Singular Vertices Research Articles

A characterization of $g_2$-minimal normal 3-pseudomanifolds with at most four singularities

Let $\Delta$ be a $g_2$-minimal normal 3-pseudomanifold. A vertex in $\Delta$ whose link is not a sphere is called a singular vertex. When $\Delta$ contains at most two singular vertices, its combinatorial characterization is known \cite{BasakSwartz}. In this article, we present a combinatorial characterization of such a $\Delta$ when it has three singular vertices, including one $\mathbb{RP}^2$-singularity, or four singular vertices, including two $\mathbb{RP}^2$-singularities. In both cases, we prove that $\Delta$ is obtained from a one-vertex suspension of a surface, and some boundary complexes of $4$-simplices by applying the combinatorial operations of types connected sums, vertex foldings, and edge foldings

The pressure-wired Stokes element: a mesh-robust version of the Scott–Vogelius element

The Scott–Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order k and a discontinuous pressure approximation of order k-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k-1$$\\end{document}. It employs a “singular distance” (measured by some geometric mesh quantity Θz≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Theta \\left( \ extbf{z}\\right) \\ge 0$$\\end{document} for triangle vertices z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{z}$$\\end{document}) and imposes a local side condition on the pressure space associated to vertices z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{z}$$\\end{document} with Θz=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta \\left( \ extbf{z}\\right) =0$$\\end{document}. The method is inf-sup stable for any fixed regular triangulation and k≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 4$$\\end{document}. However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices 0<Θz≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\Theta \\left( \ extbf{z}\\right) \\ll 1$$\\end{document}. In this paper, we introduce a very simple parameter-dependent modification of the Scott–Vogelius element with a mesh-robust inf-sup constant. To this end, we provide sharp two-sided bounds for the inf-sup constant with an optimal dependence on the “singular distance”. We characterise the critical pressures to guarantee that the effect on the divergence-free condition for the discrete velocity is negligibly small, for which we provide numerical evidence.

A locally calculable P3‐pressure in a decoupled method for incompressible Stokes equations

AbstractThis article will suggest a new finite element method to find a ‐velocity and a ‐pressure solving incompressible Stokes equations at low cost. The method solves first the decoupled equation for a ‐velocity. Then, using the calculated velocity, a locally calculable ‐pressure will be defined component‐wisely. The resulting ‐pressure is analyzed to have the optimal order of convergence. Since the pressure is calculated by local computation only, the chief time cost of the new method is on solving the decoupled equation for the ‐velocity. Besides, the method overcomes the problem of singular vertices or corners.

A vector encoding for topology finding of structured quad-based patterns for surface structures

Topology-optimisation strategies for structural design of discrete surface structures result in unstructured patterns that require post-rationalisation to limit the number and complexity of the various structural elements. Fabrication-related objectives still demand further processing. Designers need topology exploration, before topology optimisation, following a generative-design approach that is decoupled from density tuning and shape design, to produce high-quality patterns. Such an approach allows to flexibly embed multiple design constraints, benefit from state-of-the-art form-finding algorithms and explore design trade-offs of the multiple requirements related to architecture, engineering and construction. This research investigates a parameterisation strategy to encode topological exploration of structured patterns based on quad meshes. The focus is set on singular vertices, which connect an irregular number of blocks, cables, or beams. Singularities are independent from pattern density and geometry, and have a fundamental influence on the qualitative and quantitative performance of the structure. We introduce an L-system encoding a quad-mesh grammar into a string that describes topological transformations of these singularities. Design applications to a net and a gridshell demonstrate the influence of singularities on the design of surface structures, and highlight the flexibility and generality of the approach in terms of geometrical processing and performance metrics. Beyond exploration, this parameterisation strategy opens to novel applications of search and optimisation methods for generative design of singularities in structural patterns.

Conical $SL(3)$ foams

In the unoriented SL(3) foam theory, singular vertices are generic singularities of two-dimensional complexes. Singular vertices have neighborhoods homeomorphic to cones over the one-skeleton of the tetrahedron, viewed as a trivalent graph on the two-sphere. In this paper we consider foams with singular vertices with neighborhoods homeomorphic to cones over more general planar trivalent graphs. These graphs are subject to suitable conditions on their Kempe equivalence Tait coloring classes and include the dodecahedron graph. In this modification of the original homology theory it is straightforward to show that modules associated to the dodecahedron graph are free of rank 60, which is still an open problem for the original unoriented SL(3) foam theory.

