scholarly journals The N-point gravitational lens as cover and his the profile cover

2019 ◽  
Keyword(s):  
Jordan Curve ◽  
Complex Variable ◽  
Algebraic Topology ◽  
Riemann Surfaces ◽  
Complex Form ◽  
Picture Plane ◽  
Gravitational Lens ◽  
Gravitational Lenses ◽  
Critical Set ◽  
Closed Jordan Curve

The study of mathematical models of gravitational lenses are not direct observations. A special place in such studies is the visualization of the lens model. The image of the source and its images in the N-point gravitational lens, in the picture plane, visualizes the mathematical model - the algebraic equation of the lens. Recently, the number of studies of the equation of the N-point gravitational lens by algebraic methods has increased [6–8]. Such studies make it possible to consider the gravitational lens not only as an algebraic, but also as a topological object. In the work, the equation of the N-point gravitational lens in the complex form is studied. A bundle above the source plane is assigned to it. We investigated one subfamily of lens equations. A critical set of equations of this subfamily is a closed Jordan curve. To the equations of this subfamily we put in correspondence not only a vector bundle, but also a covering. A method for describing coverings is developed for equations whose caustic in the finite plane is a closed Jordan curve (Jordan caustic). A special case of such coverings is coverings for the equation of an N-point gravitational lens, the critical set of which is a closed Jordan curve. These equations, also, have Jordan caustics. The method is similar to the method for describing Riemann surfaces of algebraic functions, graphs ‒ profiles. The algorithm for constructing coverings and the developed method for describing these coverings illustrates an example of a cover given by a rational non-analytic function of a complex variable The covering surface has not only a Jordan caustic, but also a second-order branch point at an infinitely distant point. The methods of the theory of functions of a complex variable, algebraic geometry, algebraic topology and graph theory are used.

scholarly journals Fixed Points Features in N-Point Gravitational Lenses

2019 ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Convex Polygon ◽  
Center Of Mass ◽  
Complex Form ◽  
Polynomial Equation ◽  
Gravitational Lens ◽  
Gravitational Lenses ◽  
Coordinate Form ◽  
Minimal Convex

A set of fixed points in N-point gravitational lenses is studied in the paper. We use complex form of lens mapping to study fixed points. There are some merits of using a complex form over coordinate. In coordinate form gravitational lens is described by a system of two equations and in complex form is described by one equation. We transform complex equation of N-point gravitational lens into polynomial equation. It is convenient to study polynomial equation. Lens mapping presented as a linear combination of two mappings: complex analytical and identity. Analytical mapping is specified by deflection function. Fixed points are roots of deflection function. We show, that all fixed points of lens mapping appertain to the minimal convex polygon. Vertices of the polygon are points into which dimensionless point masses are. Method of construction of fixed points in N-point gravitational lens is shown. There are no fixed points in 1-point gravitational lens. We study properties of fixed points and their relation to the center of mass of the system. We obtained dependence of distribution of fixed points on center of mass. We analyzed different possibilities of distribution in N-point gravitational lens. Some cases, when fixed points merge with the center of mass are shown. We show a linear dependence of fixed point on center of mass in 2-point gravitational lens and we have built a model of this dependence. We obtained dependence of fixed point to center of mass in 3-point lens in case when masses form a triangle or line. In case of triangle, there are examples when fixed points merges. We study conditions, when there are no one-valued dependence of distribution of fixed points in case of 3-points gravitational lens and more complicated lens.

scholarly journals Dynamics of Ion Channels via Non-Hermitian Quantum Mechanics

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 125
Author(s):  
Tobias Gulden ◽  
Alex Kamenev
Keyword(s):  
Quantum Mechanics ◽  
Algebraic Topology ◽  
Riemann Surfaces ◽  
Surrounding Medium ◽  
Coulomb Systems ◽  
Dielectric Constants ◽  
Ion Interactions ◽  
Multivalent Ions ◽  
Electric Filed ◽  
Entropic Effects

We study dynamics and thermodynamics of ion transport in narrow, water-filled channels, considered as effective 1D Coulomb systems. The long range nature of the inter-ion interactions comes about due to the dielectric constants mismatch between the water and the surrounding medium, confining the electric filed to stay mostly within the water-filled channel. Statistical mechanics of such Coulomb systems is dominated by entropic effects which may be accurately accounted for by mapping onto an effective quantum mechanics. In presence of multivalent ions the corresponding quantum mechanics appears to be non-Hermitian. In this review we discuss a framework for semiclassical calculations for the effective non-Hermitian Hamiltonians. Non-Hermiticity elevates WKB action integrals from the real line to closed cycles on a complex Riemann surfaces where direct calculations are not attainable. We circumvent this issue by applying tools from algebraic topology, such as the Picard-Fuchs equation. We discuss how its solutions relate to the thermodynamics and correlation functions of multivalent solutions within narrow, water-filled channels.

