Euclidean and Lorentzian quantum gravity—lessons from two dimensions

No theory of four-dimensional quantum gravity exists as yet. In this situation the two-dimensional theory, which can be analyzed by conventional field-theoretical methods, can serve as a toy model for studying some aspects of quantum gravity. It represents one of the rare settings in a quantum-gravitational context where one can calculate quantities truly independent of any background geometry.We review recent progress in our understanding of 2d quantum gravity, and in particular the relation between the Euclidean and Lorentzian sectors of the quantum theory. We show that conventional 2d Euclidean quantum gravity can be obtained from Lorentzian quantum gravity by an analytic continuation only if we allow for spatial topology changes in the latter. Once this is done, one obtains a theory of quantum gravity where space-time is fractal: the intrinsic Hausdorff dimension of usual 2d Euclidean quantum gravity is four, and not two. However, certain aspects of quantum space-time remain two-dimensional, exemplified by the fact that its so-called spectral dimension is equal to two.     

Method for taking into account gravity in free-surface flow simulation

A numerical algorithm that correctly takes into account the force of gravity in the presence of density discontinuities is constructed using unstructured collocated grids and splitting algorithms based on SIMPLE-type methods. A correct hydrostatic pressure field is obtained by explicitly extracting the gravity force contribution to the pressure equation and computing it using the solution of the gravity equilibrium problem for a two-phase medium. To ensure that the force of gravity is balanced by the pressure gradient in the case of a medium at rest, an algorithm is proposed according to which the pressure gradient in the equations of motion is replaced by a modification allowing for the force of gravity. Well-known free-surface problems are used to show that, in contrast to previously known algorithms, the proposed ones on unstructured meshes correctly predict hydrostatic pressure fields and do not yield velocity oscillations or free-surface distortions.     

Unified description of cosmological and static solutions in affine generalized theories of gravity: Vecton-scalaron duality and its applications

We briefly describe the simplest class of affine theories of gravity in multidimensional space-times with symmetric connections and their reductions to two-dimensional dilaton-vecton gravity field theories. The distinctive feature of these theories is the presence of an absolutely neutral massive (or tachyonic) vector field (vecton) with an essentially nonlinear coupling to the dilaton gravity. We emphasize that the vecton field in dilaton-vecton gravity can be consistently replaced by a new effectively massive scalar field (scalaron) with an unusual coupling to the dilaton gravity. With this vecton-scalaron duality, we can use the methods and results of the standard dilaton gravity coupled to usual scalars in more complex dilaton-scalaron gravity theories equivalent to dilaton-vecton gravity. We present the dilaton-vecton gravity models derived by reductions of multidimensional affine theories and obtain one-dimensional dynamical systems simultaneously describing cosmological and static states in any gauge. Our approach is fully applicable to studying static and cosmological solutions in multidimensional theories and also in general one-dimensional dilaton-scalaron gravity models. We focus on general and global properties of the models, seeking integrals and analyzing the structure of the solution space. In integrable cases, it can be usefully visualized by drawing a “topological portrait” resembling the phase portraits of dynamical systems and simply exposing the global properties of static and cosmological solutions, including horizons, singularities, etc. For analytic approximations, we also propose an integral equation well suited for iterations.     

Effect of gravity on subject-specific human lung deformation

A biomechanical model of human lung is developed and used to investigate the effect of gravity on lung deformation. The lung is assumed to behave as a poro-elastic medium with spatially dependent elastic property. Finite element analysis is performed on a three-dimensional (3D) lung geometry reconstructed from a four-dimensional Computed Tomography (4DCT) scan dataset of human patient. The spatially dependent Young’s modulus (YM) values are estimated using inverse analysis from a linear elastic deformation model. The predicted deformation of selected landmarks is monitored with and without gravity, and compared with data obtained from 4DCT registration. The results show that gravity indeed significantly affects the magnitude and distribution of lung deformation with the maximum displacement enhanced by 54% in the direction of gravity, for the conditions investigated. In summary, the accuracy of predicted deformation is improved through incorporation of gravity in the biomechanical model of lung.     

Global large solutions to incompressible Navier‐Stokes equations with gravity

In this per, we consider a special class of initial data for the three‐dimensional incompressible Navier–Stokes equations with gravity. We show that, under such conditions, the incompressible Navier‐Stokes equations with gravity are globally well posed, and the velocity minus gravity term has finite energy. The important features of the initial data is that the velocity fields minus gravity term are almost parallel to the corresponding vorticity fields in a very large space domain. Copyright © 2014 John Wiley & Sons, Ltd.     

The distance-decay function of geographical gravity model: Power law or exponential law?

