Physical Sciences and Mathematics Commons^™

Classification Of Minimal Separating Sets In Low Genus Surfaces, J. J. P. Veerman, William Maxwell, Victor Rielly, Austin K. Williams

Mathematics and Statistics Faculty Publications and Presentations

Consider a surface S and let M ⊂ S. If S \ M is not connected, then we say M separates S, and we refer to M as a separating set of S. If M separates S, and no proper subset of M separates S, then we say M is a minimal separating set of S. In this paper we use computational methods of combinatorial topology to classify the minimal separating sets of the orientable surfaces of genus g = 2 and g = 3. The classification for genus 0 and 1 was done …

Variational Geometric Approach To Generalized Differential And Conjugate Calculi In Convex Analysis, Boris S. Mordukhovich, Nguyen Mau Nam, R. Blake Rector, T. Tran

Mathematics and Statistics Faculty Publications and Presentations

This paper develops a geometric approach of variational analysis for the case of convex objects considered in locally convex topological spaces and also in Banach space settings. Besides deriving in this way new results of convex calculus, we present an overview of some known achievements with their unified and simplified proofs based on the developed geometric variational schemes. Key words. Convex and variational analysis, Fenchel conjugates, normals and subgradients, coderivatives, convex calculus, optimal value functions.

Formalizing The Panarchy Adaptive Cycle With The Cusp Catastrophe, Martin Zwick, Joshua Hughes

Systems Science Faculty Publications and Presentations

The panarchy adaptive cycle, a general model for change in natural and human systems, can be formalized by the cusp catastrophe of René Thom's topological theory. Both the adaptive cycle and the cusp catastrophe have been used to model ecological, economic, and social systems in which slow and small continuous changes in two control variables produce fast and large discontinuous changes in system behavior. The panarchy adaptive cycle, the more recent of the two models, has been used so far only for qualitative descriptions of typical dynamics of such systems. The cusp catastrophe, while also often employed qualitatively, is a …

Formalizing The Panarchy Adaptive Cycle With The Cusp Catastrophe [Presentation], Martin Zwick, Joshua Hughes

Systems Science Faculty Publications and Presentations

The panarchy adaptive cycle, a general model for change in natural and human systems, can be formalized by the cusp catastrophe of René Thom's topological theory. Both the adaptive cycle and the cusp catastrophe have been used to model ecological, economic, and social systems in which slow and small continuous changes in two control variables produce fast and large discontinuous changes in system behavior. The panarchy adaptive cycle, the more recent of the two models, has been used so far only for qualitative descriptions of typical dynamics of such systems. The cusp catastrophe, while also often employed qualitatively, is a …

Random Walks On Digraphs, J.J.P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

Let V = {1, · · · n} be a vertex set and S a non-negative row-stochastic matrix (i.e. rows sum to 1). V and S define a digraph G = G(V, S) and a directed graph Laplacian L as follows. If (S)ij > 0 (in what follows we will leave out the parentheses) there is a directed edge j → i. Thus the ith row of S identifies the edges coming into vertex i and their weights. This set of vertices are collectively the neighbors of i, and is denoted by Ni . The diagonal elements Sii are chosen such …

Ideas & Graphs, Martin Zwick

Systems Science Faculty Publications and Presentations

A graph can specify the skeletal structure of an idea, onto which meaning can be added by interpreting the structure.

This paper considers graphs (but not hypergraphs) consisting of four nodes, and suggests meanings that can be associated with several different directed and undirected graphs.

Drawing on Bennett's "systematics," specifically on the Tetrad that systematics offers as a model of 'activity,' the analysis here shows that the Tetrad is versatile model of problem-solving, regulation and control, and other processes.

