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Let $U_h:\mathbb R^{d}\to \mathbb R^{d}$ be a smooth vector field and consider the associated overdamped Langevin equation $$dX_t=-U_h(X_t)\,dt+\sqrt{2h}\,dB_t$$ in the low temperature regime $h\rightarrow 0$. In this work, we study the spectrum of the associated diffusion $L=-h\Delta+U_h\cdot\nabla$ under the assumptions that $U_h=U_{0}+h\nu$, where the vector fields $U_{0}:\mathbb R^{d}\to \mathbb R^{d}$ and $\nu:\mathbb R^{d}\to \mathbb R^{d}$ are independent of $h\in(0,1]$, and that the dynamics admits $e^{-\frac Vh}$ as an invariant measure for some smooth function $V:\mathbb{R}^d\rightarrow\mathbb{R}$. Assuming additionally that $V$ is a Morse function admitting $n_0$ local minima, we prove that there exists $\epsilon>0$ such that in the limit $h\to 0$, $L$ admits exactly $n_0$ eigenvalues in the strip $\{0\leq \operatorname{Re}(z)< \epsilon\}$, which have moreover exponentially small moduli. Under a generic assumption on the potential barriers of the Morse function $V$, we also prove that the asymptotic behaviors of these small eigenvalues are given by Eyring-Kramers type formulas.

The classification of Fatou components for rational functions was concluded with Sullivan's proof of the No Wandering Domains Theorem in 1985. In 2016 it was shown, in joint work of the first and last author with Buff, Dujardin and Raissy, that wandering domains do exist in higher dimensions. In fact, wandering domains arise even for a seemingly simple class of maps: polynomial skew products. While the construction gives an infinite dimensional class of examples, and has been extended to polynomial automorphisms of $\mathbb C^4$ by Hahn and the last author, the currently known wandering domains are essentially unique. Our goal in this paper is to construct a second example, arising from similar techniques, but with distinctly different dynamical behavior. Instead of wandering domains arising from a Lavaurs map with an attracting fixed point, we construct a domain arising from a Lavaurs map with a fixed point of Siegel type. Siegel disks are not robust under perturbations, as opposed to attracting fixed points. We prove a necessary and sufficient condition for the existence of a so-called trapping domain for non-autonomous dynamical systems given by sequences of maps converging parabolically towards a Siegel type limit map. Guaranteeing that this condition is satisfied in our current construction requires a reconsideration of the proof of the original wandering domain, as more precise estimates on the rate of convergence towards the Lavaurs map are required. By adapting ideas introduced recently by Bedford, Smillie and Ueda, and by proving the existence of parabolic curves, with control on their domains of definition, we prove that the convergence rate is parabolic., 54 pages, 2 figures

To evaluate the profitability of a production before the launch of the manufacturing process, most industrial companies use numerical simulation. This allows to test virtually several configurations of the parameters of a given product and to decide on its performance (defined by the specifications). In order to measure the impact of industrial process fluctuations on product performance, we are particularly interested in estimating the probability of failure of the product. Since each simulation requires the execution of a complex and expensive calculation code, it is not possible to perform a sufficient number of tests to estimate this probability using, for example, a Monte-Carlo method. Under the constraint of a limited number of code calls, we propose two very different estimation methods.The first is based on the principles of Bayesian estimation. Our observations are the results of numerical simulation. The probability of failure is seen as a random variable, the construction of which is based on that of a random process to model the expensive calculation code. To correctly define this model, the Kriging method is used. Conditionally to the observations, the posterior distribution of the random variable, which models the probability of failure, is inaccessible. To learn about this distribution, we construct approximations of the following characteristics: expectation, variance, quantiles... We use the theory of stochastic orders to compare random variables and, more specifically, the convex order. The construction of an optimal experimental design is ensured by the implementation of a sequential experimental planning procedure, based on the principle of the SUR ("Stepwise Uncertainty Reduction") strategies.The second method is an iterative procedure, particularly adapted to the case where the probability of failure is very small, i.e. the redoubt event is rare. The expensive calculation code is represented by a function that is assumed to be Lipschitz continuous. At each iteration, this hypothesis is used to construct approximations, by default and by excess, of the probability of failure. We show that these approximations converge towards the true value with the number of iterations. In practice, they are estimated using the Monte-Carlo method known as splitting method.The proposed methods are relatively simple to implement and the results they provide can be easily interpreted. We test them on various examples, as well as on a real case from STMicroelectronics., Pour évaluer la rentabilité d'une production en amont du lancement du processus de fabrication, la plupart des entreprises industrielles ont recours à la simulation numérique. Cela permet de tester virtuellement plusieurs configurations des paramètres d'un produit donné et de statuer quant à ses performances (i.e. les spécifications imposées par le cahier des charges). Afin de mesurer l'impact des fluctuations des procédés industriels sur les performances du produit, nous nous intéressons en particulier à l'estimation de sa probabilité de défaillance. Chaque simulation exigeant l'exécution d'un code de calcul complexe et coûteux, il n'est pas possible d'effectuer un nombre de tests suffisant pour estimer cette probabilité via, par exemple, une méthode Monte-Carlo. Sous la contrainte d'un nombre limité d'appels au code, nous proposons deux méthodes d'estimation très différentes.La première s'appuie sur les principes de l'estimation bayésienne. Nos observations sont les résultats de simulation numérique. La probabilité de défaillance est vue comme une variable aléatoire, dont la construction repose sur celle d'un processus aléatoire destiné à modéliser le code de calcul coûteux. Pour définir correctement ce modèle, on utilise la méthode de krigeage. Conditionnellement aux observations, la loi a posteriori de la variable aléatoire, qui modélise la probabilité de défaillance, est inaccessible. Pour apprendre sur cette loi, nous construisons des approximations des caractéristiques suivantes: espérance, variance, quantiles... On utilise pour cela la théorie des ordres stochastiques pour la comparaison de variables aléatoires et, plus particulièrement, l'ordre convexe. La construction d'un plan d'expériences optimal est assurée par la mise en place d'une procédure de planificationd'expériences séquentielle, basée sur le principe des stratégies SUR ("Stepwise Uncertainty Reduction").La seconde méthode est une procédure itérative, particulièrement adaptée au cas où la probabilité de défaillance est très petite, i.e. l'événement redouté est rare. Le code de calcul coûteux est représenté par une fonction que l'on suppose lipschitzienne. À chaque itération, cette hypothèse est utilisée pour construire des approximations, par défaut et par excès, de la probabilité de défaillance. Nous montrons que ces approximations convergent vers la valeur vraie avec le nombre d'itérations. En pratique, on les estime grâce à la méthode Monte-Carlo dite de "splitting".Les méthodes que l'on propose sont relativement simples à mettre en œuvre et les résultats qu'elles fournissent peuvent être interprétés sans difficulté. Nous les testons sur divers exemples, ainsi que sur un cas réel provenant de la société STMicroelectronics.

