Physical Sciences and Mathematics Commons^™

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

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

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

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

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

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

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

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 …

Counting Conjugacy Classes Of Elements Of Finite Order In Lie Groups, Tamar Friedmann, Richard P. Stanley

Mathematics Sciences: Faculty Publications

Using combinatorial techniques, we answer two questions about simple classical Lie groups. Define N(G,m) to be the number of conjugacy classes of elements of finite order m in a Lie group G, and N(G,m,s) to be the number of such classes whose elements have s distinct eigenvalues or conjugate pairs of eigenvalues.What is N(G,m) for G a unitary, orthogonal, or symplectic group?What is N(G,m,s) for these groups? For some cases, the first question was answered a few decades ago via group-theoretic techniques.It appears that the second question has not been asked before; here it is inspired by questions related …

The Classification Of V -Transverse Knots And Loose Legendrians, Patricia Cahn, Vladimir Chernov

Mathematics Sciences: Faculty Publications

We classify knots in a 3-manifold M that are transverse to a nowhere zero vector field V up to the corresponding isotopy relation. When V is the coorienting vector field of a contact structure, these knots are the same as pseudo-Legendrian knots, which were introduced by Benedetti and Petronio. We show that two loose Legendrian knots with the same overtwisted disk in their complement are Legendrian isotopic if and only if they are pseudoLegendrian isotopic.

V -transverse knots are naturally framed. We show that each framed isotopy class corresponds to infinitely many V -transverse isotopy classes whose elements are pairwise …

L∞-Extremal Mappings In Amle And Teichmüller Theory, Luca Capogna

Mathematics Sciences: Faculty Publications

These lecture focus on two vector-valued extremal problems which have a common feature in that the corresponding energy functionals involve L ∞ norm of an energy density rather than the more familiar L p norms. Specifically, we will address (a) the problem of extremal quasiconformal mappings and (b) the problem of absolutely minimizing Lipschitz extensions.