Mehler's formula and functional calculus Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

We show that Mehler's formula can be used to handle several formulas involving the quantization of singular Hamiltonians. In particular, we diagonalize in the Hermite basis the Weyl quantization of the characteristic function of several domains of the phase space.     

Semi-classical analysis on H-type groups Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

In this paper, we develop semi-classical analysis on H-type groups. We define semi-classical pseudodifferential operators, prove the boundedness of their action on square integrable functions and develop a symbolic calculus. Then, we define the semi-classical measures of bounded families of square integrable functions which consist of a pair formed by a measure defined on the product of the group and its unitary dual, and by a field of trace class positive operators acting on the Hilbert spaces of the representations. We illustrate the theory by analyzing examples, which show in particular that this semi-classical analysis takes into account the finite-dimensional representations of the group, even though they are negligible with respect to the Plancherel measure.     

Behaviour of the Stokes operators under domain perturbation Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

Depending on the geometry of the domain, one can define—at least—three different Stokes operators with Dirichlet boundary conditions. We describe how the resolvents of these Stokes operators converge with respect to a converging sequence of domains.     

Enhanced dissipation for the Kolmogorov flow via the hypocoercivity method Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

In this paper, we solve Beck and Wayne's conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called the Kolmogorov flow by developing the hypocoercivity method introduced by Villani(2009).     

The NLS approximation for two dimensional deep gravity waves

Science China Mathematics - This article is concerned with infinite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic...     

Global strong solutions to 3-D Navier-Stokes system with strong dissipation in one direction Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

We consider three-dimensional incompressible Navier-Stokes equations(NS) with different viscous coefficients in the vertical and horizontal variables. In particular, when one of these viscous coefficients is large enough compared with the initial data, we prove the global well-posedness of this system. In fact, we obtain the existence of a global strong solution to(NS) when the initial data verifies an anisotropic smallness condition which takes into account the different roles of the horizontal and vertical viscosity.     

Dynamics of a rigid body in a two-dimensional incompressible perfect fluid and the zero-radius limit Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

In this survey we report some recent results on the dynamics of a rigid body immersed in a twodimensional incompressible perfect fluid, with an emphasis on the zero-radius limit.     

A note on the regularity of the holes for permeability property through a perforated domain for the 2D Euler equations Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

For equations of order two with the Dirichlet boundary condition, as the Laplace problem, the Stokes and the Navier-Stokes systems, perforated domains were only studied when the distance between the holes d_ε is equal to or much larger than the size of the holes ε. Such a diluted porous medium is interesting because it contains some cases where we have a non-negligible effect on the solution when(ε, d_ε) →(0, 0).Smaller distances were avoided for mathematical reasons and for these large distances, the geometry of the holes does not affect or rarely affect the asymptotic result. Very recently, it was shown for the 2D-Euler equations that a porous medium is non-negligible only for inter-holes distances much smaller than the sizes of the holes.For this result, the boundary regularity of holes plays a crucial role, and the permeability criterion depends on the geometry of the lateral boundary. In this paper, we relax slightly the regularity condition, allowing a corner, and we note that a line of irregular obstacles cannot slow down a perfect fluid in any regime such thatε ln d_ε→ 0.     

Exponential convergence of the deep neural network approximation for analytic functions Dedicated to Professor Ta Tsien Li on the Occasion of His 80th Birthday

We prove that for analytic functions in low dimension, the convergence rate of the deep neural network approximation is exponential.     

Generalized Riemann problem for Euler system Dedicated to Professor LI TaTsien on the Occasion of His 80th Birthday

This article is a survey on the progress in the study of the generalized Riemann problems for MD Euler system. A new result on generalized Riemann problems for Euler systems containing all three main nonlinear waves(shock, rarefaction wave and contact discontinuity) is also introduced.     

Dedicated to S.K. Godunov on the Occasion of His 90th Birthday

Computational Mathematics and Mathematical Physics -     

The Special Issue Is Dedicated to Prof Li Tatsien on the Occasion of His 70th Birthday

Professor Li has published more than 200 papers and 16 monographs and textbooks, among which 4 monographs are printed in English in U.S.A., U.K. and France respectively. He has received domestic prizes and awards, including: Shanghai Top Science and Technology Award, Hua Loo-Keng Prize of Mathematics, one Second Prize and one Third Prize of National Natural Sciences from the State, one First Prize of Scientific and Technological Progress from the State Education Commission, and one First Prize of Scientific and Technological Progress from Shanghai Municipality. Two of his works are awarded as Excellent University Textbooks from the State. One Exceptional Shanghai Award for Teaching Achievements in Higher Education and one First National Award for Teaching Achievements in Higher Education. Moreover, he has received the Science and Technology Progress Award of Ho Leug Ho Lee Foundation.     

Orlicz mixed quermassintegrals Dedicated to Professor Ren De-lin on the Occasion of his 80th Birthday

The notion of mixed quermassintegrals in the classical Brunn-Minkowski theory is extended to that of Orlicz mixed quermassintegrals in the Orlicz Brunn-Minkowski theory. The analogs of the classical CauchyKubota formula, the Minkowski isoperimetric inequality and the Brunn-Minkowski inequality are established for this new Orlicz mixed quermassintegrals.     

