Physical Sciences and Mathematics Commons^™

Decomposition Of Beatty And Complementary Sequences, Geremías Polanco

Mathematics Sciences: Faculty Publications

In this paper we express the difference of two complementary Beatty sequences, as the sum of two Beatty sequences closely related to them. In the process we introduce a new Algorithm that generalizes the well known Minimum Excluded algorithm and provides a method to generate combinatorially any pair of complementary Beatty sequences.

Modules And Representations Up To Homotopy Of Lie N-Algebroids, M. Jotz, Rajan Amit Mehta, T. Papantonis

Mathematics Sciences: Faculty Publications

This paper studies differential graded modules and representations up to homotopy of Lie n-algebroids, for general n ∈ N. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie n-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures.Moreover, theWeil algebra of a …

Frobenius Objects In The Category Of Spans, Ivan Contreras, Molly Keller, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of simplicial sets. An interesting class of examples comes from groupoids.

Our primary motivation is that Span can be viewed as a set-theoretic model for the symplectic category, and thus Frobenius objects in Span provide set-theoretic models for classical topological field theories. The paper includes an explanation of this relationship.

Given a finite commutative Frobenius object in Span, one can obtain …

Relaxed Wythoff Has All Beatty Solutions, Jon Kay, Geremías Polanco

Mathematics Sciences: Faculty Publications

We find conditions under which the P-positions of three subtraction games arise as pairs of complementary Beatty sequences. The first game is due to Fraenkel and the second is an extension of the first game to non-monotone settings. We show that the P-positions of the second game can be inferred from the recurrence of Fraenkel's paper if a certain inequality is satisfied. This inequality is shown to be necessary if the P-positions are known to be pairs of complementary Beatty sequences, and the family of irrationals for which this inequality holds is explicitly given. We highlight several games in the …

Neumann Problems For P-Harmonic Functions, And Induced Nonlocal Operators In Metric Measure Spaces, Luca Capogna, Josh Kline, Riikka Korte, Nageswari Shanmugalingam, Marie Snipes

Mathematics Sciences: Faculty Publications

Following ideas of Caffarelli and Silvestre in [20], and using recent progress in hyperbolic fillings, we define fractional p-Laplacians (−∆p) θ with 0 < θ < 1 on any compact, doubling metric measure space (Z, d, ν), and prove existence, regularity and stability for the non-homogenous non-local equation (−∆p) θu = f. These results, in turn, rest on the new existence, global Hölder regularity and stability theorems that we prove for the Neumann problem for p-Laplacians ∆p, 1 < p < ∞, in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincaré inequality. Our work also includes as special cases much of the previous results by other authors in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure spaces context, our work does not rely on the assumption that (Z, d, ν) supports a Poincaré inequality.

The Transition Matrix Between The Specht And 𝔰𝔩3 Web Bases Is Unitriangular With Respect To Shadow Containment, Heather M. Russell, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for 𝔰𝔩k ⁠. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (e.g., Kazhdan–Lusztig polynomials). By ”flattening” the braiding maps, webs can also be viewed as the basis elements of a symmetric group representation. In this paper, we define …

Lipschitz Regularity For The Parabolic P-Laplacian In The Heisenberg Group, Luca Capogna, Giovanna Citti, Xiao Zhong

Mathematics Sciences: Faculty Publications

In this paper we establish the local Lipschitz regularity of weak solutions of a certain class of quasilinear, degenerate parabolic equations in the Heisenberg group Hn , or more in general in contact subRiemannian manifolds. In particular we extend to the non-stationary setting.

