LisMath Seminar 

The word problem, the rational word problem and the regular idempotent problem for inverse semigroups will be discussed.

Bibliography

1. Brough, T. Inverse semigroups with rational word problem are finite, arXiv:1311.3955
2. Kambits, M. Anisimov's theorem for inverse semigroups, IJAC, to appear.
3. Lawson, M. V. Inverse semigroups, the theory of partial symmetries,World Scientific, 1998.
4. Munn, W.D. Free inverse semigroups, Proc. London Math. Soc. 29 (3), 385-404, 1974

We will present a family of time reversible stochastic processes known (among other names) as Bernstein processes [1,2]. These processes are much closer, in their properties, to the solutions of deterministic (Lagrangian or Hamiltonian) dynamical equations, they are in fact absolutely continuous with respect to the Wiener process, i.e. they are processes with a drift term. The relevant Feynman-Kac formula for this class of processes will be proved. Some particular cases of the formula will be given as examples: the well-known Feynman-Kac formula and Doob's relation between Wiener and Ornstein-Uhlenbeck processes [3,4].

The consequences of this probabilistic perturbation theory and the underlying time reversible processes should go beyond stochastic analysis. We will give a hint of two applications, one motivated by mathematical quantum physics and the other by a stochastic version of geometric mechanics.

The first concerns a rigorous probabilistic interpretation of Feynman informal perturbation theory in his Path Integral approach [7], which needs the development of a rigorous integration by parts formula, also inspired by Feynman, done in terms of special reversible probability measures on path space [6]. Furthermore, to investigate the consequences of this probabilistic perturbation theory in the more general context of stochasttic analysis.

The second application concerns Geometric Mechanics, that in the recent years started to investigate perturbations under random noise, preserving as much as possible the geometric content. Roughly, what has been done by this community up to now is to add noise to the Hamiltonian equation for the momentum (i.e random force) [8], or add noise to the configuration (i.e the trajectories are random processes) [5]. This goal is close to the one of stochastic perturbation, whose foundations lie in the dynamics of the above mentioned Bernstein processes, e.g. [10]. In this context, a stochastic Euler-Poincaré reduction has been recently proved [7]: the corresponding equations of motion are dissipative perturbations of Hamiltonian systems, and should be relevant, in particular, in Hydrodynamics. So far the main application regards the Navier-Stokes equations (as perturbations of Euler equations) but many other systems can be considered and studied in this perspective. This study is still in its beginnings and many mathematical questions remain to be solved.

Bibliography

1. J. C. Zambrini, Variational processes and stochastic versions of mechanics, Journal of Math. Physics 27, p. 2307-2330 (1986).
2. Albeverio, Yasue and Zambrini, Euclidean quantum mechanics: analytical approach Annales de l'I.H.P., section A, tome 50, no3 (1989), p. 259-308.
3. P. Lescot, J. C. Zambrini. Probabilistic deformation of contact geometry, difusion processes and their quadratures. Progress in Probability 59, Birkhauser Verlag Basel (2007), 203-226.
4. A. B. Cruzeiro and J. C. Zambrini. Ornstein-Uhlenbeck processes as Bernstein difusions, Proceedings of the Conference on Stochastic Analysis (Barcelona), Birkhauser, Boston, Inc. (1993) (P.P. no 32).
5. J. C. Zambrini, The research program of Stochastic Deformation (with a view toward Geometric Mechanics).
6. A. B. Cruzeiro and J. C. Zambrini, Malliavin Calculus and Euclidean Quantum Mechanics, 1. Functional calculus, Journal of Functional Analysis,91,1,p.62 (1991).
7. Richard P. Feynman, Albert R. Hibbs. Quantum Mechanics and Path Integrals: Emended Edition. Dover Publications, Incorporated, 2012.
8. A. B. Cruzeiro, M. Arnaudon and X. Chen, Stochastic Euler-Poincaré reduction.
9. Joan-Andreu, Lazaro-Cami, Juan-Pablo Ortega. Stochastic Hamiltonian dynamical systems.
10. Fernanda Cipriano, A Stochastic Variational Principle for Burgers Equation and its Symmetries. Stochastic Analysis and Mathematical Physics II, 4th International ANESTOC Workshop in Santiago, Chile. Birkhauser, R. Rebolledo (2003), p.29..

