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Articles 121 - 150 of 297
Full-Text Articles in Mathematics
Splines In Geometry And Topology, Julianna Tymoczko
Splines In Geometry And Topology, Julianna Tymoczko
Mathematics Sciences: Faculty Publications
This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject.
A Bayesian Method For Cluster Detection With Application To Five Cancer Sites In Puget Sound, Albert Y. Kim, Jon Wakefield
A Bayesian Method For Cluster Detection With Application To Five Cancer Sites In Puget Sound, Albert Y. Kim, Jon Wakefield
Mathematics Sciences: Faculty Publications
Cluster detection is an important public health endeavor and in this paper we describe and apply a recently developed Bayesian method. Commonly-used approaches are based on so-called scan statistics and suffer from a number of difficulties including how to choose a level of significance and how to deal with the possibility of multiple clusters. The basis of our model is to partition the study region into a set of areas which are either “null” or “non-null”, the latter corresponding to clusters (excess risk) or anti-clusters (reduced risk). We demonstrate the Bayesian method and compare with a popular existing approach, using …
Generalized Splines On Arbitrary Graphs, Simcha Gilbert, Julianna Tymoczko, Shira Viel
Generalized Splines On Arbitrary Graphs, Simcha Gilbert, Julianna Tymoczko, Shira Viel
Mathematics Sciences: Faculty Publications
Let G be a graph whose edges are labeled by ideals of a commutative ring. We introduce a generalized spline, which is a vertex labeling of G by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the corresponding edge ideal. Generalized splines arise naturally in combinatorics (algebraic splines of Billera and others) and in algebraic topology (certain equivariant cohomology rings, described by Goresky, Kottwitz, and MacPherson, among others). The central question of this paper asks when an arbitrary edge-labeled graph has nontrivial generalized splines. The answer is “always”, and we …
Sticky Seeding In Discrete-Time Reversible-Threshold Networks, Gwen Spencer
Sticky Seeding In Discrete-Time Reversible-Threshold Networks, Gwen Spencer
Mathematics Sciences: Faculty Publications
When nodes can repeatedly update their behavior (as in agent-based models from computational social science or repeated-game play settings) the problem of optimal network seeding becomes very complex. For a popular spreading-phenomena model of binary-behavior updating based on thresholds of adoption among neighbors, we consider several planning problems in the design of Sticky Interventions: when adoption decisions are reversible, the planner aims to find a Seed Set where temporary intervention leads to long-term behavior change. We prove that completely converting a network at minimum cost is Ω(ln(OP T ))-hard to approximate and that maximizing conversion subject to a budget is …
Equilibrium Circulation And Stress Distribution In Viscoelastic Creeping Flow, Joseph A. Biello, Becca Thomases
Equilibrium Circulation And Stress Distribution In Viscoelastic Creeping Flow, Joseph A. Biello, Becca Thomases
Mathematics Sciences: Faculty Publications
An analytic, asymptotic approximation of the nonlinear steady-state equations for viscoelastic creeping flow, modeled by the Oldroyd-B equations with polymer stress diffusion, is derived. Near the extensional stagnation point the flow stretches and aligns polymers along the outgoing streamlines of the stagnation point resulting in a stress-island, or birefringent strand. The polymer stress diffusion coefficient is used, both as an asymptotic parameter and a regularization parameter. The structure of the singular part of the polymer stress tensor is a Gaussian aligned with the incoming streamline of the stagnation point a smoothed δ-distribution whose width is proportional to the square-root of …
Unequal Edge Inclusion Probabilities In Link-Tracing Network Sampling With Implications For Respondent-Driven Sampling, Miles Q. Ott, Krista J. Gile
Unequal Edge Inclusion Probabilities In Link-Tracing Network Sampling With Implications For Respondent-Driven Sampling, Miles Q. Ott, Krista J. Gile
Statistical and Data Sciences: Faculty Publications
Respondent-Driven Sampling (RDS) is a widely adopted linktracing sampling design used to draw valid statistical inference from samples of populations for which there is no available sampling frame. RDS estimators rely upon the assumption that each edge (representing a relationship between two individuals) in the underlying network has an equal probability of being sampled. We show that this assumption is violated in even the simplest cases, and that RDS estimators are sensitive to the violation of this assumption.
