Contact topology and hydrodynamics III: knotted orbits

We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct a steady nonsingular solution to the Euler equations on a Riemannian whose flowlines trace out closed curves of all possible knot and link types. Using careful contact-topological controls, we can make such vector fields real-analytic and transverse to the tight contact structure on . Sufficient review of concepts is included to make this paper independent of the previous works in this series.

     

Regular closure

Regular closure is an operation performed on submodules of arbitrary modules over a commutative Noetherian ring. The regular closure contains the tight closure when both are defined, but in general, the regular closure is strictly larger. Regular closure is interesting, in part, because it is defined a priori in all characteristics, including mixed characteristic. We show that one can test regular closure in a Noetherian ring by considering only local maps to regular local rings. In certain cases, it is necessary only to consider maps to certain affine algebras. We also prove the equivalence of two variants of regular closure for a class of rings that includes .

     

致密储层束缚水干化反应及干化效应差异分析与评价

通过向致密储层中注入干化剂,使其与地层水发生干化反应能够消耗地层水、降低储层含水饱和度,从而提高气体渗流能力。以致密气藏干化原理为基础,对所选干化剂进行原理性干化实验发现,干化剂实际耗水量大于理论耗水量。分析认为,干化剂实际干化效果还受化学反应热效应的影响。本文以热化学理论为基础,研究了干化剂与水反应的热效应及其对干化效果的影响。并考虑到实际地层往往处于不同的温度和压力条件下,故对干化反应热效应随体系温度和压力的变化情况做了研究。结果表明,所选干化剂与水反应能够放出约5.28k J/g的热量,放出的热量能够使反应速率加快和使部分水分子蒸发成气相从而提高干化效率;干化剂化学反应热随温度和压力的升高而增加,但增加速率较小;温度和压力的变化对热蒸发耗水量影响较小。     

The tight-coupling approximation for baryon acoustic oscillations

The tight-coupling approximation (TCA) used to describe the early dynamics of the baryons–photons system is systematically built to higher orders in the inverse of the interaction rate. This expansion can be either used to grasp the physical effects by deriving simple analytic solutions or to obtain a form of the system which is stable numerically at early times. In linear cosmological perturbations, we estimate numerically its precision, and we discuss the implications for the baryons acoustic oscillations. The TCA can be extended to the second order cosmological perturbations, and in particular we recover that vorticity is not generated at lowest order of this expansion.     

Sums of Laplace eigenvalues—rotationally symmetric maximizers in the plane

The sum of the first n?1 eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio ³(area)/(moment of inertia) for the domain is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and similar conclusions are derived for Robin boundary conditions and Schrödinger eigenvalues of potentials that grow at infinity. A key ingredient in the method is the tight frame property of the roots of unity. For general convex plane domains, the disk is conjectured to maximize sums of Neumann eigenvalues.     

The Elastic Continuum Limit of the Tight Binding Model

The authors consider the simplest quantum mechanics model of solids,the tight binding model,and prove that in the continuum limit,the energy of tight binding model converges to that of the continuum elasticity model obtained using Cauchy-Born rule.The technique in this paper is based mainly on spectral perturbation theory for large matrices.