scholarly journals THE ANALYSIS OF IMAGES IN N-POINT GRAVITATIONAL LENS BY METHODS OF ALGEBRAIC GEOMETRY

2017 ◽  
Vol 57 (6) ◽  
pp. 404
Author(s):  
Albert Kotvytskiy ◽  
Semen Bronza ◽  
Vladimir Shablenko
Keyword(s):  
Algebraic Geometry ◽  
Fundamental Problem ◽  
Polynomial System ◽  
Gravitational Lens ◽  
Complex Numbers ◽  
Gravitational Lenses ◽  
Algebraic Version ◽  
Several Variables ◽  
In The Beginning ◽  
Bezout Theorem

<p>This paper is devoted to the study of images in <em>N</em>-point gravitational lenses by methods of algebraic geometry. In the beginning, we carefully define images in algebraic terms. Based on the definition, we show that in this model of gravitational lenses (for a point source), the dimensions of the images can be only 0 and 1. We reduce it to the fundamental problem of classical algebraic geometry - the study of solutions of a polynomial system of equations. Further, we use well-known concepts and theorems. We adapt known or prove new assertions. Sometimes, these statements have a fairly general form and can be applied to other problems of algebraic geometry. In this paper: the criterion for irreducibility of polynomials in several variables over the field of complex numbers is effectively used. In this paper, an algebraic version of the Bezout theorem and some other statements are formulated and proved. We have applied the theorems proved by us to study the imaging of dimensions 1 and 0.</p>

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.

Numerical irreducible decomposition over a number field

2018 ◽  
Vol 17 (10) ◽  
pp. 1850195
Author(s):  
Timothy M. McCoy ◽  
Chris Peterson ◽  
Andrew J. Sommese
Keyword(s):  
Algebraic Geometry ◽  
Number Field ◽  
Polynomial System ◽  
Field Extension ◽  
General Point ◽  
Numerical Algebraic Geometry ◽  
Irreducible Decomposition ◽  
Algebraic Set ◽  
Field Of Definition ◽  
Witness Set

Let [Formula: see text] be a set of elements in the polynomial ring [Formula: see text], let [Formula: see text] denote the ideal generated by the elements of [Formula: see text], and let [Formula: see text] denote the radical of [Formula: see text]. There is a unique decomposition [Formula: see text] with each [Formula: see text] a prime ideal corresponding to a minimal associated prime of [Formula: see text] over [Formula: see text]. Let [Formula: see text] denote the reduced algebraic set corresponding to the common zeroes of the elements of [Formula: see text]. Techniques from numerical algebraic geometry can be used to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text]. This corresponds to producing a witness set for [Formula: see text] for each [Formula: see text] together with the degree and dimension of [Formula: see text] (a point in a witness set for [Formula: see text] can be considered as a numerical approximation for a general point on [Formula: see text]). The purpose of this paper is to show how to extend these results taking into account the field of definition for the polynomial system. In particular, let [Formula: see text] be a number field (i.e. a finite field extension of [Formula: see text]) and let [Formula: see text] be a set of elements in [Formula: see text]. We show how to extend techniques from numerical algebraic geometry to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text].

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 The Hodge conjecture: The complications of understanding the shape of geometric spaces

2018 ◽  
pp. 51
Author(s):  
Vicente Muñoz Velázquez
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Natural Condition ◽  
Mathematical Physics ◽  
Complex Numbers ◽  
Hodge Conjecture ◽  
Real Numbers ◽  
Geometric Objects ◽  
Complex Submanifolds ◽  
Geometry Analysis

The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950 and is currently a stimulus for the development of several theories based on geometry, analysis, and mathematical physics. It proposes a natural condition for the existence of complex submanifolds within a complex manifold. Manifolds are the spaces in which geometric objects can be considered. In complex manifolds, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers.

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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