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We study spectra of Toeplitz operators $T_a $ with periodic symbols in Bergman spaces $A^2(\Pi)$ on unbounded periodic planar domains $\Pi$, which are defined as the union of infinitely many copies of the translated, bounded periodic cell $\varpi$. We introduce Floquet-transform techniques and prove a version of the band-gap-spectrum formula, which is well-known in the framework of periodic elliptic spectral problems and which describes the essential spectrum of $T_a$ in terms of the spectra of a family of Toepliz-type operators $T_{a,\eta}$ in the cell $\varpi$, where $\eta$ is the so-called Floquet variable. As an application, we consider periodic domains $\Pi_h$ containing thin geometric structures and show how to construct a Toeplitz operator $T_{\sf a}: A^2(\Pi_h) \to A^2(\Pi_h)$ such that the essential spectrum of $T_{\sf a}$ contains disjoint components which approximatively coincide with any given finite set of real numbers. Moreover, our method provides a systematic and illustrative way how to construct such examples by using Toeplitz operators on the unit disc $\mathbb{D}$ e.g. with radial symbols. Using a Riemann mapping one can then find a Toeplitz operator $T_a : A^2(\mathbb{D}) \to A^2(\mathbb{D})$ with a bounded symbol and with the same spectral properties as $T_{\sf a}$.

Modelling multivariate spatio-temporal data with complex dependency structures is a challenging task but can be simplified by assuming that the original variables are generated from independent latent components. If these components are found, they can be modelled univariately. Blind source separation aims to recover the latent components by estimating the unmixing transformation based on the observed data only. The current methods for spatio-temporal blind source separation are restricted to linear unmixing, and nonlinear variants have not been implemented. In this paper, we extend identifiable variational autoencoder to the nonlinear nonstationary spatio-temporal blind source separation setting and demonstrate its performance using comprehensive simulation studies. Additionally, we introduce two alternative methods for the latent dimension estimation, which is a crucial task in order to obtain the correct latent representation. Finally, we illustrate the proposed methods using a meteorological application, where we estimate the latent dimension and the latent components, interpret the components, and show how nonstationarity can be accounted and prediction accuracy can be improved by using the proposed nonlinear blind source separation method as a preprocessing method.

Invertibility of Toeplitz operators on the Bergman space and the related Douglas problem are long standing open problems. In this paper we study the invertibility problem under the novel geometric condition on the image of the symbols, which relaxes the standard positivity condition. We show that under our geometric assumption, the Toeplitz operator $T_\varphi$ is invertible if and only if the Berezin transform of $|\varphi|$ is invertible in $L^{\infty}$. It is well known that the Douglas problem is still open for harmonic functions. We study a class of rather general harmonic polynomials and characterize the invertibility of the corresponding Toeplitz operators. We also give a number of related results and examples., Comment: The authors are in the possession of pdf-slides of a talk in a workshop in Wuhan University on Dec. 1st, 2024, where Mingjin Li and J.Long of Guizhou Normal University, China, are claiming the ownership of the main results of our paper with the same notation and the same results. Such claims are utterly false and should be given no attention

Let $\omega$ be a radial weight on the unit disc of the complex plane $\mathbb{D}$ and denote $\omega_x =\int_0^1 s^x \omega(s)\,ds$, $x\ge 0$, for the moments of $\omega$ and $\widehat{\omega}(r)=\int_r^1 \omega(s)\,ds$ for the tail integrals. A radial weight $\omega$ belongs to the class $\widehat{\mathcal{D}}$ if satisfies the upper doubling condition $$\sup_{0

Photonic modes exhibiting a polarization winding akin to a vortex possess an integer topological charge. Lasing with topological charge 1 or 2 can be realized in periodic lattices of up to six-fold rotational symmetry. Higher order charges require symmetries not compatible with any two-dimensional Bravais lattice. Here, we experimentally demonstrate lasing with topological charges as high as -5, +7, -17 and +19 in quasicrystals. We discover rich ordered structures of increasing topological charges in the reciprocal space. Our quasicrystal design utilizes group theory in determining electromagnetic field nodes, where lossy plasmonic nanoparticles are positioned to maximize gain. Our results open a new path for fundamental studies of higher-order topological defects, coherent light beams of high topological charge, and realizations of omni-directional, flat-band-like lasing.

