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Amdeberhan's conjectures on the enumeration, the average size, and the largest size of $(n,n+1)$-core partitions with distinct parts have motivated many research on this topic. Recently, Straub and Nath-Sellers obtained formulas for the numbers of $(n, dn-1)$ and $(n, dn+1)$-core partitions with distinct parts, respectively. Let $X_{s,t}$ be the size of a uniform random $(s,t)$-core partition with distinct parts when $s$ and $t$ are coprime to each other. Some explicit formulas for the $k$-th moments $\mathbb{E} [X_{n,n+1}^k]$ and $\mathbb{E} [X_{2n+1,2n+3}^k]$ were given by Zaleski and Zeilberger when $k$ is small. Zaleski also studied the expectation and higher moments of $X_{n,dn-1}$ and conjectured some polynomiality properties concerning them in arXiv:1702.05634. Motivated by the above works, we derive several polynomiality results and asymptotic formulas for the $k$-th moments of $X_{n,dn+1}$ and $X_{n,dn-1}$ in this paper, by studying the beta sets of core partitions. In particular, we show that these $k$-th moments are asymptotically some polynomials of n with degrees at most $2k$, when $d$ is given and $n$ tends to infinity. Moreover, when $d=1$, we derive that the $k$-th moment $\mathbb{E} [X_{n,n+1}^k]$ of $X_{n,n+1}$ is asymptotically equal to $\left(n^2/10\right)^k$ when $n$ tends to infinity. The explicit formulas for the expectations $\mathbb{E} [X_{n,dn+1}]$ and $\mathbb{E} [X_{n,dn-1}]$ are also given. The $(n,dn-1)$-core case in our results proves several conjectures of Zaleski on the polynomiality of the expectation and higher moments of $X_{n,dn-1}$., Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematics

In this paper, we consider the first negative eigenvalue of eigenforms of half-integral weight k + 1 ∕ 2 and obtain an almost type bound.

International audience; This paper is concerned with the existence and qualitative properties of transition fronts for time periodic bistable reaction-diffusion equations in R-N. We first show that any almost-planar transition front is actually planar, regardless of the number of transition layers. Then we prove that all transition fronts admit a global mean speed gamma and it holds gamma = vertical bar c vertical bar, where c is the speed of the planar traveling front. Finally we establish the existence of a transition front in R-N that is not a standard traveling front. Such a front behaves like three moving time periodic planar fronts as time goes to -infinity and like a time periodic V-shaped traveling front as time goes to infinity.

International audience

International audience; This paper deals with the multidimensional stability of time periodic V-shaped traveling fronts in bistable reaction–diffusion equations. It is well known that time periodic V-shaped traveling fronts are asymptotically stable in two dimensional space. In the current study, we further show that such fronts are asymptotically stable under spatially decaying initial perturbations in Rn with n≥3. In particular, we show that the fronts are algebraically stable if the initial perturbations belong to L1 in a certain sense. Furthermore, we prove that there exists a solution oscillating permanently between two time periodic V-shaped traveling fronts, which implies that time periodic V-shaped traveling fronts are not always asymptotically stable under general bounded perturbations. Finally we show that time periodic V-shaped traveling fronts are only time global solutions of the Cauchy problem if the initial perturbations lie between two time periodic V-shaped traveling fronts.

This paper is concerned with entire solutions of monotone bistable reaction–diffusion systems in $$\mathbb {R}^N$$ . We first show that there is a new entire solution for bistable reaction–diffusion systems which behaves as three moving planar fronts as time goes to $$-\,\infty $$ and as a V-shaped traveling front as time goes to $$\infty $$ . Then we address the speed of the interfaces corresponding to the entire solution by appealing to the super-sub solutions technique. Finally, we apply our results to two important models in biology.

International audience; This paper is concerned with the existence and further properties of propagation speeds of transition fronts for bistable reaction-diffusion equations in exterior domains and in some domains with multiple cylindrical branches. In exterior domains we show that all transition fronts with complete propagation propagate with the same global mean speed, which turns out to be equal to the uniquely defined planar speed. In domains with multiple cylindrical branches, we show that the solutions emanating from some branches and propagating completely are transition fronts propagating with the unique planar speed. We also give some geometrical and scaling conditions on the domain, either exterior or with multiple cylindrical branches, which guarantee that any transition front has a global mean speed.

International audience; An experimental investigation is conducted on the ow around tandem cylinders for ranges of diameter ratio d/D = 0.25–1.0, spacing ratio L/d = 5.5–20, and Reynolds number Re = 0.8 × 104–2.42 × 104, where d and D are the diameters of the upstream and downstream cylinders, respectively, L is the distance from the upstream cylinder center to the forward stagnation point of the downstream one. The focus is given on examining the effects of d/D, L/d and Re on Strouhal number St, flow structures and fluid forces measured using hotwire, particle image velocimetry (PIV) and load cell measurement techniques, respectively. Changes in d/D and L/d in the ranges examined lead to five flow regimes, namely lock-in, intermittent lock-in, no lock-in, subharmonic lock-in and shear-layer reattachment regimes. Time-mean drag coefficient (CD) and fluctuating drag and lift coefficients (CD' and CL′ ) are more sensitive to L/d than d/D. The scenario is opposite for St where d/D is more prominent than L/d to change the St. The detailed facet of the dependence on d/D and L/d of CD, CD′, CL′ and St is discussed based on shear-layer velocity, approaching velocity, vortex formation length, and wake width.

