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We derive a large deviations principle for the trajectories generated by a class of ergodic Markov processes. Specifically, we work with M/M/∞ queueing processes. We study large deviations of these processes scaled equally in both space and time directions. Our main result is that the probabilities of long excursions originating at state 0 would converge to zero function with the rate proportional to the square of the scaling parameter. The rate function is expressed as an integral of a linear combination of trajectories.
Schubert calculus is a great source of beautiful identities which really ought to have bijective proofs. For instance, Macdonald (1991) proves, non-bijectively, an identity for a weighted sum over the reduced words for a permutation pi. I'll give a algorithmic bijective proof in the simplest case: when pi is a dominant permutation, the sum evaluates to l(pi)!
This talk serves as part II of a two-part elementary introduction to mathematical fluid mechanics. In this series of talks we will introduce and (at least partially) derive the Euler and Navier-Stokes equations modeling incompressible fluid flow in two and three dimensions. We will also discuss quantities of interest such as the velocity, pressure, vorticity, and particle trajectory map. For our discussion of inviscid flows modeled by the Euler equations, we will focus in particular on vortex motion, namely Kelvin's circulation theorem and the Helmholtz vortex theorems with applica
For the past 15 years I have been involved with research that seeks to understand how students can transition from more informal reasoning to more formal, mathematical reasoning. As part of this NSF funded research, we have developed curriculum to help students make this transition. In my current project our team is developing instructional materials appropriate for a first college course in linear algebra including the topics of (1) span and linear independence, (2) matrix multiplication and linear transformations, and (3) eigen theory, change of basis and diagonalization.