Proves two non-trapping obstacles coincide if scattering rays have similar travelling times or scattering length spectra.
problem Identifying non-trapping obstacles based on scattering properties.
method Proves two obstacles coincide if their scattering rays have similar travelling times or scattering length spectra under weak non-degeneracy conditions.
result Two non-trapping obstacles coincide if their scattering rays have similar travelling times or scattering length spectra.
New proof for 2D lens rigidity from billiard trajectories.
problem Determining obstacles in 2D space from billiard trajectories.
method Separate proof for 2D case, using billiard trajectories.
result Proves rigidity for 2D strictly convex bodies.
Wavelet scattering spectra model non-Gaussian time-series, proving scale invariance for self-similar processes.
problem Modeling non-Gaussian time-series with stationary increments.
method Complex wavelet transform for scale variations, joint correlation matrix for scale dependencies, second wavelet transform for diagonalization, maximum entropy models conditioned by scattering spectra coefficients.
result Scattering spectra of self-similar processes are scale invariant, allowing statistical testing and generation of new time-series.
Deep neural networks correct Mie scattering in FTIR spectra of biological samples.
problem Mie scattering obscures biochemically relevant spectral information in FTIR spectra of biological samples.
method Deep neural networks to approximate the preprocessing function that removes Mie scattering.
result The model is faster and more generalizable across different tissue types.
Study applies inverse scattering to BKM systems, linking spectra and integrable systems.
problem Applying inverse scattering to BKM systems.
method Developed methods for BKM systems, relating Schrödinger-Hill operators, Neumann system, and KdV equations.
result Initial observations indicate potential for applying inverse scattering to BKM systems.
Study sub-Riemannian metrics on compact Lie groups, finding same shortest loops.
problem Exploring sub-Riemannian length spectra on compact Lie groups.
method Restricting Killing form to root spaces of compact Lie groups.
result Same shortest loops in Riemannian and sub-Riemannian cases.
In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compar…
The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.
problem Characterizing hyperbolic surfaces by their simple length spectra.
method Proving rigidity of length identities over Teichmüller spaces.
result Simple length spectra can be used as moduli for generic hyperbolic surfaces.
We study the length, weak length and complex length spectrum of closed geodesics of a compact flat Riemannian manifold, comparing length-isospectrality with isospectrality of the Laplacian acting on p-forms. Using integral roots of the Krawtchouk polynomials, we give many pairs of p-isospectral flat manifolds having di…
The paper shows that arithmetic hyperbolic 3-orbifolds have many non-commensurable pairs with similar geodesic spectra.
problem Understanding the relationship between geodesic length spectra and commensurability of arithmetic hyperbolic 3-orbifolds.
method Using a bounded gaps result for prime ideals in number fields, the paper constructs infinitely many non-commensurable pairs of arithmetic hyperbolic 3-orbifolds with similar geodesic spectra.
result Arithmetic hyperbolic 3-orbifolds can have many non-commensurable pairs with similar geodesic spectra.
Study approximate marked length spectrum rigidity in non-positively curved groups.
problem Approximate rigidity of marked length spectra in non-positively curved groups.
method Compare marked length spectra of isometric actions of groups with non-positively curved features.
result Supremum of quotient of marked length spectra is approximately determined by restricted spectra.
The study quantifies how many questions are needed to determine a surface's length spectrum.
problem Quantifying the number of questions needed to determine a surface's length spectrum.
method Inverse spectral problems for hyperbolic surfaces, focusing on length spectra and their relation to surface geometry.
result A quantitative upper bound on the number of isospectral but non-isometric surfaces of a given genus.
We prove an analogue of Farb-Masur's theorem that the length-spectra metric on moduli space is "almost isometric" to a simple model V(S) which is induced by the cone metric over the complex of curves. As an application, we know that the Teichmüller metric and the length-spectra metric are "almost isometric…
Algorithms compute length spectra of torus graphs efficiently.
problem Computing length spectra of graphs embedded on a torus.
method Preprocessing and algorithms based on polyhedral norms.
result Efficient computation of length spectra and spectrum comparison.
