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The derivation of shallow water models from Navier–Stokes equations is revisited yielding a class of two-layer shallow water models. An improved velocity profile is proposed, based on the superposition of an inviscid fluid and a viscous layer inspired by the Interactive Boundary Layer interaction used in aeronautics. This leads to a new friction law which depends not only on velocity and depth but also on the variations of velocity and thickness of the viscous layer. The resulting system is an extended shallow water model consisting of three depth-integrated equations: the first two are mass and momentum conservation in which a slight correction on hydrostatic pressure has been made; the third one, known as von Kármán equation, describes the evolution of the viscous layer. This coupled model is shown to be conditionally hyperbolic, and a Godunov-type finite volume scheme is also proposed. Several numerical examples are provided and compared to the Multi-Layer Saint-Venant model. They emphasize the ability of the model to deal with unsteady viscous effects. They illustrate also the phase-lag between friction and topography, and even recover possible reverse flows.

International audience; Fat cells, called adipocytes, are designed to regulate energy homeostasis by storing energy in the form of lipids. Adipocyte size distribution is assumed to play a role in the development of obesity-related diseases. This population of cells that do not have a characteristic size, indeed a bimodal size distribution is observed in adipose tissue. We propose a model based on a partial differential equation to describe adipocyte size distribution. The model includes a description of the lipid fluxes and the cell size fluctuations and using a formulation of a stationary solution fast computation of bimodal distribution is achieved. We investigate the parameter identifiability and estimate parameter values with CMA-ES algorithm. We first validate the procedure on synthetic data, then we estimate parameter values with experimental data of 32 rats. We discuss the estimated parameter values and their variability within the population, as well as the relation between estimated values and their biological significance. Finally, a sensitivity analysis is performed to specify the influence of parameters on 1 cell size distribution and explain the differences between the model and the measurements. The proposed framework enables the characterization of adipocyte size distribution with four parameters and can be easily adapted to measurements of cell size distribution in different health conditions.

We address a problem of stabilization of a scalar conservation laws with fixed flux pointwiseconstraints. The PDE models the impact of two toll gates on the traffic flow and the feedbacklaws is designed to limit the number of cars passing through the two toll gates. The existenceof solutions is proved using a Schauder’s fixed point theorem, together with a coupled PDEODE. To prove stabilization results, we study the structure of the solutions using an extensionof backward generalized characteristics.

In this work, we analyse the metastability of non-reversible diffusion processes $$dX_t=\boldsymbol{b}(X_t)dt+\sqrt h\,dB_t$$ on a bounded domain $\Omega$ when $\mathbf{b}$ admits the decomposition $\mathbf{b}=-(\nabla f+\mathbf{\ell})$ and $\nabla f \cdot \mathbf{\ell}=0$. In this setting, we first show that, when $h\to 0$, the principal eigenvalue of the generator of $(X_t)_{t\ge 0}$ with Dirichlet boundary conditions on the boundary $\partial\Omega$ of $\Omega$ is exponentially close to the inverse of the mean exit time from $\Omega$, uniformly in the initial conditions $X_0=x$ within the compacts of $\Omega$. The asymptotic behavior of the law of the exit time in this limit is also obtained. The main novelty of these first results follows from the consideration of non-reversible elliptic diffusions whose associated dynamical systems $\dot X=\mathbf{b}(X)$ admit equilibrium points on $\partial\Omega$. In a second time, when in addition $\div \mathbf{\ell} =0$, we derive a new sharp asymptotic equivalent in the limit $h\to 0$ of the principal eigenvalue of the generator of the process and of its mean exit time from $\Omega$. Our proofs combine tools from large deviations theory and from semiclassical analysis, and truly relies on the notion of quasi-stationary distribution.

Biological data show that the size distribution of adipose cells follows a bimodal distribution. In this work, we introduce a Lifshitz-Slyozov type model, based on a transport partial differential equation, for the dynamics of the size distribution of adipose cells. We prove a new convergence result from the related Becker-Döring model, a system composed of several ordinary differential equations, toward mild solutions of the Lifshitz-Slyozov model using distribution tail techniques. Then, this result allows us to propose a new advective-diffusive model, the second-order diffusive Lifshitz-Slyozov model, which is expected to better fit the experimental data. Numerical simulations of the solutions to the diffusive Lifshitz-Slyozov model are performed using a well-balanced scheme and compared to solutions to the transport model. Those simulations show that both bimodal and unimodal profiles can be reached asymptotically depending on several parameters. We put in evidence that the asymptotic profile for the second-order system does not depend on initial conditions, unlike for the transport Lifshitz-Slyozov model.

