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International audience; We present algorithmic, complexity, and implementation results on the problem of sampling points from a spectrahedron, that is, the feasible region of a semidefinite program.Our main tool is geometric random walks. We analyze the arithmetic and bit complexity of certain primitive geometric operations that are based on the algebraic properties of spectrahedra and the polynomial eigenvalue problem. This study leads to the implementation of a broad collection of random walks for sampling from spectrahedra that experimentally show faster mixing times than methods currently employed either in theoretical studies or in applications, including the popular family of Hit-and-Run walks. The different random walks offer a variety of advantages, thus allowing us to efficiently sample from general probability distributions, for example the family of log-concave distributions which arise in numerous applications. We focus on two major applications of independent interest: (i) approximate the volume of a spectrahedron, and (ii) compute the expectation of functions coming from robust optimal control.We exploit efficient linear algebra algorithms and implementations to address the aforementioned computations in very high dimension. In particular, we provide a C++ open source implementation of our methods that scales efficiently, for the first time, up to dimension 200. We illustrate its efficiency on various data sets.

25 pages, 1 figure; In this paper, we investigate the structure of the saturation of ideals generated by square systems of sparse homogeneous polynomials over a toric variety $X$ with respect to the irrelevant ideal of $X$. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on $X$. In particular, we prove that toric Sylvester forms yield bases of some graded components of $I^{sat}/I$, where $I$ denotes an ideal generated by $n + 1$ generic forms, $n$ is the dimension of $X$ and $I$ sat the saturation of $I$ with respect to the irrelevant ideal of the Cox ring of $X$. Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we prove a new formula for computing toric residues of the product of two forms.

International audience; The Canny-Emiris formula gives the sparse resultant as a ratio between the determinant of a Sylvester-type matrix and a minor of it, by a subdivision algorithm. The most complete proof of the formula was given by D'Andrea et al. in [9] under general conditions on the underlying mixed subdivision. Before the proof, Canny and Pedersen had proposed a greedy algorithm which provides smaller matrices, in general. The goal of this paper is to give an explicit class of mixed subdivisions for the greedy approach such that the formula holds, and the dimensions of the matrices are reduced compared to the subdivision algorithm. We measure this reduction for the case when the Newton polytopes are zonotopes generated by n line segments (where n is the rank of the underlying lattice), and for the case of multihomogeneous systems. This article comes with a JULIA implementation of the treated cases.

International audience; We rely on aggregate separation bounds for univariate polynomials to introduce novel worst-case separation bounds for the isolated roots of zero-dimensional, positive-dimensional, and overde- termined polynomial systems. We exploit the structure of the given system, as well as bounds on the height of the sparse (or toric) resultant, by means of mixed volume, thus establishing adaptive bounds. Our bounds improve upon Canny’s Gap theorem [9]. Moreover, they exploit sparseness and they apply without any assumptions on the input polynomial system. To evaluate the quality of the bounds, we present polynomial systems whose root separation is asymptotically not far from our bounds.We apply our bounds to three problems. First, we use them to estimate the bitsize of the eigenvalues and eigenvectors of an integer matrix; thus we provide a new proof that the problem has polynomial bit complexity. Second, we bound the value of a positive polynomial over the simplex: we improve by at least one order of magnitude upon all existing bounds. Finally, we asymptotically bound the number of steps of any purely subdivision-based algorithm that isolates all real roots of a polynomial system.

International audience; Rigid graph theory is an active area with many open problems, especially regarding embeddings in R^d or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number of embeddings to that of solutions of a well-constrained algebraic system and exploit progress in the latter domain. In particular, the system's complex solutions naturally extend the notion of real embeddings, thus allowing us to employ bounds on complex roots. We focus on multihomogeneous Bézout (m-Bézout) bounds of algebraic systems since they are fast to compute and rather tight for systems exhibiting structure as in our case. We introduce two methods to relate such bounds to combinatorial properties of minimally rigid graphs in C^d and S^d. The first relates the number of graph orientations to the m-Bézout bound, while the second leverages a matrix permanent formulation. Using these approaches we improve the best known asymptotic upper bounds for planar graphs in dimension 3, and all minimally rigid graphs in dimension d ≥ 5, both in the Euclidean and spherical case. Our computations indicate that m-Bézout bounds are tight for embeddings of planar graphs in S^2 and C36. We exploit Bernstein's second theorem on the exactness of mixed volume, and relate it to the m-Bézout bound by analyzing the associated Newton polytopes. We reduce the number of checks required to verify exactness by an exponential factor, and conjecture further that it suffices to check a linear instead of an exponential number of cases overall.

By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system. Naturally, the complex solutions of such systems extend the notion of rigidity to $\mathbb{C}^d$. A major open problem has been to obtain tight upper bounds on the number of embeddings in $\mathbb{C}^d$, for a given number $|V|$ of vertices, which obviously also bound their number in $\mathbb{R}^d$. Moreover, in most known cases, the maximal numbers of embeddings in $\mathbb{C}^d$ and $\mathbb{R}^d$ coincide. For decades, only the trivial bound of $O(2^{d\cdot |V|})$ was known on the number of embeddings.Recently, matrix permanent bounds have led to a small improvement for $d\geq 5$. This work improves upon the existing upper bounds for the number of embeddings in $\mathbb{R}^d$ and $S^d$, by exploiting outdegree-constrained orientations on a graphical construction, where the proof iteratively eliminates vertices or vertex paths. For the most important cases of $d=2$ and $d=3$, the new bounds are $O(3.7764^{|V|})$ and $O(6.8399^{|V|})$, respectively. In general, the recent asymptotic bound mentioned above is improved by a factor of $1/ \sqrt{2}$. Besides being the first substantial improvement upon a long-standing upper bound, our method is essentially the first general approach relying on combinatorial arguments rather than algebraic root counts.

We discuss and extend a powerful, geometric framework to represent the set of portfolios, which identifies the space of asset allocations with the points lying in a convex polytope. Based on this viewpoint, we survey certain state-of-the-art tools from geometric and statistical computing to handle important and difficult problems in digital finance. Although our tools are quite general, in this paper, we focus on two specific questions. The first concerns crisis detection, which is of prime interest for the public in general and for policy makers in particular because of the significant impact that crises have on the economy. Certain features in stock markets lead to this type of anomaly detection: Given the assets' returns, we describe the relationship between portfolios' return and volatility by means of a copula, without making any assumption on investors' strategies. We examine a recent method relying on copulae to construct an appropriate indicator that allows us to automate crisis detection. On real data the indicator detects all past crashes in the cryptocurrency market and from the DJ600-Europe index, from 1990 to 2008, the indicator identifies correctly 4 crises and issues one false positive for which we offer an explanation. Our second contribution is to introduce an original computational framework to model asset allocation strategies, which is of independent interest for digital finance and its applications. Our approach addresses the crucial question of evaluating portfolio management, and is relevant the individual managers as well as financial institutions. To evaluate portfolio performance, we provide a new portfolio score, based on the aforementioned framework and concepts. In particular, it relies on statistical properties of portfolios, and we show how they can be computed efficiently.

