Finite Metrics Research Articles

Subdivisions of Hypersimplices: With a View Toward Finite Metric Spaces

The secondary fan Σ ( k , n ) is a polyhedral fan which stratifies the regular subdivisions of the hypersimplices Δ ( k , n ) . We find new infinite families of rays of Σ ( k , n ) , and we compute the fans Σ ( 2 , 7 ) and Σ ( 3 , 6 ) . In the special case k = 2 the fan Σ ( 2 , n ) is closely related to the metric fan MF ( n ) , which forms a natural parameter space for the metric spaces on n points. So our results yield a classification of the finite metric spaces on seven points.

Topological Properties of a Non‐Hermitian Quasi‐1D Chain with a Flat Band

AbstractThe spectral properties of a non‐Hermitian quasi‐1D lattice in two of the possible dimerization configurations are investigated. Specifically, it focuses on a non‐Hermitian diamond chain that presents a zero‐energy flat band. The flat band originates from wave interference and results in eigenstates with a finite contribution only on two sites of the unit cell. To achieve the non‐Hermitian characteristics, the system under study presents non‐reciprocal hopping terms in the chain. This leads to the accumulation of eigenstates on the boundary of the system, known as the non‐Hermitian skin effect. Despite this accumulation of eigenstates, for one of the two considered configurations, it is possible to characterize the presence of non‐trivial edge states at zero energy by a real‐space topological invariant known as the biorthogonal polarization. This work shows that this invariant, evaluated using the destructive interference method, characterizes the non‐trivial phase of the non‐Hermitian diamond chain. For the second non‐Hermitian configuration, there is a finite quantum metric associated with the flat band. Additionally, the system presents the skin effect despite the system having a purely real or imaginary spectrum. The two non‐Hermitian diamond chains can be mapped into two models of the Su‐Schrieffer‐Heeger chains, either non‐Hermitian, and Hermitian, both in the presence of a flat band. This mapping allows to draw valuable insights into the behavior and properties of these systems.

Talagrand’s influence inequality revisited

Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the uniform probability measure $\sigma_n$. Talagrand's influence inequality (1994) asserts that there exists $C\in(0,\infty)$ such that for every $n\in\mathbb{N}$, every function $f:\mathscr{C}_n\to\mathbb{C}$ satisfies $$\mathrm{Var}_{\sigma_n}(f) \leq C \sum_{i=1}^n \frac{\|\partial_if\|_{L_2(\sigma_n)}^2}{1+\log\big(\|\partial_if\|_{L_2(\sigma_n)}/\|\partial_i f\|_{L_1(\sigma_n)}\big)}.$$ In this work, we undertake a systematic investigation of this and related inequalities via harmonic analytic and stochastic techniques and derive applications to metric embeddings. We prove that Talagrand's inequality extends, up to an additional doubly logarithmic factor, to Banach space-valued functions under the necessary assumption that the target space has Rademacher type 2 and that this doubly logarithmic term can be omitted if the target space admits an equivalent 2-uniformly smooth norm. These are the first vector-valued extensions of Talagrand's influence inequality. We also obtain a joint strengthening of results of Bakry-Meyer (1982) and Naor-Schechtman (2002) on the action of negative powers of the hypercube Laplacian on functions $f:\mathscr{C}_n\to E$, whose target space $E$ has nontrivial Rademacher type via a new vector-valued version of Meyer's multiplier theorem (1984). Inspired by Talagrand's influence inequality, we introduce a new metric invariant called Talagrand type and estimate it for Banach spaces with prescribed Rademacher or martingale type, Gromov hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature. Finally, we prove that Talagrand type is an obstruction to the bi-Lipschitz embeddability of nonlinear quotients of the hypercube $\mathscr{C}_n$, thus deriving new nonembeddability results for these finite metrics.

Homeomorphisms of Finite Metric Distortion Between Riemannian Manifolds

Homeomorphisms of Finite Metric Distortion Between Riemannian Manifolds

Local Geometry of the Gromov–Hausdorff Metric Space and Totally Asymmetric Finite Metric Spaces

In the present paper, we investigate the structure of the metric space M of compact metric spaces considered up to an isometry and endowed with the Gromov–Hausdorff metric in a neighborhood of a finite metric space, whose isometry group is trivial. It is shown that a sufficiently small ball in the subspace of M consisting of finite spaces with the same number of points centered at such a space is isometric to a corresponding ball in the space ℝN endowed with the norm |(x1, . . . , xN)| = \( \underset{i}{\max}\left|{x}_i\right| \). Also an isometric embedding of a finite metric space into a neighborhood of a finite asymmetric space in M is constructed.

