Portland State University

This seminar features expository talks on topics in analysis that reflect the participants' interests, but often don't get into courses. In previous years we've heard talks on functional, complex, and harmonic analysis, probability, and dynamical systems. The atmosphere is relaxed and supportive; everyone is welcome to participate: give a talk, or just be part of the audience.

Friday, May 29 @ 2 PM in NH 346

Prof. Elaine Cozzi of Oregon State University will speak on:

"Far away effects of local change to Euler velocity"

Abstract.In this talk, we establish a weighted stability type estimate for solutions to the Euler equations in two and three dimensions. As a corollary of our estimate, we obtain an upper bound on far away effects at later times resulting from a local change to the initial velocity.

Friday, May 15 & 22 @ 2 PM in NH 346

Mau Nam Nguyen will speak on:

"Convex separation theorems and generalized differentiation of convex functions "

Abstract.Convex separation theorems play a crucial role in developing generalized differentiation theory for convex functions. In this talk we present different versions of convex separation in both finite and infinite dimensions. Then we introduce a geometric approach to obtain major subdifferential calculus rules in convex analysis

Friday, May 1 & 8 @ 2 PM in NH 346

Steve Silverman will speak on:

"Metric spaces homeomorphic to the rationals"

Abstract.Sierpinski proved (circa 1921) that:These talks will discuss the proof of this result, emphasizing the role played by several remarkable theorems characterizing the order types of the rationals and subsets of the Cantor Set.Every countable metric space with no isolated points is homemorphic to the rationals.

Friday, April 17 & 24 @ 2 PM in NH 346

Joel Shapiro will speak on:

"Finding fixed points by `walking through rooms.' "

Abstract.We’ll prove the Brouwer Fixed-Point Theorem using a remarkable combinatorial lemma due to Emanuel Sperner (c. 1928). The method works in any finite dimension, but for simplicity the discussion will take place in R^{2}. We’ll show how to use Sperner’s Lemma to provide approximate fixed points by simple algorithmic “walking through rooms” proof due to C. Lemke (c. 1965). We'll use a modification of this algorithm due to Jim Rulla to provide an alternate proof of Sperner's Lemma.

Notes for this talk here: here.

Friday, April 3 @ 2 PM in NH 346

Jim Rulla will speak on

"Levinson's Algorithm and its engineering applications"

Abstract.Norman Levinson discovered a recursive least-squares solution to Wiener's optimal filter problem (c. 1947). The proof is surprising, and the engineers' interpretation inspires further study. We'll (briefly) review the formulation of the problem and then prove Levinson's result.

Notes for this talk: here.

Friday, February 27 & March 6 @ 2 PM in NH 346

Robert Lyons, Digimarc and PSU, will speak on:

"Estimating the Phase of Discrete Frequencies"

Tuesday, February 10 @ 4 PM in NH 373 (please note change of day, time, and place)Abstract.Starting with a discussion of one of the most important results in modern signal processing, the Nyquist-Shannon sampling theorem, we will discuss a specific problem that arose from Digimarc’s work on digital watermarking. When a frequency impulse appears on an image one can estimate its phase by taking the Discrete Fourier Transform. If the frequency does not have integer coordinates then it spills into several frequency bins. How can one resolve the frequency phase from this frequency bump? We shall explore this problem and discuss why this is related to digital watermarking.

Prof. Jane Ye, University of Victoria, will speak on:

"Smoothing SQP methods for solving nonsmooth and nonconvex constrained optimization problems."

Friday, January 23 & 30 @ 2 PM in NH 346Abstract.We propose a smoothing sequential quadratic programming (SQP) algorithm for solving a nonsmooth and nonconvex constrained optimization problem. We show that any accumulation point of the iteration sequence generated by the smoothing SQP algorithm is a Clarke stationary point, provided that the sequence of the multipliers and the sequence of exact penalty parameters are bounded. Furthermore, we introduce a new constraint qualification called the weakly generalized Mangasarian-Fromovitz constraint qualification (WGMFCQ) that is weaker than the GMFCQ. We show that the extended version of the WGMFCQ guarantees the boundedness of the sequence of the multipliers and the sequence of the exact penalty parameters and thus guarantees the global convergence of the smoothing SQP algorithm. We demonstrate that the WGMFCQ can be satisfied by bilevel programs for which the GMFCQ never holds and hence our algorithm can be used to solve the bilevel program.

