Dual spaces

January 11, 2015

Suppose $latex {(A,||cdot||_A)}&fg=000000$ and $latex {(B,||cdot||_B)}&fg=000000$ are Banach space, $latex {A^*}&fg=000000$ and $latex {B^*}&fg=000000$ are their dual spaces. If $latex {Asubset B}&fg=000000$ with $latex {||cdot||_Bleq C||cdot||_A}&fg=000000$, then

$latex displaystyle i:Amapsto B&fg=000000$

$latex displaystyle quad xrightarrow x&fg=000000$

is an embedding. Let us consider the relation of two dual spaces. For any $latex {fin B^*}&fg=000000$

$latex displaystyle |langle f,xrangle|=|f(x)|leq ||f||_{B^*}||x||_Bleq C||f||_{B^*}||x||_Aquad forall, xin A&fg=000000$

Then $latex {f|_{A}}&fg=000000$ will be a bounded linear functional on $latex {A}&fg=000000$

$latex displaystyle i^*:B^*mapsto A^*&fg=000000$

$latex displaystyle qquad frightarrow f|_A&fg=000000$

is a bounded linear operator.

In a very special case that $latex {A}&fg=000000$ is a closed subset of $latex {B}&fg=000000$ under the norm $latex {||cdot||_B}&fg=000000$, one can prove $latex {i^*}&fg=000000$ is surjective. In fact $latex {forall,gin A^*}&fg=000000$ can be extended to $latex {bar{g}}&fg=000000$ on $latex {B}&fg=000000$ by Hahn-Banach thm such that $latex {i^*bar{g}=g}&fg=000000$. Then

$latex displaystyle A^*=B^*/ker i^*.&fg=000000$

Let us take $latex {A=H^1_0(Omega)}&fg=000000$ and $latex displaystyle B=H^1(Omega)&fg=000000$…

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Machine Learning School, Cambridge 2009

January 6, 2015

Old but very explanatory:

http://videolectures.net/mlss09uk_cambridge/

Topics titles

1) Introduction to Bayesian Inference

2) Graphical Models

3) Markov Chains and Monte Carlo

4) Information Theory

5) Kernel Methods

6) Approximate Inference

7) Topic Models

8) Gaussian Processes

9) Convex Optimization

10) Learning Theory

11) Computer Vision

12) Nonparametric Bayesian Models

13) Machine Learning and Cognitive Science

14) Reinforcement Learning

15) Foundations of Nonparametric Bayesian Methods

16) Deep Belief Networks

17) Particle Filters

18) Causality

19) Information Retrieval

20) Bayesian or Frequentist? Which Are You?

Paper of the day: 01/05/15

January 6, 2015

ROTH TYPE THEOREMS IN FINITE GROUPS, JOZSEF SOLYMOSI, 2012

http://arxiv.org/pdf/1201.3670.pdf

Paper of the day: 12/03/14

January 3, 2015

“ON THE MAXIMUM OF THE FUNDAMENTAL FUNCTIONS OF THE ULTRASPHERICAL POLYNOMIALS”, P. ERDOS, 1944

http://www.renyi.hu/~p_erdos/1944-05.pdf

Main theorem of the paper:
Let ${ -1 \leq x_1 < x_2 < \dots < x_n \leq 1 }$ be the roots of the ultraspherical polynomial ${P_n^{(\alpha)}(x) }$ with ${ 0 \leq \alpha \leq 3/2 }$. Let ${ l_k^{(n)}(x) = \dfrac{P_n^{(\alpha)}(x)}{P_n^{'(\alpha)}(x_k) (x-x_k)} }$ be the fundamental polynomial of the Lagrange interpolation. Then ${ max_{k = 1,2, \dots, n, -1 \leq x \leq 1} | l_k^{(n)}(x)| =l_1^{(k)}(-1) =l_n^{(n)}(-1)}$.

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How many rational distances can there be between N points in the plane?

December 30, 2014

Terry has a nice post up bout the Erdös-Ulam problem, which was unfamiliar to me.  Here’s the problem:

Let S be a subset of R^2 such that the distance between any two points in S is a rational number.  Can we conclude that S is not topologically dense?

