Weak lensing of the CMB

The cosmic microwave background (CMB) represents a unique source for the study of gravitational lensing. It is extended across the entire sky, partially polarized, located at the extreme distance of z = 1,100, and is thought to have the simple, underlying statistics of a Gaussian random field. Here we review the weak lensing of the CMB, highlighting the aspects which differentiate it from the weak lensing of other sources, such as galaxies. We discuss the statistics of the lensing deflection field which remaps the CMB, and the corresponding effect on the power spectra. We then focus on methods for reconstructing the lensing deflections, describing efficient quadratic maximum-likelihood estimators and delensing. We end by reviewing recent detections and observational prospects.     

Gravitational lensing by black holes

We review the theoretical aspects of gravitational lensing by black holes, and discuss the perspectives for realistic observations. We will first treat lensing by spherically symmetric black holes, in which the formation of infinite sequences of higher order images emerges in the clearest way. We will then consider the effects of the spin of the black hole, with the formation of giant higher order caustics and multiple images. Finally, we will consider the perspectives for observations of black hole lensing, from the detection of secondary images of stellar sources and spots on the accretion disk to the interpretation of iron K-lines and direct imaging of the shadow of the black hole.     

Cosmology with weak lensing surveys

Weak gravitational lensing is responsible for the shearing and magnification of the images of high-redshift sources due to the presence of intervening matter. The distortions are due to fluctuations in the gravitational potential, and are directly related to the distribution of matter and to the geometry and dynamics of the Universe. As a consequence, weak gravitational lensing offers unique possibilities for probing the Dark Matter and Dark Energy in the Universe. In this review, we summarise the theoretical and observational state of the subject, focussing on the statistical aspects of weak lensing, and consider the prospects for weak lensing surveys in the future.     

Gravitational Lensing of Fermion Stars

We discuss the fermion stars, the self-gravitating systems of Fermi gases, as possible gravitational lenses. It is supposed that the fermions interact with themselves and other particles only by gravity, so they are the candidates of dark matter. We calculate Einstein deflection angles, study the image configurations, and calculate the magnification factors for a number of fermion stars that range from strong relativistic configurations to nonrelativistic ones. We find that typically there are three images, one Einstein ring and one radial critical curve for both cases. Two of the images are within the Einstein ring, and the other is outside, which may be very far. All these lensing characteristics can help to identify fermion stars as potential lensing objects, thus might give direct evidence that dark fermion stars exist in the universe.     

Problems with small area surveys: lensing covariance of supernova distance measurements

While luminosity distances from type Ia supernovae (SNe) are a powerful probe of cosmology, the accuracy with which these distances can be measured is limited by cosmic magnification due to gravitational lensing by the intervening large-scale structure. Spatial clustering of foreground mass leads to correlated errors in SNe distances. By including the full covariance matrix of SNe, we show that future wide-field surveys will remain largely unaffected by lensing correlations. However, "pencil beam" surveys, and those with narrow (but possibly long) fields of view, can be strongly affected. For a survey with 30 arcmin mean separation between SNe, lensing covariance leads to a approximately 45% increase in the expected errors in dark energy parameters.     

An LMI approach to asymptotical stability of multi-delayed neural networks

In this paper, some criteria are derived for global asymptotic stability of a class of neural networks with multiple constant or time-varying delays. Based on the Lyapunov–Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) approach, some delay-independent criteria for neural networks with multiple constant delays and delay-dependent criteria for neural networks with multiple time-varying delays are provided to guarantee global asymptotic stability of these networks. The main results are generalizations of some recent results reported in the literature.     

Strong lensing of gravitational waves as seen by LISA

We discuss strong gravitational lensing of gravitational waves from the merging of massive black hole binaries in the context of the LISA mission. Detection of multiple events would provide invaluable information on competing theories of gravity, evolution and formation of structures and, possibly, constraints on H0 and other cosmological parameters. Most of the optical depth for lensing is provided by intervening massive galactic halos, for which wave optics effects are negligible. Probabilities to observe multiple events are sizable for a broad range of formation histories. For the most optimistic models, up to ? 4 multiple events with a signal to noise ratio ? 8 are expected in a 5-year mission. Chances are significant even for conservative models with either light (? 60%) or heavy (? 40%) seeds. Because of lensing amplification, some intrinsically too faint signals are brought over threshold (? 2 per year).     

Transitive ensembles of random matrices related to orthogonal polynomials

Transitive correlations of eigenvalues for random matrix ensembles intermediate between real symmetric and hermitian, self-dual quaternion and hermitian, and antisymmetric and hermitian are studied. Expressions for exact n-point correlation functions are obtained for random matrix ensembles related to general orthogonal polynomials. The asymptotic formulas in the limit of large matrix dimension are evaluated at the spectrum edges for the ensembles related to the Legendre polynomials. The results interpolate known asymptotic formulas for random matrix eigenvalues.     

Analysis of one-dimensional wave scattering by the random walk method

The random walk method is applied to a one-dimensional Helmholtz equation with a variable wave number. The solution is represented as a mathematical expectation of a specified functional on paths in a complex space. This solution degenerates to the ray-method approximation in domains where the latter method may be used, but the probabilistic formulas presented describe also backscattered waves whose existence is not explained by the standard asymptotic techniques. The numerical results confirm the efficiency of the random walk approach to the analysis of wave propagation.     

