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Relevant bibliographies by topics / Conformal mappings of special domains

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Academic literature on the topic 'Conformal mappings of special domains'

Author: Grafiati

Published: 4 June 2021

Last updated: 17 February 2022

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Journal articles on the topic "Conformal mappings of special domains"

1

Denega, Iryna. "Estimation of the products of the inner radii of domains with an additional symmetry condition." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 33 (December 27, 2019): 83–90. http://dx.doi.org/10.37069/1683-4720-2019-33-6.

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In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. Based on these elementary estimates a number of new estimates for functions realizing a conformal mapping of a disc onto domains with certain special properties are obtained. Estimates of this type are fundamental to solving some metric problems arising when considering the cor\-res\-pon\-dence of boundaries under a conformal mapping. Also, on the basis of the results concerning various extremal properties of conformal mappings, the problem of the representability of functions of a complex variable by a uniformly convergent series of polynomials is solved. In this paper, we consider the problem on maximum the products of the inner radii of $n$ disjoint domains with an additional symmetry condition that contain points of extended complex plane and the degree $\gamma$ of the inner radius of the domain that contains the zero point. An upper estimate for the maximum of this product is found for all values of $\gamma\in(0,\,n]$. The main result of the paper generalizes and strengthens the results of the predecessors [1-4] for the case of an arbitrary arrangement of points systems on $\overline{\mathbb{C}}$. In proving the main theorem, the arguments of proving of Lemma 1 [5] and the ideas of proving Theorem 1 [3] played a key role. We also established the conditions under which the structure of points and domains is not important. The corresponding results are obtained for the case when the points are placed on the unit circle and in the case of any fixed $n$-radial system of points.

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Costamagna, Eugenio, and Paolo Di Barba. "Inhomogeneous dielectrics: conformal mapping and finite-element models." Open Physics 15, no. 1 (December 29, 2017): 839–44. http://dx.doi.org/10.1515/phys-2017-0099.

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AbstractField singularities in electrostatic and magnetostatic fields require special attention in field calculations, and today finite element methods are normally used, both in homogeneous and in inhomogeneous dielectric cases. Conformal mappings are a traditional tool in the homogeneous case, but two-stage Schwarz-Christoffel + Finite Difference procedures have been proposed for a long time to solve problems in case of inhomogeneous dielectric materials too. This allowed to overcome accuracy problems caused by convex corners in the domain boundary and relevant field singularities, and to easily apply finite difference (FD) solvers in rectangular domains. In this paper, compound procedures Schwarz-Christoffel + Finite Elements Method procedures are suggested, to improve both the accuracy and the speed of second stage calculations. The results are compared to Schwarz-Christoffel + Finite Difference and to direct finite-element calculations, and the small differences analyzed considering a well know case study geometry,i.e., a shielded dielectric-supported stripline geometry.

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Kalmoun, El Mostafa, Mohamed M. S. Nasser, and Khalifa A. Hazaa. "The Motion of a Point Vortex in Multiply-Connected Polygonal Domains." Symmetry 12, no. 7 (July 16, 2020): 1175. http://dx.doi.org/10.3390/sym12071175.

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We study the motion of a single point vortex in simply- and multiply-connected polygonal domains. In the case of multiply-connected domains, the polygonal obstacles can be viewed as the cross-sections of 3D polygonal cylinders. First, we utilize conformal mappings to transfer the polygonal domains onto circular domains. Then, we employ the Schottky-Klein prime function to compute the Hamiltonian governing the point vortex motion in circular domains. We compare between the topological structures of the contour lines of the Hamiltonian in symmetric and asymmetric domains. Special attention is paid to the interaction of point vortex trajectories with the polygonal obstacles. In this context, we discuss the effect of symmetry breaking, and obstacle location and shape on the behavior of vortex motion.

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Crowdy, D. G., and A. S. Fokas. "Conformal mappings to a doubly connected polycircular arc domain." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2084 (May 31, 2007): 1885–907. http://dx.doi.org/10.1098/rspa.2007.1847.