A two-level method for sparse time-frequency representation of multiscale data Dedicated to Professor LI TaTsien on the Occasion of His 80th Birthday

Based on the recently developed data-driven time-frequency analysis(Hou and Shi, 2013), we propose a two-level method to look for the sparse time-frequency decomposition of multiscale data. In the two-level method, we first run a local algorithm to get a good approximation of the instantaneous frequency. We then pass this instantaneous frequency to the global algorithm to get an accurate global intrinsic mode function(IMF)and instantaneous frequency. The two-level method alleviates the difficulty of the mode mixing to some extent.We also present a method to reduce the end effects.     

RC-positivity and rigidity of harmonic maps into Riemannian manifolds Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

In this paper,we show that every harmonic map from a compact K?hler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant.In particular,there is no non-constant harmonic map from a compact Koahler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.     

Supersonic flow onto solid wedges,multidimensional shock waves and free boundary problems Dedicated to Professor LI Ta Tsien on the Occasion of His 80th Birthday

When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge,governed by the Euler equations,there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle—the steady weak shock with supersonic or subsonic downstream flow(determined by the wedge angle that is less than or greater than the sonic angle)and the steady strong shock with subsonic downstream flow,both of which satisfy the entropy condition.The fundamental issue—whether one or both of the steady weak and strong shocks are physically admissible solutions—has been vigorously debated over the past eight decades.In this paper,we survey some recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic regimes.For the static stability,we first show how the stability problem can be formulated as an initial-boundary value type problem and then reformulate it into a free boundary problem when the perturbation of both the upstream steady supersonic flow and the wedge boundary are suitably regular and small,and we finally present some recent results on the static stability of the steady supersonic and transonic shocks.For the dynamic stability for potential flow,we first show how the stability problem can be formulated as an initial-boundary value problem and then use the self-similarity of the problem to reduce it into a boundary value problem and further reformulate it into a free boundary problem,and we finally survey some recent developments in solving this free boundary problem for the existence of the PrandtlMeyer configurations that tend to the steady weak supersonic or transonic oblique shock solutions as time goes to infinity.Some further developments and mathematical challenges in this direction are also discussed.     

Weakly complex Einstein-Finsler vector bundle Dedicated to Professor Tongde Zhong for His 90th Birthday

Let(E, F) be a complex Finsler vector bundle over a compact Khler manifold(M, g) with Khler form Φ. We prove that if(E, F) is a weakly complex Einstein-Finsler vector bundle in the sense of Aikou(1997),then it is modeled on a complex Minkowski space. Consequently, a complex Einstein-Finsler vector bundle(E, F) over a compact Khler manifold(M, g) is necessarily Φ-semistable and(E, F) =(E_1, F_1)⊕…⊕(E_k, F_k),where Fj := F |Ej, and each(Ej, Fj) is modeled on a complex Minkowski space whose associated Hermitian vector bundle is a Φ-stable Einstein-Hermitian vector bundle with the same factor c as(E, F).     

Two families of Niho sequences having four-valued cross correlation with m-sequences Dedicated to Professor Yingbo Zhang on the Occasion of His 70th Birthday

For two odd integers m and s with 1≤s m and gcd(m, s) = 1, let h satisfy h(2~s-1) ≡1(mod 2~m+ 1) and d =(h + 1)(2~m-1) + 1. The cross correlation function between a binary m-sequence of period 22~m-1 and its d-decimation sequence is proved to take four values, and the correlation distribution is completely determined. Let n be an even integer and k be an integer with 1≤k≤n/2. For an odd prime p and a p-ary m-sequence {s(t)} of period p~n-1, define u(t) =∑(p~k-1)/2 i=0 s(d_it), where d_i = ip~(n/2) + p~k-i and i = 0, 1,...,(p~k-1)/2. It is proved that the cross correlation function between {u(t)} and {s(t)} is three-valued or four-valued depending on whether k is equal to n/2 or not, and the distribution is also determined.     

Partial Differential Equations and Continuum Mechanics: a Collection of Papers Dedicated to Professor Philippe G. Ciarlet on the Occasion of His 80th Birthday

Acta Mathematicae Applicatae Sinica, English Series -     

On the Bifurcations and Multiple Endemic States of a Single Strain HIV Model Dedicated to Professor Toshikazu Sunada on the Occasion of his 60th Birthday

The dynamics of a single strain HIV model is studied.The basic reproduction number R_0 used as a bifurcation parameter shows that the system undergoes transcritical and saddle-node bifurcations.The usual threshold unit value of R_0 does not completely determine the eradication of the disease in an HIV infected person.In particular,a sub-threshold value R_C is established which determines the system's number of endemic states:multiple if R_c Ro 1,only one if R_C = R_0 = 1,and none if R_0 R_C 1.