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In the early nineties, R. M. Aron, B. Cole, T. Gamelin and W.B. Johnson initiated the study of the maximal ideal space (spectrum) of Banach algebras of holomorphic functions defined on the open unit ball of an infinite dimensional complex Banach space. Within this framework, we investigate the fibers and Gleason parts of the spectrum of the algebra of holomorphic and uniformly continuous functions on the unit ball of $\ell_p$ ($1\le p<\infty$). We show that the inherent geometry of these spaces provides a fundamental ingredient for our results. We prove that whenever $p\in\mathbb N$ ($p\ge 2$), the fiber of every $z\in B_{\ell_p}$ contains a set of cardinal $2^{\mathfrak c}$ such that any two elements of this set belong to different Gleason parts. For the case $p=1$, we complete the known description of the fibers, showing that, for each $z\in\overline B_{\ell_1''}\setminus S_{\ell_1}$, the fiber over $z$ is not a singleton. Also, we establish that different fibers over elements in $S_{\ell_1''}$ cannot share Gleason parts., Comment: 25 pages

Given a frequency $\lambda=(\lambda_n)$, we consider the Hardy spaces $ \mathcal{H}_p^\lambda$ of $\lambda$-Dirichlet series $ D = \sum_n a_n e^{-\lambda_n s}$ and study the asymptotic behavior of the upper and lower democracy functions of its canonical basis $\mathcal B=\{e^{-\lambda_ns}\}$. For the ordinary case, $\mathcal B=\{n^{-s}\}$, we give the correct asymptotic behavior of all such functions, while in the general case we give sharp lower and upper bounds for all possible behaviors. Moreover, for $p>2$ we present examples showing that any intermediate behavior (between the extreme bounds) can occur. We also study how different properties of the frequency $\lambda$ lead to particular behaviors of the corresponding fundamental functions. Finally, we apply our results to analyze greedy-type properties of $\mathcal B=\{e^{-\lambda_ns}\}$ for some particular $\lambda$'s.

Dielectric nanostructures have demonstrated optical antenna effects due to Mie resonances. Preliminary investigations on dielectric nanoantennas have been carried out for a trifecta of enhancements, i.e., simultaneous enhancements in absorption, emission directionality and radiative decay rates of quantum emitters. However, these investigations are limited by fragile substrates or low Purcell factor, which is extremely important for exciting quantum emitters electrically. In this paper, we present a Si mix antenna array to achieve the trifecta enhancement of ~1200 fold with a Purcell factor of ~47. The antenna design incorporates ~10 nm gaps within which fluorescent molecules strongly absorb the pump laser energy through a resonant mode. In the emission process, the antenna array increases the radiative decay rates of the fluorescence molecules via Purcell effect and provides directional emission through a separate mode. This work could lead to novel CMOS compatible platforms for enhancing fluorescence for biological and chemical applications., Comment: 20 pages, 4 figures

Linear colour conversion is a process where an emitter absorbs a photon and then emits another photon with either higher or lower energy, corresponding to up- or down conversion, respectively. In this regard, the presence of a volumetric cavity plays a crucial role in enhancing absorption and photoluminescence (PL), as it allows for large volumes of interaction between the exciting photons and the emissive materials, maximising the colour conversion efficiency. Here, we present a dual resonance nanostructure made of a titanium dioxide (TiO2) subwavelength grating to enhance the colour down-conversion efficiency of green light at ~530 nm emitted by gradient alloyed CdxZn1-xSeyS1-y colloidal quantum dots (QDs) when excited with a blue light at ~460 nm. A large mode volume can be created within the QD layer by the hybridisation of the grating resonances and waveguide modes. This allows increasing mode overlap between the resonances and the QDs, resulting in large absorption and tailored emission enhancements. Particularly, we achieved polarized light emission with maximum photoluminescence enhancement of ~140 times at a specific angular direction, and a total enhancement of ~34 times within 0.55 numerical aperture (NA) of the collecting objective. The enhancement encompasses absorption enhancement, Purcell enhancement and directionality enhancement (i.e., outcoupling). We achieved total absorption of 35% for green QDs with a remarkably thin colour conversion layer of ~ 400 nm (inclusive of the TiO2 layer). This work provides a guideline for designing large-volume cavities for practical application in absorption/fluorescence enhancement, such as down colour conversion in microLED displays, detectors or photovoltaics., Comment: 37 pages, 11 figures (4 in maintext, 11 in SI)

