scholarly journals Local $L^2$ results for $\overline{\partial}$ on a singular surface

2003 ◽  
Vol 92 (2) ◽  
pp. 269 ◽  
Author(s):  
Klas Diederich ◽  
John Erik Fornæss ◽  
Sophia Vassiliadou
Keyword(s):  
Complex Analysis ◽  
Blow Up ◽  
Function Spaces ◽  
Singular Surface ◽  
Singular Surfaces ◽  
Singular Spaces ◽  
Cauchy Riemann

The Cauchy-Riemann equations are fundamental in complex analysis. This paper contributes to the understanding of these equations on singular spaces. Various methods have been used to overcome the problem of defining forms near singularities. One can blow up the singularity, restrict forms from smooth ambient spaces or work on the regular points. In this paper we use the latter approach to obtain square integrable solutions on singular surfaces. This can be briefly called the Kohn solution up to the singularity to contrast with results in terms of curvature, weights or different function spaces.

Growth Kinetics of Molecular Beam Epitaxy on Non-Singular Surfaces

1992 ◽  
Vol 280 ◽  
Author(s):  
J. F. Egler ◽  
N. Otsuka ◽  
K. Mahalingam
Keyword(s):  
Surface Roughness ◽  
Monte Carlo ◽  
Growth Kinetics ◽  
Molecular Beam ◽  
Singular Surface ◽  
Sticking Coefficient ◽  
Singular Surfaces ◽  
Vicinal Surfaces ◽  
Kinetics Of ◽  
Multiple Configurations

ABSTRACTGrowth kinetics on non-singular surfaces were studied by Monte Carlo simulations. In contrast to the growth on singular and vicinal surfaces, the sticking coefficient on the non-singular surfaces was found to decrease with increase of the surface roughness. Simulations of annealing processes showed that surface diffusion of atoms leads to a stationary surface roughness, which is explained by multiple configurations having the lowest energy in the non-singular surface.

Zero Cycles on Singular surfaces

2008 ◽  
Vol 4 (1) ◽  
pp. 101-143 ◽  
Author(s):  
Amalendu Krishna
Keyword(s):  
Singular Surface ◽  
Resolution Of Singularities ◽  
Smooth Surfaces ◽  
Projective Modules ◽  
Singular Surfaces ◽  
Bloch’S Conjecture

AbstractLet X be a reduced and projective singular surface over ℂ and let → X be a resolution of singularities of X. We show that CH2(X) ≅ CH2() if and only if for i = 0, 1. This verifies a conjecture of Srinivas.We also verify Bloch's conjecture for singular surfaces assuming it holds for smooth surfaces. As a byproduct, we give an application to projective modules on certain singular affine surfaces.

scholarly journals HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES

2010 ◽  
Vol 02 (01) ◽  
pp. 1-55 ◽  
Author(s):  
JEAN-PAUL BRASSELET ◽  
JÖRG SCHÜRMANN ◽  
SHOJI YOKURA
Keyword(s):  
Chern Class ◽  
Blow Up ◽  
Homology Class ◽  
Grothendieck Group ◽  
Characteristic Classes ◽  
Chern Classes ◽  
Singular Spaces ◽  
Coherent Sheaves ◽  
Roch Theorem ◽  
Mixed Hodge Modules

In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformationmCy: K0( var /X) → G0(X) ⊗ ℤ[y], which generalizes the total λ-class λy(T*X) of the cotangent bundle to singular spaces. Here K0( var /X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G0(X) is the Grothendieck group of coherent sheaves of [Formula: see text]-modules. A first construction of mCy is based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mCy is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito. We define a natural transformation Ty* : K0( var /X) → H*(X) ⊗ ℚ[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Ty* is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y = -1), the Todd class transformation in the singular Riemann-Roch theorem of Baum–Fulton–MacPherson (for y = 0) and the L-class transformation of Cappell-Shaneson (for y = 1). We also explain the relation among the "stringy version" of our characteristic classes, the elliptic class of Borisov–Libgober and the stringy Chern classes of Aluffi and De Fernex–Lupercio–Nevins–Uribe. All our results can be extended to varieties over a base field k of characteristic 0.

UNSTEADY DIFFUSION OF FLUIDS THROUGH SOLIDS UNDERGOING LARGE DEFORMATIONS

1991 ◽  
Vol 01 (03) ◽  
pp. 311-346 ◽  
Author(s):  
L. TAO ◽  
K.R. RAJAGOPAL ◽  
A.S. WINEMAN
Keyword(s):  
Initial Value Problem ◽  
Constitutive Equations ◽  
Large Deformations ◽  
Singular Surface ◽  
Jump Conditions ◽  
Singular Surfaces ◽  
Initial Value ◽  
Diffusion Problems

In this work, we develop a theory for studying the unsteady diffusion problems which involve the motion of singular surfaces within the context of the theory of interacting continua. The theory allows for the possibility of mixtures with different properties on either side of the surface. Constitutive equations have to be postulated for the mixtures and the singular surface. The jump conditions across the singular surface are obtained by extending techniques developed in the case of a single continuum. To evaluate the validity of the theory we solve a typical boundary-initial value problem employing the theory.

