On the Lamb vector and the hydrodynamic charge

This work is an attempt to test the concept of the hydrodynamic charge (analogous to the electric charge in electromagnetism) in the simple case of a coherent structure such as the Burgers vortex. We provide experimental measurements of both the so-called Lamb vector and its divergence (the charge) by two-dimensional particles images velocimetry. In addition, we perform a Helmholtz–Hodge decomposition of the Lamb vector in order to explore its topological features. We compare the charge with the well-known Q-criterion in order to assess its interest in detecting and characterizing coherent structure. Usefulness of this concept in studies of vortex dynamics is demonstrated.

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Obtaining fast error rates in nonconvex situations

We show that under mild assumptions on the learning problem, one can obtain a fast error rate for every reasonable fixed target function even if the base class is not convex. To that end, we show that in such cases the excess loss class satisfies a Bernstein type condition.     

Empirical minimization

We investigate the behavior of the empirical minimization algorithm using various methods. We first analyze it by comparing the empirical, random, structure and the original one on the class, either in an additive sense, via the uniform law of large numbers, or in a multiplicative sense, using isomorphic coordinate projections. We then show that a direct analysis of the empirical minimization algorithm yields a significantly better bound, and that the estimates we obtain are essentially sharp. The method of proof we use is based on Talagrand's concentration inequality for empirical processes. Research partially supported by NSF under award DMS-0434393. Research partially supported by the Australian Research Council Discovery Porject DP0343616.     

The Proalgebraic Completion of Rigid Groups

A finitely generated group is called representation rigid (briefly, rigid) if for every n, has only finitely many classes of simple representations in dimension n. Examples include higher rank S-arithmetic groups. By Margulis super rigidity, the latter have a stronger property: they are representation super rigid; i.e., their proalgebraic completion is finite dimensional. We construct examples of nonlinear rigid groups which are not super rigid, and which exhibit every possible type of infinite dimensionality. Whether linear representation rigid groups are super rigid remains an open question.     

Mixing of all orders of Lie groups actions

Summary We show that for Lie groups whose adjoint representation reflects their topology, mixing implies mixing of all orders. In particular we prove that mixing actions of a semisimple Lie group are mixing of all orders, answering a conjecture of B. Marcus.Oblatum 6-VIII-1990 & 4-IV-1991Sponsored in part by the Edmund Landau Center for research in Mathematical Analysis supported by the Minerva Foundation (Federal Republic of Germany)     

Automata and Square Complexes

We introduce a new geometric tool for analyzing groups of finite automata. To each finite automaton we associate a square complex. The square complex is covered by a product of two trees iff the automaton is bi-reversible. Using this method we give examples of free groups and of Kazhdan groups which are generated by the different states of one finite (bi-reversible) automaton. We also reprove the theorem of Macedońska, Nekrashevych, Sushchansky, on the connection between bi-reversible automata and the commensurator of a regular tree.