Length of the singular set of Schottky group

In the theory of automorphic functions for a properly discontinuous group G of linear transformations, the Poincaré theta series plays an essential role, since the convergence problem of the series occupies an important part of the theory. This problem was treated by many mathematicians such as Poincaré, Burnside [2], Fricke [4], Myrberg [6], [7] and others. Poincaré proved that the (-2m)-dimensional Poincaré theta series always converges if m is a positive integer greater than 2, and Burnside treated the problem and conjectured that ( -2)-dimensional Poincaré theta series always converges if G is a Schottky group. This conjecture was solved negatively by Myrberg. As is shown later (Theorem A), the convergence of Poincaré theta series gives an information on a metrical property of the singular set of the group.

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Schottky groups acting on homogeneous rational manifolds

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2016-0065 ◽

2019 ◽

Vol 2019 (753) ◽

pp. 23-56 ◽

Cited By ~ 1

Author(s):

Christian Miebach ◽

Karl Oeljeklaus

Keyword(s):

Deformation Theory ◽

Group Actions ◽

Picard Group ◽

Schottky Group ◽

Fundamental Groups ◽

Complex Manifolds ◽

Schottky Groups ◽

Algebraic Dimension ◽

Geometric Invariants ◽

Compact Complex Manifolds

AbstractWe systematically study Schottky group actions on homogeneous rational manifolds and find two new families besides those given by Nori’s well-known construction. This yields new examples of non-Kähler compact complex manifolds having free fundamental groups. We then investigate their analytic and geometric invariants such as the Kodaira and algebraic dimension, the Picard group and the deformation theory, thus extending results due to Lárusson and to Seade and Verjovsky. As a byproduct, we see that the Schottky construction allows to recover examples of equivariant compactifications of {{\rm{SL}}(2,\mathbb{C})/\Gamma} for Γ a discrete free loxodromic subgroup of {{\rm{SL}}(2,\mathbb{C})}, previously obtained by A. Guillot.

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Partial regularity of stable solutions to the fractional Geľfand-Liouville equation

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0177 ◽

2021 ◽

Vol 10 (1) ◽

pp. 1316-1327

Author(s):

Ali Hyder ◽

Wen Yang

Keyword(s):

Liouville Equation ◽

Weak Solutions ◽

Partial Regularity ◽

Singular Set ◽

Stable Solutions ◽

Gelfand Problem

Abstract We analyze stable weak solutions to the fractional Geľfand problem ( − Δ ) s u = e u i n Ω ⊂ R n . $$\begin{array}{} \displaystyle (-{\it\Delta})^su = e^u\quad\mathrm{in}\quad {\it\Omega}\subset\mathbb{R}^n. \end{array}$$ We prove that the dimension of the singular set is at most n − 10s.

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Singing to Emmanuel: The Wall Paintings of Sant Miquel in Terrassa and the 6th Century Artistic Reception of Byzantium in the Western Mediterranean

Arts ◽

10.3390/arts8040128 ◽

2019 ◽

Vol 8 (4) ◽

pp. 128

Author(s):

Carles Sánchez Márquez

Keyword(s):

19Th Century ◽

Eastern Mediterranean ◽

Western Mediterranean ◽

Wall Paintings ◽

Singular Set ◽

Late 19Th Century ◽

Early Date ◽

Early Christian ◽

Christian Art ◽

Technical Study

Since the late 19th century the wall paintings of Sant Miquel in Terrassa have drawn attention due to their singularity. From the early studies of Josep Puig i Cadafalch (1867–1956) to the present, both the iconographic program and the chronology of the paintings have fueled controversy among scholars. In particular, chronological estimates range from the time of Early Christian Art to the Carolingian period. However, a recent technical study of the paintings seems to confirm an early date around the 6th century. This new data allows us to reassess the question in other terms and explore a new possible context for the paintings. First, it is very likely that the choice of iconographic topics was related to the debates on the Arian heresy that took place in Visigothic Spain during the 5th and 6th centuries. Secondly, the paintings of Sant Miquel should be reconsidered as a possible reception of a larger 6th-century pictorial tradition linked to the Eastern Mediterranean, which is used in a very particular way. However, thus far we ignore which were the means for this artistic transmission as well as the reasons which led the “doers” of Terrassa to select such a peculiar and unique repertoire of topics, motifs, and inscriptions. My paper addresses all these questions in order to propose a new Mediterranean framework for the making of this singular set of paintings.

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KNOT SINGULARITIES OF HARMONIC MORPHISMS

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091598001278 ◽

2001 ◽

Vol 44 (1) ◽

pp. 71-85 ◽

Cited By ~ 1

Author(s):

Paul Baird

Keyword(s):

Riemann Surface ◽

Complex Surface ◽

Subject Classification ◽

Harmonic Morphisms ◽

Singular Set ◽

Harmonic Morphism ◽

Complex Curves ◽

Analytic Curve ◽

Primary 57M25 ◽

Straight Lines

AbstractA harmonic morphism defined on $\mathbb{R}^3$ with values in a Riemann surface is characterized in terms of a complex analytic curve in the complex surface of straight lines. We show how, to a certain family of complex curves, the singular set of the corresponding harmonic morphism has an isolated component consisting of a continuously embedded knot.AMS 2000 Mathematics subject classification: Primary 57M25. Secondary 57M12; 58E20

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Approximation Properties of the Singular Set

Theorems on Regularity and Singularity of Energy Minimizing Maps ◽

10.1007/978-3-0348-9193-6_3 ◽

1996 ◽

pp. 51-90

Author(s):

Leon Simon

Keyword(s):

Singular Set ◽

Approximation Properties

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On the structure of the singular set of a complex analytic foliation

Journal of the Mathematical Society of Japan ◽

10.2969/jmsj/05040837 ◽

1998 ◽

Vol 50 (4) ◽

pp. 837-857 ◽

Cited By ~ 3

Author(s):

Junya YOSHIZAKI

Keyword(s):

Singular Set ◽

Analytic Foliation

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KNOT POINTS IN TWO-DIMENSIONAL MAPS AND RELATED PROPERTIES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409023184 ◽

2009 ◽

Vol 19 (02) ◽

pp. 545-555 ◽

Cited By ~ 1

Author(s):

F. TRAMONTANA ◽

L. GARDINI ◽

D. FOURNIER-PRUNARET ◽

P. CHARGE

Keyword(s):

Focal Point ◽

Singular Line ◽

Two Dimensional ◽

Focal Points ◽

Singular Set ◽

One Dimensional ◽

Attracting Set ◽

Straight Line ◽

Special Character ◽

Stable And Unstable Sets

We consider the class of two-dimensional maps of the plane for which there exists a whole one-dimensional singular set (for example, a straight line) that is mapped into one point, called a "knot point" of the map. The special character of this kind of point has been already observed in maps of this class with at least one of the inverses having a vanishing denominator. In that framework, a knot is the so-called focal point of the inverse map (it is the same point). In this paper, we show that knots may also exist in other families of maps, not related to an inverse having values going to infinity. Some particular properties related to focal points persist, such as the existence of a "point to slope" correspondence between the points of the singular line and the slopes in the knot, lobes issuing from the knot point and loops in infinitely many points of an attracting set or in invariant stable and unstable sets.

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