Mathematics Commons^™

Q-Groupoids And Their Cohomology, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

We approach Mackenzie's L{script}A{script}-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of L{script}A{script}-groupoids to the category of Q-groupoids. We associate to every Qgroupoid a double complex that provides a model for the Q-cohomology of the classifying space. As examples, we obtain models for equivariant Q-and orbifold Q-cohomology, and for equivariant Lie algebroid and orbifold Lie algebroid cohomology. We obtain double complexes associated to Poisson groupoids and groupoid-algebroid "matched pairs".

Transition To Mixing And Oscillations In A Stokesian Viscoelastic Flow, Becca Thomases, Michael Shelley

Mathematics Sciences: Faculty Publications

In seeking to understand experiments on low-Reynolds-number mixing and flow transitions in viscoelastic fluids, we simulate the dynamics of the Oldroyd-B model, with a simple background force driving the flow. We find that at small Weissenberg number, flows are "slaved" to the extensional geometry imposed by forcing. For large Weissenberg number, such solutions become unstable and transit to a structurally dissimilar state dominated by a single large vortex. This new state can show persistent oscillatory behavior with the production and destruction of smaller-scale vortices that drive mixing.

Generalized Mean Curvature Flow In Carnot Groups, Luca Capogna, Giovanna Citti

Mathematics Sciences: Faculty Publications

In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting in [4] and [12]. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow.

Some Properties Of Yao Y4 Subgraphs, Joseph O'Rourke

Computer Science: Faculty Publications

The Yao graph for k = 4, Y₄, is naturally partitioned into four subgraphs, one per quadrant. We show that the subgraphs for one quadrant differ from the subgraphs for two adjacent quadrants in three properties: planarity, connectedness, and whether the directed graphs are spanners.

Sparsity-Certifying Graph Decompositions, Ileana Streinu, Louis Theran

Computer Science: Faculty Publications

We describe a new algorithm, the (k, ℓ)-pebble game with colors, and use it to obtain a characterization of the family of (k, ℓ)-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k …

Grünbaum Colorings Of Toroidal Triangulations, Michael O. Albertson, Hannah Alpert, Sarah-Marie Belcastro, Ruth Haas

Mathematics Sciences: Faculty Publications

We prove that if G is a triangulation of the torus and χ(G) 6 ≠ 5, then there is a 3-coloring of the edges of G so that the edges bounding every face are assigned three different colors.

Q-Algebroids And Their Cohomology, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

A Q-algebroid is a graded Lie algebroid equipped with a compatible homological vector field and is the infinitesimal object corresponding to a Q-groupoid. We associate to every Q-algebroid a double complex. As a special case, we define the Becchi-Rouet-Stora-Tyutin (BRST) model of a Lie algebroid, which generalizes the BRST model for equivariant cohomology. We extend to this setting the Mathai-Quillen-Kalkman isomorphism of the BRST and Weil models, and we suggest a definition of a basic subcomplex which, however, requires a choice of a connection. Other examples include Roytenberg's homological double of a Lie bialgebroid, Ginzburg's model of equivariant Lie algebroid …

Unfolding Convex Polyhedra Via Quasigeodesic Star Unfoldings, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu

Computer Science: Faculty Publications

We extend the notion of a star unfolding to be based on a simple quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple, planar polygon: shortest paths from all vertices of P to Q are cut, and all but one segment of Q is cut.

Singularities Of Hinge Structures, Ciprian Borcea, Ileana Streinu

Computer Science: Faculty Publications

Motivated by the hinge structure present in protein chains and other molecular conformations, we study the singularities of certain maps associated to body-and-hinge and panel-and-hinge chains. These are sequentially articulated systems where two consecutive rigid pieces are connected by a hinge, that is, a codimension two axis. The singularities, or critical points, correspond to a dimensional drop in the linear span of the axes, regarded as points on a Grassmann variety in its Pl¨ucker embedding. These results are valid in arbitrary dimension. The three dimensional case is also relevant in robotics.

Permutation Representations On Schubert Varieties, Julianna S. Tymoczko

Mathematics Sciences: Faculty Publications

This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over ℂ and over ℂ[t1, t2, . . . , tn]. We show these group actions are the same as an action of simple transpositions studied geometrically by M. Brion, and give topological meaning to the divided difference operators of Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and others. We analyze these representations using the combinatorial approach to equivariant cohomology introduced by Goresky-Kottwitz-MacPherson. We find that each permutation representation on equivariant cohomology produces a representation on ordinary cohomology that is trivial, though the equivariant representation is not.