Given any measurable subset $\omega$ of a closed Riemannian manifold $(M,g)$ and given any $T>0$, we define $\ell^T(\omega)\in[0,1]$ as the smallest average time over $[0,T]$ spent by all geodesic rays in $\omega$. This quantity appears naturally when studying observability properties for the wave equation on $M$, with $\omega$ as an observation subset: the condition $\ell^T(\omega)>0$ is the well known \emph{Geometric Control Condition}.In this article we establish two properties of the functional $\ell^T$, one is geometric and the other is probabilistic.The first geometric property is on the maximal discrepancy of $\ell^T$ when taking the closure. We may have $\ell^T(\mathring{\omega})1/2$ then the Geometric Control Condition is satisfied and thus the wave equation is observable on $\omega$ in time $T$. The second property is of probabilistic nature. We take $M=\mathbb{T}^2$, the flat two-dimensional torus, and we consider a regular grid on it, a regular checkerboard made of $n^2$ square white cells. We construct random subsets $\omega_\varepsilon^n$ by darkening each cell in this grid with a probability $\varepsilon$. We prove that the random law $\ell^T(\omega_\varepsilon^n)$ converges in probability to $\varepsilon$ as $n\rightarrow+\infty$.As a consequence, if $n$ is large enough then the Geometric Control Condition is satisfied almost surely and thus the wave equation is observable on $\omega_\varepsilon^n$ in time $T$.

International audience; Interrill erosion is driven by raindrops and sheet flow. Raindrop impacts cause sediment detachment and splash, but can also affect flow transport. Even if these processes have been studied for long, the actual effect of raindrop impacts on particle settling velocities has not been experimentally assessed. This leads to unconstrained adjustments in the soil erosion models, the settling velocity of particles being a freely adjustable parameter allowing for better fitting the particle flux measured at the outlet. To address the effect of raindrop impacts on the settling of particles in sheet flow, a laboratory flume experiment was designed, using an upstream feeder of sediment (100–200 μm) and simulated rainfalls. It reproduced conditions close to sheet flow, while not allowing for the detachment of particles from the flume bottom. Two series of experiments were run: a series with a high rainfall intensity (175 mm h−1) generated by an oscillating-nozzle rainfall simulator, and a series with lower rainfall intensities (10, 15, 25, 35, 55 mm h−1) generated by a drop-former rainfall simulator. When a rainfall was applied, it systematically decreased the sediment concentration at the outflow compared to the no rain condition, however no obvious relationship was found with the rainfall intensity. This shows that raindrop impacts increase particle settling velocities in sheet flow. Two underlying mechanisms are suggested, related to the momentum of the raindrops or to the turbulence caused by the raindrops into the flow. Further studies should be carried out, using computational fluid dynamics and collaboration with the fluid mechanics community.

We study random walks on relatively hyperbolic groups whose law is convergent, in the sense that the derivative of its Green function is finite at the spectral radius.When parabolic subgroups are virtually abelian, we prove that for such a random walk satisfies a local limit theorem of the form $p_n(e, e)\sim CR^{-n}n^{-d/2}$, where $p_n(e, e)$ is the probability of returning to the origin at time $n$, $R$ is the inverse of the spectral radius of the random walk and $d$ is the minimal rank of a parabolic subgroup along which the random walk is spectrally degenerate.This concludes the classification all possible behaviour for $p_n(e, e)$ on such groups.; Nous étudions les marches aléatoires sur des groupes relativement hyperboliques dont la loi est convergente, au sens où la dérivée de sa fonction de Green est finie au rayon spectral.Lorsque les sous-groupes paraboliques sont virtuellement abéliens, nous montrons qu'une telle marche vérifie un théorème de la limite locale de la forme $p_n(e, e)\sim CR^{-n}n^{-d/2}$, où $p_n(e, e)$ est la probabilité de retour à l'origine au temps $n$, $R$est l'inverse du rayon spectral de la marche et $d$ est le rang minimal d'un sous-groupe parabolique le long duquel la marche aléatoire est spectralement dégénérée.

International audience; We establish some properties of quantum limits on a product manifold, proving for instance that, under appropriate assumptions, the quantum limits on the product of manifolds are absolutely continuous if the quantum limits on each manifolds are absolutely continuous. On a product of Riemannian manifolds satisfying the minimal multiplicity property, we prove that a periodic geodesic can never be charged by a quantum limit.

We derive by the traditional algebraic Bethe ansatz method the Bethe equations for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [J. Phys. A 37 (2004), 433-440, arXiv:hep-th/0304092]. The technical difficulties due to the breaking of $\mathsf{U}(1)$ symmetry and the absence of a reference state are overcome by an algebraic construction where the two-boundary Temperley-Lieb Hamiltonian is realised in a new $U_{\mathfrak{q}}\mathfrak{sl}_2$-invariant spin chain involving infinite-dimensional Verma modules on the edges [J. High Energy Phys. 2022 (2022), no. 11, 016, 64 pages, arXiv:2207.12772]. The equivalence of the two Hamiltonians is established by proving Schur-Weyl duality between $U_{\mathfrak{q}}\mathfrak{sl}_2$ and the two-boundary Temperley-Lieb algebra. In this framework, the Nepomechie condition turns out to have a simple algebraic interpretation in terms of quantum group fusion rules.

