The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus

In 1960–1962, E. Kähler enriched É. Cartan’s exterior calculus, making it suitable for quantum mechanics (QM) and not only classical physics. His “Kähler-Dirac” (KD) equation reproduces the fine structure of the hydrogen atom. Its positron solutions correspond to the same sign of the energy as electrons. The Cartan-Kähler view of some basic concepts of differential geometry is presented, as it explains why the components of Kähler’s tensor-valued differential forms have three series of indices. We demonstrate the power of his calculus by developing for the electron’s and positron’s large components their standard Hamiltonian beyond the Pauli approximation, but without resort to Foldy-Wouthuysen transformations or ad hoc alternatives (positrons are not identified with small components in K ähler’s work). The emergence of negative energies for positrons in the Dirac theory is interpreted from the perspective of the KD equation. Hamiltonians in closed form (i.e. exact through a finite number of terms) are obtained for both large and small components when the potential is time-independent. A new but as yet modest new interpretation of QM starts to emerge from that calculus’ peculiarities, which are present even when the input differential form in the Kähler equation is scalar-valued. Examples are the presence of an extra spin term, the greater number of components of “wave functions” and the non-association of small components with antiparticles. Contact with geometry is made through a Kähler type equation pertaining to Clifford-valued differential forms.     

Multi-coupled single scattering method of solving vector radiative transfer equations

<正>A new method of multi-coupled single scattering(MCSS) for solving a vector radiative transfer equation is developed and made public on Internet.Recent solutions from Chandrasekhar’s X-Y method is used to validate the MCSS’s result,which shows high precision.The MCSS method is theoretically simple and clear,so it can be easily and credibly extended to the simulation of aerosol/cloud atmosphere’s radiative properties,which provides effective support for research into polarized remote sensing.     

三层弱循环耦合可激发介质中螺旋波动力学

采用Bär模型研究了具有循环反馈耦合的三层可激发介质中的螺旋波动力学行为,数值模拟结果显示: 在耦合强度较小时, 在各子系统中可观察到螺旋波漂移或漫游; 当耦合强度稍大时, 相互作用既可以使螺旋波漫游或漂移出系统边界而使子系统回到静息态,还可以使子系统的螺旋波态转变为靶波或湍流态, 并观察到子系统的渐近态依赖初值现象; 继续增大耦合强度, 三个子系统的螺旋波可达到近似广义同步; 当耦合强度更大时, 螺旋波演化为湍流态.     

Colliding Einstein-Maxwell waves that admit a separable wave equation

We show that the massive wave equation is separable in two particular colliding Einstein-Maxwell waves space-times. The first is the type D metric of Chandrasekhar and Xanthopoulos whereas the second is the doubly polarized Bell-Szekeres solution which is of Petrov type General.     

Dynamical evolution of spatiotemporal patterns in mammalian middle cortex

The spatiotemporal structure of brain oscillations are important in understanding neural function. We analyze oscillatory episodes from isotropic preparations from the middle layers of a mammalian cortex which display irregular and chaotic spatiotemporal wave activity, within which spontaneously emerge spiral and plane waves. The dimensionality of these dynamics shows a consistent decrease during the middle of these episodes, regardless of the presence of simple spiral or plane waves. It is important to define the relevant biological order parameters which govern these dynamical bifurcations.     

基于MST雷达垂直风速的大气温度剖面反演

本文主要利用MST(mesosphere-stratosphere-troposphere)雷达垂直风速时间序列进行频谱分析,计算B-V(Brunt-V?is?l?)频率,再根据B-V频率和温度的关系,本文建立了离散的温度反演模型,最后反演得到了全高度的温度剖面.通过计算温度剖面和探空仪实测温度剖面比较可见,二者的符合度非常高,变化趋势也完全一致.本文还进一步讨论了,利用B-V频率和MST雷达的水平风速还可以计算理查德森数,通过理查德森数可以很清晰的判断大气的稳定性,进而可以量化大气很多动力学特征以及解释大气中的很多波动现象.因此,B-V频率的获得是MST雷达对大气动力学研究的又一贡献,它可以准确地反演大气的温度剖面,同时获取大气动力学稳定性参数.     

