Some uses of Green's theorem in solving the Navier–Stokes equations

This paper gives a review of methods where Green's theorem may be employed in solving numerically the Navier–Stokes equations for incompressible fluid motion. They are based on the concept of using the theorem to transform local boundary conditions given on the boundary of a closed region in the solution domain into global, or integral, conditions taken over it. Two formulations of the Navier–Stokes equations are considered: that in terms of the streamfunction and vorticity for two-dimensional motion and that in terms of the primitive variables of the velocity components and the pressure. In the first formulation overspecification of conditions for the streamfunction is utilized to obtain conditions of integral type for the vorticity and in the second formulation integral conditions for the pressure are found. Some illustrations of the principle of the method are given in one space dimension, including some derived from two-dimensional flows using the series truncation method. In particular, an illustration is given of the calculation of surface vorticity for two-dimensional flow normal to a flat plate. An account is also given of the implementation of these methods for general two-dimensional flows in both of the mentioned formulations and a numerical illustration is given.     

A note on spin,vorticity and the deformation-rate tensor

The validity of statements in the literature concerning the angular velocity of the principal axes of the stretching tensor is examined and an alternative expression is provided for it.     

A remark on vorticity maximal function

In this note the following inequality is proved. For any nonnegative measure μH⁻¹(R²), xR² and 0<r<1, there holds

(1)

where C is a positive constant. Using (1) an estimate for the vorticity maximal function similar to the estimate in Majda [A. Majda, Remarks on weak solutions for vortex sheets with a distinguished sign, Indiana Univ. Math. J. 42 (1993) 921–939] is established without assuming the initial vorticity having compact support. From this a more simple proof of the Delort's celebrated theorem [J.M. Delort, Existence de mappes de fourbillon en dimension deux, J. Amer. Math. Soc. 4 (1991) 553–586] is presented.     

Pryce-Hoyle Tensor in a Combined Einstein-Cartan-Brans-Dicke Model

In addition to introducing matter injection through a scalar field determined by Pryce-Hoyle tensor, we also combine it with a BCDE (Brans-Dicke-Einstein-Cartan) theory with lambda-term developed earlier by Berman (Astrophys. Space Sci. 314:79–82, 2008), for inflationary scenario. It involves a variable cosmological constant, which decreases with time, jointly with energy density, cosmic pressure, shear, vorticity, and Hubble’s parameter, while the scale factor, total spin and scalar field increase exponentially. The post-inflationary fluid resembles a perfect one, though total spin grows, but not the angular speed (Berman, in Astrophys. Space Sci. 312:275, 2007). The Pryce-Hoyle tensor, which can measured by the number of injected particles per unit proper volume and time, as well as shear and vorticity, can be neglected in the aftermath of inflation (“no-hair”).     

HUMAN CHROMOSOME BANDING WITH RESTRICTION ENDONUCLEASES HaeIII,HinfI AND PstI

Human chromosome banding was carried out with restriction endonucleases (REs)HaeⅢ, HinfⅠ, PstⅠ and the effects of some factors on it were examined. HaeⅢ induced Cand G banding patterns, HinfⅠ induced negative C bands but the centromeric heterochro-matin of chromosomes 3 and 4 remained selectively dark-stained. A new heteromorphismof C banded region was discovered in chromosome 4. PstⅠ induced G-like banding pattern.The duration of enzymatic digestion, the concentration of glycerol in the reaction buffer,as well as the ageing and heating of chromosome preparations all had a significant influ-ence on the banding effects of the REs.     

Exponential growth of the vorticity gradient for the Euler equation on the torus

We prove that there are solutions to the Euler equation on the torus with C^1,α$C^{1,\alpha }$ vorticity and smooth except at one point such that the vorticity gradient grows in L^∞$L^{\infty }$ at least exponentially as t→∞$t\to \infty$. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Šverák [6].     

玉米和类玉米杂交后代异染色质纽的遗传稳定性分析

利用玉米的近缘亲属摩擦禾的基因组DNA。对玉米自交系330、二倍体多年生类玉米及其远缘杂交孤雌生殖后代异源种质纯系540的染色体进行比较基因组原位杂交，发现上述3个供试种染色体上的异染色质纽因杂交信号极强而能被清楚地显示．与亲代相比，540中出现异染色质纽数目减少的现象．并讨论了造成这种现象的可能原因．     

Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis

We present a theory of very long waves propagating on the surface of water. The waves evolve slowly, both on the scale ε (weak nonlinearity), and on the scale, σ, of the depth variation. In our model, dispersion does not affect the evolution of the wave even over the large distances that tsunamis may travel. We allow a distribution of vorticity, in addition to variable depth. Our solution is not valid for depth=O(ε^4/5); the equations here are expressed in terms of the single parameter ε^2/5σ and matched to the solution in deep water. For a slow depth variation of the background state (consistent with our model), we prove that a constant-vorticity solution exists, from deep water to shoreline, and that regions of isolated vorticity can also exist, for appropriate bottom profiles. We describe how the wave properties are modified by the presence of vorticity. Some graphical examples of our various solutions are presented.