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Series: Math Physics Seminar

The relative isoperimetric inequality inside an open, convex cone C states that under a volume constraint, the ball intersected the cone minimizes the perimeter inside C. In this talk, we will show how one can use optimal transport theory to obtain this inequality, and we will prove a corresponding sharp stability result. This is joint work with Alessio Figalli.

Series: Math Physics Seminar

The phenomenon of wave run-up has the capital importance for the beach erosion, coastal protection and flood hazard estimation. In the present talk we will discuss two particular aspects of the wave run-up problem. In this talk we focus on the wave run-up phenomena on a sloping beach. In the first part of the talk we present a simple stochastic model of the bottom roughness. Then, we quantify the roughness effect onto the maximal run-up height using Monte-Carlo simulations. A critical comparison with more conventional approaches is also performed.In the second part of the talk we study the run-up of simple wave groups on beaches of various geometries. Some resonant amplification phenomena are unveiled. The maximal run-up height in resonant cases can be 20 times higher than in regular situations. Thus, this work can provide a possible mechanism of extreme tsunami run-up conventionally ascribed to "local site effects".References:Dutykh, D., Labart, C., & Mitsotakis, D. (2011). Long wave run-up on random beaches. Phys. Rev. Lett, 107, 184504.Stefanakis, T., Dias, F., & Dutykh, D. (2011). Local Runup Amplification by Resonant Wave Interactions. Phys. Rev. Lett., 107, 124502.

Series: Math Physics Seminar

In this seminar we will show that the nonlinear mechanics of solids with distributed dislocations can be formulated as a nonlinear elasticity problem provided that the material manifold – where the body is stress-free − is chosen appropriately. Choosing a Weitzenböck manifold (a manifold with a flat and metric-compatible affine connection that has torsion) with torsion tensor identified with the given dislocation density tensor the body would be stress-free in the material manifold by construction. For classical nonlinear elastic solids in order to calculate stresses one needs to know the changes of the relative distances, i.e. a metric in the material manifold is needed. For distributed dislocations this metric is the metric compatible with the Weitzenböck connection. We will present exact solutions for the residual stress field of several distributed dislocation problems in incompressible nonlinear elastic solids using Cartan's method of moving frames. We will also discuss zero-stress dislocation distributions in nonlinear dislocation mechanics.

Series: Math Physics Seminar

An
implicit method [1, 2], TARDIS (Transient Advection Reaction
Diffusion Implicit Simulations), has been developed that successfully
couples the compressible flow to the comprehensive chemistry and
multi-component transport properties. TARDIS has been demonstrated in
application to two fundamental combustion problems of great interest.
First,
TARDIS was used to investigate stretched laminar flame velocities in
eight flame configurations: outwardly and inwardly propagating
H2/air and CH4/air in cylindrical and spherical geometries.
Fractional power laws are observed between the velocity deficit and
the flame curvature Second,
the response of transient outwardly propagating premixed H2/air and
CH4/air flames subjected to joint pressure and equivalence ratio
oscillations were investigated. A fuller version of the abstract can be obtained from http://www.math.gatech.edu/~rll6/malik_abstract-Apr-2012.docx
[1]
Malik, N.A. and Lindstedt, R.P. The response of transient
inhomogeneous flames to pressure fluctuations and stretch: planar
and outwardly propagating hydrogen/air flames.
Combust.
Sci. Tech. 82(9), 2010.
[2]
Malik,
N. A. “Fractional
powers laws in stretched flame velocities in finite thickness flames:
a numerical study using realistic chemistry”.
Under
review, (2012).
[3]
Markstein, G.H. Non-steady Flame Propagation. Pergamon Press, 1964.
[4]
Weis,M., Zarzalis, N., and Suntz, R. Experimental study of markstein
number effects on laminar flamelet velocity in turbulent premixed
flames. Combust. Flame,
154:671--691, 2008.

Series: Math Physics Seminar

Note nonstandard day and time.

Consider an N by N matrix X of complex entries with iid real and imaginary parts
with probability distribution h where h has Gaussian decay. We show that the local density of
eigenvalues of X converges to the circular law with probability 1. More precisely, if we let a
function f (z) have compact support in C and f_{\delta,z_0} (x) = f ( z-z^0 / \delta ) then the sequence of densities
(1/N\delta^2) \int f_\delta d\mu_N
converges to the circular law density (1/N\delta^2) \int f_\delta d\mu
with probability 1. Here we show
this convergence for \delta = N^{-1/8}, which is an improvement on the previously known results
with \delta = 1. As a corollary, we also deduce that for square covariance matrices the number of
eigenvalues in intervals of size in the intervals [a/N^2 , b/N^2] is smaller than log N with probability
tending to 1.

Series: Math Physics Seminar

Zeros of vibrational modes have been fascinating physicists for
several centuries. Mathematical study of zeros of eigenfunctions goes
back at least to Sturm, who showed that, in dimension d=1, the n-th
eigenfunction has n-1 zeros. Courant showed that in higher dimensions
only half of this is true, namely zero curves of the n-th eigenfunction of
the Laplace operator on a compact domain partition the domain into at
most n parts (which are called "nodal domains").
It recently transpired that the difference between this "natural" number n of
nodal domains and the actual values can be interpreted as an index of instability
of a certain energy functional with respect to suitably chosen perturbations. We
will discuss two examples of this phenomenon: (1) stability of the nodal
partitions of a domain in R^d with respect to a perturbation of the partition
boundaries and (2) stability of a graph eigenvalue with respect to a perturbation
by magnetic field. In both cases, the "nodal defect" of the eigenfunction
coincides with the Morse index of the energy functional at the corresponding
critical point.
Based on preprints arXiv:1107.3489 (joint with P.Kuchment and
U.Smilansky) and arXiv:1110.5373

Series: Math Physics Seminar

We present an improved version of commutator methods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of singularly continuous spectrum and to the local finiteness of point spectrum. Some applications for Floquet operators and for cocycles over irrational rotations will be presented.

Series: Math Physics Seminar

This talk is concerned with new symmetry results for the extremals of the Caffarelli-Kohn-Nirenberg inequalities in a range of parameters for which no explicit results of symmetry have previously been known. The method proceeds via spectral estimates. This is joint work with Jean Dolbeault and Maria Esteban.

Series: Math Physics Seminar

I will discuss local and nonlocal anisotropic heat transport along magnetic field lines in a tokamak, a device used to confine plasma undergoing fusion. I will give computational results that relate certain dynamical features of the magnetic field, e.g. resonance islands, chaotic regions, transport barriers, etc. to the asymptotic temperature profiles for heat transport along the magnetic field lines.

Series: Math Physics Seminar

We offer several perspectives on the behavior at infinity of solutions of discrete Schroedinger equations. First we study pairs of discrete Schroedinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n \rightarrow \infty. With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-of-constants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one. We use the relations between the solution sets to derive exponential dichotomy of solutions and elucidate the structure of transfer matrices.Later, we present a sharp discrete analogue of the Liouville-Green (WKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which our perturbation results apply. In addition, we point out an exact relationship connecting the diagonal part of the Green matrix to the asymptotic behavior of solutions. With both of these tools it is possible to identify an Agmon metric, in terms of which, in some situations, any decreasing solution must decrease exponentially. This is joint work with Evans Harrell.