R. Vilela Mendes

Non-commutative time-frequency tomography

Abstract The characterization of non-stationary signals requires joint time and frequency information. However, time ( t ) and frequency ( ω ) being non-commuting variables there cannot be a joint probability density in the ( t , ω ) plane and the time-frequency distributions, that have been proposed, have difficult interpretation problems arising from negative or complex values and spurious components. As an alternative, time-frequency information may be obtained by looking at the marginal distributions along rotated directions in the ( t , ω ) plane. The rigorous probability interpretation of the marginal distributions avoids all interpretation ambiguities. Applications to signal analysis and signal detection are discussed as well as an extension of the method to other pairs of non-commuting variables.

Lyapunov exponent in quantum mechanics a phase-space approach

Abstract Using the symplectic tomography map, both for the probability distributions in classical phase-space and for the Wigner functions of its quantum counterpart, we discuss a notion of Lyapunov exponent for quantum dynamics. Because the marginal distributions, obtained by the tomography map, are always well-defined probabilities, the correspondence between classical and quantum notions is very clear. Then we also obtain the corresponding expressions in Hilbert space. Some examples are worked out. Classical and quantum exponents are seen to coincide for local and non-local time-dependent quadratic potentials. For non-quadratic potentials classical and quantum exponents are different and some insight is obtained on the taming effect of quantum mechanics on classical chaos. A detailed analysis is made for the standard map. Providing an unambiguous extension of the notion of Lyapunov exponent to quantum mechanics, the method that is developed is also computationally efficient in obtaining analytical results for the Lyapunov exponent, both classical and quantum.

Tomograms and other transforms: a unified view

A general framework is presented which unifies the treatment of wavelet-like, quasidistribution and tomographic transforms. Explicit formulae relating the three types of transforms are obtained. The case of transforms associated with the symplectic and affine groups is treated in some detail. Special emphasis is given to the properties of the scale–time and scale–frequency tomograms. Tomograms are interpreted as a tool to sample the signal space by a family of curves or as the matrix element of a projector.

Hierarchical structures and asymmetric stochastic processes on p-adics and adeles

The space of states of some phenomena, in physics and other sciences, displays a hierarchical structure. When that is the case, it is natural to label the states by a p‐adic number field. Both the classification of the states and their relationships are then based on a notion of distance with ultrametric properties. The dynamics of the phenomena, that is, the transition between different states, is also a function of the p‐adic distance dp. However, because the distance is a symmetric function, probabilistic processes which depend only on dp have a uniform invariant probability measure, that is, all states are equally probable at large times. This being a severe limitation for cases of physical interest, processes with asymmetric transition functions have been studied. In addition to the dependence on the ultrametric distance, the asymmetric transition functions are allowed to depend also on the probability of the target state, leading to any desired invariant probability measure. When each state of a phy...

A fractional calculus interpretation of the fractional volatility model

Based on criteria of mathematical simplicity and consistency with empirical market data, a model with volatility driven by fractional noise has been constructed which provides a fairly accurate mathematical parametrization of the data. Here, the model is formulated in terms of a fractional integration of stochastic processes.

Language identification of controlled systems: modeling, control, and anomaly detection

Formal language techniques have been used in the past to study autonomous dynamical systems. However, for controlled systems, new features are needed to distinguish between information generated by the system and input control. We show how the modeling framework for controlled dynamical systems leads naturally to a formulation in terms of context-dependent grammars. A learning algorithm is proposed for online generation of the grammar productions, this formulation then being used for modeling, control and anomaly detection. Practical applications are described for electromechanical drives. Grammatical interpolation techniques yield accurate results, and the pattern detection capabilities of the language-based formulation makes it a promising technique for the early detection of anomalies or faulty behavior.

Quantum computation by quantumlike systems

Abstract Using a quantumlike description for light propagation in nonhomogeneous optical fibers, quantum information processing can be implemented by optical means. Quantumlike bits (qulbits) are associated to light modes in the optical fiber and quantum gates to segments of the fiber providing an unitary transformation of the mode structure along a space direction. Simulation of nonlinear quantum effects is also discussed.

Geometry, stochastic calculus, and quantum fields in a noncommutative space–time

The algebras of nonrelativistic and of classical mechanics are unstable algebraic structures. Their deformation towards stable structures leads, respectively, to relativity and to quantum mechanics. Likewise, the combined relativistic quantum mechanics algebra is also unstable. Its stabilization requires the noncommutativity of the space–time coordinates and the existence of a fundamental length constant. The new relativistic quantum mechanics algebra has important consequences on the geometry of space–time, on quantum stochastic calculus, and on the construction of quantum fields. Some of these effects are studied in this paper.

Non-commutative space–time and the uncertainty principle

Abstract The full algebra of relativistic quantum mechanics (Lorentz plus Heisenberg) is unstable. Stabilization by deformation leads to a new deformation parameter e l 2 , l being a length and e a ± sign. The implications of the deformed algebras for the uncertainty principle and the density of states are worked out and compared with the results of past analysis following from gravity and string theory.

Conditional exponents, entropies and a measure of dynamical self-organization

Abstract In dynamical systems composed of interacting parts, conditional exponents, conditional exponent entropies and cylindrical entropies are shown to be well-defined ergodic invariants which characterize the dynamical self-organization and statistical independence of the constituent parts. An example of interacting Bernoulli units is used to illustrate the nature of these invariants.