Riemann-Hilbert problems for axially symmetric monogenic functions in Rn+1\mathbb{R}^{n+1}

We focus on the Clifford-algebra valued variable coefficients Riemann-Hilbert boundary value problems \big{(}for short RHBVPs\big{)} for axially monogenic functions on Euclidean space $\mathbb{R}^{n+1},n\in \mathbb{N}$. With the help of Vekua system, we first make one-to-one correspondence between the RHBVPs considered in axial domains and the RHBVPs of generalized analytic function on complex plane. Subsequently, we use it to solve the former problems, by obtaining the solutions and solvable conditions of the latter problems, so that we naturally get solutions to the corresponding Schwarz problems. In addition, we also use the above method to extend the case to RHBVPs for axially null-solutions to \big{(}\mathcal{D}-\alpha\big{)}\phi=0,\alpha\in\mathbb{R}.Comment: 14 page

Riemann-Hilbert problems for monogenic functions in axially symmetric domains

We consider Riemann-Hilbert boundary value problems (for short RHBVPs) with variable coefficients for axially symmetric monogenic functions defined in axial symmetric domains. This is done by constructing a method to reduce the RHBVPs for axially symmetric monogenic functions defined in four-dimensional axial symmetric domains into the RHBVPs for analytic functions defined over the complex plane. Then we derive solutions to the corresponding Schwarz problem. Finally, we generalize the results obtained to null-solutions of (D−α)ϕ=0, α∈R, where R denotes the field of real numbers

Riemann–Hilbert Problems for Monogenic Functions on Upper Half Ball of R^4

In this paper we are interested in finding solutions to Riemann– Hilbert boundary value problems, for short Riemann–Hilbert problems, with variable coefficients in the case of axially monogenic functions defined over the upper half unit ball centred at the origin in four-dimensional Euclidean space. Our main idea is to transfer Riemann– Hilbert problems for axially monogenic functions defined over the up- per half unit ball centred at the origin of four-dimensional Euclidean spaces into Riemann–Hilbert problems for analytic functions defined over the upper half unit disk of the complex plane. Furthermore, we extend our results to axially symmetric null-solutions of perturbed generalized Cauchy–Riemann equations

On the global operator and Fueter mapping theorem for slice polyanalytic functions

In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball.Comment: To appear in Anal. Appl. (Singap.

Nonlinear interactions of gravitational waves

This work presents the foundations of a new solution technique for the characteristic initial value problem of colliding plane gravitational waves. It has extensive similarities to the approach of Alekseev and Griffiths in 2001, but uses the inverse scattering method with a Riemann-Hilbert problem. This allows for a further transformation to a continuous Riemann-Hilbert problem with a solution given in terms of an integral equation for a regular unknown function. Ambiguities in the solution of the initial Riemann-Hilbert problem lead to the construction of a whole family of exact spacetimes generalising the proper solution of the initial value problem. Therefore the described technique also serves as an interesting solution generating method. The procedure is exemplified by extending the Szekeres class of colliding wave spacetimes with two additional real parameters. The obtained solution features a limiting case of a new type of impulsive waves, which are circularly polarised. A semi-analytic approximation scheme for the solution to the general initial value problem of colliding plane waves is introduced.Diese Arbeit präsentiert die Grundlagen einer neuen Lösungstechnik für das charakteristische Anfangswertproblem kollidierender ebener Gravitationswellen. Sie weist deutliche Ähnlichkeiten zu einem 2001 von Alekseev und Griffiths beschriebenen Verfahren auf, verwendet jedoch die inverse Streumethode mit einem Riemann-Hilbert-Problem. Dies erlaubt eine weitere Transformation zu einem stetigen Riemann-Hilbert-Problem, dessen Lösung in Form einer Integralgleichung für eine reguläre unbekannte Funktion gegeben ist. Mehrdeutigkeiten in der Lösung des anfänglichen Riemann-Hilbert-Problems führen zur Konstruktion einer ganzen Familie exakter Raumzeiten, welche die eigentliche Lösung des Anfangswertproblems verallgemeinert. Deshalb kann die vorgestellte Technik auch als eine interessante Methode zur Lösungserzeugung dienen. Das Verfahren wird anhand der Erweiterung der Szekeres-Klasse von Wellenkollisionsraumzeiten %mit kollidierenden Gravitationswellen demonstriert. Die dadurch erhaltene Lösung beinhaltet als Grenzfall einen neuen Typ impulsiver Gravitationswellen mit zirkularer Polarisation. Ein halbanalytisches Approximationsschema für die Lösung des charakteristischen Anfangswertproblems kollidierender ebener Gravitationswellen wird vorgestellt

