Hyperbolic Length Research Articles

Decreasing the non-linearity of hybrid reluctance actuators by air gap design

This paper presents an air gap design approach to improve the linearity of rotational Hybrid Reluctance Actuators (HRAs) used in fast steering mirrors. The approach involves modeling a one-degree-of-freedom HRA with a magnetic equivalent circuit to identify and analyze sources of non-linearities. On the basis of the verified model, solutions for an improved linear system behavior are analytically searched. Two linearized HRA designs are proposed, one with linear cross-section dependency and the other with hyperbolic air gap length dependency. Finite element method simulations are employed to evaluate the performance with respect to the linearity and the motor constant of these designs, showing factor 50 improved system linearity.

The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature

The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature

Conformal capacity of hedgehogs

We discuss problems concerning the conformal condenser capacity of “hedgehogs”, which are compact sets E E in the unit disk D = { z : | z | > 1 } \mathbb {D}=\{z:\,|z|>1\} consisting of a central body E 0 E_0 that is typically a smaller disk D ¯ r = { z : | z | ≤ r } \overline {\mathbb {D}}_r=\{z:\,|z|\le r\} , 0 > r > 1 0>r>1 , and several spikes E k E_k that are compact sets lying on radial intervals I ( α k ) = { t e i α k : 0 ≤ t > 1 } I(\alpha _k)=\{te^{i\alpha _k}:\,0\le t>1\} . The main questions we are concerned with are the following: (1) How does the conformal capacity c a p ( E ) \mathrm {cap}(E) of E = ∪ k = 0 n E k E=\cup _{k=0}^n E_k behave when the spikes E k E_k , k = 1 k=1 , …, n n , move along the intervals I ( α k ) I(\alpha _k) toward the central body if their hyperbolic lengths are preserved during the motion? (2) How does the capacity c a p ( E ) \mathrm {cap}(E) depend on the distribution of angles between the spikes E k E_k ? We prove several results related to these questions and discuss methods of applying symmetrization type transformations to study the capacity of hedgehogs. Several open problems, including problems on the capacity of hedgehogs in the three-dimensional hyperbolic space, will also be suggested.

On genetic algorithm and artificial neural network combined optimization for a Mars rotorcraft blade

The Mars rotorcraft can help the Mars rover to optimize the driving route and explore more efficiently. However, the thin and cold Martian atmosphere can heavily increase the consumed power of the rotorcraft blades and reduce the generated thrust, leading to the reduction in power loading. The aerodynamic force generated by the Mars rotorcraft blade can be significantly influenced by the blade structure, which has several main structural characteristics and needs to be well designed. A multivariable optimization design method of blade structure is proposed based on neural network, genetic algorithm, and computational fluid dynamics simulations. Experiments conducted in the Martian atmosphere simulator indicate that the Mars rotorcraft blades with the hyperbolic chord length distribution and twist have higher power loading in the constant collective pitch angle. • A model was built to predict rotor hover performance at low-Reynolds-number. • Established an optimization model of the planform of the Mars rotorcraft blade. • The optimized blade consumes 29% less power than the conventional blade. • The optimization rusult were verified experimentally in Mars simulated environment.

A hyperbolic divergence based nonparametric test for two‐sample multivariate distributions

AbstractTwo‐sample hypothesis testing, as a fundamental problem in statistical inference, seeks to detect the difference between two probability measures and has numerous real‐world applications. Current test procedures for multivariate two‐sample problems typically rely on angles and lengths in a Euclidean space, or lengths in a unit hypersphere after representing data with the spherical model. This article introduces a hyperbolic divergence based on hyperbolic lengths in hyperbolic geometry, as well as a subsequent nonparametric approach to testing the multivariate two‐sample problem. We investigate the properties of our test procedure and discover that our hyperbolic divergence statistic is strongly consistent and consistent against all other alternatives; we also demonstrate that its limit distribution is an infinite mixture of distributions under the null hypothesis and a normal distribution under the alternative hypothesis. To calculate the ‐value, we employ the permutation method. Furthermore, in numerical studies, we compare our method with several nonparametric procedures under various distributional assumptions and alternatives. We discover that our test procedure has some advantages when the distributions' complex correlation structures differ. Finally, we examine one real data set to show how our method can be used to test two‐sample heterogeneity.

Euclidean Artin–Tits groups are acylindrically hyperbolic

In this paper, we prove that all Euclidean Artin–Tits groups are acylindrically hyperbolic. To any Garside group of finite type, Wiest and the author associated a hyperbolic graph called the additional length graph and they used it to show that central quotients of Artin–Tits groups of spherical type are acylindrically hyperbolic. In general, a Euclidean Artin–Tits group is not a priori a Garside group but McCammond and Sulway have shown that it embeds into an infinite-type Garside group which they call a crystallographic Garside group . We associate a hyperbolic additional length graph to this crystallographic Garside group and we exhibit elements of the Euclidean Artin–Tits group which act loxodromically and weakly properly discontinuously on this hyperbolic graph.

