Finite Triangulation Research Articles

Partitioning the projective plane and the dunce hat

We show that any finite triangulation of the real projective plane or the dunce hat is partitionable. To prove this, we introduce simple yet useful gluing tools that allow us to reduce partitionability of a given complex to that of smaller constituent relative subcomplexes such as the disk or the open Möbius strip. The gluing process generates partitioning schemes with a distinctive shelling/constructible flavor. We also give a tool to lift partitionability of relative simplicial complexes to that of the preimages of certain simplicial quotient maps.

Fermionic Topological Order on Generic Triangulations

Consider a finite triangulation of a surface M of genus g and assume that spin-less fermions populate the edges of the triangulation. The quantum dynamics of such particles takes place inside the algebra of canonical anti-commutation relations (CAR). Following Kitaev’s work on toric models, we identify a sub-algebra of CAR generated by elements associated to the triangles and vertices of the triangulation. We show that any Hamiltonian drawn from this sub-algebra displays topological spectral degeneracy. More precisely, if $${{\mathcal {P}}}$$ is any of its spectral projections, the Booleanization of the fundamental group $$\pi _1(M)$$ can be embedded inside the group of invertible elements of the corner algebra $${{\mathcal {P}}}\, \mathrm{CAR} \, {{\mathcal {P}}}$$ . As a consequence, $${{\mathcal {P}}}$$ decomposes in $$4^g$$ lower projections. Furthermore, a projective representation of $${{\mathbb {Z}}}_2^{4g}$$ is also explicitly constructed inside this corner algebra. Key to all these is a presentation of CAR as a crossed product with the Boolean group $$(2^X,\Delta )$$ , where X is the set of fermion sites and $$\Delta $$ is the symmetric difference of its sub-sets.

Verified computations for closed hyperbolic 3‐manifolds

Extending methods first used by Casson, we show how to verify a hyperbolic structure on a finite triangulation of a closed 3-manifold using interval arithmetic methods. A key ingredient is a new theoretical result (akin to a theorem by Neumann-Zagier and Moser for ideal triangulations upon which HIKMOT is based) showing that there is a redundancy among the edge equations if the edges avoid "gimbal lock". We successfully test the algorithm on known examples such as the orientable closed manifolds in the Hodgson-Weeks census and the bundle census by Bell. We also tackle a previously unsolved problem and determine all knots and links with up to 14 crossings that have a hyperbolic branched double cover.

Local limits of uniform triangulations in high genus

We prove a conjecture of Benjamini and Curien stating that the local limits of uniform random triangulations whose genus is proportional to the number of faces are the planar stochastic hyperbolic triangulations (PSHT) defined in Curien (Probab Theory Relat Fields 165(3):509–540, 2016). The proof relies on a combinatorial argument and the Goulden–Jackson recurrence relation to obtain tightness, and probabilistic arguments showing the uniqueness of the limit. As a consequence, we obtain asymptotics up to subexponential factors on the number of triangulations when both the size and the genus go to infinity. As a part of our proof, we also obtain the following result of independent interest: if a random triangulation of the plane T is weakly Markovian in the sense that the probability to observe a finite triangulation t around the root only depends on the perimeter and volume of t, then T is a mixture of PSHT.

Solution of motion parameters of natural fragments based on monte carlo subdivision projection simulation

In order to get the average windward area of natural fragments of non-metallic materials more accurately and efficiently, and to further simulate the motion parameters, a simulation calculation model of the average windward area of natural fragments is established based on the Monte Carlo subdivision projection method. By the finite element modeling, coordinate translation, random rotation, plane projection and triangulation, the average windward area of natural fragments with arbitrary irregular shape is simulated. Then, based on aerodynamics, the dispersion model of natural fragments is established. The simulation results of the two-dimensional trajectory and the farthest distance of natural fragments are obtained by Runge-Kutta method and cubic spline interpolation, which are compared with the empirical formula of the average windward area of regular fragments. The results show that the value by the Monte Carlo split projection simulation are more consistent with the experimental value, compared with the empirical formula of the windward area of spherical fragment, cube fragment, short cylinder fragment, cylinder fragment, rhombus fragment and cuboid fragment. The error of the farthest distance is between 13% and 25%. It is proved that the result of the average windward area based on Monte Carlo Subdivision Projection Simulation is more suitable for solving the dispersion model of natural fragments of low-speed and non-metallic materials.

