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This study describes and evaluates i-GRIP, a novel movement intention estimator designed to facilitate the control of assistive devices for grasping tasks in individuals with upper limb impairments. Operating within a collaborative grasping control paradigm, the users naturally move their hand towards an object they wish to grasp and i-GRIP identifies the target of the movement and selects an appropriate grip for the assistive device to perform. In an experimental study involving 11 healthy participants, i-GRIP exhibited promising estimation performances and responsiveness. The proposed approach paves the way towards more intuitive control of grasping assistive device for individuals with upper limb impairments., Comment: International Conference on Humanoid Robots, IEEE-RAS, Nov 2024, Nancy, France

This study introduces i-GRIP, an innovative movement goal estimator designed to facilitate the control of assistive devices for grasping tasks in individuals with upperlimb impairments. The algorithm operates within a collaborative control paradigm, eliminating the need for specific user actions apart from naturally moving their hand toward a desired object. i-GRIP analyzes the hand's movement in an object-populated scene to determine its target and select an appropriate grip. In an experimental study involving 11 healthy participants, i-GRIP showed promising goal estimation performances and responsiveness and the potential to facilitate the daily use of assistive devices for individuals with upper-limb impairments in the future.

Relighting, which synthesizes a novel view under a given lighting condition (unseen in training time), is a must feature for immersive photo-realistic experience. However, real-time relighting is challenging due to high computation cost of the rendering equation which requires shape and material decomposition and visibility test to model shadow. Additionally, for indirect illumination, additional computation of rendering equation on each secondary surface point (where reflection occurs) is required rendering real-time relighting challenging. We propose a novel method that executes a CNN renderer to compute primary surface points and rendering parameters, required for direct illumination. We also present a lightweight hash grid-based renderer, for indirect illumination, which is recursively executed to perform the secondary ray tracing process. Both renderers are trained in a distillation from a pre-trained teacher model and provide real-time physically-based rendering under unseen lighting condition at a negligible loss of rendering quality., Comment: Under review

Over the past few years, robotics simulators have largely improved in efficiency and scalability, enabling them to generate years of simulated data in a few hours. Yet, efficiently and accurately computing the simulation derivatives remains an open challenge, with potentially high gains on the convergence speed of reinforcement learning and trajectory optimization algorithms, especially for problems involving physical contact interactions. This paper contributes to this objective by introducing a unified and efficient algorithmic solution for computing the analytical derivatives of robotic simulators. The approach considers both the collision and frictional stages, accounting for their intrinsic nonsmoothness and also exploiting the sparsity induced by the underlying multibody systems. These derivatives have been implemented in C++, and the code will be open-sourced in the Simple simulator. They depict state-of-the-art timings ranging from 5 microseconds for a 7-dof manipulator up to 95 microseconds for 36-dof humanoid, outperforming alternative solutions by a factor of at least 100.

The penetration of renewable energies requires additional storages to deal with intermittency. Accordingly, there is growing interest in evaluating the opportunity cost (usage value) associated with stored energy in large storages, a cost obtained by solving a multistage stochastic optimization problem. Today, to compute usage values under uncertainties, an adequacy resource problem is solved using stochastic dynamic programming assuming a hazard-decision information structure. This modelling assumes complete knowledge of the coming week uncertainties, which is not adapted to the system operation as the intermittency occurs at smaller timescale. We equip the twotimescale problem with a new information structure considering planning and recourse decisions: decision-hazard-decision. This structure is used to decompose the multistage decision-making process into a nonanticipative planning step in which the on/off decisions for the thermal units are made, and a recourse step in which the power modulation decisions are made once the uncertainties have been disclosed. In a numerical case, we illustrate how usage values are sensitive as how the disclosure of information is modelled.

