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Motivated by recent work on the use of topological methods to study collections of rings between an integral domain and its quotient field, we examine spaces of subrings of a commutative ring, where these spaces are endowed with the Zariski or patch topologies. We introduce three notions to study such a space $X$: patch bundles, patch presheaves and patch algebras. When $X$ is compact and Hausdorff, patch bundles give a way to approximate $X$ with topologically more tractable spaces, namely Stone spaces. Patch presheaves encode the space $X$ into stalks of a presheaf of rings over a Boolean algebra, thus giving a more geometrical setting for studying $X$. To both objects, a patch bundle and a patch presheaf, we associate what we call a patch algebra, a commutative ring that efficiently realizes the rings in $X$ as factor rings, or even localizations, and whose structure reflects various properties of the rings in $X$.

Co-based honeycomb magnets have been actively studied recently for the potential realization of emergent quantum magnetism therein such as the Kitaev spin liquid. Here we employ density functional and dynamical mean-field theory methods to examine a family of the Kitaev magnet candidates BaCo$_2$($X$O$_4$)$_2$ ($X$ = P, As, Sb), where the compound with $X$ = Sb being not synthesized yet. Our study confirms the formation of Mott insulating phase and the $J_{\rm eff}$ = 1/2 spin moments at Co$^{2+}$ sites despite the presence of a sizable amount of trigonal crystal field in all three compounds. The pnictogen substitution from phosphorus to antimony significantly changes the in-plane lattice parameters and direct overlap integral between the neighboring Co ions, leading to the suppression of the Heisenberg interaction. More interestingly, the marginal antiferromagnetic nearest-neighbor Kitaev term changes sign into a ferromagnetic one and becomes sizable at the $X$ = Sb limit. Our study suggests that the pnictogen substitution can be a viable route to continuously tune magnetic exchange interactions and to promote magnetic frustration for the realization of potential spin liquid phases in BaCo$_2$($X$O$_4$)$_2$., Comment: 8 pages, 4 figures

We develop MultiSTOP, a Reinforcement Learning framework for solving functional equations in physics. This new methodology produces actual numerical solutions instead of bounds on them. We extend the original BootSTOP algorithm by adding multiple constraints derived from domain-specific knowledge, even in integral form, to improve the accuracy of the solution. We investigate a particular equation in a one-dimensional Conformal Field Theory., Comment: ICLR 2024 Workshop on AI4DifferentialEquations In Science

Continual Learning (CL) focuses on maximizing the predictive performance of a model across a non-stationary stream of data. Unfortunately, CL models tend to forget previous knowledge, thus often underperforming when compared with an offline model trained jointly on the entire data stream. Given that any CL model will eventually make mistakes, it is of crucial importance to build calibrated CL models: models that can reliably tell their confidence when making a prediction. Model calibration is an active research topic in machine learning, yet to be properly investigated in CL. We provide the first empirical study of the behavior of calibration approaches in CL, showing that CL strategies do not inherently learn calibrated models. To mitigate this issue, we design a continual calibration approach that improves the performance of post-processing calibration methods over a wide range of different benchmarks and CL strategies. CL does not necessarily need perfect predictive models, but rather it can benefit from reliable predictive models. We believe our study on continual calibration represents a first step towards this direction., Comment: Accepted at CLVISION workshop, CVPR 2024

We employ ab-initio electronic structure calculations to investigate the charge-density waves and periodic lattice distortions in bilayer 2H-NbSe$_2$. We demonstrate that the vertical stacking can give rise to a variety of patterns that may lower the symmetry of the charge-density waves exhibited separately by the two composing 1H-NbSe$_2$ monolayers. The general tendency to a spontaneous symmetry breaking observed in the ground state and the first excited states is shown to originate from a non-negligible inter-layer coupling. Simulated images for scanning tunnelling microscopy (STM) as well as diffraction/scattering patterns show signatures of the different stacking orders. This may not only be useful to reinterpret past experiments on surfaces and thin films, but may also be exploited to devise ad-hoc experiments for the investigation of the stacking order in 2H-NbSe$_2$. We anticipate that our analysis does not only apply to the 2H-NbSe$_2$ bilayer, but is also relevant for thin films and bulk, whose smallest centro-symmetric component is indeed the bilayer. Finally, our results illustrate clearly that the vertical stacking is not only important for 1T structures, as exemplified by the metal-to-insulator transition observed in 1T-TaS$_2$, but seems to be a general feature of metallic layered transition metal dichalcogenides as well., Comment: 15 pages, 9 figures

We explore the inelastic spectra of electrons impinging in a magnetic system. The methodology here presented is intended to highlight the charge-dependent interaction of the electron beam in a STEM-EELS experiment, and the local vector potential generated by the magnetic lattice. This interaction shows an intensity $10^{-2}$ smaller than the purely spin interaction, which is taken to be functionally the same as in the inelastic neutron experiment. On the other hand, it shows a strong scattering vector dependence ($\kappa^{-4}$) and a dependence with the relative orientation between the probe wavevector and the local magnetic moments of the solid. We present YIG as a case study due to its high interest by the community.