Toward Permutation Bases In The Equivariant Cohomology Rings Of Regular Semisimple Hessenberg Varieties, Megumi Harada, Martha Precup, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Recent work of Shareshian and Wachs, Brosnan and Chow, and Guay-Paquet connects the wellknown Stanley–Stembridge conjecture in combinatorics to the dot action of the symmetric group S_n on the cohomology rings H∗ (Hess(S, h)) of regular semisimple Hessenberg varieties. In particular, in order to prove the Stanley–Stembridge conjecture, it suffices to construct (for any Hessenberg function h) a permutation basis of H∗ (Hess(S, h)) whose elements have stabilizers isomorphic to Young subgroups. In this manuscript we give several results which contribute toward this goal. Specifically, in some special cases, we give a new, purely combinatorial construction of classes …

System Identification Through Lipschitz Regularized Deep Neural Networks, Elisa Negrini, Giovanna Citti, Luca Capogna

Mathematics Sciences: Faculty Publications

In this paper we use neural networks to learn governing equations from data. Specifically we reconstruct the right-hand side of a system of ODEs x˙(t)=f(t,x(t)) directly from observed uniformly time-sampled data using a neural network. In contrast with other neural network-based approaches to this problem, we add a Lipschitz regularization term to our loss function. In the synthetic examples we observed empirically that this regularization results in a smoother approximating function and better generalization properties when compared with non-regularized models, both on trajectory and non-trajectory data, especially in presence of noise. In contrast with sparse regression approaches, since neural networks …

A Neural Network Ensemble Approach To System Identification, Elisa Negrini, Giovanna Citti, Luca Capogna

Mathematics Sciences: Faculty Publications

We present a new algorithm for learning unknown governing equations from trajectory data, using and ensemble of neural networks. Given samples of solutions x(t) to an unknown dynamical system x˙(t) = f(t, x(t)), we approximate the function f using an ensemble of neural networks. We express the equation in integral form and use Euler method to predict the solution at every successive time step using at each iteration a different neural network as a prior for f. This procedure yields M-1 time-independent networks, where M is the number of time steps at which x(t) is observed. Finally, we obtain a …

Linking Numbers In Three-Manifolds, Patricia Cahn, Alexandra Kjuchukova

Mathematics Sciences: Faculty Publications

Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral three-fold cover of S3 branched along a knot α⊂ S3. Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot α can be derived from dihedral covers of α. The linking numbers we compute are necessary for evaluating one …

Hessenberg Varieties Of Parabolic Type, Martha Precup, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

This paper studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each parabolic Hessenberg variety is the pullback of a Steinberg variety under the projection of the flag variety to an appropriate partial flag variety and we give three applications of this result. The first application constructs an explicit paving of all Steinberg varieties in Lie type A in terms of semistandard tableaux. As a result, we obtain an elementary proof of a theorem of Steinberg and Shimomura that the well-known Kostka numbers count the maximal-dimensional irreducible components of …

Representation Stability Of The Cohomology Of Springer Varieties And Some Combinatorial Consequences, Aba Mbirika, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

A sequence of Sn-representations { Vn} is said to be uniformly representation stable if the decomposition of Vn= ⨁ μcμ,nV(μ) n into irreducible representations is independent of n for each μ—that is, the multiplicities cμ,n are eventually independent of n for each μ. Church–Ellenberg–Farb proved that the cohomology of flag varieties (the so-called diagonal coinvariant algebra) is uniformly representation stable. We generalize their result from flag varieties to all Springer fibers. More precisely, we show that for any increasing subsequence of Young diagrams, the corresponding sequence of Springer representations form a graded co-FI-module of finite type (in the sense of …

Courant Cohomology, Cartan Calculus, Connections, Curvature, Characteristic Classes, Miquel Cueca, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

We give an explicit description, in terms of bracket, anchor, and pairing, of the standard cochain complex associated to a Courant algebroid. In this formulation, the differential satisfies a formula that is formally identical to the Cartan formula for the de Rham differential. This perspective allows us to develop the theory of Courant algebroid connections in a way that mirrors the classical theory of connections. Using a special class of connections, we construct secondary characteristic classes associated to any Courant algebroid.

Conformal Equivalence Of Visual Metrics In Pseudoconvex Domains, Luca Capogna, Enrico Le Donne

Mathematics Sciences: Faculty Publications

We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between bounded, smooth strongly pseudoconvex domains in Cn are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between bounded smooth pseudoconvex domains. The proofs are inspired by Mostow’s proof of his rigidity theorem and are based on the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on the Bonk-Schramm hyperbolic fillings.