The $Z$-invariant Ising model (Baxter in Philos Trans R Soc Lond A Math Phys Eng Sci 289(1359):315--346, 1978) is defined on an isoradial graph and has coupling constants depending on an elliptic parameter $k$. When $k=0$ the model is critical, and as $k$ varies the whole range of temperatures is covered. In this paper we study the corresponding dimer model on the Fisher graph, thus extending our papers (Boutillier and de Tili{\`e}re in Probab Theory Relat Fields 147:379--413, 2010; Commun Math Phys 301(2):473--516, 2011) to the full $Z$-invariant case. One of our main results is an explicit, local formula for the inverse of the Kasteleyn operator. Its most remarkable feature is that it is an elliptic generalization of Boutillier and de Tili{\`e}re (2011): it involves a local function and the massive discrete exponential function introduced in Boutillier et al. (Invent Math 208(1):109--189, 2017). This shows in particular that $Z$-invariance, and not criticality, is at the heart of obtaining local expressions. We then compute asymptotics and deduce an explicit, local expression for a natural Gibbs measure. We prove a local formula for the Ising model free energy. We also prove that this free energy is equal, up to constants, to that of the $Z$-invariant spanning forests of Boutillier et al. (2017), and deduce that the two models have the same second order phase transition in $k$. Next, we prove a self-duality relation for this model, extending a result of Baxter to all isoradial graphs. In the last part we prove explicit, local expressions for the dimer model on a bipartite graph corresponding to the XOR version of this $Z$-invariant Ising model., Comment: 72 pages, 21 figures

A two-dimension extension of the Self-organized Gradient Percolation (SGP) method initially developed for the one-dimensional simulation is proposed. The initialization in the two directions is considered as the analytic solution of the 2D (homogeneous) diffusion equation. The evolution of the saturation front is assumed to be the evolution of both standard deviations in each direction. The validation of the implementation is done by comparisons between SGP and finite element results., Comment: CSMA 2019, May 2019, Giens (var), France

International audience

We describe a new method which allows us to obtain a result of exact controllability to trajectories of multidimensional conservation laws in the context of entropy solutions and under a mere non-degeneracy assumption on the flux and a natural geometric condition., Comptes Rendus Math{\'e}matique, Elsevier Masson, 2019

In the scalar 1D case, conservation laws and Hamilton-Jacobi equations are deeply related. For both, we characterize those profiles that can be attained as solutions at a given positive time corresponding to at least one initial datum. Then, for each of the two equations, we precisely identify all those initial data yielding a solution that coincide with a given profile at that positive time. Various topological and geometrical properties of the set of these initial data are then proved. 2000 Mathematics Subject Classification: 35L65, 35F21, 93B30, 35R30.

In this work we extend the theory of the classical Hardy space $H^1$ to the rational Dunkl setting. Specifically, let $\Delta$ be the Dunkl Laplacian on a Euclidean space $\mathbb{R}^N$. On the half-space $\mathbb{R}_+\times\mathbb{R}^N$, we consider systems of conjugate $(\partial_t^2+\Delta_{\mathbf{x}})$-harmonic functions satisfying an appropriate uniform $L^1$ condition. We prove that the boundary values of such harmonic functions, which constitute the real Hardy space $H^1$, can be characterized in several different ways, namely by means of atoms, Riesz transforms, maximal functions or Littlewood-Paley square functions.