Intrinsic mixed-integer polycubes for hexahedral meshing

Polycube mapping is an attractive approach for the generation of all-hexahedral meshes with a fully regular interior, i.e. free of internal singular edges or vertices. It is based on determining a low distortion map between the input model and a polycube domain, which then pulls back the regular voxel grid to form a hexahedral mesh for the model. Automatically finding an appropriate polycube domain for a given model, however, is a challenging problem. Existing algorithms are either very sensitive to the embedding and orientation of the input model, restricted to only subclasses of possible domains, or depend crucially on some initialization because they rely on a non-convex optimization formulation. This can easily lead to unsatisfactory and unnecessary corners and edges in the polycube structure. We present a novel approach to the problem of finding high-quality polycube domains. It is based on an entirely intrinsic formulation as a mixed integer optimization problem, which can be tackled by solving a series of simple convex problems, each of which can be solved to the global optimum. Experiments demonstrate that our method avoids many of the undesired corners and surface irregularities common to many previous methods.

Convergence of Lagrange finite elements for the Maxwell eigenvalue problem in two dimensions

AbstractWe consider finite element approximations of the Maxwell eigenvalue problem in two dimensions. We prove, in certain settings, convergence of the discrete eigenvalues using Lagrange finite elements. In particular, we prove convergence in three scenarios: piecewise linear elements on Powell–Sabin triangulations, piecewise quadratic elements on Clough–Tocher triangulations and piecewise quartics (and higher) elements on general shape-regular triangulations. We provide numerical experiments that support the theoretical results. The computations also show that, on general triangulations, the eigenvalue approximations are very sensitive to nearly singular vertices, i.e., vertices that fall on exactly two ‘almost’ straight lines.

Low-order divergence-free approximations for the Stokes problem on Worsey–Farin and Powell–Sabin splits

We derive low-order, inf–sup stable and divergence-free finite element approximations for the Stokes problem using Worsey–Farin splits in three dimensions and Powell–Sabin splits in two dimensions. The velocity space simply consists of continuous, piecewise linear polynomials, whereas the pressure space is a subspace of piecewise constants with weak continuity properties at singular edges (3D) and singular vertices (2D). We discuss implementation aspects that arise when coding the pressure space, and in particular, show that the pressure constraints can be enforced at an algebraic level.

Formation of superscar waves in plane polygonal billiards* *In memory of Charles Schmit.

Polygonal billiards constitute a special class of models. Though they have zero Lyapunov exponent their classical and quantum properties are involved due to scattering on singular vertices with angles with integer n. It is demonstrated that in the semiclassical limit the multiple singular scattering on such vertices, when optical boundaries of many scatters overlap, leads to the vanishing of quantum wave functions along straight lines built by these scatters. This phenomenon has an especially important consequence for polygonal billiards where periodic orbits (when they exist) form pencils of parallel rays restricted from the both sides by singular vertices. Due to the singular scattering on boundary vertices, waves propagated inside a periodic orbit pencil tend in the semiclassical limit to zero along pencil boundaries thus forming weakly interacting quasi-modes. Contrary to scars in chaotic systems, the discussed quasi-modes in polygonal billiards become almost exact for high-excited states and for brevity they are designated as superscars. Many pictures of eigenfunctions for a triangular billiard and a barrier billiard which have clear superscar structures are presented in the paper. Special attention is given to the development of quantitative methods of detecting and analysing such superscars. In particular, it is demonstrated that the overlap between superscar waves associated with a fixed periodic orbit and eigenfunctions of a barrier billiard is distributed according to the Breit-Wigner distribution typical for weakly interacting quasi-modes. For special sub-class of rational polygonal billiards called Veech polygons where all periodic orbits can be calculated analytically it is argued and checked numerically that their eigenfunctions are fractal in the Fourier space.

Direct Seamless Parametrization

We present a method for seamless surface parametrization. Recent popular methods first generate a cross-field, where curvature is concentrated at singular vertices. Next, in a separate step, the surface is laid out in the domain subject to derived seamlessness constraints. This decoupling of the process into two independent problems, each with its own objective, leads to suboptimal results. In contrast, our method solves both problems together using domain variables. The key ingredient to the robustness of our method is a rounding strategy based on local estimation. The insight is that testing a small patch to decide between two likely possibilities is a good estimator. Most distortion measures can be used with our method, which get minimized consistently throughout the pipeline. Our method also enables feature alignment, as well as alignment to principle curvatures, and isotropic and anisotropic scaling.