22 pages, 30 réf.; We illustrate with several new applications the power and elegance of the Bendixson–Dulac theorem to obtain upper bounds of the number of limit cycles for several families of planar vector fields. In some cases we propose to use a function related with the curvature of the orbits of the vector field as a Dulac function. We get some general results for Li´enard type equations and for rigid planar systems. We also present a remarkable phenomenon: for each integer m ≥ 2, we provide a simple 1-parametric differential system for which we prove that it has limit cycles only for the values of the parameter in a subset of an interval of length smaller that 3√2(3/m)m/2 that decreases exponentially when m grows. One of the strengths of the results presented in this work is that although they are obtained with simple calculations, that can be easily checked by hand, they improve and extend previous studies. Another one is that, for certain systems, it is possible to reduce the question of the number of limit cycles to the study of the shape of a planar curve and the sign of an associated function in one or two variables.

Neural style transfer is a deep learning technique that produces an unprecedentedly rich style transfer from a style image to a content image and is particularly impressive when it comes to transferring style from a painting to an image. It was originally achieved by solving an optimization problem to match the global style statistics of the style image while preserving the local geometric features of the content image. The two main drawbacks of this original approach is that it is computationally expensive and that the resolution of the output images is limited by high GPU memory requirements. Many solutions have been proposed to both accelerate neural style transfer and increase its resolution, but they all compromise the quality of the produced images. Indeed, transferring the style of a painting is a complex task involving features at different scales, from the color palette and compositional style to the fine brushstrokes and texture of the canvas. This paper provides a solution to solve the original global optimization for ultra-high resolution images, enabling multiscale style transfer at unprecedented image sizes. This is achieved by spatially localizing the computation of each forward and backward passes through the VGG network. Extensive qualitative and quantitative comparisons show that our method produces a style transfer of unmatched quality for such high resolution painting styles., 10 pages, 5 figures

We study spectral properties of sub-Riemannian Laplacians, which are hypoelliptic operators. The main objective is to obtain quantum ergodicity results, what we have achieved in the 3D contact case. In the general case we study the small-time asymptotics of sub-Riemannian heat kernels. We prove that they are given by the nilpotentized heat kernel. In the equiregular case, we infer the local and microlocal Weyl law, putting in light the Weyl measure in sR geometry. This measure coincides with the Popp measure in low dimension but differs from it in general. We prove that spectral concentration occurs on the shief generated by Lie brackets of length r-1, where r is the degree of nonholonomy. In the singular case, like Martinet or Grushin, the situation is more involved but we obtain small-time asymptotic expansions of the heat kernel and the Weyl law in some cases. Finally, we give the Weyl law in the general singular case, under the assumption that the singular set is stratifiable.

``Do Carroll particles move?'' The answer depends on the characteristics of the particle such as its mass, spin, electric charge, and magnetic moment. A massive Carroll particle (closely related to fractons) does not move; its immobility follows from Carroll boost symmetry which implies dipole conservation, but not conversely. A massless Carroll particle may propagate by following the Hall law, consistently with the partial breaking of the Carroll boost symmetry. The framework is extended to Carroll field theory. In $d=2$ space dimensions, the Carroll group has a two-fold central extension which allows us to generalize the dynamics to massive and massless particles, including anyons. The anyonic spin and magnetic moment combine with the doubly-extended structure parameterized by two Casimir invariants interpreted as intrinsic magnetization and non-commutativity parameter. The extended Carroll particle subjected to an electromagnetic background field moves following a generalized Hall law which includes a Zeeman force. This theory is illustrated by massless, uncharged anyons with doubly-centrally extended structure we call exotic photons, which move on the horizon of a Black Hole, giving rise to an anyonic spin-Hall Effect., Thoroughly revised and extended version. 120 pages,13 figures

International audience; Building on the idea of Tolman and Ehrenfest that heat has weight, Luttinger established a deep connection between gravitational fields and thermal transport. However, this relation does not include anomalous quantum fluctuations that become paramount in strongly curved spacetime. In this work, we revisit the celebrated Tolman-Ehrenfest and Luttinger relations and show how to incorporate the quantum energy scales associated with these fluctuations, captured by gravitational anomalies of quantum field theories. We point out that such anomalous fluctuations naturally occur in the quantum atmosphere of a black hole. Our results reveal that analogous fluctuations are also observable in thermal conductors in flat-space time provided local temperature varies strongly. As a consequence, we establish that the gravitational anomalies manifest themselves naturally in non-linear thermal response of a quantum wire. In addition, we propose a systematic way to identify thermal analogues of black hole's anomalous quantum fluctuations associated to gravitational anomalies. We identify their signatures in propagating energy waves following a thermal quench, as well as in the energy density of heating Floquet states induced by repeated thermal quenches.

