scholarly journals Deforming geometric transitions

2015 ◽  
Vol 12 (05) ◽  
pp. 1550059 ◽  
Author(s):  
Michele Rossi
Keyword(s):  
Moduli Space ◽  
Equivalence Classes ◽  
Complex Moduli ◽  
Geometric Transitions

After a quick review of the wild structure of the complex moduli space of Calabi-Yau 3-folds and the role of geometric transitions in this context (the Calabi-Yau web) the concept of deformation equivalence for geometric transitions is introduced to understand the arrows of the Gross–Reid Calabi-Yau web as deformation-equivalence classes of geometric transitions. Then the focus will be on some results and suitable examples to understand under which conditions it is possible to get simple geometric transitions, which are almost the only well-understood geometric transitions both in mathematics and in physics.

scholarly journals Effective metrics for multi-robot motion-planning

2018 ◽  
Vol 37 (13-14) ◽  
pp. 1741-1759 ◽  
Author(s):  
Aviel Atias ◽  
Kiril Solovey ◽  
Oren Salzman ◽  
Dan Halperin
Keyword(s):  
Motion Planning ◽  
Success Rate ◽  
Shape Matching ◽  
Equivalence Classes ◽  
Robot Motion ◽  
Robot Motion Planning ◽  
The Novel ◽  
Multi Robot

We study the effectiveness of metrics for multi-robot motion-planning (MRMP) when using rapidly-exploring random tree (RRT)-style sampling-based planners. These metrics play the crucial role of determining the nearest neighbors of configurations and in that they regulate the connectivity of the underlying roadmaps produced by the planners and other properties such as the quality of solution paths. After screening over a dozen different metrics we focus on the five most promising ones: two more traditional metrics, and three novel ones, which we propose here, adapted from the domain of shape-matching. In addition to the novel multi-robot metrics, a central contribution of this work are tools to analyze and predict the effectiveness of metrics in the MRMP context. We identify a suite of possible substructures in the configuration space, for which it is fairly easy: (i) to define a so-called natural distance that allows us to predict the performance of a metric, which is done by comparing the distribution of its values for sampled pairs of configurations to the distribution induced by the natural distance; and (ii) to define equivalence classes of configurations and test how well a metric covers the different classes. We provide experiments that attest to the ability of our tools to predict the effectiveness of metrics: those metrics that qualify in the analysis yield higher success rate of the planner with fewer vertices in the roadmap. We also show how combining several metrics together may lead to better results (success rate and size of roadmap) than using a single metric.

A NOTE ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF RATIONAL MAPS

2017 ◽  
Vol 60 (1) ◽  
pp. 199-207
Author(s):  
RUBEN A. HIDALGO ◽  
SAÚL QUISPE
Keyword(s):  
Moduli Space ◽  
Singular Locus ◽  
Equivalence Classes ◽  
Rational Maps ◽  
Branch Locus ◽  
Cubic Curve ◽  
Dimension 2 ◽  
Holomorphic Automorphisms

AbstractMilnor proved that the moduli space Md of rational maps of degree d ≥ 2 has a complex orbifold structure of dimension 2(d − 1). Let us denote by ${\mathcal S}$d the singular locus of Md and by ${\mathcal B}$d the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify M2 with ℂ2 and, within that identification, that ${\mathcal B}$2 is a cubic curve; so ${\mathcal B}$2 is connected and ${\mathcal S}$2 = ∅. If d ≥ 3, then it is well known that ${\mathcal S}$d = ${\mathcal B}$d. In this paper, we use simple arguments to prove the connectivity of ${\mathcal S}$d.

Moduli of Rank 2 Stable Bundles and Hecke Curves

2016 ◽  
Vol 59 (4) ◽  
pp. 865-877
Author(s):  
Sarbeswar Pal
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Short Note ◽  
Equivalence Classes ◽  
Complex Numbers ◽  
Stable Bundles ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Arbitrary Genus ◽  
Rank 2

AbstractLet X be a smooth projective curve of arbitrary genus g > 3 over the complex numbers. In this short note we will show that the moduli space of rank 2 stable vector bundles with determinant isomorphic to Lx , where Lx denotes the line bundle corresponding to a point x ∊ X, is isomorphic to a certain variety of lines in the moduli space of S-equivalence classes of semistable bundles of rank 2 with trivial determinant.

scholarly journals On the Connectedness of Moduli Spaces of Flat Connections over Compact Surfaces

2004 ◽  
Vol 56 (6) ◽  
pp. 1228-1236 ◽  
Author(s):  
Nan-Kuo Ho ◽  
Chiu-Chu Melissa Liu
Keyword(s):  
Lie Groups ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Orientable Surface ◽  
Equivalence Classes ◽  
Nonorientable Surface ◽  
Flat Connections ◽  
Gauge Equivalence ◽  
Compact Orientable Surface ◽  
Simply Connected

AbstractWe study the connectedness of the moduli space of gauge equivalence classes of flat G-connections on a compact orientable surface or a compact nonorientable surface for a class of compact connected Lie groups. This class includes all the compact, connected, simply connected Lie groups, and some non-semisimple classical groups.

