Probabilistic Methods on Erdos Problems

The study of perfect numbers dates back to Euler and Mersenne. A perfect number is a number that is equal to the sum of its proper divisors which are said to include the multiplicative unit 1. The following theorem is a classical number theory result. All even numbers are of the form $2^k(2^k-1)$ where $2^k-1$ is a Mersenne prime, that is, a prime where $k=P$ and the number $P$ is prime. One interesting conjecture is that there are no odd perfect numbers

Comparing probabilistic methods for outlier detection

This paper compares the use of two posterior probability methods to deal with outliers in linear models. We show that putting together diagnostics that come from the mean-shift and variance-shift models yields a procedure that seems to be more effective than the use of probabilities computed from the posterior distributions of actual realized residuals. The relation of the suggested procedure to the use of a certain predictive distribution for diagnostics is derived

Comparing probabilistic methods for outlier detection.

This paper compares the use of two posterior probability methods to deal with outliers in linear models. We show that putting together diagnostics that come from the mean-shift and variance-shift models yields a procedure that seems to be more effective than the use of probabilities computed from the posterior distributions of actual realized residuals. The relation of the suggested procedure to the use of a certain predictive distribution for diagnostics is derived.Diagnostic; Posterior and Predictive distributions; Leverage; Linear models;

Probabilistic methods for discrete nonlinear Schr\"odinger equations

We show that the thermodynamics of the focusing cubic discrete nonlinear Schrodinger equation are exactly solvable in dimensions three and higher. A number of explicit formulas are derived.Comment: 30 pages, 2 figures. To appear in Comm. Pure Appl. Mat

Four lectures on probabilistic methods for data science

Methods of high-dimensional probability play a central role in applications for statistics, signal processing theoretical computer science and related fields. These lectures present a sample of particularly useful tools of high-dimensional probability, focusing on the classical and matrix Bernstein's inequality and the uniform matrix deviation inequality. We illustrate these tools with applications for dimension reduction, network analysis, covariance estimation, matrix completion and sparse signal recovery. The lectures are geared towards beginning graduate students who have taken a rigorous course in probability but may not have any experience in data science applications.Comment: Lectures given at 2016 PCMI Graduate Summer School in Mathematics of Data. Some typos, inaccuracies fixe

Probabilistic methods in the analysis of protein interaction networks

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