We introduce new $U_q\mathfrak{sl}_2$-invariant boundary conditions for the open XXZ spin chain. For generic values of $q$ we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $q=e^{\frac{i\pi}{p}}$ a $2p$-th root of unity $ - $ to its $p$-dimensional analogue. Both cases are parametrised by a continuous "spin" $\alpha\in\mathbb{C}$. To motivate our construction, we first specialise to $q=i$, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the $(\eta,\xi)$ ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic $q$ we investigate the algebraic properties of the underlying lattice algebras. We show that if $q^\alpha\notin\pm q^{\mathbb{Z}}$, the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to $U_q\mathfrak{sl}_2$. Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of $q$ and $\alpha$. Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains., Comment: 54 pages

International audience; We establish small-time asymptotic expansions for heat kernels of hypoelliptic Hörmander operators in a neighborhood of the diagonal, generalizing former results obtained in particular by Métivier and by Ben Arous. The coefficients of our expansions are identified in terms of the nilpotentization of the underlying sub-Riemannian structure. Our approach is purely analytic and relies in particular on local and global subelliptic estimates as well as on the local nature of small-time asymptotics of heat kernels. The fact that our expansions are valid not only along the diagonal but in an asymptotic neighborhood of the diagonal is the main novelty, useful in view of deriving Weyl laws for subelliptic Laplacians. In turn, we establish a number of other results on hypoelliptic heat kernels that are interesting in themselves, such as Kac's principle of not feeling the boundary, asymptotic results for singular perturbations of hypoelliptic operators, global smoothing properties for selfadjoint heat semigroups.

International audience; Enteric pathogens colonize the digestive tract of farm livestock, such as chickens or pigs and potentially infect food products, which results in a threat for human health and important economic losses. It has been shown that the ability to excrete the pathogen in the environment and contaminate other animals is variable. This heterogeneity in pathogen carriage and shedding results from interactions between the host’s immune response, the pathogen and the commensal intestinal microbiota and plays and important role in epidemic propagation and control. The objective here is to propose a theoretical model of pathogen excretion and transmission linking within and between host dynamic.

International audience; The Gauss map g of a surface ⌃ in R 4 takes its values in the Grassmannian of oriented 2-planes of R 4 : G + (2, 4). We give geometric criteria of stability for minimal surfaces in R 4 in terms of g. We show in particular that if the spherical area of the Gauss map |g(⌃)| of a minimal surface is smaller than 2⇡ then the surface is stable by deformations which fix the boundary of the surface. This answers the question of [BDC3] in R 4 .

In this paper, we prove sharp pointwise kernel estimates and dispersive properties for the linear wave equation on noncompact Riemannian symmetric spaces G/K of any rank with G complex. As a consequence, we deduce Strichartz inequalities for a large family of admissible pairs and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on hyperbolic spaces., Comment: To appear in Pure and Applied Analysis

Let $(X_t)_{t\ge 0}$ be the stochastic process solution to the overdamped Langevin dynamics $$dX_t=-\nabla f(X_t) \, dt +\sqrt h \, dB_t$$ and let $\Omega \subset \mathbb R^d $ be the basin of attraction of a local minimum of $f: \mathbb R^d \to \mathbb R$. Up to a small perturbation of $\Omega$ to make it smooth, we prove that the exit rates of $(X_t)_{t\ge 0}$ from $\Omega$ through each of the saddle points of $f$ on $\partial \Omega$ can be parametrized by the celebrated Eyring-Kramers laws, in the limit $h \to 0$. This result provides firm mathematical grounds to jump Markov models which are used to model the evolution of molecular systems, as well as to some numerical methods which use these underlying jump Markov models to efficiently sample metastable trajectories of the overdamped Langevin dynamics.

Using the fact that the horizon of black holes is a Carroll manifold, we show that an ``exotic photon'' i.e. a particle without mass and charge but with anyonic spin, magnetic moment and ``exotic'' charges associated with the 2-parameter central extension of the 2-dimensional Carroll group moves on the horizon of a Kerr-Newman Black Hole consistently with the Hall law., 17 pages, 1 figure. Published version

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function., 42 pages, 8 figures

Kinetic and hydrodynamic theories are widely employed for describing the collective behaviour of active matter systems. At the fluctuating level, these have been obtained from explicit coarse-graining procedures in the limit where each particle interacts weakly with many others, so that the total forces and torques exerted on each of them is of order unity at all times. Such limit is however not relevant for dilute systems that mostly interact via alignment; there, collisions are rare and make the self-propulsion direction to change abruptly. We derive a fluctuating kinetic theory, and the corresponding fluctuating hydrodynamics, for aligning self-propelled particles in the limit of dilute systems. We discover that fluctuations at kinetic level are not Gaussian and depend on the interactions among particles, but that only their Gaussian part survives in the hydrodynamic limit. At variance with fluctuating hydrodynamics for weakly interacting particles, we find that the noise variance at hydrodynamic level depends on the interaction rules among particles and is proportional to the square of the density, reflecting the binary nature of the aligning process. The results of this paper, which are derived for polar self-propelled particles with polar alignment, could be straightforwardly extended to polar particles with nematic alignment or to fully nematic systems., 23 pages, 4 figures

