/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Quantum phase engineering is demonstrated with two techniques that allow the spatial phase distribution of a Bose-Einstein condensate (BEC) to be written and read out. A quantum state was designed and produced by optically imprinting a phase pattern onto a BEC of sodium atoms, and matter-wave interferometry with spatially resolved imaging was used to analyze the resultant phase distribution. An appropriate phase imprint created solitons, the first experimental realization of this nonlinear phenomenon in a BEC. The subsequent evolution of these excitations was investigated both experimentally and theoretically.

https://science.sciencemag.org/content/sci/287/5450/97.full.pdf

Generating solitons by phase engineering of a bose-einstein condensate

In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of Rn, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

/pdf/the-brunn-minkowski-inequality-3ympoqc7tt.pdf

The Brunn-Minkowski inequality

Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a "highlight application" such as Arrow's theorem from economics, the Goldreich-Levin algorithm from cryptography/learning theory, Hstad's NP-hardness of approximation results, and "sharp threshold" theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students, and researchers in computer science theory and related mathematical fields.

/pdf/analysis-of-boolean-functions-4kueie1j8m.pdf

Analysis of Boolean Functions

Zeros of orthogonal polynomials

/pdf/analysis-second-edition-52bk2wkw1x.pdf

Analysis, Second edition

The excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay, but such a particle should have a true ground state — one that minimizes the energy and satisfies the Schrodinger equation. We prove quite generally that this state exists for all values of the fine-structure constant and the ultraviolet cutoff. We also show the same thing for a many-particle system under physically natural conditions.

Ground states in non-relativistic quantum electrodynamics

The sharp version of the logarithmic Hardy-Littlewood-Sobolev inequality including the cases of equality is established. We then show that this implies Beckner's generalization of Onofri's inequality to arbitrary dimensions and determines the cases of equality.

Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on sn

Extremals of functionals with competing symmetries

We prove a sharp analog of Young’s inequality on SN, and deduce from it certain sharp entropy inequalities. The proof turns on constructing a nonlinear heat flow that drives trial functions to optimizers in a monotonic manner. This strategy also works for the generalization of Young’s inequality on RN to more than three functions, and leads to significant new information about the optimizers and the constants.

Michael Loss

Papers

Analysis, Second edition

Ground states in non-relativistic quantum electrodynamics

Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on sn

Extremals of functionals with competing symmetries

A Sharp analog of Young's Inequality on $S^N$ and Related Entropy Inequalities