We focus on deep learning algorithms, improving upon the weather research and forecasting (WRF) model, and we show that the combination of these methods produces day-ahead wind speed predictions of high accuracy, with no need for previous-day measurements. We also show that previous-day data offer a significant enhancement in a short-term neural network for hour-ahead predictions, assuming that they are available on a daily basis. Our main contribution is the design and testing of original neural networks that capture both spatial and temporal characteristics of the wind, by combining convolutional (CNN) as well as recurrent (RNN) neural networks. The input predictions are obtained by a WRF model that we appropriately parameterize; we also specify a grid adapted to each park so as to capture its topography. Training uses historical data from five wind farms in Greece, and the 5-month testing period includes winter months, which exhibit the highest wind speed values. Our models improve WRF accuracy on average by 19.4%, and the improvement occurs in every month; expectedly, the improvement is lowest for the park where WRF performs best. Our neural network is competitive to state-of-the-art models, achieving an average MAE of 1.75 m/s. Accuracy improves for speed values up to 20 m/s, which are important in wind energy prediction. We also develop an RNN model and show that MAE reduces to less than 1 m/s for short-term predictions if actual data is employed.

International audience; We offer a closed form bound on the m-Bézout bound for multi-homogeneous systems whose equations include two variable subsets of the same degree. Our bound is expectedly not tight, since computation of the m-Bézout number is P-hard by reduction to the permanent. On the upside, our bound is tighter than the existing closed-form bound derived from the permanent, which applies only to systems characterized by further structure. Our work is inspired by the application of the m-Bézout bound to counting Euclidean embeddings of distance graphs. Distance geometry and rigidity theory study graphs with a finite number of configurations, up to rigid transformations, which are prescribed by the edge lengths. Counting embeddings is an algebraic question once one constructs a system whose solutions correspond to the different embeddings. Surprisingly, the best asymptotic bound on the number of embeddings had for decades been Bézout's, applied to the obvious system of quadratic equations expressing the length constraints. This is essentially , for graphs of n vertices in d dimensions, and implies a bound of for the most famous case of Laman graphs in the plane. However, the best lower bound is about , which follows by numerically solving appropriate instances. In [3], the authors leverage the m-Bézout bound and express it by the number of certain constrained orientations of simple graphs. A combinatorial process on these graphs has recently improved the bound on orientations and, therefore, has improved the bounds on the number of distance graph embeddings [4]. For Laman graphs the new bound is inferior to thus improving upon Bézout's bound for the first time. In this paper, we obtain a closed-form bound on the m-Bézout number of a class of multi-homogeneous systems that subsumes the systems encountered in distance graph embeddings.

International audience; We exploit a recent computational framework to model and detect financial crises in stock markets, as well as shock events in cryptocurrency markets, which are characterized by a sudden or severe drop in prices. Our method manages to detect all past crises in the French industrial stock market starting with the crash of 1929, including financial crises after 1990 (e.g. dot-com bubble burst of 2000, stock market downturn of 2002), and all past crashes in the cryptocurrency market, namely in 2018, and also in 2020 due to covid-19. We leverage copulae clustering, based on the distance between probability distributions, in order to validate the reliability of the framework; we show that clusters contain copulae from similar market states such as normal states, or crises. Moreover, we propose a novel regression model that can detect successfully all past events using less than 10% of the information that the previous framework requires. We train our model by historical data on the industry assets, and we are able to detect all past shock events in the cryptocurrency market. Our tools provide the essential components of our software framework that offers fast and reliable detection, or even prediction, of shock events in stock and cryptocurrency markets of hundreds of assets.

We present algorithmic, complexity, and implementation results on the problem of sampling points from a spectrahedron, that is the feasible region of a semidefinite program. Our main tool is geometric random walks. We analyze the arithmetic and bit complexity of certain primitive geometric operations that are based on the algebraic properties of spectrahedra and the polynomial eigenvalue problem. This study leads to the implementation of a broad collection of random walks for sampling from spectrahedra that experimentally show faster mixing times than methods currently employed either in theoretical studies or in applications, including the popular family of Hit-and-Run walks. The different random walks offer a variety of advantages , thus allowing us to efficiently sample from general probability distributions, for example the family of log-concave distributions which arise in numerous applications. We focus on two major applications of independent interest: (i) approximate the volume of a spectrahedron, and (ii) compute the expectation of functions coming from robust optimal control. We exploit efficient linear algebra algorithms and implementations to address the aforemen-tioned computations in very high dimension. In particular, we provide a C++ open source implementation of our methods that scales efficiently, for the first time, up to dimension 200. We illustrate its efficiency on various data sets.

International audience; Approximate Nearest Neighbor (ANN) search is a fundamental computational problem that has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets, whereas complex shapes have not been sufficiently addressed. Here, we focus on distance functions between discretized curves in Euclidean space: They appear in a wide range of applications, from road segments and molecular backbones to time-series in general dimension. For ℓp-products of Euclidean metrics, for any constant p, we propose simple and efficient data structures for ANN based on randomized projections: These data structures are of independent interest. Furthermore, they serve to solve proximity questions under a notion of distance between discretized curves, which generalizes both discrete Fréchet and Dynamic Time Warping distance functions. These are two very popular and practical approaches to comparing such curves. We offer, for both approaches, the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our methods; our algorithm is especially efficient when the length of the curves is bounded. Finally, we focus on discrete Fréchet distance when the ambient space is high dimensional and derive complexity bounds in terms of doubling dimension as well as an improved approximate near neighbor search.

We study the problem of estimating the volume of convex polytopes, focusing on zonotopes. Although a lot of effort is devoted to practical algorithms for polytopes given as an intersection of halfspaces, there is no such method for zonotopes. Our algorithm is based on Multiphase Monte Carlo (MMC) methods, and our main contributions include: (i) a new uniform sampler employing Billiard Walk for the first time in volume computation, (ii) a new simulated annealing generalizing existing MMC by making use of adaptive convex bodies which fit to the input, thus drastically reducing the number of phases. Extensive experiments on zonotopes show our algorithm requires sub-linear number of oracle calls in the dimension, while the best theoretical bound is cubic. Moreover, our algorithm can be easily generalized to any convex body. We offer an open-source, optimized C++ implementation, and analyze its performance. Our code tackles problems intractable so far, offering the first efficient algorithm for zonotopes which scales to high dimensions (e.g. one hundred dimensions in less than 1 h).