Propagation in polymer parameterised field theory

The Hamiltonian constraint operator in loop quantum gravity acts ultralocally. Smolin has argued that this ultralocality seems incompatible with the existence of a quantum dynamics which propagates perturbations between macroscopically seperated regions of quantum geometry. We present evidence to the contrary within an LQG type ‘polymer’ quantization of two dimensional parameterised field theory (PFT). PFT is a generally covariant reformulation of free field propagation on flat spacetime. We show explicitly that while, as in LQG, the Hamiltonian constraint operator in PFT acts ultralocally, states in the joint kernel of the Hamiltonian and diffeomorphism constraints of PFT necessarily describe propagation effects. The particular structure of the finite triangulation Hamiltonian constraint operator plays a crucial role, as does the necessity of imposing (the continuum limit of) its kinematic adjoint as a constraint. Propagation is seen as a property encoded by physical states in the kernel of the constraints rather than that of repeated actions of the finite triangulation Hamiltonian constraint on kinematic states. The analysis yields robust structural lessons for putative constructions of the Hamiltonian constraint in LQG for which ultralocal action co-exists with a description of propagation effects by physical states.

The finite body triangulation: algorithms, subgraphs, homogeneity estimation and application.

The concept of a finite body Dirichlet tessellation has been extended to that of a finite body Delaunay 'triangulation' to provide a more meaningful description of the spatial distribution of nonspherical secondary phase bodies in 2- and 3-dimensional images. A finite body triangulation (FBT) consists of a network of minimum edge-to-edge distances between adjacent objects in a microstructure. From this is also obtained the characteristic object chords formed by the intersection of the object boundary with the finite body tessellation. These two sets of distances form the basis of a parsimonious homogeneity estimation. The characteristics of the spatial distribution are then evaluated with respect to the distances between objects and the distances within them. Quantitative analysis shows that more physically representative distributions can be obtained by selecting subgraphs, such as the relative neighbourhood graph and the minimum spanning tree, from the finite body tessellation. To demonstrate their potential, we apply these methods to 3-dimensional X-ray computed tomographic images of foamed cement and their 2-dimensional cross sections. The Python computer code used to estimate the FBT is made available. Other applications for the algorithm - such as porous media transport and crack-tip propagation - are also discussed.

The Ising Model on the Random Planar Causal Triangulation: Bounds on the Critical Line and Magnetization Properties

We investigate a Gibbs (annealed) probability measure defined on Ising spin configurations on causal triangulations of the plane. We study the region where such measure can be defined and provide bounds on the boundary of this region (critical line). We prove that for any finite random triangulation the magnetization of the central spin is sensitive of the boundary conditions. Furthermore, we show that in the infinite volume limit, the magnetization of the central spin vanishes for values of the temperature high enough.

A TQFT of Turaev–Viro Type on Shaped Triangulations

A shaped triangulation is a finite triangulation of an oriented pseudo-three-manifold where each tetrahedron carries dihedral angles of an ideal hyperbolic tetrahedron. To each shaped triangulation, we associate a quantum partition function in the form of an absolutely convergent state integral which is invariant under shaped 3–2 Pachner moves and invariant with respect to shape gauge transformations generated by total dihedral angles around internal edges through the Neumann–Zagier Poisson bracket. Similarly to Turaev–Viro theory, the state variables live on edges of the triangulation but take their values on the whole real axis. The tetrahedral weight functions are composed of three hyperbolic gamma functions in a way that they enjoy a manifest tetrahedral symmetry. We conjecture that for shaped triangulations of closed three-manifolds, our partition function is twice the absolute value squared of the partition function of Techmuller TQFT defined by Andersen and Kashaev. This is similar to the known relationship between the Turaev–Viro and the Witten–Reshetikhin–Turaev invariants of three-manifolds. We also discuss interpretations of our construction in terms of three-dimensional supersymmetric field theories related to triangulated three-dimensional manifolds.

Constraint algebra in loop quantum gravity reloaded. I. Toy model of aU(1)3gauge theory

We analyze the issue of anomaly-free representations of the constraint algebra in Loop Quantum Gravity (LQG) in the context of a diffeomorphism-invariant gauge theory in three spacetime dimensions. We construct a Hamiltonian constraint operator whose commutator matches with a quantization of the classical Poisson bracket involving structure functions. Our quantization scheme is based on a geometric interpretation of the Hamiltonian constraint as a generator of phase space-dependent diffeomorphisms. The resulting Hamiltonian constraint at finite triangulation has a conceptual similarity with the "mu-bar"-scheme in loop quantum cosmology and highly intricate action on the spin-network states of the theory. We construct a subspace of non-normalizable states (distributions) on which the continuum Hamiltonian constraint is defined which leads to an anomaly-free representation of the Poisson bracket of two Hamiltonian constraints in loop quantized framework.

