DOCSLIB.ORG

Hamiltonian Group Actions Today's Plan: • O(L) (Bundle Over CP 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hamiltonian Group Actions Today's Plan: • O(L) (Bundle Over CP 1 Hamiltonian group actions Wednesday Mar.3, 2010; 12-2 pm Today’s plan: •O(`) (bundle over CP1.) • Blowups and their effect on topology and combinatorics • Cutting • First Chern class The bundle O(`). (C∞) complex line bundles over CP1, up to isomorphism, are classified by the integers. In fact, the natural map 1 1 holomorphic line bundles over CP → complex line bundles over CP induces a bijection on equivalence classes. The bundle that corresponds to the integer ` is often called O(`), though strictly speaking this name denotes its space of holomorphic sections. The bundle can be constructed as follows. 2 ` × (C r {0}) ×C C−` 3 [z, w, v] = [zλ, wλ, λ v] CP1 A holomorphic section of this bundle can be written as ` X j `−j v = f(z, w) = ajz w . j=0 Thus, the dimension of the space of holomorphic sections is ` + 1. A generic holomorphic section has ` simple zeros, of which each contributes 1 to the intersection number of this section with the zero section. Thus: ` = the self intersection number of the zero section. The symplectic picture. Recall that the symplectic Delzant construction and the complex Delzant (Audin!) con- struction yield isomorphic manifolds. (We stated this without proof.) [Picture: the bottom part of a Hirzebruch trapezoid with right edge of slope −1/`.] We found a neighborhood of the preimage of the bottom edge that is isomorphict to O(`). Examples: [Picture of Hirzebruch trapezoid with each edge labelled by the self intersection of its preimage: from the right edge, counterclockwise: 0; −`; 0; `.] 1 2 [Same picture, with the bottom left corner chopped off. Labels of the edges, from the one on the right, counterclockwise: 0, −`, −1, −1, ` − 1. by the self intersection of its preimage: bottom edge – `; top edge – −`; left edge – 0; right edge – 0.] A concrete example: The tautological bundle over CP1, which we denote Taut, is O(−1). Let us see how. The fiber of Taut over [z, w] is C · (z, w). We have an isomorphism 2 × ∼ C r {0} ×C C1 = Taut via [z, w, v] 7→ the vector v · (z, w) in the fibre over [z, w]. (This is well defined: [zλ, wλ, λ−1v] maps to the same thing.) The map [z, w, v] 7→ v · (z, w) from Taut to C2 is a diffeomorphism above C2 r {0} and ∼ 1 2 takes the zero section (= CP ) to the origin. Through this map, Taut is the blowup of C at the origin. [Picture: a region bounded on the left by a vertical line, on the bottom by a horizontal line, and on the right by a line of slope 1.] Another concrete example: The tangent bundle of CP1 is O(2). Let us see how. Note that 1 2 T[z,w]CP = T(z,w)C /C · (z, w) 3 (u1, u2) O 2 = T(zλ,wλ)C /C · (zλ, wλ) 3 (λu1, λu2). Define 2 1 × C r {0} ×C C−2 → T CP by ( 2 (0, v/z) in T(z,w)C /C · (z, w) if z 6= 0 [z, w, v] 7→ 2 (−v/w, 0) in T(z,w)C /C · (z, w) if w 6= 0. Note: • This is unambiguous if both z and w are 6= 0; • This gets multiplied by λ (as it should) if we replace (z, w, v) by (zλ, wλ, λ2v). Symplectic picture: [Piece of a region bounded on the left by a vertical line, on the bottom by a horizontal line, and on the right by a line of slope = −1/2. Only shade a neighborhood of the “floor”.] or [similar, but the left line has slope 1 and the right line has slope −1.] Vectors along the edges that emanate from a corner are the weight vectors for the torus action. The standard S1 action on CP1 has weights 1 at the south pole and −1 at the north pole (with appropriate sign conventions). On the tangent bundle, at the origin of the tangent at the south and north poles, the weights are 1 and −1 respectively along the tangent to the zero section, and they are the same – 1 and −1 – along the fibre. (Indeed, 3 the