Open Access. Powered by Scholars. Published by Universities.®
- Discipline
- Keyword
-
- Sub-Riemannian geometry (7)
- Phyllotaxis (5)
- Carnot groups (4)
- Machine learning (4)
- Mean curvature flow (4)
-
- Oldroyd-B (4)
- Simplicial set (4)
- Topological quantum field theory (4)
- Actigraphy (3)
- Circadian rhythms (3)
- Fibonacci (3)
- Frobenius algebra (3)
- Graded manifold (3)
- Lie algebroid (3)
- Subelliptic PDE (3)
- 2-Segal set (2)
- Category of relations (2)
- Chlamydomonas reinhardtii (2)
- Circadian rhythm (2)
- Combinatorics (2)
- Complex fluids (2)
- Complex geometry (2)
- Computational biofluid dynamics (2)
- Convexity (2)
- Courant algebroids (2)
- Deep learning (2)
- Dirac (2)
- Disc-stacking model (2)
- FENE-P (2)
- Groupoid (2)
Articles 151 - 180 of 196
Full-Text Articles in Mathematics
A Positive Monk Formula In The S1-Equivariant Cohomology Of Type A Peterson Varieties, Megumi Harada, Julianna Tymoczko
A Positive Monk Formula In The S1-Equivariant Cohomology Of Type A Peterson Varieties, Megumi Harada, Julianna Tymoczko
Mathematics Sciences: Faculty Publications
Peterson varieties are a special class of Hessenberg varieties that have been extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley-Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety Y in type An−1, with respect to a natural S1 -action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculus beyond the realm of …
Symmetric Factorization Of The Conformation Tensor In Viscoelastic Fluid Models, Nusret Balci, Becca Thomases, Michael Renardy, Charles R. Doering
Symmetric Factorization Of The Conformation Tensor In Viscoelastic Fluid Models, Nusret Balci, Becca Thomases, Michael Renardy, Charles R. Doering
Mathematics Sciences: Faculty Publications
The positive-definite symmetric polymer conformation tensor possesses a unique symmetric square root that satisfies a closed evolution equation in the Oldroyd-B and FENE-P models of viscoelastic fluid flow. When expressed in terms of the velocity field and the symmetric square root of the conformation tensor, these models' equations of motion formally constitute an evolution in a Hilbert space with a total energy functional that defines a norm. Moreover, this formulation is easily implemented in direct numerical simulations resulting in significant practical advantages in terms of both accuracy and stability.
On Homotopy Poisson Actions And Reduction Of Symplectic Q-Manifolds, Rajan Amit Mehta
On Homotopy Poisson Actions And Reduction Of Symplectic Q-Manifolds, Rajan Amit Mehta
Mathematics Sciences: Faculty Publications
We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field.
The Geometric And Dynamic Essence Of Phyllotaxis, Pau Atela
The Geometric And Dynamic Essence Of Phyllotaxis, Pau Atela
Mathematics Sciences: Faculty Publications
We present a dynamic geometric model of phyllotaxis based on two postulates, primordia formation and meristem expansion. We find that Fibonacci, Lucas, bijugate and multijugate are all variations of the same unifying phenomenon and that the difference lies on small changes in the position of initial primordia. We explore the set of all initial positions and color-code its points depending on the phyllotactic type of the pattern that arises.
Orbifold Singularities, Lie Algebras Of The Third Kind (Latkes), And Pure Yang-Mills With Matter, Tamar Friedmann
Orbifold Singularities, Lie Algebras Of The Third Kind (Latkes), And Pure Yang-Mills With Matter, Tamar Friedmann
Mathematics Sciences: Faculty Publications
We discover the unique, simple Lie Algebra of the Third Kind, or LATKe, that stems from codimension 6 orbifold singularities and gives rise to a new kind of YangMills theory which simultaneously is pure and contains matter. The root space of the LATKe is 1-dimensional and its Dynkin diagram consists of one point. The uniqueness of the LATKe is a vacuum selection mechanism.