Steve Silverman will speak on: "They Joy of Flipping a Fair Coin"

Friday, January 9 @ 2 PM in NH 346Abstract.I'll show how to simulate an arbitrary biased coin with 2 (expected) flips of a fair coin,how to obtain a random digit with the minimum (expected) number of flips of a fair coin, and will examine the “two envelope problem.”

Joel Shapiro will speak on: "The Banach-Schröder-Bernstein Theorem"

Abstract.The Schröder-Bernstein Theorem says that if a set A is mapped 1-1 into B and B is mapped 1-1 into A then A can be mapped 1-1 onto B. Banach’s generalization says that the same is true for the more general class of “piecewise congruence” maps; it plays an important role in the study of paradoxical decompositions. In this talk I’ll point out that Banach’s generalization of Schröder-Bernstein follows, word for word, from an elegant proof for the original theorem that John Erdman presented in this seminar a couple of years ago. All prerequisites will be reviewed in detail.

Friday, December 5 @ 2 PM in NH 346

Jim Rulla will speak on: "Optimal Filters for Analysts"

Abstract.Optimal filters are just convolution operators minimizing some measure of error, but they are often enshrined in the language of statistics. This talk will investigate—with a minimum of statistics—the following topics:

- The structure of optimal digital filter problems
- The role of stationarity
- Levinson's algorithm
- Stability
- Burg's algorithm

Friday, November 21 @ 2 PM in NH 346

Jim Rulla will speak on: "Filters for Analysts"

Abstract.Physicists and engineers use filters in everything from hardware to data analysis. For many of our engineering friends, filters are the ``hammers'' that make ``every problem look like a nail'' --- they see filters everywhere. They use Wiener's optimal filters do everything from removing the noise corrupting the signal they want, to compressing your voice for transmission over the internet. They use Kalman's filter to control your plane's landing.

Analysts use filters, too, but we know them by different names: convolutions or mollifiers or multiplier transformations. If you work with engineers, it helps to understand their filters in the language of analysis. Even if you don't, filters provide examples of applied differential and difference equations --- with a twist. I'll use the language of analysis to talk about (and demonstrate) some applications of filters, what we can learn from the engineers' approach, and some pitfalls to avoid.

Friday, November 7, 14 @ 2 PM in NH 346

George Nicol will speak on: "Closure Systems"

Abstract.A closure system, made up of an non-empty set and a closure function, generates an intersection space: a topological space in which the arbitrary intersection of open sets is open. Example: any topology on a finite set arises in this manner.

This talk will:

- Discuss the relationship between closure systems, dual spaces, set relations, and σ−algebras,
- Present a conjecture regarding the number of pre-closure functions that exist on a finite set, and
- Give an example of an intersection space for a family of sets pair-wise equivalent by finite decomposition.

Friday, October 24 and 31 @ 2 PM in NH 346

Steve Silverman will speak on: "A Potpourri of Questions about Sets Equivalent by Finite Decompositions in Dimensions 1, 2, and 3 (and some answers)"

Abstract. From the potpourri:

- Are there subsets of the line or plane that admit paradoxical decompositions?
- Is there a bounded subset, E, of the line or plane with two points p and q such that E - {p} and E - {q} are congruent to E?
- Are the reals and irrationals equivalent by finite decomposition? The rationals and algebraics? [0,1] and a proper subset? The rationals in [0,1] and a proper subset?
- Is 3-space the disjoint union of unit circles?

Friday, October 3, 10, 17 @ 2 PM in NH 346

Joel Shapiro will speak on: "The Banach-Tarski Paradox"

Abstract.The Banach-Tarski Paradox, perhaps the most perplexing result in all of mathematics, says that you can cut a solid three dimensional ball into finitely many pieces, then reassemble these pieces, using only rigid motions, into two copies of the same ball. Although the result lies deep in the foundations of mathematics, its proof is surprisingly accessible, requiring only basic set theory and linear algebra.

Notes for these lectures