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.
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Quantum resonances for tensors on hyperbolic spaces are studied.
problem Quantum resonances of symmetric tensors on asymptotically hyperbolic spaces.
method Analyzes the Lichnerowicz Laplacian on manifolds with even Riemannian conformally compact Einstein metrics and quotients of hyperbolic space.
result Resolvent of the Lichnerowicz Laplacian has meromorphic continuation to the complex plane, defining quantum resonances.
Study of spectral properties of Lorentzian quasi-Fuchsian manifolds.
problem Understanding the spectral properties of Lorentzian quasi-Fuchsian manifolds.
method Analyzing the geodesic flow, Ruelle resonances, and pseudo-Riemannian Laplacian.
result Meromorphic extension of the resolvent of the pseudo-Riemannian Laplacian with poles of finite rank.
Procedure maps quantum systems to curved spacetimes with resonant frequencies.
problem Mapping quantum mechanics to curved spacetimes with resonant frequencies.
method Klein-Gordonization procedure, reducing to nonlinear elliptic equation.
result Large family of spacetimes with resonant spectra for massless wave equations.
We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family $X_\eps$ of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances o…
Classical-quantum correspondence on hyperbolic manifolds established.
problem Classical-quantum spectrum correspondence on hyperbolic manifolds.
method Direct correspondence between geodesic flow and Laplacian spectra.
result Established correspondence for convex cocompact hyperbolic manifolds.
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,…
Holomorphic vector bundles on Hopf manifolds admit flat connections.
problem Understanding flat connections on holomorphic vector bundles over Hopf manifolds.
method Defining resonant and non-resonant Mall bundles, proving the existence of flat connections on non-resonant bundles, and applying the Poincare-Dulac theorem.
result Non-resonant Hopf manifolds are linearizable, generalizing Kodaira's result.
New resonance theory for Anosov flows connects spectral properties to mixing measures.
problem Defining and analyzing Ruelle-Taylor resonances for Anosov actions.
method Combining microlocal methods and J. Taylor's cohomological theory, defining Ruelle-Taylor resonances and proving Fredholm theory.
result Ruelle-Taylor resonances form a discrete subset of with being a leading resonance.
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…
Inverse problem solved for rotationally symmetric manifolds using eigenvalues and resonances.
problem Determining the rotation radius of a manifold from its eigenvalues and resonances.
method Unitary equivalence to one-dimensional Schrödinger operators, non-linear real analytic isomorphism between Hilbert spaces.
result The rotation radius is uniquely determined by its eigenvalues and resonances.
Resonator Networks outperform optimization methods at solving high-dimensional vector factorizationcs.NE
Resonator Networks solve high-dimensional vector factorization better than optimization methods.
problem High-dimensional vector factorization problem in Vector Symbolic Architectures.
method Recurrent neural network (Resonator Networks) that combines nonlinear dynamics and superposition search.
result Resonator Networks outperform optimization methods in solving high-dimensional vector factorization.
Study reveals a link between Ruelle-Pollicott resonances and cohomology eigenvalues for Anosov diffeomorphisms.
problem Understanding the speed of mixing in Anosov diffeomorphisms.
method Investigates Ruelle-Pollicott resonances on manifolds of any dimension, connecting them to cohomology eigenvalues of a quasi-compact transfer operator.
result Established a cohomological bound for the speed of mixing of Anosov diffeomorphisms.
Proves new fixed point formulae for complex manifolds with boundary.
problem Fixed points on complex manifolds with boundary conditions.
method Logarithmic Lefschetz fixed point formulae, normal rescaling, relative duality.
result Resonant boundary terms record normal contact and tangential multiplicity.
The study finds resonance points in polarised curves with polynomial conserved quantities.
problem Finding resonance points in polarised curves with polynomial conserved quantities.
method Using the non-orthogonality assumption on the conserved quantity, the study deduces the existence of resonance points.
result Every finite type polarised curve in the conformal 2-sphere with a polynomial conserved quantity admits a resonance point.
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,)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of . The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue op…
Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.
problem Spectral gap for convex cocompact hyperbolic surfaces and their covers.
method Using thermodynamic formalism for twisted Selberg zeta functions.
result Uniform resonance-free regions for convex cocompact hyperbolic surfaces and expanders.
We find a resonance free region polynomially close to the critical line on Conformally compact manifolds with polyhomogeneous metric.
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 , we complete the proof of the optimal 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.
problem SGDm with fixed step-size diverges under covariate shift.
method Approximated learning system as a time-varying system of ODEs and characterized divergence/convergence modes.
result SGDm with fixed step-size can diverge under covariate shift, similar to resonance in oscillators.
Study resonant forms for dissipative Anosov flows on 3-manifolds.
problem Determine resonant forms and their cohomology classes for dissipative Anosov flows.
method General theory including horocyclic invariance and local geometry analysis.
result Explicit computation of resonant forms and helicity for quasi-Fuchsian flows.
This paper details the techniques and algorithms implemented in Kahler, a Python library that implements discrete exterior calculus on arbitrary Hermitian manifolds. Borrowing techniques and ideas first implemented in PyDEC, Kahler provides a uniquely general framework for computation using discrete exterior calculus. …
GKP codes connect quantum gates to algebraic curves, enabling fault-tolerant quantum computation.
problem Implementing fault-tolerant quantum computation in quantum harmonic oscillator systems.
method Exploring the topological and algebraic structure of GKP codes, showing how gates correspond to symplectic automorphisms and mapping class groups of surfaces.
result GKP Clifford gates are identified with symplectic automorphisms of GKP lattices and mapping class groups of surfaces, providing a topological interpretation of fault tolerance.