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The explicit construction of the conformal mapping of a concentric annulus to a doubly connected polygonal domain was first reported by Akhiezer in 1928. The construction of an analogous formula for the case of a polycircular arc domain, i.e. for a doubly connected domain whose boundaries are a union of circular arc segments, has remained an important open problem. In this paper, we present this explicit formula. We first introduce a new method for deriving the classical formula of Akhiezer and then show how to generalize the method to the case of a doubly connected polycircular arc domain. As an analytical check of the formula, a special exact solution for a doubly connected polycircular arc mapping is derived and compared with that obtained from the more general construction. As an illustrative example, a doubly connected polycircular arc domain arising in a classic potential flow problem considered in the last century by Lord Rayleigh is considered in detail.

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CROWDY, DARREN. "CONFORMAL SLIT MAPS IN APPLIED MATHEMATICS." ANZIAM Journal 53, no. 3 (January 2012): 171–89. http://dx.doi.org/10.1017/s1446181112000119.

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AbstractConformal slit maps play a fundamental theoretical role in analytic function theory and potential theory. A lesser-known fact is that they also have a key role to play in applied mathematics. This review article discusses several canonical conformal slit maps for multiply connected domains and gives explicit formulae for them in terms of a classical special function known as the Schottky–Klein prime function associated with a circular preimage domain. It is shown, by a series of examples, that these slit mapping functions can be used as basic building blocks to construct more complicated functions relevant to a variety of applied mathematical problems.

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Dittmar, Bodo. "The Robin function and conformal welding – A new proof of the existence." Georgian Mathematical Journal 27, no. 1 (March 1, 2020): 43–51. http://dx.doi.org/10.1515/gmj-2017-0057.

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AbstractGreen’s function of the mixed boundary value problem for harmonic functions is sometimes named the Robin function {R(z,\zeta\/)} after the French mathematical physicist Gustave Robin (1855–1897). The aim of this paper is to provide a new proof of the existence of the Robin function for planar n-fold connected domains using a special version of the well-known Koebe’s uniformization theorem and a conformal mapping which is closely related to the Robin function in the simply connected case.

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7

Crowdy, Darren. "Stress fields around two pores in an elastic body: exact quadrature domain solutions." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2180 (August 2015): 20150240. http://dx.doi.org/10.1098/rspa.2015.0240.

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Analytical solutions are given for the stress fields, in both compression and far-field shear, in a two-dimensional elastic body containing two interacting non-circular pores. The two complex potentials governing the solutions are found by using a conformal mapping from a pre-image annulus with those potentials expressed in terms of the Schottky–Klein prime function for the annulus. Solutions for a three-parameter family of elastic bodies with two equal symmetric pores are presented and the compressibility of a special family of pore pairs is studied in detail. The methodology extends to two unequal pores. The importance for boundary value problems of plane elasticity of a special class of planar domains known as quadrature domains is also elucidated. This observation provides the route to generalization of the mathematical approach here to finding analytical solutions for the stress fields in bodies containing any finite number of pores.

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8

Maitani, Fumio. "Conformal slit mappings from periodic domains." Kodai Mathematical Journal 28, no. 2 (June 2005): 265–74. http://dx.doi.org/10.2996/kmj/1123767007.

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Herron, David A., and Pekka Koskela. "Quasiextremal distance domains and conformal mappings onto circle domains." Complex Variables, Theory and Application: An International Journal 15, no. 3 (September 1990): 167–79. http://dx.doi.org/10.1080/17476939008814448.

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Bourchtein, Andrei, and Ludmila Bourchtein. "On Conformal Conic Mappings of Spherical Domains." Scientific World Journal 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/840280.

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The problem of the generation of homogeneous grids for spherical domains is considered in the class of conformal conic mappings. The equivalence between secant and tangent projections is shown and splitting the set of conformal conic mappings into equivalence classes is presented. The problem of minimization of the mapping factor variation is solved in the class of conformal conic mappings. Obtained results can be used in applied sciences, such as geophysical fluid dynamics and cartography, where the flattening of the Earth surface is required.

