Riemann’s On the Hypotheses Which Lie at the Foundations of Geometry

One must be truly enamored of the internet that it allows one to find and read a copy of Bernhard Riemann’s doctoral thesis Habilitation Lecture (in English translation) at the University of Göttingen from 1854!

His brief paper has created a tsunami of mathematical work and research in the ensuing 156 years. It has ultimately become one of the seminal works in the development of the algebra and calculus of n-dimensional manifolds.

Global classical solutions of the Boltzmann equation with long-range interactions

Global classical solutions of the Boltzmann equation with long-range interactions (pnas.org)

Finally, after 140 years, Robert Strain and Philip Gressman at the University of Pennsylvania have found a mathematical proof of Boltzmann’s equation, which predicts the motion of gas molecules.

Abstract

This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential r-(p-1) with p > 2, and more generally. We present here a mathematical framework for unique global in time solutions for all of these potentials. We consider it remarkable that this equation, derived by Boltzmann (1) in 1872 and Maxwell (2) in 1867, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects due to grazing collisions.

via pnas.org

 

In Law Schools, Grades Go Up, Just Like That

In Law Schools, Grades Go Up, Just Like That (New York Times)
In the last two years, at least 10 law schools have made their grading systems more lenient to give their students a better chance in a soft job market.

Is GPA tampering and grade inflation going too far with changes like this?