On Edge Exchangeable Random Graphs

We study a recent model for edge exchangeable random graphs introduced by Crane and Dempsey; in particular we study asymptotic properties of the random simple graph obtained by merging multiple edges. We study a number of examples, and show that the model can produce dense, sparse and extremely sparse random graphs. One example yields a power-law degree distribution. We give some examples where the random graph is dense and converges a.s. in the sense of graph limit theory, but also an example where a.s. every graph limit is the limit of some subsequence. Another example is sparse and yields convergence to a non-integrable generalized graphon defined on \((0,\infty )\).     

On the expected number of distinct points in a subset visited by anN-step random walk

Many investigators have calculated asymptotically valid expressions for the expected number of distinct points visited by ann-step random walk on a lattice. In this note we point out that the same formalism can be used to study the expected number of distinct points in a subset of lattice points. We also calculate the expected occupancy of the subset and give sufficient conditions for the ratio of the two calculated quantities to have the same asymptotic time dependence as for the full lattice. Specific examples are considered.     

Gravitational Lensing by Charged Black Holes

We formulate the lensing effects of a spherically symmetric electrically charged black hole using thin lens equations. The charged black hole leads to three images and could lead to three Einstein rings provided the parameters such as the mass, charge and the distances satisfy certain constraints. We have computed the exact positions of images and magnification properties for a super-massive black hole with electric charge.     

并矢格林函数下的球形超透镜

应用并矢量格林函数法完成了球形超透镜的数值模拟,证明了球形超镜的亚波长成像能力.并矢格林函数法是处理电磁场问题的一种系统理论和有效方法,它弥补了坐标变换法的不足.与平板透镜相比,球形超透镜有几个优点,包括有限的横截面,能成放大或者缩小的像,很高的分辨率,进行二维成像等. 关键词： 球形超透镜 并矢量格林函数法 表面模 高分辨率     

Asymptotics and Dimensional Dependence of the Number of Critical Points of Random Holomorphic Sections

We prove two conjectures from [DSZ2,DSZ3] concerning the expected number of critical points of random holomorphic sections of a positive line bundle. We show that, on average, the critical points of minimal Morse index are the most plentiful for holomorphic sections of \({\mathcal {O}(N) \to \mathbb {CP}^m}\) and, in an asymptotic sense, for those of line bundles over general Kähler manifolds. We calculate the expected number of these critical points for the respective cases and use these to obtain growth rates and asymptotic bounds for the total expected number of critical points in these cases. This line of research was motivated by landscape problems in string theory and spin glasses.     

Strong Field Gravitational Lensing in a Magnetic Charged Reissner-Nordstr?m Black Hole Pierced by a Cosmic String

The principal focus of this paper is to study the strong field gravitational lensing in a magnetic charged Reissner-Nordstr?m black hole based on the method of cosmic string. We obtain the new coefficients including the tension of the cosmic strings, the strong field deflection limit coefficients, the deflection angle and the magnification, and obtain the relationship between the cosmic string parameter and the new coefficients. The result shows that the cosmic strings have some important effect on the gravitational lensing in a black hole when they pierce it.     

Behavior near the focal points of asymptotic solutions to the Cauchy problem for the linearized shallow water equations with initial localized perturbations

We study the behavior of the wave part of asymptotic solutions to the Cauchy problem for linearized shallow water equations with initial perturbations localized near the origin. The global representation for these solutions based on the generalized Maslov canonical operator was given earlier. The asymptotic solutions are also localized in the neighborhood of certain curves (fronts). The simplification of general formulas and the behavior of asymptotic solutions in a neighborhood of the regular part of fronts was also given earlier. Here the behavior of asymptotic solutions in a neighborhood of the focal point of the fronts is discussed in detail and the proof of formulas announced earlier for the wave equation is given. This paper can be regarded as a continuation of the paper in Russiian Journal of Mathematical Physics 15 (2), 192–221 (2008). In memoriam V.A. Borovikov     

Strong Field Limit of Black Hole Gravitational Lensing

We give the formulation of the gravitational lensing theory in the strong field limit for a Schwarzschild black hole as a counterpart to the weak field approach. It is possible to expand the full black hole lens equation to work a simple analytical theory that describes the physics in the strong field limit at a high accuracy degree. In this way, we derive compact and reliable mathematical formulae for the position of additional critical curves, relativistic images and their magnification, arising in this limit.     

Increase of the Number of Detectable Gravitational Waves Signals Due to Gravitational Lensing

This article deals with the gravitational lensing (GL) of gravitational waves (GW). We compute the increase in the number of detected GW events due to GL. First, we check that geometrical optics is valid for the GW frequency range on which Earth-based detectors are sensitive, and that this is also partially true for what concerns the future space-based interferometer LISA. To infer this result, both the diffraction parameter and a cut-off frequency are computed. Then, the variation in the number of GW signals is estimated in the general case, and applied to some lens models: point mass lens and singular isothermal sphere (SIS profile). An estimation of the magnification factor has also been done for the softened isothermal sphere and for the King profile. The results appear to be strongly model-dependent, but in all cases the increase in the number of detected GW signals is negligible. The use of time delays among images is also investigated.     

Synchronization in power-law networks

We consider realistic power-law graphs, for which the power-law holds only for a certain range of degrees. We show that synchronizability of such networks depends on the expected average and expected maximum degree. In particular, we find that networks with realistic power-law graphs are less synchronizable than classical random networks. Finally, we consider hybrid graphs, which consist of two parts: a global graph and a local graph. We show that hybrid networks, for which the number of global edges is proportional to the number of total edges, almost surely synchronize.     

Study on FPS Optical Interconnection Realized by 2×2 Holographic lens Array

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