Motivated by crowdsourcing, we consider a problem where we partially observe the correctness of the answers of $n$ experts on $d$ questions. In this paper, we assume that both the experts and the questions can be ordered, namely that the matrix $M$ containing the probability that expert $i$ answers correctly to question $j$ is bi-isotonic up to a permutation of it rows and columns. When $n=d$, this also encompasses the strongly stochastic transitive (SST) model from the tournament literature. Here, we focus on the relevant problem of deciphering small entries of $M$ from large entries of $M$, which is key in crowdsourcing for efficient allocation of workers to questions. More precisely, we aim at recovering a (or several) level set $p$ of the matrix up to a precision $h$, namely recovering resp. the sets of positions $(i,j)$ in $M$ such that $M_{ij}>p+h$ and $M_{i,j}

In this paper, we consider the problem of seriation of a permuted structured matrix based on noisy observations. The entries of the matrix relate to an expected quantification of interaction between two objects: the higher the value, the closer the objects. A popular structured class for modelling such matrices is the permuted Robinson class, namely the set of matrices whose coefficients are decreasing away from its diagonal, up to a permutation of its lines and columns. We consider in this paper two submodels of Robinson matrices: the T{\oe}plitz model, and the latent position model. We provide a computational lower bound based on the low-degree paradigm, which hints that there is a statistical-computational gap for seriation when measuring the error based on the Frobenius norm. We also provide a simple and polynomial-time algorithm that achives this lower bound. Along the way, we also characterize the information-theory optimal risk thereby giving evidence for the extent of the computation/information gap for this problem., Comment: 39 pages, 2 figures

In this paper we realize the supersymmetric classical $W$-algebras $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$ and $\mathcal{W}(\overline{\mathfrak{gl}}(n|n))$ as differential algebras generated by the coefficients of a monic superdifferential operator $L$. In the case of $\mathcal{W}(\overline{\mathfrak{gl}}(n|n))$ (resp. $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$) this operator is even (resp. odd). We show that the supersymmetric Poisson vertex algebra bracket on these supersymmetric W-algebras is the supersymmetric analogue of the quadratic Gelfand-Dickey bracket associated to the operator $L$. Finally, we construct integrable hierarchies of evolutionary Hamiltonian PDEs on both W-algebras. A key observation is that to construct these hierarchies on the algebra $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$ one needs to introduce a new concept of even supersymmetric Poisson vertex algebras.

The development of hydrogen infrastructures requires to reduce their costs. In this paper, we develop a multistage stochastic optimization model for the management of a hydrogen infrastructure which consists of an electrolyser, a compressor and a storage to serve a transportation demand. This infrastructure is powered by three different sources: on-site photovoltaic panels (PV), renewable energy through a power purchase agreement (PPA) and the power grid. We consider uncertainties affecting on-site photovoltaic production and hydrogen demand. Renewable energy sources are emphasized in the hydrogen production process to ensure eligibility for a subsidy, which is awarded if the proportion of nonrenewable electricity usage stays under a predetermined threshold. We solve the multistage stochastic optimization problem using a decomposition method based on Lagrange duality. The numerical results indicate that the solution to this problem, formulated as a policy, achieves a small duality gap, thus proving the effectiveness of this approach.

In this paper, we study the problem of noisy, convex, zeroth order optimisation of a function $f$ over a bounded convex set $\bar{\mathcal X}\subset \mathbb{R}^d$. Given a budget $n$ of noisy queries to the function $f$ that can be allocated sequentially and adaptively, our aim is to construct an algorithm that returns a point $\hat x\in \bar{\mathcal X}$ such that $f(\hat x)$ is as small as possible. We provide a conceptually simple method inspired by the textbook center of gravity method, but adapted to the noisy and zeroth order setting. We prove that this method is such that the $f(\hat x) - \min_{x\in \bar{\mathcal X}} f(x)$ is of smaller order than $d^2/\sqrt{n}$ up to poly-logarithmic terms. We slightly improve upon existing literature, where to the best of our knowledge the best known rate is in [Lattimore, 2024] is of order $d^{2.5}/\sqrt{n}$, albeit for a more challenging problem. Our main contribution is however conceptual, as we believe that our algorithm and its analysis bring novel ideas and are significantly simpler than existing approaches.