In a recent work, a contribution to the Nernst current of a Dirac or Weyl semimetal coming from the conformal anomaly was reported. Being originated from an anomaly - a vacuum contribution -, a non-zero transport coefficient was predicted at zero temperature and chemical potential. In this work we perform a Kubo formula calculation of the thermoelectrical coefficient and confirm that the result agrees with the quantum field theory estimation in the limit of zero temperature and chemical potential. For finite chemical potential {\mu} around {\mu} = 0 the transverse Seebeck coefficient shows a plateau indicating that only the zero conformal Landau levels contributes to this intrinsic effect. This result opens the way to the experimental observation of a geometric anomaly which is much harder to explore than the standard chiral anomaly., Comment: 7 pages, 5 figures

Spin-orbit interaction (SOI) plays a key role in creating Majorana zero modes in semiconductor nanowires proximity coupled to a superconductor. We track the evolution of the induced superconducting gap in InSb nanowires coupled to a NbTiN superconductor in a large range of magnetic field strengths and orientations. Based on realistic simulations of our devices, we reveal SOI with a strength of 0.15-0.35 eV$\require{mediawiki-texvc}\AA$. Our approach identifies the direction of the spin-orbit field, which is strongly affected by the superconductor geometry and electrostatic gates., Comment: 7 pages, 4 figures. Supplemental Material as ancillary file

A phenomenological model is constructed, that captures the effects of coupling magnetic and elastic degrees of freedom, in the presence of external, stochastic perturbations, in terms of the interaction of magnetic moments with a bath, whose individual degrees of freedom cannot be resolved and only their mesoscopic properties are relevant. In the present work, the consequences of identifying the effects of dissipation as resulting from interactions with a bath of spins are explored, in addition to elastic, degrees of freedom. The corresponding stochastic differential equations are solved numerically and the moments of the magnetization are computed. The stochastic equations implicitly define a measure on the space of spin configurations, whose moments at equal times satisfy a hierarchy of deterministic, ordinary differential equations. Closure assumptions are used to truncate the hierarchy and the same moments are computed. We focus on the advantages and problems that each approach presents, for the approach to equilibrium and, in particular, the emergence of longitudinal damping., 20 pages, 3 figure panels. Improved presentation. Version to appear in Eur. Phys. J. B

International audience; The gut microbiota, mainly located in the colon, is engaged in a complex dialogue with the large intestinal epithelium through which important regulatory processes for the health and well-being of the host take place. Imbalances of the microbial populations, called dysbiosis, are related to several pathological status, emphasizing the importance of understanding the gut bacterial ecology. Among the ecological drivers of the microbiota, the spatial structure of the colon is of special interest: spatio-temporal mechanisms can lead to the constitution of spatial interactions among the bacterial populations and of en-vironemental niches that impact the overall colonization of the colon. In the present study, we introduce a mathematical model of the colon microbiota in its fluid environment, based on the explicit coupling of a population dynamics model of microbial populations involved in fibre degradation with a fluid dynamics model of the luminal content. This modelling framework is used to study the main drivers of the spatial structure of the microbiota, specially focusing on the dietary fibres, the epithelial motility, the microbial active swimming and viscosity gradients in the digestive track. We found 1) that the viscosity gradients allow the creation of favourable niches in the vicinity of the mucus layer; 2) that very low microbial active swimming in the radial direction is enough to promote bacterial growth, which shed a new light on microbial motility in the colon and 3) that dietary fibres are the main driver of the spatial structure of the microbiota in the distal bowel whereas epithelial motility is preponderant for the colonization of the proximal colon; in the transverse colon, fibre levels and chemotaxis have the strongest impact on the distribution of the microbial communities.

Quantum polarization effects associated with the conformal anomaly in a static magnetic field background may generate a transverse electric current in the vacuum. The current may be produced either in an unbounded curved spacetime or in a flat spacetime in a physically bounded system. In both cases, the magnitude of the electric current is proportional to the beta-function associated with renormalization of the electric charge. In our article, we investigate the electric current density induced by the magnetic field in the vicinity of a Dirichlet boundary in the scalar QED. Using first-principle lattice simulations we show that the electric current, generated by this `conformal magnetic effect at the edge' (CMEE), is well described by the conformal anomaly provided the conformal symmetry is classically unbroken. Outside of the conformal limit, the current density is characterized by an anomalous power law near the edge of the system and by an exponential suppression of the current far away from the edge., Comment: 7 pages, 5 figures

Kepler's rescaling becomes, when "Eisenhart-Duval lifted" to $5$-dimensional "Bargmann" gravitational wave spacetime, an ordinary spacetime symmetry for motion along null geodesics, which are the lifts of Keplerian trajectories. The lifted rescaling generates a well-behaved conserved Noether charge upstairs, which takes an unconventional form when expressed in conventional terms. This conserved quantity seems to have escaped attention so far. Applications include the Virial Theorem and also Kepler's Third Law. The lifted Kepler rescaling is a Chrono-Projective transformation. The results extend to celestial mechanics and Newtonian Cosmology., 13 pages, 1 figure. Dedicated to the memory of Christian Duval (1947-2018)

International audience; We consider XXX spin-$1/2$ Heisenberg chain with non-diagonal boundary conditions. We obtain a compact determinant representation for the scalar product of on-shell and off-shell Bethe vectors. In the particular case when both Bethe vectors are on shell, we obtain a determinant representation for the norm of on-shell Bethe vector and prove orthogonality of the on-shell vectors corresponding to the different eigenvalues of the transfer matrix.