Artificial Intelligence; International audience; Osteoporosis is characterized by decrease of bone mineral density and alterations of the bone microarchitecture, leading to bone fragility and an increased risk of fractures. Discrimination between osteoporotic and healthy subjects using X-ray images presents a major challenge for medical image analysis and classification. This work introduces a new approach based on multifractal texture analysis for the characterization of the trabecular bone microarchitecture to distinguish between Control Subjects (CS) and Osteoporotic Patients (OP). The proposed method enables a description of both local and global regularity and roughness of the trabecular bone on radiographic images. To this end, the multifractal spectrum is exploited to extract new texture attributes and reveal bone microarchitecture changes due to osteoporosis. To assess the proposed approach effectiveness, tests were carried out for the classification of two populations (CS and OP) using a logistic regression model. A classification rate of 98.01% was reached showing that the proposed approach presents a promising potential as a complementary tool for osteoporosis diagnosis.

International audience

We prove that, given a constant $K> 2$ and a bounded linear operator $T$ from a JB$^*$-triple $E$ into a complex Hilbert space $H$, there exists a norm-one functional $\psi\in E^*$ satisfying $$\|T(x)\| \leq K \, \|T\| \, \|x\|_{\psi},$$ for all $x\in E$. Applying this result we show that, given $G > 8 (1+2\sqrt{3})$ and a bounded bilinear form $V$ on the Cartesian product of two JB$^*$-triples $E$ and $B$, there exist norm-one functionals $\varphi\in E^{*}$ and $\psi\in B^{*}$ satisfying $$|V(x,y)| \leq G \ \|V\| \, \|x\|_{\varphi} \, \|y\|_{\psi}$$ for all $(x,y)\in E \times B$. These results prove a conjecture pursued during almost twenty years., Comment: 23 pages. We corrected a misprint, added some comments and precised one reference

We propose to "gauge" the group of similarity transformations that acts on a space of non-Hermitian scalar theories. We introduce the "similarity gauge field", which acts as a gauge connection on the space of non-Hermitian theories characterized by (and equivalent to a Hermitian) real-valued mass spectrum. This extension leads to new effects: if the mass matrix is not the same in distant regions of space, but its eigenvalues coincide pairwise in both regions, the particle masses stay constant in the whole spacetime, making the model indistinguishable from a standard, low-energy and scalar Hermitian one. However, contrary to the Hermitian case, the high-energy scalar particles become unstable at a particular wavelength determined by the strength of the emergent similarity gauge field. This instability corresponds to momentum-dependent exceptional points, whose locations cannot be identified from an analysis of the eigenvalues of the coordinate-dependent squared mass matrix in isolation, as one might naively have expected. For a doublet of scalar particles with masses of the order of 1 MeV and a similarity gauge rotation of order unity at distances of 1 meter, the corrections to the masses are about 10^{-7}eV, which makes no experimentally detectable imprint on the low-energy spectrum. However, the instability occurs at 10^{18} eV, suggestively in the energy range of detectable ultra-high-energy cosmic rays, thereby making this truly non-Hermitian effect and its generalizations of phenomenological interest for high-energy particle physics., 14 pages, 4 figures; v2 matches the published version

We present a spatial model describing the growth of a photosynthetic microalgae biofilm. In this 2D-model we consider photosynthesis, cell carbon accumulation, extracellular matrix excretion, and mortality. The rate of each of these mechanisms is given by kinetic laws regulated by light, nitrate, oxygen and inorganic carbon. The model is based on mixture theory and the behaviour of each component is defined on one hand by mass conservation, which takes into account biological features of the system, and on the other hand by conservation of momentum, which expresses the physical properties of the components. The model simulates the biofilm structural dynamics following an initial colonization phase. It shows that a 75 μm thick active region drives the biofilm development. We then determine the optimal harvesting period and biofilm height which maximize productivity. Finally, different harvesting patterns are tested and their effect on biofilm structure are discussed. The optimal strategy differs whether the objective is to recover the total biofilm or just the algal biomass.

We consider the symplectic representation $\rho_n$ of a braid group $B(n)$ in $Sp(2l,\mathbb{Z})$, for $l=\Big[\dfrac{n-1}{2}\Big]$. If $P$ is a $4l^2$ polynomial on the coefficients of the matrices in $Sp(2l,\mathbb{Z})$, we show that the set $\{\beta\in B(n): P(\rho_n(\beta))=0\}$ is transient for non degenerate random walks on $B(n)$. If $n$ is odd, we derive that the $n$-braids $\beta$ verifying $|\Delta_{\hat{\beta}}(-1)|\leq C$ for some constant $C$ form a transient set: here $\Delta_{\hat{\beta}}$ denotes the Alexander polynomial of the closure of $\beta$. We also derive that for a random $3$-braid, the quasipositive links $(\beta\sigma_i\beta^{-1}\sigma_j)^p$ have zero signature for every integer $p$ and $1\leq i,j\leq 2$. \\ As an example of such braids, we investigate the signature of the Lissajous toric knots with $3$ strands.

