Vortex structures with complex points singularities in two-dimensional Euler equations. New exact solutions

In this work we present a new class of exact stationary solutions for two-dimensional (2D) Euler equations. Unlike already known solutions, the new ones contain complex singularities. We consider point singularities which have a vector field index greater than 1 as complex. For example, the dipole singularity is complex because its index is equal to 2. We present in explicit form a large class of exact localized stationary solutions for 2D Euler equations with a singularity whose index is equal to 3. The solutions obtained are expressed in terms of elementary functions. These solutions represent a complex singularity point surrounded by a vortex satellite structure. We also discuss the motion equation of singularities and conditions for singularity point stationarity which provide the stationarity of the complex vortex configuration.     

Bohmian mechanics at space–time singularities: II. Spacelike singularities

We develop an extension of Bohmian mechanics by defining Bohm-like trajectories for quantum particles in a curved background space–time containing a spacelike singularity. As an example of such a metric we use the Schwarzschild metric, which contains two spacelike singularities, one in the past and one in the future. Since the particle world lines are everywhere timelike or lightlike, particles can be annihilated but not created at a future spacelike singularity, and created but not annihilated at a past spacelike singularity. It is argued that in the presence of future (past) spacelike singularities, there is a unique natural Bohm-like evolution law directed to the future (past). This law differs from the one in non-singular space–times mainly in two ways: it involves Fock space since the particle number is not conserved, and the wave function is replaced by a density matrix. In particular, we determine the evolution equation for the density matrix, a pure-to-mixed evolution equation of a quasi-Lindblad form. We have to leave open whether a curvature cut-off needs to be introduced for this equation to be well defined.     

Horizonless,singularity-free,compact shells satisfying NEC

Gravitational collapse singularities are undesirable, yet inevitable to a large extent in General Relativity. When matter satisfying null energy condition (NEC) collapses to the extent a closed trapped surface is formed, a singularity is inevitable according to Penrose’s singularity theorem. Since positive mass vacuum solutions are generally black holes with trapped surfaces inside the event horizon, matter cannot collapse to an arbitrarily small size without generating a singularity. However, in modified theories of gravity where positive mass vacuum solutions are naked singularities with no trapped surfaces, it is reasonable to expect that matter can collapse to an arbitrarily small size without generating a singularity. Here we examine this possibility in the context of a modified theory of gravity with torsion in an extra dimension. We study singularity-free static shell solutions to evaluate the validity of NEC on the shell. We find that with sufficiently high pressure, matter can be collapsed to arbitrarily small size without violating NEC and without producing a singularity.     

The osgood criterion and finite‐time cosmological singularities

In this paper, we apply Osgood's criterion from the theory of ordinary differential equations to detect finite‐time singularities in a spatially flat FLRW universe in the context of a perfect fluid, a perfect fluid with bulk viscosity, and a Chaplygin and anti‐Chaplygin gas. In particular, we applied Osgood's criterion to demonstrate singularity behaviour for Type 0/big crunch singularities as well as Type II/sudden singularities. We show that in each case the choice of initial conditions is important as a certain number of initial conditions leads to finite‐time, Type 0 singularities, while other precise choices of initial conditions which depend on the cosmological matter parameters and the cosmological constant can avoid such a finite‐time singularity. Osgood's criterion provides a powerful and yet simple way of deducing the existence of these singularities, and also interestingly enough, provides clues of how to eliminate singularities from certain cosmological models.     

Lifshitz tails and long-time decay in random systems with arbitrary disorder

In random systems, the density of states of various linear problems, such as phonons, tight-binding electrons, or diffusion in a medium with traps, exhibits an exponentially small Liftshitz tail at band edges. When the distribution of the appropriate random variables (atomic masses, site energies, trap depths) has a delta function at its lower (upper) bound, the Lifshitz singularities are pure exponentials. We study in a quantitative way how these singularities are affected by a universal logarithmic correction for continuous distributions starting with a power law. We derive an asymptotic expansion of the Lifshitz tail to all orders in this logarithmic variable. For distributions starting with an essential singularity, the exponent of the Lifshitz singularity itself is modified. These results are obtained in the example of harmonic chains with random masses. It is argued that analogous results hoid in higher dimensions. Their implications for other models, such as the long-time decay in trapping problems, are also discussed.     