31 pages, 7 figures; International audience; In this paper, the extinction problem for a class of distylous plant populations is considered within the framework of certain nonhomogeneous nearest-neighbor random walks in the positive quadrant. For the latter, extinction means absorption at one of the axes. Despite connections with some classical probabilistic models (standard two-type Galton-Watson process, two-urn model), exact formulae for the probabilities of absorption seem to be difficult to come by and one must therefore resort to good approximations. In order to meet this task, we develop potential-theoretic tools and provide various sub- and super-harmonic functions which, for large initial populations, provide bounds which in particular improve those that have appeared earlier in the literature.

Lemma 4.8 in the article [Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions, Electronic J. Probability 21 (18):1-48 (2016), arXiv:1504.02953] contains a mistake, which implies a weaker regularity estimate than the one stated in Proposition 4.11. This does not affect the proof of Theorem 2.1, but Theorems 2.2 and 2.3 only follow from the given proof if either the space dimension $d$ is equal to $2$, or the nonlinearity $F(U,V)$ is linear in $V$. To fix this problem and provide a proof of Theorems 2.2 and 2.3 valid in full generality, we consider an alternative formulation of the fixed-point problem, involving a modified integration operator with nonlocal singularity and a slightly different regularity structure. We provide the multilevel Schauder estimates and renormalisation-group analysis required for the fixed-point argument in this new setting., Comment: 24 pages, corrigendum for arXiv:1504.02953, revised version

We derive the general conditions for a large family of shift symmetry breaking degenerate higher order scalar-tensor (DHOST) theories to admit stealth black hole solutions. Such black hole configurations correspond to vacuum solutions of general relativity and admit a scalar hair which does not gravitate, revealing itself only at the perturbative level. We focus our investigation on hairy Schwarzschild-(A)dS or pure Schwarzschild solutions, dressed with a linear time-dependent scalar hair, and assuming a constant kinetic term. We also discuss subclasses of this family which satisfy the observational constraint ${c}_{\mathrm{grav}}={c}_{\text{light}}$, as well as the recent constraint ensuring the absence of graviton decay. We provide at the end concrete examples of DHOST Lagrangians satisfying our conditions. This work provides a first analysis of exact black hole solutions in shift symmetry breaking DHOST theories beyond Horndeski.

In this work we introduce a statistical framework in order to analyze the spatial redundancy in natural images. This notion of spatial redundancy must be defined locally and thus we give some examples of functions (auto-similarity and template similarity) which, given one or two images, computes a similarity measurement between patches. Two patches are said to be similar if the similarity measurement is small enough. To derive a criterion for taking a decision on the similarity between two patches we present an a contrario model. Namely, two patches are said to be similar if the associated similarity measurement is unlikely to happen in a background model. Choosing Gaussian random fields as background models we derive non-asymptotic expressions for the probability distribution function of similarity measurements. We introduce a fast algorithm in order to assess redundancy in natural images and present applications in denoising, periodicity analysis and texture ranking., Comment: Submitted to SIIMS

International audience; We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing -invariant R-matrix. We study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of Bethe vectors are identical up to normalization and reshuffling of the Bethe parameters. To prove this correspondence we use the current approach. This identity gives new combinatorial relations for the scalar products of the Bethe vectors. The q-deformed case, as well as the superalgebra case, are also evoked in the conclusion.

In this paper we focus on a general optimal control problem involving a dynamical system described by a nonlinear Caputo fractional differential equation of order 0 α≤ 1, associated to a general Bolza cost written as the sum of a standard Mayer cost and a Lagrange cost given by a Riemann-Liouville fractional integral of order β ≥α. In addition the present work handles general control and mixed initial/final state constraints. Adapting the standard Filippov's approach based on appropriate compactness assumptions and on the convexity of the set of augmented velocities, we give an existence result for at least one optimal solution. Then, the major contribution of this paper is the statement of a Pontryagin maximum principle which provides a first-order necessary optimality condition that can be applied to the fractional framework considered here. In particular, Hamiltonian maximization condition and transversality conditions on the adjoint vector are derived. Our proof is based on the sensitivity analysis of the Caputo fractional state equation with respect to needle-like control perturbations and on Ekeland's variational principle. The paper is concluded with two illustrating examples and with a list of several perspectives for forthcoming works.

International audience; In this paper we study the ergodic problem for viscous Hamilton-Jacobi equations with superlinear Hamiltonian and inward drift. We investigate (i) existence and uniqueness of eigenfunctions associated with the generalized principal eigenvalue of the ergodic problem, (ii) relationships with the corresponding stochastic control problem of both finite and infinite time horizon, and (iii) the precise growth exponent of the generalized principal eigenvalue with respect to a perturbation of the potential function.

International audience; Motivated by experiments on cell segregation, we present a two-species model of interacting particles, aiming at a quantitative description of this phenomenon. Under precise scaling hypothesis, we derive from the microscopic model a macroscopic one and we analyze it. In particular, we determine the range of parameters for which segregation is expected. We compare our analytical results and numerical simulations of the macroscopic model to direct simulations of the particles, and comment on possible links with experiments.