The massive Dirac equation in Kerr geometry: separability in Eddington–Finkelstein-type coordinates and asymptotics

The separability of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington–Finkelstein-type coordinates is shown. To this end, Kerr geometry is described by a Carter tetrad and the Dirac spinors and matrices are given in a chiral Newman–Penrose dyad representation. Applying Chandrasekhar’s mode ansatz, the Dirac equation is separated into systems of radial and angular ordinary differential equations. Asymptotic radial solutions at infinity, the event horizon, and the Cauchy horizon are explicitly derived. Their decay is analyzed by means of error estimates. Moreover, the eigenfunctions and eigenvalues of the angular system are discussed. Finally, as an application, the scattering of Dirac waves by the gravitational field of a Kerr black hole is studied. This work provides the basis for a Hamiltonian formulation of the massive Dirac equation in Kerr geometry in horizon-penetrating coordinates and for the construction of a functional analytic integral representation of the Dirac propagator.     

Energy spectrum of the ensemble of weakly nonlinear gravity-capillary waves on a fluid surface

We consider nonlinear gravity-capillary waves with the nonlinearity parameter ? ～ 0.1–0.25. For this nonlinearity, time scale separation does not occur and the kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in [15]. We for the first time compute an analytic expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function of the ratio of surface tension to the gravity acceleration. We show that its two limits—pure capillary and pure gravity waves on a fluid surface—coincide with the previously obtained results. We also discuss relations of the D-cascade model with a few known models used in the theory of nonlinear waves such as Zakharov’s equation, resonance of modes with nonlinear Stokes-corrected frequencies, and the Benjamin-Feir index. These connections are crucial in understanding and forecasting specifics of the energy transport in a variety of multicomponent wave dynamics, from oceanography to optics, from plasma physics to acoustics.     

Hydrodynamic stability and stellar oscillations

Chandrasekhar’s monograph on Hydrodynamic and hydromagnetic stability, published in 1961, is a standard reference on linear stability theory. It gives a detailed account of stability of fluid flow in a variety of circumstances, including convection, stability of Couette flow, Rayleigh–Taylor instability, Kelvin–Helmholtz instability as well as the Jean’s instability for star formation. In most cases he has extended these studies to include effects of rotation and magnetic field. In a later paper he has given a variational formulation for equations of non-radial stellar oscillations. This forms the basis for helioseismic inversion techniques as well as extension to include the effect of rotation, magnetic field and other large-scale flows using a perturbation treatment.     

Scattering of Dirac Waves off Reissner-Nordström Black Holes

Dirac equation in Kerr geometry was separated by Chandrasekhar in 1976. In the present paper, the radial parts of Dirac equation in Reissner-Nordström (RN) geometry are solved. We concentrate on two cases. The first one is that the energies of the waves are greater than the height of the potential barrier and the second one is that the waves hit on the potential barrier. In each case, the reflection and transmission coefficients and the wave function are computed. We compare the solutions with several parameters to show how the properties of the scattered wave depend on these parameters.     

On Bianchi’s Bäcklund transformation of quadrics

We investigate basic features of Bianchi’s Bäcklund transformation of quadrics and obtain it under weaker minimal assumptions: the only Bäcklund transformation with defining surface is Bianchi’s Bäcklund transformation of quadrics.     

The metre

A musical wind instrument transforms a constant pressure input from the player's mouth into a fluctuating pressure output in the form of a radiating sound wave. In reed woodwind and brass instruments, this transformation is achieved through a nonlinear coupling between two vibrating systems: the flow control valve formed by the mechanical reed or the lips of the player, and the air column contained by the pipe. Although the basic physics of reed wind instruments was developed by Helmholtz in the nineteenth century, the application of ideas from the modern theory of nonlinear dynamics has led to recent advances in our understanding of some musically important features of wind instrument behaviour. As a first step, the nonlinear aspects of the musical oscillator can be considered to be concentrated in the flow control valve; the air column can be treated as a linear vibrating system, with a set of natural modes of vibration corresponding to the standing waves in the pipe. Recent models based on these assumptions have had reasonable success in predicting the threshold blowing pressure and sounding frequency of a clarinet, as well as explaining at least qualitatively the way in which the timbre of the sound varies with blowing pressure. The situation is more complicated for brass instruments, in which the player's lips provide the flow valve. Experiments using artificial lips have been important in permitting systematic studies of the coupling between lips and air column; the detailed nature of this coupling is still not fully understood. In addition, the assumption of linearity in the air column vibratory system sometimes breaks down for brass instruments. Nonlinear effects in the propagation of high amplitude sound waves can lead to the development of shock waves in trumpets and trombones, with important musical consequences.     

