Un método para sacar raíces cuadradas exactas

En este art\'\i culo se presenta, con una gran variedad de ejemplos, unm\'etodo para sacar ra\'\i ces cuadradas exactas. Este m\'etodo se present\'opor primera vez hace 15 a\~nos con el nombre de ley Costeana, pero adiferencia de ahora se enfatiza en el hecho que puede ser implementadoen el curso de cuarto de primaria, al cual asiste la autora (primer autor)de este articulo.

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity

In two previous papers [J. Differential Equations, 228 (2006), pp. 530 579; Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261 1300] we have developed fast algorithms for the computations of invariant tori in quasi‐periodic systems and developed theorems that assess their accuracy. In this paper, we study the results of implementing these algorithms and study their performance in actual implementations. More importantly, we note that, due to the speed of the algorithms and the theoretical developments about their reliability, we can compute with confidence invariant objects close to the breakdown of their hyperbolicity properties. This allows us to identify a mechanism of loss of hyperbolicity and measure some of its quantitative regularities. We find that some systems lose hyperbolicity because the stable and unstable bundles approach each other but the Lyapunov multipliers remain away from 1. We find empirically that, close to the breakdown, the distances between the invariant bundles and the Lyapunov multipliers which are natural measures of hyperbolicity depend on the parameters, with power laws with universal exponents. We also observe that, even if the rigorous justifications in [J. Differential Equations, 228 (2006), pp. 530-579] are developed only for hyperbolic tori, the algorithms work also for elliptic tori in Hamiltonian systems. We can continue these tori and also compute some bifurcations at resonance which may lead to the existence of hyperbolic tori with nonorientable bundles. We compute manifolds tangent to nonorientable bundles.

Sierpinski curve Julia sets for quadratic rational maps

We investigate under which dynamical conditions the Julia set of a quadratic rational map is a Sierpiński curve.

Persistent random walk model for transport trough thin slabs

We present a model for transport in multiply scattering media based on a three-dimensional generalization of the persistent random walk. The model assumes that photons move along directions that are parallel to the axes. Although this hypothesis is not realistic, it allows us to solve exactly the problem of multiple scattering propagation in a thin slab. Among other quantities, the transmission probability and the mean transmission time can be calculated exactly. Besides being completely solvable, the model could be used as a benchmark for approximation schemes to multiple light scattering.

SQualTrack: A Tool for Robust Fault Detection

One of the techniques used to detect faults in dynamic systems is analytical redundancy. An important difficulty in applying this technique to real systems is dealing with the uncertainties associated with the system itself and with the measurements. In this paper, this uncertainty is taken into account by the use of intervals for the parameters of the model and for the measurements. The method that is proposed in this paper checks the consistency between the system's behavior, obtained from the measurements, and the model's behavior; if they are inconsistent, then there is a fault. The problem of detecting faults is stated as a quantified real constraint satisfaction problem, which can be solved using the modal interval analysis (MIA). MIA is used because it provides powerful tools to extend the calculations over real functions to intervals. To improve the results of the detection of the faults, the simultaneous use of several sliding time windows is proposed. The result of implementing this method is semiqualitative tracking (SQualTrack), a fault-detection tool that is robust in the sense that it does not generate false alarms, i.e., if there are false alarms, they indicate either that the interval model does not represent the system adequately or that the interval measurements do not represent the true values of the variables adequately. SQualTrack is currently being used to detect faults in real processes. Some of these applications using real data have been developed within the European project advanced decision support system for chemical/petrochemical manufacturing processes and are also described in this paper

Full instability behavior of N-dimensional dynamical systems with a one-directional nonlinear vector field

We show how certain N-dimensional dynamical systems are able to exploit the full instability capabilities of their fixed points to do Hopf bifurcations and how such a behavior produces complex time evolutions based on the nonlinear combination of the oscillation modes that emerged from these bifurcations. For really different oscillation frequencies, the evolutions describe robust wave form structures, usually periodic, in which selfsimilarity with respect to both the time scale and system dimension is clearly appreciated. For closer frequencies, the evolution signals usually appear irregular but are still based on the repetition of complex wave form structures. The study is developed by considering vector fields with a scalar-valued nonlinear function of a single variable that is a linear combination of the N dynamical variables. In this case, the linear stability analysis can be used to design N-dimensional systems in which the fixed points of a saddle-node pair experience up to N21 Hopf bifurcations with preselected oscillation frequencies. The secondary processes occurring in the phase region where the variety of limit cycles appear may be rather complex and difficult to characterize, but they produce the nonlinear mixing of oscillation modes with relatively generic features

Chaotic, memory, and cooling rate effects in spin glasses: Evaluation of the Edwards-Anderson model

We investigate chaotic, memory, and cooling rate effects in the three-dimensional Edwards-Anderson model by doing thermoremanent (TRM) and ac susceptibility numerical experiments and making a detailed comparison with laboratory experiments on spin glasses. In contrast to the experiments, the Edwards-Anderson model does not show any trace of reinitialization processes in temperature change experiments (TRM or ac). A detailed comparison with ac relaxation experiments in the presence of dc magnetic field or coupling distribution perturbations reveals that the absence of chaotic effects in the Edwards-Anderson model is a consequence of the presence of strong cooling rate effects. We discuss possible solutions to this discrepancy, in particular the smallness of the time scales reached in numerical experiments, but we also question the validity of the Edwards-Anderson model to reproduce the experimental results.