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Dissertations / Theses on the topic "Conformal mappings of special domains"

1

Bauer, Ulrich Josef [Verfasser], Stephan [Gutachter] Ruscheweyh, and Wolfgang [Gutachter] Lauf. "Conformal Mappings onto Simply and Multiply Connected Circular Arc Polygon Domains / Ulrich Josef Bauer. Gutachter: Stephan Ruscheweyh ; Wolfgang Lauf." Würzburg : Universität Würzburg, 2016. http://d-nb.info/1111784388/34.

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Bauer, Ulrich Josef. "Conformal Mappings onto Simply and Multiply Connected Circular Arc Polygon Domains." Doctoral thesis, 2015. https://nbn-resolving.org/urn:nbn:de:bvb:20-opus-123914.

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The goal of this thesis is to investigate conformal mappings onto circular arc polygon domains, i.e. domains that are bounded by polygons consisting of circular arcs instead of line segments. Conformal mappings onto circular arc polygon domains contain parameters in addition to the classical parameters of the Schwarz-Christoffel transformation. To contribute to the parameter problem of conformal mappings from the unit disk onto circular arc polygon domains, we investigate two special cases of these mappings. In the first case we can describe the additional parameters if the bounding circular arc polygon is a polygon with straight sides. In the second case we provide an approximation for the additional parameters if the circular arc polygon domain satisfies some symmetry conditions. These results allow us to draw conclusions on the connection between these additional parameters and the classical parameters of the mapping. For conformal mappings onto multiply connected circular arc polygon domains, we provide an alternative construction of the mapping formula without using the Schottky-Klein prime function. In the process of constructing our main result, mappings for domains of connectivity three or greater, we also provide a formula for conformal mappings onto doubly connected circular arc polygon domains. The comparison of these mapping formulas with already known mappings allows us to provide values for some of the parameters of the mappings onto doubly connected circular arc polygon domains if the image domain is a polygonal domain. The different components of the mapping formula are constructed by using a slightly modified variant of the Poincaré theta series. This construction includes the design of a function to remove unwanted poles and of different versions of functions that are analytic on the domain of definition of the mapping functions and satisfy some special functional equations. We also provide the necessary concepts to numerically evaluate the conformal mappings onto multiply connected circular arc polygon domains. As the evaluation of such a map requires the solution of a differential equation, we provide a possible configuration of curves inside the preimage domain to solve the equation along them in addition to a description of the procedure for computing either the formula for the doubly connected case or the case of connectivity three or greater. We also describe the procedures for solving the parameter problem for multiply connected circular arc polygon domains
Das Ziel dieser Arbeit ist die Untersuchung von konformen Abbildungen auf Zirkularpolygongebiete, d.h. Gebiete, die von Polygonen berandet werden, die sich aus Kreisbögen statt Geradenstücken zusammensetzen. Konforme Abbildungen auf Zirkularpolygongebiete enthalten Parameter zusätzlich zu den klassischen Parametern der Schwarz-Christoffel-Transformation. Um zum Parameterproblem der konformen Abbildungen der Einheitskreisscheibe auf Zirkularpolygongebiete beizutragen, werden zwei Spezialfälle dieser Abbildungen untersucht. Im ersten Fall können die zusätzlichen Parameter angeben werden, falls das berandende Zirkularpolygon ein Polygon mit geraden Seiten ist. Im zweiten Fall wird eine Approximation für die zusätzlichen Parameter angegeben, falls das Zirkularpolygongebiet gewisse Symmetriebedingungen erfüllt. Diese Ergebnisse erlauben es Schlüsse zu ziehen in Bezug auf die Verbindung zwischen den zusätzlichen Parametern und den klassischen Parametern der Abbildung. Für Konforme Abbildungen auf mehrfach zusammenhängende Zirkularpolygongebiete wird eine alternative Konstruktion der Abbildungsformel angegeben, welche nicht die "Schottky-Klein prime function" verwendet. Während der Konstruktion des Hauptergebnisses, der Abbildungsformel für Gebiete mit einem Zusammenhang von drei oder mehr, wird auch eine Formel für die konformen Abbildungen auf zweifach zusammenhängende Zirkularpolygongebiete angegeben. Der Vergleich dieser Abbildungsformeln mit bereits bekannten Abbildungen erlaubt es Werte für einige der Parameter der Abbildungen auf zweifach zusammenhängende Zirkularpolygongebiete anzugeben, falls das Bildgebiet ein Polygonalgebiet ist. Die unterschiedlichen Komponenten der Abbildungsformel sind unter Verwendung einer leicht modifizierten Form der Poincaré-Theta-Reihe konstruiert. Diese Konstruktion enthält die Gestaltung einer Funktion um unerwünschte Polstellen zu entfernen und von unterschiedlichen Versionen von Funktionen, die analytisch auf dem Definitionsgebiet der Abbildungsfunktion sind und spezielle Funktionalgleichungen erfüllen. Es werden auch die notwendigen Konzepte angegeben um die konformen Abbildungen auf mehrfach zusammenhängende Zirkularpolygongebiete numerisch auszuwerten. Prozedurbeschreibungen für die Berechnung der Formel für zweifach zusammenhängende Bildgebiete und für den Fall eines Zusammenhangs von drei oder mehr werden angegeben. Da für das Auswerten solcher Abbildungen das Lösen einer Differentialgleichung notwendig ist, werden mögliche Konfigurationen von Kurven im Urbildgebiet angegeben, an denen entlang die Gleichung gelöst werden kann. Es werden weiterhin Prozeduren beschrieben um das Parameterproblem für mehrfach zusammenhängende Zirkularpolygongebiete zu lösen

