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In this work, we construct different classes of coherent states related to a quantum system, recently studied in [1], of an electron moving in a plane in uniform external magnetic and electric fields which possesses both discrete and continuous spectra. The eigenfunctions are realized as an orthonormal basis of a suitable Hilbert space appropriate for building the related coherent states. These latter are achieved in the context where we consider both spectra purely discrete obeying the criteria that a family of coherent states must satisfy.

Prony series representations have been extensively applied to characterizing the time-domain linear viscoelastic (LVE) material functions for asphalt concrete. However, existing methods that can generate high-quality Prony series parameters (i.e., discrete spectra) mostly involve complicated programming algorithms, which poses a challenge for quick access of Prony series parameters. Also, very limited research has been devoted to establishing methods for simultaneously determining both retardation and relaxation spectra. To resolve these issues, this study presented a practical approach to fast acquiring high-quality Prony series parameters for both relaxation modulus and creep compliance of asphalt concrete by using the complex modulus test data. The approach adopts the analytical representations of the continuous relaxation and retardation spectra from the Havriliak-Negami (HN) and 2S2P1D complex modulus models to directly determine the discrete spectra, and the elastic constants, Ee and Dg, for both LVE modulus and compliance functions are further calculated by fitting the corresponding generalized Maxwell model representations to smoothed data from the storage modulus representations of the HN and 2S2P1D complex modulus models. In this way, all the procedures in the proposed method can be easily implemented in Microsoft Excel. The results showed that the HN and 2S2P1D models yielded slightly different continuous spectral patterns at shorter relaxation times and longer retardation times. However, at the region covered by the test data, the continuous spectra of the two complex modulus models were very close to each other. Thus, the two models can generate comparable Prony series parameters within the time or frequency range covered by the test data. Considering that the quality of the resulting Prony series parameters are closely related to the master curve models used for presmoothing, the HN and 2S2P1D models were compared with the conventional Sigmoidal model. Additionally, the Black diagram was recommended for examining the quality of the complex modulus test data before constructing the master curves.

Identifying coordinate transformations that make strongly nonlinear dynamics approximately linear has the potential to enable nonlinear prediction, estimation, and control using linear theory. The Koopman operator is a leading data-driven embedding, and its eigenfunctions provide intrinsic coordinates that globally linearize the dynamics. However, identifying and representing these eigenfunctions has proven challenging. This work leverages deep learning to discover representations of Koopman eigenfunctions from data. Our network is parsimonious and interpretable by construction, embedding the dynamics on a low-dimensional manifold. We identify nonlinear coordinates on which the dynamics are globally linear using a modified auto-encoder. We also generalize Koopman representations to include a ubiquitous class of systems with continuous spectra. Our framework parametrizes the continuous frequency using an auxiliary network, enabling a compact and efficient embedding, while connecting our models to decades of asymptotics. Thus, we benefit from the power of deep learning, while retaining the physical interpretability of Koopman embeddings. It is often advantageous to transform a strongly nonlinear system into a linear one in order to simplify its analysis for prediction and control. Here the authors combine dynamical systems with deep learning to identify these hard-to-find transformations.

We study some features of a warped five-dimensional model that solves the hierarchy problem and exhibits a continuum of Kaluza-Klein (KK) modes with a mass gap at the TeV scale. We compute the propagators and spectral functions for massless bulk gauge bosons, and study how the continuum can be reached as the limit of a set of models with discrete spectrum. Finally, we study the low energy effective theory and provide explicit results for the Wilson coefficients.

Dynamic mode decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a significant challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes and dealing with continuous spectra, which both occur regularly in turbulent flows. Residual dynamic mode decomposition (ResDMD), introduced by Colbrook & Townsend (Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. 2021. arXiv:2111.14889), overcomes such challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in various fluid dynamic situations at varying Reynolds numbers from both numerical and experimental data. Examples include vortex shedding behind a cylinder, hot-wire data acquired in a turbulent boundary layer, particle image velocimetry data focusing on a wall-jet flow and laser-induced plasma acoustic pressure signals. We present some advantages of ResDMD: the ability to resolve nonlinear and transient modes verifiably; the verification of learnt dictionaries; the verification of Koopman mode decompositions; and spectral calculations with reduced broadening effects. We also discuss how a new ordering of modes via residuals enables greater accuracy than the traditional modulus ordering (e.g. when forecasting) with a smaller dictionary. This result paves the way for more significant dynamic compression of large datasets without sacrificing accuracy.