We show that the scalar products of on-shell and off-shell Bethe vectors in the algebra1ic Bethe ansatz solvable models satisfy a system of linear equations. We find solutions to this system for a wide class of integrable models. We also apply our method to the XXX spin chain with broken $U(1)$ symmetry., 16 pages, no figures

We provide an explicit isomorphism between a quotient of the Bannai--Ito algebra and the Brauer algebra. We clarify also the connection with the action of the Lie superalgebra osp(1|2) on the threefold tensor product of its fundamental representation. Finally, a conjecture is proposed to describe the centralizer of osp(1|2) acting on three copies of an arbitrary finite irreducible representation in terms of a quotient of the Bannai-Ito algebra., Comment: Dedicated to the memory of Peter G. O. Freund

The conformal anomaly in curved spacetime generates a nontrivial anomalous vertex, given by the three-point correlation function $TTT$ of the energy-momentum tensor $T^{\mu\nu}$. We show that a temperature inhomogeneity in a gas of charged massless particles generates, via the $TTT$ vertex, a pressure anisotropy with respect to the axis of the temperature variation. This very particular signature may provide an experimental access to the elusive gravitational coefficient $b$ which determines the anomaly contribution of the Weyl tensor to the trace of the energy-momentum tensor in curved spacetime. We present an estimate of the pressure anisotropy both for fermionic quasiparticles in the solid-state environment of Dirac semimetals as well as for a quark-gluon plasma in relativistic heavy-ion collisions. In both cases, the pressure anisotropy is small compared to the mean thermal pressure., Comment: 9 pages, 2 figures. Revised final version accepted for publication on Physics Letters B

We study existence and stability of solutions of (E 1) --$\Delta$u + $\mu$ |x| 2 u + g(u) = $\nu$ in $\Omega$, u = 0 on $\partial$$\Omega$, where $\Omega$ is a bounded, smooth domain of R N , N $\ge$ 2, containing the origin, $\mu$ $\ge$ -- (N --2) 2 4 is a constant, g is a nondecreasing function satisfying some integral growth assumption and $\nu$ is a Radon measure on $\Omega$. We show that the situation differs according $\nu$ is diffuse or concentrated at the origin. When g is a power we introduce a capacity framework to find necessary and sufficient condition for solvability.

We study linear perturbations of a rotating black hole solution that has been recently discovered in degenerate higher-order scalar-tensor (DHOST) theories. We find a parametrization which permits the explicit resolution of the scalar perturbation while the tensor perturbation is obtained as a Teukolsky equation supplemented by an effective source term. The effective source term is related to the black hole hair and can be computed exactly for any value of the black hole spin. We discuss how the perturbations of the geometry and thus the emitted gravitational waves could be modified in comparison with general relativity., Comment: 6 pages

We study cosmology in a class of minimally modified gravity (MMG) with two local gravitational degrees of freedom. We classify modified gravity theories into type-I and type-II: theories of type-I have an Einstein frame and can be recast by change of variables as general relativity (GR) with a non-minimal matter coupling, while theories of type-II have no Einstein frame. Considering a canonical transformation of the lapse, the 3-dimensional induced metric and their conjugate momenta we generate type-I MMG. We then show that phenomenological deviations from GR, such as the speed of gravitational waves $c_T$ and the effective gravitational constant for scalar perturbations $G_{\rm eff}$, are characterized by two functions of an auxiliary variable. We study the phenomenology of several models all having $c_T=1$. We obtain a scenario with $c_T=1$ in which the effective equation-of-state parameter of dark energy is different from $-1$ even though the cosmic acceleration is caused by a bare cosmological constant, and we find that it is possible to reconstruct the theory on choosing a selected time-evolution for the effective dark energy component., 14 pages, 1 figure, updated to match the version to appear in JCAP

International audience; In the context of connections between algebras coming from quantum integrable systems and algebras associated to the orthogonal polynomials of the Askey scheme, we prove that the truncated reflection algebra attached to the Yangian of sl2 is isomorphic to the Hahn algebra. As a by-product, we provide a general setup based on Euler polynomials to study truncations of reflection algebras.

International audience; We study the universal Hopf algebra L of Majid and Lyubashenko in the case that the underlying ribbon category is the category of representations of a finite dimensional ribbon quasi-Hopf algebra A . We show that L=A⁎ with coadjoint action and compute the Hopf algebra structure morphisms of L in terms of the defining data of A . We give explicitly the condition on A which makes Rep A factorisable and compute Lyubashenko's projective SL(2,Z) -action on the centre of A in this case.

Minimally modified gravity theories are modifications of general relativity with two local gravitational degrees of freedom in four dimensions. Their construction relies on the breaking of 4D diffeomorphism invariance keeping however the symmetry under 3D diffeomorphisms. Here, we construct these theories from a Hamiltonian point of view. We start with the phase space of general relativity in the ADM formalism. Then, we find the conditions that the Hamiltonian must satisfy for the theory to propagate (up to) two gravitational degrees of freedom with the assumptions that the lapse and the shift are not dynamical, and the theory remains invariant under 3D diffeomorphisms. This construction enables us to recover the well-known "cuscuton" class of scalar-tensor theories in the unitary gauge. We also exhibit a new class of interesting theories, that we dubb $f({\cal H})$ theories, where the usual Hamiltonian constraint $\cal H$ of general relativity is replaced by $f({\cal H})$ where $f$ is an arbitrary function., 20 pages

