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We present an elementary construction of a (highly degenerate) Hopf pairing between the universal enveloping algebra $U(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$ over arbitrary field $\mathbf{k}$ and the Hopf algebra $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$ of regular functions on the automorphism group of $\mathfrak{g}$. This pairing induces a Hopf action of $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$ on $U(\mathfrak{g})$ which together with an explicitly given coaction makes $U(\mathfrak{g})$ into a braided commutative Yetter--Drinfeld $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$-module algebra. From these data one constructs a Hopf algebroid structure on the smash product algebra $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))\sharp U(\mathfrak{g})$ retaining essential features from earlier constructions of a Hopf algebroid structure on infinite-dimensional versions of Heisenberg double of $U(\mathfrak{g})$, including a noncommutative phase space of Lie algebra type, while avoiding the need of completed tensor products. We prove a slightly more general result where algebra $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$ is replaced by $\mathcal{O}(\mathrm{Aut}(\mathfrak{h}))$ and where $\mathfrak{h}$ is any finite-dimensional Leibniz algebra having $\mathfrak{g}$ as its maximal Lie algebra quotient., Comment: v2: 18 pages, title slightly changed, presentation thoroughly revised

We introduce a natural generalization of the definition of a symmetric Hopf algebroid, internal to any symmetric monoidal category with coequalizers that commute with the monoidal product. Motivation for this is the study of Heisenberg doubles of countably dimensional Hopf algebras $A$ as internal Hopf algebroids over a (noncommutative) base $A$ in the category $\mathrm{indproVect}$ of filtered cofiltered vector spaces introduced by the author. One example of such Heisenberg double is internal Hopf algebroid $U(\mathfrak{g}) \sharp U(\mathfrak{g})^*$ over universal enveloping algebra $U(\mathfrak{g})$ of a finite-dimesional Lie algebra $\mathfrak{g}$ that is a properly internalized version of a completed Hopf algebroid previously studied as a Lie algebra type noncommutative phase space., Comment: 17 pages

A class of left bialgebroids whose underlying algebra $A\sharp H$ is a smash product of a bialgebra $H$ with a braided commutative Yetter--Drinfeld $H$-algebra $A$ has recently been studied in relation to models of field theories on noncommutative spaces. In [A. Borowiec, A. Pachol, ``Twisted bialgebroids versus bialgebroids from a Drinfeld twist'', J. Phys. A50 (2017) 055205] a proof has been presented that the bialgebroid $A_F\sharp H^F$ where $H^F$ and $A_F$ are the twists of $H$ and $A$ by a Drinfeld 2-cocycle $F = \sum F^1\otimes F^2$ is isomorphic to the twist of the bialgebroid $A\sharp H$ by the bialgebroid 2-cocycle $\sum 1\sharp F^1\otimes 1\sharp F^2$ induced by $F$. They assume $H$ is quasitriangular, which is reasonable for many physical applications. However the proof and the entire paper take for granted that the coaction and the prebraiding are both given by special formulas involving the R-matrix. There are counterexamples of Yetter--Drinfeld modules over quasitriangular Hopf algebras which are not of this special form. Nevertheless, the main result essentially survives. We present a proof with general coaction and the correct prebraiding, and even without the assumption of quasitriangularity., Comment: v2: 10 pages, improved presentation, in particular more structured exposition in Sec. 2 and 3. Added Concluding remarks (Sec. 4) and few references. To appear in J. Phys. A

The MERTIS (MErcury Radiometer and Thermal Infrared Spectrometer) instrument onboard the ESA/JAXA BepiColombo mission will provide mid-infrared data, which will be crucial to characterize the surface mineralogy of Mercury. In order to interpret the results, we are creating a database of mid infrared spectra. As part of a study of synthetic glasses which are to serve as analog materials for the interpretation of remote sensing and modeling data, we present mid infrared data for analog materials of Mercury regolith, surface and mantle compositions. In addition, we provide data for similar analogs of Earth, Moon, Venus, and Mars rocks for a coherent picture. The analog samples have been first characterized by optical microscopy, Raman spectroscopy and EMPA. Powdered size fractions (0-25 micron, 25-63 micron, 63-125 micron, and 125 -250 micron) were studied in reflectance in the mid-infrared range from 2.5 to 18 micron (550 to 2000cm-1), additional micro FTIR analyses were also obtained. Results for the size fractions of the surface and regolith analogs for Mercury show typical features for amorphous material with Christiansen Features (CF) at 8 to 8.1 micron, Reststrahlen Bands (RB) at 9.8 to 9.9 micron, and the Transparency Feature (TF) at 12 micron. The six bulk silicate Mercury analogs have varying CF positions from 8.1 to 9 micron, with RB crystalline features of various olivines dominating in most samples. Similarly, bulk silicate analogs of the other planetary bodies show glassy features for the surface analogs with CF from 7.9 micron (Earth Continental Crust) to 8.3 micron (Lunar Mare), strong RB from 9.5 micron (Earth Continental Crust ) to 10.6 micron (Lunar Mare and Highlands). TF are usually very weak for the glassy analogs.

