Length Spectrum Research Articles

The energy spectrum of metrics on surfaces

Let (N,rho ) be a Riemannian manifold, S a surface of genus at least two and let f:S rightarrow N be a continuous map. We consider the energy spectrum of (N,rho ) (and f) which assigns to each point [J]in mathcal {T}(S) in the Teichmüller space of S the infimum of the Dirichlet energies of all maps (S,J)rightarrow (N,rho ) homotopic to f. We study the relation between the energy spectrum and the simple length spectrum. Our main result is that if N=S, f={{,mathrm{id},}} and rho is a metric of non-positive curvature, then the energy spectrum determines the simple length spectrum. Furthermore, we prove that the converse does not hold by exhibiting two metrics on S with equal simple length spectrum but different energy spectrum. As corollaries to our results we obtain that the set of hyperbolic metrics and the set of singular flat metrics induced by quadratic differentials satisfy energy spectrum rigidity, i.e. a metric in these sets is determined, up to isotopy, by its energy spectrum. We prove that analogous statements also hold true for Kleinian surface groups.

Footprints of Geodesics in Persistent Homology

Given a metric space X and a subspace $$A\subset X$$ , we prove that A can generate various algebraic elements in persistent homology of X. We call such elements (algebraic) footprints of A. Our results imply that footprints typically appear in dimensions above $$\dim (A)$$ . Higher dimensional persistent homology thus encodes lower dimensional geometric features of X. We pay special attention to a specific type of geodesics in a geodesic surface X called geodesic circles. We explain how they may generate non-trivial odd-dimensional and two-dimensional footprints. In particular, we can detect even some contractible geodesics using two- and three-dimensional persistent homology. This provides a link between persistent homology and the length spectrum in Riemannian geometry.

Stable random fields, Patterson–Sullivan measures and extremal cocycle growth

We study extreme values of group-indexed stable random fields for discrete groups G acting geometrically on spaces X in the following cases: (1) G acts properly discontinuously by isometries on a CAT(-1) space X, (2) G is a lattice in a higher rank Lie group, acting on a symmetric space X, and (3) G is the mapping class group of a surface acting on its Teichmüller space. The connection between extreme values and the geometric action is mediated by the action of the group G on its limit set equipped with the Patterson–Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space X and show that its non-vanishing is equivalent to finiteness of the Bowen–Margulis measure for the associated unit tangent bundle U(X/G) provided X/G has non-arithmetic length spectrum. As a consequence, we establish a dichotomy for the growth-rate of a partial maxima sequence of stationary symmetric \(\alpha \)-stable (\(0< \alpha < 2\)) random fields indexed by groups acting on such spaces. We also establish analogous results for normal subgroups of free groups.

Quantum signatures of chaos in relativistic quantum billiards with shapes of circle- and ellipse-sectors* *Dedicated to the memory of Fritz Haake. Fritz introduced me (Barbara) into the field of quantum chaos. He supervised my diploma and doctoral thesis and since then has been a mentor, friend and helpful advisor during all the stages of my scientific career. I take his

According to the Berry–Tabor conjecture, the spectral properties of typical nonrelativistic quantum systems with an integrable classical counterpart agree with those of Poissonian random numbers. We investigate to what extend it applies to relativistic neutrino billiards (NBs) consisting of a spin-1/2 particle confined to a bounded planar domain by imposing suitable boundary conditions (BCs). In distinction to nonrelativistic quantum billiards (QBs), NBs do not have a well-defined classical counterpart. However, the peaks in the length spectra, that is, the modulus of the Fourier transform of the spectral density from wave number to length, of NBs are just like for QBs at the lengths of periodic orbits of the classical billiard (CB). This implies that there must be a connection between NBs and the dynamic of the CB. We demonstrate that NBs with shapes of circle- and ellipse-sectors with an integrable classical dynamic, obtained by cutting the circle and ellipse NB along symmetry lines, have no common eigenstates with the latter and that, indeed, their spectral properties can be similar to those of classically chaotic QBs. These features orginate from the intermingling of symmetries of the spinor components and the discontinuity in the BCs leading to contradictory conditional equations at corners connecting curved and straight boundary parts. To corroborate the necessity of the curved boundary part in order to generate GOE-like behavior, we furthermore consider the right-angled triangle NB constructed by halving the equilateral-triangle NB along a symmetry axis. For an understanding of these findings in terms of purely classical quantities we use the semiclassical approach recently developed for massive NBs, and Poincaré–Husimi distributions of the eigenstates in classical phase space. The results indicate, that in the ultrarelativistic limit these NBs do not show the behavior expected for classically chaotic QBs.

