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Berwald J.,Institute for Mathematics and Its Applications | Gidea M.,Yeshiva University | Gidea M.,Institute for Advanced Study
Mathematical Biosciences and Engineering | Year: 2014

We consider a model for substrate-depletion oscillations in genetic systems, based on a stochastic differential equation with a slowly evolving external signal. We show the existence of critical transitions in the system. We apply two methods to numerically test the synthetic time series generated by the system for early indicators of critical transitions: a detrended fluctuation analysis method, and a novel method based on topological data analysis (persistence diagrams).


Billey S.,University of Washington | Hamaker Z.,Institute for Mathematics and its Applications | Roberts A.,Highline College | Young B.,University of Oregon
Electronic Journal of Combinatorics | Year: 2014

We define an analog of David Little’s algorithm for reduced words in type B, and investigate its main properties. In particular, we show that our algorithm preserves the recording tableau of Kraśkiewicz insertion, and that it provides a bijective realization of the Type B transition equations in Schubert calculus. Many other aspects of type A theory carry over to this new setting. Our primary tool is a shifted version of the dual equivalence graphs defined by Assaf and further developed by Roberts. We provide an axiomatic characterization of shifted dual equivalence graphs, and use them to prove a structure theorem for the graph of Type B Coxeter-Knuth relations. © 2014, Australian National University. All rights reserved.


Kuttler K.L.,Brigham Young University | Li J.,Institute for Mathematics and Its Applications | Shillor M.,University of Maryland Baltimore County
Nonlinear Analysis: Real World Applications | Year: 2015

This work presents and analyzes a model for the vibrations of a viscoelastic Gao Beam, which may come in contact with a deformable random foundation and allows for stochastic inputs. The body force involves a stochastic integral that includes Brownian motion. In addition, the gap between the beam and the foundation is a stochastic process, which is one of the novelties in the paper, and contact is described with the normal compliance condition. The existence and uniqueness of strong solutions to the model is established and it is shown that the solutions are adapted to the filtration determined by a given Wiener process for the stochastic force noise term. © 2014 Published by Elsevier Ltd.


Lesnick M.,Institute for Mathematics and its Applications
Foundations of Computational Mathematics | Year: 2015

In 2009, Chazal et al. introduced ϵ-interleavings of persistence modules. ϵ-interleavings induce a pseudometric dI on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of ϵ-interleavings and dI generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view toward applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, dI is equal to the bottleneck distance dB. This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the ϵ-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two ϵ-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, dI satisfies a universality property. This universality result is the central result of the paper. It says that dI satisfies a stability property generalizing one which dB is known to satisfy, and that in addition, if d is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then d≤dI. We also show that a variant of this universality result holds for dB, over arbitrary fields. Finally, we show that dI restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules. © 2015, SFoCM.

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