Smoothness Of Lipschitz Minimal Intrinsic Graphs In Heisenberg Groups ℍN, N > 1, Luca Capogna, Giovanna Citti, Maria Manfredini
Smoothness Of Lipschitz Minimal Intrinsic Graphs In Heisenberg Groups ℍN, N > 1, Luca Capogna, Giovanna Citti, Maria Manfredini
Mathematics Sciences: Faculty Publications
We prove that Lipschitz intrinsic graphs in the Heisenberg groups ℍn, with n > 1, which are vanishing viscosity solutions of the minimal surface equation, are smooth and satisfy the PDE in a strong sense.
Convexity And Horizontal Second Fundamental Forms For Hypersurfaces In Carnot Groups, Luca Capogna, Scott D. Pauls, Jeremy T. Tyson
Convexity And Horizontal Second Fundamental Forms For Hypersurfaces In Carnot Groups, Luca Capogna, Scott D. Pauls, Jeremy T. Tyson
Mathematics Sciences: Faculty Publications
We use a Riemannian approximation scheme to give a characterization for smooth convex functions on a Carnot group (in the sense of Danielli-Garofalo- Nhieu or Lu-Manfredi-Stroffolini) in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph.
Poset Pinball, Gkm-Compatible Subspaces, And Hessenberg Varieties, Megumi Harada, Julianna Tymoczko
Poset Pinball, Gkm-Compatible Subspaces, And Hessenberg Varieties, Megumi Harada, Julianna Tymoczko
Mathematics Sciences: Faculty Publications
This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem of building module bases, we introduce a …
Schubert Polynomials And Classes Of Hessenberg Varieties, Dave Anderson, Julianna Tymoczko
Schubert Polynomials And Classes Of Hessenberg Varieties, Dave Anderson, Julianna Tymoczko
Mathematics Sciences: Faculty Publications
Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a " Giambelli formula" expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, …
Lie Algebroid Structures On Double Vector Bundles And Representation Theory Of Lie Algebroids, Alfonso Gracia-Saz, Rajan Amit Mehta
Lie Algebroid Structures On Double Vector Bundles And Representation Theory Of Lie Algebroids, Alfonso Gracia-Saz, Rajan Amit Mehta
Mathematics Sciences: Faculty Publications
A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.
A Sharp Diameter Bound For Unipotent Groups Of Classical Type Overℤ /Pℤ, Jordan S. Ellenberg, Julianna Tymoczko
A Sharp Diameter Bound For Unipotent Groups Of Classical Type Overℤ /Pℤ, Jordan S. Ellenberg, Julianna Tymoczko
Mathematics Sciences: Faculty Publications
The unipotent subgroup of a finite group of Lie type over a prime field Fp comes equipped with a natural set of generators; the properties of the Cayley graph associated to this set of generators have been much studied. In the present paper, we show that the diameter of this Cayley graph is bounded above and below by constant multiples of np + n2 log p, where n is the rank of the associated Lie group. This generalizes the result of Ellenberg, A sharp diameter bound for an upper triangular matrix group, Harvard University, 1993, which treated the case of …
On Baryon Number Non-Conservation In Two-Dimensional O(2n+1) Qcd, Tamar Friedmann
On Baryon Number Non-Conservation In Two-Dimensional O(2n+1) Qcd, Tamar Friedmann
Mathematics Sciences: Faculty Publications
We construct a classical dynamical system whose phase space is a certain infinite dimensional Grassmannian manifold, and propose that it is equivalent to the large N limit of two-dimensional QCD with an O(2N + 1) gauge group. In this theory, we find that baryon number is a topological quantity that is conserved only modulo 2. We also relate this theory to the master field approach to matrix models.
