arXiv subjects

R. Vilela Mendes

Publications and source records attributed to R. Vilela Mendes.

At least 37 records · Page 2Linked to original sources

arXiv2019

Non-commutative probability and non-commutative processes

A probability space is a pair ($\mathcal{A},ϕ$) where $\mathcal{A}$ is an algebra and $ϕ$ a state on the algebra. In classical probability $\mathcal{A}$ is the algebra of linear combinations of indicator functions on the sample space and in quantum probability $\mathcal{A}$ is the Heisenberg or Clifford algebra. However, other algebras are of interest in non-commutative probability. Here one discusses some other non-commutative probability spaces, in particular those associated to non-commutative space-time.

math.PR↗

arXiv2019

Space-times over normed division algebras, revisited

Normed division and Clifford algebras have been extensively used in the past as a mathematical framework to accommodate the structures of the standard model and grand unified theories. Less discussed has been the question of why such algebraic structures appear in Nature. One possibility could be an intrinsic complex, quaternionic or octonionic nature of the spacetime manifold. Then, an obvious question is why spacetime appears nevertheless to be simply parametrized by the real numbers. How the real slices of an higher dimensional spacetime manifold might be almost independent from each other is discussed here. This comes about as a result of the different nature of the representations of the real kinematical groups and those of the extended spaces. Some of the internal symmetry transformations might however appear as representations on homogeneous spaces of the extended group transformations that cannot be implemented on the elementary states.

hep-th↗

arXiv2019

Stochastic stability of invariant measures: The 2D Euler equation

In finite-dimensional dynamical systems, stochastic stability provides the selection of physical relevant measures from the myriad invariant measures of conservative systems. That this might also apply to infinite-dimensional systems is the inspiration for this work. As an example the 2D Euler equation is studied. Among other results this study suggests that the coherent structures observed in 2D hydrodynamics are associated to configurations that maximize stochastically stable measures uniquely determined by the boundary conditions in mode space.

math.DS↗

arXiv2019

Space-time: Commutative or noncommutative ?

Noncommutativity of the spacetime coordinates has been explored in several contexts, mostly associated to phenomena at the Planck length scale. However, approaching this question through deformation theory and the principle of stability of physical theories, one concludes that the scales of noncommutativity of the coordinates and noncommutativity of the generators of translations are independent. This suggests that the scale of the spacetime coordinates noncommutativity could be larger than the Planck length. This paper attempts to explore the experimental perspectives to settle this question, either on the lab or by measurements of phenomena of cosmological origin.

hep-th↗

arXiv2018

Fractional networks, the new structure

Real world networks have, for a long time, been modelled by scale-free networks, which have many sparsely connected nodes and a few highly connected ones (the hubs). However, both in society and in biology, a new structure must be acknowledged, the fractional networks. These networks are characterized by the existence of very many long-range connections, display superdiffusion, Lévy flights and robustness properties different from the scale-free networks.

nlin.AO↗

arXiv2017

Current algebra, statistical mechanics and quantum models

Results obtained in the past for free boson systems at zero and nonzero temperature are revisited to clarify the physical meaning of current algebra reducible functionals which are associated to systems with density fluctuations, leading to observable effects on phase transitions. To use current algebra as a tool for the formulation of quantum statistical mechanics amounts to the construction of unitary representations of diffeomorphism groups. Two mathematical equivalent procedures exist for this purpose. One searches for quasi-invariant measures on configuration spaces, the other for a cyclic vector in Hilbert space. Here, one argues that the second approach is closer to the physical intuition when modelling complex systems. An example of application of the current algebra methodology to the pairing phenomenon in two-dimensional fermion systems is discussed.

quant-ph↗

arXiv2017

A consistent measure for lattice Yang-Mills

The construction of a consistent measure for Yang-Mills is a precondition for an accurate formulation of non-perturbative approaches to QCD, both analytical and numerical. Using projective limits as subsets of Cartesian products of homomorphisms from a lattice to the structure group, a consistent interaction measure and an infinite-dimensional calculus has been constructed for a theory of non-abelian generalized connections on a hypercubic lattice. Here, after reviewing and clarifying past work, new results are obtained for the mass gap when the structure group is compact.

hep-th↗

arXiv2017

The geometry of noncommutative space-time

Stabilization, by deformation, of the Poincaré-Heisenberg algebra requires both the introduction of a fundamental lentgh and the noncommutativity of translations which is associated to the gravitational field. The noncommutative geometry structure that follows from the deformed algebra is studied both for the non-commutative tangent space and the full space with gravity. The contact points of this approach with the work of David Finkelstein are emphasized.

math-ph↗

arXiv2017

Quantum control of quasi-collision states: A protocol for hybrid fusion

When confined to small regions quantum systems exhibit electronic and structural properties different from their free space behavior. These properties are of interest, for example, for molecular insertion, hydrogen storage and the exploration of new pathways for chemical and nuclear reactions. Here, a confined three-body problem is studied, with emphasis on the study of the "quantum scars" associated to dynamical collisions. For the particular case of nuclear reactions it is proposed that a molecular cage might simply be used as a confining device with the collision states accessed by quantum control techniques.