Computing Ribbon Obstructions For Colored Knots, Patricia Cahn, Alexandra Kjuchukova

Mathematics Sciences: Faculty Publications

Kjuchukova's p invariant gives a ribbon obstruction for Fox p-colored knots. The invariant is derived from dihedral branched covers of 4-manifolds, and can be used to calculate the signatures of these covers when singularities on the branching sets are present. In this note, we give an algorithm for evaluating p from a colored knot diagram, and compute a couple of examples.

Regularity For A Class Of Quasilinear Degenerate Parabolic Equations In The Heisenberg Group, Luca Capogna, Giovanna Citti, Nicola Garofalo

Mathematics Sciences: Faculty Publications

We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in [31] of the Hölder regularity of p-harmonic functions in the Heisenberg group Hn. Given a number p ≥ 2, in this paper we establish the C1 smoothness of weak solutions of a class of quasilinear PDE in Hn modeled on the equation.

Phyllotaxis As Geometric Canalization During Plant Development, Christophe Godin, Christophe Golé, Stéphane Douady

Mathematics Sciences: Faculty Publications

Why living forms develop in a relatively robust manner, despite various sources of internal or external variability, is a fundamental question in developmental biology. Part of the answer relies on the notion of developmental constraints: at any stage of ontogenesis, morphogenetic processes are constrained to operate within the context of the current organism being built. One such universal constraint is the shape of the organism itself, which progressively channels the development of the organism toward its final shape. Here, we illustrate this notion with plants, where strikingly symmetric patterns (phyllotaxis) are formed by lateral organs. This Hypothesis article aims first …

“Playing The Whole Game”: A Data Collection And Analysis Exercise With Google Calendar, Albert Y. Kim, Johanna Hardin

Statistical and Data Sciences: Faculty Publications

We provide a computational exercise suitable for early introduction in an undergraduate statistics or data science course that allows students to “play the whole game” of data science: performing both data collection and data analysis. While many teaching resources exist for data analysis, such resources are not as abundant for data collection given the inherent difficulty of the task. Our proposed exercise centers around student use of Google Calendar to collect data with the goal of answering the question “How do I spend my time?” On the one hand, the exercise involves answering a question with near universal appeal, but …

Integrating Data Science Ethics Into An Undergraduate Major, Benjamin Baumer, Randi L. Garcia, Albert Y. Kim, Katherine M. Kinnaird, Miles Q. Ott

Statistical and Data Sciences: Faculty Publications

We present a programmatic approach to incorporating ethics into an undergraduate major in statistical and data sciences. We discuss departmental-level initiatives designed to meet the National Academy of Sciences recommendation for weaving ethics into the curriculum from top-to-bottom as our majors progress from our introductory courses to our senior capstone course, as well as from side-to-side through co-curricular programming. We also provide six examples of data science ethics modules used in five different courses at our liberal arts college, each focusing on a different ethical consideration. The modules are designed to be portable such that they can be flexibly incorporated …

Frobenius Objects In The Category Of Relations, Rajan Amit Mehta, Ruoqi Zhang

Mathematics Sciences: Faculty Publications

We give a characterization, in terms of simplicial sets, of Frobenius objects in the category of relations. This result generalizes a result of Heunen, Contreras, and Cattaneo showing that special dagger Frobenius objects in the category of relations are in correspondence with groupoids. As an additional example, we construct a Frobenius object in the category of relations whose elements are certain cohomology classes in a compact oriented Riemannian manifold.

Preparing For A Career At A Liberal Arts College, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

No abstract provided.

