An extension of the Loop theorem and resolutions of generalized 3-manifolds with 0-dimensional singular set

1984 ◽  
Vol 78 (2) ◽  
pp. 161-222 ◽  
Author(s):  
T. L. Thickstun
Keyword(s):  
Singular Set ◽  
Loop Theorem

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

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.

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

Arts ◽  
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.

scholarly journals KNOT SINGULARITIES OF HARMONIC MORPHISMS

2001 ◽  
Vol 44 (1) ◽  
pp. 71-85 ◽  
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

KNOT POINTS IN TWO-DIMENSIONAL MAPS AND RELATED PROPERTIES

2009 ◽  
Vol 19 (02) ◽  
pp. 545-555 ◽  
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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