The large $j$ asymptotic behavior of $4$-dimensional spin foam amplitude is investigated for the extended spin foam model (Conrady-Hnybida extension) on a simplicial complex. We study the most general situation in which timelike tetrahedra with timelike triangles are taken into account. The large $j$ asymptotic behavior is determined by critical configurations of the amplitude. We identify the critical configurations that correspond to the Lorentzian simplicial geometries with timelike tetrahedra and triangles. Their contributions to the amplitude are phases asymptotically, whose exponents equal to Regge action of gravity. The amplitude may also contains critical configurations corresponding to non-degenerate split signature $4$-simplices and degenerate vector geometries. But for vertex amplitudes containing at least one timelike tetrahedron and one spacelike tetrahedron, critical configurations only give Lorentzian $4$-simplices, while the split signature and degenerate $4$-simplices do not appear., Comment: 24+28 pages, 2 figures

In this short note we wonder about the explicit expression of the expanding of the $p$-th Lovelock product. We use the 1990's works of S. A. Fulling et al. on the symmetries of the Riemann tensor, and we show that the number of independent scalars appearing in this expanding is equal to the number of Young diagrams with all row lengths even in the decomposition of the $p$-th plethysm of the Young diagram representing the symmetries of the Riemann tensor., Comment: 5 pages

The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey-Wilson algebra. The operators that these polynomials respectively diagonalize are identified within the Askey-Wilson algebra generated by twisted primitive elements of $\mathfrak U_q(sl(2))$. The little q-Jacobi operator and a tridiagonalization of it are shown to realize the equitable embedding of the Askey-Wilson algebra into $\mathfrak U_q(sl(2))$., v2: References added. 15 pp

We point out that results of Shimizu on internal characters imply a useful non-semisimple variant of the categorical Verlinde formula for factorisable finite tensor categories. When combined with results on pseudo-trace functions by Miyamoto and Arike-Nagatomo, one can make a precise conjecture for a non-semisimple modular Verlinde formula which relates modular properties of pseudo-trace functions and the product in the Grothendieck ring of the corresponding vertex operator algebra., Comment: 37 pages

We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For $\mathfrak{su}(2)$, this leads to the Heun-Krawtchouk algebra. The corresponding Heun-Krawtchouk operator is identified as the Hamiltonian of the quantum analogue of the Zhukovski-Voltera gyrostat. For $\mathfrak{su}(1,1)$, one obtains the Heun algebras attached to the Meixner, Meixner-Pollaczek and Laguerre polynomials. These Heun algebras are shown to be isomorphic the the Hahn algebra. Focusing on the harmonic oscillator algebra $\mathfrak{ho}$ leads to the Heun-Charlier algebra. The connections to orthogonal polynomials are achieved through realizations of the underlying Lie algebras in terms of difference and differential operators. In the $\mathfrak{su}(1,1)$ cases, it is observed that the Heun operator can be transformed into the Hahn, Continuous Hahn and Confluent Heun operators respectively., Comment: 15 pages

We present a simple construction for a tridiagonal matrix $T$ that commutes with the hopping matrix for the entanglement Hamiltonian ${\cal H}$ of open finite free-Fermion chains associated with families of discrete orthogonal polynomials. It is based on the notion of algebraic Heun operator attached to bispectral problems, and the parallel between entanglement studies and the theory of time and band limiting. As examples, we consider Fermionic chains related to the Chebychev, Krawtchouk and dual Hahn polynomials. For the former case, which corresponds to a homogeneous chain, the outcome of our construction coincides with a recent result of Eisler and Peschel; the latter cases yield commuting operators for particular inhomogeneous chains. Since $T$ is tridiagonal and non-degenerate, it can be readily diagonalized numerically, which in turn can be used to calculate the spectrum of ${\cal H}$, and therefore the entanglement entropy., 17 pages, final version

We first establish a sharp relation between the order of vanishing of a Dirichlet eigenfunction at a point and the leading term of the asymptotic expansion of the Dirichlet eigenvalue variation, as a removed compact set concentrates at that point. Then we apply this spectral stability result to the study of the asymptotic behaviour of eigenvalues of Aharonov-Bohm operators with two colliding poles moving on an axis of symmetry of the domain.

An operator of Heun-Askey-Wilson type is diagonalized within the framework of the algebraic Bethe ansatz using the theory of Leonard pairs. For different specializations and the generic case, the corresponding eigenstates are constructed in the form of Bethe states, whose Bethe roots satisfy Bethe ansatz equations of homogeneous or inhomogenous type. For each set of Bethe equations, an alternative presentation is given in terms of `symmetrized' Bethe roots. Also, two families of on-shell Bethe states are shown to generate two explicit bases on which a Leonard pair acts in a tridiagonal fashion. In a second part, the (in)homogeneous Baxter T-Q relations are derived. Certain realizations of the Heun-Askey-Wilson operator as second q-difference operators are introduced. Acting on the Q-polynomials, they produce the T-Q relations. For a special case, the Q-polynomial is identified with the Askey-Wilson polynomial, which allows one to obtain the solution of the associated Bethe ansatz equations. The analysis presented can be viewed as a toy model for studying integrable models generated from the Askey-Wilson algebra and its generalizations. As examples, the q-analog of the quantum Euler top and various types of three-sites Heisenberg spin chains in a magnetic field with inhomogeneous couplings, three-body and boundary interactions are solved. Numerical examples are given. The results also apply to the time-band limiting problem in signal processing., 49 pp; v2: Compared with published version, typos corrected in eqs. (1.7), (3.75), eq. between (3.76)-(3.77) and Prop. 3.7

