3,652 research outputs found
Entropy of unimodular Lattice Triangulations
Triangulations are important objects of study in combinatorics, finite
element simulations and quantum gravity, where its entropy is crucial for many
physical properties. Due to their inherent complex topological structure even
the number of possible triangulations is unknown for large systems. We present
a novel algorithm for an approximate enumeration which is based on calculations
of the density of states using the Wang-Landau flat histogram sampling. For
triangulations on two-dimensional integer lattices we achive excellent
agreement with known exact numbers of small triangulations as well as an
improvement of analytical calculated asymptotics. The entropy density is
consistent with rigorous upper and lower bounds. The presented
numerical scheme can easily be applied to other counting and optimization
problems.Comment: 6 pages, 7 figure
Crumpled triangulations and critical points in 4D simplicial quantum gravity
This is an expanded and revised version of our geometrical analysis of the
strong coupling phase of 4D simplicial quantum gravity. The main differences
with respect to the former version is a full discussion of singular
triangulations with singular vertices connected by a subsingular edge. In
particular we provide analytical arguments which characterize the entropical
properties of triangulations with a singular edge connecting two singular
vertices. The analytical estimate of the location of the critical coupling at
k_2\simeq 1.3093 is presented in more details. Finally we also provide a model
for pseudo-criticality at finite N_4(S^4).Comment: 44 page
There are 174 Subdivisions of the Hexahedron into Tetrahedra
This article answers an important theoretical question: How many different
subdivisions of the hexahedron into tetrahedra are there? It is well known that
the cube has five subdivisions into 6 tetrahedra and one subdivision into 5
tetrahedra. However, all hexahedra are not cubes and moving the vertex
positions increases the number of subdivisions. Recent hexahedral dominant
meshing methods try to take these configurations into account for combining
tetrahedra into hexahedra, but fail to enumerate them all: they use only a set
of 10 subdivisions among the 174 we found in this article.
The enumeration of these 174 subdivisions of the hexahedron into tetrahedra
is our combinatorial result. Each of the 174 subdivisions has between 5 and 15
tetrahedra and is actually a class of 2 to 48 equivalent instances which are
identical up to vertex relabeling. We further show that exactly 171 of these
subdivisions have a geometrical realization, i.e. there exist coordinates of
the eight hexahedron vertices in a three-dimensional space such that the
geometrical tetrahedral mesh is valid. We exhibit the tetrahedral meshes for
these configurations and show in particular subdivisions of hexahedra with 15
tetrahedra that have a strictly positive Jacobian
Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations
One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology
Volumes of Polytopes Without Triangulations
The geometry of the dual amplituhedron is generally described in reference to
a particular triangulation. A given triangulation manifests only certain
aspects of the underlying space while obscuring others, therefore understanding
this geometry without reference to a particular triangulation is desirable. In
this note we introduce a new formalism for computing the volumes of general
polytopes in any dimension. We define new "vertex objects" and introduce a
calculus for expressing volumes of polytopes in terms of them. These
expressions are unique, independent of any triangulation, manifestly depend
only on the vertices of the underlying polytope, and can be used to easily
derive identities amongst different triangulations. As one application of this
formalism, we obtain new expressions for the volume of the tree-level,
-point NMHV dual amplituhedron.Comment: 32 pages, 12 figure
New higher-order transition in causal dynamical triangulations
We reinvestigate the recently discovered bifurcation phase transition in
Causal Dynamical Triangulations (CDT) and provide further evidence that it is a
higher order transition. We also investigate the impact of introducing matter
in the form of massless scalar fields to CDT. We discuss the impact of scalar
fields on the measured spatial volumes and fluctuation profiles in addition to
analysing how the scalar fields influence the position of the bifurcation
transition.Comment: 15 pages, 11 figures. Conforms with version accepted for publication
in Phys. Rev.
Signature Change of the Metric in CDT Quantum Gravity?
We study the effective transfer matrix within the semiclassical and
bifurcation phases of CDT quantum gravity. We find that for sufficiently large
lattice volumes the kinetic term of the effective transfer matrix has a
different sign in each of the two phases. We argue that this sign change can be
viewed as a Wick rotation of the metric. We discuss the likely microscopic
mechanism responsible for the bifurcation phase transition, and propose an
order parameter that can potentially be used to determine the precise location
and order of the transition. Using the effective transfer matrix we
approximately locate the position of the bifurcation transition in some region
of coupling constant space, allowing us to present an updated version of the
CDT phase diagram.Comment: 16 pages, 9 figure
- …