Simulating the damped vibrations of a fractional oscillator with fuzzy initial conditions.

A Picard-like scheme using quadrature and differential quadrature rules, formerly introduced to solve integro-differential equations, is herein adapted to solve the problem of an oscillator with damping defined by the Riemann- Liouville fractional derivative and with fuzzy initial conditions. Considering fuzzy initial conditions has the meaning of a fuzzification of the problem via the Zadeh’s extension principle. Following Zadeh, fuzziness is a way to take into account an uncertainty which cannot be identified as randomness. In the crisp domain, the proposed approach is able to approximate the reference analytical solutions with high accuracy and a relatively low computational cost. In the linear regime, the technique proposed becomes a non-recursive scheme, providing semi-analytical solutions by means of operational matrices and vectors of known quantities. In this sense, an example of application is given by the free damped vibrations of a linear oscillator in a medium with small viscosity, usually solved by using the method of multiple scales (in the crisp domain)

A Note On Asymptotic Smoothness Of The Extensions Of Zadeh

The concept of asymptotic smooth transformation was introduced by J. Hale [10]. It is a very important property for a transformation between complete metric spaces to have a global attractor. This property has also consequences on asymptotic stability of attractors. In our work we study the conditions under which the Zadeh's extension of a continuous map f : R n → R n is asymptotically smooth in the complete metric space JF(R n) of normal fuzzy sets with the induced Hausdorff metric d ∞ (see Kloeden and Diamond [8]).212141153Barros, L.C., Bassanezi, R.C., Tonelli, P.A., On the continuity of Zadeh's extension (1997) Proceedings Seventh IFSA World Congress, 2, pp. 3-8. , PragueBarros, L.C., Bassanezi, R.C., Tonelli, P.A., Fuzzy modeling in populations dynamics (2000) Ecological Modeling, 128, pp. 27-33Brumley, W.E., On the asymptotic behavior of solutions of differential difference equations of neutral type (1970) J. of Differential Equations, 7, pp. 175-188Cabrelli, C.A., Forte, B., Molter, U., Vrscay, E., Iterated Fuzzy Sets Systems: A new approach to the inverse for fractals and other sets (1992) J. of Math. Anal, and Appl., 171, pp. 79-100Cooperman, G., (1978) α-Condensing Maps and Dissipative Processes, , Ph. D. Thesis, Brown University, Providence, R. IDiamond, P., Chaos in iterated fuzzy systems (1994) J. of Mathematical Analysis and Applications, 184, pp. 472-484Diamond, P., Time Dependent Differential Inclusions, Cocycle Attractors and Fuzzy Differential Equations (1999) IEEE Trans. on Fuzzy Systems, 7, pp. 734-740Diamond, P., Kloeden, P., (1994) Metric Spaces of Fuzzy Sets: Theory and Applications, , World Scientific PubFriedmann, M., Ma, M., Kandel, A., Numerical solutions of fuzzy differential and integral equations (1999) Fuzzy Sets and Systems, 106, pp. 35-48Hale, J.K., Asymptotic Behavior of Dissipative Systems (1988) Math. Surveys and Monographs, 25. , American Mathematical Society, ProvidenceHüllermeier, E., An Approach to Modeling and Simulation of Uncertain Dynamical Systems (1997) J. Uncertainty, Fuzziness, Know Ledge-Bases Syst., 5, pp. 117-137Kloeden, P.E., Fuzzy dynamical systems (1982) Fuzzy Sets and Systems, 7, pp. 275-296Kloeden, P.E., Chaotic iterations of fuzzy sets (1991) Fuzzy Sets and Systems, 42, pp. 37-42Nguyen, H.T., A note on thé extension principle for fuzzy sets (1978) J. Math. Anal. Appl., 64, pp. 369-380Puri, M.L., Ralescu, D.A., Fuzzy Random Variables (1986) J. of Mathematical Analysis and Applications, 114, pp. 409-422Roman-Flores, H., Barros, L.C., Bassanezzi, R., A note on Zadeh's Extensions (2001) Fuzzy Sets and Systems, 117, pp. 327-331Roman-Flores, H., On the Compactness of E(X) (1998) Appl. Math. Lett., 11, pp. 13-17Zadeh, L.A., Fuzzy sets (1965) Inform. Control, 8, pp. 338-35

Quantum mechanics on non commutative spaces and squeezed states: a functional approach

We review here the quantum mechanics of some noncommutative theories in which no state saturates simultaneously all the non trivial Heisenberg uncertainty relations. We show how the difference of structure between the Poisson brackets and the commutators in these theories generically leads to a harmonic oscillator whose positions and momenta mean values are not strictly equal to the ones predicted by classical mechanics. This raises the question of the nature of quasi classical states in these models. We propose an extension based on a variational principle. The action considered is the sum of the absolute values of the expressions associated to the non trivial Heisenberg uncertainty relations. We first verify that our proposal works in the usual theory i.e we recover the known Gaussian functions. Besides them, we find other states which can be expressed as products of Gaussians with specific hyper geometrics. We illustrate our construction in two models defined on a four dimensional phase space: a model endowed with a minimal length uncertainty and the non commutative plane. Our proposal leads to second order partial differential equations. We find analytical solutions in specific cases. We briefly discuss how our proposal may be applied to the fuzzy sphere and analyze its shortcomings.Comment: 15 pages revtex. The title has been modified,the paper shortened and misprints have been corrected. Version to appear in JHE

Determination of economic systems behaviour under uncertainty

The paper discuses systems of difference equations with fuzzy parameters and presents some solution procedures with the purpose to study the dynamic behaviour of economic systems in case of uncertainty. The trajectories of the endogenous variables are evaluated firstly at contiguous moments of time, and then, simultaneously. The relations between different solutions are shown. The author also consider essential to provide an algorithm for computing the exact α-cuts of the obtained solution

Tracking uncertainty in a spatially explicit susceptible-infected epidemic model

In this paper we conceive an interval-valued continuous cellular automaton for describing the spatio-temporal dynamics of an epidemic, in which the magnitude of the initial outbreak and/or the epidemic properties are only imprecisely known. In contrast to well-established approaches that rely on probability distributions for keeping track of the uncertainty in spatio-temporal models, we resort to an interval representation of uncertainty. Such an approach lowers the amount of computing power that is needed to run model simulations, and reduces the need for data that are indispensable for constructing the probability distributions upon which other paradigms are based

Branes, Quantization and Fuzzy Spheres

We propose generalized quantization axioms for Nambu-Poisson manifolds, which allow for a geometric interpretation of n-Lie algebras and their enveloping algebras. We illustrate these axioms by describing extensions of Berezin-Toeplitz quantization to produce various examples of quantum spaces of relevance to the dynamics of M-branes, such as fuzzy spheres in diverse dimensions. We briefly describe preliminary steps towards making the notion of quantized 2-plectic manifolds rigorous by extending the groupoid approach to quantization of symplectic manifolds.Comment: 18 pages; Based on Review Talk at the Workshop on "Noncommutative Field Theory and Gravity", Corfu Summer Institute on Elementary Particles and Physics, September 8-12, 2010, Corfu, Greece; to be published in Proceedings of Scienc