The Mixed Problem In L P For Some Two-Dimensional Lipschitz Domains, Loredana Lanzani, Luca Capogna, Russell M. Brown

Mathematics Sciences: Faculty Publications

We consider the mixed problem, {Δ u = 0 in Ω ∂u = f N on N u = fD on D in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, f D , has one derivative in L p (D) of the boundary and the Neumann data, f N , is in L p (N). We find a p 0 > 1 so that for p in an interval (1, p 0), we may find a unique solution to the mixed problem and the gradient of the solution …

Enumerating Constrained Non-Crossing Minimally Rigid Frameworks, David Avis, Naoki Katoh, Makoto Ohsaki, Ileana Streinu, Shin-Ichi Tanigawa

Computer Science: Faculty Publications

In this paper we present an algorithm for enumerating without repetitions all the non-crossing generically minimally rigid bar-and-joint frameworks under edge constraints, which we call constrained non-crossing Laman frameworks, on a given set of n points in the plane. Our algorithm is based on the reverse search paradigm of Avis and Fukuda. It generates each output graph in O(n4) time and O(n) space, or, with a slightly different implementation, in O(n3) time and O(n2) space. In particular, we obtain that the set of all the constrained non-crossing Laman …

Unfolding Manhattan Towers, Mirela Damian, Robin Flatland, Joseph O'Rourke

Computer Science: Faculty Publications

We provide an algorithm for unfolding the surface of any orthogonal polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar orthogonal polygon. The algorithm cuts along edges of a 4×5×1 refinement of the vertex grid.

A Note On Ill-Posedness Of The Cauchy Problem For Heisenberg Wave Maps, Luca Capogna, Jalal Shatah

Mathematics Sciences: Faculty Publications

We introduce a notion of wave maps with a target in the sub- Riemannian Heisenberg group and study their relation with Riemannian wave maps with range in Lagrangian submanifolds. As an application we establish existence and eventually ill-posedness of the corresponding Cauchy problem.

Pebble Game Algorithms And Sparse Graphs, Audrey Lee, Ileana Streinu

Computer Science: Faculty Publications

A multi-graph G on n vertices is (k,ℓ)-sparse if every subset of n^′⩽n vertices spans at most kn^′-ℓ edges. G is tight if, in addition, it has exactly kn-ℓ edges. For integer valuesk and ℓ∈[0,2k), we characterize the (k,ℓ)-sparse graphs via a family of simple, elegant and efficient algorithms called the (k,ℓ)-pebble games. [A. Lee, I. Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (8) (2008) 1425–1437] from graphs to hypergraphs.

Cauchy’S Arm Lemma On A Growing Sphere, Zachary Abel, David Charlton, Sébastien Collette, Erik D. Demaine, Martin L. Demaine, Stefan Langerman, Joseph O'Rourke, Val Pinciu, Godfried Toussaint

Computer Science: Faculty Publications

We propose a variant of Cauchy's Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases. The main focus of this work is a comparison of three alternate proofs, to show the links between Toponogov's Comparison Theorem, Legendre's Theorem and Cauchy's Arm Lemma.

Grid Vertex-Unfolding Orthogonal Polyhedra, Mirela Damian

Computer Science: Faculty Publications

No abstract provided.

A Class Of Convex Polyhedra With Few Edge Unfoldings, Alex Benton, Joseph O'Rourke

Computer Science: Faculty Publications

We construct a sequence of convex polyhedra on n vertices with the property that, as n -> infinity, the fraction of its edge unfoldings that avoid overlap approaches 0, and so the fraction that overlap approaches 1. Nevertheless, each does have (several) nonoverlapping edge unfoldings.

Calculus In Context, James Callahan, David Cox, Kenneth Hoffman, Donal O'Shea, Harriet Pollatsek, Lester Senechal

Open Educational Resources: Textbooks

Designing the curriculum

We believe that calculus can be for students what it was for Euler and the Bernoullis: a language and a tool for exploring the whole fabric of science. We also believe that much of the mathematical depth and vitality of calculus lies in connections to other sciences. The mathematical questions that arise are compelling in part because the answers matter to other disciplines. We began our work with a "clean slate," not by asking what parts of the traditional course to include or discard. Our starting points are thus our summary of what calculus is really about. …

Band Unfoldings And Prismatoids: A Counterexample, Joseph O'Rourke

Computer Science: Faculty Publications

This note shows that the hope expressed in [ADL+07]--that the new algorithm for edge-unfolding any polyhedral band without overlap might lead to an algorithm for unfolding any prismatoid without overlap--cannot be realized. A prismatoid is constructed whose sides constitute a nested polyhedral band, with the property that every placement of the prismatoid top face overlaps with the band unfolding.

