Number Of Simplices Research Articles

Tensor network approach to 2D Lorentzian quantum Regge calculus

We demonstrate a tensor renormalization group (TRG) calculation for a two-dimensional Lorentzian model of quantum Regge calculus (QRC). This model is expressed in terms of a tensor network by discretizing the continuous edge lengths of simplicial manifolds and identifying them as tensor indices. The expectation value of space-time area, which is obtained through the higher-order TRG method, nicely reproduces the exact value. The Lorentzian model does not have the spike configuration that was an obstacle in the Euclidean QRC, but it still has a length-divergent configuration called a pinched geometry. We find a possibility that the pinched geometry is suppressed by checking the average edge length squared in the limit where the number of simplices is large. This implies that the Lorentzian model may describe smooth geometries, although the investigation of the higher moments is required to make the statement more conclusive. Our results also indicate that TRG is a promising approach to numerical study of simplicial quantum gravity.

Augmented Simplicial Combinatorics through Category Theory: Cones, Suspensions and Joins

In this work, we analyze the combinatorial properties of the category of augmented semi-simplicial sets. We consider various monoidal structures induced by the co-product, the product, and the join operator in this category. In addition, we also consider monoidal structures on augmented sequences of integers induced by the sum and product of integers and by the join of augmented sequences. The cardinal functor that associates to each finite set X its cardinal |X| induces the sequential cardinal that associates to each augmented semi-simplicial finite set X an augmented sequence |X|n of non-negative integers. We prove that the sequential cardinal functor is monoidal for the corresponding monoidal structures. This allows us to easily calculate the number of simplices of cones and suspensions of an augmented semi-simplicial set as well as other augmented semi-simplicial sets which are built by joins. In this way, the monoidal structures of the augmented sequences of numbers may be thought of as an algebraization of the augmented semi-simplicial sets that allows us to do a simpler study of the combinatorics of the augmented semi-simplicial finite sets.

Optimizing the Simplicial-Map Neural Network Architecture.

Simplicial-map neural networks are a recent neural network architecture induced by simplicial maps defined between simplicial complexes. It has been proved that simplicial-map neural networks are universal approximators and that they can be refined to be robust to adversarial attacks. In this paper, the refinement toward robustness is optimized by reducing the number of simplices (i.e., nodes) needed. We have shown experimentally that such a refined neural network is equivalent to the original network as a classification tool but requires much less storage.

$S$-Hypersimplices, Pulling Triangulations, and Monotone paths

An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this paper we study faces and dissections of $S$-hypersimplices. Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order.

On the Number of Coloured Triangulations of d-Manifolds

We give superexponential lower and upper bounds on the number of coloured d-dimensional triangulations whose underlying space is an oriented manifold, when the number of simplices goes to infinity and $$d\ge 3$$ is fixed. In the special case of dimension 3, the lower and upper bounds match up to exponential factors, and we show that there are $$2^{O(n)} n^{\frac{n}{6}}$$ coloured triangulations of 3-manifolds with n tetrahedra. Our results also imply that random coloured triangulations of 3-manifolds have a sublinear number of vertices. The upper bounds apply in particular to coloured d-spheres for which they seem to be the best known bounds in any dimension $$d\ge 3$$ , even though it is often conjectured that exponential bounds hold in this case. We also ask a related question on regular edge-coloured graphs having the property that each 3-coloured component is planar, which is of independent interest.

Homotopy type of skeleta of the flag complex over a finite vector space and generalized Galois numbers

Let $${\mathscr {F}}$$ stand for the flag complex associated to the lattice of proper subspaces of a finite-dimensional vector space V. This paper aims at giving a (discrete) Morse theoretical proof of the fact that the k-th skeleton of $${\mathscr {F}}$$ is homotopy equivalent to a wedge of spheres of dimension $$\min \{k,\dim ({\mathscr {F}})\}$$. The tight control provided by Morse theoretic methods (through an explicit discrete gradient field) allows us to give a formula for the number of spheres appearing in each of these wedge summands. As an application, we derive an explicit formula for the number of flags on V of a given dimension, i.e., the number of simplices in $${\mathscr {F}}$$ of the given dimension. Rather than depending on generalized Galois numbers, our formula for flags is given in terms of weighted inversion statistics of the symmetric group.

Quantum search on simplicial complexes

In this paper, we propose an extension of quantum searches on graphs driven by quantum walks to simplicial complexes. To this end, we newly define a quantum walk on simplicial complex which is an alternative of preceding studies by authors. We show that the quantum search on the specific simplicial complex corresponding to the triangulation of $n$-dimensional unit square driven by this new simplicial quantum walk works well, namely, a marked simplex can be found with probability $1+o(1)$ with in a time $O(\sqrt{N})$, where $N$ is the number of simplices with the dimension of marked simplex.

Expected sizes of Poisson–Delaunay mosaics and their discrete Morse functions

AbstractMapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝn, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensionsn≤ 4.

An improvement on the number of simplices in [formula omitted

Let E be a set of points in Fqd. Bennett et al. (2016) proved that if |E|≫qd−d−1k+1 then E determines a positive proportion of all k-simplices. In this paper, we give an improvement of this result in the case when E is the Cartesian product of sets. Namely, we show that if E is the Cartesian product of sets and qkdk+1−1∕d=o(|E|), the number of congruence classes of k-simplices determined by E is at least (1−o(1))qk+12, and in some cases our result is sharp.