Designs in Finite Metric Spaces: A Probabilistic Approach

A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in Q-polynomial distance-regular graphs. An approximation of their cumulative distribution function, based on the notion of Christoffel function in approximation theory is given. As an application we derive limit laws on the weight distributions of binary orthogonal arrays of strength going to infinity. An analogous result for combinatorial designs of strength going to infinity is given.

Cylinder curves in finite holonomy flat metrics

For an orientable surface of finite type equipped with a flat metric with holonomy of finite order [Formula: see text], the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when [Formula: see text] is well-studied as such surfaces are (semi-)translation surfaces. Not only is the set always infinite, the core curves form an infinite diameter subset of the curve complex. In this paper we focus on the case [Formula: see text] and construct examples illustrating a range of behaviors for the embedded cylinder curves. We prove that if [Formula: see text] and the surface is fully punctured, then the embedded cylinder curves form a finite diameter subset of the curve complex. The same analysis shows that the embedded cylinder curves can only have infinite diameter when the metric has a very specific form. Using this we characterize precisely when the embedded cylinder curves accumulate on a point in the Gromov boundary.

Characterizations of canonically compactifiable graphs via intrinsic metrics and algebraic properties

We consider infinite graphs and the associated energy forms. We show that a graph is canonically compactifiable (i.e. all functions of finite energy are bounded) if and only if the underlying set is totally bounded with respect to any finite measure intrinsic metric. Furthermore, we show that a graph is canonically compactifiable if and only if the space of functions of finite energy is an algebra. These results answer questions in a recent work of Georgakopoulos, Haeseler, Keller, Lenz, and Wojciechowski.

Ranking and Rating Bicycle Helmet Safety Performance in Oblique Impacts Using Eight Different Brain Injury Models

Bicycle helmets are shown to offer protection against head injuries. Rating methods and test standards are used to evaluate different helmet designs and safety performance. Both strain-based injury criteria obtained from finite element brain injury models and metrics derived from global kinematic responses can be used to evaluate helmet safety performance. Little is known about how different injury models or injury metrics would rank and rate different helmets. The objective of this study was to determine how eight brain models and eight metrics based on global kinematics rank and rate a large number of bicycle helmets (n=17) subjected to oblique impacts. The results showed that the ranking and rating are influenced by the choice of model and metric. Kendall’s tau varied between 0.50 and 0.95 when the ranking was based on maximum principal strain from brain models. One specific helmet was rated as 2-star when using one brain model but as 4-star by another model. This could cause confusion for consumers rather than inform them of the relative safety performance of a helmet. Therefore, we suggest that the biomechanics community should create a norm or recommendation for future ranking and rating methods.

Exact Controllability of the 1-D Wave Equation on Finite Metric Tree Graphs

In this paper, we consider initial boundary value problems and control problems for the wave equation on finite metric graphs with Dirichlet boundary controls. We propose new constructive algorithms for solving initial boundary value problems on general graphs and boundary control problems on tree graphs. We demonstrate that the wave equation on a tree is exactly controllable if and only if controls are applied at all or all but one of the boundary vertices. We find the minimal controllability time and prove that our result is optimal in the general case. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the full controllability utilizes both dynamical and moment method approaches.

The polytopal structure of the tight-span of a totally split-decomposable metric

The tight-span of a finite metric space is a polytopal complex that has appeared in several areas of mathematics. In this paper we determine the polytopal structure of the tight-span of a totally split-decomposable (finite) metric. These metrics are a generalization of tree-metrics and have importance within phylogenetics. In previous work, we showed that the cells of the tight-span of such a metric are zonotopes that are polytope isomorphic to either hypercubes or rhombic dodecahedra. Here, we extend these results and show that the tight-span of a totally split-decomposable metric can be broken up into a canonical collection of polytopal complexes whose polytopal structures can be directly determined from the metric. This allows us to also completely determine the polytopal structure of the tight-span of a totally split-decomposable metric. We anticipate that our improved understanding of this structure may lead to improved techniques for phylogenetic inference.