Unfolding Restricted Convex Caps, Joseph O'Rourke

Computer Science: Faculty Publications

This paper details an algorithm for unfolding a class of convex polyhedra, where each polyhedron in the class consists of a convex cap over a rectangular base, with several restrictions: the cap’s faces are quadrilaterals, with vertices over an underlying integer lattice, and such that the cap convexity is "radially monotone," a type of smoothness constraint. Extensions of Cauchy’s arm lemma are used in the proof of non-overlap.

Paving Hessenberg Varieties By Affines, Julianna S. Tymoczko

Mathematics Sciences: Faculty Publications

Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety arising in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all classical types, generalizing results of De Concini-Lusztig-Procesi and Kostant. This paving is in fact the intersection of a particular Bruhat decomposition with the Hessenberg variety. The nonempty cells of the paving and their dimensions are identified by combinatorial conditions on roots. We use the paving to prove these Hessenberg varieties have no odd-dimensional homology.

Characterizing Sparse Graphs By Map Decompositions, Ruth Haas, Audrey Lee, Ileana Streinu, Louis Theran

Mathematics Sciences: Faculty Publications

A map is a graph that admits an orientation of its edges so that each vertex has out-degree exactly 1. We characterize graphs which admit a decomposition into k edge-disjoint maps after: (1) the addition of any ℓ edges; (2) the addition of some ℓ edges. These graphs are identified with classes of sparse graphs; the results are also given in matroidal terms.

Epsilon-Unfolding Orthogonal Polyhedra, Mirela Damian, Robin Flatland, Joseph O'Rourke

Computer Science: Faculty Publications

An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron may be unfolded. Here we prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at right angles) of genus zero may be unfolded. Our cuts are not necessarily along edges of the polyhedron, but they are always parallel to polyhedron edges. For a polyhedron of n vertices, portions of the unfolding will be rectangular strips which, in the worst case, …

A New Lower Bound On Guard Placement For Wireless Localization, Mirela Damian, Robin Flatland, Joseph O'Rourke, Suneeta Ramswami

Computer Science: Faculty Publications

The problem of wireless localization asks to place and orient stations in the plane, each of which broadcasts a unique key within a fixed angular range, so that each point in the plane can determine whether it is inside or outside a given polygonal region. The primary goal is to minimize the number of stations. In this paper we establish a lower bound of ⌊2n/3⌋−1 stations for polygons in general position, for the case in which the placement of stations is restricted to polygon vertices, improving upon the existing ⌈n/2⌉ lower bound.

Graded Sparse Graphs And Matroids, Audrey Lee, Ileana Streinu, Louis Theran

Computer Science: Faculty Publications

Sparse graphs and their associated matroids play an important role in rigidity theory, where they capture the combinatorics of some families of generic minimally rigid structures. We define a new family called graded sparse graphs, arising from generically pinned bar-and-joint frameworks, and prove that they also form matroids. We also address several algorithmic problems on graded sparse graphs: Decision, Spanning, Extraction, Components, Optimization, and Extension. We sketch variations on pebble game algorithms to solve them.

Emergence Of Singular Structures In Oldroyd-B Fluids, Becca Thomases, Michael Shelley

Mathematics Sciences: Faculty Publications

Numerical simulations reveal the formation of singular structures in the polymer stress field of a viscoelastic fluid modeled by the Oldroyd-B equations driven by a simple body force. These singularities emerge exponentially in time at hyperbolic stagnation points in the flow and their algebraic structure depends critically on the Weissenberg number. Beyond a first critical Weissenberg number the stress field approaches a cusp singularity, and beyond a second critical Weissenberg number the stress becomes unbounded exponentially in time. A local approximation to the solution at the hyperbolic point is derived from a simple ansatz, and there is excellent agreement between …

Connecting Polygonizations Via Stretches And Twangs, Mirela Damian, Robin Flatland, Joseph O'Rourke, Suneeta Ramswami

Computer Science: Faculty Publications

We show that the space of polygonizations of a fixed planar point set S of n points is connected by O(n² ) “moves” between simple polygons. Each move is composed of a sequence of atomic moves called “stretches” and "twangs". These atomic moves walk between weakly simple "polygonal wraps" of S. These moves show promise to serve as a basis for generating random polygons.

Lessons Learned From The 1918-1919 Influenza Pandemic In Minneapolis And St. Paul, Minnesota, Miles Q. Ott, Shelly F. Shaw, Richard N. Danila, Ruth Lynfield

Statistical and Data Sciences: Faculty Publications

No abstract provided.

Conformality And Q-Harmonicity In Carnot Groups, Luca Capogna, Michael Cowling

Mathematics Sciences: Faculty Publications

We show that if f is a 1-quasiconformal map defined on an open subset of a Carnot group G, then composition with f preserves Q-harmonic functions. We combine this with a regularity theorem for Q-harmonic functions and an algebraic regularity theorem for maps between Carnot groups to show that f is smooth. We give some applications to the study of rigidity.