scholarly journals From ALE to ALF gravitational instantons

2018 ◽  
Vol 154 (6) ◽  
pp. 1159-1221
Author(s):  
Hugues Auvray
Keyword(s):  
Dihedral Group ◽  
Gravitational Instantons ◽  
Analytic Construction ◽  
Analytic Methods ◽  
Hyperkähler Metrics ◽  
Monge Ampere

In this article, we give an analytic construction of ALF hyperkähler metrics on smooth deformations of the Kleinian singularity $\mathbb{C}^{2}/{\mathcal{D}}_{k}$, with ${\mathcal{D}}_{k}$ the binary dihedral group of order $4k$, $k\geqslant 2$. More precisely, we start from the ALE hyperkähler metrics constructed on these spaces by Kronheimer, and use analytic methods, e.g. resolution of a Monge–Ampère equation, to produce ALF hyperkähler metrics with the same associated Kähler classes.

scholarly journals THE SPACE OF HYPERKÄHLER METRICS ON A 4-MANIFOLD WITH BOUNDARY

2017 ◽  
Vol 5 ◽  
Author(s):  
JOEL FINE ◽  
JASON D. LOTAY ◽  
MICHAEL SINGER
Keyword(s):  
Boundary Value Problem ◽  
Moduli Space ◽  
Dimensional Manifold ◽  
Gravitational Instantons ◽  
Infinite Dimensional ◽  
Manifold With Boundary ◽  
Isometric Actions ◽  
Infinitesimal Deformations ◽  
Hyperkähler Metrics ◽  
Infinite Dimensional Manifold

Let $X$ be a compact 4-manifold with boundary. We study the space of hyperkähler triples $\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3}$ on $X$, modulo diffeomorphisms which are the identity on the boundary. We prove that this moduli space is a smooth infinite-dimensional manifold and describe the tangent space in terms of triples of closed anti-self-dual 2-forms. We also explore the corresponding boundary value problem: a hyperkähler triple restricts to a closed framing of the bundle of 2-forms on the boundary; we identify the infinitesimal deformations of this closed framing that can be filled in to hyperkähler deformations of the original triple. Finally we study explicit examples coming from gravitational instantons with isometric actions of $\text{SU}(2)$.

Flat Lorentz surfaces in anti-de Sitter 3-space and Gravitational Instantons

2016 ◽  
Vol 13 (02) ◽  
pp. 1650012
Author(s):  
Jun-Ichi Inoguchi ◽  
Marianty Ionel ◽  
Sungwook Lee
Keyword(s):  
Representation Formula ◽  
Gauss Map ◽  
Conformal Structure ◽  
Weierstrass Representation ◽  
De Sitter ◽  
Gravitational Instantons ◽  
Hyperbolic Gauss Map ◽  
Monge Ampere ◽  
The Relationship ◽  
Lorentz Surface

In this paper, we study flat Lorentz surfaces in anti-de Sitter 3-space [Formula: see text] in terms of the second conformal structure. Those flat Lorentz surfaces can be represented in terms of a Lorentz holomorphic and a Lorentz anti-holomorphic data similarly to Weierstraß representation formula. An analogue of hyperbolic Gauß map is considered for timelike surfaces in [Formula: see text] and the relationship between the conformality (or the holomorphicity) of hyperbolic Gauß map and the flatness of a Lorentz surface is discussed. It is shown that flat Lorentz surfaces in [Formula: see text] are associated with a hyperbolic Monge–Ampère equation. It is also known that Monge–Ampére equation may be regarded as a 2-dimensional reduction of the Einstein’s field equation. Using this connection, we construct a class of anti-self-dual gravitational instantons from flat Lorentz surfaces in [Formula: see text].

Analytical Study on Big Data

2018 ◽  
Vol 8 (5) ◽  
pp. 75
Author(s):  
Vivek Raich ◽  
Pankaj Maurya
Keyword(s):  
Information Technology ◽  
Big Data ◽  
Decision Maker ◽  
Analytical Study ◽  
Large Data ◽  
Decision Makers ◽  
Continuous Increase ◽  
Analytic Methods ◽  
Data Store ◽  
Business Engineering

in the time of the Information Technology, the big data store is going on. Due to which, Huge amounts of data are available for decision makers, and this has resulted in the progress of information technology and its wide growth in many areas of business, engineering, medical, and scientific studies. Big data means that the size which is bigger in size, but there are several types, which are not easy to handle, technology is required to handle it. Due to continuous increase in the data in this way, it is important to study and manage these datasets by adjusting the requirements so that the necessary information can be obtained.The aim of this paper is to analyze some of the analytic methods and tools. Which can be applied to large data. In addition, the application of Big Data has been analyzed, using the Decision Maker working on big data and using enlightened information for different applications.

scholarly journals A note on two-term exponential sum and the reciprocal of the quartic Gauss sums

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wenpeng Zhang ◽  
Xingxing Lv
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Gauss Sums ◽  
Power Mean ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

AbstractThe main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

scholarly journals Some characterizatsion of coprime graph of dihedral group D 2n

2021 ◽  
Vol 1722 ◽  
pp. 012051
Author(s):  
A G Syarifudin ◽  
Nurhabibah ◽  
D P Malik ◽  
I G A W Wardhana
Keyword(s):  
Dihedral Group ◽  
Group D

Examining how meta‐analytic methods perform in the presence of bias: A simulation study

2021 ◽  
Author(s):  
Paul Bramley ◽  
José A. López‐López ◽  
Julian P. T. Higgins
Keyword(s):  
Simulation Study ◽  
Analytic Methods

scholarly journals The exterior Dirichlet problems of Monge–Ampère equations in dimension two

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Unit Ball ◽  
Bounded Domain ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Necessary And Sufficient ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ det D 2 u = f in dimension two with f being a perturbation of $f_{0}$ f 0 at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

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