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Series: Analysis Seminar

The Horn inequalities give a characterization of eigenvalues of self-adjoint n by n matrices A, B, C with A+B+C=0. The proof requires powerful tools from algebraic geometry. In this talk I will talk about our recent result of these inequalities that are indeed valid for self-adjoint operators of an arbitrary finite factors. Since in this setting there is no readily available machinery from algebraic geometry, we are forced to look for an analysts friendly proof. A (complete) matricial form of our result is known to imply an affirmative answer to the Connes' embedding problem. Geometers in town especially welcome!

Series: Analysis Seminar

Variable transformations are used to enhance the normally poor performance of trapezoidal rule approximations of finite-range integrals I[f]=\int^1_0f(x)dx. Letting x=\psi(t), where \psi(t) is an increasing function for 0 < t < 1 and \psi(0)=0 and \psi(1)=1, the trapezoidal rule is applied to the transformed integral I[f]=\int^1_0f(\psi(t))\psi'(t)dt. By choosing \psi(t) appropriately, approximations of very high accuracy can be obtained for I[f] via this approach. In this talk, we survey the various transformations that exist in the literature. In view of recent generalizations of the classical Euler-Maclaurin expansion, we show how some of these transformations can be tuned to optimize the numerical results. If time permits, we will also discuss some recent asymptotic expansions for Gauss-Legendre integration rules in the presence of endpoint singularities and show how their performance can be optimized by tuning variable transformations. The variable transformation approach presents a very flexible device that enables one to write his/her own high-accuracy numerical integration code in a simple way without the need to look up tables of abscissas and weights for special Gaussian integration formulas.

Series: Analysis Seminar

In any standard course of Analytical Mechanics students are indoctrinated that a Lagrangian have a profound physical meaning (kinetic energy minus potential energy) and that Lagrangians do not exist in the case of nonconservative system. We present an old and regretfully forgotten method by Jacobi which allows to find many nonphysical Lagrangians of simple physical models (e.g., the harmonic oscillator) and also of nonconservative systems (e.g., the damped oscillator). The same method can be applied to any equation of second-order, and extended to fourth-order equations as well as systems of second and first order. Examples from Physics, Number Theory and Biology will be provided.

Series: Analysis Seminar

It is a conjecture of Zygmund that the averages of a square integrable
function over line segments oriented along a Lipschitz vector field on
the plane converge pointwise almost everywhere. This statement is
equivalent to the weak L^2 boundedness of the directional maximal
operator along the vector field. A related conjecture, attributed to
Stein, is the weak L^2 boundedness of the directional Hilbert transform
taken along a Lipschitz vector field. In this talk, we will discuss
recent partial progress towards Stein’s conjecture obtained in
collaboration with I. Parissis, and separately with S. Guo, C. Thiele
and P. Zorin-Kranich. In particular, I will discuss the recently
obtained sharp bound for the Hilbert transform along finite order
lacunary sets in all dimensions, the singular integral counterpart of
the Parcet-Rogers characterization of L^p boundedness for the
directional maximal function in higher dimensions.