A Weighted Möbius Function, Derek Garton

Mathematics and Statistics Faculty Publications and Presentations

Fix an odd prime ℓ and let G be the poset of isomorphism classes of finite abelian ℓ-groups, ordered by inclusion. If ξ:G→R^≥0 is a discrete probability distribution on G and A ∈ G, define the Ath moment of ξ to be . The question of determining conditions that ensure ξ is completely determined by its moments has been of recent interest in many problems of Cohen–Lenstra type. Furthermore, recovering ξ from its moments requires a new Möbius-type inversion formula on G. In this paper, we define this function, relate it to the classical Möbius …

A Finite Difference Method For Off-Fault Plasticity Throughout The Earthquake Cycle, Brittany A. Erickson, Eric M. Dunham, Arash Khosravifar

Mathematics and Statistics Faculty Publications and Presentations

We have developed an efficient computational framework for simulating multiple earthquake cycles with off-fault plasticity. The method is developed for the classical antiplane problem of a vertical strike-slip fault governed by rate-and-state friction, with inertial effects captured through the radiationdamping approximation. Both rate-independent plasticity and viscoplasticity are considered, where stresses are constrained by a Drucker-Prager yield condition. The off-fault volume is discretized using finite differences and tectonic loading is imposed by displacing the remote side boundaries at a constant rate. Time-stepping combines an adaptive Runge-Kutta method with an incremental solution process which makes use of an elastoplastic tangent stiffness tensor …

Dynamically Distinguishing Polynomials, Andrew Bridy, Derek Garton

Mathematics and Statistics Faculty Publications and Presentations

A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: For any prime p, reduce its coefficients mod p and consider its action on the field FpFp. We say a subset of Z[x]Z[x] is dynamically distinguishable mod p if the associated mod pdynamical systems are pairwise non-isomorphic. For any k,M∈Z>1k,M∈Z>1, we prove that there are infinitely many sets of integers MM of size M such that {xk+m∣m∈M}{xk+m∣m∈M} is dynamically distinguishable mod p for most p (in the sense of natural density). Our proof uses the Galois theory of dynatomic polynomials largely developed …

Subgradients Of Minimal Time Functions Without Calmness, Nguyen Mau Nam, Dang Van Cuong

Mathematics and Statistics Faculty Publications and Presentations

In recent years there has been great interest in variational analysis of a class of nonsmooth functions called the minimal time function. In this paper we continue this line of research by providing new results on generalized differentiation of this class of functions, relaxing assumptions imposed on the functions and sets involved for the results. In particular, we focus on the singular subdifferential and the limiting subdifferential of this class of functions.

Shift-Symmetric Configurations In Two-Dimensional Cellular Automata: Irreversibility, Insolvability, And Enumeration, Peter Banda, John S. Caughman Iv, Martin Cenek, Christof Teuscher

Mathematics and Statistics Faculty Publications and Presentations

The search for symmetry as an unusual yet profoundly appealing phenomenon, and the origin of regular, repeating configuration patterns have been for a long time a central focus of complexity science, and physics.

Here, we introduce group-theoretic concepts to identify and enumerate the symmetric inputs, which result in irreversible system behaviors with undesired effects on many computational tasks. The concept of so-called configuration shift-symmetry is applied on two-dimensional cellular automata as an ideal model of computation. The results show the universal insolvability of “non-symmetric” tasks regardless of the transition function. By using a compact enumeration formula and bounding the number …

Some Remarks On Interpolation And Best Approximation, Randolph E. Bank, Jeffrey S. Ovall

Mathematics and Statistics Faculty Publications and Presentations

Sufficient conditions are provided for establishing equivalence between best approximation error and projection/interpolation error in finite-dimensional vector spaces for general (semi)norms. The results are applied to several standard finite element spaces, modes of interpolation and (semi)norms, and a numerical study of the dependence on polynomial degree of constants appearing in our estimates is provided.

Space-Time Cfosls Methods With Amge Upscaling, Martin Neumüller, Panayot Vassilevski, Umberto E. Villa

Mathematics and Statistics Faculty Publications and Presentations

This work considers the combined space-time discretization of time-dependent partial differential equations by using first order least square methods. We also impose an explicit constraint representing space-time mass conservation. To alleviate the restrictive memory demand of the method, we use dimension reduction via accurate element agglomeration AMG coarsening, referred to as AMGe upscaling. Numerical experiments demonstrating the accuracy of the studied AMGe upscaling method are provided.