Regularity Of Non-Characteristic Minimal Graphs In The Heisenberg Group ℍ1, Luca Capogna, Giovanna Citti, Maria Manfredini
Regularity Of Non-Characteristic Minimal Graphs In The Heisenberg Group ℍ1, Luca Capogna, Giovanna Citti, Maria Manfredini
Mathematics Sciences: Faculty Publications
Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are apriori estimates on the solutions of the approximating Riemannian PDE and the ensuing C∞ regularity of the sub-Riemannian minimal surface along its Legendrian foliation.
Q-Groupoids And Their Cohomology, Rajan Amit Mehta
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
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
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.
Grünbaum Colorings Of Toroidal Triangulations, Michael O. Albertson, Hannah Alpert, Sarah-Marie Belcastro, Ruth Haas
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
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 …
Permutation Representations On Schubert Varieties, Julianna S. Tymoczko
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
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 …
A Note On Ill-Posedness Of The Cauchy Problem For Heisenberg Wave Maps, Luca Capogna, Jalal Shatah
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.
Paving Hessenberg Varieties By Affines, Julianna S. Tymoczko
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
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.
Emergence Of Singular Structures In Oldroyd-B Fluids, Becca Thomases, Michael Shelley
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 …
Conformality And Q-Harmonicity In Carnot Groups, Luca Capogna, Michael Cowling
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.
Ahlfors Type Estimates For Perimeter Measures In Carnot-Carathéodory Spaces, Luca Capogna, Nicola Garofalo
Ahlfors Type Estimates For Perimeter Measures In Carnot-Carathéodory Spaces, Luca Capogna, Nicola Garofalo
Mathematics Sciences: Faculty Publications
We study the relationship between the geometry of hypersurfaces in a Carnot-Carathéodory (CC) space and the Ahlfors regularity of the corresponding perimeter measure. To this end we establish comparison theorems for perimeter estimates between an hypersurface and its tangent space, and between a CC geometry and its "tangent" Carnot group structure.
Local Energy Decay For Solutions Of Multi-Dimensional Isotropic Symmetric Hyperbolic Systems, Thomas C. Sideris, Becca Thomases
Local Energy Decay For Solutions Of Multi-Dimensional Isotropic Symmetric Hyperbolic Systems, Thomas C. Sideris, Becca Thomases
Mathematics Sciences: Faculty Publications
The local decay of energy is established for solutions to certain linear, multidimensional symmetric hyperbolic systems, with constraints. The key assumptions are isotropy and nondegeneracy of the associated symbols. Examples are given, including Maxwell's equations and linearized elasticity. Such estimates prove useful in treating nonlinear perturbations.
Linear Conditions Imposed On Flag Varieties, Julianna S. Tymoczko
Linear Conditions Imposed On Flag Varieties, Julianna S. Tymoczko
Mathematics Sciences: Faculty Publications
We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenberg varieties over GLn(ℂ) and show that they have no odd-dimensional homology. We provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space. We characterize these affine pieces by fillings of Young tableaux and show that the dimension of the affine piece can be computed by combinatorial rules generalizing the Eulerian numbers. We give an equivalent formulation of …
A Note On The Engulfing Property And The R 1+Α -Regularity Of Convex Functions In Carnot Groups, Luca Capogna, Diego Maldonado
A Note On The Engulfing Property And The R 1+Α -Regularity Of Convex Functions In Carnot Groups, Luca Capogna, Diego Maldonado
Mathematics Sciences: Faculty Publications
We study the engulfing property for convex functions in Carnot groups. As an application we show that the horizontal gradient of functions with this property is Hölder continuous.
Exponents For B-Stable Ideals, Eric Sommers, Julianna Tymoczko
Exponents For B-Stable Ideals, Eric Sommers, Julianna Tymoczko
Mathematics Sciences: Faculty Publications
Let G be a simple algebraic group over the complex numbers containing a Borel subgroup B. Given a B-stable ideal I in the nilradical of the Lie algebra of B, we define natural numbers m 1, m 2,. . .,m k which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types A n, B n, C n and some other types. When I = 0, we recover the usual exponents of G by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the …