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.

175 – Quiz 7

November 25, 2009

Happy Thanksgiving!

Here is quiz 7.

175 – Quiz 6

November 13, 2009

Here is quiz 6.

175 – Self test and Extra credit problems

November 12, 2009

Here are the two parts of the test (part 1, part 2), in case you want a blank copy. As I mentioned, they do not include all topics; they mostly cover material related to chapter 7 of the book, and even then, they are not comprehensive (for example, the second part does not include parts or word problems), but I hope you found them helpful in identifying topics that may require further study or review.

Here are two extra credit problems. Please let me know if you need me to clarify either one.

502 – Equivalents of the axiom of choice

November 11, 2009

The goal of this note is to show the following result:

Theorem 1 The following statements are equivalent in ${{\sf ZF}:}$

1. The axiom of choice: Every set can be well-ordered.
2. Every collection of nonempty set admits a choice function, i.e., if ${x\ne\emptyset}$ for all ${x\in I,}$ then there is ${f:I\rightarrow\bigcup I}$ such that ${f(x)\in x}$ for all ${x\in I.}$
3. Zorn’s lemma: If ${(P,\le)}$ is a partially ordered set with the property that every chain has an upper bound, then ${P}$ has maximal elements.
4. Any family of pairwise disjoint nonempty sets admits a selector, i.e., a set ${S}$ such that ${|S\cap x|=1}$ for all ${x}$ in the family.
5. Any set is a well-ordered union of finite sets of bounded size, i.e., for every set ${x}$ there is a natural ${m,}$ an ordinal ${\alpha,}$ and a function ${f:\alpha\rightarrow{\mathcal P}(x)}$ such that ${|f(\beta)|\le m}$ for all ${\beta<\alpha,}$ and ${\bigcup_{\beta<\alpha}f(\beta)=x.}$
6. Tychonoff’s theorem: The topological product of compact spaces is compact.
7. Every vector space (over any field) admits a basis.

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.

502 – Cantor-Bendixson derivatives

November 8, 2009

Given a topological space $X$ and a set $B\subseteq X,$ let $B'$ be the set of accumulation points of $B,$ i.e., those points $p$ of $X$ such that any open neighborhood of $p$ meets $B$ in an infinite set.

Suppose that $B$ is closed. Then $B'\subseteq B.$ Define $B^\alpha$ for $B$ closed compact by recursion: $B^0=B,$ $B^{\alpha+1}=(B^\alpha)',$ and $B^\lambda=\bigcap_{\alpha<\lambda}B^\alpha$ for $\lambda$ limit. Note that this is a decreasing sequence, so that if we set $B^\infty=\bigcap_{\alpha\in{\sf ORD}}B^\alpha,$ there must be an $\alpha$ such that $B^\infty=B^\beta$ for all $\beta\ge\alpha.$

[The sets $B^\alpha$ are the Cantor-Bendixson derivatives of $B.$ In general, a derivative operation is a way of associating to sets $B$ some kind of “boundary.”]