Nesterov's Smoothing Technique And Minimizing Differences Of Convex Functions For Hierarchical Clustering, Mau Nam Nguyen, Wondi Geremew, Sam Raynolds, Tuyen Tran

Mathematics and Statistics Faculty Publications and Presentations

A bilevel hierarchical clustering model is commonly used in designing optimal multicast networks. In this paper we will consider two different formulations of the bilevel hierarchical clustering problem -- a discrete optimization problem which can be shown to be NP-hard. Our approach is to reformulate the problem as a continuous optimization problem by making some relaxations on the discreteness conditions. This approach was considered by other researchers earlier, but their proposed methods depend on the square of the Euclidian norm because of its differentiability. By applying the Nesterov smoothing technique and the DCA -- a numerical algorithm for minimizing differences …

Identifying Clouds With Convolutional Neural Networks, Jeff Mullins, Sean Richardson, Peter Drake

Portland Institute for Computational Science Publications

The greatest source of uncertainty in model estimates of projected climate change involve clouds and aerosols. Photographic images of clouds in the sky are simple to acquire and archive, but climate scientists need an automated process for identifying clouds in these images. We bring machine learning to bear on this problem. Specifically, we use convolutional neural networks, which to our knowledge have not previously been applied to this task. We trained a network to identify clear sky, thin cloud, thick cloud, and non-sky pixels in photos taken by the Total Sky Imager. The trained network is capable of classifying 91.9% …

Equators Have At Most Countable Many Singularities With Bounded Total Angle, Pilar Herreros, Mario Ponce, J.J.P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

For distinct points p and q in a two-dimensional Riemannian manifold, one defines their mediatrix Lpq as the set of equidistant points to p and q. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. In the case of a topological sphere, mediatrices are called equators and it can benoticed that there are no branching points, thus an equator is a topological circle with possibly many Lipschitz singularities. This paper establishes that mediatrices have the radial …

Convergence Analysis Of A Proximal Point Algorithm For Minimizing Differences Of Functions, Thai An Nguyen, Mau Nam Nguyen

Mathematics and Statistics Faculty Publications and Presentations

Several optimization schemes have been known for convex optimization problems. However, numerical algorithms for solving nonconvex optimization problems are still underdeveloped. A significant progress to go beyond convexity was made by considering the class of functions representable as differences of convex functions. In this paper, we introduce a generalized proximal point algorithm to minimize the difference of a nonconvex function and a convex function. We also study convergence results of this algorithm under the main assumption that the objective function satisfies the Kurdyka– ᴌojasiewicz property.

Reorganizing Algebraic Thinking: An Introduction To Dynamic System Modeling, Diana Fisher

Mathematics and Statistics Faculty Publications and Presentations

System Dynamics (SD) modeling is a powerful analytical method used by professional scientists, academics, and governmental officials to study the behavior patterns of complex systems. Specifically through use of the Stella software, it is a method that I and others have used for over two decades with high school, and even middle school, math and science students. In this paper I describe an introduction to SD modeling intended for an algebra class (in either middle or high school). In the body of the paper, a nested sequence of simple bank account examples, increasing in complexity, is used to demonstrate a …

On The Uniformity Of (3/2)N Modulo 1, Paula Neeley, Daniel Taylor-Rodriguez, J.J.P. Veerman, Thomas Roth

Mathematics and Statistics Faculty Publications and Presentations

It has been conjectured that the sequence (3/2)n modulo 1 is uniformly distributed. The distribution of this sequence is significant in relation to unsolved problems in number theory including the Collatz conjecture. In this paper, we describe an algorithm to compute (3/2)n modulo 1 to n = 108 . We then statistically analyze its distribution. Our results strongly agree with the hypothesis that (3/2)n modulo 1 is uniformly distributed.

Predicting Risk Of Adverse Outcomes In Knee Replacement Surgery With Reconstructability Analysis, Cecily Corrine Froemke, Martin Zwick

Systems Science Faculty Publications and Presentations

Reconstructability Analysis (RA) is a data mining method that searches for relations in data, especially non-linear and higher order relations. This study shows that RA can provide useful predictions of complications in knee replacement surgery.