In this work, we give sharp asymptotic equivalents in the small temperature regime of the smallest eigenvalues of the generator of some piecewise deterministic Markov processes (including the ZigZag process and the Bouncy Particle Sampler process) with refreshment rate $\alpha$ on the one-dimensional torus T. These asymptotic equivalents are usually called Eyring-Kramers type formulas in the literature. The case when the refreshment rate $\alpha$ vanishes on T is also considered.

A unified framework for the Chevalley and equitable presentation of $U_q(sl_2)$ is introduced. It is given in terms of a system of Freidel-Maillet type equations satisfied by a pair of quantum K-operators ${\mathcal K}^\pm$, whose entries are expressed in terms of either Chevalley or equitable generators. The Hopf algebra structure is reconsidered in light of this unified framework, and interwining relations for each pair of ${\mathcal K}^\pm$ are obtained. A K-operator solving a spectral parameter dependent Freidel-Maillet type equation is also considered. Different specializations of this K-operator are shown to admit a decomposition in terms of ${\mathcal K}^\pm$ of Chevalley or equitable type. Explicit examples of K-matrices without/with spectral parameter are derived by specializing the K-operators previously obtained., Comment: 20 pages; v2: Abstract improved, footnote 2 added

We consider the elliptic quasilinear equation - Δ m ⁢ u = u p ⁢ | ∇ ⁡ u | q {-\Delta_{m}u=u^{p}\lvert\nabla u\rvert^{q}} in ℝ N {\mathbb{R}^{N}} , q ≥ m {q\geq m} and p > 0 {p>0} , 1 < m < N {1 . Our main result is a Liouville-type property, namely, all the positive C 1 {C^{1}} solutions in ℝ N {\mathbb{R}^{N}} are constant. We also give their asymptotic behaviour; all the solutions in an exterior domain ℝ N ∖ B r 0 {\mathbb{R}^{N}\setminus B_{r_{0}}} are bounded. The solutions in B r 0 ∖ { 0 } {B_{r_{0}}\setminus\{0\}} can be extended as continuous functions in B r 0 {B_{r_{0}}} . The solutions in ℝ N ∖ { 0 } {\mathbb{R}^{N}\setminus\{0\}} has a finite limit l ≥ 0 {l\geq 0} as | x | → ∞ {\lvert x\rvert\to\infty} . Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a precise Osserman-type estimate for the equation satisfied by the gradient.

We estimate the bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces in terms of the critical exponents of appropriate Poincar\'e series. Our main result is the higher rank analog of a characterization due to Elstrodt, Patterson, Sullivan and Corlette in rank one. It improves upon previous results obtained by Leuzinger and Weber in higher rank., Comment: To appear in Geometriae Dedicata

We consider Fokker-Planck type differential operators associated with general Langevin processes admitting a Gibbs stationary distribution. Under assumptions insuring suitable resolvent estimates, we prove Eyring-Kramers formulas for the bottom of the spectrum of these operators in the low temperature regime. Our approach is based on the construction of sharp Gaussian quasimodes which avoids supersymmetry or PT-symmetry assumptions., 45 pages, 3 figures

The paper is concerned with a new approach for the recurrence property of the oscillating process on Z in Kemperman's sense. In the case when the random walk is ascending on Z − and descending on Z + , we determine the invariant measure of the embedded process of successive crossing times and then prove a necessary and sufficient condition for recurrence. Finally, we make use of this result to show that the general oscillating process is recurrent under some Hölder-typed moment assumptions.

This work is dedicated to the analysis of a class of energy stable and linearly well-balanced numerical schemes dedicated to the non-linear Shallow Water equations with Coriolis force. The proposed algorithms rely on colocated finite volume approximations formulated on cartesian geometries. They involve appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic equilibrium. We show that the resulting methods ensure semi-discrete energy estimates and numerical results show a very clear improvement around the nonlinear geostrophic equilibrium when compared to those of classic Godunov-type schemes.

43 pages, 4 fig.; We establish sharp pointwise kernel estimates and dispersive properties for the wave equation on noncompact symmetric spaces of general rank. This is achieved by combining the stationary phase method and the Hadamard parametrix, and in particular, by introducing a subtle spectral decomposition, which allows us to overcome a well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general. As consequences, we deduce the Strichartz inequality for a large family of admissible pairs and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on hyperbolic spaces.

We present new exact results for the complex generalized integral means spectrum (in the sense of [DHLZ18]) for two kinds of whole-plane Loewner evolutions driven by a Lévy process: (1) The case of a Lévy process with continuous trajectories, which corresponds to Schramm-Loewner evolution SLE κ with a drift term in the Brownian driving function. There is no known result for its standard integral means spectrum, and we show that a natural path to access it goes through the introduction of the complex generalized integral means spectrum, which is obtained via the so-called Liouville quantum gravity. (2) The case of symmetric Lévy processes for which we generalize results by Loutsenko and Yermolayeva ([Lou12, LY13, LY14, LY19]).