Automatic Domain Partitioning of Piecewise-Affine Simplicial Functions Implementing Model Predictive Controllers

This brief is concerned with the problem of mitigating the “curse of dimensionality” in the piecewise-affine simplicial approximation of multivariate model predictive control (MPC) functions by resorting to nonuniform domain partitions, in view of their implementation in high-speed embedded digital architectures. Two dual strategies are used to reduce the number of simplices with respect to uniform partitions, without reducing approximation accuracy. Both circuit implementation purposes and the MPC technique impose additional constraints, with respect to other techniques already available for multiresolution domain partitioning. The algorithms are tested on a benchmark example concerned with adaptive cruise control.

Generalizations of the Kolmogorov–Barzdin embedding estimates

We consider several ways to measure the `geometric complexity' of an embedding from a simplicial complex into Euclidean space. One of these is a version of `thickness', based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved for graphs. We also consider the distortion of knots. We give an alternate proof of a theorem of Pardon that there are isotopy classes of knots requiring arbitrarily large distortion. This proof is based on the expander-like properties of arithmetic hyperbolic manifolds.

New Bounds on the Size of Optimal Meshes

AbstractThe theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing ϕ, then the number of vertices in an optimal mesh will be O(ϕdn), where d is the input dimension. We give a new analysis of this integral showing that the output size is only θ(n+nlogϕ). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O(n).

FITTING A STEP FUNCTION TO A POINT SET WITH OUTLIERS BASED ON SIMPLICIAL THICKNESS DATA STRUCTURES

Given a real ∊ > 0, an integer g ≥ 0 and a set of points in the plane, we study the problem of computing a piecewise linear functional curve with minimum number of line segments to approximate all points after removing g outliers such that the approximation error is at most ∊. We give an improved algorithm over the previous work. The algorithm is based on two dynamic data structures developed in this paper for the simplicial thickness queries, which are of independent interest. For a set S of simplices in the d-dimensional space ℝd(d ≥ 2 is a constant), the simplicial thickness of a point p is defined as the number of simplices in S that contain p. Given a set P of n points in ℝd, we develop two dynamic data structures to support the following operations. (1) Simplex insertion: Insert a simplex into S. (2) Simplex deletion: Delete a simplex from S. (3) Simplicial thickness query: Given a query simplex σ, compute the minimum simplicial thickness among all points in σ∩P. The first data structure is constructed in O(n1+δ) time, for any constant δ > 0, and can support each operation in O(n1-1/d) time; the second one is con-structed in O(n log n) time and can support each operation in O(n1-1/d( log n)O(1)) time. Both data structures use O(n). space. These data structures may find other applications as well.

A property that characterizes Euler characteristic among invariants of combinatorial manifolds

If a real valued invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension in the manifold, then the invariant is completely determined by the Euler characteristic of the manifold and its boundary. So essentially, the Euler characteristic is the unique invariant of this type.

Example of a fermion-nondoubling amorphous lattice

An example of an amorphous lattice featuring no fermion doubling is constructed. The lattice represents a bounded two-dimensional simplicial complex with a disk topology, every internal 0-simplex of which belongs to boundaries of an odd number of 1-simplices. The total number of simplices in the complex tends to infinity.

Volume formulae for regular hyperbolic cubes

We express the volume of a regular cube in hyperbolic n n -space as an integral on [ 0 , ∞ ) [0, \infty ) , and derive from this an asymptotic volume formula for the regular ideal hyperbolic n n -cube. This in turn is applied to finding an asymptotic lower bound for the least number of simplices into which a Euclidean n n -cube can be triangulated.

Linear Conditions on the Number of Faces of Manifolds with Boundary

The Euler equation and the Dehn–Sommerville equations are known to be the only (rational) linear conditions forf-vectors (number of simplices at various dimensions) of triangulations of spheres. We generalize this fact to arbitrary triangulations, linear triangulations of manifolds, and polytopal triangulations of Euclidean balls. We prove that for closed manifolds, the Euler equation and the Dehn–Sommerville equations remain the only linear conditions. We also prove that for manifolds with nonempty boundary, the Euler equation is the only linear condition. These results are proved not only over Q, but also over Z and Z/kZ.

F-Vectors of Polyhedra

The f-vector of a triangulation of a polyhedron X is the numbers of simplices at various dimensions. We prove that the affine span of f-vectors of X has dimension (n+s+1)/2, where n is the dimension of X, and s is the dimension of the part of X that is singular with respect to the local Euler characteristic.

Subdivision of simplices relative to a cutting plane and finite concave minimization

In this article, our primary concern is the classical problem of minimizing globally a concave function over a compact polyhedron (Problem (P)). We present a new simplicial branch and bound approach, which combines triangulations of intersections of simplices with halfspaces and ideas from outer approximation in such a way, that a class of finite algorithms for solving (P) results. For arbitrary compact convex feasible sets one obtains a not necessarily finite but convergent algorithm. Theoretical investigations include determination of the number of simplices in each applied triangulation step and bounds on the number of iterations in the resulting algorithms. Preliminary numerical results are given, and additional applications are sketched.

An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere

A general and direct method for computing the Betti numbers of a finite simplicial complex in S d is given. This method is complete for d ⩽ 3, where versions of this method run in time O( nα( n)) and O( n), n the number of simplices. An implementation of the algorithm is applied to alpha shapes, which is a novel geometric modeling tool.