502 – Cantor-Bendixson derivatives

Given a topological space and a set let be the set of accumulation points of i.e., those points of such that any open neighborhood of meets in an infinite set.

Suppose that is closed. Then Define for closed compact by recursion: and for limit. Note that this is a decreasing sequence, so that if we set there must be an such that for all  

[The sets are the Cantor-Bendixson derivatives of In general, a derivative operation is a way of associating to sets some kind of ``boundary.'']

502 – The Löwenheim-Skølem theorem

In this note I sketch the proof of the Löwenheim-Skølem (or Löwenheim-Skølem-Tarski) theorem for first order theories. This basic result of model theory is really a consequence of a set theoretic combinatorial lemma, as the proof will demonstrate.

Let be a first order language, understood as a set of constant, function, and relation symbols. Let

so is unless is finite, in which case we take Talking about rather than simplifies the presentation slightly.

The Löwenheim-Skølem theorem is concerned with the possible infinite sizes of models of first order theories. Of course, a theory could only have finite models; the result does not say anything about if that is the case.

Theorem 1 If is a first order theory in a language and there is at least one infinite model of then there are models of of size for all

We will prove a more precise statement. Before stating it, note that it is possible to have a theory in some uncountable language such that has models of certain infinite sizes, but not all. Theorem 1 does not say anything about infinite models of of size What cardinals in this range are the possible sizes of models of is actually a rather difficult problem, and we will not address it.

175 – Quiz 5

Here is quiz 5.

598 – Upcoming talk: Leming Qu

Leming Qu, Wed. November 11, 2:40-3:30 pm, MG 120.

Wavelet Image Restoration and Regularization Parameters Selection

For the restoration of an image based on its noisy distorted observations, we propose wavelet domain restoration by a scale-dependent penalized regularization method (WaveRSL1). The data-adaptive choice of the regularization parameters is based on the Akaike Information Criterion (AIC) and the degrees of freedom (df) are estimated by the number of nonzero elements in the solution. Experiments on some commonly used testing images illustrate that the proposed method possesses good empirical properties.

175 – Midterm 2

Here is the second midterm.

598 – Upcoming talk: Jodi Mead

Jodi Mead, Wed. November 4, 2:40-3:30 pm, MG 120.

Non-smooth Solutions to Least Squares Problems

In an attempt to overcome the ill-posedness or ill-conditioning of inverse problems, regularization methods are implemented by introducing assumptions on the solution.  Common regularization methods include total variation, L-curve, Generalized Cross Validation (GCV), and the discrepancy principle. It is generally accepted that all of these approaches except total variation unnecessarily smooth solutions, mainly because the regularization operator is in . Alternatively, statistical approaches to ill-posed problems typically involve specifying a priori information about the parameters in the form of Bayesian inference. These approaches can be more accurate than typical regularization methods because the regularization term is weighted with a matrix rather than a constant. The drawback is that the matrix weight requires information that is typically not available or is expensive to calculate.

The method developed by the author and colleagues can be viewed as a regularization method that uses statistical information to find matrices to weight the regularization term.  We will demonstrate that unique and simple  solutions found by this method do not unnecessarily smooth solutions when the regularization term is accurately weighted with a diagonal matrix.

502 – Cancellation laws

Two homework problems. The first one is easier, so you can consider the second one to be extra credit. A proof of these results can be found in different places, for example, the paper Division by three, by Conway and Doyle. (Please don’t look at the paper while working on the homework, of course.) Unfortunately, the paper could use a serious trimming and editing, so I cannot really recommend it, but the proof is carefully written there.

1. Without using the axiom of choice, show that if and are sets, and then
2. Same as 1., but now with instead of  

598 – Upcoming talk: Jens Harlander

Jens Harlander, Wed. October 28, 2:40-3:30 pm, MG 120.

Introduction to Computational Complexity

Complexity theory provides ways of measuring the difficulty of computational mathematics problems. Some problems are indeed impossibly difficult (your Math 108 and 143 students are right after all!). For example, there does not exist an algorithm that decides whether a polynomial (in an arbitrary number of variables) with integer coefficients has integer roots. However for many difficult problems, simple strategies work well in practice as long as one is willing to ignore a hopefully sparse set of inputs. I will discuss basic features of the theory, give you more examples of impossibly hard problems and tell you about the relevance of all of this to Internet security.

598 – Schedule of talks

Here is the list of speakers for the rest of the term. 

• Jens Harlander, October 28.
• Jodi Mead, November 4.
• Leming Qu, November 11.
• Grady Wright, November 18.
• Marion Scheepers, December 2. 
• Laurie Cavey, December 9.

175 – Quiz 4

Here is quiz 4.