Friday, February 26th, 2016

Van Cyr – Cyr, Van and Kra, Bryna. “Complexity of Short Rectangles and Periodicity.” European Journal of Combinatorics 52, no. Pt. A (2016) : 146-173.

Van Cyr, Assistant Professor of Mathematics

The Morse-Hedlund Theorem states that a bi-infinite sequence eta in a finite alphabet is periodic if and only if there exists n is an element of N such that the block complexity function P-eta(n) satisfies P-eta(n) <= n. In dimension two, Nivat conjectured that if there exist n, k is an element of N such that the n x k rectangular complexity P-eta(n, k) satisfies P-eta(n, k) <= nk, then eta is periodic. Sander and Tijdeman showed that this holds for k <= 2. We generalize their result, showing that Nivat's Conjecture holds for k <= 3. The method involves translating the combinatorial problem to a question about the nonexpansive subspaces of a certain Z(2) dynamical system, and then analyzing the resulting system. (C) 2015 Elsevier Ltd. All rights reserved.

Cyr, Van and Kra, Bryna. “Complexity of Short Rectangles and Periodicity.” European Journal of Combinatorics 52, no. Pt. A (2016) : 146-173.

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Friday, February 26th, 2016

Pamela Gorkin – Gorkin, Pamela and Akeroyd, John. “Uniform Approximation by Indestructible Blaschke Products.” Journal of Mathematical Analysis and its Applications 434, no. 2 (2016) : 1419-1434.

Pamela Gorkin, Professor of Mathematics

We address the question: Are the inner functions in the uniform closure in the algebra of bounded analytic functions of the indestructible Blaschke products? We show, in particular, that every inner function with countable spectrum is in the closure of the indestructible Blaschke products, that every Blaschke product is a product of two indestructible Blaschke products and we study approximation in modulus.

Gorkin, Pamela and Akeroyd, John. “Uniform Approximation by Indestructible Blaschke Products.” Journal of Mathematical Analysis and its Applications 434, no. 2 (2016) : 1419-1434.

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Friday, February 26th, 2016

Van Cyr – Cyr, Van and Kra, Bryna. “Nonexpansive Z(2)-Subdynamics and Nivat’s Conjecture.”Transactions of the American Mathematical Society 367, no. 9 (2015) : 6487-6537.

Van Cyr, Assistant Professor of Mathematics

For a finite A, and eta: Z -> A the Morse-Hedlund Theorem states that eta is periodic if and only if there exists n E N such that the block complexity function P-eta(n) satisfies P-eta(n) <= n, and this statement is naturally studied by analyzing the dynamics of a Z-action associated with ai. In dimension two, we analyze the subdynamics of a Z(2)-action associated with eta: Z(2) -> A and show that if there exist n,k is an element of N such that the ii x k rectangular complexity (n, k) satisfies P,7(n k) < nk, then the periodicity of eta is equivalent to a statement about the expansive subspaces of this action. As a corollary, we show that if there exist n, k E N such that P-eta(n, k) <= nk/2, then eta is periodic. This proves a weak form of a conjecture of Nivat in the combinatorics of words.

Cyr, Van and Kra, Bryna. “Nonexpansive Z(2)-Subdynamics and Nivat’s Conjecture.”Transactions of the American Mathematical Society 367, no. 9 (2015) : 6487-6537.

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Friday, February 26th, 2016

Jeffrey J. Langford – Langford, Jeffrey J. “Neumann Comparison Theorems in Elliptic PDEs.” Potential Analysis 43, no. 3 (2015) : 415-459.

Jeffrey J. Langford, Assistant Professor of Mathematics

In this paper we prove a spherical comparison result for the (k,n)-spherical rearrangement using the spherical Green’s function and a rearrangement inequality of A. Baernstein. We next use a simple reflection argument to obtain a Neumann comparison result on a hemisphere for the (k,n)-hemispherical rearrangement. Using the Lambert equal-area projection and stereographic projection, we obtain Neumann comparison results in Euclidean space. We end with a discussion of open problems in the unit disk involving heat flow.

Langford, Jeffrey J. “Neumann Comparison Theorems in Elliptic PDEs.” Potential Analysis 43, no. 3 (2015) : 415-459.

