Abstract. I can't turn the proof of the Brouwer Fixed Point Theorem into an algorithm—yet—but it ``feels'' as if there is an interesting algorithm lurking behind the proof. I shall attempt to explain why.
Abstract. In this talk, about closure systems, an algebraic and topological approach is used in the construction of intersection topologies, in this case, for an infinite subset of the multiplicative semigroup of an algebra-based function space. The constructions and theorems are applicable to knot theory, array-path theory, and neural nets.
Abstract. This talk will explain the eponymous one-page 1950 paper in Proc. of the National Academy of Sciences for which John Nash won the Nobel Prize in Economics.
Abstract. A fixed point theorem for set valued maps will be proved, and employed to prove a fundamental theorem of game theory. Sufficient time will be allotted for motivation and illuminating examples accompanied with arm waving in the style of L. Bernstein.
Abstract. Given a continuous vector field in R^n that always points `in the direction' of a convex region Omega. We prove that under certain conditions the convex domain Omega is a trapping region for the flow (joint work with Pablo Baldivieso).Friday, May 3 @ 1PM in NH 346
Abstract. We describe a theory we are developing to quantitatively understand transients of large oscillator networks. The theory is asymptotic in the number N of agents, where N tends to infinity. The application we have in mind is automatic control of traffic on crowded highways. However transients in large networks have wide applicability.Friday, April 12 @ 3PM in NH 346
- Brief review of the fixed point theorems & applications covered so far.
- Overview of what's (hopefully) to come.
- Time permitting: Haar measure for compact abelian groups.
Abstract. The implicit Euler algorithm gives numerical approximations to solutions to both the heat and the wave equations, but the approximations converge faster for the heat than the wave equation. This talk explains how the heat equation's subgradient structure gives faster rates than the wave equation's structure. The talk relies heavily on pictures, with a little first semester calculus and expansion of squares of vector norms. All are welcome.
Abstract. Sigma-Delta modulators are electronic circuits that generate binary digits (bits) which are identified as subgradients of the ramp function
r(x) = 0 if x ≤ 0Thinking of the bits as subgradients motivates us to use ramp functions to investigate questions of stability and performance of Sigma-Delta modulators. In turn, these ramp functions' subgradients motivate improved architectures for Sigma-Delta modulators, so the picture
= x if x > 0.
subgradient -> ramp function -> subgradientbrings us full-circle. The talk is elementary and self-contained.
Abstract. Minimization of functions is a recurring theme in mathematics, with applications to statistics, engineering, and the natural and social sciences. It is natural, therefore, to seek a mathematical structure providing (easy-to-apply) algorithms for minimizing functions. Subgradients are the appropriate structure for a large class of minimization problems. This (expository) talk will explore why the picture
minimization problem <--> subgradientis interesting in both directions. Topics of exploration will include
- center of mass—vs—the Fermat-Torricelli equilibrium point
- why equi-ripple?
- Sigma-Delta modulators (in electrical circuits)
- maximal monotone operators
- optimal convergence rates for Euler's backward scheme
Abstract. The Markov-Kakutani theorem leads to an invariant form of the Hahn-Banach theorem, which leads to surprising extensions of measures and linear functionals.
Abstract. Convergence of the Weiszfeld's iterative solution to the generalized Fermat-Toricelli problem. Discussion of previous efforts to prove this, and of some open problems.Friday February 8 @ 1 PM in NH 346
Lecture notes available from the "Lecture Notes" link at: http://web.pdx.edu/~mnn3/
Abstract. We will revisit the Fermat-Torricelli problem from both theoretical and numerical viewpoints using modern tools of convex analysis and optimization. We will also discuss new developments on generalized versions of the Fermat-Torricelli problem that involve distances to convex sets.
Abstract. In the early 17th century, Pierre de Fermat proposed the following problem:Given three points in the plane, find a point such that the sum of its Euclidean distances to the three given points is minimal.This problem was solved by Evangelista Torricelli and was named the Fermat-Torricelli problem.
A more general version of the Fermat-Torricelli problem asks for a point that minimizes the sum of the distances to a finite number of given points in Rn. Based on fixed point iterations, Weiszfeld introduced in 1937 the first numerical algorithm for solving the Fermat-Torricelli problem in its general form. However, a correct statement for the convergence of the algorithm along with the proof were given by Kuhn in 1972. Kuhn also pointed out an example in which the Weiszfeld algorithm fails to converge. Several new algorithms have been introduced recently to improve the Weiszfeld algorithm.
In this talk, we will revisit the Fermat-Torricelli problem from both theoretical and numerical viewpoints using modern tools of convex analysis and optimization. We will also discuss new developments on generalized versions of the Fermat-Torricelli problem that involve distances to convex sets.
Abstract. I'll review what we've done concerning the Markov-Kakutani fixed point theorem, and its generalization to certain non-commutative situations. In particular I want to highlight the connection between algebra, fixed-point theory, and functional analysis, with emphasis on the necessity of having to work with:(1) Transformation groups that may not be commutative (e.g. solvable groups), andIf time permits, I'll return to "paradoxical" matters, giving a simple example of a paradoxical group, and indicating how such groups can provide paradoxical decompositions of the spaces upon which they act.
(2) Linear topological spaces more general than Banach spaces (especially, dual spaces in the weak star topology).
Abstract. I'll review the main points of Steve's talk of last week, indicate how the main result leads to an isometry-invariant finitely additive probability measure on the closed unit disc, and show how this measure makes impossible the sort of "paradoxical decomposition" that the Banach-Tarski Theorem promises for the unit ball of 3-space.
Abstract. The Markov/Kakutani Fixed Point Theorem states: If K is a compact, convex subset of a topological linear space, and F is a commuting family of continuous linear maps from K to K, then there is a point p in K such that f(p) = p for all f in F.
A proof will be given of a generalization of this theorem that includes the case when F is a solvable group.
Time permitting we will apply this to show the existence of a finitely additive, isometry-invariant measure on all subsets of the plane.
If you're interested but can't attend, please let me know by email (
email@example.com) what days/times will work for you. We'll try to work something out.
Abstract. This will be the first of several lectures I plan to give this term on some classical fixed-point theorems. In this first lecture, I'll give a couple examples that illustrate the importance of fixed points, and will discuss—mostly without proof—the fixed-point theorems of Brouwer and Banach.
Abstract. Discussion of the Banach Contraction Mapping Principle and its application to situations as diverse as Newton's Method and the solution of initial value problems.
Abstract. Application of the Banach Contraction Mapping Principle to the iterative solution of initial value problems.
Abstract. A fixed-point theoretic argument for the Schroeder-Bernstein theorem.
"The Schroeder-Bernstein property in *-rings of operators."
Abstract. I'll show how the Brouwer Fixed Point Theorem is implied by the "No Retraction Theorem" for the closed unit ball of Rn, and will further discuss the notion of "retraction" (a.k.a. "projection"), and its intimate connection with fixed points.
Abstract. Last time we showed how the No-Retraction Theorem implied the Brouwer Fixed Point Theorem. This time we'll prove a C1 version of the No-Retraction Theorem, and show how it implies the Brouwer Fixed Point Theorem.Friday Nov. 23: Thanksgiving Break: No Meeting
Abstract. The Schauder Fixed Point Theorem says that every compact convex subset of a normed vector space has the fixed point property. A generalization will be proved; some interesting examples, and counterexamples will be discussed.