The Cauchy Riemann equation on singular spaces

1998 ◽  
Vol 93 (3) ◽  
pp. 453-477 ◽  
Author(s):  
John Erik Forn�ss ◽  
Estela A. Erik
Keyword(s):  
Singular Spaces ◽  
Riemann Equation ◽  
Cauchy Riemann

scholarly journals The nodal cubic surfaces and the surfaces from which they are derived by projection

1928 ◽  
Vol 119 (782) ◽  
pp. 213-248 ◽  
Keyword(s):  
Detailed Examination ◽  
Singular Surface ◽  
Ruled Surfaces ◽  
Singular Surfaces ◽  
Cubic Surfaces ◽  
The Subject ◽  
Geometrical Significance

The cubic surfaces have been classified according to the character of their singularities by Schlafli and by Cayley, who find that there are 21 types in addition to ruled surfaces. In their treatment of the matter each case is considered separately by algebraical methods, and there is a marked lack of any simple unifying principle, which it is the object of this paper to supply. A means whereby this can be done is suggested by the theorem that every surface is the projection of a non-singular surface in higher space. The considerations employed in the proof of this result are somewhat abstruse, and the purely geometrical significance is obscure, so that the more detailed examination of particular cases is of genuine interest. Accordingly, the subject of this paper is the generation of the various nodal cubic surfaces by the projection of non-singular surfaces, specifically the non-ruled surfaces of order n in space of n dimensions (denoted throughout by F n ); it will be shown that these arise by the projection of the same surface, F 9 .

Some Types of Irregular Surfaces

1935 ◽  
Vol 31 (2) ◽  
pp. 159-173 ◽  
Author(s):  
L. Roth
Keyword(s):  
Singular Surface ◽  
Sectional Genus ◽  
Singular Surfaces ◽  
Minimum Order ◽  
Irregular Surfaces ◽  
Hyperelliptic Surfaces ◽  
Hyperelliptic Surface ◽  
Special Case

It is a problem of considerable interest in the theory of surfaces to determine the irregular non-singular surface of minimum order, not referable to a scroll; in previous investigations the author has discussed the regularity or referability of surfaces in higher space, reaching the conclusion that all non-singular surfaces of order n ≤ 10 in S4 are regular or referable, with the possible exception of the surface of order n = 10 and sectional genus π = 6, which may be elliptic (pg = 0, pa = −1) or hyperelliptic (pg = 1, pa = −1). In their memoir on hyperelliptic surfaces, Enriques and Severi have obtained for the irregular hyperelliptic surface of general moduli a model 6F10 of minimum order, situated in S4, with the characters n = 10, π = 6. Using transcendental methods, Comessatti has constructed a class of irregular hyperelliptic surfaces the properties of which he has examined in detail; this class includes a member 6Π10 which is a special case of 6F10; and since Comessatti has shown that Π10 is without singularities, so also is F10, whence it follows that F10 is a solution of the proposed problem.

Kinematic Modules for Singularity-Free Movement With Three Cartesian Freedoms

1993 ◽  
Vol 115 (2) ◽  
pp. 207-213 ◽  
Author(s):  
G. L. Long ◽  
J. M. McCarthy ◽  
R. P. Paul
Keyword(s):  
Arbitrary Point ◽  
Singular Surface ◽  
Free Movement ◽  
Terminal Link ◽  
Singular Surfaces ◽  
Third Order ◽  
Structural Requirements ◽  
Revolute Joint ◽  
Singular Configurations ◽  
Serial Chain

The singularity conditions of three-revolute-joint serial chain manipulators are investigated by considering the motion of an arbitrary point on a terminal link. A distinction is made between singular configurations (every point on the terminal link is singular) and singular surfaces (a chosen point on the terminal link lies on a singular surface). Of particular interest is a class of 3R manipulators—each manipulator within this class forms a third-order screw system. We outline the structural requirements for this 3R class, and then determine the singularity conditions for several cases. For several manipulators within this 3R class, we describe a strategy to develop four-revolute-joint kinematic modules for singularity-free movement with three Cartesian freedoms. The algorithm to control the kinematic modules is simple and can be implemented in real-time.

Positive solutions for a quasilinear system Via blow up

1993 ◽  
Vol 18 (12) ◽  
pp. 2071-2106
Author(s):  
Philippe Clément ◽  
Raúl Manásevich ◽  
Enzo Mitidieri
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
Quasilinear System
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