Week of March 15 - 19

Monday, March 15

 2:30 GROUPS AND GEOMETRIES SEMINAR  - Room 459 MSC
      Curt Bennett, Mathematics and Statistics, BGSU 
      "Orthogonal and symplectic groups and geometries"

 3:30 ANALYSIS SEMINAR  - Room 459 MSC
      Gordon Wade, Mathematics and Statistics, BGSU
      "An abstract Poincare inequality"

Tuesday, March 16

 4:00 STATISTICS SEMINAR  - Room 459 MSC
      Hanfeng Chen, Mathematics and Statistics, BGSU 
      "Nonregular models"
      Abstract: Standard statistical procedures often require that the
        set-up model satisfy some regularity conditions. I will
        discuss the consequences and difficulties in statistical
        inference when these regularity conditions are not satisfied.

Wednesday, March 17

 3:30 ALGEBRA SEMINAR  - Room 459 MSC
      Warren McGovern, Mathematics and Statistics, BGSU 
      "Algebraic properties of rings of continuous functions"

 3:30 STATISTICS SEMINAR  - Room 304 MSC **** Note room ****
      G. P. Patil, Distinguished Lukacs Professor, BGSU, and
      C. Taillie, Senior Research Associate, Center for Statistical
        Ecology and Environmental Statistics, Department of
        Statistics, Penn State
      "Statistical issues and approaches for multiscale modeling and
       assessment of landscapes based on single-resolution thematic
       raster maps"

      Abstract: Landscape pattern as represented in a thematic raster
        map is the joint result of two ingredients: (i) the marginal
        landcover distribution and (ii) the spatial arrangement of the
        landcover categories across the pixels.  Although the
        landcover distribution has no explicit spatial content, it
        nonetheless affects the apparent spatial pattern as perceived
        by a human studying the map or as measured by most landscape
        metrics.  To separate the perceptual from the "real" spatial
        pattern, landscape models should explicitly include the
        marginal landcover distribution as one set of parameters with
        additional parameters to regulate and summarize the "real"
        spatial pattern.  Vanishing of these additional parameters
        then signifies an absence of spatial pattern, i.e., a random
        assignment of landcover categories to pixels subject to the
        given marginal landcover distribution.

        One such parametric family of landscape models employs a
        hierarchical sequence of Markov transition matrices to
        generate the raster maps at successively finer resolutions
        until the resolution of the data map is reached.  Fitting of
        the model is based on linking the hierarchical transitions in
        the model to spatial transitions across the data map.  The
        fitting of this model for the last transition matrix has been
        discussed in the earlier seminar.  It is briefly reviewed to
        establish notation.  We then show how transition matrices at
        earlier hierarchical levels can be estimated using spatial
        transition matrices at broader spatial scales in the data map.
        Self-similarity of the hierarchical model (i.e., constancy of
        the transition matrices) is characterized in terms of the
        spatial adjacency matrices and a graphical test of
        self-similarity is described.

        We then study the eigen-decomposition of the transition
        matrices, with a view toward using the eigenvectors and
        eigenvalues for landscape characterization.  Eigenvectors are
        interpreted in terms of the spatial persistence of
        perturbations to the stationary distribution.  Ordered
        eigenvalues can be plotted against rank order and roughly
        indicate a fitted model's placement in a spectrum of spatial
        pattern ranging from "no pattern" to "very patchy" or "highly
        segregated."  For the 102 watersheds of Pennsylvania, the
        eigenvalue plots are nearly straight lines and cover a small
        portion of parameter space.  This suggests the desirability of
        more parsimonious modeling of the transition matrices.

        We accordingly describe several parametric subfamilies of
        transition matrices using the notion of diagonal dominance as
        guiding principle.  Diagonal dominance determines the degree
        to which daughters are like their mothers or, in the spatial
        domain, pixels are like their neighbors.  One model, in
        particular, involves parameters p, q, and c, where p is the
        marginal landcover distribution, c is a scalar parameter that
        is an inverse measure of diagonal dominance, and q is a
        probability vector that determines the landcover category of
        daughter pixels conditional on the daughter being different
        from the mother.  The square of the parameter (1-c) is found
        to be strongly correlated with the Kullback-Liebler distance
        between the actual model and the model with the same marginal
        p but with no spatial structure, and becomes a measure of
        spatial complexity for the model.  Upper and lower bounds are
        obtained for the eigenvalues of this pqc model.  These bounds
        show that when q is related to p in an appropriate way, the
        eigenvalue plots for the model are similar to those observed
        for the 102 watersheds of Pennsylvania.

        The spatial data matrices have more degrees of freedom than
        the submodels so the eigen-decomposition (which gives a
        perfect fit) cannot be used for fitting.  Instead we minimize
        some criterion function measuring the distance between
        observed and expected frequencies, such as chi-square.
        However, due to the spatial dependence, the criterion function
        cannot be benchmarked against the chi-square distribution for
        goodness-of-fit tests.  Goodness-of-fit for landscape models
        is a major open problem.

Thursday, March 18

 4:00 STATISTICS SEMINAR  - Room 459 MSC
      G. P. Patil, Distinguished Lukacs Professor, BGSU, and
      C. Taillie, Senior Research Associate, Center for Statistical
        Ecology and Environmental Statistics, Department of
        Statistics, Penn State
      "Statistical issues and approaches for multiscale modeling and
       assessment of landscapes based on single-resolution thematic
       raster maps"
      Abstract:  See above.

Friday, March 19

 3:30 Coffee
 3:45 COLLOQUIUM  - Room 459 MSC
      Barbara Moses and Waldemar Weber, Mathematics and Statistics, BGSU 
      "The Seven-Eleven Problem"
      Abstract: A customer purchased four items at a Seven-Eleven Store.
        At first, the clerk multiplied their prices together and charged
        $7.11, but the customer protested, saying that the prices should
        have been added instead.  What was the cost of each item, if the
        price remained unchanged after making this correction?

        While solving this problem and thereby answering this question, we
        will have an opportunity to review general problem-solving
        heuristics and to utilize popular symbol-manipulation software for
        narrowing large search-spaces through a vigorous combination of
        theory and practice.  Hopefully, some appropriate representations
        of symmetric functions as well as interesting comparisons of
        additive and multiplicative operations will be obtained along the
        way.  Indeed, as we are careful to observe in our Summer Workshop
        on Problem Solving (Math 470/586), "since problem solving depends
        upon a theoretical contribution, it does more than answer getting."
        This workshop, offered July 25 to August 1, not only considers the
        educational uses of problems, but also explores effective ways to
        approach a general variety of them.  Though most of the workshop
        illustrations are drawn from precalculus mathematics, the
        particular solution for the present example also requires some
        differential calculus.