We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any bifurcation parameter, one can find parameters where $k$ critical points bifurcate \emph{independently}, with $k$ up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an analytical criterion for the non-vanishing of the self-intersections of the bifurcation current, we deduce the equality of the supports of the bifurcation current and the bifurcation measure for such families. Combined with results by Dujardin and Taflin, this also implies that the support of the bifurcation measure in these families has non-empty interior.As part of our proof we construct, in these families, subfamilies of codimension 1 where the bifurcation locus has non empty interior. This provides a new independent proof of the existence of holomorphic families of arbitrarily large dimension whose bifurcation locus has non empty interior. Finally, it shows that the Hausdorff dimension of the support of the bifurcation measure is maximal at any point of its support., 26 pages

We establish some algebraic properties of the group $\mathrm{Diff}(\mathbb{C}^n,0)$ of germs of analytic diffeomorphisms of $\mathbb{C}^n$, and its formal completion $\widehat{\mathrm{Diff}}(\mathbb{C}^n,0)$. For instance we describe the commutator of $\mathrm{Diff}(\mathbb{C}^n,0)$, but also prove that any finitely generated subgroup of $\mathrm{Diff}(\mathbb{C}^n,0)$ is residually finite; we thus obtain some constraints of groups that embed into $\mathrm{Diff}(\mathbb{C}^n,0)$. We show that $\widehat{\mathrm{Diff}}(\mathbb{C}^n,0)$ is an Hopfian group, and that $\mathrm{Diff}(\mathbb{C}^n,0)$ and $\widehat{\mathrm{Diff}}(\mathbb{C}^n,0)$ are not co-Hopfian. We end by the description of the automorphisms groups of $\widehat{\mathrm{Diff}}(\mathbb{C},0)$, and $\mathrm{Diff}(\mathbb{C},0)$.

We present new results for the complex generalized integral means spectrum for two kinds of whole-plane Loewner evolutions driven by L\'evy processes: - L\'evy processes with continuous trajectories, which correspond to Schramm-Loewner evolutions (SLE) with a drift term in the Brownian driving function. A natural path to access the standard integral means spectrum in the presence of drift goes through the introduction of the complex generalized integral means spectrum, which is obtained via the so-called Liouville quantum gravity. -Symmetric L\'evy processes for which we generalize recent results by Loutsenko and Yermolayeva., Comment: 48 pages, 13 figures. To be published in the special issue of Annales Henri Poincar\'e dedicated to the memory of Krzysztof Gawedzki

Building on classical invariant theory, it is observed that the polarised traces generate the centraliser $Z_L(sl(N))$ of the diagonal embedding of $U(sl(N))$ in $U(sl(N))^{\otimes L}$. The paper then focuses on $sl(3)$ and the case $L=2$. A Calabi--Yau algebra $\mathcal{A}$ with three generators is introduced and explicitly shown to possess a PBW basis and a certain central element. It is seen that $Z_2(sl(3))$ is isomorphic to a quotient of the algebra $\mathcal{A}$ by a single explicit relation fixing the value of the central element. Upon concentrating on three highest weight representations occurring in the Clebsch--Gordan series of $U(sl(3))$, a specialisation of $\mathcal{A}$ arises, involving the pairs of numbers characterising the three highest weights. In this realisation in $U(sl(3))\otimes U(sl(3))$, the coefficients in the defining relations and the value of the central element have degrees that correspond to the fundamental degrees of the Weyl group of type $E_6$. With the correct association between the six parameters of the representations and some roots of $E_6$, the symmetry under the full Weyl group of type $E_6$ is made manifest. The coefficients of the relations and the value of the central element in the realisation in $U(sl(3))\otimes U(sl(3))$ are expressed in terms of the fundamental invariant polynomials associated to $E_6$. It is also shown that the relations of the algebra $\mathcal{A}$ can be realised with Heun type operators in the Racah or Hahn algebra., 26 pages

International audience; We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice H having a Uq(sln) quantum group symmetry. We focus here on the n=3 case of cubic webs, based on the Kuperberg A2 spider, and illustrate its properties by comparisons with the well-known dilute loop model (the n=2 case) throughout. A local vertex-model reformulation is exhibited, analogous to the correspondence between the loop model and a three-state vertex model. The n=3 representation uses seven states per link of H, displays explicitly the geometrical content of the webs and their Uq(sl3) symmetry, and permits us to study the model on a cylinder via a local transfer matrix. A numerical study of the effective central charge reveals that for q=−eiγ, in the range γ∈[0,π/3], the web model possesses a dense and a dilute critical point, just like its loop model counterpart. In the dense q=−eiπ/4 case, the n=3 webs can be identified with spin interfaces of the critical three-state Potts model defined on the triangular lattice dual to H. We also provide another mapping to a Z3 spin model on H itself, using a high-temperature expansion. We then discuss the sector structure of the transfer matrix, for generic q, and its relation to defect configurations in both the strip and the cylinder geometries. These defects define the finite-size precursors of electromagnetic operators. This discussion paves the road for a Coulomb gas description of the conformal properties of defect webs, which will form the object of a subsequent paper. Finally, we identify the fractal dimension of critical webs in the q=−eiπ/3 case, which is the n=3 analogue of the polymer limit in the loop model.