In the perspective of manipulating geometric objects, there exists two main representations of curves and surfaces: parametric and implicit representations. Both are useful for different purposes and thus complement each other. Parametric representations are efficient in sampling points on an object; implicit representations are efficient in determining whether a point belongs to an object or not. Because of that, having both representations of the same objects at the same time maximizes the range of operations one can do with geometric objects. Switching from one representation to another is not an easy task. It usually requires the use of algebraic properties. Thus, there is a strong link between algebra and geometry, symbolised by the algebraic varieties: they are geometric objects described by an algebraic structure.This thesis explores new kinds of implicit representations and algorithms for computing implicit representations. We show that different methods are adapted to different situations even when it comes to the choice of an implicit representation amongst several possibilities. Space curves can thus be described implicitly by conical surfaces, moving lines and/or moving quadrics... each description having different geometrical properties and practical usage. As there is not one implicit representation or implicitization algorithm that would be the best in any situation, we develop methods that fit to different kinds of informations known about the object we want to represent. As we show, objects constructed by sweeping a rigid body can be represented using the knowledge of that nature. Similarly, very particular curves may have a complicated algebraic structure. Depending on our tolerance to approximation, such curves can thus be perturbed to simplify greatly their algebraic structure or, on the contrary, be represented by a rich implicit representation format.; Dans l'optique de manipuler des objets géométriques, il existe deux méthodes principales de représentation de courbes et de surfaces : les paramétriques et les implicites. Chacune de ces méthode est utile pour différents objectifs et sont donc complémentaires. Les représentations paramétriques sont aptes à générer des points de l'objet ; les représentations implicites sont aptes à déterminer si un point est sur un objet ou non. De ce fait, il est utile d'avoir accès à ces deux types de représentations et ainsi maximiser les possibilités d'effectuer divers types d'opérations sur les objets géométriques. Passer d'une méthode de représentation à une autre n'est pas une tâche aisée. Cela requiert généralement l'utilisation de méthodes algébriques. Il y a donc un lien fort entre l'algèbre et la géométrie qui est symbolisé par la notion de variété algébrique : il s'agit d'objets géométriques décrites par une structure algébrique.Cette thèse explore de nouveaux types de représentations implicites et d'algorithmes produisant des représentations implicites. Nous montrons que différentes méthodes sont adaptées à différentes situations même pour ce qui est du choix de la représentation implicite à choisir parmi plusieurs possibilités. Les courbes dans l'espace tri-dimensionnel peuvent ainsi être décrites implicitement par des surfaces coniques, par des surfaces mobiles de différents degrés... chaque type de description ayant des propriétés géométriques et des usages pratiques différents. Comme il n'y a pas d'unique représentation implicite ou d'unique algorithme d'implicitisation qui serait le meilleur dans toutes les situations, nous développons aussi des méthodes qui s'adaptent à différents types d'informations connues à propos de l'objet que nous souhaitons représenter. Ainsi que nous le montrons, les objets construits en tant que rémanence d'une structure rigide au cours d'un mouvement continu peuvent être représentés en employant cette information sur la nature de leur construction. Similairement, certaines courbes très particulières possèdent une structure algébrique compliquée. Selon notre tolérance à l'approximation, de telles courbes peuvent être légèrement modifiées pour simplifier grandement leur structure algébrique ou, au contraire, être représentées par une forme riche de représentation implicite.

International audience; We examine the capability of near-spherical-shaped particles to reproduce the triple-wavelength particle linear depolarization ratio (PLDR) and lidar ratio (LR) values measured over Europe for stratospheric smoke originating from Canadian wildfires. The smoke layers were detected both in the troposphere and the stratosphere, though in the latter case the particles presented PLDR values of almost 18 % at 532 nm as well as a strong spectral dependence from the UV to the near-IR wavelength. Although recent simulation studies of rather complicated smoke particle morphologies have shown that heavily coated smoke aggregates can produce large PLDR, herein we propose a much simpler model of compact near-spherical smoke particles. This assumption allows for the reproduction of the observed intensive optical properties of stratospheric smoke, as well as their spectral dependence. We further examine whether an extension of the current Aerosol Robotic Network (AERONET) scattering model to include the near-spherical shapes could be of benefit to the AERONET retrieval for stratospheric smoke cases associated with enhanced PLDR. Results of our study illustrate the fact that triple-wavelength PLDR and LR lidar measurements can provide us with additional insight when it comes to particle characterization.; Nous examinons la capacité des particules de forme quasi-sphérique à reproduire le PLDR (Particle Linear Depolarization Ratio) mesurées en Europe pour la fumée stratosphérique provenant des incendies de forêt au Canada. Des couches de fumée ont été détectées à la fois dans la troposphère et la stratosphère, bien que dans ce dernier cas, les particules présentaient des valeurs PLDR de près de 18% à 532 nm ainsi qu'une forte dépendance spectrale de l'UV à proche infrarouge. L'hypothèse selon laquelle les particules de fumée ont une forme presque sphérique permet la reproduction du PLDR et du rapport Lidar (LR) observés, alors que 20% n'était pas possible lors de l'utilisation de formes plus compliquées. Les résultats présentés ici sont étayés par des découvertes récentes dans la littérature, montrant que jusqu'à présent la forme quasi sphérique (ou des formes étroitement similaires) est la seule morphologie trouvée capable de reproduire les propriétés optiques intensives observées de la fumée stratosphérique, ainsi que leur dépendance spectrale.

We consider the approximation of two NP-hard problems: Minkowski decomposition (MinkDecomp) of lattice polygons in the plane and the closely related problem of multidimensional subset sum (kD-SS) in arbitrary dimension. In kD-SS, a multiset S of k-dimensional vectors is given, along with a target vector t, and one must decide whether there exists a subset of S that sums up to t. We prove, through a gap-preserving reduction from Set Cover that, for general dimension k, the corresponding optimization problem kD-SS-opt is not in APX, although the classic 1D-SS-opt has a PTAS. Our approach relates kD-SS with the well studied closest vector problem. On the positive side, we present a \(O(n^3/\epsilon ^2)\) approximation algorithm for 2D-SS-opt, where n is the cardinality of the multiset and \(\epsilon >0\) bounds the additive error in terms of some property of the input. We state two variations of this algorithm, which are more suitable for implementation. Employing a reduction of the optimization version of MinkDecomp to 2D-SS-opt we approximate the former: For an input polygon Q and parameter \(\epsilon >0\), we compute summand polygons A and B, where \(Q' = A+B\) is such that some geometric function differs on Q and \(Q'\) by \(O(\epsilon \, D)\), where D is the diameter of Q, or the Hausdorff distance between Q and \(Q'\) is also in \(O(\epsilon \, D)\). We conclude with experimental results based on our implementations.

International audience; Randomized dimensionality reduction has been recognized as one of the fundamental techniques in handling high-dimensional data. Starting with the celebrated Johnson-Lindenstrauss Lemma, such reductions have been studied in depth for the Euclidean (L2) metric, but much less for the Manhattan (L1) metric. Our primary motivation is the approximate nearest neighbor problem in L1. We exploit its reduction to the decision-with-witness version, called approximate near neighbor, which incurs a roughly logarithmic overhead. In 2007, Indyk and Naor, in the context of approximate nearest neighbors, introduced the notion of nearest neighbor-preserving embeddings. These are randomized embeddings between two metric spaces with guaranteed bounded distortion only for the distances between a query point and a point set. Such embeddings are known to exist for both L2 and L1 metrics, as well as for doubling subsets of L2. The case that remained open were doubling subsets of L1. In this paper, we propose a dimension reduction by means of a near neighbor-preserving embedding for doubling subsets of L1. Our approach is to represent the pointset with a carefully chosen covering set, then randomly project the latter. We study two types of covering sets: c-approximate r-nets and randomly shifted grids, and we discuss the tradeoff between them in terms of preprocessing time and target dimension. We employ Cauchy variables: certain concentration bounds derived should be of independent interest.

