1,715 research outputs found
Bivariate polynomial mappings associated with simple complex Lie algebras
There are three families of bivariate polynomial maps associated with the
rank- simple complex Lie algebras and . It is
known that the bivariate polynomial map associated with induces a
permutation of if and only if for . In this paper, we give similar criteria for the other two families. As an
application, a counterexample is given to a conjecture posed by Lidl and Wells
about the generalized Schur's problem.Comment: 16 pages, 6 figure
Spectral method for matching exterior and interior elliptic problems
A spectral method is described for solving coupled elliptic problems on an
interior and an exterior domain. The method is formulated and tested on the
two-dimensional interior Poisson and exterior Laplace problems, whose solutions
and their normal derivatives are required to be continuous across the
interface. A complete basis of homogeneous solutions for the interior and
exterior regions, corresponding to all possible Dirichlet boundary values at
the interface, are calculated in a preprocessing step. This basis is used to
construct the influence matrix which serves to transform the coupled boundary
conditions into conditions on the interior problem. Chebyshev approximations
are used to represent both the interior solutions and the boundary values. A
standard Chebyshev spectral method is used to calculate the interior solutions.
The exterior harmonic solutions are calculated as the convolution of the
free-space Green's function with a surface density; this surface density is
itself the solution to an integral equation which has an analytic solution when
the boundary values are given as a Chebyshev expansion. Properties of Chebyshev
approximations insure that the basis of exterior harmonic functions represents
the external near-boundary solutions uniformly. The method is tested by
calculating the electrostatic potential resulting from charge distributions in
a rectangle. The resulting influence matrix is well-conditioned and solutions
converge exponentially as the resolution is increased. The generalization of
this approach to three-dimensional problems is discussed, in particular the
magnetohydrodynamic equations in a finite cylindrical domain surrounded by a
vacuum
Quasiconformal mappings, from Ptolemy's geography to the work of Teichmüller
The origin of quasiconformal mappings, like that of conformal mappings, can be traced back to old cartography where the basic problem was the search for mappings from the sphere onto the plane with minimal deviation from conformality, subject to certain conditions which were made precise. In this paper, we survey the development of cartography, highlighting the main ideas that are related to quasiconformality. Some of these ideas were completely ignored in the previous historical surveys on quasiconformal mappings. We then survey early quasiconformal theory in the works of Grötzsch, Lavrentieff, Ahlfors and Teichmüller, which are the 20th-century founders of the theory
On the Arithmetic Exceptionality of Polynomial Mappings
In this note we prove that certain polynomial mappings
in -variables
obtained from simple complex Lie algebras of arbitrary rank , are exceptional.Comment: 5 page
Nicolas-Auguste Tissot: A link between cartography and quasiconformal theory
Nicolas-Auguste Tissot (1824--1897) published a series of papers on
cartography in which he introduced a tool which became known later on, among
geographers, under the name of the "Tissot indicatrix." This tool was broadly
used during the twentieth century in the theory and in the practical aspects of
the drawing of geographical maps. The Tissot indicatrix is a graphical
representation of a field of ellipses on a map that describes its distortion.
Tissot studied extensively, from a mathematical viewpoint, the distortion of
mappings from the sphere onto the Euclidean plane that are used in drawing
geographical maps, and more generally he developed a theory for the distorsion
of mappings between general surfaces. His ideas are at the heart of the work on
quasiconformal mappings that was developed several decades after him by
Gr{\"o}tzsch, Lavrentieff, Ahlfors and Teichm{\"u}ller. Gr{\"o}tzsch mentions
the work of Tissot and he uses the terminology related to his name (in
particular, Gr{\"o}tzsch uses the Tissot indicatrix). Teichm{\"u}ller mentions
the name of Tissot in a historical section in one of his fundamental papers
where he claims that quasiconformal mappings were used by geographers, but
without giving any hint about the nature of Tissot's work. The name of Tissot
is also missing from all the historical surveys on quasiconformal mappings. In
the present paper, we report on this work of Tissot. We shall also mention some
related works on cartography, on the differential geometry of surfaces, and on
the theory of quasiconformal mappings. This will place Tissot's work in its
proper context. The final version of this paper will appear in the journal
Arch. Hist. Exact Sciences
Rigidity and absence of line fields for meromorphic and Ahlfors islands maps
In this note, we give an elementary proof of the absence of invariant line
fields on the conical Julia set of an analytic function of one variable. This
proof applies not only to rational as well as transcendental meromorphic
functions (where it was previously known), but even to the extremely general
setting of Ahlfors islands maps as defined by Adam Epstein.
In fact, we prove a more general result on the absence of
invariant_differentials_, measurable with respect to a conformal measure that
is supported on the (unbranched) conical Julia set. This includes the study of
cohomological equations for , which are relevant to a number of
well-known rigidity questions.Comment: 17 page
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