We consider the first exit point distribution from a bounded domain $$\Omega $$ of the stochastic process $$(X_t)_{t\ge 0}$$ solution to the overdamped Langevin dynamics $$\begin{aligned} d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t \end{aligned}$$ starting from deterministic initial conditions in $$\Omega $$ , under rather general assumptions on f (for instance, f may have several critical points in $$\Omega $$ ). This work is a continuation of the previous paper [14] where the exit point distribution from $$\Omega $$ is studied when $$X_0$$ is initially distributed according to the quasi-stationary distribution of $$(X_t)_{t\ge 0}$$ in $$\Omega $$ . The proofs are based on analytical results on the dependency of the exit point distribution on the initial condition, large deviation techniques and results on the genericity of Morse functions.

We study the $p$-tuples of holomorphic vector fields $(X_1,X_2,\ldots,X_p)$ satisfying the barycentric property $\displaystyle\sum_k\exp tX_k=p\cdot\mathrm{id}$, where $\exp tX$ denotes the flow of $X$.

International audience; This paper describes a novel approach for on demand volumetric texture synthesis based on a deep learning framework that allows for the generation of high quality 3D data at interactive rates. Based on a few example images of textures, a generative network is trained to synthesize coherent portions of solid textures of arbitrary sizes that reproduce the visual characteristics of the examples along some directions. To cope with memory limitations and computation complexity that are inherent to both high resolution and 3D processing on the GPU, only 2D textures referred to as "slices" are generated during the training stage. These synthetic textures are compared to exemplar images via a perceptual loss function based on a pre-trained deep network. The proposed network is very light (less than 100k parameters), therefore it only requires sustainable training (i.e. few hours) and is capable of very fast generation (around a second for 256^3 voxels) on a single GPU. Integrated with a spatially seeded PRNG the proposed generator network directly returns an RGB value given a set of 3D coordinates. The synthesized volumes have good visual results that are at least equivalent to the state-of-the-art patch based approaches. They are naturally seamlessly tileable and can be fully generated in parallel.

Let $L_n(k)$ denote the least common multiple of $k$ independent random integers uniformly chosen in $\{1,2,\ldots ,n\}$. In this note, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as $n\to\infty$ of $\frac{f(L_n(k))}{n^{rk}}$ for a wide class of multiplicative arithmetic functions~$f$ with polynomial growth $r>-1$. Furthermore, we identify the limit as an infinite product of independent random variables indexed by prime numbers. Along the way, we compute the generating function of a trimmed sum of independent geometric laws, occurring in the above infinite product. This generating function is rational; we relate it to the generating function of a certain max-type Diophantine equation, of which we solve a generalized version. Our results extend theorems by Erd\H{o}s and Wintner (1939), Fern\'{a}ndez and Fern\'{a}ndez (2013) and Hilberdink and T\'{o}th (2016)., Comment: 19 pages

Within the Russo-Susskind-Thorlacius (RST) two-dimensional model that includes a scalar (dilaton) field we address the important question of how the classical black hole geometry is modified in a semiclassical gravitational theory. It is the principle goal of this paper to analyze what is the back-reacted geometry that corresponds to a given quantum state. The story is shown to be dramatically different for the Hartle-Hawking state (HH) and for the Boulware state. In the HH case the back-reacted geometry is a modification of the classical black hole metric that still has a smooth horizon with a regular curvature. On the other hand, for the Boulware state the classical horizon is replaced by a throat in which the $(tt)$ component of the metric (while non-zero) is extremely small. The value of the metric at the throat is bounded by the inverse of the classical black hole entropy. On the other side of the throat the spacetime is ended at a null singularity. More generally, we identify a family of quantum states and their respective back-reacted geometries. We also identify a certain duality in the space of states. Finally, we study a hybrid set-up where both physical and non-physical fields, such as the ghosts, could be present. We suggest that it is natural to associate ghosts with the Boulware state, while the physical fields can be in any quantum state. In particular, if the physical fields are in the HH state, then the corresponding semiclassical geometry is horizonless. Depending on the balance between the number of physical fields and ghosts, it generically has a throat that may join with another asymptotically flat region on the other side of the throat., Comment: 38 pages, 7 figures. Version 2. Published version. Minor typos corrected

Recent experiments [Grinenko $\textit{et al.}~\href{http://doi.org/10.1038/s41567-021-01350-9}{\text{Nature Physics}~\textbf{17},~1254\!-\!1259~(2021)}$] reported the observation of a condensate of four-fermion composites. This is a resistive state that spontaneously breaks the time-reversal symmetry, leading to unconventional magnetic properties, detected in muon spin rotation experiments and by the appearance of a spontaneous Nernst effect. In this work, we derive an effective model that describes the observed spontaneous magnetic fields in this state. We show that this model, which is alike to the Faddeev-Skyrme model for the four-fermion order parameter, hosts skyrmions: magnetic-flux-carrying topological excitations.