双压电材料三维界面端部力电耦合场奇异性

基于切口根部物理场的幂级数渐近展开假设,从三维应力平衡方程和麦克斯韦方程组出发,导出关于双压电材料楔形界面切口端部奇性指数的特征微分方程组,并将切口的力电学边界条件表达为奇性指数和特征角函数的组合,从而将双压电材料楔形界面切口端部奇性指数的计算转化为相应边界条件下常微分方程组特征值的求解,运用插值矩阵法求解界面端部若干阶应力奇性指数和相应特征函数.计算结果与已有结果对比表明本文方法的有效性和具有较高的计算精度.     

Singularities in Einstein–conformally coupled Higgs cosmological models

The dynamics of Einstein–conformally coupled Higgs field (EccH) system is investigated near the initial singularities in the presence of Friedman–Robertson–Walker symmetries. We solve the field equations asymptotically up to fourth order near the singularities analytically, and determine the solutions numerically as well. We found all the asymptotic, power series singular solutions, which are (1) solutions with a scalar polynomial curvature singularity but the Higgs field is bounded (‘Small Bang’), or (2) solutions with a Milne type singularity with bounded spacetime curvature and Higgs field, or (3) solutions with a scalar polynomial curvature singularity and diverging Higgs field (‘Big Bang’). Thus, in the present EccH model there is a new kind of physical spacetime singularity (‘Small Bang’). We also show that, in a neighbourhood of the singularity in these solutions, the Higgs sector does not have any symmetry breaking instantaneous vacuum state, and hence then the Brout–Englert–Higgs mechanism does not work. The large scale behaviour of the solutions is investigated numerically as well. In particular, the numerical calculations indicate that there are singular solutions that cannot be approximated by power series.     

Random Wavelet Series Based on a Tree-Indexed Markov Chain

We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Hölder exponent form a set with large intersection.     

Singularities and spectral changes of Gaussian beams focused by a lens with spherical aberration

We address the effect of the truncation parameter and spherical aberration (SA) on the singularity transformation and spectral behavior of the polychromatic Gaussian beams focused by an aperture lens with SA in detail. The numerical simulation results, based on the derived equations of the intensity and the spectral density, are given. It is found that the axial singularities vanished with the change of the truncated parameter. The intensity and drastic spectral change fade away with an annihilation process of the phase singularities, and the drastic spectral change does not disappear immediately at the moment the phase singularity annihilates. The singularities in the focal region will redistribute with the increment of SA coefficient, some singularities will vanish, some will spilt into two new singularities, and other off-axial singularities will appear and split into two new singularities as well. When SA coefficient changed, we can find that the axial singularities disappear as well with the decreasing value of truncation parameter. These new splitted singularities due to the change of SA coefficient will converge into one singularity again and disappear gradually.     

Amplitude equations for non-linear Rayleigh waves

We investigate asymptotic equations describing small amplitude surface elastic waves in the half-plane (Rayleigh waves). For hyperelastic materials such model equations are Hamiltonian systems, and are seen to lead to the formation of singularities in the surface elastic displacement. At the time of singularity formation the Fourier spectra of the solutions exhibit power law decay, and the observed exponents suggest the existence of both differentiable and non-differentiable singular profiles.     

Singularities and conjugate points in FLRW spacetimes

Conjugate points play an important role in the proofs of the singularity theorems of Hawking and Penrose. We examine the relation between singularities and conjugate points in FLRW spacetimes with a singularity. In particular we prove a theorem that when a non-comoving, non-spacelike geodesic in a singular FLRW spacetime obeys conditions (39) and (40), every point on that geodesic is part of a pair of conjugate points. The proof is based on the Raychaudhuri equation. We find that the theorem is applicable to all non-comoving, non-spacelike geodesics in FLRW spacetimes with non-negative spatial curvature and scale factors that near the singularity have power law behavior or power law behavior times a logarithm. When the spatial curvature is negative, the theorem is applicable to a subset of these spacetimes.     

