598 – Upcoming talk: Laurie Cavey

December 4, 2009

Laurie Cavey, Wed. December 9, 2:40-3:30 pm, MG 120.

Developing Students’ Understanding of Mathematical Definitions: Why Bother?

Definitions are a fundamental part of doing mathematics, yet studies indicate that many students struggle to learn and apply definitions. In fact, many instructors wonder (myself included) how students can misapply definitions that are so clearly stated. Part of the issue is that a student’s previous mathematical experiences influence how she thinks, even when encountering a new idea that is seemingly unrelated. Not knowing what these experiences might entail, it can be difficult to know how to help students develop a better understanding of a particular definition. So, why bother? I will provide a brief overview of the research in this area including an instructional strategy (student generated examples) that may influence the way we think about developing students’ understanding of definitions.

598 – Upcoming talk: Marion Scheepers

November 30, 2009

Marion Scheepers, Wed. December 2, 2:40-3:30 pm, MG 120.

Cryptography

Online shopping and banking, Wireless communication and remote control devices have become common place. Nontrivial computing power and scanning devices of high power have become readily available. This creates an environment in which information in transit can be easily accessed or changed by unknown parties.

Cryptography is the main tool used to keep information secure. In this talk we will give a brief, motivated, outline of some of the mathematical foundations of cryptography. We also give an example to illustrate that mere possession of a good crypto-system does not guarantee security – one must also use it right.

598 – Upcoming talk: Grady Wright

November 11, 2009

Grady Wright, Wed. November 18, 2:40-3:30 pm, MG 120.

Scattered Node Finite Difference-Type Formulas Generated from Radial Basis Functions with Applications

In the finite difference (FD) method for solving partial differential equations (PDEs), derivatives at a node are approximated by a weighted sum of function values at some surrounding nodes. In the one dimensional case, the weights of the FD formulas are conveniently computed using polynomial interpolation. These one dimensional formulas can be combined to create FD formulas for partial derivatives in two and higher dimensions. This strategy, however, requires that the nodes of the FD “stencils” are situated on some kind of structured grid (or collection of structured grids), which severely limits the application of the FD method to PDEs in irregular geometries.  In this talk, we present a novel approach that resolves this problem by allowing the nodes of the FD stencils to be placed freely and by using radial basis function (RBF) interpolation for computing the corresponding weights in the scattered node FD-type formulas. We show how this RBF approach can exactly reproduce all classical FD formulas and how compact FD formulas can be generalized to scattered nodes and RBFs. This latter result is important in that it allows the number of nodes in the stencils to remain relatively low without sacrificing accuracy.  For the Poisson equation, these new compact scattered node schemes can also be made diagonally dominant, which ensures both a high degree of robustness and applicability of iterative methods. We conclude the talk with some numerical examples and future applications of the method for geophysical problems.

598 – Upcoming talk: Leming Qu

November 3, 2009

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 $L^1$ 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.

598 – Upcoming talk: Jodi Mead

October 28, 2009

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 $L^2$. 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 $\chi^2$ 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 $L^2$ solutions found by this method do not unnecessarily smooth solutions when the regularization term is accurately weighted with a diagonal matrix.

598 – Upcoming talk: Jens Harlander

October 21, 2009

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

October 20, 2009

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

598 – Structuring mathematical proofs

September 27, 2009

Here is a link to Uri Leron’s paper Structuring Mathematical Proofs, The American Mathematical Monthly 90 (3), (Mar., 1983), 174-185. Dr. Leron talks here about the non-linear nature of proofs (remember the examples I mentioned from Szemeredi and Shelah) and discusses what he calls the “structural method.” It is worth keeping in mind his ideas as you continue through graduate school, and especially when faced with the tasks of giving a talk or writing up your results (even if you disagree with him).

The basic idea of the structural method is that proofs should perhaps be presented in levels, each giving at least an outline of a complete argument. As you descend through the levels, you fill in details. It is a fairly natural approach (like dividing a result into a series of lemmas), and it has the advantage that it helps the audience understand where the argument is going and have a better global picture of what is going on. It is also harder than one would think, in actual practice, to organize one’s arguments according to this method and, even if just for practice, I find it useful every now and then to see how to present a result, even one whose proof I understand fairly well, following this approach rather than the more standard linear technique.

598 – Dates for presentations and written reports

September 9, 2009

The (tentative) schedule of talks is as follows; remember that the talks should be about 15 minutes long. Blackboard talks are fine; if you want to use slides let me know ahead of time (meaning two days before, at least), to make sure we have the necessary equipment.

• September 30: Arnold, Bailey, Balmer.
• October 7: Borthakur, Davidson, Droesch.
• October 14: Dummar, Griffin, Hensley.
• October 21: Lohmeier.

I may make a change so that we have a slightly more symmetric schedule (two days with two talks rather than one day with only one), but for now these are the dates.

The deadline for turning in your written report is two weeks after your presentation, but feel free to turn it in earlier. I do not have much in terms of requirements here; it should be in $\LaTeX$ and ideally it is a summary of the paper you chose to present, but I think it makes more sense to be of the part of the paper that your presentation actually used, in case not the whole paper was covered. Again, ideally, follow the standard outline: An abstract (if applicable), introduction, body of work, conclusion and discussion of future work (if applicable), and references. If you see that a slightly different format may suit better your report, go with it. Email me or talk to me to make sure it makes sense to deviate from the path. If you are not too sure of what the report should have/not have, email me or talk to me as well.

I’ll give you some feedback, both on the content and on the $\LaTeX$. If you would like to make changes afterwards so I can give you some additional feedback, feel free. Let me know so I can make sure I’ll have time to look at it.

598 – Papers for presentation

September 3, 2009

Here is the list of papers you will be making presentations on. I will be updating the list as I am being informed. (This list may change, if you decide to work on a different paper, this is mostly so we can keep track of the papers that have been chosen so far.)