A Generalization Of Turaev’S Virtual String Cobracket And Self-Intersections Of Virtual Strings, Patricia Cahn
A Generalization Of Turaev’S Virtual String Cobracket And Self-Intersections Of Virtual Strings, Patricia Cahn
Mathematics Sciences: Faculty Publications
Previously we defined an operation µ that generalizes Turaev’s cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding question for virtual strings, and conjecture that µ gives a formula for the minimum number of self-intersection points of a virtual string in a given virtual homotopy class. To support the conjecture, we show that µ gives a bound on the minimal self-intersection number of a virtual string which is …
A Coin Vibrational Motor Swimming At Low Reynolds Number, Alice C. Quillen, Hesam Askari, Douglas H. Kelley, Tamar Friedmann, Patrick W. Oakes
A Coin Vibrational Motor Swimming At Low Reynolds Number, Alice C. Quillen, Hesam Askari, Douglas H. Kelley, Tamar Friedmann, Patrick W. Oakes
Mathematics Sciences: Faculty Publications
Low-cost coin vibrational motors, used in haptic feedback, exhibit rotational internal motion inside a rigid case. Because the motor case motion exhibits rotational symmetry, when placed into a fluid such as glycerin, the motor does not swim even though its oscillatory motions induce steady streaming in the fluid. However, a piece of rubber foam stuck to the curved case and giving the motor neutral buoyancy also breaks the rotational symmetry allowing it to swim. We measured a 1 cm diameter coin vibrational motor swimming in glycerin at a speed of a body length in 3 seconds or at 3 mm/s. …
Geometric Deformations Of Sodalite Frameworks, Ciprian Borcea, Ileana Streinu
Geometric Deformations Of Sodalite Frameworks, Ciprian Borcea, Ileana Streinu
Computer Science: Faculty Publications
In mathematical crystallography and computational materials science, it is important to infer flexibility properties of framework materials from their geometric representation. We study combinatorial, geometric and kinematic properties for frameworks modeled on sodalite.
Geometric Auxetics, Ciprian Borcea, Ileana Streinu
Geometric Auxetics, Ciprian Borcea, Ileana Streinu
Computer Science: Faculty Publications
We formulate a mathematical theory of auxetic behavior based on one-parameter deformations of periodic frameworks. Our approach is purely geometric, relies on the evolution of the periodicity lattice and works in any dimension. We demonstrate its usefulness by predicting or recognizing, without experiment, computer simulations or numerical approximations, the auxetic capabilities of several well-known structures available in the literature. We propose new principles of auxetic design and rely on the stronger notion of expansive behavior to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures.
Hypercube Unfoldings That Tile R3 And R2, Giovanna Diaz, Joseph O'Rourke
Hypercube Unfoldings That Tile R3 And R2, Giovanna Diaz, Joseph O'Rourke
Computer Science: Faculty Publications
We show that the hypercube has a face-unfolding that tiles space, and that unfolding has an edge-unfolding that tiles the plane. So the hypercube is a "dimension-descending tiler." We also show that the hypercube cross unfolding made famous by Dali tiles space, but we leave open the question of whether or not it has an edge-unfolding that tiles the plane.
Regularity Of Mean Curvature Flow Of Graphs On Lie Groups Free Up To Step 2, Luca Capogna, Giovanna Citti, Maria Manfredini
Regularity Of Mean Curvature Flow Of Graphs On Lie Groups Free Up To Step 2, Luca Capogna, Giovanna Citti, Maria Manfredini
Mathematics Sciences: Faculty Publications
We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics σ ε collapsing to a subRiemannian metric σ0 as ε → 0. We establish Ckα estimates for this flow, that are uniform as ε → 0 and as a consequence prove long time existence for the subRiemannian mean curvature flow of the graph. Our proof extend to the setting of every step two Carnot group (not necessarily free) and can be adapted following …
Spiral Unfoldings Of Convex Polyhedra, Joseph O'Rourke
Spiral Unfoldings Of Convex Polyhedra, Joseph O'Rourke
Computer Science: Faculty Publications
The notion of a spiral unfolding of a convex polyhedron, resulting by flattening a special type of Hamiltonian cut-path, is explored. The Platonic and Archimedian solids all have nonoverlapping spiral unfoldings, although among generic polyhedra, overlap is more the rule than the exception. The structure of spiral unfoldings is investigated, primarily by analyzing one particular class, the polyhedra of revolution.