Coupled dynamic analysis of a single gimbal control moment gyro cluster integrated with an isolation system

Control moment gyros (CMGs) are widely used as actuators for attitude control in spacecraft. However, micro-vibrations produced by CMGs will degrade the pointing performance of high-sensitivity instruments on-board the spacecraft. This paper addresses dynamic modelling and performs an analysis on the micro-vibration isolation for a single gimbal CMG (SGCMG) cluster. First, an analytical model was developed to describe both the coupled SGCMG cluster and the multi-axis isolation system that can express the dynamic outputs. This analytical model accurately reflects the mass and inertia properties, the gyroscopic effects and flexible modes of the coupled system, which can be generalized for isolation applications of SGCMG clusters. Second, the analytical model was validated using MSC.NASTRAN software based on the finite element technique. The dynamic characteristics of the coupled system are affected by the mass distribution and the gyroscopic effects of the SGCMGs. The gyroscopic effects produced by the rotary flywheel will stiffen or soften several of the structural modes of the coupled system. In addition, the gyroscopic effect of each SGCMG can interact with or counteract that of others, which induce vibration modes coupled together. Finally, the performance of the passive isolation was analysed. It was demonstrated that the gyroscopic effects should be considered in isolation studies on SGCMG clusters; otherwise, the isolation performance will be underestimated if they are ignored.     

On the genericity of spacetime singularities

We consider here the genericity aspects of spacetime singularities that occur in cosmology and in gravitational collapse. The singularity theorems (that predict the occurrence of singularities in general relativity) allow the singularities of gravitational collapse to be either visible to external observers or covered by an event horizon of gravity. It is shown that the visible singularities that develop as final states of spherical collapse are generic. Some consequences of this fact are discussed.      

Einstein equation at singularities

Einstein’s equation is rewritten in an equivalent form, which remains valid at the singularities in some major cases. These cases include the Schwarzschild singularity, the Friedmann-Lemaître-Robertson-Walker Big Bang singularity, isotropic singularities, and a class of warped product singularities. This equation is constructed in terms of the Ricci part of the Riemann curvature (as the Kulkarni-Nomizu product between Einstein’s equation and the metric tensor).     

Fermionic greybody factors of two and five-dimensional dilatonic black holes

We analyze the persistence of curvature singularities when analyzed using quantum theory. First, quantum test particles obeying the Klein–Gordon and Chandrasekhar–Dirac equation are used to probe the classical timelike naked singularity. We show that the classical singularity is felt even by our quantum probes. Next, we use loop quantization to resolve a singularity hidden beneath the horizon. The singularity is resolved in this case.     

Black holes without singularities

The conventional interpretation of the Hawking-Penrose singularity theorems is that gravitational collapse, signified by the presence of a closed trapped surface, generally leads to the formation of a singularity. Consideration is given here to an alternative interpretation according to which collapse scenarios may give rise, not to singularities, but to chronology violation instead. An example is given of a singularity-free, chronology-violating space-time with a (non-achronal) closed trapped surface. In a large class of singularity-free space-times, the presence of a closed trapped surface, achronal or not, necessitates a violation of chronology. Moreover, all closed trapped surfaces and chronology violations are confined to black holes; weak cosmic censorship must hold in the sense that the region of the space-time visible from infinity is globally hyperbolic.Expanded version of essay given Honorable Mention in the 1986 Gravity Foundation Award.     

The structure of naked singularity in self-similar gravitational collapse

We study the structure and formation of naked singularities in selfsimilar gravitational collapse for an adiabatic perfect fluid. Conditions are obtained for the singularity to be either locally or globally naked and for the families of non-spacelike geodesics to terminate at the singularity in past. This is shown to be a strong curvature naked singularity in a powerful sense and an interesting relationship is pointed out between positivity of energy and occurrence of naked singularity.