differential elimination can be viewed as a preparation to numerical treatments. Together with the applications, the paper introduces two implementations of the differential elimination algorithms: the diffalg package and the BLAD libraries

Alien Registration- Boulier, Joseph (Caribou, Aroostook County)

https://digitalmaine.com/alien_docs/26005/thumbnail.jp

A Differential Algebra Introduction For Tropical Differential Geometry

Doctora

On the role of differential algebra in biological modeling

Extended abstract of an invited talk at Differential Algebra and related Computer Algebra (Catania, Italy, March 28th, 2008)International audienceDifferential algebra is an algebraic theory for studying systems of polynomial ordinary differential equations (ODE). Among all the methods developed for system modeling in cellular biology, it is particularly related to the well-established approach based on nonlinear ODE. A subtheory of the differential algebra, the differential elimination, has proved to be useful in the parameters estimation problem. It seems however still more promising in the quasi-steady state approximation theory, recent results show

Measuring Consensus in Binary Forecasts: NFL Game Predictions

Previous research on defining and measuring consensus (agreement) among forecasters has been concerned with evaluation of forecasts of continuous variables. This previous work is not relevant when the forecasts involve binary decisions: up-down or win-lose. In this paper we use Cohen¡¯s kappa coefficient, a measure of inter-rater agreement involving binary choices, to evaluate forecasts of National Football League games. This statistic is applied to the forecasts of 74 experts and 31 statistical systems that predicted the outcomes of games during two NFL seasons. We conclude that the forecasters, particularly the systems, displayed significant levels of agreement and that levels of agreement in picking game winners were higher than in picking against the betting line. There is greater agreement among statistical systems in picking game winners or picking winners against the line as the season progresses, but no change in levels of agreement among experts. High levels of consensus among forecasters are associated with greater accuracy in picking game winners, but not in picking against the line.binary forecasts, NFL, agreement, consensus, kappa coefficient