Goodness-of-fit procedures are provided to test the validity of compound models for the total claims, involving specific laws for the constituent components, namely the claim frequency distribution and the distribution of individual claim sizes. This is done without the need for observations on these two component variables. Goodness-of-fit tests that utilize the Laplace transform as well as classical tools based on the distribution function, are proposed and compared. These methods are validated by simulations and then applied to insurance data. MSC 2010: 60G55, 60G40, 12E10.

International audience; A new technique for finding implicit matrix-based representations of rational curves in arbitrary dimension is introduced. It relies on the use of moving quadrics following curve parameterizations, providing a high-order extension of the implicit matrix representations built from their linear counterparts, the moving planes. The matrices we obtain offer new, more compact, implicit representations of rational curves. Their entries are filled by linear and quadratic forms in the space variables and their ranks drop exactly on the curve. Typically, for a general rational curve of degree d we obtain a matrix whose size is half of the size of the corresponding matrix obtained with the moving planes method. We illustrate the advantages of these new matrices with some examples, including the computation of the singularities of a rational curve.

International audience; Implicitization usually focuses on plane curves and (hyper)surfaces, in other words, varieties of codimension 1. In this paper we shift the focus on space curves and, more generally, on varieties of codimension larger than 1, and discuss approaches that are not sensitive to base points.Our first contribution is a direct generalization of an implicitization method based on interpolation matrices for objects of high codimension given parametrically or as point clouds. Our result shows the completeness of this approach which, furthermore, reduces geometric operations and predicates to linear algebra computations.Our second, and main contribution is an implicitization method of parametric space curves and varieties of codimension > 1, which exploits the theory of Chow forms to obtain the equations of conical (hyper)surfaces intersecting precisely at the given object. We design a new, practical, randomized algorithm that always produces correct output but possibly with a non-minimal number of surfaces. For space curves, which is the most common case, our algorithm returns 3 surfaces whose polynomials are of near-optimal degree; moreover, computation reduces to a Sylvester resultant. We illustrate our algorithm through a series of examples and compare our Maple code with other methods implemented in Maple. Our prototype is not faster but yields fewer equations and is more robust than Maple's implicitize. Although not optimized, it is comparable with Gröbner bases and matrix representations derived from syzygies, for degrees up to 6.

We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear inequalities. We implement and evaluate practical randomized algorithms for accurately approximating the polytope's volume in high dimensions (e.g. one hundred). To carry out this efficiently we experimentally correlate the effect of parameters, such as random walk length and number of sample points, on accuracy and runtime. Moreover, we exploit the problem's geometry by implementing an iterative rounding procedure, computing partial generations of random points and designing fast polytope boundary oracles. Our publicly available code is significantly faster than exact computation and more accurate than existing approximation methods. We provide volume approximations for the Birkhoff polytopes B_11,...,B_15, whereas exact methods have only computed that of B_10., Comment: 15 pages, 2 figures, 8 tables, in Proc. of SoCG'14

International audience; Spheres are known to be rigid geometric objects: they cannot be deformed isometrically, i.e. while preserving the length of curves, in a twice differentiable way. An unexpected result by J. Nash (Ann. of Math. 60: 383-396, 1954) and N. Kuiper (Indag. Math. 17: 545-555, 1955) shows that this is no longer the case if one requires the deformations to be only continuously differentiable. A remarkable consequence of their result makes possible the isometric reduction of a unit sphere inside an arbitrarily small ball. In particular, if one views the Earth as a round sphere the theory allows to reduce its diameter to that of a terrestrial globe while preserving geodesic distances. Here we describe the first explicit construction and visualization of such a reduced sphere. The construction amounts to solve a non-linear PDE with boundary conditions. The resulting surface consists of two unit spherical caps joined by a C 1 fractal equatorial belt. An intriguing question then arises about the transition between the smooth and the C 1 fractal geometries. We show that this transition is similar to the one observed when connecting a Koch curve to a line segment.

International audience; We describe the algebraic-geometric modeling platform Axl, which provides tools for the manipulation, computation and visualisation of semi-algebraic models. This includes meshes, basic geometric objects such as spheres, cylinders, cones, ellipsoids, torus, piecewise polynomial parameterisations of curves, surfaces or volumes such as b-spline pa-rameterisations, as well as algebraic curves and surfaces defined by polynomial equations. Moreover, Axl provides algorithms for processing these geometric representations, such as computing intersection loci (points, curves) of parametric models, singularities of algebraic curves or surfaces, certified topology of curves and surfaces, etc. We present its main features and describe its generic extension mechanism, which allows one to define new data types and new processes on the data, which benefit from automatic visualisation and interaction facilities. The application capacities of the software are illustrated by short descriptions of plugins on algebraic curves and surfaces and on splines for Isogeometric Analysis.

We examine volume computation of general-dimensional polytopes and more general convex bodies, defined as the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be constructed by computing volumes of convex bodies. We design and implement practical algorithms in the exact and approximate setting, we experimentally juxtapose them and study the tradeoff of exactness and accuracy for speed. We analyze the following methods in order of increasing generality: rejection sampling relying on uniformly sampling the simplex, which is the fastest approach, but inaccurate for small volumes; exact formulae based on the computation of integrals of probability distribution functions; an optimized Lawrence sign decomposition method, since the polytopes at hand are shown to be simple; Markov chain Monte Carlo algorithms using random walks based on the hit-and-run paradigm generalized to nonlinear convex bodies and relying on new methods for computing a ball enclosed; the latter is experimentally extended to non-convex bodies with very encouraging results. Our C++ software, based on CGAL and Eigen and available on github, is shown to be very effective in up to 100 dimensions. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises., 22 pages, 6 figures, Symposium on Computational Geometry 2018

The approximate nearest neighbor problem ($\epsilon$-ANN) in high dimensional Euclidean space has been mainly addressed by Locality Sensitive Hashing (LSH), which has polynomial dependence in the dimension, sublinear query time, but subquadratic space requirement. In this paper, we introduce a new definition of "low-quality" embeddings for metric spaces. It requires that, for some query point $q$, there exists an approximate nearest neighbor among the pre-images of the $k>1$ approximate nearest neighbors in the target space. Focusing on Euclidean spaces, we employ random projections in order to reduce the original problem to one in a space of dimension inversely proportional to $k$. The $k$ approximate nearest neighbors can be efficiently retrieved by a data structure such as BBD-trees. The same approach is applied to the problem of computing an approximate near neighbor, where we obtain a data structure requiring linear space, and query time in $O(d n^{\rho})$, for $\rho\approx 1-\epsilon^2/\log(1/\epsilon)$. This directly implies a solution for $\epsilon$-ANN, while achieving a better exponent in the query time than the method based on BBD-trees. Better bounds are obtained in the case of doubling subsets of $\ell_2$, by combining our method with $r$-nets. We implement our method in C++, and present experimental results in dimension up to $500$ and $10^6$ points, which show that performance is better than predicted by the analysis. In addition, we compare our ANN approach to E2LSH, which implements LSH, and we show that the theoretical advantages of each method are reflected on their actual performance., Comment: 15 pages, 3 figures