1. Joint species distribution models (JSDMs) have gained considerable traction among ecologists over the past decade, due to their capacity to answer a wide range of questions at both the species- and the community-level. The family of generalized linear latent variable models in particular has proven popular for building JSDMs, being able to handle many response types including presence-absence data, biomass, overdispersed and/or zero-inflated counts. 2. We extend latent variable models to handle percent cover data, with vegetation, sessile invertebrate, and macroalgal cover data representing the prime examples of such data arising in community ecology. 3. Sparsity is a commonly encountered challenge with percent cover data. Responses are typically recorded as percentages covered per plot, though some species may be completely absent or present, i.e., have 0% or 100% cover respectively, rendering the use of beta distribution inadequate. 4. We propose two JSDMs suitable for percent cover data, namely a hurdle beta model and an ordered beta model. We compare the two proposed approaches to a beta distribution for shifted responses, transformed presence-absence data, and an ordinal model for percent cover classes. Results demonstrate the hurdle beta JSDM was generally the most accurate at retrieving the latent variables and predicting ecological percent cover data.

In spatial blind source separation the observed multivariate random fields are assumed to be mixtures of latent spatially dependent random fields. The objective is to recover latent random fields by estimating the unmixing transformation. Currently, the algorithms for spatial blind source separation can only estimate linear unmixing transformations. Nonlinear blind source separation methods for spatial data are scarce. In this paper we extend an identifiable variational autoencoder that can estimate nonlinear unmixing transformations to spatially dependent data and demonstrate its performance for both stationary and nonstationary spatial data using simulations. In addition, we introduce scaled mean absolute Shapley additive explanations for interpreting the latent components through nonlinear mixing transformation. The spatial identifiable variational autoencoder is applied to a geochemical dataset to find the latent random fields, which are then interpreted by using the scaled mean absolute Shapley additive explanations. Finally, we illustrate how the proposed method can be used as a pre-processing method when making multivariate predictions.

We theoretically predict the full quantum geometric tensor, comprising the quantum metric and the Berry curvature, for a square lattice of plasmonic nanoparticles. The gold nanoparticles act as dipole or multipole antenna radiatively coupled over long distances. The photonic-plasmonic eigenfunctions and energies of the system depend on momentum and polarization (pseudospin), and their topological properties are encoded in the quantum geometric tensor. By T-matrix numerical simulations, we identify a TE-TM band splitting at the diagonals of the first Brillouin zone, that is not predicted by the empty lattice band structure nor by the highly symmetric nature of the system. Further, we find quantum metric around these regions of the reciprocal space, and even a non-zero Berry curvature despite the trivial lattice geometry and absence of magnetic field. We show that this non-zero Berry curvature arises exclusively from non-Hermitian effects which break the time-reversal symmetry. The quantum metric, in contrast, originates from a pseudospin-orbit coupling given by the polarization and directional dependence of the radiation., Comment: Revised version

We experimentally observe the quantum geometric tensor, namely the quantum metric and the Berry curvature, for a square lattice of radiatively coupled plasmonic nanoparticles. We observe a non-zero Berry curvature and show that it arises solely from non-Hermitian effects. The quantum metric is found to originate from a pseudospin-orbit coupling. The long-range nature of the radiative interaction renders the behavior distinct from tight-binding systems: Berry curvature and quantum metric are centered around high-symmetry lines of the Brillouin zone instead of high-symmetry points. Our results inspire new pathways in the design of topological systems by tailoring losses or gain., Comment: Revised version

We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some symmetries such as the so-called Fichera layer. The geometry we consider depends on two parameters gathered in some vector $\kappa=(\kappa_1,\kappa_2)$ which characterizes the domain at the edges. By exchanging the axes and/or modifying their orientations if necessary, it is sufficient to restrict the analysis to the cases $\kappa_1\ge0$ and $\kappa_2\in[-\kappa_1,\kappa_1]$. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to $\kappa$. In particular, we show that for a given $\kappa_1>0$, there is some $h(\kappa_1)>0$ such that discrete spectrum exists for $\kappa_2\in[-\kappa_1,0)\cup(h(\kappa_1),\kappa_1]$ whereas it is empty for $\kappa_2\in[0,h(\kappa_1)]$. The proofs rely on classical arguments of spectral theory such as the max-min principle. The main originality lies rather in the delicate use of the features of the geometry.

