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## Seminars, Colloquia, and Conferences
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## Seminars, Colloquia, and Conferences## ColloquiumThe colloquium meets on ## Fall 2014
September 19, 2014
September 26, 2014 The seminar starts by reviewing experimental data to illustrate the large deformation stress-strain response of nonlinear elastic materials. This is followed by a summary of the main ingredients of the nonlinear theory of elasticity and of suitable strain-energy functions to describe the isotropic and anisotropic responses of highly deformable materials. The second part of the seminar focuses on the coupling of mechanical and magnetic effects and on the development of constitutive equations for magnetoelastic materials. These smart materials typically consist of an elastomeric matrix and a distribution of nanoscale ferromagnetic particles and have the capability to change their mechanical properties by the application of a magnetic field. We summarize the relevant equations and propose a coupled free-energy formulation, which depends on the deformation gradient and on the magnetic induction. Finally, we discuss how constitutive equations are specialized to isotropic incompressible magneto-sensitive elastomers in either Lagrangian or Eulerian forms. October 3, 2014 A gas can be considered either as a large system of microscopic interacting particles, or as a continuous medium governed by fluid equations. A natural question is therefore to understand whether both kinds of models give consistent predictions of the dynamics. October 24, 2014 The Maxwell-Boltzmann ergodic hypothesis aimed to lay a foundation under statistical mechanics, which is at a microscopic scale a deterministic system. Similar complexity was discovered by PoincarĂ© in celestial mechanics and by Hadamard in the motion of a free particle in a negatively curved space. We start with a guided tour of the history of the subject from various perspectives and then discuss the central mechanism that produces pseudorandom behavior in these deterministic systems, the Hopf argument. It has been known to extend well beyond the scope of its initial application in 1939, and we show that it also leads to much stronger conclusions: Not only do time averages of observables coincide with space averages (which was the purpose for making the ergodic hypothesis), but any finite number of observables will become decorrelated with time. That is, the Hopf argument does not only yield ergodicity but mixing, and often mixing of all orders. October 31, 2014 The Central Limit Theorem (CLT) is one of the most fundamental theorems in probability theory. The CLT states that a sequence of appropriately scaled sums of i.i.d. random variables converges weakly to the standard normal distribution. Although mathematicians had worked on the CLT as early as the 1600s, William Feller gave a proof of the CLT in 1935 by employing L\'evy's continuity theorem. L\'evy's continuity theorem, a nontrivial result, establishes the equivalence of weak convergence for a sequence of random variables and the convergence of the characteristic functions for those random variables. In this talk, I will present the main ideas of our direct proof of the CLT which does not employ L\'evy's continuity theorem. In our proof, we transform a random variable into an i.i.d. sequence on [0,1] and then expand this sequence with respect to the Haar wavelet basis. |
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