This paper investigates applications of the Tsirelson spectral measures to noise filtering problems within the classical Wiener noise framework. We particularly focus on those among those measures associated to square integrable Radon-Nikodym derivatives of laws of observation signals absolutely continuous with respect to the Wiener measure. Explicit formulas are provided to compute the values of the Tsirelson spectral measure of specific sets of interests. Then, several applications to filtering problems are considered, notably to the so-called innovation problem of filtering.

Metalenses, in order to compete with conventional bulk optics in commercial imaging systems, often require large field of view (FOV) and broadband operation simultaneously. However, strong chromatic and coma aberrations present in common metalens designs have so far limited their widespread use. Stacking of metalenses as one of the possible solutions increases the overall complexity of the optical system and hinders the main benefit of reduced thickness and light weight. To tackle both issues, here we propose a single-layer imaging system utilizing a recently developed class of metalenses providing large field of view. Using it, we demonstrate full-color imaging with a FOV of 100 degrees. This approach, empowered by computational imaging techniques, produce high quality images, both in terms of color reproduction and sharpness. Suitable for real-time unpolarized light operation with the standard color filters present in prevalent camera systems, our results might enable a pathway for consumer electronics applications of this emerging technology., Comment: 22 pages (main), 5 figures (main)

Optical tweezers have become essential tools to manipulate atoms or molecules at a single particle level. However, using standard diffracted-limited optical systems, the transverse size of the trap is lower bounded by the optical wavelength, limiting the application range of optical tweezers. Here we report trapping of single ultracold atom in an optical trap that can be continuously tuned from a standard Airy focus to a subwavelength hotspot smaller than the usual Abbe's diffraction limit. The hotspot was generated using the effect of superoscillations, by the precise interference of multiple free-space coherent waves. We argue that superoscillatory trapping and continuous potential tuning offer not only a way to generate compact and tenable ensembles of trapped atoms for quantum simulators but will also be useful in single molecule quantum chemistry and the study of cooperative atom-photon interaction within subwavelength arrays of quantum emitters., Comment: Published in Communications Physics 13 pages including supplementary information, 5 figures

In this work, we investigate an optimization problem over adapted couplings between pairs of real valued random variables, possibly describing random times. We relate those couplings to a specific class of causal transport plans between probabilities on the set of real numbers endowed with a filtration, for which their provide a specific representation, which is motivated by a precise characterization of the corresponding deterministic transport plans. From this, under mild hypothesis, the existence of a solution to the problem investigated here is obtained. Furthermore, several examples are provided, within this explicit framework., Comment: Document of work in progess

The enhancement of the photoluminescence of quantum dots induced by an optical nanoantenna has been studied considerably, but there is still significant interest in optimizing and miniaturizing such structures, especially when accompanied by an experimental demonstration. Most of the realizations use plasmonic platforms, and some also use all-dielectric nanoantennas, but hybrid dielectric-plasmonic (subwavelength) nanostructures have been very little explored. In this paper, we propose and demonstrate single subwavelength hybrid dielectric-plasmonic optical nanoantennas coupled to localized quantum dot emitters that constitute efficient and bright unidirectional photon sources under optical pumping. To achieve this, we devised a silicon nanoring sitting on a gold mirror with a 10 nm gap in-between, where an assembly of colloidal quantum dots is embedded. Such a structure supports both (radiative) antenna mode and (nonradiative) gap mode resonances, which we exploit for the dual purpose of out-coupling the light emitted by the quantum dots into the far-field with out-of-plane directivity, and for enhancing the excitation of the dots by the optical pump. Moreover, almost independent control of the resonance spectral positions can be achieved by simple tuning of geometrical parameters such as the ring inner and outer diameters, allowing us to conveniently adjust these resonances with respect to the quantum dots emission and absorption wavelengths. Using the proposed architecture, we obtain experimentally average fluorescence enhancement factors up to $654\times$ folds mainly due to high radiative efficiencies, and associated with a directional emission of the photoluminescence into a cone of $\pm 17\degree$ in the direction normal to the sample plane. We believe the solution presented here to be viable and relevant for the next generation of light-emitting devices., Comment: 39 pages, 4 figures