Quadrilateral mesh generation II: Meromorphic quartic differentials and Abel–Jacobi condition

This work discovers the equivalence relation between quadrilateral meshes and meromorphic quartic differentials. Each quad-mesh induces a conformal structure of the surface, and a meromorphic quartic differential, where the configuration of singular vertices corresponds to the configurations of the poles and zeros (divisor) of the meromorphic differential. Due to Riemann surface theory, the configuration of singularities of a quad-mesh satisfies the Abel–Jacobi condition. Inversely, if a divisor satisfies the Abel–Jacobi condition, then there exists a meromorphic quartic differential whose divisor equals the given one. Furthermore, if the meromorphic quartic differential is with finite trajectories, then it also induces a quad-mesh, the poles and zeros of the meromorphic differential correspond to the singular vertices of the quad-mesh.Besides the theoretic proofs, the computational algorithm for verification of Abel–Jacobi condition is also explained in detail. Furthermore, constructive algorithm of meromorphic quartic differential on genus zero surfaces is proposed, which is based on the global algebraic representation of meromorphic differentials.Our experimental results demonstrate the efficiency and efficacy of the algorithm. This opens up a novel direction for quad-mesh generation using algebraic geometric approach.

A boundary element-based automatic domain partitioning approach for semi-structured quad mesh generation

In this paper, a boundary element-based approach is proposed for partitioning a planar domain automatically into a set of 4-sided regions, which is suitable for semi-structured quad mesh generation. The basic idea is to generate a smooth cross-field via solving PDEs with the boundary element method and then partition the input domain by extracting singular structures of the cross-field. Firstly, a cross is configured at each boundary vertex of the input domain. Then, two Laplacian equations are selected as the governing equation to smoothly propagate the crosses defined on the boundary into the interior of the domain. To obtain an accurate cross-field, the boundary element method is used to solve the governing equations. Then the singular vertices are identified by analyzing the structure of the cross-field, and the streamlines emanating from these points are traced and simplified as segmentation curves to partition the domain. Finally, to demonstrate the efficiency and effectiveness of the proposed approach, some quad mesh generation examples and comparison with previous approaches are presented.

Spurious pressure in Scott–Vogelius elements

We will analyze the characteristics of Scott–Vogelius finite elements on singular vertices, which cause spurious pressures on solving Stokes equations. A simple postprocessing will be suggested to remove those spurious pressures.

Orbigraphs: a graph-theoretic analog to Riemannian orbifolds

A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold that is locally modeled on $R^n$ modulo the action of a finite group. Orbifolds have proven interesting in a variety of settings. Spectral geometers have examined the link between the Laplace spectrum of an orbifold and the singularities of the orbifold. One open question in this field is whether or not a singular orbifold and a manifold can be Laplace isospectral. Motivated by the connection between spectral geometry and spectral graph theory, we define a graph theoretic analogue of an orbifold called an orbigraph. We obtain results about the relationship between an orbigraph and the spectrum of its adjacency matrix. We prove that the number of singular vertices present in an orbigraph is bounded above and below by spectrally determined quantities, and show that an orbigraph with a singular point and a regular graph cannot be cospectral. We also provide a lower bound on the Cheeger constant of an orbigraph.

Dynamic structure function of some singular Fermi liquids

The density correlations of some singular Fermi liquids with anomalous properties such as resistivity varying linearly with T at low temperatures, a $T \log T$ contribution to the entropy and thermopower, etc., are expected to be quite different from that in Landau Fermi liquids. A possible statistical mechanical model for the quantum critical fluctuations in diverse systems with such properties is the 2D dissipative quantum XY model. Exact relations between the density correlations and singular irreducible vertices due to coupling of Fermions to the topological excitations of the 2D dissipative quantum XY model are used to derive results which were proposed phenomenologically long ago but are measurable only recently due to advances in experimental techniques. The density correlations are unusual at all momenta ${\bf q}$ and energy $\nu$, from the hydrodynamic limit to that for large momenta and energy. The hydrodynamic limit together with the continuity equation gives the linear in T resistivity. The density correlations are almost independent of frequency up to a high frequency cut-off for $q_{ZB} \gtrsim q >> \nu/v_F$; $q_{ZB}$ is the Brillouin zone boundary and $v_F$ is the Fermi-velocity. The results should be applicable to loop-current quantum criticality in cuprates, and to 2D Fe based compounds near their antiferromagnetic quantum-criticality. The relation of the results to the temperature and frequency dependent conductivity and to Raman response is also discussed.