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Friday, February 26th, 2016

Peter R.W. McNamara – McNamara, Peter R. W. and Steingrímsson, Einar. “On the Topology of the Permutation Pattern Poset.” Journal of Combinatorial Theory, Series A 134, (2015) : 1-35.

Peter R.W. McNamara, Associate Professor of Mathematics

The set of all permutations, ordered by pattern containment, forms a poset. This paper presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the Möbius function of decomposable permutations given by Burstein et al.

McNamara, Peter R. W. and Steingrímsson, Einar. “On the Topology of the Permutation Pattern Poset.” Journal of Combinatorial Theory, Series A 134, (2015) : 1-35.

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Friday, February 26th, 2016

Ueli Daepp – Daepp, Ueli; Gorkin, Pamela; Shaffer, Andrew; Sokolowsky, Benjamin; and Voss, Karl. “Decomposing Finite Blaschke Products.” Journal of Mathematical Analysis and Applications 426, no. 2 (2015) : 1201-1216.

Ueli Daepp, Professor of Mathematics

We present four algorithms to determine whether or not a Blaschke product is a composition of two non-trivial Blaschke products and, if it is, the algorithms suggest what the composition must be. The initial algorithm is a naive counting argument, the second considers critical values and the counting argument, the third is a geometric argument that exploits the relationship between Blaschke products and curves with the Ponce let property, and it can also be expressed in terms of a group associated with the Blaschke product. The final algorithm looks at inverse images under the Blaschke product. Our algorithms are accompanied by an applet that implements them. (C) 2015 Elsevier Inc. All rights reserved.

Daepp, Ueli; Gorkin, Pamela; Shaffer, Andrew; Sokolowsky, Benjamin; and Voss, Karl. “Decomposing Finite Blaschke Products.” Journal of Mathematical Analysis and Applications 426, no. 2 (2015) : 1201-1216.

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Friday, February 26th, 2016

Mark J. Meyer – Meyer, Mark J.; Coull, Brent A.; Versace, Francesco; Cinciripini, Paul; and Morris, Jeffrey S. “Bayesian Function-on-Function Regression for Multilevel Functional Data.” Biometrics 71, no. 3 (2015) : 563-574.

Mark J. Meyer, Assistant Professor of Mathematics

Medical and public health research increasingly involves the collection of complex and high dimensional data. In particular, functional data-where the unit of observation is a curve or set of curves that are finely sampled over a grid-is frequently obtained. Moreover, researchers often sample multiple curves per person resulting in repeated functional measures. A common question is how to analyze the relationship between two functional variables. We propose a general function-on-function regression model for repeatedly sampled functional data on a fine grid, presenting a simple model as well as a more extensive mixed model framework, and introducing various functional Bayesian inferential procedures that account for multiple testing. We examine these models via simulation and a data analysis with data from a study that used event-related potentials to examine how the brain processes various types of images.

Meyer, Mark J.; Coull, Brent A.; Versace, Francesco; Cinciripini, Paul; and Morris, Jeffrey S. “Bayesian Function-on-Function Regression for Multilevel Functional Data.” Biometrics 71, no. 3 (2015) : 563-574.

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Friday, February 26th, 2016

Emily Dryden – Bahuaud, Eric; Dryden, Emily; and Vertman, Boris. “Mapping Properties of the Heat Operator on Edge Manifolds.” Mathematische Nachrichten 288, (2015 ).

Emily Dryden, Associate Professor of Mathematics

We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity. We establish the mapping properties of the heat operator, recovering and extending the classical results from smooth manifolds and conical spaces. The estimates, together with strong continuity of the heat operator, yield short-time existence of solutions to certain semilinear parabolic equations. Our discussion reviews and generalizes earlier work by Jeffres and Loya.

Bahuaud, Eric; Dryden, Emily; and Vertman, Boris. “Mapping Properties of the Heat Operator on Edge Manifolds.” Mathematische Nachrichten 288, (2015 ).

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Friday, February 26th, 2016

Adam Piggott – Piggott, Adam. “On Groups Presented by Monadic Rewriting Systems with Generators of Finite Order.” Bulletin of the Australian Mathematical Society 91, no. 3 (2015) : 426-434.