We present Trieste, an open-source Python package for Bayesian optimization and active learning benefiting from the scalability and efficiency of TensorFlow. Our library enables the plug-and-play of popular TensorFlow-based models within sequential decision-making loops, e.g. Gaussian processes from GPflow or GPflux, or neural networks from Keras. This modular mindset is central to the package and extends to our acquisition functions and the internal dynamics of the decision-making loop, both of which can be tailored and extended by researchers or engineers when tackling custom use cases. Trieste is a research-friendly and production-ready toolkit backed by a comprehensive test suite, extensive documentation, and available at https://github.com/secondmind-labs/trieste.

In a study to provide ground truth data for mid infrared observations of the surface of Mercury with the MERTIS (Mercury Radiometer and Thermal Infrared Spectrometer) instrument onboard the ESA/JAXA BepiColombo mission, we have studied 17 synthetic glasses. These samples have the chemical compositions of characteristic Hermean surface areas based on MESSENGER data. The samples have been characterized using optical microscopy, EMPA and Raman spectroscopy. Mid infrared spectra have been obtained from polished thin sections using Micro FTIR, and of powdered size fractions of bulk material (0-25, 25-63, 93-125 and 125-250 micron) in the 2.5 to 18 micron range. The synthetic glasses display mostly spectra typical for amorphous materials with a dominating, single Reststrahlen Band (RB) at 9.5 micron to 10.7 micron. RB Features of crystalline forsterite are found in some cases at 9.5 to 10.2 micron, 10.4 to 11.2 micron, and at 11.9 micron. Dendritic crystallization starts at a MgO content higher than 23 wt.% MgO. The Reststrahlen Bands, Christiansen Features (CF), and Transparency Features (TF) shift depending on the SiO2 and MgO contents. Also a shift of the Christiansen Feature of the glasses compared with the SCFM (SiO2/(SiO2+CaO+FeO+MgO)) index is observed. This shift could potentially help distinguish crystalline and amorphous material in remote sensing data. A comparison between the degree of polymerization of the glass and the width of the characteristic strong silicate feature shows a weak positive correlation. A comparison with a high-quality mid-IR spectrum of Mercury shows some moderate similarity to the results of this study, but does not explain all features.

We have analyzed 14 impact melt glass samples, covering the compositional range from highly felsic to mafic/basaltic, as part of our effort to provide mid-infrared spectra (7-14 micron) for MERTIS (Mercury Radiometer and Thermal Infrared Spectrometer), an instrument onboard of the ESA/JAXA BepiColombo mission. Since Mercury was exposed to many impacts in its history, and impact glasses are also common on other bodies, powders of tektites (Irghizite, Libyan Desert Glass, Moldavite, Muong Nong, Thailandite) and impact glasses (from the Dellen, El'gygytgyn, Lonar, Mien, Mistastin, and Popigai impact structures) were analyzed in four size fractions of (0-25, 25-63, 93-125 and 125-250 micron) from 2.5 to 19 micron in bi-directional reflectance. The characteristic Christiansen Feature (CF) is identified between 7.3 micron (Libyan Desert Glass) and 8.2 micron (Dellen). Most samples show mid-infrared spectra typical of highly amorphous material, dominated by a strong Reststrahlen Band (RB) between 8.9 micron (Libyan Desert Glass) and 10.3 micron (Dellen). Even substantial amounts of mineral fragments hardly affect this general band shape. Comparisons of the SiO2 content representing the felsic/mafic composition of the samples with the CF shows felsic/intermediate glass and tektites forming a big group, and comparatively mafic samples a second one. An additional sign of a highly amorphous state is the lack of features at wavelengths longer than about 15 micron. The tektites and two impact glasses, Irghizite and El'gygytgyn respectively, have much weaker water features than most of the other impact glasses.

