You have arrived at this post with a single click. There is a lot of information here, not to mention on other sites, for example your Facebook news feed, where you probably found this link. Although Facebook has reached a very high level of complexity compared to early webpages or even early computer programs, you navigate through these sites and programs like second nature. But, if you think about it, a computer does nothing else than process a flow of zeros and ones. Same is true for the twitch of your finger executing the click that brought you here, which is just an orchestrated stream of electric pulses in your brain. At the core, computers and living things share a very similar design principle. In this post, we shall take a trip to the domain of these living things, seeing them through the eyes of an engineer.
It is intuitive and easy to see that if is a countable set and , where , is a disjoint collection of subsets, then is countable. (Although one has to be careful with saying “intuitive” and “easy to see” in set theory.) A natural question is, what happens if we allow the sets to have nonempty but finite intersections? This question was posed in Halmos’ classic problem book Problems for Mathematicians: Young and Old, and this short post is dedicated to a few solutions.
1. Prologue: simplexes are simple. One day, after having lunch with my friend, we were discussing mathematics while walking back to the university. He told me that after making progress with a difficult problem, an n-dimensional simplex appeared, whose vertices played an important role. Those vertices were defined by a complicated expression deduced from some parameters of a dynamical system. The following conversation unraveled.
“It would be interesting to determine the number of neighbouring vertices for a given vertex of the simplex.”
“Wait, what? Are you kidding me?”
“Because every pair of vertices are neighbors in a simplex.”
Sometimes it happens that we are so confused amidst the complex relations and convoluted ideas that such a simple pattern eludes us. This post is dedicated to a few problems, where it is easy to get lost in details and miss these obvious (or not so obvious) patterns. Although the solutions are explained, I urge everyone to wander into the forest (and maybe get lost in the woods) before exploring the paths I laid out.
One of the most fundamental theorems in mathematics is the famous Baire category theorem, which never ceases to amaze me. If you are a modern analysis enthusiast like me you probably know the feeling when you are working on a seemingly difficult problem and the solution appears to elude you every time you think about it, and then all of a sudden, snap! “Oh, I can just use the Baire category theorem.”
This post is dedicated to this feeling.
This time we shall explore a topic which really fits the name of the blog. (Note: This was posted on my blog Nonempty Spaces, from which I migrated the old posts here.) Let be an complex matrix. It is a well known fact that the spectrum of defined as
is a nonempty set. However, if we look at the as a linear transformation rather than a matrix, we can see quickly that the spectrum does not depend on (on which acts), we may as well take operators acting on an arbitrary Hilbert space . Can it happen that the spectrum of is empty?
This post is devoted to exploring this question. The answer will be surprising and the methods developed are very interesting and beautiful.
This week I gave an introductory talk at the Tenth Summer School on Potential Theory about potential theoretic methods in the study of orthogonal polynomials. This post is based upon my notes for that lecture. The aim of that talk was to prove a theorem about the weak-* convergence of normalized counting measures for zeros of orthogonal polynomials.
A small note: This is a post which was migrated from my old blog. Actually, I gave this talk almost a year ago, but the mathematics still holds. (I sincerely hope so.)
There is a phenomenon in physics and mathematics which has been captivating the minds of scientists for a long time and it is called simply “universality”. In brief, a phenomenon exhibits universality if, be as wild and diverse in microscopic scale as possible, a clear pattern emerges if one looks at it from macrosopic scale. I first encountered this phenomena from a technical point of view while studying asymptotic behavior of orthogonal polynomials, and it got me too. In this post we will see some examples of systems exhibiting universal behavior and we shall see how do orthogonal polynomials enter the picture.