AN APPLICATION OF THE TOPOLOGICAL DEGREE TO GRAVITATIONAL LENSES

1998 ◽  
Vol 13 (02) ◽  
pp. 83-86 ◽  
Author(s):  
MARCO LOMBARDI
Keyword(s):  
Point Source ◽  
Mass Distribution ◽  
Topological Degree ◽  
General Theorem ◽  
Gravitational Lens ◽  
Gravitational Lenses ◽  
General Proof ◽  
Special Case

In this letter we provide a new proof of a general theorem on gravitational lenses, first proven by Burke (1981) for the special case of thin lenses. The theorem states that a transparent gravitational lens with non-singular mass distribution produces an odd number of images of a point source. Our general proof shows that the topological degree finds natural and interesting applications in the theory of gravitational lenses.

scholarly journals Gaia GraL: Gaia DR2 gravitational lens systems

2018 ◽  
Vol 618 ◽  
pp. A56 ◽  
Author(s):  
C. Ducourant ◽  
O. Wertz ◽  
A. Krone-Martins ◽  
R. Teixeira ◽  
J.-F. Le Campion ◽  
...  
Keyword(s):  
Spatial Resolution ◽  
Space Mission ◽  
Gravitational Lens ◽  
Model Parameters ◽  
Gravitational Lenses ◽  
Bayesian Modelling ◽  
Lens System ◽  
Data Release ◽  
Gaia Dr2 ◽  
Exhaustive List

Context. Thanks to its spatial resolution, the ESA/Gaia space mission offers a unique opportunity to discover new multiply imaged quasars and to study the already known lensed systems at sub-milliarcsecond astrometric precisions. Aims. In this paper, we address the detection of the known multiply imaged quasars from the Gaia Data Release 2 (DR2) and determine the astrometric and photometric properties of the individually detected images found in the Gaia DR2 catalogue. Methods. We have compiled an exhaustive list of quasar gravitational lenses from the literature to search for counterparts in the Gaia DR2. We then analysed the astrometric and photometric properties of these Gaia’s detections. To highlight the tremendous potential of Gaia at the sub-milliarcsecond level we finally performed a simple Bayesian modelling of the well-known gravitational lens system HE0435-1223, using Gaia DR2 and HST astrometry. Results. From 481 known multiply imaged quasars, 206 have at least one image found in the Gaia DR2. Among the 44 known quadruply imaged quasars of the list, 29 have at least one image in the Gaia DR2, 12 of which are fully detected (2MASX J01471020+4630433, HE 0435-1223, SDSS1004+4112, PG1115+080, RXJ1131-1231, 2MASS J11344050-2103230, 2MASS J13102005-1714579, B1422+231, J1606-2333, J1721+8842, WFI2033-4723, WGD2038-4008), eight have three counterparts, eight have two and one has only one. As expected, the modelling of HE0435-1223 shows that the model parameters are significantly better constrained when using Gaia astrometry compared to HST astrometry, in particular the relative positions of the background quasar source and the centroid of the deflector. The Gaia sub-milliarcsecond astrometry also significantly reduces the parameter correlations. Conclusions. Besides providing an up-to-date list of multiply imaged quasars and their detection in the Gaia DR2, this paper shows that more complex modelling scenarios will certainly benefit from Gaia sub-milliarcsecond astrometry.

scholarly journals Gravitational Lensing

1992 ◽  
Vol 9 ◽  
pp. 3-32 ◽  
Author(s):  
Sjur Refsdal ◽  
Jean Surdej
Keyword(s):  
Gravitational Lensing ◽  
Gravitational Lens ◽  
Medium Size ◽  
Gravitational Lenses ◽  
Optical Lens ◽  
Basic Principles ◽  
History Of ◽  
Photometric Monitoring ◽  
A Site ◽  
Near Future

AbstractAtmospheric lensing effects deform our view of distant objects; similarly, without any doubt, gravitational lensing perturbs our view of the distant Universe and affects our physical understanding of various classes of extragalactic objects. We summarize here part of the theoretical and observational evidences supporting these claims.After briefly reviewing the history of gravitational lenses, we recall the basic principles underlying the formation of gravitationally lensed images of distant cosmic sources. We describe a simple optical lens experiment, which was actually shown during the oral discourse, and which accounts for all types of presently known gravitational lens systems.The various optical and radio searches for new gravitational lens systems that are being carried out at major observatories are reviewed. State-of-the-art observations of selected gravitational lens systems, obtained with highly performing ground-based telescopes, are then presented. These include several examples of multiply imaged QSO images, radio rings and giant luminous arcs.Through the modeling of these enigmatic objects, we show how it is possible to weigh the mass of distant lensing galaxies as well as to probe the distribution of luminous and dark matter in the Universe. Among the astrophysical and cosmological interests of observing and studying gravitational lenses, we also discuss the possibility of deriving the value of the Hubble parameter Ho from the measurement of a time delay, and how to determine the size and structure of distant quasars via the observational study of micro-lensing effects.At the end of this paper, we conclude on how to possibly achieve major astro-physical and cosmological goals in the near future by dedicating, on a site with good atmospheric seeing conditions, a medium size (2-3 m) telescope to the photometric monitoring of the multiple images of known and suspected gravitational lens systems.

