arXiv research

Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.

New resonance theory for Anosov flows connects spectral properties to mixing measures.

Study of spectral properties of Lorentzian quasi-Fuchsian manifolds.

We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by countting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspe…

We study the spectral theory of asymptotically hyperbolic manifolds with ends of warped product type. Our main result is an upper bound on the resonance counting function with a geometric constant expressed in terms of the respective Weyl constants for the core of the manifold and the base manifold defining the ends.

Paper proposes JDR to denoise graph features and rewire graphs for better node classification.

Study proves projective Anosov subgroups lead to mixing flows in specific spaces.

Heat flow on lens spaces settles into Morse functions with four critical points.

Researchers compute the full spectrum of Laplace operator on distance spheres in symmetric spaces.

This work uses Sylvester normalizing flows for more accurate metabolite quantification in MRS.

We establish a splitting formula for the spectral flow of the odd signature operator on a closed 3-manifold M coupled to a path of SU(2) connections, provided M = S cup X, where S is the solid torus. It describes the spectral flow on M in terms of the spectral flow on S, the spectral flow on X (with certain Atiyah-Pato…

Study reveals how to determine area and curvature from fluid flow resonances.

We provide two examples of spectral analysis techniques of Schroedinger operators applied to geometric Laplacians. In particular we show how to adapt the method of analytic dilation to Laplacians on complete manifolds with corners of codimension 2 finding the absence of singular continuous spectrum for these operators,…

iCVI-ARTMAP accelerates clustering with adaptive resonance theory and validity indices.

For hyperbolic Riemann surfaces of finite geometry, we study Selberg's zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsion-free, discrete subgroup of SL(2,R) is determined by its trace sp…

Let $Γ<\mathrm{SL}_2(\mathbb{Z})$ be a non-elementary finitely generated subgroup and let $Γ(q)$ be its congruence subgroup of level $q$ for each $q\in \mathbb{N}$. We obtain an asymptotic formula for the matrix coefficients of $L^2(Γ(q) \backslash \mathrm{SL}_2(\mathbb{R}))$ with a {\it uniform} exponential error term…

In this paper, we use Chas-Sullivan theory on loop homology and Leray-Serre spectral sequence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the resonance identity of non-contractible ho…

The paper shows how to uniquely determine a connection up to gauge.

Holomorphic vector bundles on Hopf manifolds admit flat connections.

Infrared spectra obtained from cell or tissue specimen have commonly been observed to involve a significant degree of (resonant) Mie scattering, which often overshadows biochemically relevant spectral information by a non-linear, non-additive spectral component in Fourier transformed infrared (FTIR) spectroscopic measu…

Study reveals a link between Ruelle-Pollicott resonances and cohomology eigenvalues for Anosov diffeomorphisms.

For compact and for convex co-compact oriented hyperbolic surfaces, we prove an explicit correspondence between classical Ruelle resonant states and quantum resonant states, except at negative integers where the correspondence involves holomorphic sections of line bundles.

We develop theoretical foundations of Resonator Networks, a new type of recurrent neural network introduced in Frady et al. (2020) to solve a high-dimensional vector factorization problem arising in Vector Symbolic Architectures. Given a composite vector formed by the Hadamard product between a discrete set of high-dim…

We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold $M = (0,\infty) \times Y$ whose rotation radius is constant outside some compact interval. The Laplacian on $M$ is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potenti…

The study finds resonance points in polarised curves with polynomial conserved quantities.

The resonant band is a useful notion for the computation of the nontrivial monodromy eigenspaces of the Milnor fiber of a real line arrangement. In this article, we develop the resonant band description for the cohomology of the Aomoto complex. As an application, we prove that real 4-nets do not exist.

We show that the resolvent of the Laplacian on SL(3,$\mathbb{R}$)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of $\mathbb{C}$. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue op…

We find a resonance free region polynomially close to the critical line on Conformally compact manifolds with polyhomogeneous metric.

CycleQSM uses deep learning to accurately map tissue magnetic susceptibility without needing paired data.

We investigate the resonance varieties, lower central series ranks, and Chen ranks of the pure virtual braid groups and their upper-triangular subgroups. As an application, we give a complete answer to the 1-formality question for this class of groups. In the process, we explore various connections between the Alexande…

For a conformally compact manifold that is hyperbolic near infinity and of dimension $n+1$, we complete the proof of the optimal $O(r^{n+1})$ upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we es…

SGDm with fixed step-size diverges under covariate shift, similar to a parametric oscillator.

Study resonant forms for dissipative Anosov flows on 3-manifolds.

Study geometric structures on LVM threefolds, focusing on resonant structures.

The isoresidual fibration maps Riemann sphere strata to resonance arrangements.

Study of resonances and residue operators for hyperbolic spaces.

We prove the meromorphic extension to C for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the geometrically finite quotients with rational non-maximal rank cusps previous…

On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric ten…

As a consequence of a result of Cardoso and Vodev, we show that the resolvent of the Laplacian on asymptotically hyperbolic manifolds is analytic in an exponential neighbourhood of the critical line. The case of non-trapping metrics with constant curvature near infinity is also considered: there exists a strip with at …

On an asymptotically hyperbolic manifold (X,g), we show that the resolvent resonances coincide, with multiplicities, with the poles of the renormalized scattering operator, except for the special points n/2-k (with k>0 integer) where an additional term appears: this is the dimension of the kernel of the k-conformal Lap…

On an asymptotically conic manifold $(M,g)$, we analyze the asymptotics of the integral kernel of the resolvent $R_q(k):=(Δ_q+k^2)^{-1}$ of the Hodge Laplacian $Δ_q$ on $q$-forms as the spectral parameter $k$ approaches zero, assuming that 0 is not a resonance. The first application we give is an $L^p$ Sobolev estimate…

We consider actions of Z^k, k \ge 2, by Anosov diffeomorphisms which are uniformly quasiconformal on each coarse Lyapunov distribution. These actions generalize Cartan actions for which coarse Lyapunov distributions are one-dimensional. We show that, under certain non-resonance assumptions on the Lyapunov exponents, a …

A machine learning framework predicts self-induced stochastic resonance in neurons.

Suppose that $(X, g)$ is a conformally compact $(n+1)$-dimensional manifold that is hyperbolic at infinity in the sense that outside of a compact set $K \subset X$ the sectional curvatures of $g$ are identically equal to minus one. We prove that the counting function for the resolvent resonances has maximal order of gr…

In this paper, we establish first the resonance identity for non-contractible homologically visible prime closed geodesics on Finsler $n$-dimensional real projective space $(\mathbb{R}P^n,F)$ when there exist only finitely many distinct non-contractible closed geodesics on $(\mathbb{R}P^n,F)$, where the integer $n\geq2…

On geometrically finite hyperbolic manifolds $Γ\backslash H^{d}$, including those with non-maximal rank cusps, we give upper bounds on the number $N(R)$ of resonances of the Laplacian in disks of size $R$ as $R\to \infty$. In particular, if the parabolic subgroups of $Γ$ satisfy a certain Diophantine condition, the bou…

SBI provides more accurate pole positions than chi-squared minimization in model misspecification.

We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalue…