partialdifferentialequations.wordpress.com

freeboundaries.wordpress.com

beni22sof | Free Boundaries

https://freeboundaries.wordpress.com/author/beni22sof

Shape optimization problems and methods. Introduction to Shape Optimization. September 25, 2012. Shape Optimization Introductory Course. This course was given by Dorin Bucur. In November 2010 in Timisoara, and was my first contact with the field of shape optimization. As an outsider, at that time, I was really impressed by this course and by the problems presented here. The course does not go into deep details, and it is aimed to be a presentation of the field. 1 The Dido Problem. Beni Bogosel's blog.

mercmath.wordpress.com

The circle method 1: Sums of 5 or more squares | Musings of the Mercurial Mathematician

https://mercmath.wordpress.com/2011/05/05/the-circle-method-1-sums-of-5-or-more-squares

Musings of the Mercurial Mathematician. The circle method 1: Sums of 5 or more squares. This is our first step in solving Waring’s problem. In particular, we’ll study the problem for degree 2. Later, we’ll combine these methods with Weyl’s inequality for higher degree exponential sums to solve Waring’s problem. We want to understand the asymptotics of. The number of ways to represent a number. As a sum of. Squares, i.e. We will solve this problem in dimensions. Using the Hardy-Littlewood circle method.

mercmath.wordpress.com

The circle method 0: Introducing Waring’s problem | Musings of the Mercurial Mathematician

https://mercmath.wordpress.com/2011/04/13/the-circle-method-0-introducing-warings-problem

Musings of the Mercurial Mathematician. The circle method 0: Introducing Waring’s problem. It’s a classical problem in number theory to ask which integers. Can be represented by a sum of squares. After a little bit of playing with sums of two squares, one might be led to conjecture the following theorem. Can be represented as a sum of two squares precisely when. Is of the form. Is a prime congruent to 1 mod 4. Every natural number is the sum of at most 4 squares. Later, the 3 squares question was solved.

mercmath.wordpress.com

Update: New Year’s Resolution 2011 | Musings of the Mercurial Mathematician

https://mercmath.wordpress.com/2011/07/01/update-new-years-resolution-2011

Musings of the Mercurial Mathematician. Update: New Year’s Resolution 2011. I’m working on my third installment of my series of Hardy–Littlewood posts. Currently, there will be at least 6 posts in this series including the 0th post. I’ve understood better the Euclidean problem of spherical averages, but luckily, T. Tao wrote an excellent blog post. OK, back to work. Working is much harder now that Google has been released =). Leave a Reply Cancel reply. Enter your comment here. Packing planes in R 3.

mercmath.wordpress.com

The circle method 2: Elementary estimates for the singular series for sums of 5 or more squares | Musings of the Mercurial Mathematician

https://mercmath.wordpress.com/2011/07/01/the-circle-method-2-elementary-estimates-for-the-singular-series-for-sums-of-5-or-more-squares

Musings of the Mercurial Mathematician. The circle method 2: Elementary estimates for the singular series for sums of 5 or more squares. This is the long-awaited second step in solving Waring’s problem. Continuing our study of the sum of squares problem. In this post, we’ll analyze the singular series to show that we have a genuine asymptotic when. And what we’ve already shown. We want to understand the asymptotics of. The number of ways to represent a number. As a sum of. Squares, i.e. Is the normalized...

mercmath.wordpress.com

Summary of the Guth’s endpoint multilinear Kakeya conjecture | Musings of the Mercurial Mathematician

https://mercmath.wordpress.com/2011/09/26/summary-of-the-guths-endpoint-multilinear-kakeya-conjecture

Musings of the Mercurial Mathematician. Summary of the Guth’s endpoint multilinear Kakeya conjecture. I recently went to a summer school and presented Guth’s paper. In doing so, I wrote a MLK summary. And two talks: MLK talk 1. In a subsequent seminar, I recycled this talk adding a little about the polynomial method: Polynomial method talk. I hope this partially explains why I haven’t posted lately…. Leave a Reply Cancel reply. Enter your comment here. Email (Address never made public).

mercmath.wordpress.com

Notations | Musings of the Mercurial Mathematician

https://mercmath.wordpress.com/notations

Musings of the Mercurial Mathematician. This page will keep a record of notations that I frequently use. Will often be a prime or an exponent depending on the context (usually if the matter at hand is arithmetic or analytic respectively). Leave a Reply Cancel reply. Enter your comment here. Fill in your details below or click an icon to log in:. Email (Address never made public). You are commenting using your WordPress.com account. ( Log Out. You are commenting using your Twitter account. ( Log Out.

mercmath.wordpress.com

Packing planes in R^3 | Musings of the Mercurial Mathematician

https://mercmath.wordpress.com/2011/10/04/packing-planes-in-r3

Musings of the Mercurial Mathematician. Packing planes in R 3. In these notes I’ll discuss Marstrand’s work on packing planes in. I follow very closely Marstrand’s approach and at the moment, I’m not making any effort to keep the exposition self-contained – sorry. In a later post, I plan to describe Falconer’s theorem on packing k-planes in. All measures will refer to Lebesgue or (in the case of the 2-sphere. Has measure 0, then for almost all directions. In a set of planar measure 0. With measure 0; let.