We introduce and study the notion of weak weight-semi-greedy Markushevich bases - which extends the concepts of weight semi-greedy and weak semi-greedy Markushevich bases. In particular, we study conditions under which such bases are weight almost greedy. We also define the notion of weak weight almost greedy bases, and prove that this formally weaker concept is equivalent to that concept of weight almost greedy bases. Finally, we study some parameters involving the weak thresholding and Chebyshevian greedy algorithms., Comment: May 20, 2023: Fixed a line in the proof of Proposition 3.24. September 29, 2022: Added a missing condition in the statement of Example 7.3; fixed some typos, and made general style improvements; November 1, 2022 styleim provements April 7, 2023: The proof of Prop. 3.24 was modified so that only finitely supported vectors are used, so only approximations by projections are needed

In this paper we study a new class of bases, weaker than quasi-greedy bases, which retain their unconditionality properties and can provide the same optimality for the thresholding greedy algorithm. We measure how far these bases are from being unconditional and use this concept to give a new characterization of nearly unconditional bases.

In nonlinear greedy approximation theory, bidemocratic bases have traditionally played the role of dualizing democratic, greedy, quasi-greedy, or almost greedy bases. In this article we shift the viewpoint and study them for their own sake, just as we would with any other kind of greedy-type bases. In particular we show that bidemocratic bases need not be quasi-greedy, despite the fact that they retain a strong unconditionality flavor which brings them very close to being quasi-greedy. Our constructive approach gives that for each $1

This paper investigates the problem to determine whether a given stochastic process generates a sampled Brownian filtration. A fairly general sufficient condition is obtained by applying the Frank H. Clarke contraction criteria to a functional whose construction relies on the quasi-invariance properties of a sampled Wiener measure. The latter are investigated from the precise analytic structure underlying this sampled Brownian motion. In particular, the Cameron-Martin space is identified from usual Lebesgue integrals over a Guseinov measure. As an application we obtain sufficient conditions for the existence of a solution to a class of not necessarily Markovian stochastic differential equations driven by a sampled Brownian motion. By taking a sampling times set which coincides with the unit interval, the results apply to continuous time models., Comment: This is a working paper

Dielectric metasurfaces, extremely thin nanostructured dielectric surfaces, hold promise to replace conventional refractive optics, such as lenses, due to their high performance and compactness. However, designing large field-of-view (FOV) metalenses, which are of particular importance when imaging relatively big objects at short distances, remains one of the most critical challenges. Recently, metalenses implementing a quadratic phase profile have been put forward to solve this problem with a single element, but despite their theoretical ability to provide $180^\circ\,$FOV, imaging over very large FOV has not been demonstrated yet. In this work, we provide an in-depth analysis of the imaging properties of quadratic metalenses and, in particular, show that due to their intrinsic barrel distortion or fish-eye effect, there is a fundamental trade-off between the FOV achievable in a given imaging configuration and the optical resolution of the metalens and/or the detector resolution. To illustrate how to harness the full potential of quadratic metalenses, we apply these considerations to the fingerprint detection problem, and demonstrate experimentally the full imaging of a $5\,$mm fingerprint with features of the order of $100\,\mu$m, with a metalens ten times smaller in size and located at a distance of only $2.5\,$mm away from the object. This constitutes the most compact system reported so far for the fingerprint detection., Comment: 6 figures

