Northern China harbored the world's earliest complex societies based on millet farming, in two major centers in the Yellow (YR) and West Liao (WLR) River basins. Until now, their genetic histories have remained largely unknown. Here we present 55 ancient genomes dating to 7500-1700 BP from the YR, WLR, and Amur River (AR) regions. Contrary to the genetic stability in the AR, the YR and WLR genetic profiles substantially changed over time. The YR populations show a monotonic increase over time in their genetic affinity with present-day southern Chinese and Southeast Asians. In the WLR, intensification of farming in the Late Neolithic is correlated with increased YR affinity while the inclusion of a pastoral economy in the Bronze Age was correlated with increased AR affinity. Our results suggest a link between changes in subsistence strategy and human migration, and fuel the debate about archaeolinguistic signatures of past human migration.
Abstract. In this paper, we present a regularization to the 1D Grad's moment system to achieve global hyperbolicity. The regularization is based on the observation that the characteristic polynomial of the Jacobian of the flux in Grad's moment system is independent of the intermediate coefficients in the Hermite expansion. The method does not rely on the form of the collision at all, thus this regularization is applicable to the system without collision terms. Moreover, the proposed approach is proved to be the unique one if only the last moment equation is allowed to be altered to match the condition that the characteristic speeds coincide with the Gauss-Hermite interpolation points. The hyperbolic structure of the regularized system, including the signal speeds, Riemann invariants, and the properties of the characteristic waves including the rarefaction wave, contact discontinuity, and shock are provided in the perfect formations.
In this paper, we propose a globally hyperbolic regularization to the general Grad's moment system in multidimensional spaces. Systems with moments up to an arbitrary order are studied. The characteristic speeds of the regularized moment system can be analytically given and depend only on the macroscopic velocity and the temperature. The structure of the eigenvalues and eigenvectors of the coefficient matrix is fully clarified. The regularization together with the properties of the resulting moment systems is consistent with the simple onedimensional case discussed in [1]. In addition, all characteristic waves are proven to be genuinely nonlinear or linearly degenerate, and the studies on the properties of rarefaction waves, contact discontinuities, and shock waves are included. GLOBALLY HYPERBOLIC MOMENT SYSTEM 465 Ã C PROOF. Let i D N D .˛/, with j˛j Ä M . Then we need only to verify that (3.20) z A M .i; 1WN / r D r w i is always valid. Since A M is determined by (3.2), (3.3), and (3.4), and y A M and z A M are defined as in (3.11) and (3.13), respectively, we can write all entries of z A M . Now let us verify equation (3.20) case by case: (1) For˛D 0, z A M .1; N D .e 1 // D 1 is the only nonzero entry of z A M .1; 1WN /; hence, z A M .i; 1WN / r D 1 r u 1 D r D r w i :(2) For˛D e 1 , z A M .i; 1WN / r D 2 r p 2e 1 =2 D 2 r D r u 1 D r w i :(3) For˛D e k , k D 2; : : : ; D, z A M .i; 1WN / r D 1 r p e 1 Ce k =2 D r u k D r w i : ÃHe M 1 . / ZHENNING CAI
Abstract. In this paper, sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems. These error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach. Our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work.Key words. mesh adaptivity, optimal control, a posteriori error estimate, finite element method AMS subject classifications. 49J20, 65N30PII. S0363012901389342 1. Introduction. Finite element approximation of optimal control problems has long been an important topic in engineering design work and has been extensively studied in the literature. There have been extensive theoretical and numerical studies for finite element approximation of various optimal control problems; see [2,12,13,15,20,23,37,44]. For instance, for the optimal control problems governed by some linear elliptic or parabolic state equations, a priori error estimates of the finite element approximation were established long ago; see, for example, [12,13,15,20,23,37]. Furthermore, a priori error estimates were established for the finite element approximation of some important flow control problems in [17] and [11]. A priori error estimates have also been obtained for a class of state constrained control problems in [43], though the state equation is assumed to be linear. In [29], this assumption has been removed by reformulating the control problem as an abstract optimization problem in some Banach spaces and then applying nonsmooth analysis. In fact, the state equation there can be a variational inequality.In recent years, the adaptive finite element method has been extensively investigated. Adaptive finite element approximation is among the most important means to boost the accuracy and efficiency of finite element discretizations. It ensures a higher density of nodes in a certain area of the given domain, where the solution is more difficult to approximate. At the heart of any adaptive finite element method is an a posteriori error estimator or indicator. The literature in this area is extensive. Some of the techniques directly relevant to our work can be found in [1,5,6,7,28,32,34,42,47]. It is our belief that adaptive finite element enhancement is one of the future directions to pursue in developing sophisticated numerical methods for optimal design problems.
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