Towards an anomaly-free quantum dynamics for a weak coupling limit of Euclidean gravity

The ${G}_{\mathrm{Newton}}\ensuremath{\rightarrow}0$ limit of Euclidean gravity introduced by Smolin is described by a generally covariant $U(1{)}^{3}$ gauge theory. The Poisson-bracket algebra of its Hamiltonian and diffeomorphism constraints is isomorphic to that of gravity. Motivated by recent results in parametrized field theory and by the search for an anomaly-free quantum dynamics for loop quantum gravity, the quantum Hamiltonian constraint of density weight $4/3$ for this $U(1{)}^{3}$ theory is constructed so as to produce a nontrivial loop quantum gravity type representation of its Poisson brackets through the following steps. First, the constraint at finite triangulation and the commutator between a pair of such constraints are constructed as operators on the ``charge'' network basis. Next, the continuum limit of the commutator is evaluated with respect to an operator topology defined by a certain space of ``vertex smooth'' distributions. Finally, the operator corresponding to the Poisson bracket between a pair of Hamiltonian constraints is constructed at finite triangulation in such a way as to generate a ``generalized'' diffeomorphism and its continuum limit is shown to agree with that of the commutator between a pair of finite-triangulation Hamiltonian constraints. Our results, in conjunction with the recent work of Henderson, Laddha and Tomlin in a ($2+1$)-dimensional context, constitute the necessary first steps toward a satisfactory treatment of the quantum dynamics of this model.

The diffeomorphism constraint operator in loop quantum gravity

We construct the diffeomorphism constraint operator at finite triangulation from the basic holonomy- flux operators of Loop Quantum Gravity, and show that the action of its continuum limit provides an anomaly free representation of the Lie algebra of diffeomorphisms of the 3- manifold. Key features of our analysis include: (i) finite triangulation approximants to the curvature, Fiab of the Ashtekar- Barbero connection which involve not only small loop holonomies but also small surface fluxes as well as an explicit dependence on the edge labels of the spin network being acted on (ii) the dependence of the small loop underlying the holonomy on both the direction and magnitude of the shift vector field (iii) continuum constraint operators which do not have finite action on the kinematic Hilbert space, thus implementing a key lesson from recent studies of parameterised field theory by the authors. Features (i) and (ii) provide the first hints in LQG of a conceptual similarity with the so called "mu- bar" scheme of Loop Quantum Cosmology. We expect our work to be of use in the construction of an anomaly free quantum dynamics for LQG.We highlight the main steps and results of our construction while suppressing most of the technical details. This work was done jointly with Alok Laddha.

The diffeomorphism constraint operator in loop quantum gravity

We construct the smeared diffeomorphism constraint operator at finite triangulation from the basic holonomy-flux operators of loop quantum gravity (LQG), evaluate its continuum limit on the Lewandowski–Marolf habitat and show that the action of the continuum operator provides an anomaly-free representation of the Lie algebra of diffeomorphisms of the 3-manifold. Key features of our analysis include (i) finite triangulation approximants to the curvature, Fiab, of the Ashtekar–Barbero connection which involve not only small loop holonomies but also small surface fluxes as well as an explicit dependence on the edge labels of the spin network being acted on (ii) the dependence of the small loop underlying the holonomy on both the direction and magnitude of the shift vector field, (iii) continuum constraint operators which do not have finite action on the kinematic Hilbert space, thus implementing a key lesson from recent studies of parameterized field theory by the authors. Features (i) and (ii) provide the first hints in LQG of a conceptual similarity with the so-called mu-bar scheme of loop quantum cosmology. We expect our work to be of use in the construction of an anomaly-free quantum dynamics for LQG.

Adaptive finite element mesh triangulation using self-organizing neural networks

The finite element method is a computationally intensive method. Effective use of the method requires setting up the computational framework in an appropriate manner, which typically requires expertise. The computational cost of generating the mesh may be much lower, comparable, or in some cases higher than the cost associated with the numeric solver of the partial differential equations, depending on the application and the specific numeric scheme at hand. The aim of this paper is to present a mesh generation approach using the application of self-organizing artificial neural networks through adaptive finite element computations. The problem domain is initially constructed using the self-organizing neural networks. This domain is used as the background mesh which forms the input for finite element analysis and from which adaptive parameters are calculated through adaptivity analysis. Subsequently, self-organizing neural network is used again to adjust the location of randomly selected mesh nodes as is the coordinates of all nodes within a certain neighborhood of the chosen node. The adjustment is a movement of the selected nodes toward a specific input point on the mesh. Thus, based on the results obtained from the adaptivity analysis, the movement of nodal points adjusts the element sizes in a way that the concentration of elements will occur in the regions of high stresses. The methods and experiments developed here are for two-dimensional triangular elements but seem naturally extendible to quadrilateral elements.