A Formula For The Cohomology And K-Class Of A Regular Hessenberg Variety, Erik Insko, Julianna Tymoczko, Alexander Woo

Mathematics Sciences: Faculty Publications

Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator X and a nondecreasing function h. The family of Hessenberg varieties for regular X is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and K-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on h. Our formula and our methods are different from a recent result …

Circada: Shiny Apps For Exploration Of Experimental And Synthetic Circadian Time Series With An Educational Emphasis, Lisa Cenek, Liubou Klindziuk, Cindy Lopez, Eleanor Mccartney, Blanca Martin Burgos, Selma Tir, Mary E. Harrington, Tanya L. Leise

Psychology: Faculty Publications

Circadian rhythms are daily oscillations in physiology and behavior that can be assessed by recording body temperature, locomotor activity, or bioluminescent reporters, among other measures. These different types of data can vary greatly in waveform, noise characteristics, typical sampling rate, and length of recording. We developed 2 Shiny apps for exploration of these data, enabling visualization and analysis of circadian parameters such as period and phase. Methods include the discrete wavelet transform, sine fitting, the Lomb-Scargle periodogram, autocorrelation, and maximum entropy spectral analysis, giving a sense of how well each method works on each type of data. The apps also …

How Emergent Social Patterns In Allogrooming Combat Parasitic Infections, Shelby N. Wilson, Suzanne S. Sindi, Heather Z. Brooks, Maryann E. Hohn, Candice R. Price, Ami E. Radunskaya, Nakeya D. Williams, Nina H. Fefferman

Mathematics Sciences: Faculty Publications

Members of social groups risk infection through contact with those in their social network. Evidence that social organization may protect populations from pathogens in certain circumstances prompts the question as to how social organization affects the spread of ectoparasites. The same grooming behaviors that establish social bonds also play a role in the progression of ectoparasitic outbreaks. In this paper, we model the interactions between social organization and allogrooming efficiency to consider how ectoparasitic threats may have shaped the evolution of social behaviors. To better understand the impacts of social grooming on organizational structure, we consider several dynamic models of …

A Permutation Test And Spatial Cross-Validation Approach To Assess Models Of Interspecific Competition Between Trees, David Allen, Albert Y. Kim

Statistical and Data Sciences: Faculty Publications

Measuring species-specific competitive interactions is key to understanding plant communities. Repeat censused large forest dynamics plots offer an ideal setting to measure these interactions by estimating the species-specific competitive effect on neighboring tree growth. Estimating these interaction values can be difficult, however, because the number of them grows with the square of the number of species. Furthermore, confidence in the estimates can be overestimated if any spatial structure of model errors is not considered. Here we measured these interactions in a forest dynamics plot in a transitional oak-hickory forest. We analytically fit Bayesian linear regression models of annual tree radial …

Analysis In Metric Spaces, Mario Bonk, Luca Capogna, Piotr Hajłasz, Nageswari Shanmugalingam, Jeremy Tyson

Mathematics Sciences: Faculty Publications

No abstract provided.

Three-Dimensional Viscoelastic Instabilities In A Four-Roll Mill Geometry At The Stokes Limit, Paloma Gutierrez-Castillo, Adam Kagel, Becca Thomases

Mathematics Sciences: Faculty Publications

Three-dimensional numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll mill background force (extended to the third dimension) were performed. Both the Oldroyd-B model and FENE-P model of viscoelastic fluids were used. Different temporal behaviors were observed depending on the Weissenberg number (non-dimensional relaxation time), model, and initial conditions. Temporal dynamics evolve on long time scales, and simulations were accelerated by using a Graphics Processing Unit (GPU). Previously, parameter explorations and long-time simulations in 3D were prohibitively expensive. For a small Weissenberg number, all the solutions are constant in the third dimension, displaying strictly two-dimensional temporal evolutions. …

A Filtration On The Cohomology Rings Of Regular Nilpotent Hessenberg Varieties, Megumi Harada, Tatsuya Horiguchi, Satoshi Murai, Martha Precup, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Let n be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in GL(n, C) / B such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in GL(n- 1 , C) / B, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce …

Graphs Admitting Only Constant Splines, Katie Anders, Alissa S. Crans, Briana Foster-Greenwood, Blake Mellor, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

We study generalized graph splines, introduced by Gilbert, Tymoczko, and Viel (2016). For a large class of rings, we characterize the graphs that only admit constant splines. To do this, we prove that if a graph has a particular type of cutset (e.g., a bridge), then the space of splines naturally decomposes as a certain direct sum of submodules. As an application, we use these results to describe splines on a triangulation studied by Zhou and Lai, but over a different ring than they used.