We investigate the Active Clustering Problem (ACP). A learner interacts with an $N$-armed stochastic bandit with $d$-dimensional subGaussian feedback. There exists a hidden partition of the arms into $K$ groups, such that arms within the same group, share the same mean vector. The learner's task is to uncover this hidden partition with the smallest budget - i.e., the least number of observation - and with a probability of error smaller than a prescribed constant $\delta$. In this paper, (i) we derive a non-asymptotic lower bound for the budget, and (ii) we introduce the computationally efficient ACB algorithm, whose budget matches the lower bound in most regimes. We improve on the performance of a uniform sampling strategy. Importantly, contrary to the batch setting, we establish that there is no computation-information gap in the active setting., Comment: 50 pages

Since 2019, GRAVITY has provided direct observations of giant planets and brown dwarfs at separations of down to 95 mas from the host star. Some of these observations have provided the first direct confirmation of companions previously detected by indirect techniques (astrometry and radial velocities). We want to improve the observing strategy and data reduction in order to lower the inner working angle of GRAVITY in dual-field on-axis mode. We also want to determine the current limitations of the instrument when observing faint companions with separations in the 30-150 mas range. To improve the inner working angle, we propose a fiber off-pointing strategy during the observations to maximize the ratio of companion-light-to-star-light coupling in the science fiber. We also tested a lower-order model for speckles to decouple the companion light from the star light. We then evaluated the detection limits of GRAVITY using planet injection and retrieval in representative archival data. We compare our results to theoretical expectations. We validate our observing and data-reduction strategy with on-sky observations; first in the context of brown dwarf follow-up on the auxiliary telescopes with HD 984 B, and second with the first confirmation of a substellar candidate around the star Gaia DR3 2728129004119806464. With synthetic companion injection, we demonstrate that the instrument can detect companions down to a contrast of $8\times 10^{-4}$ ($\Delta \mathrm{K}= 7.7$ mag) at a separation of 35 mas, and a contrast of $3\times 10^{-5}$ ($\Delta \mathrm{K}= 11$ mag) at 100 mas from a bright primary (K<6.5), for 30 min exposure time. With its inner working angle and astrometric precision, GRAVITY has a unique reach in direct observation parameter space. This study demonstrates the promising synergies between GRAVITY and Gaia for the confirmation and characterization of substellar companions., Comment: 16 pages, 14 figures. Submitted to A&A

Several protocols are proposed to build continuous energy surfaces of many-body quantum systems, regarding both energy and states. The standard variational principle is augmented with constraints on state overlap, ensuring arbitrary precision on continuity. As an illustration, the lowest energy and excited state paths relevant for the $^{240}$Pu asymmetric fission are studied. The scission is clearly signed, with a neutron excess in the neck, the ultimate glue before its rupture. Our approach can potentially connect any couple of Hilbert space states, which opens up new horizons for various applications., Comment: 6 pages, 2 figures

Whether rigid or compliant, contact interactions are inherent to robot motions, enabling them to move or manipulate things. Contact interactions result from complex physical phenomena, that can be mathematically cast as Nonlinear Complementarity Problems (NCPs) in the context of rigid or compliant point contact interactions. Such a class of complementarity problems is, in general, difficult to solve both from an optimization and numerical perspective. Over the past decades, dedicated and specialized contact solvers, implemented in modern robotics simulators (e.g., Bullet, Drake, MuJoCo, DART, Raisim) have emerged. Yet, most of these solvers tend either to solve a relaxed formulation of the original contact problems (at the price of physical inconsistencies) or to scale poorly with the problem dimension or its numerical conditioning (e.g., a robotic hand manipulating a paper sheet). In this paper, we introduce a unified and efficient approach to solving NCPs in the context of contact simulation. It relies on a sound combination of the Alternating Direction Method of Multipliers (ADMM) and proximal algorithms to account for both compliant and rigid contact interfaces in a unified way. To handle ill-conditioned problems and accelerate the convergence rate, we also propose an efficient update strategy to adapt the ADMM hyperparameters automatically. By leveraging proximal methods, we also propose new algorithmic solutions to efficiently evaluate the inverse dynamics involving rigid and compliant contact interactions, extending the approach developed in MuJoCo. We validate the efficiency and robustness of our contact solver against several alternative contact methods of the literature and benchmark them on various robotics and granular mechanics scenarios. Our code is made open-source at https://github.com/Simple-Robotics/Simple.