tangent to the fibre at its origin is just another copy of the tangent to the zero section at that point.) There is also the fibrewise S1 action; its weights are +1 on each fiber and 0 in the tangent directions. Symplectic toric blowup in (real) dimension 4 (Terminology: “toric surface”.) [Picture of “corner chopping” of size δ.] Topologically: remove an open ball and collapse the boundary. Remarks. Under this operation, (1) The area decreases by δ2/2. (2) The perimeter decreases by δ. Before the blowup, there must have been two adjacent edges of lengths > δ. It follows that 1 the perimeter after the blowup ≥ the perimeter before the blowup. 2 (3) The length of an edge before the blowup is a nonnegative integer linear combination of the lengths of edges after the blowup. Corollary: (Lemma 2.16 of the paper by Karshon-Kessler-Pinsonnault) Let ∆0,..., ∆s be the polygons obtained by a sequence of blowups of sizes δ1, . , δs. P 1 2 (1) area ∆s = area ∆0 − δ . 2 i P (2) perimeter ∆s = perimeter ∆0 − δi. s (3) For every i, δi < perimeter ∆i < 2 perimeter ∆s. (4) The length of every edge of ∆i is a nonnegative integer combination of the lengths of the edges of ∆s. (Didn’t show in class today; will discuss later: Let M = M∆s and ω = ω∆s . + – Item (3), plus a lemma that uses the fact that b2 (M) = 1, give an a-priori bound on the numbers δi in terms of the symplectic manifold (M, ω). – This and the fact that δi is the area of the exceptional divisor imply that there exists a finite set D(M, ω) such that every δi must be in this set. – If ∆0 is a Hirzebruch trapezoid corresponding to integers a and b and slope `, from the numbers δi and from (M, ω) we can determine a + b and a · b and hence determine a and b. Given a and b, there are finitely many posisbilities for the slope `. This gives “finiteness”.) (A note made while the notes are being typed in the summer: recently, Dusa McDuff has a proof of “finiteness” of toric actions in *any* dimensions.) Blowup - topologically: [Picture: region bounded by vertical line on left, segment of slope -1 on left/bottom, horizonal line on bottom.] 4 A neighborhood of the “exceptional divisor” is O(−1) (same picture). It is a disc bundle over CP1. This is the same as O(1) with the orientation reversed. In turn, O(1) can be identified with CP2 r {point}. So ... topologically, the blowup of M = CP2]M, where CP2 denotes CP2 with its orientation reversed and where ] denotes connected sum. This 1 = disc bundle over CP ∪ (M r {point}) , glued along an annulus bundle over CP1 (which is a neighbourhood of S3 in C2). Thus, Mf := the blowup of M = U ∪ V where 1 3 U ≈ CP ,V ≈ M r {point} ,U ∩ V ≈ S . By Van Kampen: π1(U ∪ V ) = π1(U) ?π1(U∩V ) π1(V ) is the amalgamated free product = (trivial) ?trivial π1(M r {point}) = π1(M). Thus, π1(Mf) = π1(M). By Mayer Vietoris: / Hn(U ∩ V ) / Hn(U) ⊕ Hn(V ) / Hn(U ∪ V ) BCED / Hn−1(U ∩ V ) / ... GF When n = 2,@A we get 3 1 3 H2(S ) −−−→ H2(CP ) ⊕ H2(M r {point}) −−−→ H2(Mf) −−−→ H1(S ) . | {z } | {z } | {z } | {z } =0 =Z =H2(M) =0 We deduce that H2(Mf) = Z ⊕ H2(M) where the inclusion H2(M) ,→ H2(Mf) is obtained by choosing cycles away from the surgery ∼ 1 and where the new generator is the exceptional divisor = [CP ]. Symplectic cutting [Picture: momentum map for an S1 action ....] Topologically, 1 (1) Mcut = {Φ > 0} t {Φ = 0}/S . 