Weak attractors for the dissipative fractional Heisenberg equation

In this short paper, we consider the long time behaviors of the fractional Heisenberg equation and the existence of a global weak attractor is proved for the shift dynamics in the path space. The key ingredient is some new type of commutator structure introduced in this paper, which seems indispensable in proving the compactness of the dynamics. The technique introduced in this paper may also be useful to other fractional order partial differential equations.     

Local-in-time adjoint-based method for design optimization of unsteady flows

We present a new local-in-time discrete adjoint-based methodology for solving design optimization problems arising in unsteady aerodynamic applications. The new methodology circumvents storage requirements associated with the straightforward implementation of a global adjoint-based optimization method that stores the entire flow solution history for all time levels. This storage cost may quickly become prohibitive for large-scale applications. The key idea of the local-in-time method is to divide the entire time interval into several subintervals and to approximate the solution of the unsteady adjoint equations and the sensitivity derivative as a combination of the corresponding local quantities computed on each time subinterval. Since each subinterval contains relatively few time levels, the storage cost of the local-in-time method is much lower than that of the global methods, thus making the time-dependent adjoint optimization feasible for practical applications. Another attractive feature of the new technique is that the converged solution obtained with the local-in-time method is a local extremum of the original optimization problem. The new method carries no computational overhead as compared with the global implementation of adjoint-based methods. The paper presents a detailed comparison of the global- and local-in-time adjoint-based methods for design optimization problems governed by the unsteady compressible 2-D Euler equations.     

Uniform limit laws of the logarithm for nonparametric estimators of the regression function in presence of censored data

In this paper, we establish uniform-in-bandwidth limit laws of the logarithm for nonparametric Inverse Probability of Censoring Weighted (I.P.C.W.) estimators of the multivariate regression function under random censorship. A similar result is deduced for estimators of the conditional distribution function. The uniform-in-bandwidth consistency for estimators of the conditional density and the conditional hazard rate functions are also derived from our main result. Moreover, the logarithm laws we establish are shown to yield almost sure simultaneous asymptotic confidence bands for the functions we consider. Examples of confidence bands obtained from simulated data are displayed.      

A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces

On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo–Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber–Krahn inequalities or localised Sobolev or Nash inequalities. We are able to treat more general pointwise estimates, where the heat kernel rate of decay is not necessarily governed by the volume growth. A crucial role is played by the finite propagation speed property for the associated wave equation, and our main result holds for an abstract semigroup of operators satisfying the Davies–Gaffney estimates.     

一类Hermitian鞍点矩阵的特征值估计

本文研究了一类大型稀疏Hermitian鞍点线性系统Az=(B E E* 0)(x y)=(f g)=b系数矩阵的特征值,其中B∈C~(p×p)是Hermitian正定阵矩阵,E∈C~(p×q)是列降秩.本文分别给出了该系数矩阵正特征值与负特征值界的一个估计式,同时通过数值算例验证本文所给出的特征值界的估计是合理且有效的.