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Books on the topic "Conformal mappings of special domains"

1

Semmes, Stephen. A generalization of Riemann mappings and geometric structures on a space of domains in Cn̳. Providence, RI: American Mathematical Society, 1992.

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2

1941-, Hag Kari, and Broch Ole Jacob, eds. The ubiquitous quasidisk. Providence, Rhode Island: American Mathematical Society, 2012.

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Book chapters on the topic "Conformal mappings of special domains"

1

Anderson, G. D., M. K. Vamanamurthy, and M. Vuorinen. "Conformal invariants, quasiconformal maps, and special functions." In Quasiconformal Space Mappings, 1–19. Berlin, Heidelberg: Springer Berlin Heidelberg, 1992. http://dx.doi.org/10.1007/bfb0094235.

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2

Czapla, Roman. "Symbolic Computation of Conformal Mappings onto Slit Domains." In Trends in Mathematics, 777–83. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-12577-0_85.

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3

"Domain Decomposition for Special Quadrilaterals." In Numerical Conformal Mapping, 105–39. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814289535_0003.

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4

"Conformal mappings of simply connected domains." In Translations of Mathematical Monographs, 25–94. Providence, Rhode Island: American Mathematical Society, 1992. http://dx.doi.org/10.1090/mmono/106/02.

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5

"Conformal mappings of multiply connected domains." In Translations of Mathematical Monographs, 95–136. Providence, Rhode Island: American Mathematical Society, 1992. http://dx.doi.org/10.1090/mmono/106/03.

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6

Suffridge, T. J. "Some Special Classes of Conformal Mappings." In Handbook of Complex Analysis, 309–38. Elsevier, 2005. http://dx.doi.org/10.1016/s1874-5709(05)80011-3.

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7

"Approximate Block Method of Conformal Mapping of Polygons onto Canonical Domains." In Block Method for Solving the Laplace Equation and for Constructing Conformal Mappings, 118–33. CRC Press, 2017. http://dx.doi.org/10.1201/9781315150321-7.