Topological recursion for Masur–Veech volumes

We study the Masur–Veech volumes M V g , n $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus g $g$ with n $n$ punctures. We show that the volumes M V g , n $MV_{g,n}$ are the constant terms of a family of polynomials in n $n$ variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of [Delecroix, Goujard, Zograf, Zorich, Duke J. Math 170 (2021), no. 12, math.GT/1908.08611] proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in [Andersen, Borot, Orantin, Geometric recursion, math.GT/1711.04729, 2017]. We also obtain an expression of the area Siegel–Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur–Veech volumes, and thus of area Siegel–Veech constants, for low g $g$ and n $n$ , which leads us to propose conjectural formulae for low g $g$ but all n $n$ . We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

Comparison theorems for Lorentzian length spaces with lower timelike curvature bounds

In this article we introduce a notion of normalized angle for Lorentzian pre-length spaces. This concept allows us to prove some equivalences to the definition of timelike curvature bounds from below for Lorentzian pre-length spaces. Specifically, we establish some comparison theorems known as the local Lorentzian version of the Toponogov theorem and the Alexandrov convexity property. Finally, as an application we obtain a first variation formula for non-negatively curved globally hyperbolic Lorentzian length spaces.

Counting hyperbolic multigeodesics with respect to the lengths of individual components and asymptotics of Weil–Petersson volumes

Given a connected, oriented, complete, finite area hyperbolic surface $X$ of genus $g$ with $n$ punctures, Mirzakhani showed that the number of multi-geodesics on $X$ of total hyperbolic length $\leq L$ in the mapping class group orbit of a given simple or filling closed multi-curve is asymptotic as $L \to \infty$ to a polynomial in $L$ of degree $6g-6+2n$. We establish asymptotics of the same kind for countings of multi-geodesics in mapping class group orbits of simple or filling closed multi-curves that keep track of the hyperbolic lengths of individual components, proving and generalizing a conjecture of Wolpert. In the simple case we consider more precise countings that also keep track of the class of the multi-geodesics in the space of projective measured geodesic laminations. We provide a unified geometric and topological description of the leading terms of the asymptotics of all the countings considered. Our proofs combine techniques and results from several papers of Mirzakhani as well as ideas introduced by Margulis in his thesis.

Maximum Power Point Tracking Control of PV-TE Hybrid Power Generation System in Greenhouse

The application of photovoltaic-thermoelectric (PV-TE) combined power generation system in greenhouse is an effective way to solve the problems of high energy consumption and high pollution. In order to improve the efficiency of the PV-TE system, maximum power point tracking (MPPT) control is required. Aiming at the shortcomings of the traditional incremental conductance (INC) method, which is fixed in step size, a hyperbolic tangent-type adaptive variable step length INC method is proposed. The method takes advantage of the monotonically increasing and the fast changing speed of the hyperbolic tangent function, so that the step length can be adjusted rapidly and adaptively according to the change of external environmental conditions such as light intensity. The simulation results show that the proposed method can rapidly track the maximum power point when the illumination intensity changes drastically, and meanwhile it has smaller steady-state error and can realize MPPT control well.

The type problem for Riemann surfaces via Fenchel–Nielsen parameters

A Riemann surface X $X$ is said to be of parabolic type if it does not support a Green's function. Equivalently, the geodesic flow on the unit tangent bundle of X $X$ (equipped with the hyperbolic metric) is ergodic. Given a Riemann surface X $X$ of arbitrary topological type and a hyperbolic pants decomposition of X $X$ , we obtain sufficient conditions for parabolicity of X $X$ in terms of the Fenchel–Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of nonstandard half-collar about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two nonstandard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as for half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (also known as the infinite Loch–Ness monster), the ladder surface, and abelian covers of compact surfaces are also studied.

An estimate for the hyperbolic lengths of closed geodesics on Riemann surfaces II

Let be a Riemann surface obtained by removing distinct points from a hyperbolic Riemann surface R. For any closed geodesic on R, there is the shortest closed geodesic on which is freely homotopic to on R. In our previous paper [Yamamoto. An estimate for the hyperbolic length of closed geodesics on Riemann surfaces. Complex Var Elliptic Equ. 2019;64(9):1582–1607], we gave an upper bound for the hyperbolic length of on depending only on and the hyperbolic length of on R in the case when R is of finite type. The purpose of this paper is to improve our previous result.