The objective of this study is to estimate the duration, frequency and intensity of precipitation extreme episodes in Cyprus, in relation with the two phases of the Eastern Mediterranean teleconnection Pattern (EMP), during winter for the period 1958–2005. A standardised teleconnection index was employed to determine the phases (positive and negative) and the strength of the EMP. The identification of the precipitation extremes was performed with the aid of four climatic indices. It was found that during the positive phase of the pattern, the length of dry periods reduces while that of wet periods increases, being followed by increase of frequency of extreme wet days and precipitation intensity. On the contrary, during the negative phase, the dry spells become longer in accordance with shortening of the wet spells, decrease of the number of extreme wet days and precipitation intensity.

International audience; An interpolation problem is defined by a set of linear forms on the (multivariate) polynomial ring and values to be achieved by an interpolant. For Lagrange interpolation the linear forms consist of evaluations at some nodes,while Hermite interpolation also considers the values of successive derivatives. Both are examples of ideal interpolation in that the kernels of the linear forms intersect into an ideal. For an ideal interpolation problem with symmetry, we address the simultaneous computation of a symmetry adapted basis of the least interpolation space and the symmetry adapted H-basis of the ideal. Beside its manifest presence in the output, symmetry is exploited computationally at all stages of the algorithm. For an ideal invariant, under a group action, defined by a Groebner basis, the algorithm allows to obtain a symmetry adapted basis of the quotient and of the generators. We shall also note how it applies surprisingly but straightforwardly to compute fundamental invariants and equivariants of a reflection group.

International audience; It has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact formulae, including older and recent results, in sparse elimination. We start with root bounds and juxtapose two recent formulae: a generating function of the m-Bézout bound and a closed-form expression for the mixed volume by means of a matrix permanent. For the sparse resultant, a bevy of results have established determinantal or rational formulae for a large class of systems, starting with Macaulay. The discriminant is closely related to the resultant but admits no compact formula except for very simple cases. We offer a new determinantal formula for the discriminant of a sparse multilinear system arising in computing Nash equilibria. We introduce an alternative notion of compact formula, namely the Newton polytope of the unknown polynomial. It is possible to compute it efficiently for sparse resultants, discriminants, as well as the implicit equation of a parameterized variety. This leads us to consider implicit matrix representations of geometric objects.

International audience; We examine implicit representations of parametric or point cloud models, based on interpolation matrices, which are not sensitive to base points. We show how interpolation matrices can be used for ray shooting of a parametric ray with a surface patch, including the case of high-multiplicity intersections. Most matrix operations are executed during pre-processing since they solely depend on the surface. For a given ray, the bottleneck is equation solving. Our Maple code handles bicubic patches in ≤ 1 sec, though numerical issues might arise. Our second contribution is to extend the method to parametric space curves and, generally, to codimension > 1, by computing the equations of (hyper)surfaces intersecting precisely at the given object. By means of Chow forms, we propose a new, practical, randomized algorithm that always produces correct output but possibly with a non-minimal number of surfaces. For space curves, we obtain 3 surfaces whose polynomials are of near-optimal degree; in this case, computation reduces to a Sylvester resultant. We illustrate our algorithm through a series of examples and compare our Maple prototype with other methods implemented in Maple i.e., Groebner basis and implicit matrix representations. Our Maple prototype is not faster but yields fewer equations and seems more robust than Maple's implicitize; it is also comparable with the other methods for degrees up to 6.

The contribution of the thesis is threefold. The first Problem is computing the discriminant, when the system’s dimension varies. Thus solving polynomial equations and systems by exploiting the structure and sparseness of them have been studied. Specifically, closed formulas for the degree of the sparse (mixed) discriminant and the sparse resultant of polynomial equations have been studied, as well as relationships between them. Closed formulas when one of the polynomials factors in the context of the theory of sparse elimination using the Newton polytope have been proposed. The purpose is to facilitate the computation of the sparse (or mixed) discriminant of a well-constrained polynomial system and to generalize the formula that connects the mixed discriminant with the sparse resultant. Further, we are given a system of n ⩾ 2 homogeneous polynomials in n variables which is equivariant with respect to the symmetric group of n symbols. We then prove that its resultant can be decomposed into a product of several resultants that are given in terms of some divided differences. As an application, we obtain a decomposition formula for the discriminant.The second Problem is computing the Minkowski decomposition of a polytope and the third one was the problem of Multidimensional Subset Sum (kD-SS) in arbitrary dimension. We present an algorithm for computing all Minkowski Decompositions (MinkDecomp) of a given convex, integral d-dimensional polytope. Further, we consider the approximation of two NP-hard problems: Minkowski Decomposition (MinkDecomp) of lattice polygons in the plane and the closely related problem of Multidimensional Subset Sum (kD-SS) in arbitrary dimension. In kD-SS, a multiset S of k-dimensional vectors is given, along with a target vector t, and one must decide whether there exists a subset of S that sums up to t. We prove, through a gap-preserving reduction from Set Cover that, for general dimension k, the corresponding optimization problem kD-SS-opt is not in APX, although the classic 1D-SS-opt has a PTAS. On the positive side, we present an approximation algorithm for 2D-SS-opt, where ε > 0 bounds the additive error in terms of some property of the input. By a reduction of the optimization version of MinkDecomp to 2D-SS-opt we approximate the former.; La contribution de la thèse est triple. Le premier problème est le calcul du discriminant, lorsque la dimension du système varie. Ainsi, la résolution d'équations et de systèmes polynomiaux en exploitant la structure de ceux-ci a été étudiée. Des formules fermées pour le degré du discriminant clairsemé (mixte) et la résultante des équations polynomiales ont été étudiées, ainsi que des relations entre eux, en utilisant le polytope de Newton. Le but est de faciliter le calcul du discriminant creuse (ou mixte) d'un système polynomial bien contraint et de généraliser la formule qui relie le discriminant mixte au résultant creux. De plus, etant donne' un système de n ⩾ 2 polynômes homogènes en n variables qui est équivariant par rapport au groupe symétrique de n symboles, nous montrons que son résultant peut être décomposé en un produit de plusieurs résultants en termes de différences divisées. En tant qu'application, nous obtenons une formule de décomposition pour le discriminant.Le deuxième problème est le calcul de la décomposition de Minkowski d'un polytope et le troisième était le problème de la somme des sous-ensembles multidimensionnels (kD-SubsetSum) dans une dimension arbitraire. Nous présentons un algorithme pour calculer toutes les décompositions de Minkowski (MinkDecomp) d'un polytope d-dimensionnel convexe et entier. Nous considérons l'approximation de deux problèmes NP-hard: la décomposition de Minkowski (MinkDecomp) des polygones dans le plan et le problème de la somme des sous-ensembles multidimensionnels (kD-SS) dans une dimension arbitraire. En kD-SS, un multiset S de vecteurs k-dimensionnels est donné, avec un vecteur cible t, et il faut décider s'il existe un sous-ensemble de S qui somme jusqu'à t. Nous prouvons, à travers une réduction gap-preserving de Set Cover, que le problème d'optimisation correspondant kD-SS-opt n'est pas dans APX, bien que le classique 1D-SS-opt ait un PTAS. Du côté positif, nous présentons un algorithme d'approximation pour 2D-SS-opt, où ε> 0 limite l'erreur additive.