Recent strides in nonlinear model predictive control (NMPC) underscore a dependence on numerical advancements to efficiently and accurately solve large-scale problems. Given the substantial number of variables characterizing typical whole-body optimal control (OC) problems - often numbering in the thousands - exploiting the sparse structure of the numerical problem becomes crucial to meet computational demands, typically in the range of a few milliseconds. Addressing the linear-quadratic regulator (LQR) problem is a fundamental building block for computing Newton or Sequential Quadratic Programming (SQP) steps in direct optimal control methods. This paper concentrates on equality-constrained problems featuring implicit system dynamics and dual regularization, a characteristic of advanced interiorpoint or augmented Lagrangian solvers. Here, we introduce a parallel algorithm for solving an LQR problem with dual regularization. Leveraging a rewriting of the LQR recursion through block elimination, we first enhanced the efficiency of the serial algorithm and then subsequently generalized it to handle parametric problems. This extension enables us to split decision variables and solve multiple subproblems concurrently. Our algorithm is implemented in our nonlinear numerical optimal control library ALIGATOR. It showcases improved performance over previous serial formulations and we validate its efficacy by deploying it in the model predictive control of a real quadruped robot., Comment: new version after title change, camera-ready version sent to R:SS 2024

Oscillations of conductance observed in strong magnetic fields are a striking manifestation of the quantum dynamics of charge carriers in solids. The large charge carrier density in typical metals sets the scale of oscillations in both electrical and thermal conductivity, which characterize the Fermi surface. In semimetals, thermal transport at low-charge carrier density is expected to be phonon dominated, yet several experiments observe giant quantum oscillations in thermal transport. This raises the question of whether there is an overarching mechanism leading to sizable oscillations that survives in phonon-dominated semimetals. In this work, we show that such a mechanism exists. It relies on the peculiar phase-space allowed for phonon scattering by electrons when only a few Landau levels are filled. Our measurements on the Dirac semimetal ZrTe5 support this counter-intuitive mechanism through observation of pronounced thermal quantum oscillations, since they occur in similar magnitude and phase in directions parallel and transverse to the magnetic field. Our phase-space argument applies to all low-density semimetals, topological or not, including graphene and bismuth. Our work illustrates that phonon absorption can be leveraged to reveal degrees of freedom through their imprint on longitudinal thermal transport.

The Casimir effect is a macroscopic evidence of the quantum nature of the vacuum. On a ring, it leads to a finite size correction to the vacuum energy. In this work, we show that this vacuum's energy and pressure acquire additional, sizable corrections, when the ring's radius is increased fast enough, an experimentally accessible model of an expanding universe. This effect is distinct from the dynamical Casimir effect: it is a manifestation of the conformal anomaly, originating from the spacetime curvature induced by the increase of the ring's radius. This anomalous dynamical Casimir effect is measurable through the work necessary to increase the ring size, which becomes non-monotonous in time., Comment: 8 pages, 4 figures

This paper builds upon our recent work, published in Lett. Math. Phys., 112: 94, 2022, where we established that the integrable Volterra lattice on a free associative algebra and the whole hierarchy of its symmetries admits a quantisation dependent on a parameter $\omega$. We also uncovered an intriguing aspect: all odd-degree symmetries of the hierarchy admits an alternative, non-deformation quantisation, resulting in a non-commutative algebra for any choice of the quantisation parameter $\omega$. In this study, we demonstrate that each equation within the quantum Volterra hierarchy can be expressed in the Heisenberg form. We provide explicit expressions for all quantum Hamiltonians and establish their commutativity. In the classical limit, these quantum Hamiltonians yield explicit expressions for the classical ones of the commutative Volterra hierarchy. Furthermore, we present Heisenberg equations and their Hamiltonians in the case of non-deformation quantisation. Finally, we discuss commuting first integrals, central elements of the quantum algebra, and the integrability problem for periodic reductions of the Volterra lattice in the context of both quantisations.

