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We describe a promising approach to efficiently morph spherical graphs, extending earlier approaches of Awartani and Henderson [Trans. AMS 1987] and Kobourov and Landis [JGAA 2006]. Specifically, we describe two methods to morph…
Geometric integration theory can be employed when numerically solving ODEs or PDEs with constraints. In this paper, we present several one-step algorithms of various orders for ODEs on a collection of spheres. To demonstrate the versatility…
Heegaard splittings provide a natural representation of closed 3-manifolds by gluing two handlebodies along a common surface. These splittings can be equivalently given by two finite sets of meridians lying on the surface, which define a…
In this paper, we study the uniqueness of the direct decomposition of a toric manifold. We first observe that the direct decomposition of a toric manifold as \emph{algebraic varieties} is unique up to order of the factors. An algebraically…
We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold…
In this paper, we present new incremental algorithms for maintaining data structures that represent all connectivity cuts of size one in directed graphs (digraphs), and the strongly connected components that result by the removal of each of…
Many physical questions in fluid dynamics can be recast in terms of norm constrained optimisation problems; which in-turn, can be further recast as unconstrained problems on spherical manifolds. Due to the nonlinearities of the governing…
We provide algorithms involving edge slides, for a connected simple graph to evolve in a finite number of steps to another connected simple graph in a prescribed configuration, and for the regularization of such a graph by the minimization…
One approach to produce a pair of homeomorphic-but-not-diffeomophic closed 4-manifolds is to find a knot which is smoothly slice in one but not the other. This approach has never been run successfully. We give the first examples of a pair…
The flow and deformation of macromolecules is ubiquitous in nature and industry, and an understanding of this phenomenon at both macroscopic and microscopic length scales is of fundamental and practical importance. Here we present the…
There are two main approaches to building locally flat embedded surfaces in 4-manifolds: direct methods which geometrically manipulate a given map of a surface, and more indirect methods using surgery theory. Both rely on Freedman-Quinn's…
A theorem on the existence of the unique minimal topologic handle decomposition of differentiable simply connected five-dimensional manifolds is proved. For a decomposition of this sort, the number of handles of each index is given.
We prove that any knot or link in any 3-manifold can be nicely decomposed (splitted) by a filling Dehn sphere. This has interesting consequences in the study of branched coverings over knots and links. We give an algorithm for computing…
In this paper, we propose an incremental algorithm for computing cylindrical algebraic decompositions. The algorithm consists of two parts: computing a complex cylindrical tree and refining this complex tree into a cylindrical tree in real…
We propose an algorithm for the computational homogenization of locally periodic hyperelastic structures undergoing large deformations due to external quasi-static loading. The algorithm performs clustering of macroscopic deformations into…
A variational approach to the reconstruction of a shape (2D simple manifolds) as triangulated surface from given level set using shape gradients is presented. It involves an energy functional that depends on the local shape characteristics…
Manifold models provide low-dimensional representations that are useful for processing and analyzing data in a transformation-invariant way. In this paper, we study the problem of learning smooth pattern transformation manifolds from image…
We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial…
We present a new deep learning approach for matching deformable shapes by introducing {\it Shape Deformation Networks} which jointly encode 3D shapes and correspondences. This is achieved by factoring the surface representation into (i) a…
We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is…