This study is part of an effort to build a mid-infrared database (7-14micron) of spectra for MERTIS (Mercury Radiometer and Thermal Infrared Spectrometer), an instrument onboard of the ESA/JAXA BepiColombo space probe to be launched to Mercury in 2017. Mercury was exposed to abundant impacts throughout its history. This study of terrestrial impactites can provide estimates of the effects of shock metamorphism on the mid-infrared spectral properties of planetary materials. In this study, we focus on the Noerdlinger Ries crater in Southern Germany, a well preserved and easily accessible impact crater with abundant suevite impactites. Suevite and melt glass bulk samples from Otting and Aumuehle, as well as red suevite from Polsingen were characterized and their reflectance spectra in mid-infrared range obtained. In addition, in-situ mid-infrared spectra were made from glasses and matrix areas in thin sections. The results show similar, but distinguishable spectra for both bulk suevite and melt glass samples, as well as in-situ measurements. Impact melt glass from Aumuehle and Otting have spectra dominated by a Reststrahlen band at 9.3-9.6 micron. Bulk melt rock from Polsingen and bulk suevite and fine-grained matrix have their strongest band between 9.4 to 9.6 micron. There are also features between 8.5 and 9 micron, and 12.5 - 12.8 micron associated with crystalline phases. There is evidence of weathering products in the fine-grained matrix, such as smectites. Mercury endured many impacts with impactors of all sizes over its history. So spectral characteristics observed for impactites formed only in a single impact like in the Ries impact event can be expected to be very common on planetary bodies exposed to many more impacts in their past. We conclude that in mid-infrared remote sensing data the surface of Mercury can be expected to be dominated by features of amorphous materials.

Given a Hopf algebra $H$, Brzezi\'nski and Militaru have shown that each braided commutative Yetter-Drinfeld $H$-module algebra $A$ gives rise to an associative $A$-bialgebroid structure on the smash product algebra $A \sharp H$. They also exhibited an antipode map making $A\sharp H$ the total algebra of a Lu's Hopf algebroid over $A$. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair $A$ and $A^{\mathrm{op}}$ of braided commutative Yetter-Drinfeld $H$-module algebras, and output is a symmetric Hopf algebroid $A\sharp H \cong H\sharp A^{\mathrm{op}}$ over $A$. This construction does not require that the antipode of $H$ is invertible., Comment: 34 pages; in version 2: change of numbering of theorems, propositions, etc. to agree with the version published in Journal of Algebra and Its Applications, minor corrections

The newly discovered two-dimensional magnetic semiconductors such as CrI$_3$ have triggered a surge of interest stemming from their exotic spin-dependent properties and potential applications in spintronics and magneto-optoelectronics. Using first-principle density-functional-theory calculations, we investigate the properties of the spin-polarization texture in momentum space in the prototype 2D centrosymmetric ferromagnetic (FM) bilayer of CrI$_3$, with perpendicular magnetization. The FM bilayer displays a rich in-plane spin texture in its highest valence bands. We show the existence of two distinct spin canting effects, which result from the coupling of FM order with structural chirality and electric polarization, in establishing the in-plane spin texture. The first effect is generated by the chirality of the layer stacking and the spin-orbit-polarized nature of the valence states, and yields the same canting on both layers. The second effect is a Rashba-related effect, which in a centrosymmetric ferromagnet induces in a single electronic state two opposite spin-canted components on the two layers. Furthermore, using the FM bilayer as an example, we provide general rules imposed by magnetic-space-group symmetries on spin-polarization vectors, serving as guidelines for the shape of the spin texture in the Brillouin zone of magnetic crystals. Finally, we show that the above effects can be effectively used to manipulate the spin texture via compressive strain, which induces in the FM bilayer valence-band-edge states with canted spins., Comment: 31 pages, 9 main figures, 3 figures in appendix

The antiferromagnetic (AFM) CrI$_3$ bilayer is a particularly interesting representative of van der Waals 2D semiconductors, which are currently being studied for their magnetism and for their potential in spintronics. Using ab initio density-functional theory calculations, we investigate the spin texture in momentum space of the states of the (doubly degenerate) highest valence band of the AFM CrI$_3$ bilayer with Cr-spin moments perpendicular to the layers. We find the existence, in the main central part of the Brillouin zone, of a Rashba in-plane spin texture of opposite signs on the two layers, resulting from the intrinsic local electric fields acting on each layer. To study the layer segregation of the wavefunctions, we apply a small electric field that splits the degenerate states according to their layer occupancy. We find that the wavefunctions of the highest valence band are layer-segregated, belonging to only one of the two layers with opposite in-plane spin textures, and the segregation occurs over nearly the whole Brillouin zone. The corresponding layer locking of the in-plane-canted spin is related to the separation in energy of the highest AFM band from the rest of the valence bands. We explain how the band interactions destroy the layer locking at the $K$, $K'$, and $\Gamma$ points. Furthermore, we compare the layer locking of the in-plane-canted spin in our AFM bilayer system with the hidden spin polarization in centrosymmetric nonmagnetic materials, pointing out the differences in segregation mechanisms and their consequences for the layer locking. We note that a similar Rashba effect with layer locking of in-plane-canted spin could occur in other van der Waals AFM bilayers with strong spin-orbit coupling and an isolated energy band., Comment: 20 pages, 4 figures in main text, 1 figure in Appendix, and 6 figures in Supporting Information