Blending of hyperbolic closed curves

AbstractIn recent years, game developers are interested in developing games in the hyperbolic space. Shape blending is one of the fundamental techniques to produce animation and videos games. This paper presents two algorithms for blending between two closed curves in the hyperbolic plane in a manner that guarantees that the intermediate curves are closed. We deal with hyperbolic discrete curves on Poincaré disc which is a famous model of the hyperbolic plane. We use the linear interpolation approach of the geometric invariants of hyperbolic polygons namely hyperbolic side lengths, exterior angles and geodesic discrete curvature. We formulate the closing condition of a hyperbolic polygon in terms of its geodesic side lengths and exterior angles. This is to be able to generate closed intermediate curves. Finally, some experimental results are given to illustrate that the proposed methods generate aesthetic blending of closed hyperbolic curves.

The pluripotential compactification of Teichmüller space

In this paper, we show some new plurisubharmonic functions on Teichmüller space, related to extremal length and hyperbolic length. And we give a new compactification of Teichmüller space: the pluripotential compactification.

Algorithm for filling curves on surfaces

Let varSigma be a compact, orientable surface of negative Euler characteristic, and let h be a complete hyperbolic metric on varSigma . A geodesic curve gamma in varSigma is filling if it cuts the surface into topological disks and annuli. We propose an efficient algorithm for deciding whether a geodesic curve, represented as a word in some generators of pi _1(varSigma ), is filling. In the process, we find an explicit bound for the combinatorial length of a curve given by its Dehn–Thurston coordinate, in terms of the hyperbolic length. This gives us an efficient method for producing a collection which is guaranteed to contain all words corresponding to simple geodesics of bounded hyperbolic length.

SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES

Given integers$g,n\geqslant 0$satisfying$2-2g-n<0$, let${\mathcal{M}}_{g,n}$be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus$g$with$n$cusps. We study the global behavior of the Mirzakhani function$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$which assigns to$X\in {\mathcal{M}}_{g,n}$the Thurston measure of the set of measured geodesic laminations on$X$of hyperbolic length${\leqslant}1$. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of${\mathcal{M}}_{g,n}$and deduce that$B$is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of$B$to statistics of counting problems for simple closed hyperbolic geodesics.

An estimate for the hyperbolic length of closed geodesics on Riemann surfaces

ABSTRACTLet R be a Riemann surface of finite type , and a Riemann surface obtained by removing distinct points from R. We assume that the Riemann surfaces are equipped with the hyperbolic metric. For a closed geodesic on R, let be the shortest closed geodesic on which is freely homotopic to on R. In this paper, we give an explicit upper bound for the hyperbolic length of on depending only on and the hyperbolic length of on R.

Length and Area Estimates for (Hyperbolically) Convex Conformal Mappings

Let f be a conformal map on \({{\mathrm{\mathbb {D}}}}\). Keogh (J Lond Math Soc 29:121–123, 1954) obtained an upper bound for the length of the curve \(f(|z|=r)\), \(r \in (0,1)\), when f is a convex map. With the use of this upper bound, we prove a monotonicity result regarding the length of \(f(|z|=r)\). A similar kind of monotonicity result is proved for the area of \(f(|z|<r)\). Considering the case where \(f({{\mathrm{\mathbb {D}}}}) \subset {{\mathrm{\mathbb {D}}}}\) and f is a hyperbolically convex map, we present hyperbolic analogues of the above bounds. We prove two monotonicity theorems regarding the length and area in the hyperbolic geometry of the unit disk. In particular, the ratio of the hyperbolic length of the curve \(f(|z|=r)\) to the hyperbolic length of the curve \(|z|=r\) is a decreasing function of r. Furthermore, we prove that the ratio of the hyperbolic area of \(f(|z|<r)\) to the hyperbolic area of the disk \(|z|<r\) is also a decreasing function of r.

The effect of Fenchel-Nielsen coordinates under elementary moves

We describe the effect of Fenchel-Nielsen coordinates under elementary move for hyperbolic surfaces with geodesic boundaries, punctures and cone points, which generalize Okai's result for surfaces with geodesic boundaries. The proof relies on the parametrization of the Teichmüller space of surface of type (1,1) or (0,4) as a sub-locus of an algebraic equation in $\mathbf{R}^3$. As an application, we show that the hyperbolic length functions of closed curves are asymptotically piecewise linear functions with respect to the Fenchel Nielsen coordinates in the Teichmüller spaces of surfaces with cone points.

Central limit theorems for mapping class groups and Out(FN)

We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on $\text{Out}(F_N)$, each time under a finite second moment condition on the measure (either with respect to the Teichm\"uller metric, or with respect to the Lipschitz metric on outer space). In the mapping class group case, this describes the spread of the hyperbolic length of a simple closed curve on the surface after applying a random product of mapping classes. In the case of $\text{Out}(F_N)$, this describes the spread of the length of primitive conjugacy classes in $F_N$ under random products of outer automorphisms. Both results are based on a general criterion for establishing a central limit theorem for the Busemann cocycle on the horoboundary of a metric space, applied to either the Teichm\"uller space of the surface, or to Culler--Vogtmann's outer space.