scholarly journals Gravitational Lenses and VLBI Structures

1984 ◽  
Vol 110 ◽  
pp. 249-250
Author(s):  
K. Subramanian ◽  
D. Narasimha ◽  
S. M. Chitre
Keyword(s):  
Time Delay ◽  
Elliptical Galaxy ◽  
Gravitational Lens ◽  
Gravitational Lenses ◽  
Low Mass Stars ◽  
Effective Time ◽  
Low Mass ◽  
Intensity Fluctuations

The double quasar 00957 + 561 A,B along with bright radio arches and VLBI structures is modelled using a gravitational lens consisting of an elliptical galaxy and a cluster. The effective time-delay between components A and B comes out to be about a year and this enables one to distinguish between intensity fluctuations resulting from intrinsic quasar variations and minilensing by low mass stars.

GRAVITATIONAL LENS AMPLIFICATION OF GRAVITATIONAL RADIATION

2002 ◽  
Vol 11 (07) ◽  
pp. 1067-1074 ◽  
Author(s):  
ALEXANDER F. ZAKHAROV ◽  
YURI V. BARYSHEV
Keyword(s):  
Gravitational Wave ◽  
Gravitational Lensing ◽  
Gravitational Radiation ◽  
Gravitational Lens ◽  
Wave Form ◽  
Gravitational Lenses ◽  
Approximation Model ◽  
Radiation Sources ◽  
Wave Space ◽  
Stellar Masses

In a recent paper by Wang, Turner and Stebbins (1996) an influence of gravitational lensing on increasing an estimated rate of gravitational radiation sources was considered. We show that the authors used the geometrical optics approximation model for gravitational lensing and thus they gave overestimated rate of possible events for possible sources of gravitational radiation for the advanced LIGO detector. We show also that if we would use a more correct model of gravitational lensing, one could conclude that more strong influence on increasing rate of estimated events of gravitational radiation for advanced LIGO detector could give gravitational lenses of galactic masses but not gravitational lenses of stellar masses as Wang et al. concluced. Moreover, binary gravitational lenses could give essential distortion of gravitational wave form template, especially gravitational wave template of periodic sources and the effect could be significant for templates of quasi-periodic sources which could be detected by a future gravitational wave space detector like LISA.

scholarly journals SOME COROLLARY FACTS OF THE N-POINT GRAVITATIONAL LENS EQUATION IN A COMPLEX FORM

2019 ◽  
Vol 32 (0) ◽  
pp. 29-32
Author(s):  
O. A. Osmayev ◽  
Yu. S. Shuvalova ◽  
E. S. Bronza ◽  
K. I. Matvienko
Keyword(s):  
Complex Form ◽  
Gravitational Lens

scholarly journals A simple simulation technique for gravitational lenses

2011 ◽  
Vol 20 (2) ◽  
pp. 183-189
Author(s):  
CRISTIAN EDUARD RUSU ◽  
◽  
BETRIA SILVANA ROSSA ◽  
Keyword(s):  
Present Article ◽  
Gravitational Lensing ◽  
Weak Field ◽  
Simulation Technique ◽  
Gravitational Lens ◽  
Gravitational Lenses ◽  
Direct Simulation ◽  
Single Plane ◽  
Advantages And Disadvantages ◽  
Simple Simulation

The present article discusses the simulation of Gravitational Lensing with an algorithm developed in C++ and using the EasyBMP library. The algorithm numerically solves the general gravitational lens equation in the astrophysically significant weak field case, for any single-plane lens configuration. Examples of execution are considered, and a discussion is carried out on the advantages and disadvantages of the direct simulation technique employed.

scholarly journals Using convolutional neural networks to identify gravitational lenses in astronomical images

2019 ◽  
Vol 487 (4) ◽  
pp. 5263-5271 ◽  
Author(s):  
Andrew Davies ◽  
Stephen Serjeant ◽  
Jane M Bromley
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Roc Curve ◽  
Hubble Space Telescope ◽  
Gravitational Lens ◽  
Gravitational Lenses ◽  
Reasonable Time ◽  
Strong Lensing ◽  
Astronomical Images ◽  
Galaxy Classification

Abstract The Euclid telescope, due for launch in 2021, will perform an imaging and slitless spectroscopy survey over half the sky, to map baryon wiggles and weak lensing. During the survey, Euclid is expected to resolve 100 000 strong gravitational lens systems. This is ideal to find rare lens configurations, provided they can be identified reliably and on a reasonable time-scale. For this reason, we have developed a convolutional neural network (CNN) that can be used to identify images containing lensing systems. CNNs have already been used for image and digit classification as well as being used in astronomy for star-galaxy classification. Here, our CNN is trained and tested on Euclid-like and KiDS (Kilo-Degree Survey)-like simulations from the Euclid Strong Lensing Group, successfully classifying 77 per cent of lenses, with an area under the ROC curve of up to 0.96. Our CNN also attempts to classify the lenses in COSMOS Hubble Space Telescope F814W-band images. After convolution to the Euclid resolution, we find we can recover most systems that are identifiable by eye. The python code is available on Github.

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