Trajectory Optimization (TO) and Reinforcement Learning (RL) are powerful and complementary tools to solve optimal control problems. On the one hand, TO can efficiently compute locally-optimal solutions, but it tends to get stuck in local minima if the problem is not convex. On the other hand, RL is typically less sensitive to non-convexity, but it requires a much higher computational effort. Recently, we have proposed CACTO (Continuous Actor-Critic with Trajectory Optimization), an algorithm that uses TO to guide the exploration of an actor-critic RL algorithm. In turns, the policy encoded by the actor is used to warm-start TO, closing the loop between TO and RL. In this work, we present an extension of CACTO exploiting the idea of Sobolev learning. To make the training of the critic network faster and more data efficient, we enrich it with the gradient of the Value function, computed via a backward pass of the differential dynamic programming algorithm. Our results show that the new algorithm is more efficient than the original CACTO, reducing the number of TO episodes by a factor ranging from 3 to 10, and consequently the computation time. Moreover, we show that CACTO-SL helps TO to find better minima and to produce more consistent results.

In this paper, we introduce a class of super Adler-type operators associated with the Lie superalgebra $\mathfrak{gl}(m|n)$. We show that these operators generate Poisson vertex superalgebras which are isomorphic to the classical $\mathcal{W}$-superalgebras associated with $\mathfrak{gl}(m|n)$ and some rectangular nilpotent elements. We use this isomorphism to construct integrable hierarchies on these rectangular $\mathcal{W}$-superalgebras., Comment: 33pages

Decentralized finance (DeFi) has been the target of numerous profit-driven crimes, but the prevalence and cumulative impact of these crimes have not yet been assessed. This study provides a comprehensive assessment of profit-driven crimes targeting the DeFi sector. We collected data on 1153 crime events from 2017 to 2022. Of these, 1,048 were related to DeFi (the main focus of this study) and 105 to centralized finance (CeFi). The findings show that the entire cryptoasset industry has suffered a minimum loss of US$30B, with two thirds related to CeFi and one third to DeFi. Focusing on DeFi, a taxonomy was developed to clarify the similarities and differences among these crimes. All events were mapped onto the DeFi stack to assess the impacted technical layers, and the financial damages were quantified to gauge their scale. The results highlight that during an attack, a DeFi actor (an entity developing a DeFi technology) can serve as a direct target (due to technical vulnerabilities or exploitation of human risks), as a perpetrator (through malicious uses of contracts or market manipulations), or as an intermediary (by being imitated through, for example, phishing scams). The findings also show that DeFi actors are the first victims of crimes targeting the DeFi industry: 52.2% of events targeted them, primarily due to technical vulnerabilities at the protocol layer, and these events accounted for 83% of all financial damages. Alternatively, in 40.7% of events, DeFi actors were themselves malicious perpetrators, predominantly misusing contracts at the cryptoasset layer (e.g., rug pull scams). However, these events accounted for only 17% of all financial damages. The study offers a preliminary assessment of the size and scope of crime events within the DeFi sector and highlights the vulnerable position of DeFi actors in the ecosystem.

We consider a ranking problem where we have noisy observations from a matrix with isotonic columns whose rows have been permuted by some permutation $\pi$ *. This encompasses many models, including crowd-labeling and ranking in tournaments by pair-wise comparisons. In this work, we provide an optimal and polynomial-time procedure for recovering $\pi$ * , settling an open problem in [7]. As a byproduct, our procedure is used to improve the state-of-the art for ranking problems in the stochastically transitive model (SST). Our approach is based on iterative pairwise comparisons by suitable data-driven weighted means of the columns. These weights are built using a combination of spectral methods with new dimension-reduction techniques. In order to deal with the important case of missing data, we establish a new concentration inequality for sparse and centered rectangular Wishart-type matrices.