The distance-decay function of the geographical gravity model is originally an inverse power law, which suggests a scaling process in spatial interaction. However, the distance exponent of the model cannot be reasonably explained with the ideas from Euclidean geometry. This results in a dimension dilemma in geographical analysis. Consequently, a negative exponential function was used to replace the inverse power function to serve for a distance-decay function. But a new puzzle arose that the exponential-based gravity model goes against the first law of geography. This paper is devoted for solving these kinds of problems by mathematical reasoning and empirical analysis. New findings are as follows. First, the distance exponent of the gravity model is demonstrated to be a fractal dimension using the geometric measure relation. Second, the similarities and differences between the gravity models and spatial interaction models are revealed using allometric relations. Third, a four-parameter gravity model possesses a symmetrical expression, and we need dual gravity models to describe spatial flows. The observational data of China's cities and regions (29 elements indicative of 841 data points) in 2010 are employed to verify the theoretical inferences. A conclusion can be reached that the geographical gravity model based on power-law decay is more suitable for analyzing large, complex, and scale-free regional and urban systems. This study lends further support to the suggestion that the underlying rationale of fractal structure is entropy maximization. Moreover, it suggests that many dimensional dilemmas of spatial modeling can be solved using the concepts from fractal geometry.     

Thermal stability of a fluid layer under variable body forces

The instability of a layer of fluid confined between two horizontal parallel planes and heated from above or below under the action of a gravitational field varying with height is investigated analytically. It is found that if gravity remains downward (upward) throughout the flow domain, neutral modes do not exist. A sufficient condition for stability of a layer heated from above is that gravity remain directed downward over a sufficiently large part of the flow domain. A circle theorem limiting the growth rate of a arbitrary oscillatory mode (whether stable, neutral or unstable) is established. A sufficient condition for instability of a layer heated from below is found to be that gravity remain directed downward and the gravity profile have concave curvature throughout most of the flow domain. For a linearly varying gravity field with the downward gravitational acceleration increasing with height or decreasing with height slowly, (i) a layer heated from above is stable, (ii) a layer heated from below is unstable, and (iii) the circle theorem is modified depending on whether the gravitational acceleration is increasing or decreasing with height.     

重力输水管道优化设计方法研究

针对重力输水管道设计中存在的问题，文章以管道造价最小为目标，应用动态规划，提出了一种新的重力输水管道优化设计方法，对输水管的当量管径、流量分配、连接管设置与可靠性校核计算等问题作了探讨。该方法可用于沿线有节点流量流出的并联输水管道优化设计，能明显节省投资，有良好的实用性     

The study and applications of photochemical-dynamical gravity wave model II

A nonlinear, compressible, non-isothermal gravity wave model that involves photochemistry is used to study the effects of gravity wave on atmospheric chemical species distributions in this paper. The changes in the distributions of oxygen compound and hydrogen compound density induced by gravity wave propagation are simulated. The results indicate that when a gravity wave propagates through a mesopause region, even if it does not break, it can influence the background distributions of chemical species. The effect of gravity wave on chemical species at night is larger than in daytime.     

The study and applications of photochemical-dynamical gravity wave model Ⅱ-- The effects of stable gravity wave on chemical species distribution in mesosphere

A nonlinear, compressible, non-isothermal gravity wave model that involves photochemistry is used to study the effects of gravity wave on atmospheric chemical species distributions in this paper. The changes in the distributions of oxygen compound and hydrogen compound density induced by gravity wave propagation are simulated. The results indicate that when a gravity wave propagates through a mesopause region, even if it does not break, it can influence the background distributions of chemical species. The effect of gravity wave on chemical species at night is larger than in daytime.     

Convection in a variable gravity field

The problem of convection in a variable gravity field is studied by using methods of linear instability theory and nonlinear energy theory. It is shown that the decreasing or increasing of gravity in a specific direction can be stabilizing or destabilizing and it is further shown how to quantify this effect. Specific results are presented for the situation where gravity decreases linearly throughout a plane layer. The nonlinear results are found to be very close to the linear ones and define a small band where possible subcritical instabilities may arise.     

炉液倾动重心近似计算的数学分析

为给转炉设计提供依据,需要计算炉液倾动的重心.利用数学方法将实际问题进行简化,通过分析炉液倾动过程中变量间的相互关系,来确定每个倾动角度对应情况下的液面位置.利用数学中三重积分的有关应用,进一步得出转炉在每个倾动角度为α∈(0,π2)时的重心计算方法及相关结论,在理论上为工程计算重心的方法提供参考.     

Classical Gauge Theory of Gravity

The classical theory of gravity is formulated as a gauge theory on a frame bundle with spontaneous symmetry breaking caused by the existence of Dirac fermionic fields. The pseudo-Reimannian metric (tetrad field) is the corresponding Higgs field. We consider two variants of this theory. In the first variant, gravity is represented by the pseudo-Reimannian metric as in general relativity theory; in the second variant, it is represented by the effective metric as in Logunov's relativistic theory of gravity. The configuration space, Dirac operator, and Lagrangians are constructed for both variants.     