Bibliography

1. Michael Rathjen, arXiv:1405.4481v1

Computational Fluid Dynamics has become an indispensable tool in the scientific research as well as in the modern engineering design and development. The increasing computer power during the last decades allowed for more and more detailed numerical simulations, which deepened the insight and understanding of physically highly complex flow problems involving complicated geometries, chemical reactions, phase change, etc.

Nanofluid, which is a mixture of nano-sized particles (nanoparticles) suspended in a base fluid, is used to enhance the rate of heat transfer via its higher thermal conductivity compared to the base fluid. Due to this enhancement of the material properties, nanofluids are widely used in our life such as in transportation (engine cooling and vehicle thermal management), electronics cooling, nuclear systems cooling, heat exchanger, biomedicine, heat pipes, fuel cell, Solar water heating, etc.

In this seminar, we shall present some fluid models, following [1] and [2], and then we will discuss applications to nanofluids, based on references [3], [4], and [5].

Bibliography

1. D.J Acheson, Elementary fluid dynamics, Oxford University Press (1990).
2. W. Layton, Introduction to the numerical analysis of incompressible viscous flows, SIAM, Philadelphia (2008).H.M. Elshehabey, F.M. Hady,
3. S.E. Ahmed, R.A. Mohamed, Numerical investigation for natural convection of a nanofluid in an inclined L-shaped cavity in the presence of an inclined magnetic field. International Communications in Heat and Mass Transfer: (2014) 57, 228-238.
4. F.M. Hady, S.E. Ahmed, H.M. Elshehabey, R.A.Mohamed, Natural Convection of a Nanofluid in Inclined, Partially Open Cavities: Thermal Effects. Journal of Thermophysics and Heat Transfer, (2015) (in press).
5. H.M. Elshehabey, S.E. Ahmed, Buongiorno's mathematical model for MHD mixed convection of a fluid driven cavity with sinusoidal temperature distributions on both sides filled with nanofluid, (2014) (submitted).

Vapnik-Chervonenkis dimension is a measure of complexity of a family of sets. Informally, it measures how many subsets of an arbitrary finite set the family can recognise. VC-theory was developed as a foundation for statistical learning theory in the early 1970's, and still provides the theoretical basis for the classical approach to classification problems in machine learning. At heart of VC-theory lies a simple but surprising combinatorial lemma, that was proved independently and simultaneously in three different papers [1],[2],[3].

Model theory provides a different perspective on VC-theory. The compactness theorem allows to investigate asymptotic behaviour of finite sets via infinite sets with a particularly nice homogeneous structure (indiscernible sequences). This allowed Shelah to prove in his original paper [3] results that were thought by combinatorists to be open for over ten years afterwards (the connection between Shelah's paper [3] and the work [1],[2] was only made in mid-80's).

The goal of the seminar would be to introduce the basics of VC-theory, and to explain the connection between this subject and model theory. The seminar will follow mostly a recent account [4].

Bibiography

1. V. Vapnik and A. Chervonenkis. "On the uniform convergence of relative frequencies of events to their probabilities." Theory of Probability and its Applications, 16(2):264–280, 1971.

2. Shelah, Saharon, "A combinatorial problem; stability and order for models and theories in infinitary languages", Pacific Journal of Mathematics 41: 247–26, 1972.

3. Sauer, N., "On the density of families of sets", Journal of Combinatorial Theory, Series A 13: 145–147, 1972.

4. Hans Adler, "An introduction to theories without the independence property", Archive for Math Logic, to appear.

Bibliography

1. Marvin Greenberg, The American Mathematical Monthly, vol. 117, no. 3, March 2010, pp. 198-219
2. Michael Rathjen, arXiv:1405.4481v1, section 3

Bibliography

1. S. Hawking, The occurrence of singularities in cosmology. iii. causality and singularities, Proc. Roy. Soc. Lon. A 300 (1967), 187–201
2. G. Naber, Spacetime and singularities – an introduction, Cambridge University Press, 1988
3. L. Godinho and J. Natário, An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity (Universitext), Springer, 2014