The contribution of the thesis is threefold. The first Problem is computing the discriminant, when the system’s dimension varies. Thus solving polynomial equations and systems by exploiting the structure and sparseness of them have been studied. Specifically, closed formulas for the degree of the sparse (mixed) discriminant and the sparse resultant of polynomial equations have been studied, as well as relationships between them. Closed formulas when one of the polynomials factors in the context of the theory of sparse elimination using the Newton polytope have been proposed. The purpose is to facilitate the computation of the sparse (or mixed) discriminant of a well-constrained polynomial system and to generalize the formula that connects the mixed discriminant with the sparse resultant. Further, we are given a system of n ⩾ 2 homogeneous polynomials in n variables which is equivariant with respect to the symmetric group of n symbols. We then prove that its resultant can be decomposed into a product of several resultants that are given in terms of some divided differences. As an application, we obtain a decomposition formula for the discriminant.The second Problem is computing the Minkowski decomposition of a polytope and the third one was the problem of Multidimensional Subset Sum (kD-SS) in arbitrary dimension. We present an algorithm for computing all Minkowski Decompositions (MinkDecomp) of a given convex, integral d-dimensional polytope. Further, we consider the approximation of two NP-hard problems: Minkowski Decomposition (MinkDecomp) of lattice polygons in the plane and the closely related problem of Multidimensional Subset Sum (kD-SS) in arbitrary dimension. In kD-SS, a multiset S of k-dimensional vectors is given, along with a target vector t, and one must decide whether there exists a subset of S that sums up to t. We prove, through a gap-preserving reduction from Set Cover that, for general dimension k, the corresponding optimization problem kD-SS-opt is not in APX, although the classic 1D-SS-opt has a PTAS. On the positive side, we present an approximation algorithm for 2D-SS-opt, where ε > 0 bounds the additive error in terms of some property of the input. By a reduction of the optimization version of MinkDecomp to 2D-SS-opt we approximate the former.; La contribution de la thèse est triple. Le premier problème est le calcul du discriminant, lorsque la dimension du système varie. Ainsi, la résolution d'équations et de systèmes polynomiaux en exploitant la structure de ceux-ci a été étudiée. Des formules fermées pour le degré du discriminant clairsemé (mixte) et la résultante des équations polynomiales ont été étudiées, ainsi que des relations entre eux, en utilisant le polytope de Newton. Le but est de faciliter le calcul du discriminant creuse (ou mixte) d'un système polynomial bien contraint et de généraliser la formule qui relie le discriminant mixte au résultant creux. De plus, etant donne' un système de n ⩾ 2 polynômes homogènes en n variables qui est équivariant par rapport au groupe symétrique de n symboles, nous montrons que son résultant peut être décomposé en un produit de plusieurs résultants en termes de différences divisées. En tant qu'application, nous obtenons une formule de décomposition pour le discriminant.Le deuxième problème est le calcul de la décomposition de Minkowski d'un polytope et le troisième était le problème de la somme des sous-ensembles multidimensionnels (kD-SubsetSum) dans une dimension arbitraire. Nous présentons un algorithme pour calculer toutes les décompositions de Minkowski (MinkDecomp) d'un polytope d-dimensionnel convexe et entier. Nous considérons l'approximation de deux problèmes NP-hard: la décomposition de Minkowski (MinkDecomp) des polygones dans le plan et le problème de la somme des sous-ensembles multidimensionnels (kD-SS) dans une dimension arbitraire. En kD-SS, un multiset S de vecteurs k-dimensionnels est donné, avec un vecteur cible t, et il faut décider s'il existe un sous-ensemble de S qui somme jusqu'à t. Nous prouvons, à travers une réduction gap-preserving de Set Cover, que le problème d'optimisation correspondant kD-SS-opt n'est pas dans APX, bien que le classique 1D-SS-opt ait un PTAS. Du côté positif, nous présentons un algorithme d'approximation pour 2D-SS-opt, où ε> 0 limite l'erreur additive.

The problem of Approximate Nearest Neighbor (ANN) search is fundamental in computer science and has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets whereas complex shapes have not been sufficiently treated. Here, we focus on distance functions between discretized curves in Euclidean space: they appear in a wide range of applications, from road segments to time-series in general dimension. For $\ell_p$-products of Euclidean metrics, for any $p$, we design simple and efficient data structures for ANN, based on randomized projections, which are of independent interest. They serve to solve proximity problems under a notion of distance between discretized curves, which generalizes both discrete Fr\'echet and Dynamic Time Warping distances. These are the most popular and practical approaches to comparing such curves. We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms, our algorithm is especially efficient when the length of the curves is bounded., Comment: 18 pages

The construction of $r$-nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate $r$-nets with respect to Euclidean distance. For any fixed $\epsilon>0$, the approximation factor is $1+\epsilon$ and the complexity is polynomial in the dimension and subquadratic in the number of points. The algorithm succeeds with high probability. More specifically, the best previously known LSH-based construction of Eppstein et al.\ \cite{EHS15} is improved in terms of complexity by reducing the dependence on $\epsilon$, provided that $\epsilon$ is sufficiently small. Our method does not require LSH but, instead, follows Valiant's \cite{Val15} approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which $r$-nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the $(1+\epsilon)$-approximate $k$th nearest neighbor distance in time subquadratic in the size of the input., Comment: 20 pages

International audience; We address the description of the tropicalization of families of rational varieties under parametrizations with prescribed support, via curve valuations. We recover and extend results by Sturmfels, Tevelev and Yu for generic coefficients, considering rational parametrizations with non-trivial denominator. The advantage of our point of view is that it can be generalized to deal with non-generic parametrizations. We provide a detailed analysis of the degree of the closed image, based on combinatorial conditions on the relative positions of the supports of the polynomials defining the parametrization. We obtain a new formula and finer bounds on the degree, when the supports of the polynomials are different. We also present a new formula and bounds for the order at the origin in case the closed image is a hypersurface.

