scholarly journals Upper bounds for the essential dimension of the moduli stack of SL n -bundles over a curve

2009 ◽  
Vol 14 (4) ◽  
pp. 747-770 ◽  
Author(s):  
A. Dhillon ◽  
N. Lemire
Keyword(s):  
Upper Bounds ◽  
Essential Dimension ◽  
Moduli Stack

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

An Asymptotic Statistical Theory of Polynomial Kernel Methods

2004 ◽  
Vol 16 (8) ◽  
pp. 1705-1719 ◽  
Author(s):  
Kazushi Ikeda
Keyword(s):  
Kernel Methods ◽  
Feature Space ◽  
Upper Bounds ◽  
High Dimensional ◽  
Polynomial Kernel ◽  
Generalization Error ◽  
Essential Dimension ◽  
Input Surface ◽  
Separating Hyperplane ◽  
The Relationship

The generalization properties of learning classifiers with a polynomial kernel function are examined. In kernel methods, input vectors are mapped into a high-dimensional feature space where the mapped vectors are linearly separated. It is well-known that a linear dichotomy has an average generalization error or a learning curve proportional to the dimension of the input space and inversely proportional to the number of given examples in the asymptotic limit. However, it does not hold in the case of kernel methods since the feature vectors lie on a submanifold in the feature space, called the input surface. In this letter, we discuss how the asymptotic average generalization error depends on the relationship between the input surface and the true separating hyperplane in the feature space where the essential dimension of the true separating polynomial, named the class, is important. We show its upper bounds in several cases and confirm these using computer simulations.

scholarly journals Upper Bounds for the Essential Dimension of E7

2013 ◽  
Vol 56 (4) ◽  
pp. 795-800 ◽  
Author(s):  
Mark L. MacDonald
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Connected Group ◽  
Simply Connected

Abstract.This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type E7 over a field of characteristic not 2 or 3. In particular, ed(E7) ≤ 29, and ed(E7, 2) ≤ 27.

scholarly journals On the essential dimension of coherent sheaves

2018 ◽  
Vol 2018 (735) ◽  
pp. 265-285 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Norbert Hoffmann
Keyword(s):  
Upper Bound ◽  
Vector Bundles ◽  
Coherent Sheaf ◽  
Projective Modules ◽  
Essential Dimension ◽  
Endomorphism Algebra ◽  
Smooth Projective Curve ◽  
Coherent Sheaves ◽  
Finite Dimensional ◽  
Moduli Stack

AbstractWe characterize all fields of definition for a given coherent sheaf over a projective scheme in terms of projective modules over a finite-dimensional endomorphism algebra. This yields general results on the essential dimension of such sheaves. Applying them to vector bundles over a smooth projective curveC, we obtain an upper bound for the essential dimension of their moduli stack. The upper bound is sharp if the conjecture of Colliot-Thélène, Karpenko and Merkurjev holds. We find that the genericity property proved for Deligne–Mumford stacks by Brosnan, Reichstein and Vistoli still holds for this Artin stack, unless the curveCis elliptic.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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