Gaussian free fields and Riemannian rigidity

In the present paper, we show that on a compact Riemannian manifold $(M,g)$ of dimension $d\leqslant 4$ whose metric has negative curvature, the renormalized partition function $Z_g(\lambda)$ of a massive Gaussian Free Field determines the length spectrum of $(M,g)$ and imposes some strong geometric constraints on the Riemannian structure of $(M,g)$. In any finite dimensional family of Riemannian metrics of negative sectional curvature bounded from below and above and whose isometry group is trivial, there is only a \textbf{finite number of isometry classes} of metrics with given partition function $Z_g(\lambda)$. When $d<4$, the same result holds true if the random variable $\int_M:\phi^2:dv$ has given probability distribution and without the lower bound on the sectional curvatures.

Domination results in n$n$‐Fuchsian fibers in the moduli space of Higgs bundles

In this article, we show some domination results on the Hitchin fibration, mainly focusing on the $n$-Fuchsian fibers. More precisely, we show the energy density of associated harmonic map of an $n$-Fuchsian representation dominates the ones of all other representations in the same Hitchin fiber, which implies the domination of topological invariants: translation length spectrum and entropy. As applications of the energy density domination results, we obtain the existence and uniqueness of equivariant minimal (or maximal) surfaces in certain product Riemannian (or pseudo-Riemannian) manifold. Our proof is based on establishing an algebraic inequality generalizing a GIT theorem of Ness on the nilpotent orbits to general orbits.

Studies on spectro photophysical properties of PBBO-laser dye

Examine the effects of solvents of different polarities on the electronic absorption and fluorescence spectra of 2-(4-Biphenylyl)-6-phenylbenzoxazol (PBBO). The singlet state excited dipole moments (µe) and ground-state dipole moments (µg) were evaluated from the equations of Lippert-Mataga and Billot. The detected singlet state excited dipole moments were found to be larger than the ground‐state ones. Moreover, the effect of silver nanoparticles on the emission spectrum of PBBO has been observed. The energy transfer between PBBO and some laser dyes will be studied in addition to some parameters, including the determination of the critical transfer distance (R0) and the evaluation of the energy transfer rate constant (kET). The theoretical absorption spectra of the PBBO compound are calculated using Density function theory (DFT) calculations. In addition, the energy gap (Eg), bond length, and IR spectrum are carried out too.

Teichmüller spaces of non-discrete groups

The concept of Teichmüller space of a Fuchsian group can be extended to any non-discrete group of conformal homeomorphisms of the hyperbolic plane. In this paper, we first present three models of Teichmüller space fulfilling that goal. Then we use them to study the Teichmüller spaces T(G) of the non-discrete subgroups G of PSL(2,R). We show that T(G) is not trivial if and only if G is a subgroup consisting of hyperbolic elements with two common fixed points and accumulating to at least one non-identity element. Furthermore, we show that if T(G) is not trivial, then (1) T(G) is conformally equivalent to the open unit disk, (2) the Teichmüller metric on T(G) is equal to the hyperbolic metric on the disk, and (3) the length spectrum is just a pseudometric on T(G) and when restricted to a one-dimensional real slice, it is a metric coinciding with the Teichmüller metric.

Spectral rigidity for spherically symmetric manifolds with boundary

We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: Under a “clean intersection hypothesis” and assuming an injectivity hypothesis associated to the length spectrum, the wave trace is singular at the lengths of periodic broken rays. In particular, the Neumann spectrum of the Laplace–Beltrami operator uniquely determines the length spectrum. The trace formula also applies for the toroidal modes of the free oscillations in the earth. Under this hypothesis and the Herglotz condition, we then prove that the length spectrum is rigid: Deformations preserving the length spectrum and spherical symmetry are necessarily trivial in any dimension, provided the Herglotz condition and a geometrical condition are satisfied. Combining the two results shows that the Neumann spectrum of the Laplace–Beltrami operator is rigid in this class of manifolds with boundary.

Limits of rational maps, [formula omitted]-trees and barycentric extension

In this paper, we show that one can naturally associate a limiting dynamical system F:T⟶T on an R-tree to any degenerating sequence of rational maps fn:Cˆ⟶Cˆ of fixed degree. The construction of F is in 2 steps: first, we use barycentric extension to get Efn:H3⟶H3; second, we take appropriate limit on rescalings of hyperbolic space. An important ingredient we prove here is that the Lipschitz constant depends only on the degree of the rational map. We show that the dynamics of F records the limiting length spectra of the sequence fn.

Detection of Blood Glucose With Hemoglobin Content Using Compact Photonic Crystal Fiber.