As highlighted in a series of recent papers by Tringali and the author, fundamental aspects of the classical theory of factorization can be significantly generalized by blending the languages of monoids and preorders. Specifically, the definition of a suitable preorder on a monoid allows for the exploration of decompositions of its elements into (more or less) arbitrary factors. We provide an overview of the principal existence theorems in this new theoretical framework. Furthermore, we showcase additional applications beyond classical factorization, emphasizing its generality. In particular, we recover and refine a classical result by Howie on idempotent factorizations in the full transformation monoid of a finite set.

Continual learning is a subfield of machine learning, which aims to allow machine learning models to continuously learn on new data, by accumulating knowledge without forgetting what was learned in the past. In this work, we take a step back, and ask: "Why should one care about continual learning in the first place?". We set the stage by examining recent continual learning papers published at four major machine learning conferences, and show that memory-constrained settings dominate the field. Then, we discuss five open problems in machine learning, and even though they might seem unrelated to continual learning at first sight, we show that continual learning will inevitably be part of their solution. These problems are model editing, personalization and specialization, on-device learning, faster (re-)training and reinforcement learning. Finally, by comparing the desiderata from these unsolved problems and the current assumptions in continual learning, we highlight and discuss four future directions for continual learning research. We hope that this work offers an interesting perspective on the future of continual learning, while displaying its potential value and the paths we have to pursue in order to make it successful. This work is the result of the many discussions the authors had at the Dagstuhl seminar on Deep Continual Learning, in March 2023.

Using first-principles techniques, we study the structural, magnetic and electronic properties of (111)-oriented (LaMnO$_3$)$_{2n}$$\vert$(SrMnO$_3$)$_{n}$ superlattices of varying thickness ($n=2,4,6$). We find that the properties of the thinnest superlattice ($n=2$) are similar to the celebrated half-metallic ferromagnetic alloy La$_{2/3}$Sr$_{1/3}$MnO$_3$, with quenched Jahn-Teller distortions. At intermediate thickness ($n=4$), the $a^{-}a^{-}a^{-}$ tilting pattern transitions to the $a^{-}a^{-}c^{+}$ tilting pattern, driven by the lattice degrees of freedom in the LaMnO$_3$ region. The emergence of the Jahn-Teller modes and the spatial extent needed for their development play a key role in this structural transition. For the largest thickness considered ($n=6$), we unveil an emergent separation of Jahn-Teller and volume-breathing orders in the ground-state structure with the $a^{-}a^{-}c^{+}$ tilting pattern, whereas it vanishes in the antiferromagnetic configurations. The ground state of all superlattices is half-metallic ferromagnetic, not affected by the underlying series of structural transitions. Overall, these results outline a thickness-induced crossover between the physical properties of bulk La$_{2/3}$Sr$_{1/3}$MnO$_3$ and bulk LaMnO$_3$., Comment: 7 pages excluding references, 8 figures

Online continual learning aims to get closer to a live learning experience by learning directly on a stream of data with temporally shifting distribution and by storing a minimum amount of data from that stream. In this empirical evaluation, we evaluate various methods from the literature that tackle online continual learning. More specifically, we focus on the class-incremental setting in the context of image classification, where the learner must learn new classes incrementally from a stream of data. We compare these methods on the Split-CIFAR100 and Split-TinyImagenet benchmarks, and measure their average accuracy, forgetting, stability, and quality of the representations, to evaluate various aspects of the algorithm at the end but also during the whole training period. We find that most methods suffer from stability and underfitting issues. However, the learned representations are comparable to i.i.d. training under the same computational budget. No clear winner emerges from the results and basic experience replay, when properly tuned and implemented, is a very strong baseline. We release our modular and extensible codebase at https://github.com/AlbinSou/ocl_survey based on the avalanche framework to reproduce our results and encourage future research., Comment: ICCV Visual Continual Learning Workshop 2023 accepted paper

NEWFOCUS is an EU COST Action targeted at exploring radical solutions that could influence the design of future wireless networks. The project aims to address some of the challenges associated with optical wireless communication (OWC) and to establish it as a complementary technology to the radio frequency (RF)-based wireless systems in order to meet the demanding requirements of the fifth generation (5G) and the future sixth generation (6G) backhaul and access networks. Only 6G will be able to widely serve the exponential growth in connected devices (i.e., more than 500 billion) in 2030, real-time holographic communication, future virtual reality, etc. Space is emerging as the new frontier in 5 and 6G and beyond communication networks, where it offers high-speed wireless coverage to remote areas both in lands and sees. This activity is supported by the recent development of low-altitude Earth orbit satellite mega-constellations. The focus of this 2nd White Paper is on the use of OWC as an enabling technology for medium- and long-range links for deployment in (i) smart-cities and intelligent transportation systems; (ii) first- and last-mile access and backhaul/fronthaul wireless networks; (iii) hybrid free-space optics/RF adaptive wireless connections; (iv) space-to-ground, inter-satellite, ground-to-air, and air-to-air communications; and (v) underwater communications.

Continual Learning trains models on a stream of data, with the aim of learning new information without forgetting previous knowledge. Given the dynamic nature of such environments, explaining the predictions of these models can be challenging. We study the behavior of SHAP values explanations in Continual Learning and propose an evaluation protocol to robustly assess the change of explanations in Class-Incremental scenarios. We observed that, while Replay strategies enforce the stability of SHAP values in feedforward/convolutional models, they are not able to do the same with fully-trained recurrent models. We show that alternative recurrent approaches, like randomized recurrent models, are more effective in keeping the explanations stable over time., Comment: ESANN 2023, 6 pages, added link to code