S doesn’t have to be finite; one could have S be the set of rational points on a line, for instance.  But this appears to be almost the only screwy case.  One can ask, more ambitiously:

Is it the case that there exists a curve X of degree <= 2 containing all but 4 points of S?

Terry explains in his post how to show something like this conditional on the Bombieri-Lang conjecture.  The idea:  lay down 4 points in general position.  Then the condition that the 5th point has rational distances from x1,x2,x3, and x4 means that point lifts to a rational point on…

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How NOT to reference papers

September 16, 2014

In this post, I am going to tell a story of one paper and its authors which misrepresented my paper and refused to acknowledge the fact. It’s also a story about the section editor of Journal of Algebra which published that paper and then ignored my complaints. In my usual wordy manner, I do not get to the point right away, and cover some basics first. If you want to read only the juicy parts, just scroll down…

What’s the deal with the references?

First, let’s talk about something obvious. Why do we do what we do? I mean, why do we study for many years how to do research in mathematics, read dozens or hundreds of papers, think long thoughts until we eventually figure out a good question. We then work hard, trial-and-error, to eventually figure out a solution. Sometimes we do this in a matter of hours and sometimes…

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Some Strange Math Facts

September 15, 2014

Stanislaw Ulam was a Polish-American mathematician whose work spanned many areas of both continuous and discrete mathematics. He did pioneering research in chaos theory and Monte Carlo algorithms, and also invented the concept of a measurable cardinal in set theory. His essential modification of Edward Teller’s original H-bomb design is used by nearly all the world’s thermonuclear weapons, while he co-originated the Graph Reconstruction conjecture. His name is also associated with the equally notorious 3n+1 conjecture. Thus he was involved in some strange corners of math.

Today Ken and I want to talk about some strange facts observed by Ulam and others that we did not know or fully appreciate.

Perhaps you can use them, perhaps you may enjoy them, but they are all kind of fun. At least we think so. Ulam’s autobiographyAdventures of a Mathematician shows his sense of fun, and he was…

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ICM2014 — Kollár, Conlon, Katz, Krivelevich, Milnor

September 4, 2014

As the ICM recedes further into the past, these posts start to feel less and less fresh. I’ve had an enforced break from them as over the course of three days I drove my family from the south of France back to Cambridge. So I think I’ll try to do what I originally intended to do with all these posts, and be quite a lot briefer about each talk.

As I’ve already mentioned, Day 3 started with Jim Arthur’s excellent lecture on the Langlands programme. (In a comment on that post, somebody questioned my use of “Jim” rather than “James”. I’m pretty sure that’s how he likes to be known, but I can’t find any evidence of that on the web.) The next talk was by Demetrios Christodoulou, famous for some extraordinarily difficult results he has proved in general relativity. I’m not going to say anything about the talk, other…

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Hales-Jewett and a generalized van der Warden Theorems

August 27, 2014

The Hales-Jewett theorem is one of the most fundamental results in Ramsey theory, implying the celebrated van der Waerden theorem on arithmetic progressions, as well an its multidimensional and IP versions. One interesting property of the Hales-Jewett’s theorem is that it is a set theoretical statement, having no structure, and hence making it versatile for applications. Recently I realized that there exists an equivalent formulation of this theorem using some algebraic structure, and indeed it can be seen as an analogue of van der Waerden’s theorem. The main purpose of this post is to establish that equivalence. In an initial section I present the deduction of the multidimensional van der Waerden from the Hales-Jewett theorem, both to set the mood and to set the stage to later establish the analogy between van der Waerden’s theorem and the equivalent formulation of the Hales-Jewett theorem.

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August 11, 2014

Gian Carlo Rota

Rota’s conjecture

I just saw in the Notices of the AMS a paper by Geelen, Gerards, and Whittle where they announce and give a high level description of their recent proof of Rota’s conjecture. The 1970 conjecture asserts that for every finite field, the class of matroids representable over the field can be described by a finite list of forbidden minors. This was proved by William Tutte in 1938 for binary matroids (namely those representable over the field of two elements). For binary matroids Tutte found a single forbidden minor.  The ternary case was settled by by Bixby and by Seymour in the late 70s (four forbidden minors).  Geelen, Gerards and Kapoor proved recently that there are seven forbidden minors over a field of four elements.  The notices paper gives an excellent self-contained introduction to the conjecture.

This is a project that started in 1999 and it…

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