We have proposed Twin Elliptical Core Photonic Crystal Fiber (TEC-PCF) sensor for the detection of blood glucose level under the influence of hemoglobin components. The main featuring of the proposed biosensor is to detect the wide range of blood glucose content with enhanced sensitivity, by utilizing small length of the fiber. In order to achieve this, we have constructed asymmetric TEC-PCF where the elliptical core is filled by blood sample. The numerical sensing characteristics are evaluated using Finite Element Method (FEM). By varying hemoglobin concentrations as 120 g/L, 140 g/L and 160 g/L, we realize enhanced blood glucose sensing with detection range from 0 g/L to 100 g/L. The sensing performance of the proposed biosensor is studied through the coupling length and transmission power spectrum by calculation of effective index of the coupling mode. The obtained maximum wavelength sensitivity under the influence of 160 g/L hemoglobin content is 2.4 nm/(g/L) and 2.42 nm/(g/L) with fiber length of 0.245 mm and 0.215 mm for X and Y polarization, respectively. Further, limit of detection (LOD) is calculated under the influence of 160 g/L hemoglobin content is 0.375 mg/L and 0.372 mg/L for X and Y polarization, respectively. The proposed miniaturized sensing device can be integrated with microfluidic systems for the development of next-generation biosensor applications as point of- care and lab-on-a-chip.

BOUNDARY LAYER WIND PROFILE MEASUREMENT BASED ON A SIX-ROTOR UAV ANEMOMETER

They studied the feasibility of boundary layer wind field measurement based on a six-rotor drone equipped with an anemometer, and analyzed the measured wind characteristics including average wind speed, wind direction, turbulence intensity, gust factor, the turbulence integral length and power spectrum. The results are compared with the measured results of anemometers fixed on a wind tower. The results show that the wind speed and wind direction values measured by the UAV are basically the same as the measured values on the wind tower after correction, and the error between the two is within ±3%. The accuracy of fitted wind profile by the UAV is high. The wind profile is close to the B-type geomorphic wind profile specified by the Chinese standard. The average ground roughness calculated by data fitting is 0.148. The various fluctuating wind characteristic parameters measured by the UAV are close to the measured values on the wind tower, and the error between the two is within ±5%. There is an exponential correlation between the turbulence intensity and the gust factor. The wind power spectrum measured by the UAV is basically consistent with that measured on the wind tower and is consistent with the Karman spectrum.

Geodesic stretch, pressure metric and marked length spectrum rigidity

AbstractWe refine the recent local rigidity result for the marked length spectrum obtained by the first and third author in [GL19] and give an alternative proof using the geodesic stretch between two Anosov flows and some uniform estimate on the variance appearing in the central limit theorem for Anosov geodesic flows. In turn, we also introduce a new pressure metric on the space of isometry classes, which reduces to the Weil–Petersson metric in the case of Teichmüller space and is related to the works [BCLS15, MM08].

Periodic magnetic geodesics on Heisenberg manifolds

We study the dynamics of magnetic flows on Heisenberg groups, investigating the extent to which properties of the underlying Riemannian geometry are reflected in the magnetic flow. Much of the analysis, including a calculation of the Mañé critical value, is carried out for $$(2n+1)$$ -dimensional Heisenberg groups endowed with any left invariant metric and any exact, left-invariant magnetic field. In the three-dimensional Heisenberg case, we obtain a complete analysis of left-invariant, exact magnetic flows. This is interesting in and of itself, because of the difficulty of determining geodesic information on manifolds in general. We use this analysis to establish two primary results. We first show that the vectors tangent to periodic magnetic geodesics are dense for sufficiently large energy levels and that the lower bound for these energy levels coincides with the Mañé critical value. We then show that the marked magnetic length spectrum of left-invariant magnetic systems on compact quotients of the Heisenberg group determines the Riemannian metric. Both results confirm that this class of magnetic flows carries significant information about the underlying geometry. Finally, we provide an example to show that extending this analysis of magnetic flows to the Heisenberg-type setting is considerably more difficult.

Proposed Standard Weight Equation and Standard Length Categories for Lake Chubsucker

AbstractStandard weight equations have been proposed for a variety of fishes, including many nongame species. Here, we developed standard weight (Ws) equations for Lake Chubsucker Erimyzon succetta using the traditional regression‐line‐percentile technique and the newer empirical‐percentile method. Length and weight data for 19,727 Lake Chubsuckers from 8 to 42 cm TL and 129 populations were used in the final analysis. These fish were collected from 1977 to 2020 by various agency and university personnel from most known populations across the species’ range in North America. This species is considered vulnerable, imperiled, or extirpated in several Midwestern and northern states, but it appears to be secure in the Southeast. The resulting Ws equations were log10W = −5.0962 + 3.1125 × log10L and log10W = –5.9735 + 3.8828 × log10L −0.1701 × (log10L)2 for the RLP and quadratic empirical‐percentile method techniques, respectively, where W is weight (g) and L is total length (TL, mm). The regression‐line‐percentile equation␣exhibited a greater length‐related bias especially at the ends of the length spectrum, so we recommend the quadratic empirical‐percentile equation␣for this species. We also developed standard length categories for assessing fish population size structure. We propose minimum lengths for five standard length categories of 11, 18, 24, 28, and 35 cm TL for stock, quality, preferred, memorable, and trophy sizes, respectively. Development of a Ws equation␣and standard length categories will aid biologists in assessing condition and size structure of wild Lake Chubsucker populations and those being propagated for supplemental stocking into trophy bass ponds for native forage or in areas with depressed native populations. Additionally, by evaluating this nongame species that occupies the same environments as higher profile game fish species, a more holistic approach of evaluating the overall health of systems can be examined.