We examine matrix representations of curves and surfaces based on syzygies and constructed by interpolation through points. They are implicit representations of objects given as point clouds. The corresponding theory, including moving lines, curves and surfaces, has been developed for parametric models. Our contribution is to show how to compute the required syzygies by interpolation, when the geometric object is given by a point cloud whose sampling satisfies mild assumptions. We focus on planar and space curves, where the theory of syzygies allows us to design an exact algorithm yielding the optimal implicit expression. The method extends readily to surfaces without base points defined over triangular patches. Our Maple implementation has served to produce the examples in this paper and is available upon demand by the authors.

International audience; This paper proposes a deep learning approach for parameterizing an unorganized or scattered point cloud in R 3 with graph convolutional neural networks. It builds upon a graph convolutional neural network that predicts the weights (called parameterization weights) of certain convex combinations that lead to a mapping of the 3D points into a planar parameter domain. First, we compute a radius neighbours graph that yields proximity information to each 3D point in the cloud. This radius graph is then converted to its line graph, which encodes edge adjacencies, and is equipped with appropriate weights. The line graph is used as input to a graph convolutional neural network trained to predict optimal parameterizations. The proposed model outperforms closed-form choices of the parameterization weights and produces high quality parameterizations for surface reconstruction schemes.

International audience; Inspired by the strengths of quadric error metrics initially designed for mesh decimation, we propose a concise mesh reconstruction approach for 3D point clouds. Our approach proceeds by clustering the input points enriched with quadric error metrics, where the generator of each cluster is the optimal 3D point for the sum of its quadric error metrics. This approach favors the placement of generators on sharp features, and tends to equidistribute the error among clusters. We reconstruct the output surface mesh from the adjacency between clusters and a constrained binary solver. We combine our clustering process with an adaptive refinement driven by the error. Compared to prior art, our method avoids dense reconstruction prior to simplification and produces immediately an optimized mesh.

We analytically derive from first physical principles the functional dependence of wireless channels on the RIS configuration for generic (i.e., potentially complex-scattering) RIS-parametrized radio environments. The wireless channel is a linear input-output relation that depends non-linearly on the RIS configuration because of two independent mechanisms: i) proximity-induced mutual coupling between close-by RIS elements; ii) reverberation-induced long-range coupling between all RIS elements. Mathematically, this "structural" non-linearity originates from the inversion of an "interaction" matrix that can be cast as the sum of an infinite Born series [for i)] or Born-like series [for ii)] whose $K$th term physically represents paths involving $K$ bounces between the RIS elements [for i)] or wireless entities [for ii)]. We identify the key physical parameters that determine whether these series can be truncated after the first and second term, respectively, as tacitly done in common cascaded models of RIS-parametrized wireless channels. Numerical results obtained with the physics-compliant PhysFad model and experimental results obtained with a RIS prototype in an anechoic (echo-free) chamber and rich-scattering reverberation chambers corroborate our analysis. Our findings raise doubts about the reliability of existing performance analysis and channel-estimation protocols for cases in which cascaded models poorly describe the physical reality., Comment: 30 pages, 5 figures, submitted to an IEEE Journal

Pre-eruptive processes and their timescales are critical information for risk management at explosive volcanoes, and Santorini caldera (Greece) provides an excellent context in which to approach this subject. We ask two questions. First, are pre-eruptive processes the same for small and big eruptions? To investigate, we performed a multi-mineral diffusion timescale study of a small explosive eruption of Kameni Volcano and compared the results with those published for larger caldera-forming eruptions at Santorini. The Kameni dacite resembles products of larger eruptions in being crystal-poor, containing plagioclase with antecrystic cores and autocrystic rims, bearing orthopyroxene with sector zoning and phantom skeletal morphologies, and showing evidence for mixing of different silicic magmas prior to eruption. Diffusion timescales from Mg-Fe profiles in orthopyroxene and clinopyroxene phenocrysts are

International audience; Given a system of n homogeneous polynomials in n variables which is equivariant with respect to the canonical actions of the symmetric group of n symbols on the variables and on the polynomials, it is proved that its resultant can be decomposed into a product of several smaller resultants that are given in terms of some divided differences. As an application, we obtain a decomposition formula for the discriminant of a multivariate homogeneous symmetric polynomial.

We propose a new data-structure, the generalized randomized k-d forest, or k-d GeRaF, for approximate nearest neighbor searching in high dimensions. In particular, we introduce new ran-domization techniques to specify a set of independently constructed trees where search is performed simultaneously, hence increasing accuracy. We omit backtracking, and we optimize distance computations , thus accelerating queries. We release public domain software GeRaF and we compare it to existing implementations of state-of-the-art methods including BBD-trees, Locality Sensitive Hashing, randomized k-d forests, and product quantization. Experimental results indicate that our method would be the method of choice in dimensions around 1,000, and probably up to 10,000, and pointsets of cardinality up to a few hundred thousands or even one million; this range of inputs is encountered in many critical applications today. For instance, we handle a real dataset of 10 6 images represented in 960 dimensions with a query time of less than 1sec on average and 90% responses being true nearest neighbors.

International audience; Large scale duplicate detection, clustering and mining of documents or images has been conventionally treated with seed detection via hashing, followed by seed growing heuristics using fast search. Principled clustering methods , especially kernelized and spectral ones, have higher complexity and are difficult to scale above millions. Under the assumption of documents or images embedded in Eu-clidean space, we revisit recent advances in approximate k-means variants, and borrow their best ingredients to introduce a new one, inverted-quantized k-means (IQ-means). Key underlying concepts are quantization of data points and multi-index based inverted search from centroids to cells. Its quantization is a form of hashing and analogous to seed detection, while its updates are analogous to seed growing, yet principled in the sense of distortion minimization. We further design a dynamic variant that is able to determine the number of clusters k in a single run at nearly zero additional cost. Combined with powerful deep learned representations , we achieve clustering of a 100 million image collection on a single machine in less than one hour.

Background Few data are available regarding changes in mitral regurgitation (MR) severity with guideline-recommended medical therapy (GRMT) in heart failure (HF). Our aim was to evaluate the evolution and impact of MR after GRMT in the Biology study to Tailored treatment in chronic heart failure (BIOSTAT-CHF). Methods A retrospective post-hoc analysis was performed on HF patients from BIOSTAT-CHF with available data on MR status at baseline and at 9-month follow-up after GRMT optimization. The primary endpoint was a composite of all-cause death or HF hospitalization. Results Among 1022 patients with data at both time-points, 462 (45.2%) had moderate-severe MR at baseline and 360 (35.2%) had it at 9-month follow-up. Regression of moderate-severe MR from baseline to 9 months occurred in 192/462 patients (41.6%) and worsening from baseline to moderate-severe MR at 9 months occurred in 90/560 patients (16.1%). The presence of moderate-severe MR at 9 months, independent from baseline severity, was associated with an increased risk of the primary endpoint (unadjusted hazard ratio [HR], 2.03; 95% confidence interval [CI], 1.57–2.63; p p Conclusions Among patients with HF undergoing GRMT optimization, ACEi/ARB up-titration and HFpEF were associated with MR improvement, and the presence of moderate-severe MR after GRMT was associated with worse outcome. Graphical abstract

Adult PAP patients experience similar #COVID19 rates to the general population, and high rates of hospitalisation and deaths, underscoring their vulnerability and the need for measures to prevent infection. The impact of iGM-CSF must be considered. https://bit.ly/3M0wKnZ.