We derive both Azuma-Hoeffding and Burkholder-type inequalities for partial sums over a rectangular grid of dimension $d$ of a random field satisfying a weak dependency assumption of projective type: the difference between the expectation of an element of the random field and its conditional expectation given the rest of the field at a distance more than $\delta$ is bounded, in $L^p$ distance, by a known decreasing function of $\delta$. The analysis is based on the combination of a multi-scale approximation of random sums by martingale difference sequences, and of a careful decomposition of the domain. The obtained results extend previously known bounds under comparable hypotheses, and do not use the assumption of commuting filtrations., Comment: 20 pages, 3 figures

Assistive devices for indivuals with upper-limb movement often lack controllability and intuitiveness, in particular for grasping function. In this work, we introduce a novel user interface for grasping movement control in which the user delegates the grasping task decisions to the device, only moving their (potentially prosthetic) hand toward the targeted object.

Sequential learning with feedback graphs is a natural extension of the multi-armed bandit problem where the problem is equipped with an underlying graph structure that provides additional information - playing an action reveals the losses of all the neighbors of the action. This problem was introduced by \citet{mannor2011} and received considerable attention in recent years. It is generally stated in the literature that the minimax regret rate for this problem is of order $\sqrt{\alpha T}$, where $\alpha$ is the independence number of the graph, and $T$ is the time horizon. However, this is proven only when the number of rounds $T$ is larger than $\alpha^3$, which poses a significant restriction for the usability of this result in large graphs. In this paper, we define a new quantity $R^*$, called the \emph{problem complexity}, and prove that the minimax regret is proportional to $R^*$ for any graph and time horizon $T$. Introducing an intricate exploration strategy, we define the \mainAlgorithm algorithm that achieves the minimax optimal regret bound and becomes the first provably optimal algorithm for this setting, even if $T$ is smaller than $\alpha^3$.

We consider the problem of ranking n experts based on their performances on d tasks. We make a monotonicity assumption stating that for each pair of experts, one outperforms the other on all tasks. We consider the sequential setting where in each round, the learner has access to noisy evaluations of actively chosen pair of expert-task, given the information available up to the actual round. Given a confidence parameter $\delta$ $\in$ (0, 1), we provide strategies allowing to recover the correct ranking of experts and develop a bound on the total number of queries made by our algorithm that hold with probability at least 1 -- $\delta$. We show that our strategy is adaptive to the complexity of the problem (our bounds are instance dependent), and develop matching lower bounds up to a poly-logarithmic factor. Finally, we adapt our strategy to the relaxed problem of best expert identification and provide numerical simulation consistent with our theoretical results.

In robotics, designing robust algorithms in the face of estimation uncertainty is a challenging task. Indeed, controllers often do not consider the estimation uncertainty and only rely on the most likely estimated state. Consequently, sudden changes in the environment or the robot's dynamics can lead to catastrophic behaviors. In this work, we present a risk-sensitive Extended Kalman Filter that allows doing output-feedback Model Predictive Control (MPC) safely. This filter adapts its estimation to the control objective. By taking a pessimistic estimate concerning the value function resulting from the MPC controller, the filter provides increased robustness to the controller in phases of uncertainty as compared to a standard Extended Kalman Filter (EKF). Moreover, the filter has the same complexity as an EKF, so that it can be used for real-time model-predictive control. The paper evaluates the risk-sensitive behavior of the proposed filter when used in a nonlinear model-predictive control loop on a planar drone and industrial manipulator in simulation, as well as on an external force estimation task on a real quadruped robot. These experiments demonstrate the abilities of the approach to improve performance in the face of uncertainties significantly.