Fermi's golden rule describes the decay dynamics of unstable quantum systems coupled to a reservoir, and predicts a linear decay in time. Although it arises at relatively short times, the Fermi regime does not take hold in the earliest stages of the quantum dynamics. The standard criterion in the literature for the onset time of the Fermi regime is $t_F\sim1/\Delta\omega$, with $\Delta\omega$ the frequency interval around the resonant transition frequency $\omega_0$ of the system, over which the coupling to the reservoir does not vary appreciably. In this work, this criterion is shown to be inappropriate in general for broadband reservoirs, where the reservoir coupling spectrum takes the form $R\left(\omega\right)\propto\omega^\eta$, and for which it is found that for $\eta>1$, the onset time of the Fermi regime is given by $t_F\propto\left(\omega_{\mathrm{X}}/\omega_0\right)^{\eta-1}\times1/\omega_0$ where $\omega_{\mathrm{X}}$ is the high-frequency cutoff of the reservoir. Therefore, the onset of the Fermi regime can take place at times orders of magnitude larger than those predicted by the standard criterion. This phenomenon is shown to be related to the excitation of the off-resonant frequencies of the reservoir at short times. For broadband reservoirs with $\eta\leq1$, and for narrowband reservoirs, it is shown that the standard criterion is correct. Our findings revisit the conditions of applicability of Fermi's golden rule and improve our understanding of the dynamics of unstable quantum systems.

In the field of remote sensing and more specifically in Earth Observation, new data are available every day, coming from different sensors. Leveraging on those data in classification tasks comes at the price of intense labelling tasks that are not realistic in operational settings. While domain adaptation could be useful to counterbalance this problem, most of the usual methods assume that the data to adapt are comparable (they belong to the same metric space), which is not the case when multiple sensors are at stake. Heterogeneous domain adaptation methods are a particular solution to this problem. We present a novel method to deal with such cases, based on a modified cycleGAN version that incorporates classification losses and a metric space alignment term. We demonstrate its power on a land use classification tasks, with images from both Google Earth and Sentinel-2.

We introduce and study the notion of weak semi-greedy systems -which is inspired in the concepts of semi-greedy and Branch semi-greedy systems and weak thresholding sets-, and prove that in the context Markushevich bases in infinite dimensional Banach spaces, the notions of \textit{ semi-greedy, branch semi-greedy, weak semi-greedy, and almost greedy} Markushevich bases are all equivalent. This completes and extends some results from \cite{Berna2019}, \cite{Dilworth2003b}, and \cite{Dilworth2012}. We also exhibit an example of a semi-greedy system that is neither almost greedy nor a Markushevich basis, showing that the Markushevich condition cannot be dropped from the equivalence result. In some cases, we obtain improved upper bounds for the corresponding constants of the systems., Comment: - The title is updated. - Definition 1.1 is fixed. - Several typos are fixed. - "the weak thresholding set" -> "the only weak thresholding set" (proof of Theorem 4.2). - An inaccuracy after equation (17) is corrected. - Some references are corrected, and some are added

We extend the AROW regression algorithm developed by Vaits and Crammer in [VC11] to handle synchronous mini-batch updates and apply it to stock return prediction. By design, the model should be more robust to noise and adapt better to non-stationarity compared to a simple rolling regression. We empirically show that the new model outperforms more classical approaches by backtesting a strategy on S\&P500 stocks.

In this paper we continue the study of Lebesgue-type inequalities for greedy algorithms. We introduce the notion of strong partially greedy Markushevich bases and study the Lebesgue-type parameters associated with them. We prove that this property is equivalent to that of being conservative and quasi-greedy, extending a similar result given in [9] for Schauder bases. We also give a characterization of 1-strong partial greediness, following the study started in [1,3]., Comment: Corrected typos and improved redaction; results unchanged