We consider the propagation of acoustic waves in a waveguide containing a penetrable dissipative inclusion. We prove that as soon as the dissipation, characterized by some coefficient $\eta$, is non zero, the scattering solutions are uniquely defined. Additionally, we give an asymptotic expansion of the corresponding scattering matrix when $\eta\to0^+$ (small dissipation) and when $\eta\to+\infty$ (large dissipation). Surprisingly, at the limit $\eta\to+\infty$, we show that no energy is absorbed by the inclusion. This is due to the so-called skin-effect phenomenon and can be explained by the fact that the field no longer penetrates into the highly dissipative inclusion. These results guarantee that in monomode regime, the amplitude of the reflection coefficient has a global minimum with respect to $\eta$. The situation where this minimum is zero, that is when the device acts as a perfect absorber, is particularly interesting for certain applications. However it does not happen in general. In this work, we show how to perturb the geometry of the waveguide to create 2D perfect absorbers in monomode regime. Asymptotic expansions are justified by error estimates and theoretical results are supported by numerical illustrations.

We consider weighted Bergman spaces $A_\mu^1$ on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces we characterize the solid core of $A_\mu^1$. Also, as a consequence of a characterization of solid $A_\mu^1$-spaces we show that, in the case of entire functions, there indeed exist solid $A_\mu^1$-spaces. The second part of the paper is restricted to the case of the unit disc and it contains a characterization of the solid hull of $A_\mu^1$, when $\mu $ equals the weighted Lebesgue measure with weight $v$. The results are based on a duality relation of weighted $A^1$- and $H^\infty$-spaces, the validity of which requires the assumption that $- \log v$ belongs to the class $\mathcal{W}_0$, studied in a number of publications; moreover, $v$ has to satisfy condition $(b)$, introduced by the authors. The exponentially decreasing weight $v(z) = \exp( -1 /(1-|z|)$ provides an example satisfying both assumptions.

We study Bergman kernels $K_\Pi$ and projections $P_\Pi$ in unbounded planar domains $\Pi$, which are periodic in one dimension. In the case $\Pi$ is simply connected we write the kernel $K_\Pi$ in terms of a Riemann mapping $\varphi$ related to the bounded periodic cell $\varpi$ of the domain $\Pi$. We also introduce and adapt to the Bergman space setting the Floquet transform technique, which is a standard tool for elliptic spectral problems in periodic domains. We investigate the boundedness properties of the Floquet transform operators in Bergman spaces and derive a general formula connecting $P_\Pi$ to a projection on a bounded domain. We show how this theory can be used to reproduce the above kernel formula for $K_\Pi$. Finally, we consider weighted $L^p$-estimates for $P_\Pi$ in periodic domains.

Generalized linear latent variable models (GLLVMs) are a class of methods for analyzing multi-response data which has garnered considerable popularity in recent years, for example, in the analysis of multivariate abundance data in ecology. One of the main features of GLLVMs is their capacity to handle a variety of responses types, such as (overdispersed) counts, binomial responses, (semi-)continuous, and proportions data. On the other hand, the introduction of underlying latent variables presents some major computational challenges, as the resulting marginal likelihood function involves an intractable integral for non-normally distributed responses. This has spurred research into approximation methods to overcome this integral, with a recent and particularly computationally scalable one being that of variational approximations (VA). However, research into the use of VA of GLLVMs and related models has been hampered by the fact that closed-form approximations have only been obtained for certain pairs of response distributions and link functions. In this article, we propose an extended variational approximations (EVA) approach which widens the set of VA-applicable GLLVMs drastically. EVA draws inspiration from the underlying idea of Laplace approximations: by replacing the complete-data likelihood function with its second order Taylor approximation about the mean of the variational distribution, we can obtain a closed-form approximation to the marginal likelihood of the GLLVM for any response type and link function. Through simulation studies and an application to testate amoebae data set in ecology, we demonstrate how EVA results in a universal approach to fitting GLLVMs, which remains competitive in terms of estimation and inferential performance relative to both standard VA and a Laplace approximation approach, while being computationally more scalable than both in practice.

Inspired by our previous work on the boundedness of Toeplitz operators, we introduce weak BMO and VMO type conditions, denoted by BWMO and VWMO, respectively, for functions on the open unit disc of the complex plane. We show that the average function of a function $f$ in BWMO is boundedly oscillating, and the analogous result holds for $f$ in VWMO. The result is applied for generalizations of known results on the essential spectra and norms of Toeplitz operators. Finally, we provide examples of functions satisfying the VWMO condition which are not in the classical VMO or even in BMO.

