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University of Alaska Fairbanks
Department of Mathematics and Statistics

2006-2007 Colloquium Series

Colloquiums are usually held on Thursdays at 1:00 in Chapman 106.

Spring Schedule

Fall Schedule

Abstracts

Limit Sets in Graph Directed Constructions

Andrei Ghenciu

University of Alaska Fairbanks

We introduce the concept of a Graph Directed Markov System (GDMS) and we define what is its limit set. In general these limit sets are of Lebesgue measure zero, so we try to study them by looking at their Hausdorff dimension. We present several results that connect the Hausdorff dimension of the limit set and the zero of the topological pressure function. We start and end with several examples and applications.

Explicit solutions to the Korteweg-de Vries equation on the half line

Tuncay Aktosun

University of Texas at Arlington

We analyze the Korteweg-de Vries equation on the half line ut + ηux -6uux +uxxx = 0, where η is a nonnegative constant, x ≥0, and t ≥ 0. We present certain explicit solutions in terms of elementary function; such solutions contain those that are global in time (i.e. valid for all t ∈ [0, +∞)) and also those local in time (i.e. valid for t ∈ [0, τ ) for some positive τ ). The initial values of these solutions are associated with rational scattering data. This is joint work with C. van der Mee of University of Cagliari, Italy.

Andrei Ghenciu

University of Alaska Fairbanks

A Classical Result Applied to Fractals

We start with the set of all subsets of the positive integers, which we denote A. We define a metric on A and we analyze the induced topology. Then we use this ideas to analyze the Hausdorff dimension spectrum in graph directed constructions.

Flexible variogram models: a survey of approximation methods

Ron Barry

University of Alaska Fairbanks

Geostatistical (GS) modeling is modeling a variable that can theoretically measured at all locations on a map, but is only actually observed at a few locations. Often GS models are used to produce maps (through a process called Kriging), but they are also of interest in their own right. The usual first step in a GS analysis is fitting a variogram function to the data. Under the assumption of intrinsic stationarity, the variogram V(h) is the variance of the difference of two measurements at locations s1, s2 such that h = s1 - s2.

A problem with selecting variogram functions is that variograms have to be conditionally nonnegative definite (cnnd). This property is difficult to check, so that researchers usually use only a few families of functions that are known to be cnnd. These families of functions do not fit many data sets well, so there is incentive to find flexible variogram families that can fit many more data sets successfully. I will discuss the major approaches used to derive flexible variogram families.

Quantum graphs: controllability and inverse problems

Victor Mihkaylov

University of Alaska Fairbanks

Differential equations on graphs are used to describe many physical processes such as mechanical vibrations of multi-linked flexible structures usually composed of flexible beams or strings, propagation of electro-magnetic waves in networks of optical fibers, heat flow in a wire mesh, and also electron flow in quantum mechanical circuits.

In the talk we discuss boundary controllability and inverse problems for the wave and heat equations on graphs. We suppose that the differential equation is defined on each edge of the graph, and standard compatibility conditions are satisfied at the internal vertices. We prove that the hyperbolic system is exactly controllable (and the parabolic system is null controllable) if the graph is a tree and the control is applied to all (or to all but one) boundary vertices. Otherwise the system is generally not exactly controllable but may be spectrally controllable. The latter means that the space of reachable states contains all finite linear combinations of the eigenfunctions of the system.

We describe connections between controllability and inverse problems and show how to recover a tree (its connectivity and the lengths of the edges together with coefficients of the wave equation) by given response operator or Weyl matrix function.

The talk is based on joint work with S. Avdonin and P. Kurasov.

The Set Covering Problem and Related Topics

John Gimbel

University of Alaska Fairbanks

The Set Covering Problem is the following. Given F, a finite family of finite sets, find a subset S ⊆ F where | S | is minimized subject to ∪S = ∪F. We will discuss the computational complexity of this problem and some topics which, at first, may seem unrelated. The talk is meant for non-specialists.

'05 - '06 Colloquium Schedule