We construct the general solution of a class of Fuchsian systems of rank $N$ as well as the associated isomonodromic tau functions in terms of semi-degenerate conformal blocks of $W_N$-algebra with central charge $c=N-1$. The simplest example is given by the tau function of the Fuji-Suzuki-Tsuda system, expressed as a Fourier transform of the 4-point conformal block with respect to intermediate weight. Along the way, we generalize the result of Bowcock and Watts on the minimal set of matrix elements of vertex operators of the $W_N$-algebra for generic central charge and prove several properties of semi-degenerate vertex operators and conformal blocks for $c=N-1$., 31 pages, 1 figure

Defining a suitable erosive factor for interrill erosion has been a long-going discussion that was not resolved by experimental results.: Insights about the choice of an erosive factor.[br/] In this paper, by using computational fluid simulations of individual raindrop impacts, the relevance of four erosive factors was assessed for a range of soil resistance, rainfall type and water layer depth.[br/] Computation results show that the erosivity exponent associated to the erosive factors had a very low sensitivity to soil resistance.[br/] This confirms the common practice of separating rainfall erosivity from soil erodibility. The erosivity exponent was found to be very sensitive to the rainfall type. Therefore, an universal exponent could not be found, its value depending on the rainfall type.[br/] As a result of the balance between the shear stress development and the protective effect of the water layer, the soil detachment rate was maximum for a specific water depth. This supports to better account for the water layer depth in erosion models. Overall, the computational approach shows that the choice of an erosive factor may not be of practical importance. It also encourages for new experimental designs, able to evaluate soil detachment by raindrops — and not only the amount of splashed soil.

We propose in this article a model describing the dynamic of a system of adipocytes, struc-tured by their sizes. This model takes into account the differentiation of a population of mesenchymal cells into preadipocytes and of preadipocytes into adipocytes; the differentiation rates depend on the mean adipocyte radius. The considered equations are therefore ordinary differential equations, coupled with an advection equation, the growth rate of which depends on food availability and on the total surface of adipocytes. Since this velocity is discontinuous, we need to introduce a convenient notion of solutions coming from Filippov theory. We are consequently able to determine the stationary solutions of the system, to prove the existence and uniqueness of solutions and to describe the asymptotic behavior of solutions in some simple cases. Finally, the parameters of the model are fitted thanks to some experimental data and numerical simulations are displayed; a spatial extension of the model is studied numerically., Nous proposons dans cet article un modèle décrivant la dynamique d'un ensemble d'adipocy-tes structurés en taille. Ce modèle tient compte de la différentiation d'une population de cellules mésenchymales en préadipocytes et des préadipocytes en adipocytes; les taux de différentiation dépen-dent du rayon moyen des adipocytes. LeséquationsLeséquations considérées sont, par conséquence, deséquationsdeséquations différentielles ordinaires, couplées avec uné equation d'advection, dont le taux de croissance dépend de la nourriture disponible et de la surface totale des adipocytes. Comme la vitesse est discontinue, nous introduisons une notion appropriée des solutions, provenant de la théorie de Filippov. Nous sommes alors en mesure de déterminer les solutions stationnaires du système, de prouver l'existence et l'unicité de solutions et de décrire le comportement asymptotique des solutions dans certains cas simples. Finalement, les paramètres du modèles sont estimés grâcè a des données expérimentales et des simulations numériques sont présentées; une extension spatiale du modèles estétudiéeestétudiée numériquement.

We have recently introduced a new and very simple action for three-dimensional massive gravity. This action is written in a first order formulation where the triad and the connection play a manifestly symmetric role, but where internal Lorentz gauge symmetry is broken. The absence of Lorentz invariance, which in this model is the mechanism underlying the propagation of a massive graviton, does however prevent from writing a purely metric non-linear action for the theory. Nonetheless, in this letter, we explain how to disentangle, at the non-linear level, the metric and non-metric degrees of freedom in the equations of motion. Focusing on the metric part, we show that it satisfies modified Einstein equations with higher derivative terms. As a particular case, these equations reproduce a well-studied model known as minimal massive gravity. In the general case, we obtain new metric field equations for massive gravity in three dimensions starting from the simple first order action. These field equations are consistent through a mechanism known as "third way consistency", which our theory therefore provides a new example of., 5 pages, published version

The consequences of coupling magnetic and elastic degrees of freedom, where spins and deformations are carried by point-like objects subject to local interactions, are studied, theoretically and by detailed numerical simulations. From the constrained Lagrangians we derive consistent equations of motion for the coupled dynamical variables. In order to probe the dynamics of such a system, we consider external perturbations, such as spin transfer torques for the magnetic part, and homogeneous stresses for the elastic part, associated to their corresponding damping. This approach is applied to the study of ultrafast switching processes in anti-ferromagnetic systems, which have recently attracted attention as candidates for anti-ferromagnetic spintronic devices. Our strategy is then checked in simple, but instructive, situations. We carried out numerical experiments to study, in particular, how the magnetostrictive coupling and external stresses affect the nature of the switching processes in a prototype anti-ferromagnetic material., 6 pages, LaTeX2e, 3 figures. Highlighted the relation between the degrees of freedom that appear in the Lagrangians and the macroscopic magnetic quantities