Notebooks provide an interactive environment for programmers to develop code, analyse data and inject interleaved visualizations in a single environment. Despite their flexibility, a major pitfall that data scientists encounter is unexpected behaviour caused by the unique out-of-order execution model of notebooks. As a result, data scientists face various challenges ranging from notebook correctness, reproducibility and cleaning. In this paper, we propose a framework that performs static analysis on notebooks, incorporating their unique execution semantics. Our framework is general in the sense that it accommodate for a wide range of analyses, useful for various notebook use cases. We have instantiated our framework on a diverse set of analyses and have evaluated them on 2211 real world notebooks. Our evaluation demonstrates that the vast majority (98.7%) of notebooks can be analysed in less than a second, well within the time frame required by interactive notebook clients

In the past decade, model-free reinforcement learning (RL) has provided solutions to challenging domains such as robotics. Model-based RL shows the prospect of being more sample-efficient than model-free methods in terms of agent-environment interactions, because the model enables to extrapolate to unseen situations. In the more recent past, model-based methods have shown superior results compared to model-free methods in some challenging domains with non-linear state transitions. At the same time, it has become apparent that RL is not market-ready yet and that many real-world applications are going to require model-based approaches, because model-free methods are too sample-inefficient and show poor performance in early stages of training. The latter is particularly important in industry, e.g. in production systems that directly impact a company's revenue. This demonstrates the necessity for a toolbox to push the boundaries for model-based RL. While there is a plethora of toolboxes for model-free RL, model-based RL has received little attention in terms of toolbox development. Bellman aims to fill this gap and introduces the first thoroughly designed and tested model-based RL toolbox using state-of-the-art software engineering practices. Our modular approach enables to combine a wide range of environment models with generic model-based agent classes that recover state-of-the-art algorithms. We also provide an experiment harness to compare both model-free and model-based agents in a systematic fashion w.r.t. user-defined evaluation metrics (e.g. cumulative reward). This paves the way for new research directions, e.g. investigating uncertainty-aware environment models that are not necessarily neural-network-based, or developing algorithms to solve industrially-motivated benchmarks that share characteristics with real-world problems.

Optoelectronic tweezer-driven microrobots (OETdMs) are a versatile micromanipulation technology based on the use of light induced dielectrophoresis to move small dielectric structures (microrobots) across a photoconductive substrate. The microrobots in turn can be used to exert forces on secondary objects and carry out a wide range of micromanipulation operations, including collecting, transporting and depositing microscopic cargos. In contrast to alternative (direct) micromanipulation techniques, OETdMs are relatively gentle, making them particularly well suited to interacting with sensitive objects such as biological cells. However, at present such systems are used exclusively under manual control by a human operator. This limits the capacity for simultaneous control of multiple microrobots, reducing both experimental throughput and the possibility of cooperative multi-robot operations. In this article, we describe an approach to automated targeting and path planning to enable open-loop control of multiple microrobots. We demonstrate the performance of the method in practice, using microrobots to simultaneously collect, transport and deposit silica microspheres. Using computational simulations based on real microscopic image data, we investigate the capacity of microrobots to collect target cells from within a dissociated tissue culture. Our results indicate the feasibility of using OETdMs to autonomously carry out micromanipulation tasks within complex, unstructured environments., Comment: 7 pages, 5 figures

A key feature of sequential decision making under uncertainty is a need to balance between exploiting--choosing the best action according to the current knowledge, and exploring--obtaining information about values of other actions. The multi-armed bandit problem, a classical task that captures this trade-off, served as a vehicle in machine learning for developing bandit algorithms that proved to be useful in numerous industrial applications. The active inference framework, an approach to sequential decision making recently developed in neuroscience for understanding human and animal behaviour, is distinguished by its sophisticated strategy for resolving the exploration-exploitation trade-off. This makes active inference an exciting alternative to already established bandit algorithms. Here we derive an efficient and scalable approximate active inference algorithm and compare it to two state-of-the-art bandit algorithms: Bayesian upper confidence bound and optimistic Thompson sampling. This comparison is done on two types of bandit problems: a stationary and a dynamic switching bandit. Our empirical evaluation shows that the active inference algorithm does not produce efficient long-term behaviour in stationary bandits. However, in the more challenging switching bandit problem active inference performs substantially better than the two state-of-the-art bandit algorithms. The results open exciting venues for further research in theoretical and applied machine learning, as well as lend additional credibility to active inference as a general framework for studying human and animal behaviour.