The exploration of nonlinear elasticity and its efficient parameterization for crystalline materials

Conventional approaches to analyzing the very large coherency strains that can occur during solid-state phase transformations are founded in linear elasticity and rely on infinitesimal strain metrics. Despite this, there are many technologically important examples where misfit strains of multi-phase mixtures are very large during their synthesis and/or application. In this paper, we present a framework for constructing strain-energy expressions and stress-strain relationships beyond the linear-elastic limit for crystalline solids. This approach utilizes group theoretical concepts to minimize both the number of free parameters in the strain-energy expression and amount of first-principles training data required to parameterize strain-energy models that are invariant to all crystal symmetries. Within this framework, the strain-energy and elastic stiffness can be described to high accuracy in terms of a set of conventional symmetry-adapted finite strain metrics that we define independent of crystal symmetry. As an illustration, we use first-principles electronic structure data to parameterize strain energy polynomials and employ them to explore the strain-energy surfaces of HCP Zr and Mg, as well as several important Zr-H and Mg-Nd phases that are known to precipitate coherently within the HCP matrices of Zr and Mg.

On Eigen's Quasispecies Model, Two-Valued Fitness Landscapes, and Isometry Groups Acting on Finite Metric Spaces.

A two-valued fitness landscape is introduced for the classical Eigen's quasispecies model. This fitness landscape can be considered as a direct generalization of the so-called single- or sharply peaked landscape. A general, non-permutation invariant quasispecies model is studied, and therefore the dimension of the problem is [Formula: see text], where N is the sequence length. It is shown that if the fitness function is equal to [Formula: see text] on a G-orbit A and is equal to w elsewhere, then the mean population fitness can be found as the largest root of an algebraic equation of degree at most [Formula: see text]. Here G is an arbitrary isometry group acting on the metric space of sequences of zeroes and ones of the length N with the Hamming distance. An explicit form of this exact algebraic equation is given in terms of the spherical growth function of the G-orbit A. Motivated by the analysis of the two-valued fitness landscapes, an abstract generalization of Eigen's model is introduced such that the sequences are identified with the points of a finite metric space X together with a group of isometries acting transitively on X. In particular, a simplicial analog of the original quasispecies model is discussed, which can be considered as a mathematical model of the switching of the antigenic variants for some bacteria.

Convergence of freely decomposable Kleinian groups

We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination. We prove that if this limit projective lamination is doubly incompressible, then the sequence has compact closure in the deformation space. As a consequence we generalise Thurston’s double limit theorem and solve his conjecture on convergence of function groups affirmatively.

Classification of Finite Metric Spaces and Combinatorics of Convex Polytopes

We describe the canonical correspondence between finite metric spaces and symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those polytopes.

Probabilistic Properties of Topologies of Minimal Fillings of Finite Metric Spaces

In this work, we provide a way to introduce a probability measure on the space of minimal fillings of finite additive metric spaces as well as an algorithm for its computation. The values of probability, obtained from the analytical solution, coincide with the computer simulation for the computed cases. Also the developed technique makes it possible to find the asymptotic of the ratio for families of graph structures.

Diversities and the Geometry of Hypergraphs

Special issu PRIMA 2013 The embedding of finite metrics in 1 has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems as there is close relation between max-flow min-cut theorems and the minimal distortion embeddings of metrics into 1. Here we show that this theory can be generalized to a larger set of combinatorial optimization problems on both graphs and hypergraphs. This theory is not built on metrics and metric embeddings, but on diversities, a type of multi-way metric introduced recently by the authors. We explore diversity embeddings, 1 diversities, and their application to Steiner Tree Packing and Hypergraph Cut problems.

Smith theory, L2-cohomology, isometries of locally symmetric manifolds, and moduli spaces of curves

We investigate periodic diffeomorphisms of noncompact aspherical manifolds (and orbifolds) and describe a class of spaces that have no homotopically trivial periodic diffeomorphisms. Prominent examples are moduli spaces of curves and aspherical locally symmetric spaces with nonzero Euler characteristic. In the irreducible locally symmetric case, we show that no complete metric has more symmetry than the locally symmetric metric. In the moduli space case, we build on work of Farb and Weinberger and we prove an analogue of Royden’s theorem for complete finite volume metrics.

Periodic flats and group actions on locally symmetric spaces

We use maximal periodic flats to show that on a finite volume irreducible locally symmetric manifold of dimension $\geq 3$, no metric $g$ has more symmetry than the locally symmetric metric. We also show that if $g$ is a finite volume metric that is not locally symmetric, then its lift to the universal cover has discrete isometry group.

Equivalence conditions for the finite volume and finite element methods in spherical coordinates

A numerical technique for the solution of the compressible flow equations over unstructured grids in a spherical reference is presented. The proposed approach is based on a mixed finite volume/finite element discretization in space. Equivalence conditions relating the finite volume and the finite element metrics in spherical coordinates are derived. Numerical simulations of the explosion and implosion problems for inviscid compressible flows are carried out to evaluate the correctness of the numerical scheme and compare fairly well to one-dimensional simulations over very fine grids.