Foundations of General Relativity

This book, dedicated to Roger Penrose, is a second, mathematically oriented course in general relativity. It contains extensive references and occasional excursions in the history and philosophy of gravity, including a relatively lengthy historical introduction. The book is intended for all students of general relativity of any age and orientation who have a background including at least first courses in special and general relativity, differential geometry, and topology. The material is developed in such a way that through the last two chapters the reader may acquire a taste of the modern mathematical study of black holes initiated by Penrose, Hawking, and others, as further influenced by the initial-value or PDE approach to general relativity. Successful readers might be able to begin reading research papers on black holes, especially in mathematical physics and in the philosophy of physics. The chapters are: Historical introduction, General differential geometry, Metric differential geometry, Curvature, Geodesics and causal structure, The singularity theorems of Hawking and Penrose, The Einstein equations, The 3+1 split of space-time, Black holes I: Exact solutions, and Black holes II: General theory. These are followed by two appendices containing background on Lie groups, Lie algebras, & constant curvature, and on Formal PDE theory

Foundations of General Relativity

This book, dedicated to Roger Penrose, is a second, mathematically oriented course in general relativity. It contains extensive references and occasional excursions in the history and philosophy of gravity, including a relatively lengthy historical introduction. The book is intended for all students of general relativity of any age and orientation who have a background including at least first courses in special and general relativity, differential geometry, and topology. The material is developed in such a way that through the last two chapters the reader may acquire a taste of the modern mathematical study of black holes initiated by Penrose, Hawking, and others, as further influenced by the initial-value or PDE approach to general relativity. Successful readers might be able to begin reading research papers on black holes, especially in mathematical physics and in the philosophy of physics. The chapters are: Historical introduction, General differential geometry, Metric differential geometry, Curvature, Geodesics and causal structure, The singularity theorems of Hawking and Penrose, The Einstein equations, The 3+1 split of space-time, Black holes I: Exact solutions, and Black holes II: General theory. These are followed by two appendices containing background on Lie groups, Lie algebras, & constant curvature, and on Formal PDE theory

Foundations of General Relativity

This book, dedicated to Roger Penrose, is a second, mathematically oriented course in general relativity. It contains extensive references and occasional excursions in the history and philosophy of gravity, including a relatively lengthy historical introduction. The book is intended for all students of general relativity of any age and orientation who have a background including at least first courses in special and general relativity, differential geometry, and topology. The material is developed in such a way that through the last two chapters the reader may acquire a taste of the modern mathematical study of black holes initiated by Penrose, Hawking, and others, as further influenced by the initial-value or PDE approach to general relativity. Successful readers might be able to begin reading research papers on black holes, especially in mathematical physics and in the philosophy of physics. The chapters are: Historical introduction, General differential geometry, Metric differential geometry, Curvature, Geodesics and causal structure, The singularity theorems of Hawking and Penrose, The Einstein equations, The 3+1 split of space-time, Black holes I: Exact solutions, and Black holes II: General theory. These are followed by two appendices containing background on Lie groups, Lie algebras, & constant curvature, and on Formal PDE theory

The 2nd International Conference on Mathematical Modelling in Applied Sciences, ICMMAS’19, Belgorod, Russia, August 20-24, 2019 : book of abstracts

The proposed Scientific Program of the conference is including plenary lectures, contributed oral talks, poster sessions and listeners. Five suggested special sessions / mini-symposium are also considered by the scientific committe