Electrical control of magnetism has great potential for low-power spintronics applications and the newly discovered two-dimensional van-der-Waals magnetic materials are promising systems for this type of applications. In fact, it has been recently shown experimentally that upon electrostatic doping by electrons bilayer CrI3 undergoes an antiferromagnetic-ferromagnetic (AFM-FM) phase transition, even in the absence of magnetic field. Doping by holes, on the other hand, does not induce the same transition in the experiment, which points to an intrinsic asymmetry in the hole and electron doping that limits the control of the transition by doping. We here show, based on first-principles calculations, that the asymmetry originates in the relativistic nature of the valence-band-edge states of the pristine bilayer, which inhibits the magnetic transition upon hole doping. Based on this finding, we propose an approach to engineer this system so that it displays the AFM-FM transition for both hole and electron doping by using moderate uniaxial strain along the soft direction of the bilayer., Comment: 27 pages, 7 figures in the main text, 2 figures in appendix and 11 figures in supplementary information The previous version has been replaced with this version to correct some typo in the Introduction section

Using density-functional theory (DFT), we investigate the selectivity of adsorption of CO2 over N2 and CH4 on planar-type B clusters, based on our previous finding of strong chemisorption of CO2 on the B10-13 planar and quasiplanar clusters. We consider the prototype B8 and B12 planar-type clusters and perform a comparative study of the adsorption of the three molecules on these clusters. We find that, at room temperature, CO2 can be separated from N2 by selective binding to the B12 cluster and not to the B8 cluster. Selective adsorption of CO2 over CH4 at room temperature is possible for both clusters. Based on our DFT-adsorption data (including also a semi-infinite Boron sheet) and the available literature-adsorption value for N2 on the planar-type B36 cluster, we discuss the selectivity trend of CO2 adsorption over N2 and CH4 with planar-cluster size, showing that it extends over sizes including B10-13 clusters and significantly larger., Comment: 4 figures, 20 pages

We study a discrete-time Markov process $X_n\in\mathbb{R}^d$, for which the distribution of the future increments depends only on the relative ranking of its components (descending order by value). We endow the process with a rich-get-richer assumption and show that, together with a finite second moments assumption, it is enough to guarantee almost sure convergence of $X_n$ / $n$. We characterize the possible limits if one is free to choose the initial state, and give a condition under which the initial state is irrelevant. Finally, we show how our framework can account for ranking-based P\'olya urns and can be used to study ranking-algorithms for web interfaces.

A recent theoretical study reported large effects of perpendicular electric fields on the atomic structure of a monolayer CrI3, which could be related to the microscopic origin of the technologically promising and experimentally observed electrical switching of magnetization in bilayer CrI3. However, those theoretical results are not in line with a previous theoretical finding of only slight changes under a strong electric field in CrI3. Given the important consequences that the presence of large structural distortions in an electric field might have, we investigated the effects of external electric fields on the CrI3 monolayer using density functional theory for a wide range of field strengths. Conclusively, we find that the structural response of CrI3 to the applied perpendicular electric field is extremely small due to a very efficient electronic screening of the electric field within the monolayer. Therefore it cannot be the origin of the observed electrical switching of magnetization in the bilayer CrI3. Furthermore, we find that the very small linear dependence of the structural changes on the electric field persists up to a field value of 0.45 V/Angstrom, while the Cr magnetic moment remains constant for the same strengths of electric field., Comment: 15 pages, 4 figures

An ab initio density functional study was performed investigating the adsorption of CO$_2$ on the neutral boron B$_{n}$ ($n = 10-13$) clusters, characterized by planar and quasiplanar ground-state atomic structures. For all four clusters, we found strong chemisorption energy of CO$_2$ reaching 1.6 eV for B$_{12}$ at the cluster edge sites with the adsorbed molecule in the plane of the cluster. A configuration with chemisorbed dissociated CO$_2$ molecule also exists for B$_{11}$ and B$_{13}$ clusters. The strong adsorption is due to the bending of the CO$_2$ molecule, which provides energetically accessible fully in-plane frontier molecular orbitals matching the edge states of the clusters. At the same time, the intrinsic dipole moment of a bent CO$_2$ molecule facilitates the transfer of excess electronic charge from the cluster edges to the molecule., Comment: 18 pages, 7 figures