Self-Organized Modularization in Evolutionary Algorithms

2005 ◽  
Vol 13 (3) ◽  
pp. 303-328 ◽  
Author(s):  
Peter Dauscher ◽  
Thomas Uthmann
Keyword(s):  
Evolutionary Computation ◽  
Building Blocks ◽  
Search Space ◽  
Equivalence Classes ◽  
Self Organized ◽  
Wide Range ◽  
Simple Genetic Algorithms ◽  
Evolutionary Operators ◽  
Selection Factor

The principle of modularization has proven to be extremely successful in the field of technical applications and particularly for Software Engineering purposes. The question to be answered within the present article is whether mechanisms can also be identified within the framework of Evolutionary Computation that cause a modularization of solutions. We will concentrate on processes, where modularization results only from the typical evolutionary operators, i.e. selection and variation by recombination and mutation (and not, e.g., from special modularization operators). This is what we call Self-Organized Modularization. Based on a combination of two formalizations by Radcliffe and Altenberg, some quantitative measures of modularity are introduced. Particularly, we distinguish Built-in Modularityas an inherent property of a genotype and Effective Modularity, which depends on the rest of the population. These measures can easily be applied to a wide range of present Evolutionary Computation models. It will be shown, both theoretically and by simulation, that under certain conditions, Effective Modularity (as defined within this paper) can be a selection factor. This causes Self-Organized Modularization to take place. The experimental observations emphasize the importance of Effective Modularityin comparison with Built-in Modularity. Although the experimental results have been obtained using a minimalist toy model, they can lead to a number of consequences for existing models as well as for future approaches. Furthermore, the results suggest a complex self-amplification of highly modular equivalence classes in the case of respected relations. Since the well-known Holland schemata are just the equivalence classes of respected relations in most Simple Genetic Algorithms, this observation emphasizes the role of schemata as Building Blocks (in comparison with arbitrary subsets of the search space).

scholarly journals Relations between ageing and dependence for exchangeable lifetimes with an extension for the IFRA/DFRA property

2020 ◽  
Vol 8 (1) ◽  
pp. 1-33
Author(s):  
Giovanna Nappo ◽  
Fabio Spizzichino
Keyword(s):  
Equivalence Classes ◽  
Stochastic Dependence ◽  
The Past ◽  
Dependence Concepts ◽  
Kendall Distribution ◽  
Mutual Relations ◽  
Bivariate Ageing ◽  
Dependence Properties ◽  
Ageing Notions

AbstractWe first review an approach that had been developed in the past years to introduce concepts of “bivariate ageing” for exchangeable lifetimes and to analyze mutual relations among stochastic dependence, univariate ageing, and bivariate ageing.A specific feature of such an approach dwells on the concept of semi-copula and in the extension, from copulas to semi-copulas, of properties of stochastic dependence. In this perspective, we aim to discuss some intricate aspects of conceptual character and to provide the readers with pertinent remarks from a Bayesian Statistics standpoint. In particular we will discuss the role of extensions of dependence properties. “Archimedean” models have an important role in the present framework.In the second part of the paper, the definitions of Kendall distribution and of Kendall equivalence classes will be extended to semi-copulas and related properties will be analyzed. On such a basis, we will consider the notion of “Pseudo-Archimedean” models and extend to them the analysis of the relations between the ageing notions of IFRA/DFRA-type and the dependence concepts of PKD/NKD.

The Mumford-Tate Group of a Variation of Hodge Structure

2012 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Complex Manifold ◽  
Moduli Space ◽  
Hodge Structure ◽  
Integral Manifold ◽  
Equivalence Classes ◽  
Hodge Structures ◽  
Tate Group ◽  
Variation Of Hodge Structure ◽  
Definition Of ◽  
Variations Of Hodge Structure

This chapter deals with the Mumford-Tate group of a variation of Hodge structure (VHS). It begins by presenting a definition of VHS, which consists of a connected complex manifold and a locally liftable, holomorphic mapping that is an integral manifold of the canonical differential ideal. The moduli space of Γ‎-equivalence classes of polarized Hodge structures is also considered, along with a generic point for the VHS and the monodromy group of the VHS. Associated to a VHS is its Mumford-Tate group. The chapter proceeds by discussing the structure theorem for VHS, where S is a quasi-projective algebraic variety, referred to as global variations of Hodge structure. It concludes by describing an application of Mumford-Tate groups, along with the Noether-Lefschetz locus.

scholarly journals HIGHER RANK BRILL–NOETHER THEORY ON SECTIONS OF K3 SURFACES

2012 ◽  
Vol 23 (07) ◽  
pp. 1250075 ◽  
Author(s):  
GAVRIL FARKAS ◽  
ANGELA ORTEGA
Keyword(s):  
Moduli Space ◽  
K3 Surfaces ◽  
Moduli Space Of Curves ◽  
Picard Number ◽  
Detailed Proof ◽  
Rank 2 ◽  
Higher Rank

We discuss the role of K3 surfaces in the context of Mercat's conjecture in higher rank Brill–Noether theory. Using liftings of Koszul classes, we show that Mercat's conjecture in rank 2 fails for any number of sections and for any gonality stratum along a Noether–Lefschetz divisor inside the locus of curves lying on K3 surfaces. Then we show that Mercat's conjecture in rank 3 fails even for curves lying on K3 surfaces with Picard number 1. Finally, we provide a detailed proof of Mercat's conjecture in rank 2 for general curves of genus 11, and describe explicitly the action of the Fourier–Mukai involution on the moduli space of curves.

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