The isoresidual fibration maps Riemann sphere strata to resonance arrangements.
problem Mapping Riemann sphere strata to resonance arrangements.
method Defining isoresidual fibration and studying its properties using tree structures.
result The isoresidual fibration is an unramified cover of degree a!/(a+2-p)! above the complement of a hyperplane arrangement.
Study geometric structures on LVM threefolds, focusing on resonant structures.
problem Understanding deformations of geometric structures on LVM threefolds.
method Using the Ehresmann-Thurston principle and Kuranishi family construction.
result Construction of a family containing all LVM threefolds and complete at every point.
Study of resonances and residue operators for hyperbolic spaces.
problem Understanding resonances and residue operators for pseudo-Riemannian hyperbolic spaces.
method Analyzing the resolvent of the Laplace-Beltrami operator on pseudo-Riemannian hyperbolic spaces.
result Explicit determination of resonances and identification of residue representations.
Machine learning models simulate molecular spectra and reactions in solvents.
problem Accurate simulation of molecular spectra and reactions in solvent environments.
method Introduced FieldSchNet, a deep neural network for modeling molecular interactions with external fields.
result Demonstrated significant lowering of Claisen rearrangement reaction activation barrier using FieldSchNet.
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.
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…
Resonant machine learning uses electrical network dynamics to optimize learning efficiently.
problem Traditional energy-based learning models are dissipative and inefficient.
method Proposes a new learning framework with two energy components (active and reactive) to ensure active-power dissipation during learning.
result Support vectors in resonant SVMs correspond to self-sustained oscillations in an LC network.
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…
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.
problem Predicting coherent oscillations in slow-fast excitable systems driven by noise.
method Physics-informed machine learning with a Noise-Augmented State Predictor architecture and Kramers' escape theory constraints.
result Trained PINN accurately predicts spike-train coherence on noise intensity, excitability, and timescale separation.
In this paper, we establish first the resonance identity for non-contractible homologically visible prime closed geodesics on Finsler -dimensional real projective space when there exist only finitely many distinct non-contractible closed geodesics on , where the integer $n\geq2…
Suppose that is a conformally compact -dimensional manifold that is hyperbolic at infinity in the sense that outside of a compact set the sectional curvatures of are identically equal to minus one. We prove that the counting function for the resolvent resonances has maximal order of gr…
The existence of two non-contractible closed geodesics on every bumpy Finsler compact space formmath.DS
Paper proves existence of at least two non-contractible geodesics on Finsler space forms.
problem Existence of non-contractible closed geodesics on Finsler compact space forms.
method Established resonance identity and proved existence of geodesics.
result Existence of at least two non-contractible closed geodesics on Finsler compact space forms.
On geometrically finite hyperbolic manifolds , including those with non-maximal rank cusps, we give upper bounds on the number of resonances of the Laplacian in disks of size as . In particular, if the parabolic subgroups of satisfy a certain Diophantine condition, the bou…
SBI provides more accurate pole positions than chi-squared minimization in model misspecification.
problem Accurate pole position estimation in pi-pi scattering models.
method Simulation Based Inference (SBI) method compared to chi-squared minimization.
result SBI leads to more robust predictions of pole positions in models of pi-pi scattering.
Analyzes how transient conditions affect first-passage times in random walks.
problem Understanding first-passage times in random walks under transient conditions.
method Solves the generalized master equation analytically for a linear chain of states.
result The average first-passage time decreases with a power law dependence on the relaxation rate.
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…
iCVI-ARTMAP accelerates clustering with adaptive resonance theory and validity indices.
problem Improving clustering efficiency and accuracy using adaptive resonance theory.
method Integrates adaptive resonance theory (ARTMAP) with incremental cluster validity indices (iCVIs) for clustering.
result Significantly reduces clustering time and outperforms other methods on synthetic and real-world data.
This study assesses the reproducibility of 1H-MRS scans across different vendors and sessions.
problem Lack of harmonization in magnetic resonance spectroscopy protocols among vendors.
method Analysis of CV and ICC for within- and between-sessions, and correlation coefficients for across machines.
result Metabolite concentrations are highly reproducible across different vendors and sessions.
Survey of ART neural networks for engineering applications.
problem Understanding and utilizing ART neural networks for various machine learning tasks.
method Comprehensive review of classic and modern ART models, describing learning dynamics and engineering properties.
result Compilation of ART models and their properties for engineering applications.
Reflective Hamiltonian Monte Carlo struggles with high-dimensional sampling.
problem Slow mixing in reflective Hamiltonian Monte Carlo with inexact reflections.
method Quantifying instantaneous non-uniformity with Sinkhorn divergence; analyzing particle motion in spheres and cubes; constructing low-dimensional toy models.
result Particles spontaneously unmix, leading to resonances in particle density.
Study path spaces and their homology, extending loop products and coproducts.
problem Understanding the homology of path spaces in closed manifolds.
method Morse-Bott theory and homology operations.
result Complete computation of extended loop product and coproduct on spheres.