The Robinson-Schensted Correspondence And A2-Web Bases, Matthew Housley, Heather M. Russell, Julianna Tymoczko
The Robinson-Schensted Correspondence And A2-Web Bases, Matthew Housley, Heather M. Russell, Julianna Tymoczko
Mathematics Sciences: Faculty Publications
We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to [n, n, n]: the reduced web basis associated to Kuperberg’s combinatorial description of the spider category; and the left cell basis for the left cell construction of Kazhdan and Lusztig. In the case of [n, n], the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the image of these bases under classical maps: the Robinson–Schensted algorithm between permutations and Young tableaux and Khovanov–Kuperberg’s bijection between Young tableaux and reduced …
Quantum Mechanical Derivation Of The Wallis Formula For Π, Tamar Friedmann, C. R. Hagen
Quantum Mechanical Derivation Of The Wallis Formula For Π, Tamar Friedmann, C. R. Hagen
Mathematics Sciences: Faculty Publications
A famous pre-Newtonian formula for π is obtained directly from the variational approach to the spectrum of the hydrogen atom in spaces of arbitrary dimensions greater than one, including the physical three dimensions.
Modular Classes Of Lie Groupoid Representations Up To Homotopy, Rajan Amit Mehta
Modular Classes Of Lie Groupoid Representations Up To Homotopy, Rajan Amit Mehta
Mathematics Sciences: Faculty Publications
We describe a construction of the modular class associated to a representation up to homotopy of a Lie groupoid. In the case of the adjoint representation up to homotopy, this class is the obstruction to the existence of a volume form, in the sense of Weinstein’s “The volume of a differentiable stack”.
Liftings And Stresses For Planar Periodic Frameworks, Ciprian Borcea, Ileana Streinu
Liftings And Stresses For Planar Periodic Frameworks, Ciprian Borcea, Ileana Streinu
Computer Science: Faculty Publications
We formulate and prove a periodic analog of Maxwell’s theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove rigidity-theoretic properties for planar periodic pseudo-triangulations, generalizing their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.
The Atiyah Class Of A Dg-Vector Bundle, Rajan Amit Mehta, Mathieu Stiénon, Ping Xu
The Atiyah Class Of A Dg-Vector Bundle, Rajan Amit Mehta, Mathieu Stiénon, Ping Xu
Mathematics Sciences: Faculty Publications
We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields X(M) on a dg-manifold M with homological vector field Q admits a structure of L∞[1]-algebra with the Lie derivative LQ as unary bracket λ1, and the Atiyah cocycle AtM corresponding to a torsion-free affine connection as binary bracket λ2.
Periodic Body-And-Bar Frameworks, Ciprian Borcea, Ileana Streinu, Shin-Ichi Tanigawa
Periodic Body-And-Bar Frameworks, Ciprian Borcea, Ileana Streinu, Shin-Ichi Tanigawa
Computer Science: Faculty Publications
Periodic body-and-bar frameworks are abstractions of crystalline structures made of rigid bodies connected by fixed-length bars and subject to the action of a lattice of translations. We give a Maxwell–Laman characterization for minimally rigid periodic body-and-bar frameworks in terms of their quotient graphs. As a consequence we obtain efficient polynomial time algorithms for their recognition based on matroid partition and pebble games.
Triangle-Free Uniquely 3-Edge Colorable Cubic Graphs, Sarah-Marie Belcastro, Ruth Haas
Triangle-Free Uniquely 3-Edge Colorable Cubic Graphs, Sarah-Marie Belcastro, Ruth Haas
Mathematics Sciences: Faculty Publications
This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs. The only such graph previously known was given by Tutte in 1976.
Vassiliev Invariants Of Virtual Legendrian Knots, Patricia Cahn, Asa Levi
Vassiliev Invariants Of Virtual Legendrian Knots, Patricia Cahn, Asa Levi
Mathematics Sciences: Faculty Publications
We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destabilization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of virtual Legendrian knots and of virtual framed knots are isomorphic. In particular, Vassiliev invariants cannot be used to distinguish virtual Legendrian knots that are isotopic as virtual framed knots and have equal virtual Maslov numbers.
Expansive Periodic Mechanisms, Ciprian Borcea, Ileana Streinu
Expansive Periodic Mechanisms, Ciprian Borcea, Ileana Streinu
Computer Science: Faculty Publications
A one-parameter deformation of a periodic bar-and-joint framework is expansive when all distances between joints increase or stay the same. In dimension two, expansive behavior can be fully explained through our theory of periodic pseudo-triangulations. However, higher dimensions present new challenges. In this paper we study a number of periodic frameworks with expansive capabilities in dimension d ≥ 3 and register both similarities and contrasts with the two-dimensional case.
The Number Of Framings Of A Knot In A 3-Manifold, Patricia Cahn, Vladimir Chernov, Rustam Sadykov
The Number Of Framings Of A Knot In A 3-Manifold, Patricia Cahn, Vladimir Chernov, Rustam Sadykov
Mathematics Sciences: Faculty Publications
In view of the self-linking invariant, the number |K| of framed knots in S3 with given underlying knot K is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that |K| is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of S1 × S2; the knot K need not be zero-homologous and the manifold is not required to be compact. We show that when M is orientable, the number |K| is infinite unless K intersects a nonseparating sphere at exactly one point, in which case …
Lie Algebroid Modules And Representations Up To Homotopy, Rajan Amit Mehta
Lie Algebroid Modules And Representations Up To Homotopy, Rajan Amit Mehta
Mathematics Sciences: Faculty Publications
We establish a relationship between two different generalizations of Lie algebroid representations: representation up to homotopy and Vaĭntrob’s Lie algebroid modules. Specifically, we show that there is a noncanonical way to obtain a representation up to homotopy from a given Lie algebroid module, and that any two representations up to homotopy obtained in this way are equivalent in a natural sense. We therefore obtain a one-to-one correspondence, up to equivalence.
A Module Isomorphism Between H∗T (G/P)⊗H∗T (P/B) And H∗T (G/B), Elizabeth Drellich, Julianna Tymoczko
A Module Isomorphism Between H∗T (G/P)⊗H∗T (P/B) And H∗T (G/B), Elizabeth Drellich, Julianna Tymoczko
Mathematics Sciences: Faculty Publications
We give an explicit (new) morphism of modules between H∗T (G/P)⊗H∗T (P/B) and H∗T (G/B) and prove (the known result) that the two modules are isomorphic. Our map identifies submodules of the cohomology of the flag variety that are isomorphic to each of H∗T (G/P) and H∗ T (P/B). With this identification, the map is simply the product within the ring H∗T (G/B). We use this map in two ways. First we describe module bases for H∗T (G/B) that are different from traditional Schubert classes and from …
Mechanisms Of Elastic Enhancement And Hindrance For Finite-Length Undulatory Swimmers In Viscoelastic Fluids, Becca Thomases, Robert D. Guy
Mechanisms Of Elastic Enhancement And Hindrance For Finite-Length Undulatory Swimmers In Viscoelastic Fluids, Becca Thomases, Robert D. Guy
Mathematics Sciences: Faculty Publications
A computational model of finite-length undulatory swimmers is used to examine the physical origin of the effect of elasticity on swimming speed. We explore two distinct target swimming strokes: one derived from the motion of Caenorhabditis elegans, with large head undulations, and a contrasting stroke with large tail undulations. We show that both favorable stroke asymmetry and swimmer elasticity contribute to a speed-up, but a substantial boost results only when these two effects work together. We reproduce conflicting results from the literature simply by changing relevant physical parameters.