Adam Piggott, Associate Professor of Mathematics

We prove that the groups presented by finite convergent monadic rewriting systems with generators of finite order are exactly the free products of finitely many finite groups, thereby confirming Gilman’s Conjecture in a special case. We also prove that the finite cyclic groups of order at least three are the only finite groups admitting a presentation by more than one finite convergent monadic rewriting system (up to relabeling), and these admit presentation by exactly two such rewriting systems.

Piggott, Adam. “On Groups Presented by Monadic Rewriting Systems with Generators of Finite Order.” Bulletin of the Australian Mathematical Society 91, no. 3 (2015) : 426-434.

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Friday, February 26th, 2016

George R. Exner – Exner, George R.; Jinh, Joo Young; Jung, Ii Bong; and Lee, Mi Ryeong. “Weighted Shifts Induced by Hamburger Moment Sequences.” Journal of Mathematical Analysis and Applications 427, no. 2 (2015) : 581-599.

George R. Exner, Professor of Mathematics

We indicate how our subject emerges from the confluence of several streams of analysis, including the classical moment problems, the theory of positive matrices and subnormal operator theory. Some new properties H(n (n, = 1,2, ..) and a Hamburger-type weighted shift are considered via a Hamburger moment sequence. We discuss examples to show the various H(n) are distinct; study flatness, backward n-step extensions and perturbations of weighted shifts; and, given three initial weights alpha(0), alpha(1), alpha(2) with alpha(0) <= alpha(2) < alpha(1), we produce a completion: a weighted shift of Hamburger type but not subnormal, extending a (subnormal) completion by Stampfli in the case alpha(0) < alpha(1) < alpha(2) (C) 2015 Elsevier Inc. All rights reserved.

Exner, George R.; Jinh, Joo Young; Jung, Ii Bong; and Lee, Mi Ryeong. “Weighted Shifts Induced by Hamburger Moment Sequences.” Journal of Mathematical Analysis and Applications 427, no. 2 (2015) : 581-599.

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Friday, February 26th, 2016

Abby Flynt – Flynt, Abby and Daepp, Madeleine I.G. “Diet-Related Chronic Disease in the Northeastern United States: a Model-Based Clustering Approach.” International Journal of Health Geographics 14, no. 1 (2015) : 25.

Abby Flynt, Assistant Professor of Mathematics

Background: Obesity and diabetes are global public health concerns. Studies indicate a relationship between socioeconomic, demographic and environmental variables and the spatial patterns of diet-related chronic disease. In this paper, we propose a methodology using model-based clustering and variable selection to predict rates of obesity and diabetes. We test this method through an application in the northeastern United States. Methods: We use model-based clustering, an unsupervised learning approach, to find latent clusters of similar US counties based on a set of socioeconomic, demographic, and environmental variables chosen through the process of variable selection. We then use Analysis of Variance and Post-hoc Tukey comparisons to examine differences in rates of obesity and diabetes for the clusters from the resulting clustering solution. Results: We find access to supermarkets, median household income, population density and socioeconomic status to be important in clustering the counties of two northeastern states. The results of the cluster analysis can be used to identify two sets of counties with significantly lower rates of diet-related chronic disease than those observed in the other identified clusters. These relatively healthy clusters are distinguished by the large central and large fringe metropolitan areas contained in their component counties. However, the relationship of socio-demographic factors and diet-related chronic disease is more complicated than previous research would suggest. Additionally, we find evidence of low food access in two clusters of counties adjacent to large central and fringe metropolitan areas. While food access has previously been seen as a problem of inner-city or remote rural areas, this study offers preliminary evidence of declining food access in suburban areas. Conclusions: Model-based clustering with variable selection offers a new approach to the analysis of socioeconomic, demographic, and environmental data for diet-related chronic disease prediction. In a test application to two northeastern states, this method allows us to identify two sets of metropolitan counties with significantly lower diet-related chronic disease rates than those observed in most rural and suburban areas. Our method could be applied to larger geographic areas or other countries with comparable data sets, offering a promising method for researchers interested in the global increase in diet-related chronic disease.

Flynt, Abby and Daepp, Madeleine I.G. “Diet-Related Chronic Disease in the Northeastern United States: a Model-Based Clustering Approach.” International Journal of Health Geographics 14, no. 1 (2015) : 25.

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Friday, February 26th, 2016

Pamela Gorkin – Gorkin, Pamela and Akeroyd, John. “Constructing Frostman-Blaschke Products and Applications to Operators on Weighted Bergman Spaces.” J. Operator Theory 74, no. 1 (2015) : 149-175.