Latin trades in groups defined on planar triangulations

For a finite triangulation of the plane with faces properly coloured white and black, let A be the abelian group constructed by labelling the vertices with commuting indeterminates and adding relations which say that the labels around each white triangle add to the identity. We show that A has free rank exactly two. Let A* be the torsion subgroup of A, and B* the corresponding group for the black triangles. We show that A* and B* have the same order, and conjecture that they are isomorphic. For each spherical latin trade W, we show there is a unique disjoint mate B such that (W,B) is a connected and separated bitrade. The bitrade (W,B) is associated with a two-colourable planar triangulation and we show that W can be embedded in A*, thereby proving a conjecture due to Cavenagh and Drapal. The proof involves constructing a (0,1) presentation matrix whose permanent and determinant agree up to sign. The Smith Normal Form of this matrix determines A*, so there is an efficient algorithm to construct the embedding. Contrasting with the spherical case, for each genus g>0 we construct a latin trade which is not embeddable in any group and another that is embeddable in a cyclic group. We construct a sequence of spherical latin trades which cannot be embedded in any family of abelian groups whose torsion ranks are bounded. Also, we show that any trade that can be embedded in a finitely generated abelian group can be embedded in a finite abelian group. As a corollary, no trade can be embedded in a free abelian group.

EMBEDDING POINT SETS INTO PLANE GRAPHS OF SMALL DILATION

Let S be a set of points in the plane. What is the minimum possible dilation of all plane graphs that contain S? Even for a set S as simple as five points evenly placed on the circle, this question seems hard to answer; it is not even clear if there exists a lower bound > 1. In this paper we provide the first upper and lower bounds for the embedding problem. 1. Each finite point set can be embedded into the vertex set of a finite triangulation of dilation ≤ 1.1247. 2. Each embedding of a closed convex curve has dilation ≥ 1.00157. 3. Let P be the plane graph that results from intersecting n infinite families of equidistant, parallel lines in general position. Then the vertex set of P has dilation [Formula: see text].

A gravitational effective action on a finite triangulation

We construct a function of the edge-lengths of a triangulated surface whose variation under a rescaling of all the edges that meet at a vertex is the defect angle at that vertex. We interpret this function as a gravitational effective action on the triangulation, and the variation as a trace anomaly.

Growth and percolation on the uniform infinite planar triangulation

A construction as a growth process for sampling of the uniform in- finite planar triangulation (UIPT), defined in [AnS], is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the UIPT.By analyzing the progress rate of the growth process we show that a.s. the UIPT has growth rate r 4 up to polylogarithmic factors, in accordance with heuristic results from the physics literature. Additionally, the boundary component of the ball of radius r separating it from infinity a.s. has growth rate r 2 up to polylogarithmic factors. It is also shown that the properly scaled size of a variant of the free triangulation of an m-gon (also defined in [AnS]) converges in distribution to an asymmetric stable random variable of type 1/2.By combining Bernoulli site percolation with the growth process for the UIPT, it is shown that a.s. the critical probability p c = 1/2 and that at p c percolation does not occur.

Trends in automatic three-dimensional mesh generation

Some recent efforts on the development of methods to ensure the robustness of automatic three-dimensional mesh generation techniques are discussed. The topic areas considered are mesh entity classification, finite octree cell triangulation, and coarse mesh generation by element removal.

擬多様体の向き付け可能性とスペルナーの補題の一般化

We propose a combinatorial framework for fixed point algorithms and constructive proofs of combi· natoriallemmas in topology. The framework consists of two sets of pseudomanifolds and an operator relating them. They have lattice structures which are dual to each other. We show that the set of joins of pseudomanifolds related by the operator is a homogeneous and orientable pseudomanifold under several conditions. By exploiting this framework we generalize Sperner's lemma on convex polytope. Namely, let C be a convex polytope with m facets Fl> Fm' S be a finite triangulation of C and S = {ol 0 is a face of some simplex of S}. Given a nondenerate vertex v of C and a labelling function Q from the set of verticI!s of S to { 1, ... , m}, the set of indices of facets, there is an odd number of simplices 0 of S such that Q(0)U{iI1 ;'i i ;'i rn, 0 C Fj} strictly includes {ill ;'i i ;'i rn, v E Fi} We also prove the generalization of Sperner's lemma by Fan and van der Laan·Talrnan·Van der Heyden's lemma as corollaries to the result.