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8

"Approximate Conformal Mapping of Domains with a Periodic Structure by the Block Method." In Block Method for Solving the Laplace Equation and for Constructing Conformal Mappings, 198–229. CRC Press, 2017. http://dx.doi.org/10.1201/9781315150321-9.

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"Development and Application of the Approximate Block Method for Conformal Mapping of Simply-Connected and Doubly-Connected Domains." In Block Method for Solving the Laplace Equation and for Constructing Conformal Mappings, 134–97. CRC Press, 2017. http://dx.doi.org/10.1201/9781315150321-8.

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Conference papers on the topic "Conformal mappings of special domains"

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Bourchtein, Andrei, Ludmila Bourchtein, Ilias Kotsireas, Roderick Melnik, and Brian West. "On Conformal Mappings of Spherical Domains." In ADVANCES IN MATHEMATICAL AND COMPUTATIONAL METHODS: ADDRESSING MODERN CHALLENGES OF SCIENCE, TECHNOLOGY, AND SOCIETY. AIP, 2011. http://dx.doi.org/10.1063/1.3663447.

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Kiosak, V., A. Savchenko, and O. Gudyreva. "On the conformal mappings of special quasi-Einstein spaces." In APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: 11th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences - AMiTaNS’19. AIP Publishing, 2019. http://dx.doi.org/10.1063/1.5130793.

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Akleman, Ergun, and Tevfik Akgun. "Caricature Creation with Conformal Mapping in Complex Domain." In SIGGRAPH '21: Special Interest Group on Computer Graphics and Interactive Techniques Conference. New York, NY, USA: ACM, 2021. http://dx.doi.org/10.1145/3450618.3469149.

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Panta Pazos, Rube´n. "Hybrid Methods Approach for Solving Problems in Transport Theory." In 16th International Conference on Nuclear Engineering. ASMEDC, 2008. http://dx.doi.org/10.1115/icone16-48400.

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In this work the hybrid methods approach is introduced in order to solve some problems in Transport Theory for different geometries. The transport equation is written as: ∂ψ∂t(x,v,t)+v·∇ψ(x,v,t)+h(x,μ)ψ(x,v,t)==∫Vk(x,v,v′)ψ(x,v′,t)dv′+q(x,v,t),inΩTψ(x,v,0)=φ0(x,v),in∂Ω×Vψ(x,v,t)=φ(x,v,t),in∂Ω×V×R(1) where x represents the spatial variable in a domain D, v an element of a compact set V, ψ is the angular flux, h(x, v) the collision frequency, k(x, v, v’) the scattering kernel function and q(x, v) the source function. If ψ does not depend on the time, it is said that the problem (1) is a steady transport problem. Once the problem is defined, including the boundary conditions, it is disposed a set of chained methods in order to solve the problem. Between the different alternatives, an optimal scheme for the resolution is chosen. Two illustrations are given. For two-dimensional geometries it is employed a hybrid analytical and numerical method, for transport problems: conformal mapping first, then the solution in a proper geometry (rectangular for example). Each of the following two techniques is then applied, Krylov subspaces method or spectral-LTSN method. For three-dimensional problems also it is used a hybrid analytical and numerical method, for problems with more complex geometries: a homotopy between the original boundaries (piecewise surfaces) and another (a parallelepiped for example). Then each of two techniques are applied, Krylov subspaces method or nodal-LTSN method. In this case, the design of new geometries for reactors is a straightforward task. En each case, the domain consist of three regions, one of the source, other is the void region and the third one is a shield domain. The results are obtained both with an algebraic computer system and with a language of high level. An important extension is the study and treatment of transport problems for domains with irregular geometries, between them Lipschitzian domains. One remarkable fact of this work is the combination of different modeling and resolution techniques to solve some transport problems.

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Reports on the topic "Conformal mappings of special domains"

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Ozen, Fusun. Conformal Mappings and Special Networks of Weyl Spaces. GIQ, 2012. http://dx.doi.org/10.7546/giq-4-2003-239-247.

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