One of the utmost interesting properties of matter is magnetism. This property, which is a macroscopic consequence of quantum physics, is subjected and couples to several reservoirs. Among them, two are most relevant, namely the thermal and mechanical reservoirs. We build a Hamiltonian model for the coupling between -- classical -- magnetism and elasticity, which relies on the -- underlying -- anticommuting nature of spin, so as to describe the coupled dynamics of these degrees of freedom.The first step is to study the behavior of the classical spin -- or magnetic moment -- when coupled to different -- stochastic -- baths. First a spin bath, so as to investigate if and how such an effective model can mimic the couplings, to different magnetic moments but also to the elastic structure of the compound. A different approach is then followed where, through a Nambu dynamics model for spin precession, the ways in which this spin can be coupled to a bath, additively or multiplicatively, are studied in order to make out which is better suited to describe coupling phenomena in magnetism. Those are then studied numerically, initially stochastically, with the appropriate averaging procedure over different realizations and then deterministically, by building an effective model for the moments of the statistical distributions. This model is obtained by truncating the thus derived hierarchy of moments, so as to construct a quicker and deterministic method to deduce magnetic properties of a system.The second step is to construct models for magnetoelastic coupling, which we do via ``virtual'' particles carrying both localized magnetic moment and mechanical strain tensor. We begin by a Lagrangian formulation for the precession of spin, which is coupled to a dynamical elastic solid by a magnetoelastic coupling term. This enables us to study their coupled dynamics in a way that is fully consistent with all the symmetries, which ensures a consistent description.We then shift to a Hamiltonian description where spin is interpreted as a composite -- commuting -- variable, which is a product of underlying and not observable -- anticommuting -- variables. Such a spin interacts with a couple of canonically conjugate variables representing the elastic medium, in an extended Poisson structure. Finally, for each of these two models, we numerically study the influence of an external stress on the switching behavior of the Néel order parameter and spin accumulation for a NiO toy model antiferromagnet, induced by an external spin-transfer-torque., Le magnétisme est l'un des plus anciens phénomènes rapportés de l'histoire des sciences naturelles et probablement l'un des plus fascinants. Véritable manifestation macroscopique de la physique quantique, il subit en s'y couplant, l'influence de nombreux réservoirs énergétiques et statistiques, dont ceux de la thermique et de la mécanique.En remarquant qu'un moment magnétique élémentaire est un objet composite formé grâce à des variables anticommutantes inobservables, on peut engendrer une dynamique Hamiltonienne couplant ce degré de liberté à ceux provenant des autres réservoirs, eux-mêmes décrits par la dynamique de variables aléatoires.La première étape est d'étudier la dynamique d'un moment magnétique, vu comme un spin classique dans de tels bains. \`{A} cette fin on considère un bain magnétique afin d'évaluer la possibilité de mimer les effets de couplage entre moments magnétiques ainsi que le couplage magnétoélastique par un tel modèle effectif.Par la suite, nous montrons que la précession d'un spin classique peut être modélisée par une dynamique de Nambu qui facilite la description de la nature, additive ou multiplicative, des couplages stochastiques. La dynamique ainsi produite est d'abord étudiée numériquement de façon stochastique en moyennant les différentes réalisations obtenues; ensuite, un modèle déterministe sur la hiérarchie des moments statistiques est établi puis fermé afin de développer une méthode à la fois plus rapide, mais également déterministe de déduction des propriétés magnétiques.Finalement, pour illustrer la pertinence tangible de toutes ces notions, nous construisons une dynamique étendue de particules ``fictives'' portant à la fois un moment magnétique et une déformation mécanique locaux exprimant la magnétoélasticité, d'une part dans une approche Lagrangienne puis Hamiltonienne. Pour chacune des deux approches nous étudierons la dynamique du retournement ultrarapide d'aimantation pour NiO, oxyde antiferromagnétique prototype, sous sollicitations mécanique et électrique.Le formalisme, exposé ici, aussi bien conceptuel qu’informatique, ne sert pas, seulement, comme un exemple de l’état de l’art, mais permet une description des propriétés des milieux magnétiques, qui est fondamentale aussi bien pour la conception de nouveaux matériaux, que comme modèle pour aborder d’autres questions portant sur l’interaction entre bruit et variables dynamiques, plus généralement.

The recently discovered conserved quantity associated with Kepler rescaling is generalised by an extension of Noether's theorem which involves the classical action integral as an additional term. For a free particle the familiar Schroedinger dilations are recovered. A general pattern arises for homogeneous potentials. The associated conserved quantity allows us to derive the virial theorem. The relation to the Bargmann framework is explained and illustrated by exact plane gravitational waves., Comment: Extended version. 18 pages, 3 figures. Published version

The minimal theory of quasidilaton massive gravity with or without a Horndeski-type kinetic term for the quasidilaton field propagates only three physical modes: the two massive tensor polarizations and one scalar mode. This reduced number of degrees of freedom is realized by a Lorentz symmetry violation at cosmological scales and the presence of appropriate constraints that remove unwanted modes. Vacuum cosmological solutions have been considered in a previous work, and it has been shown that the late-time de Sitter attractor is stable under inhomogeneous perturbations. In this work, we explore the stability of cosmological solutions in the presence of matter fields. Assuming for simplicity that the quasidilaton scalar is on an attractor at the level of the background, we derive stability conditions in the subhorizon limit, and find the scalar sound speeds, as well as the modification with respect to general relativity to the gravitational potential in the quasistatic approximation. We also find that the speed limit of gravitational waves coincides with the speed of light for any homogeneous and isotropic cosmological background, on or away from the attractor., Comment: 14 pages