quant-ph↗

arXiv2016

Superprocesses on ultradistributions

Stochastic solutions provide new rigorous results for nonlinear PDE's and, through its local non-grid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions: McKean's and superprocesses. In favour of superprocesses is the fact that they handle arbitrary boundary conditions. However, when restricted to measures, superprocesses can only be used to generate solutions for a limited class of nonlinear PDE's. A new class of superprocesses, namely superprocesses on ultradistributions, is proposed to extend the stochastic solution approach to a wider class of PDE's.

math.PR↗

arXiv2015

Measure and mass gap for generalized connections on hypercubic lattices

Using projective limits as subsets of Cartesian products of homomorphisms from a lattice to the structure group, a consistent interaction measure and an infinite-dimensional calculus has been constructed for a theory of non-abelian generalized connections on a hypercubic lattice. Here, after reviewing and clarifying past work, new results are obtained for the mass gap when the structure group is compact.

math-ph↗

arXiv2015

An extended Dirac equation in noncommutative space-time

Stabilizing, by deformation, the algebra of relativistic quantum mechanics a non-commutative space-time geometry is obtained. The exterior algebra of this geometry leads to an extended massless Dirac equation which has both a massless and a large mass solution. The nature of the solutions is discussed, as well as the effects of coupling the two solutions.

quant-ph↗

arXiv2014

Signal processing on graphs: Transforms and tomograms

Using projections on the (generalized) eigenvectors associated to matrices that characterize the topological structure, several authors have constructed generalizations of the Fourier transform on graphs. By exploring mappings of the spectrum of these matrices we show how to construct more general transforms, in particular wavelet-like transforms on graphs. For time-series, tomograms, a generalization of the Radon transforms to arbitrary pairs of non-commuting operators, are positive bilinear transforms with a rigorous probabilistic interpretation which provide a full characterization of the signals and are robust in the presence of noise. Here the notion of tomogram transform is also extended to signals on arbitrary graphs

physics.data-an↗

arXiv2014

Long range dependence and the dynamics of exploited fish populations

Long range dependence or long memory is a feature of many processes in the natural world, which provides important insights on the underlying mechanisms that generate the observed data. The usual tools available to characterize the phenomenon are mostly based on second order correlations. However, the long memory effects may not be evident at the level of second order correlations and may require a deeper analysis of the nature of the stochastic processes. After a short review of the notions and tools used to characterize long range dependence, we analyse data related to the abundance of exploited fish populations which provides an example of higher order long range dependence.

q-bio.PE↗

arXiv2013

Learning ambiguous functions by neural networks

It is not, in general, possible to have access to all variables that determine the behavior of a system. Having identified a number of variables whose values can be accessed, there may still be hidden variables which influence the dynamics of the system. The result is model ambiguity in the sense that, for the same (or very similar) input values, different objective outputs should have been obtained. In addition, the degree of ambiguity may vary widely across the whole range of input values. Thus, to evaluate the accuracy of a model it is of utmost importance to create a method to obtain the degree of reliability of each output result. In this paper we present such a scheme composed of two coupled artificial neural networks: the first one being responsible for outputting the predicted value, whereas the other evaluates the reliability of the output, which is learned from the error values of the first one. As an illustration, the scheme is applied to a model for tracking slopes in a straw chamber and to a credit scoring model.

cs.NE↗

arXiv2012

Signal recognition and adapted filtering by non-commutative tomography

Tomograms, a generalization of the Radon transform to arbitrary pairs of non-commuting operators, are positive bilinear transforms with a rigorous probabilistic interpretation which provide a full characterization of the signal and are robust in the presence of noise. Tomograms based on the time-frequency operator pair, were used in the past for component separation and denoising. Here we show how, by the construction of an operator pair adapted to the signal, meaningful information with good time resolution is extracted even in very noisy situations.

physics.data-an↗

arXiv2012

The fractional volatility model: No-arbitrage, leverage and completeness

Based on a criterion of mathematical simplicity and consistency with empirical market data, a stochastic volatility model has been obtained with the volatility process driven by fractional noise. Depending on whether the stochasticity generators of log-price and volatility are independent or are the same, two versions of the model are obtained with different leverage behavior. Here, the no-arbitrage and completeness properties of the models are studied.

q-fin.ST↗

arXiv2012

Stochastic solutions of nonlinear pde's: McKean versus superprocesses

Stochastic solutions not only provide new rigorous results for nonlinear pde's but also, through its local non-grid nature, are a natural tool for parallel computation. There are two methods to construct stochastic solutions: the McKean method and superprocesses. Here a comparison is made of these two approaches and their strenghts and limitations are discussed.

math.PR↗