Characteristics of the new phase in CDT

The approach of Causal Dynamical Triangulations (CDT), a candidate theory of nonperturbative quantum gravity in 4D, turns out to have a rich phase structure. We investigate the recently discovered bifurcation phase C_{b} and relate some of its characteristics to the presence of singular vertices of very high order. The transition lines separating this phase from the “time-collapsed” B-phase and the de Sitter phase C_{dS} are of great interest when searching for physical scaling limits. The work presented here sheds light on the mechanisms behind these transitions. First, we study how the B–C_{b} transition signal depends on the volume fixing implemented in the simulations, and find results compatible with the previously determined second-order character of the transition. The transition persists in a transfer matrix formulation, where the system’s time extension is taken to be minimal. Second, we relate the new C_{b}–C_{dS} transition to the appearance of singular vertices, which leads to a direct physical interpretation in terms of a breaking of the homogeneity and isotropy observed in the de Sitter phase when crossing from C_{dS} to the bifurcation phase C_{b}.

Classification of unital simple Leavitt path algebras of infinite graphs

We prove that if E and F are graphs with a finite number of vertices and an infinite number of edges, if K is a field, and if LK(E) and LK(F) are simple Leavitt path algebras, then LK(E) is Morita equivalent to LK(F) if and only if K0alg(LK(E))≅K0alg(LK(F)) and the graphs E and F have the same number of singular vertices, and moreover, in this case one may transform the graph E into the graph F using basic moves that preserve the Morita equivalence class of the associated Leavitt path algebra. We also show that when K is a field with no free quotients, the condition that E and F have the same number of singular vertices may be replaced by K1alg(LK(E))≅K1alg(LK(F)), and we produce examples showing this cannot be done in general. We describe how we can combine our results with a classification result of Abrams, Louly, Pardo, and Smith to get a nearly complete classification of unital simple Leavitt path algebras — the only missing part is determining whether the “sign of the determinant condition” is necessary in the finite graph case. We also consider the Cuntz splice move on a graph and its effect on the associated Leavitt path algebra.

Complex twist flows on surface group representations and the local shape of the deformation space of hyperbolic cone–3–manifolds

In two former papers, the authors independently proved that the space of hyperbolic cone-3-manifolds with cone angles less than 2{\pi} and fixed singular locus is locally parametrized by the cone angles. In this sequel, we investigate the local shape of the deformation space when the singular locus is no longer fixed, i.e. when the singular vertices can be split. We show that the different possible splittings correspond to specific pair-of-pants decompositions of the smooth parts of the links of the singular vertices, and that under suitable assumptions the corresponding subspace of deformations is parametrized by the cone angles of the original edges and the lengths of the new ones.

Lyapunov graphs of nonsingular Smale flows on 𝑆¹×𝑆²

In this paper, following J. Franks’ work on Lyapunov graphs of nonsingular Smale flows on S 3 S^3 , we study Lyapunov graphs of nonsingular Smale flows on S 1 × S 2 S^1 \times S^2 . More precisely, we determine necessary and sufficient conditions on an abstract Lyapunov graph to be associated with a nonsingular Smale flow on S 1 × S 2 S^1 \times S^2 . We also study the singular type vertices in Lyapunov graphs of nonsingular Smale flows on 3-manifolds.

On the Dimension of Bivariate &lt;i&gt;C&lt;/i&gt;&lt;sup&gt;1&lt;/sup&gt; Cubic Spline Space with Homogeneous Boundary Conditions over a CT Triangulation

A bivariate spline is a piecewise polynomial with some smoothness de ned on a parti- tion. In this paper, we mainly study the dimensions of bivariate C1 cubic spline spaces S1;0 3 (CT ) and S1;1 3 (CT ) with homogeneous boundary conditions over CT by using interpolating technique, where CT stands for a CT triangulation. The dimensions are related with the numbers of the inter vertices and the singular boundary vertices. The results of this paper can be applied in many elds such as the nite element method for partial di erential equation, computer aided design, numerical approximation, and so on.