Paper proves non-arithmetic Teichmüller length spectra for subgroup of mapping class groups.
problem Proving non-arithmetic Teichmüller length spectra for subgroups of mapping class groups.
method Introducing cross-ratios on Teichmüller and projectable mapping classes, studying their geometric and dynamical properties.
result Every non-elementary subgroup of the mapping class group has non-arithmetic Teichmüller length spectrum.
Spaces with similar long paths have similar shapes.
problem Comparing shapes of Gromov hyperbolic spaces.
method Examining asymptotic marked length spectra.
result Spaces with identical spectra are roughly isometric.
The study examines how non-commensurable surfaces can share length spectra.
problem Understanding how non-commensurable arithmetic hyperbolic surfaces can share length spectra.
method Investigates quantitative results on the maximum cardinality of non-commensurable surfaces sharing a fixed set of length spectra.
result Proves a number of quantitative results about the maximum cardinality of a family of pairwise non-commensurable arithmetic hyperbolic surfaces whose length spectra contain a fixed set of nonnegative real numbers.
Study of group actions on CAT(0) cube complexes, focusing on marked length spectra.
problem Comparing marked length spectra of group actions on CAT(0) cube complexes.
method Use of finite-state automata and thermodynamic formalism for suspension flows over subshifts of finite type.
result Prove that the Manhattan curve is analytic and convex, and a straight line if and only if marked length spectra are homothetic.
A new method predicts future paths using a Monte-Carlo approach.
problem Predicting future financial paths given historical data.
method Path Shadowing Monte-Carlo method using maximum entropy model.
result Yields state-of-the-art predictions for future volatility and option smiles.
The paper connects Riemann surface length spectra to Brownian loop measures.
problem Understanding the length spectra of Riemann surfaces with additional cusps.
method Using the Brownian loop measure to relate length spectra of Riemann surfaces with and without additional cusps.
result Expressed the total mass of Brownian loops in terms of the length of geodesic representatives.
New spectra defined for metric spaces, extending existing covering spectrum.
problem Characterizing and comparing spectra for metric spaces.
method Defining and measuring 'entourage covers' to derive new spectra.
result New spectra (ECS, ES) extend existing covering spectrum (CS) and have useful properties.
The scattering data of a Riemannian manifold with boundary record the incoming and outgoing directions of each geodesic passing through. We show that the scattering data of a generic Riemannian surface with no trapped geodesics and no conjugate points determine the lengths of geodesics. Counterexamples exists when trap…
Authors derive McShane identity for super tori.
problem Deriving McShane identity for super tori.
method Super Teichmüller theory and supergeometry.
result Asymptotic growth rate of length spectra established.
New method interpolates high-dimensional scattered data using kernel theory.
problem Scattered data in high-dimensional spaces defy traditional distributional assumptions.
method Kernel interpolation framework based on integral operator theory.
result Spectra of kernel matrices predict performance of interpolation methods.
New proof shows surfaces can have identical length spectra but not simple ones.
problem Identifying when two covers of a surface have identical length spectra but not simple ones.
method Characterized isomorphism of covers and constructed surfaces with identical spectra but different simple length spectra.
result Found surfaces with identical length spectra but not simple length isospectral covers.
New method uses broken scattering to uniquely identify Finsler manifolds.
problem Identifying Finsler manifolds from scattering data.
method Uses broken scattering relation to compare geodesics.
result Two reversible Finsler manifolds with the same broken scattering relation are isometric.
The paper describes correlations of spectra for higher rank Anosov representations.
problem Understanding correlations of spectra for Anosov representations of higher rank groups.
method Relates correlation problem to counting projections in truncated hypertubes.
result Extends previous work on rank one representations to higher rank.
New curves generalize flat metrics from quadratic to q-differentials.
problem Determining flat metrics from curve lengths.
method Introduced q-simple curves to generalize results from quadratic to q-differentials.
result Lengths of q-simple curves uniquely determine non-positively curved Euclidean cone metrics induced by q-differentials.