Robust fault detection using consistency techniques for uncertainty handling

Often practical performance of analytical redundancy for fault detection and diagnosis is decreased by uncertainties prevailing not only in the system model, but also in the measurements. In this paper, the problem of fault detection is stated as a constraint satisfaction problem over continuous domains with a big number of variables and constraints. This problem can be solved using modal interval analysis and consistency techniques. Consistency techniques are then shown to be particularly efficient to check the consistency of the analytical redundancy relations (ARRs), dealing with uncertain measurements and parameters. Through the work presented in this paper, it can be observed that consistency techniques can be used to increase the performance of a robust fault detection tool, which is based on interval arithmetic. The proposed method is illustrated using a nonlinear dynamic model of a hydraulic system

SQualTrack: A Tool for Robust Fault Detection

One of the techniques used to detect faults in dynamic systems is analytical redundancy. An important difficulty in applying this technique to real systems is dealing with the uncertainties associated with the system itself and with the measurements. In this paper, this uncertainty is taken into account by the use of intervals for the parameters of the model and for the measurements. The method that is proposed in this paper checks the consistency between the system's behavior, obtained from the measurements, and the model's behavior; if they are inconsistent, then there is a fault. The problem of detecting faults is stated as a quantified real constraint satisfaction problem, which can be solved using the modal interval analysis (MIA). MIA is used because it provides powerful tools to extend the calculations over real functions to intervals. To improve the results of the detection of the faults, the simultaneous use of several sliding time windows is proposed. The result of implementing this method is semiqualitative tracking (SQualTrack), a fault-detection tool that is robust in the sense that it does not generate false alarms, i.e., if there are false alarms, they indicate either that the interval model does not represent the system adequately or that the interval measurements do not represent the true values of the variables adequately. SQualTrack is currently being used to detect faults in real processes. Some of these applications using real data have been developed within the European project advanced decision support system for chemical/petrochemical manufacturing processes and are also described in this paper

Constraint satisfaction techniques under uncertain conditions for fault diagnosis in nonlinear dynamic systems

The speed of fault isolation is crucial for the design and reconfiguration of fault tolerant control (FTC). In this paper the fault isolation problem is stated as a constraint satisfaction problem (CSP) and solved using constraint propagation techniques. The proposed method is based on constraint satisfaction techniques and uncertainty space refining of interval parameters. In comparison with other approaches based on adaptive observers, the major advantage of the presented method is that the isolation speed is fast even taking into account uncertainty in parameters, measurements and model errors and without the monotonicity assumption. In order to illustrate the proposed approach, a case study of a nonlinear dynamic system is presented

Fault diagnosis by refining the parameter uncertainty space of nonlinear dynamic systems

This paper deals with fault detection and isolation problems for nonlinear dynamic systems. Both problems are stated as constraint satisfaction problems (CSP) and solved using consistency techniques. The main contribution is the isolation method based on consistency techniques and uncertainty space refining of interval parameters. The major advantage of this method is that the isolation speed is fast even taking into account uncertainty in parameters, measurements, and model errors. Interval calculations bring independence from the assumption of monotony considered by several approaches for fault isolation which are based on observers. An application to a well known alcoholic fermentation process model is presented

An Interval Intelligent-based Approach for Fault Detection and Modelling

Not considered in the analytical model of the plant, uncertainties always dramatically decrease the performance of the fault detection task in the practice. To cope better with this prevalent problem, in this paper we develop a methodology using Modal Interval Analysis which takes into account those uncertainties in the plant model. A fault detection method is developed based on this model which is quite robust to uncertainty and results in no false alarm. As soon as a fault is detected, an ANFIS model is trained in online to capture the major behavior of the occurred fault which can be used for fault accommodation. The simulation results understandably demonstrate the capability of the proposed method for accomplishing both tasks appropriately

Necessary and sufficient conditions for robust stability using modal intervals

In this paper, robustness of parametric systems is analyzed using a new approach to interval mathematics called Modal Interval Analysis. Modal Intervals are an interval extension that, instead of classic intervals, recovers some of the properties required by a numerical system. Modal Interval Analysis not only simplifies the computation of interval functions but allows semantic interpretation of their results. Necessary, sufficient and, in some cases, necessary and sufficient conditions for robust performance are presented

Application of modal interval analysis to the simulation of the behaviour of dynamic systems with uncertain parameters

Els models matemàtics quantitatius són simplificacions de la realitat i per tant el comportament obtingut per simulació d'aquests models difereix dels reals. L'ús de models quantitatius complexes no és una solució perquè en la majoria dels casos hi ha alguna incertesa en el sistema real que no pot ser representada amb aquests models. Una forma de representar aquesta incertesa és mitjançant models qualitatius o semiqualitatius. Un model d'aquest tipus de fet representa un conjunt de models. La simulació del comportament de models quantitatius genera una trajectòria en el temps per a cada variable de sortida. Aquest no pot ser el resultat de la simulació d'un conjunt de models. Una forma de representar el comportament en aquest cas és mitjançant envolupants. L'envolupant exacta és complete, és a dir, inclou tots els possibles comportaments del model, i correcta, és a dir, tots els punts dins de l'envolupant pertanyen a la sortida de, com a mínim, una instància del model. La generació d'una envolupant així normalment és una tasca molt dura que es pot abordar, per exemple, mitjançant algorismes d'optimització global o comprovació de consistència. Per aquesta raó, en molts casos s'obtenen aproximacions a l'envolupant exacta. Una aproximació completa però no correcta a l'envolupant exacta és una envolupant sobredimensionada, mentre que una envolupant correcta però no completa és subdimensionada. Aquestes propietats s'han estudiat per diferents simuladors per a sistemes incerts.