Pamela Gorkin, Professor of Mathematics

We give an example of a uniform Frostman–Blaschke product B, whose spectrum is a Cantor set, such that the composition operator C_B is not closed-range on any weighted Bergman space A_{alpha}^p, answering two questions posed in recent papers. We include some general observations about these Blaschke products. Using methods developed in our first example, we improve upon a theorem of V.I. Vasjunin concerning the rate at which the zeros of a uniform Frostman–Blaschke product approach the unit circle.

Gorkin, Pamela and Akeroyd, John. “Constructing Frostman-Blaschke Products and Applications to Operators on Weighted Bergman Spaces.” J. Operator Theory 74, no. 1 (2015) : 149-175.

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Friday, February 26th, 2016

Peter A. Brooksbank – Brooksbank, Peter A. and Wilson, James B. “The Module Isomorphism Problem Reconsidered.”Journal of Algebra 421, no. SI (2015) : 541-559.

Peter A. Brooksbank, Professor of Mathematics

Algorithms to decide isomorphism of modules have been honed continually over the last 30 years, and their range of applicability has been extended to include modules over a wide range of rings. Highly efficient computer implementations of these algorithms form the bedrock of systems such as GAP and MAGMA, at least in regard to computations with groups and algebras. By contrast, the fundamental problem of testing for isomorphism between other types of algebraic structures – such as groups, and almost any type of algebra – seems today as intractable as ever. What explains the vastly different complexity status of the module isomorphism problem? This paper argues that the apparent discrepancy is explained by nomenclature. Current algorithms to solve module isomorphism, while efficient and immensely useful, are actually solving a highly constrained version of the problem. We report that module isomorphism in its general form is as hard as algebra isomorphism and graph isomorphism, both well-studied problems that are widely regarded as difficult. On a more positive note, for cyclic rings we describe a polynomial-time algorithm for the general module isomorphism problem. We also report on a MAGMA implementation of our algorithm. (C) 2014 Elsevier Inc. All rights reserved.

Brooksbank, Peter A. and Wilson, James B. “The Module Isomorphism Problem Reconsidered.”Journal of Algebra 421, no. SI (2015) : 541-559.

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Friday, February 26th, 2016

Pamela Gorkin – Gorkin, Pamela; Daepp, Ulrich, Voss, Karl; Shaffer, Andrew; and Sokolowsky, Benjamin. “Decomposing Finite Blaschke Products.” Journal of Mathematical Analysis and its Applications426, no. 2 (2015) : 1201-1216.

Pamela Gorkin, Professor of Mathematics

We present four algorithms to determine whether or not a Blaschke product is a composition of two non-trivial Blaschke products and, if it is, the algorithms suggest what the composition must be. The initial algorithm is a naive counting argument, the second considers critical values and the counting argument, the third is a geometric argument that exploits the relationship between Blaschke products and curves with the Poncelet property, and it can also be expressed in terms of a group associated with the Blaschke product. The final algorithm looks at inverse images under the Blaschke product. Our algorithms are accompanied by an applet that implements them.

Gorkin, Pamela; Daepp, Ulrich, Voss, Karl; Shaffer, Andrew; and Sokolowsky, Benjamin. “Decomposing Finite Blaschke Products.” Journal of Mathematical Analysis and its Applications426, no. 2 (2015) : 1201-1216.

Continue reading »

Friday, February 26th, 2016

Peter A. Brooksbank – Brooksbank, Peter A. “The Module Isomorphism Problem Reconsidered.” Journal of Algebra 421, (2015) : 541-559.

Peter A. Brooksbank, Professor of Mathematics

Algorithms to decide isomorphism of modules have been honed continually over the last 30 years, and their range of applicability has been extended to include modules over a wide range of rings. Highly efficient computer implementations of these algorithms form the bedrock of systems such as GAP and MAGMA, at least in regard to computations with groups and algebras. By contrast, the fundamental problem of testing for isomorphism between other types of algebraic structures — such as groups, and almost any type of algebra — seems today as intractable as ever. What explains the vastly different complexity status of the module isomorphism problem?

