Ergodic Theory, Dynamical systems, Fractals and Applications
Org: Shafiqul Islam
(University of Prince Edward island) and Franklin Mendivil
- ARNO BERGER, University of Alberta
- MATT BETTI, Mount Allison University
- ILLIA BINDER, University of Toronto
- ERIK BOLLT, Clarkson University
On Matching, and Even Rectifying, Dynamical Systems through Koopman Operator Eigenfunctions [PDF]
Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of non- linear dynamic behavior (e.g. through normal forms). In this presentation we will argue that the use of the Koopman operator and its spectrum is particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven algorithm developments. We believe, and document through illustrative examples, that this can nontrivially extend the use and applicability of the Koopman spectral theoretical and computational machinery beyond modeling and prediction, towards what can be considered as a systematic discovery of “Cole-Hopf-type” transformations for dynamics.
- CHRISTOPHER BOSE, University of Victoria
- TRUBEE DAVISON, University of Colorado, Boulder
- PEYMEN ESLAMI, University of Warwick
- CHRISTOPHER ESSEX, Western University
- MARLENE FRIGON, Universite de Montreal
Existence and multiplicity results for systems of differential equations [PDF]
In this talk, we will present existence and multiplicity results for systems of first order differential equations. One should point out that one can find very few multiplicity results for systems of differential equations in the literature. Our results will rely on a new notion called the method of solution-region. This notion generalizes the method of upper and lower solutions in the scalar case.
- IGNACIO GARCIA, University of Waterloo
- SHAFIQUL ISLAM, University of Prince Edward Island
- KAMRAN KAVEH, Harvard University
- ZHENYANG LI, Honghe University, China
- FRANKLIN MENDIVIL, Acadia University
- DORETTE PRONK, Dalhousie University
- KAZI RAHMAN, York University
- S M ASHRAFUR RAHMAN, York University
- TORU SERA, Kyoto University
- SASCHA TROSCHEIT, University of Waterloo
Self-conformal sets with positive Hausdorff measure [PDF]
We investigate the Hausdorff measure and content on a class of quasi self-similar sets that include self-conformal sets. We show that any Hausdorff measurable subset of such sets has comparable Hausdorff measure and Hausdorff content. In particular, we show that graph-directed and sub self-conformal sets with positive Hausdorff measure are Ahlfors regular, irrespective of separation conditions. We use this to resolve a self-conformal extension of the dimension drop conjecture for self-conformal subsets of the line with positive Hausdorff measure by showing that its Hausdorff dimension falls below the expected value if and only if there are exact overlaps.
- SHARIF ULLAH, Kitami Institute of Technology, Japan
Application of Different Affine Mappings for Design and Manufacturing of Complex Shapes [PDF]
Affine maps can be used to model complex shapes wherein the shapes are represented by some point clouds. For example, a set of predefined affine maps known as Iterative Function Systems (IFS) can be used recursively to create fractals (complex shapes). Sometimes, the parameters associated with the IFS needs fine-tuning for the sake of modeling. However, we revisit the concept of the affine map (including IFS) from the viewpoint of the ease of design and manufacturing of complex shapes. In this talk, we thus present the mathematical settings regarding affine mapping for sake of design and manufacturing of complex shapes. In particular, we show some examples of CAD models of complex shapes created from some affine-mapping-driven point clouds. In addition, we show the physical models of the respective CAD, which are manufactured by additive manufacturing (3D printing). Moreover, we highlight the challenges that lie ahead.
- MICHAEL YAMPOLSKY, University of Toronto