One of the simplest non-Hermitian Hamiltonians, first proposed by Schwartz in 1960, that may possess a spectral singularity is analysed from the point of view of the non-Hermitian generalization of quantum mechanics. It is shown that the η operator, being a second-order differential operator, has supersymmetric structure. Asymptotic behaviour of the eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result, the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible presence of a spectral singularity in the spectrum of the non-Hermitian Hamiltonian may be detected as a resonance in the scattering cross section of its Hermitian counterpart. Nevertheless, just at the singular point, the equivalent Hermitian Hamiltonian becomes undetermined.

A bstract We study a precise and computationally tractable notion of operator complexity in holographic quantum theories, including the ensemble dual of Jackiw-Teitelboim gravity and two-dimensional holographic conformal field theories. This is a refined, “microcanonical” version of K-complexity that applies to theories with infinite or continuous spectra (including quantum field theories), and in the holographic theories we study exhibits exponential growth for a scrambling time, followed by linear growth until saturation at a time exponential in the entropy — a behavior that is characteristic of chaos. We show that the linear growth regime implies a universal random matrix description of the operator dynamics after scrambling. Our main tool for establishing this connection is a “complexity renormalization group” framework we develop that allows us to study the effective operator dynamics for different timescales by “integrating out” large K-complexities. In the dual gravity setting, we comment on the empirical match between our version of K-complexity and the maximal volume proposal, and speculate on a connection between the universal random matrix theory dynamics of operator growth after scrambling and the spatial translation symmetry of smooth black hole interiors.

Water vapour continuum absorption is an important contributor to the Earth's radiative cooling and energy balance. Here, we describe the development and status of the MT_CKD (MlawerTobinCloughKneizysDavies) water vapour continuum absorption model. The perspective adopted in developing the MT_CKD model has been to constrain the model so that it is consistent with quality analyses of spectral atmospheric and laboratory measurements of the foreign and self continuum. For field measurements, only cases for which the characterization of the atmospheric state has been highly scrutinized have been used. Continuum coefficients in spectral regions that have not been subject to compelling analyses are determined by a mathematical formulation of the spectral shape associated with each water vapour monomer line. This formulation, which is based on continuum values in spectral regions in which the coefficients are well constrained by measurements, is applied consistently to all water vapour monomer lines from the microwave to the visible. The results are summed-up (separately for the foreign and self) to obtain continuum coefficients from 0 to 20 000 cm−1. For each water vapour line, the MT_CKD line shape formulation consists of two components: exponentially decaying far wings of the line plus a contribution from a water vapour molecule undergoing a weak interaction with a second molecule. In the MT_CKD model, the first component is the primary agent for the continuum between water vapour bands, while the second component is responsible for the majority of the continuum within water vapour bands. The MT_CKD model should be regarded as a semi-empirical model with strong constraints provided by the known physics. Keeping the MT_CKD continuum consistent with current observational studies necessitates periodic updates to the water vapour continuum coefficients. In addition to providing details on the MT_CKD line shape formulation, we describe the most recent update to the model, MT_CKD_2.5, which is based on an analysis of satellite- and ground-based observations from 2385 to 2600 cm−1 (approx. 4 μm).

Using two different warped five-dimensional (5D) models with two branes along the extra dimension, we study the Green’s functions and the spectral properties of some of the fields propagating in the bulk. While the first model has a discrete spectrum of Kaluza-Klein (KK) modes, the second one has a continuous spectrum above a mass gap. We also study the positivity of the spectral functions, as well as the coupling of the graviton and the radion with SM matter fields.

We consider a nonlinear eigenvalue problem driven by the sum of pp and qq-Laplacians. We show that the problem has a continuous spectrum. Our result reveals a discontinuity property for the spectrum of a parametric (p,qp,q)-differential operator as the parameter β\\beta goes to 1−1^-.