Physics simulation is ubiquitous in robotics. Whether in model-based approaches (e.g., trajectory optimization), or model-free algorithms (e.g., reinforcement learning), physics simulators are a central component of modern control pipelines in robotics. Over the past decades, several robotic simulators have been developed, each with dedicated contact modeling assumptions and algorithmic solutions. In this article, we survey the main contact models and the associated numerical methods commonly used in robotics for simulating advanced robot motions involving contact interactions. In particular, we recall the physical laws underlying contacts and friction (i.e., Signorini condition, Coulomb's law, and the maximum dissipation principle), and how they are transcribed in current simulators. For each physics engine, we expose their inherent physical relaxations along with their limitations due to the numerical techniques employed. Based on our study, we propose theoretically grounded quantitative criteria on which we build benchmarks assessing both the physical and computational aspects of simulation. We support our work with an open-source and efficient C++ implementation of the existing algorithmic variations. Our results demonstrate that some approximations or algorithms commonly used in robotics can severely widen the reality gap and impact target applications. We hope this work will help motivate the development of new contact models, contact solvers, and robotic simulators in general, at the root of recent progress in motion generation in robotics.

Topological materials can host edge and corner states that are protected from disorder and material imperfections. In particular, the topological edge states of mechanical structures present unmatched opportunities for achieving robust responses in wave guiding, sensing, computation, and filtering. However, determining whether a mechanical structure is topologically nontrivial and features topologically-protected modes has hitherto relied on theoretical models. This strong requirement has limited the experimental and practical significance of topological mechanics to laboratory demonstrations. Here, we introduce and validate an experimental method to detect the topologically protected zero modes of mechanical structures without resorting to any modeling step. Our practical method is based on a simple electrostatic analogy: topological zero modes are akin to electric charges. To detect them, we identify elementary mechanical molecules and measure their chiral polarization, a recently introduced marker of topology in chiral phases. Topological zero modes are then identified as singularities of the polarization field. Our method readily applies to any mechanical structure and effectively detects the edge and corner states of regular and higher-order topological insulators. Our findings extend the reach of chiral topological phases beyond designer materials, and allow their direct experimental investigation.

In energy management, it is common that strategic investment decisions (storage capacity, production units) are made at a slow time scale, whereas operational decisions (storage, production) are made at a fast time scale: for such problems, the total number of decision stages may be huge. In this paper, we consider multistage stochastic optimization problems with two time-scales, and we propose a time block decomposition scheme to address them numerically. More precisely, our approach relies on two assumptions. On the one hand, we suppose slow time scale stagewise independence of the noise process: the random variables that occur during a slow time scale interval are independent of those at another slow time scale interval. This makes it possible to use Dynamic Programming at the slow time scale. On the other hand, we suppose a dynamically monotone property for the problem under consideration, which makes it possible to obtain bounds. Then, we present two algorithmic methods to compute upper and lower bounds for slow time scale Bellman value functions. Both methods rely respectively on primal and dual decomposition of the Bellman equation applied at the slow time scale. We assess the methods tractability and validate their efficiency by solving a battery management problem where the fast time scale operational decisions have an impact on the storage current capacity, hence on the strategic decisions to renew the battery at the slow time scale.

Markov Decision Processes (Mdps) form a versatile framework used to model a wide range of optimization problems. The Mdp model consists of sets of states, actions, time steps, rewards, and probability transitions. When in a given state and at a given time, the decision maker's action generates a reward and determines the state at the next time step according to the probability transition function. However, Mdps assume that the decision maker knows the state of the controlled dynamical system. Hence, when one needs to optimize controlled dynamical systems under partial observation, one often turns toward the formalism of Partially Observed Markov Decision Processes (Pomdp). Pomdps are often untractable in the general case as Dynamic Programming suffers from the curse of dimensionality. Instead of focusing on the general Pomdps, we present a subclass where transitions and observations mappings are deterministic: Deterministic Partially Observed Markov Decision Processes (Det-Pomdp). That subclass of problems has been studied by (Littman, 1996) and (Bonet, 2009). It was first considered as a limit case of Pomdps by Littman, mainly used to illustrate the complexity of Pomdps when considering as few sources of uncertainties as possible. In this paper, we improve on Littman's complexity bounds. We then introduce and study an even simpler class: Separated Det-Pomdps and give some new complexity bounds for this class. This new class of problems uses a property of the dynamics and observation to push back the curse of dimensionality.