Symplectic fibrations

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Riemann Surfaces ◽  
Final Section ◽  
Total Space ◽  
Fibre Bundles ◽  
Symplectic Forms ◽  
Dehn Twists ◽  
Universal Construction ◽  
Symplectic Connection

This chapter begins with a general discussion of symplectic fibrations and symplectic forms on the total space. The next section describes in detail symplectic 2-sphere bundles over Riemann surfaces. Subsequent sections develop the notions of symplectic connection and holonomy, explain the Sternberg–Weinstein universal construction for fibre bundles, discuss Seidel’s construction of generalized Dehn twists, and introduce the Guillemin–Lerman–Sternberg coupling form. The final section studies Hamiltonian fibrations.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

scholarly journals ON PRODUCTS OF PSEUDO-ANOSOV MAPS AND DEHN TWISTS OF RIEMANN SURFACES WITH PUNCTURES

2010 ◽  
Vol 88 (3) ◽  
pp. 413-428 ◽  
Author(s):  
C. ZHANG
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Dehn Twists

AbstractLet S be a Riemann surface of type (p,n) with 3p+n>4 and n≥1. We investigate products of some pseudo-Anosov maps θ and Dehn twists tα on S, and prove that under certain conditions the products tkα∘θ are pseudo-Anosov for all integers k. We also give examples that show that tkα∘θ are not pseudo-Anosov for some integers k.

Triangulation of Fibre Bundles

1967 ◽  
Vol 19 ◽  
pp. 499-513 ◽  
Author(s):  
H. Putz
Keyword(s):  
Vector Bundles ◽  
Fibre Bundle ◽  
Base Space ◽  
Total Space ◽  
Fibre Bundles ◽  
Differentiable Manifolds ◽  
Geometric Methods

In this paper we consider the following problem. Let (E, M, N, π) be a differentiable fibre bundle, whereEis the total space,Mthe base space,Nthe fibre, andπ: E→Mthe projection map. Then, given aCrtriangulation (ƒ, D) ofM,can one obtain aCrtriangulation (F, K) ofEsuch that the induced mapƒ–1πF:K→ D is linear? R. Lashof and M. Rothenberg (3) have obtained this result for vector bundles.Using methods quite different from theirs, we obtain a solution in the general case. The methods we use are the geometric methods developed by J. H. C. Whitehead. (7) in his triangulation of differentiable manifolds, as found in (5). In fact, our solution consists of generalizing his techniques in a fibre bundle setting.

MARS — Its Tenth Anniversary of Operation and its Future

1980 ◽  
Vol 19 (03) ◽  
pp. 125-132
Author(s):  
G. S. Lodwick ◽  
C. R. Wickizer ◽  
E. Dickhaus
Keyword(s):  
Conceptual Development ◽  
Administrative Support ◽  
Final Section ◽  
Economic Benefits ◽  
Hospital Environment ◽  
Computer Scientist ◽  
University Of Missouri ◽  
Complete Bibliography ◽  
The University ◽  
Economic Considerations

The Missouri Automated Radiology System recently passed its tenth year of clinical operation at the University of Missouri. This article presents the views of a radiologist who has been instrumental in the conceptual development and administrative support of MARS for most of this period, an economist who evaluated MARS from 1972 to 1974 as part of her doctoral dissertation, and a computer scientist who has worked for two years in the development of a Standard MUMPS version of MARS. The first section provides a historical perspective. The second deals with economic considerations of the present MARS system, and suggests those improvements which offer the greatest economic benefits. The final section discusses the new approaches employed in the latest version of MARS, as well as areas for further application in the overall radiology and hospital environment. A complete bibliography on MARS is provided for further reading.

Introduction

2018 ◽  
Author(s):  
Peter Hopkins
Keyword(s):  
Final Section ◽  
Short Note ◽  
Current Situation ◽  
Research Projects ◽  
Everyday Lives ◽  
Early Settlement ◽  
The Everyday ◽  
Initial Research

The chapters in this collection explore the everyday lives, experiences, practices and attitudes of Muslims in Scotland. In order to set the context for these chapters, in this introduction I explore the early settlement of Muslims in Scotland and discuss some of the initial research projects that charted the settlement of Asians and Pakistanis in Scotland’s main cities. I then discuss the current situation for Muslims in Scotland through data from the 2011 Scottish Census. Following a short note about the significance of the Scottish context, in the final section, the main themes and issues that have been explored in research about Muslims in Scotland.

The Foundation of St Andrews Cathedral Priory, 1140.

2005 ◽  
Vol 84 (1) ◽  
pp. 1-37 ◽  
Author(s):  
A. A. M. Duncan
Keyword(s):  
Final Section ◽  
Serious Illness ◽  
Marian Devotion

The hitherto accepted date of the priory's foundation, 1144, was copied on the bishop's diploma from the bull of Lucius II, and is impossible; Bower's 1140 is to be preferred. The foundation narrative (FN) probably by Robert, the first prior, ascribes to a Pictish king the grant to St Andrew of the Boar's Raik, but that was ignored by Wyntoun and Bower and is probably wrong. It seems that Alexander I made this gift, renegued on it, and restored it towards the end of his life. Though intended to found an Augustinian priory, the Raik was kept by the bishop until in 1138-9 David I obtained from Nostell a prior, Robert; Robert was unable to advance the foundation through his reluctance to recruit canons from elsewhere, perhaps resisting Scone and/or Holyrood. He and clerics of his resided in a ‘parsonage’, the vacant house of one of the seven ‘parsons’ who represented the earliest clerics of St Andrews, and are uniquely described in FN; they developed the hospital. In 1140 David I and Earl Henry at St Andrews compelled the bishop to disgorge the Raik and thereby establish the priory. The date was probably St Andrew's day, 1140, a month after the foundation of the abbey of St Mary at Newbattle. Both foundations should be seen as thanksgiving for Henry's recovery from serious illness. A narrower dating is suggested for some St Andrews charters, the endowments showing a closer relationship with those of Holyrood abbey than with those of Scone priory. Prior Robert probably wished from the beginning to recruit the céli Dé (Culdees) as canons and to obtain their endowments, succeeding at Lochleven but, despite papal and royal approval, failing at St Andrews. A final section asks why David I was so generous to the regular orders, suggesting that he was much influenced by the development of Marian devotion in his lifetime, when the Virgin had become head and most powerful of the hierarchy of saints.

Public Hearings about Same-Sex Marriage

2012 ◽  
Vol 1 (1) ◽  
pp. 83-107 ◽  
Author(s):  
Karen Tracy
Keyword(s):  
Discourse Analysis ◽  
Final Section ◽  
Democratic Process ◽  
Same Sex ◽  
Same Sex Marriage ◽  
Public Hearings ◽  
Context Features

This study analyzes public hearings about same-sex marriage to show how the contexts that are established for citizens' and legislators' talk make arguments about the issue being disputed. Situated within the traditions of argument studies and discourse analysis, the article explores different meanings of “context.” The study evidences how two sets of context features created positive (or negative) stances toward the issue of same-sex marriage, and shows that how the controversy was formulated and how participation was designed gave distinct advantages to speakers advocating for (or against) same-sex marriage. The final section draws out implications of these legislative choices for citizen presenters and for the officials themselves as the enactors and guardians of democratic process.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

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