Background The sodium-glucose cotransporter-2 inhibitor empagliflozin decreases the risk of cardiovascular death or hospitalization for heart failure (HF) in patients with HF with reduced ejection fraction. Empagliflozin reduces serum uric acid (SUA), but the relevance of this effect in patients with HF is unclear. This study aimed to investigate the effect of empagliflozin on SUA levels and the therapeutic efficacy of empagliflozin in relation to SUA. Methods The association between SUA and the composite primary outcome of cardiovascular death or hospitalization for worsening HF, its components, and all-cause mortality was investigated in 3676 patients of the EMPEROR-Reduced trial (98.6% of the study cohort). The treatment effect of empagliflozin was studied in relation to SUA as continuous variable, to clinical hyperuricaemia (SUA >5.7 mg/dL for women, >7.0 mg/dL for men) and in subgroups of patients of tertiles of SUA. Results Hyperuricaemia was prevalent in 53% of patients with no sex differences. Elevated SUA (highest tertile, mean SUA 9.38 ± 1.49 mg/dL) was associated with advanced severity of HF and with worst outcome [composite outcome, hazard ratio (HR) 1.64 (95% confidence interval, CI 1.28–2.10); cardiovascular mortality, HR 1.98 (95% CI 1.35–2.91); all-cause mortality, HR 1.8 (95% CI 1.29–2.49), all P Conclusion Hyperuricaemia is common in HF and is an independent predictor of advanced disease severity and increased mortality. Empagliflozin induced a rapid and sustained reduction of SUA levels and of clinical events related to hyperuricaemia. The benefit of empagliflozin on the primary outcome was observed independently of SUA.

Radial diffusion has been established as one of the most important mechanisms contributing to both the acceleration and loss of relativistic electrons in the outer radiation belt, as well as to the supply of particles to the inner radiation belt. In the framework of the “SafeSpace” project, we have used 9 years (2011–2019) of multi-point magnetic and electric field measurements from THEMIS A, D and E satellites to create a database of radial diffusion coefficients (DLL) and ultra-low-frequency (ULF) wave power spectral densities (PSDs) spanning an L∗ range from 3 to 8. In this work we investigate the dependence of the DLL on the various solar wind parameters, geomagnetic indices and coupling functions, as well as the L-shell, during the solar cycle 24. Moreover, we discuss the uncertainties introduced on the estimation of DLL time series by the partial azimuthal coverage provided by in situ measurements. Furthermore, we investigate, via a superposed analysis, the dependence of the DLL on solar wind drivers. We show, for the first time to the best of our knowledge, that the interplanetary coronal mass ejection (ICME)-driven disturbances accompanied by high solar wind pressure values combined with intense magnetospheric compression can produce DLLB values comparable to or even greater than the ones of DLLE. This feature cannot be captured by semi-empirical models and introduces a significant energy dependence on the DLL. Finally, we show the advantages of using DLL time series by means of numerical simulations of relativistic electron fluxes performed with the Salammbô code and significant deviations in the predictions of several semi-empirical models depending on the level of geomagnetic activity and L-shell.

Geometrically continuous splines are piecewise polynomial functions defined on a collection of patches which are stitched together through transition maps. They are called $G^{r}$-splines if, after composition with the transition maps, they are continuously differentiable functions to order $r$ on each pair of patches with stitched boundaries. This type of splines has been used to represent smooth shapes with complex topology for which (parametric) spline functions on fixed partitions are not sufficient. In this article, we develop new algebraic tools to analyze $G^r$-spline spaces. We define $G^{r}$-domains and transition maps using an algebraic approach, and establish an algebraic criterion to determine whether a piecewise function is $G^r$-continuous on the given domain. In the proposed framework, we construct a chain complex whose top homology is isomorphic to the $G^{r}$-spline space. This complex generalizes Billera-Schenck-Stillman homological complex used to study parametric splines. Additionally, we show how previous constructions of $G^r$-splines fit into this new algebraic framework, and present an algorithm to construct a bases for $G^r$-spline spaces. We illustrate how our algebraic approach works with concrete examples, and prove a dimension formula for the $G^r$-spline space in terms of invariants to the chain complex. In some special cases, explicit dimension formulas in terms of the degree of splines are also given., 46 pages, 5 figures, 1 table

International audience; Introduction: Previous meta-analyses have linked social connections and mild cognitive impairment, dementia, and mortality. However, these used aggregate data from North America and Europe and examined a limited number of social connection markers. Methods: We used individual participant data (N = 39271, M age = 70.67 (40-102), 58.86% female, M education = 8.43 years, M follow-up = 3.22 years) from 13 longitudinal ageing studies. A two-stage meta-analysis of Cox regression models examined the association between social connection markers with our primary outcomes. Results: We found associations between good social connections structure and quality and lower risk of incident mild cognitive impairment (MCI); between social structure and function and lower risk of incident dementia and mortality. Only in Asian cohorts, being married/in a relationship was associated with reduced risk of dementia, and having a confidante was associated with reduced risk of dementia and mortality. Discussion: Different aspects of social connections-structure, function, and qualityare associated with benefits for healthy aging internationally.

BACKGROUND: Hearing impairment is common among older adults and affects cognitive assessments for identification of dementia which rely on good hearing function. We developed and validated a version of the Montreal Cognitive Assessment (MoCA) for people with hearing impairment.METHODS: We adapted existing MoCA 8.1 items for people with hearing impairment by presenting instructions and stimuli in written rather than spoken format. One Attention domain and two Language domain items required substitution by alternative items. Three and four candidate items respectively were constructed and field-tested along with the items adapted to written form. We used a combination of individual item analysis and item substitution to select the set of alternative items to be included in the final form of the MoCA-H in place of the excluded original items. We then evaluated the performance and reliability of the final tool, including making any required adjustments for demographic factors.RESULTS: One hundred and fifty-nine hearing-impaired participants, including 76 with normal cognition and 83 with dementia, completed the adapted version of the MoCA. A further 97 participants with normal hearing completed the standard MoCA as well as the novel items developed for the MoCA-H to assess score equivalence between the existing and alternative MoCA items and for independence from hearing impairment. Twenty-eight participants were retested between 2-4 weeks after initial testing. After the selection of optimal item set, the final MoCA-H had an area under the curve of 0.973 (95% CI 0.952-0.994). At a cut-point of 24 points or less sensitivity and specificity for dementia was 92.8% and 90.8%, respectively. The intraclass correlation for test-retest reliability was 0.92 (95%CI 0.78-0.97).CONCLUSION: The MoCA-H is a sensitive and reliable means of identifying dementia among adults with acquired hearing impairment.