The covering spectrum is a geometric invariant of a Riemannian manifold, more generally of a metric space, that measures the size of its one-dimensional holes by isolating a portion of the length spectrum. In a previous paper we demonstrated that the covering spectrum is not a spectral invariant of a manifold in dimens…
We prove stability and exponential convergence of the Perfectly Matched Layer (PML) method for acoustic scattering on manifolds with axial analytic quasicylindrical ends. These manifolds model long-range geometric perturbations (e.g. bending or stretching) of tubular waveguides filled with homogeneous or inhomogeneous …
New series analyze manifold homology, proving a generalized Gromov inequality.
problem Analyzing homology spectra of manifolds and polyhedra.
method Defining Dirichlet series and investigating their properties.
result Established an inequality involving the entire homology spectrum.
We prove that the length spectrum metric and the arc-length spectrum metric are almost-isometric on the ε0-relative part of Teichmuller spaces of surfaces with boundary.
The paper studies dynamics of rational maps using barycentric extensions and Berkovich spaces.
problem Classifying rational maps with bounded length spectra.
method Geometric and algebraic constructions on R-trees.
result Two constructions for limiting dynamics on R-trees are equivalent.
We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.
The study finds sparse sets that uniquely determine metrics on negatively curved manifolds.
problem Determining metrics on negatively curved manifolds using spectral data.
method Analyzing conjugacy classes and marked length spectra.
result Sparse sets exist that uniquely determine metrics on negatively curved manifolds.
We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We relate the covering spectrum to the (marked) shift spectrum of such a space. We de…
Closed manifolds with close marked spectra are approximately isometric.
problem Closed manifolds with close marked length spectra are approximately isometric.
method Using Hamenstädt's methods and Gromov compactness theorem, we show diffeomorphism and volume equality.
result Closed manifolds with close marked spectra are approximately isometric.
Scattering rigidity of a Riemannian manifold allows one to tell the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to tell the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results…
Proposes a method for training Bayesian neural networks using synthetic data from Raman and CARS spectra.
problem Limited real observations in Raman and CARS spectroscopy.
method Log-Gaussian Gamma Processes and Bayesian Neural Networks.
result Trained Bayesian neural networks provide accurate estimates of Raman and CARS spectra with uncertainty quantification.
Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution ena…
New method disentangles sources of different timescales in planetary seismic data.
problem Unsupervised source separation of multi-scale seismic data from planetary missions.
method Wavelet scattering spectra for multi-scale clustering and variational autoencoder for source separation.
result Disentangles sources with different timescales in InSight mission seismic data.
New proof shows Fuchsian groups have irrational length spectra.
problem Irrationality of the length spectrum in Fuchsian groups.
method Elementary proof of linear independence of group elements' lengths.
result Non-elementary Fuchsian groups contain elements with linearly independent lengths over Q.
Deep learning solves wave-based inverse problems, including super-resolution imaging.
problem Solving inverse wave scattering problems across all length scales.
method Wide-band butterfly network coupled with dynamic noise injection.
result Framework successfully solves super-resolution imaging problems.
Study shows surfaces with similar length spectra are smoothly deformable.
problem Quantifying how similar the length spectra of two negatively curved surfaces are.
method Analyzes marked length spectra of closed negatively curved surfaces and proves smooth deformations.
result Smooth diffeomorphisms exist between surfaces with close length spectra.
The paper explores how magnetic systems' spectra can identify metrics and 1-forms.
problem Can the marked magnetic action spectrum of magnetic systems with Anosov flow determine the metric and 1-form?
method The paper addresses this question in two settings: locally for systems with close metrics and 1-forms, and for metrics in the same conformal class.
result The paper answers the question affirmatively in both settings.
For a compact manifold, which has a part isometric to a cylinder of finite length, we consider an adiabatic limit procedure, in which the length of the cylinder tends to infinity. We study the asymptotic of the spectrum of Hodge-Laplacian and the asymptotic of the L2-metric on de Rham cohomology. As an application, …
The action of the mapping class group of the thrice-punctured projective plane on its GL(2,C) character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corres…
The study examines arithmetic orbifolds and their length spectra, proving uniform discreteness and linear dependence of geodesic lengths.
problem Uniform discreteness and linear dependence of geodesic lengths in arithmetic orbifolds.
method Analyzes Salem numbers and Lie groups to prove uniform discreteness, and uses geometric properties to show linear dependence of geodesic lengths.
result Existence of a positive constant δ(X) such that squares of lengths of closed geodesics shorter than δ must be pairwise linearly dependent over Q.