Coherence of wall pressure fluctuations in zero and adverse pressure gradients

The wall pressure fluctuations induced by turbulent boundary layers on a flat plate model were measured with an L-shaped array of Kulite miniature pressure sensors. By installing a NACA airfoil above the plate with an adjustable angle of attack, different adverse pressure gradients were produced on the plate. The streamwise and spanwise coherence and coherence lengths of the wall pressure fluctuations are calculated for zero and adverse pressure gradient flows. The effect of the pressure gradient on the coherence and coherence lengths in both streamwise and spanwise directions is quantified. Based on the results of the present test cases and selected published datasets in zero pressure gradients, covering the range of Reynolds numbers based on the momentum thickness between 3300 and 42000, the Reynolds number dependence of the streamwise wall pressure coherence is discussed and mathematically expressed. A coherence length model for zero pressure gradient boundary layers, taking into account the Reynolds number effect, is proposed based on scaled spectra of the coherence lengths. In comparison with the other published models, the present model achieves a considerable improvement of the prediction accuracy for boundary layer flows, covering a large range of Reynolds numbers. Furthermore, an evaluation of the Corcos model and the Smol’yakov and Tkachenko model for prediction of the off-axis coherence of the fluctuating wall pressure field is made.

Minimal length, maximal momentum and stochastic gravitational waves spectrum generated from cosmological QCD phase transition

We investigate thoroughly the temporal evolution of the universe temperature as a function of the Hubble parameter associated with the Stochastic Gravitational Wave (SGW), that formed at the cosmological QCD phase transition epoch to the current epoch, within the Generalized Uncertainty Principle (GUP) framework. Here we use GUP version which provide constraints on the minimum measurable length and the maximum observable momentum, that characterized by a free parameter α. We study the effect of this parameter on the SGW background. We show that the effect can slightly enhance the SGW frequency in the lower frequency regime which might be important in the detection of SGW in the future GW detection facilities.

Long-term evolution of the three-dimensional structure of string-fluid complex plasmas in the PK-4 experiment.

Microparticle suspensions in a polarity-switched discharge plasma of the Plasmakristall-4 facility on board the International Space Station exhibit string-like order. As pointed out in [Phys. Rev. Research 2, 033314 (2020)2643-156410.1103/PhysRevResearch.2.033314], the string-order is subject to evolution on the timescale of minutes at constant gas pressure and constant parameters of polarity switching. We perform a detailed analysis of this evolution using the pair correlations and length spectrum of the string-like clusters (SLCs). Average exponential decay rate of the SLC length spectrum is used as a measure of string order. The analysis shows that the improvement of the string-like order is accompanied by the decrease of the thickness of the microparticle suspension, microparticle number density, and total amount of microparticles in the field of view. This suggests that the observed long-term evolution of the string-like order is caused by the redistribution of the microparticles, which significantly modifies the plasma conditions.

Isospectral finiteness on convex cocompact hyperbolic 3-manifolds

In this paper, we show that given a set of lengths of closed geodesics, there are only finitely many convex cocompact hyperbolic 3-manifolds with that specified length spectrum with multiplicity, homotopy equivalent to a given 3-manifold without a handlebody factor, up to orientation preserving isometries.

Invariant weakly convex cocompact subspaces for surface groups in $A_2$-buildings

This paper deals with non-Archimedean representations of punctured surface groups in $\operatorname {PGL}_3$, associated actions on (not necessarily discrete) Euclidean buildings of type $A_2$, and degenerations of convex real projective structures on surfaces. The main result is that, under good conditions on Fock-Goncharov generalized shear parameters, non-Archimedean representations acting on the Euclidean building preserve a cocompact weakly convex subspace, which is part flat surface and part tree. In particular the eigenvalue and length(s) spectra are given by an explicit finite $A_2$-complex. We use this result to describe degenerations of convex real projective structures on surfaces for an open cone of parameters. The main tool is a geometric interpretation of Fock-Goncharov parameters in $A_2$-buildings.