This paper argues that the apparent discrepancy is explained by nomenclature. Current algorithms to solve module isomorphism, while efficient and immensely useful, are actually solving a highly constrained version of the problem. We report that module isomorphism in its general form is as hard as algebra isomorphism and graph isomorphism, both well-studied problems that are widely regarded as difficult. On a more positive note, for cyclic rings we describe a polynomial-time algorithm for the general module isomorphism problem. We also report on a MAGMA implementation of our algorithm.

Brooksbank, Peter A. “The Module Isomorphism Problem Reconsidered.” Journal of Algebra 421, (2015) : 541-559.

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Friday, February 26th, 2016

Pamela Gorkin – Gorkin, Pamela; Wick, Brett; and Pott, Sandra. “Thin Sequences and Their Role in Hp Theory, Model Spaces, and Uniform Algebras.” Revista Matemática Iberoamericana 31, no. 3 (2015) : 841-864.

Pamela Gorkin, Professor of Mathematics

In this paper we revisit some facts about thin interpolating sequences in the unit disc from three perspectives: uniform algebras, model spaces, and Hp spaces. We extend the notion of asymptotic interpolation to Hp spaces, for p between 1 and infinity, providing several new ways to think about these sequences.

Gorkin, Pamela; Wick, Brett; and Pott, Sandra. “Thin Sequences and Their Role in Hp Theory, Model Spaces, and Uniform Algebras.” Revista Matemática Iberoamericana 31, no. 3 (2015) : 841-864.

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Friday, February 26th, 2016

Thomas Cassidy – Cassidy, Thomas. “Modules with Pure Resolutions over Koszul Algebras Revisited.” Journal of Algebra and Its Applications 15, no. 2 (2016) : 50035-50035.

Thomas Cassidy, Professor of Mathematics

I construct a Koszul algebra A and a finitely generated graded A-module M that together form a counterexample to a recently published claim. M is generated in degree 0 and has a pure resolution, and the graded Jacobson radical of the Yoneda algebra of A does not annihilate the Ext module of M, but nonetheless M is not a Koszul module.

Cassidy, Thomas. “Modules with Pure Resolutions over Koszul Algebras Revisited.” Journal of Algebra and Its Applications 15, no. 2 (2016) : 50035-50035.

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Friday, February 26th, 2016

Pamela Gorkin – Gorkin, Pamela and Wick, Brett. “Thin Sequences and Their Role in Model Spaces and Douglas Algebras.” Journal of Fourier Analysis and Applications 22, no. 1 (2016) : 137-158.

Pamela Gorkin, Professor of Mathematics

We study thin interpolating sequences (z_n) and their relationship to interpolation in the Hardy space H^2 and the model spaces K_Theta = H^2 ominus Theta H^2, where Theta is an inner function. Our results, phrased in terms of the functions that do the interpolation as well as Carleson measures, show that under the assumption that Theta (z _n) tends to 0. The interpolation properties in H^2 are essentially the same as those in K_Theta.

Gorkin, Pamela and Wick, Brett. “Thin Sequences and Their Role in Model Spaces and Douglas Algebras.” Journal of Fourier Analysis and Applications 22, no. 1 (2016) : 137-158.

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Friday, February 26th, 2016

Kelly Bickel – Bickel, Kelly and Wick, Brett D. “A Study of the Matrix Carleson Embedding Theorem with Applications to Sparse Operators.” Journal of Mathematical Analysis and Applications 435, no. 1 (2016) : 229-243.

Kelly Bickel, Assistant Professor of Mathematics

In this paper, we study the dyadic Carleson Embedding Theorem in the matrix weighted setting. We provide two new proofs of this theorem, which highlight connections between the matrix Carleson Embedding Theorem and both maximal functions and H-1-BMO duality. Along the way, we establish boundedness results about maximal functions associated to matrix A(2) weights and duality results concerning H-1 and BMO sequence spaces in the matrix setting. As an application, we then use this Carleson Embedding Theorem to show that if S is a sparse operator, then the operator norm of S on L-2 (W) satisfies parallel to S parallel to(2)(L2(W)-> L)((W)) less than or similar to [W](A2)(3/2), for every matrix A(2) weight W. (C) 2015 Elsevier Inc. All rights reserved.

Bickel, Kelly and Wick, Brett D. “A Study of the Matrix Carleson Embedding Theorem with Applications to Sparse Operators.” Journal of Mathematical Analysis and Applications 435, no. 1 (2016) : 229-243.

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