Counting Edge-Kempe-Equivalence Classes For 3-Edge-Colored Cubic Graphs, Sarah-Marie Belcastro, Ruth Haas
Counting Edge-Kempe-Equivalence Classes For 3-Edge-Colored Cubic Graphs, Sarah-Marie Belcastro, Ruth Haas
Mathematics Sciences: Faculty Publications
Two edge colorings of a graph are edge-Kempe equivalent if one can be obtained from the other by a series of edge-Kempe switches. This work gives some results for the number of edge-Kempe equivalence classes for cubic graphs. In particular we show every 2-connected planar bipartite cubic graph has exactly one edge-Kempe equivalence class. Additionally, we exhibit infinite families of nonplanar bipartite cubic graphs with a range of numbers of edge-Kempe equivalence classes. Techniques are developed that will be useful for analyzing other classes of graphs as well.
Harnack Estimates For Degenerate Parabolic Equations Modeled On The Subelliptic P-Laplacian, Benny Avelin, Luca Capogna, Giovanna Citti, Kaj Nyström
Harnack Estimates For Degenerate Parabolic Equations Modeled On The Subelliptic P-Laplacian, Benny Avelin, Luca Capogna, Giovanna Citti, Kaj Nyström
Mathematics Sciences: Faculty Publications
We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype∂tu=-∑i=1mXi*(|Xu|p-2Xiu) where p ≥ 2, X = (X1, . . . , X m) is a system of Lipschitz vector fields defined on a smooth manifold M endowed with a Borel measure μ, and Xi* denotes the adjoint of X i with respect to μ. Our estimates are derived assuming that (i) the control distance d generated by X induces the same topology on M; (ii) a doubling condition for the μ-measure of d-metric balls; and (iii) the validity of a Poincaré inequality involving X and …
Qcd Vs. The Centrifugal Barrier: A New Qcd Effect, Tamar Friedmann
Qcd Vs. The Centrifugal Barrier: A New Qcd Effect, Tamar Friedmann
Mathematics Sciences: Faculty Publications
We propose an extended schematic model for hadrons in which quarks and diquarks alike serve as building blocks. The outcome is a reclassification of the hadron spectrum in which there are no radially excited hadrons: all mesons and baryons previously believed to be radial excitations are orbitally excited states involving diquarks. Also, there are no exotic hadrons: all hadrons previously believed to be exotic are states involving diquarks and are an integral part of the model. We discuss the implications of this result for a new understanding of confinement and its relation to asymptotic freedom, as well as its implications …
Kif11 Dependent Cell Cycle Progression In Radial Glial Cells Is Required For Proper Neurogenesis In The Zebrafish Neural Tube, Kimberly Johnson, Chelsea Moriarty, Nessy Tania, Alissa Ortman, Kristina Dipietrantonio, Brittany Edens, Jean Eisenman, Deborah Ok, Sarah Krikorian, Jessica Barragan, Christophe Golé, Michael J.F. Barresi
Kif11 Dependent Cell Cycle Progression In Radial Glial Cells Is Required For Proper Neurogenesis In The Zebrafish Neural Tube, Kimberly Johnson, Chelsea Moriarty, Nessy Tania, Alissa Ortman, Kristina Dipietrantonio, Brittany Edens, Jean Eisenman, Deborah Ok, Sarah Krikorian, Jessica Barragan, Christophe Golé, Michael J.F. Barresi
Mathematics Sciences: Faculty Publications
Radial glia serve as the resident neural stem cells in the embryonic vertebrate nervous system, and their proliferation must be tightly regulated to generate the correct number of neuronal and glial cell progeny in the neural tube. During a forward genetic screen, we recently identified a zebrafish mutant in the kif11 loci that displayed a significant increase in radial glial cell bodies at the ventricular zone of the spinal cord. Kif11, also known as Eg5, is a kinesin-related, plus-end directed motor protein responsible for stabilizing and separating the bipolar mitotic spindle. We show here that Gfap+ radial glial cells express …