Solitary deflection waves on the supersonic domain wall in yttrium orthoferrite

The moving antiferromagnetic vortices are accompanied by solitary deflection waves. These waves allow to investigate generation and nonlinear dynamics of the antiferromagnetic vortices on the moving domain wall with the help of the two- and three-fold digital high speed photography. On the quasi-relativistic domain wall the vortex dynamics is quasi-relativistic with the limiting velocity c=20 km/s, which is equal to the spin-wave velocity. The solitary deflection waves dynamics can be explained assuming existence of the gyroscopic force. A theory for the gyroscopic force in the orthoferrite domain wall is elaborating by A.K. Zvezdin et al. currently. We present a comparison of the theoretical and experimental results on the dynamics of the solitary deflection waves, which accompany the antiferromagnetic vortices in the domain wall of orthoferrites.     

Non-Birational Twisted Derived Equivalences in Abelian GLSMs

In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between non-birational spaces, and realizing geometries in novel fashions. Examples of gauged linear sigma models with non-birational Kähler phases are a relatively new phenomenon. Most of our examples involve gauged linear sigma models for complete intersections of quadric hypersurfaces, though we also discuss some more general cases and their interpretation. We also propose a more general understanding of the relationship between Kähler phases of gauged linear sigma models, namely that they are related by (and realize) Kuznetsov’s ‘homological projective duality.’ Along the way, we shall see how ‘noncommutative spaces’ (in Kontsevich’s sense) are realized physically in gauged linear sigma models, providing examples of new types of conformal field theories. Throughout, the physical realization of stacks plays a key role in interpreting physical structures appearing in GLSMs, and we find that stacks are implicitly much more common in GLSMs than previously realized.     

Complex/Symplectic Mirrors

We construct a class of symplectic non-Kähler and complex non-Kähler string theory vacua, extending and providing evidence for an earlier suggestion by Polchinski and Strominger. The class admits a mirror pairing by construction. Comparing hints from a variety of sources, including ten-dimensional supergravity and KK reduction on SU(3)-structure manifolds, suggests a picture in which string theory extends Reid’s fantasy to connect classes of both complex non-Kähler and symplectic non-Kähler manifolds.     

Ableitung einer kontinuierlichen Mannigfaltigkeit von Erhaltungssätzen der modifizierten Korteweg-de-Vries-Gleichung nach dem Noetherschen Satz

Derivation of a Continuous Set of Conservation Laws for the Modified Korteweg-de Vries Equation by Noether's Theorem A method developed recently to derive a continuous set of conservation laws from extended Bäcklund transformations by means of Noether's theorem is applied to the modified Korteweg-de Vries (m. KdV) equation that describes Alfvén waves in a plasma. The corresponding conserved currents are equivalent to those found by WADATI , SANUKI and KONNO . It is shown that the extended Bäcklund transformation B?_α for the m. KdV equation, which coincides with that for the sine-Gordon equation, by MIURA'S transformation becomes the extended Bäcklund transformation β_x for the Korteweg-de Vries equation where x = 1/2α.     

The Geometry of Recursion Operators

We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser’s theorem for simultaneous isotopies of two families of symplectic forms. We also consider the geometric structures defined by pairs and triples of symplectic forms for which the squares of the intertwining endomorphisms are plus or minus the identity. For pairs of forms we recover the notions of symplectic pairs and of holomorphic symplectic structures. For triples we recover the notion of a hypersymplectic structure, and we also find three new structures that have not been considered before. One of these is the symplectic formulation of hyper-Kähler geometry, which turns out to be a strict generalization of the usual definition in terms of differential or Kähler geometry.     

Fragmentation of rods by cascading cracks: why spaghetti does not break in half

When thin brittle rods such as dry spaghetti pasta are bent beyond their limit curvature, they often break into more than two pieces, typically three or four. With the aim of understanding these multiple breakings, we study the dynamics of a bent rod that is suddenly released at one end. We find that the sudden relaxation of the curvature at this end leads to a burst of flexural waves, whose dynamics are described by a self-similar solution with no adjustable parameters. These flexural waves locally increase the curvature in the rod, and we argue that this counterintuitive mechanism is responsible for the fragmentation of brittle rods under bending. A simple experiment supporting the claim is presented.