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Björn Gustafsson - One of the best experts on this subject based on the ideXlab platform.

  • Vortex motion and Geometric Function Theory: the role of connections
    Philosophical Transactions of the Royal Society A: Mathematical Physical and Engineering Sciences, 2019
    Co-Authors: Björn Gustafsson
    Abstract:

    We formulate the equations for point vortex dynamics on a closed two-dimensional Riemannian manifold in the language of affine and other kinds of connections. This can be viewed as a relaxation of ...

  • Vortex motion and Geometric Function Theory: the role of connections
    arXiv: Mathematical Physics, 2018
    Co-Authors: Björn Gustafsson
    Abstract:

    We formulate the equations for point vortex dynamics on a closed two dimensional Riemann manifold in the language of affine and other kinds of connections. The speed of a vortex is then expressed in terms of the difference between an affine connection derived from the coordinate Robin Function and the Levi-Civita connection associated to the Riemannian metric. A Hamiltonian formulation of the same dynamics is also given. The relevant Hamiltonian Function consists of two main terms. One of the terms is the well-known quadratic form based on a matrix whose entries are Green and Robin Functions, while the other term describes the energy contribution from those circulating flows which are not implicit in the Green Functions. One main issue of the paper is the clarification of the somewhat intricate exchanges of energy between these two terms of the Hamiltonian.

  • Critical points of Green's Function and Geometric Function Theory
    arXiv: Complex Variables, 2009
    Co-Authors: Björn Gustafsson, Ahmed Sebbar
    Abstract:

    We study questions related to critical points of the Green's Function of a bounded multiply connected domain in the complex plane. The motion of critical points, their limiting positions as the pole approaches the boundary and the differential geometry of the level lines of the Green's Function are main themes in the paper. A unifying role is played by various affine and projective connections and corresponding M\"obius invariant differential operators. In the doubly connected case the three Eisenstein series $E_2$, $E_4$, $E_6$ are used. A specific result is that a doubly connected domain is the disjoint union of the set of critical points of the Green's Function, the set of zeros of the Bergman kernel and the separating boundary limit positions for these. At the end we consider the projective properties of the prepotential associated to a second order differential operator depending canonically on the domain.

Vladimir Nikolaevich Dubinin - One of the best experts on this subject based on the ideXlab platform.

Leonid V. Kovalev - One of the best experts on this subject based on the ideXlab platform.

  • Monotonicity of the Generalized Reduced Module
    Journal of Mathematical Sciences, 2003
    Co-Authors: Leonid V. Kovalev
    Abstract:

    In terms of polar sets, we complement some Dubinin's inequalities expressing the monotonicity of the generalized reduced module for open subsets of $${\mathbb{C}}^n$$ and $$\mathbb{R}^n$$ by equality cases. These results are applied to extremal problems of Geometric Function Theory. Bibliography: 19 titles.

  • The Reduced Module of the Complex Sphere
    Journal of Mathematical Sciences, 2001
    Co-Authors: Vladimir Nikolaevich Dubinin, Leonid V. Kovalev
    Abstract:

    The paper continues studies of reduced modules of open sets. A notion of reduced module of the complex sphere is introduced, a formula for this module is obtained, and a number of its properties are proved. Applications of the reduced module of the sphere to various problems in Geometric Function Theory are given. Bibliography: 12 titles.

Matti Vuorinen - One of the best experts on this subject based on the ideXlab platform.

  • Generalized Hyperbolic Geometries
    Springer Monographs in Mathematics, 2020
    Co-Authors: Parisa Hariri, Riku Klén, Matti Vuorinen
    Abstract:

    In Geometric Function Theory, invariance properties of metrics are important. In our work below, two notions of invariance are most important; invariance with respect to the group of Mobius transformations and invariance with respect to the group of similarity transformations.

  • Trends in Classical Analysis, Geometric Function Theory, and Geometry of Conformal Invariants
    Abstract and Applied Analysis, 2013
    Co-Authors: Árpád Baricz, Saminathan Ponnusamy, Matti Vuorinen, Karl-joachim Wirths
    Abstract:

    The present special issue of the Journal of Abstract and Applied Analysis is devoted to trends in classical analysis, Geometric Function Theory, and geometry of conformal mappings. In the sense of the title of this journal, we wanted to present a spectrum of research themes reaching from applied analysis to pure analysis and from applications of analysis in geometry to applications in differential equations and integral equations. Further, our aim was to find articles on classical Function Theory as well as on its generalizations in several directions. One additional aspect, that was paid attention to, is the tendency of mathematics to use computers to solve problems by new and effective algorithms. In detail, we addressed to the following themes. That we did not forget classical pure analysis is proved by an article that considers harmonic Functions on Riemann manifolds and by an article on geometry and topology of Banach spaces. The classical Geometric Function Theory is represented by an article on univalence criterions associated with the nth derivative. The relationships between conformal mappings and integral equations are in the scope of two articles, where algorithms are proved to compute the mappings of unbounded multiply connected as well as bounded multiply connected regions onto slit regions. Other old themes of classical Function Theory are entire Functions for which we publish a paper on uniqueness theorems for monomials of entire Functions. Concerning the generalizations of classical analysis, we incorporated a paper on the stability of solutions of fractional differential equations and two papers on harmonic mappings, specially one on the general Theory of log-harmonic mappings and one on certain classes of harmonic mappings defined by convolutions. The papers on the generalizations of holomorphic Functions are completed by a longer article on the distribution of zeros and poles of the rational approximants of a nonholomorphic Function on an interval. The aspect of new algorithms is addressed in an article on Hermite interpolation using Mobius transformations of planar Pythagorean-hodograph cubics and in a paper where algorithms to calculate inverse Z-transforms on the unit disc by number-theoretical methods are proved. We hope that in this broad variety of analytic themes many researchers can find something interesting and new.

  • Topics in special Functions III
    arXiv: Classical Analysis and ODEs, 2012
    Co-Authors: G. D. Anderson, Matti Vuorinen, Xiaohui Zhang
    Abstract:

    The authors survey recent results in special Functions of classical analysis and Geometric Function Theory, in particular the circular and hyperbolic Functions, the gamma Function, the elliptic integrals, the Gaussian hyperGeometric Function, power series, and mean values.

  • On Moduli of Rings and Quadrilaterals: Algorithms and Experiments
    SIAM Journal on Scientific Computing, 2011
    Co-Authors: Harri Hakula, Antti Rasila, Matti Vuorinen
    Abstract:

    Moduli of rings and quadrilaterals are frequently applied in Geometric Function Theory; see, e.g., the handbook by Kuhnau [Handbook of Complex Analysis: Geometric Function Theory, Vols. 1 and 2, North-Holland, Amsterdam, 2005]. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new $hp$-FEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the Schwarz-Christoffel Toolbox of Driscoll and Trefethen. We also demonstrate that the $hp$-FEM algorithm applies to the case of nonpolygonal boundary and report results with concrete error bounds.

  • Topics in Special Functions III
    Conformal Geometry and Dynamics of The American Mathematical Society, 2007
    Co-Authors: G. D. Anderson, Matti Vuorinen, Xiaohui Zhang
    Abstract:

    The authors provide a survey of recent results in special Functions of classical analysis and Geometric Function Theory, in particular, the circular and hyperbolic Functions, the gamma Function, the elliptic integrals, the Gaussian hyperGeometric Function, power series, and mean values.

Sorin G. Gal - One of the best experts on this subject based on the ideXlab platform.