Three-dimensional simulation of the runup of nonlinear surface gravity waves

The runup of nonlinear surface gravity waves is numerically simulated in two and three dimensions on the basis of the Navier-Stokes equations. The three-dimensional problem is formulated, and the boundary and initial conditions are described. The splitting method over physical processes is used to construct a discrete model taking into account the cell occupation coefficient. The runup of nonlinear surface gravity waves is simulated in two dimensions for slopes of various geometries, and the numerical results are analyzed. The structural features of the simulated three-dimensional basin are described. Three-dimensional models for the staged runup of nonlinear surface gravity waves breaking on coastal slopes in shallow water areas are considered.     

Jump Conditions for Hyperbolic Systems of Forced Conservation Laws with an Application to Gravity Currents

Weak solutions to systems of nonlinear hyperbolic conservation laws admit discontinuities that result from either an initial value or as part of the temporally developing solution itself. The propagation of such shocks or jumps is affected by forcing terms for the nonlinear system in a way that has not been investigated fully in standard references. Jump conditions for systems of conservation laws with discontinuous forcing terms are derived herein, following the method used to derive the Rankine–Hugoniot jump conditions, and the generalized results are illustrated for the one-dimensional inviscid Burger's equation with discontinuous forcing. The main application of this type of jump condition, and the primary motivation for its study, is its application to a shallow-water model of gravity currents previously described by the authors. Specifically, a new result relation between the front and height at a gravity current front is obtained by using the existing model. Front speeds for gravity currents resulting from instantaneous release are calculated numerically and used to determine the suitability of the jump conditions, which are then compared with existing theoretical expressions and experimental observations. New numerical results are portrayed for the gravity current model, suggesting that the standard method of modeling shallow-water gravity currents with a simple Froude number front condition may tend to suppress some of the finer details of the flow resolved by the numerical scheme used by the authors.     

改进的云重心评判法在高技术企业信用评价中的应用

提出一种改进的云重心评判法,并应用于高技术企业信用评价.首先,借鉴TOPSIS法的基本思想,基于理想状态和负理想状态,对综合云重心向量进行归一化,并采用修正的加权偏离度来衡量云重心的变化,从而克服了传统云重心评判法的不足.其次,针对高技术企业信用评价中的不确定性,运用改进的云重心评判法对高技术企业信用状态进行实证测评,结果证明了该方法的合理性和适用性.     

Averaging over fast gravity waves for geophysical flows with arbitary

Here a mathematically rigorous framework is developed for deriving new reduced simplified dynamical equations for geophysical flows with arbitrary potential vorticity interacting with fast gravity waves. The examples include the rotating Boussinesq and rotating shallow water equations in the quasigeostrophic limit with vanishing Rossby number. For the spatial periodic case the theory implies that the quasi—geostrophic equations are valid limiting equations in the weak topology for arbitrary initial data. Furthermore, simplified reduced equations are developed for the fashion in which the vortical waves influence the gravity waves through averaging over specific gravity wave/vortical resonances.     

4—导数引力的量子化及其重整化

该文求得了4-导数引力的场方程，用带导数算子且带权的规范固定计算出了引力子自由与三顶角等树图传播子，用带4动量的引力三项角求出了鬼外线和引力外线二类圈图的发散度．如上作法及结果与已有文献不同．通过对各种圈图发散的分析，指出一些文献用解通常重整化方程的方法得以消除发散的抵消项，是需要进一步研究的．     

Complex Ashtekar Variables and Reality Conditions for Holst’s Action

From the Holst action in terms of complex valued Ashtekar variables, additional reality conditions mimicking the linear simplicity constraints of spin-foam gravity are found. In quantum theory with the results of Ding and Rovelli, we are able to implement these constraints weakly, that is in the sense of Gupta and Bleuler. The resulting kinematical Hilbert space matches the original one of loop quantum gravity, that is for real valued Ashtekar connection. Our results perfectly fit with recent developments of Rovelli and Speziale concerning Lorentz covariance within spin-form gravity.     

Lie group analysis of gravity currents

We consider shallow water theory to study the self-similar gravity currents that describe the motion of a heavy fluid flowing into another lighter ambient fluid. Gratton and Vigo investigated the shallow water theory representing the self-similar gravity currents by using dimensional analysis [J. Gratton, C. Vigo, Self-similarity gravity currents with variable inflow revisited: Plane currents, J. Fluid. Mech. 258 (1994) 77–104]. But in this study, the self-similarity solutions of the one-layer shallow-water equations representing gravity currents are investigated by using Lie group analysis and it is shown that Lie group analysis is the generalization of the dimensional analysis for investigating the self-similarity solutions of the one-layer shallow-water equations. Applying Lie group theory, reduced equations of the shallow water equations are found. Therefore, it becomes possible to obtain the similarity forms depending on the Lie group parameters and also the self-similarity solutions for the special values of these group parameters.