Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities

We prove a family of $L^p$ uncertainty inequalities on fairly general groups and homogeneous spaces, both in the smooth and in the discrete setting. The crucial point is the proof of the $L^1$ endpoint, which is derived from a general weak isoperimetric inequality.Comment: 17 page

New monotonicity formulas for Ricci curvature and applications. I

Original manuscript November 21, 2011We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov–Hausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop–Gromov volume comparison theorem and Perelman’s celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these. Moreover, we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the Green function that we prove. This is parallel to the fact that Perelman’s monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li–Yau. In [CM4] one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross-sections of Einstein manifolds. Finally, there are obvious parallelisms between our monotonicity and the positive mass theorem of Schoen–Yau and Witten.National Science Foundation (U.S.) (Grant DMS-11040934)National Science Foundation (U.S.). Focused Research Group (Grant DMS 0854774)National Science Foundation (U.S.) (Grant 0932078

A nonlinear PDE in mathematical finance

We study a nonlinear degenerate partial Differential Equation arising in Mathematical Finance. We prove the existence of a solution to the Cauchy problem by using the PDE's regularity theory on Lie group

Hierro plasmático en recién nacidos

Since he began studying the plasma iron have been countless jobs that have been made about it, but few in the world literature, which refer to the study of iron in the plasma of newborns and their evolution in the first days of life, in which, as is known, rapid changes are taking place in the blood picture, the detailed aspects are found in the work of C. Reynafarje H, E. Picon, Loret de Mola, in our midst.Desde que se empezó a estudiar el hierro plasmático han sido innumerables los trabajos que sobre él se han hecho pero, son pocos, en la literatura mundial, los que se refieren al estudio del hierro en el plasma de recién nacidos y su evolución en los primeros días de vida, en el cual, como es sabido, se operan cambios rápidos en el cuadro hemático, cuyos aspectos detallados se encuentran en los trabajos de C. Reynafarje H, E. Picón, Loret de Mola, en nuestro medio