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Articles 1 - 13 of 13
Full-Text Articles in Physical Sciences and Mathematics
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.