In this paper, we present a novel method for assimilating geometric information from observed images. Image assimilation technology fully utilizes structural information from the dynamics of the images to retrieve the state of a system and thus to better predict its evolution. Level-set method describing the evolution of the geometry shapes of a given system is taken into account to include the dynamics of the images. This method takes advantage of Lagrangian information in an Eulerian numerical framework. In our numerical experiments, we apply this state-of-the-art technique to a pollutant transport problem, to calibrate the initial contours of pollutants and to identify diffusion coefficients of the model. It can be shown a potential approach for oil spills, because topological merging and breaking of oil slicks are well defined and easily performed by this proposed approach. Numerical results show that the proposed method is visibly efficient compared with the classical method based on the concentration map when the concentration measurements and the background fields are not well available.

This paper is concerned with the multidimensional stability of planar traveling fronts in bistable reaction–diffusion systems. It is first shown that planar traveling fronts are asymptotically stable under spatially decaying initial perturbations by appealing to the comparison principle and super-subsolution method. In particular, if the perturbations belong to L 1 ( R n − 1 ) in a certain sense, we obtain a convergence rate like t − n − 1 2 . Then we show that the solution of the Cauchy problem converges to the planar traveling front with rate t − n + 1 4 for a spatially non-decaying perturbation with the help of semigroup theory. Finally, we prove that there exists a solution oscillating permanently between two planar traveling fronts, which indicates that planar traveling fronts are not always asymptotically stable in multidimensional space under general bounded perturbations.

This paper is concerned with the existence and stability of time periodic traveling curved fronts for reaction–diffusion equations with bistable nonlinearity in \({\mathbb {R}}^3\). We first study the existence and other qualitative properties of time periodic traveling fronts of polyhedral shape. Furthermore, for any given \(g\in C^{\infty }(S^1)\) with \(\min \nolimits _{0\le \theta \le 2\pi }g(\theta )=0\) that gives a convex bounded domain with smooth boundary of positive curvature everywhere, which is included in a sequence of convex polygons, we show that there exists a three-dimensional time periodic traveling front by taking the limit of the solutions corresponding to the convex polyhedrons as the number of the lateral surfaces goes to infinity.

International audience; We present in this article a very adapted finite volume numerical scheme for transport type-equation. Thescheme is an hybrid one combining an anti-dissipative method with down-winding approach for the flux(Després and Lagoutière, C R Acad Sci Paris Sér I Math 328(10) (1999), 939–944; Goudon, Lagoutière,and Tine, Math Method Appl Sci 23(7) (2013), 1177–1215) and an high accurate method as the WENO5one (Jiang and Shu, J Comput Phys 126 (1996), 202–228). The main goal is to construct a scheme ableto capture in exact way the numerical solution of transport type-equation without artifact like numericaldiffusion or without “stairs” like oscillations and this for any regular or discontinuous initial distribution.This kind of numerical hybrid scheme is very suitable when properties on the long term asymptotic behaviorof the solution are of central importance in the modeling what is often the case in context of populationdynamics where the final distribution of the considered population and its mass preservation relation arerequired for prediction.

International audience; This work combines a level-set approach and the optimal transport-based Wasserstein distance in a data assimilation framework. The primary motivation of this work is to reduce assimilation artifacts resulting from the position and observation error in the tracking and forecast of pollutants present on the surface of oceans or lakes. Both errors lead to spurious effect on the forecast that need to be corrected. In general, the geometric contour of such pollution can be retrieved from observation while more detailed characteristics such as concentration remain unknown. Herein, level sets are tools of choice to model such contours and the dynamical evolution of their topology structures. They are compared with contours extracted from observation using the Wasserstein distance. This allows to better capture position mismatches between both sources compared with the more classical Euclidean distance. Finally, the viability of this approach is demonstrated through academic test cases and its numerical performance is discussed.

International audience; We study the existence of traveling wave solutions and spreading properties for single-layer delayed neural field equations. We focus on the case where the kinetic dynamics are of monos-table type and characterize the invasion speeds as a function of the asymptotic decay of the connectivity kernel. More precisely, we show that for exponentially bounded kernels the minimal speed of traveling waves exists and coincides with the spreading speed, which further can be explicitly characterized under a KPP type condition. We also investigate the case of algebraically decaying kernels where we prove the non-existence of traveling wave solutions and show the level sets of the solutions eventually locate in between two exponential functions of time. The uniqueness of traveling waves modulo translation is also obtained.