We report first-order spatial and temporal correlations in strongly coupled plasmonic Bose-Einstein condensates. The condensate is large, more than twenty times the spatial coherence length of the polaritons in the uncondensed system and hundred times the healing length, making plasmonic lattices an attractive platform for studying long-range spatial correlations in two dimensions (2D). We find that both spatial and temporal coherence display non-exponential decay; the results suggest power-law or stretched exponential behaviour with different exponents for spatial and temporal correlation decays., Comment: 13 pages, 16 figures

In stationary subspace analysis (SSA) one assumes that the observable p-variate time series is a linear mixture of a k-variate nonstationary time series and a (p-k)-variate stationary time series. The aim is then to estimate the unmixing matrix which transforms the observed multivariate time series onto stationary and nonstationary components. In the classical approach multivariate data are projected onto stationary and nonstationary subspaces by minimizing a Kullback-Leibler divergence between Gaussian distributions, and the method only detects nonstationarities in the first two moments. In this paper we consider SSA in a more general multivariate time series setting and propose SSA methods which are able to detect nonstationarities in mean, variance and autocorrelation, or in all of them. Simulation studies illustrate the performances of proposed methods, and it is shown that especially the method that detects all three types of nonstationarities performs well in various time series settings. The paper is concluded with an illustrative example.

Polarization textures of light may reflect fundamental phenomena such as topological defects, and can be utilized in engineering light beams. Three main routes are applied during their creation: spontaneous appearance in phase transitions, steering by an excitation beam, or structural engineering of the medium. We present an approach that uses all three in a platform offering advantages that are not simultaneously provided in any previous system: advanced structural engineering, strong-coupling condensate with effective photonic interactions, as well as room temperature and sub-picosecond operation. We demonstrate domain wall polarization textures in a plasmonic lattice Bose-Einstein condensate, by combining the dipole structure of the lattice with a non-trivial condensate phase revealed by phase retrieval. These results open new prospects for fundamental studies of non-equilibrium condensation and sources of polarization-structured beams., Comment: 29 pages, 7 figures

We study the boundedness of Toeplitz operators $T_\psi$ with locally integrable symbols on weighted harmonic Bergman spaces over the unit ball of $\mathbb{R}^n$. Generalizing earlier results for analytic function spaces, we derive a general sufficient condition for the boundedness of $T_\psi$ in terms of suitable averages of its symbol. We also obtain a similar "vanishing" condition for compactness. Finally, we show how these results can be transferred to the setting of the standard weighted Bergman spaces of analytic functions., Comment: 24 pages; added an example

We consider the linear water-wave problem in a periodic channel $\Pi^h \subset \mathbb{R}^3$, which is shallow except for a periodic array of deep potholes in it. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the essential spectrum in the linear water-wave system, which includes the spectral Steklov boundary condition posed on the free water surface. We apply methods of asymptotic analysis, where the most involved step is the construction and analysis of an appropriate boundary layer in a neighborhood of the joint of the potholes with the thin part of the channel. Consequently, the existence of a spectral gap for small enough $h$ is proven.

We establish absolute continuity of the spectrum of a periodic Schr\"odiner operator in R^n with periodic perforations. We also prove analytic dependece of the dispersion relation on the shape of the perforation.

We prove sufficient conditions for the boundedness and compactness of Toeplitz operators $T_a$ in weighted sup-normed Banach spaces $H_v^\infty$ of holomorphic functions defined on the open unit disc $\mathbb{D}$ of the complex plane; both the weights $v$ and symbols $a$ are assumed to be radial functions on $\mathbb{D}$. In an earlier work by the authors it was shown that there exists a bounded, harmonic (thus non-radial) symbol $a$ such that $T_a$ is not bounded in any space $H_v^\infty$ with an admissible weight $v$. Here, we show that a mild additional assumption on the logarithmic decay rate of a radial symbol $a$ at the boundary of $\mathbb{D} $ guarantees the boundedness of $T_a$. The sufficient conditions for the boundedness and compactness of $T_a$, in a number of variations, are derived from the general, abstract necessary and sufficient condition recently found by the authors. The results apply for a large class of weights satisfying the so called condition$(B)$, which includes in addition to standard weight classes also many rapidly decreasing weights.

We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given.

A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with time periodic coefficients. Our main results are a construction of a pointwise projector and a spectral splitting of the system into a finite dimensional system of ordinary differential equations with constant coefficients and an infinite dimensional part whose solutions have better properties in a certain sense. This complements the well-known asymptotic results for periodic hypoelliptic problems in cylinders (Kuchment) and for elliptic problems in quasicylinders (Nazarov). As an application we give a center manifold reduction for a class of non-linear ordinary differential equations in Hilbert spaces with periodic coefficients. This result generalizes the known case with constant coefficients (Mielke).