5 The first term on the right, {Φ > 0}, with its symplectic structure restricted from M, is 1 a symplectic open subset of Mcut. The second term on the right, {Φ = 0}/S , with the reduced symplectic structure induced from M, is a symplectic submanifold of Mcut. – What is the smooth structure on this disjoint union? Take the S1 momentum map M × C / R (m, z) / Φ(m) − |z|2/2. Set 2 1 Mcut = (M × C)red = Φ(m) = |z| /2 /S . We will now show how to express this as the disjoint union of an open subset and a codimension two submanifold as in (1), with their symplectic structures induced from M as required. We start with the inclusion in Mcut of the open subset of M. {Φ(m) = |z|2/2} 6 ei+ nnn nnn nnn nnn {Φ > 0} / Mcut i+ i+ p m / [m, 2Φ(m)]. We have ∗ ∗ i+ωcut = ei+(ω ⊕ ωC) = ω. We now show the inclusion in Mcut of the symplectic reduction of M. eired {Φ = 0} / {Φ(m) = |z|2/2} m7→(m,0) π πcut Mred / Mcut. ired We have ∗ ∗ ∗ ∗ π iredωcut = eiredπcutωcut ∗ = eired(ω ⊕ ωC) from the definition of the cut space as a reduction of M × C = i∗ω ∗ = π ωred from the definition of ωred as a reduction of M. A symplectic blowup is a special case of symplectic cutting. 6 Reminders on the first Chern class. Let L → Σ be a complex line bundle over an oriented closed two-manifold. Then 2 c1(L) = its Euler class ∈ H (Σ) = the Poincare dual of the zero set of a generic section or trivialize the bundle over every half of the base and take the degree of the transition function.
Load more

Recommended publications
  • MIPS IV Instruction Set
    MIPS IV Instruction Set Revision 3.2 September, 1995 Charles Price MIPS Technologies, Inc. All Right Reserved RESTRICTED RIGHTS LEGEND Use, duplication, or disclosure of the technical data contained in this document by the Government is subject to restrictions as set forth in subdivision (c) (1) (ii) of the Rights in Technical Data and Computer Software clause at DFARS 52.227-7013 and / or in similar or successor clauses in the FAR, or in the DOD or NASA FAR Supplement. Unpublished rights reserved under the Copyright Laws of the United States. Contractor / manufacturer is MIPS Technologies, Inc., 2011 N. Shoreline Blvd., Mountain View, CA 94039-7311. R2000, R3000, R6000, R4000, R4400, R4200, R8000, R4300 and R10000 are trademarks of MIPS Technologies, Inc. MIPS and R3000 are registered trademarks of MIPS Technologies, Inc. The information in this document is preliminary and subject to change without notice. MIPS Technologies, Inc. (MTI) reserves the right to change any portion of the product described herein to improve function or design. MTI does not assume liability arising out of the application or use of any product or circuit described herein. Information on MIPS products is available electronically: (a) Through the World Wide Web. Point your WWW client to: http://www.mips.com (b) Through ftp from the internet site “sgigate.sgi.com”. Login as “ftp” or “anonymous” and then cd to the directory “pub/doc”. (c) Through an automated FAX service: Inside the USA toll free: (800) 446-6477 (800-IGO-MIPS) Outside the USA: (415) 688-4321 (call from a FAX machine) MIPS Technologies, Inc.
    [Show full text]
  • Stability of Tautological Bundles on Symmetric Products of Curves
    STABILITY OF TAUTOLOGICAL BUNDLES ON SYMMETRIC PRODUCTS OF CURVES ANDREAS KRUG Abstract. We prove that, if C is a smooth projective curve over the complex numbers, and E is a stable vector bundle on C whose slope does not lie in the interval [−1, n − 1], then the associated tautological bundle E[n] on the symmetric product C(n) is again stable. Also, if E is semi-stable and its slope does not lie in the interval (−1, n − 1), then E[n] is semi-stable. Introduction Given a smooth projective curve C over the complex numbers, there is an interesting series of related higher-dimensional smooth projective varieties, namely the symmetric products C(n). For every vector bundle E on C of rank r, there is a naturally associated vector bundle E[n] of rank rn on the symmetric product C(n), called tautological or secant bundle. These tautological bundles carry important geometric information. For example, k-very ampleness of line bundles can be expressed in terms of the associated tautological bundles, and these bundles play an important role in the proof of the gonality conjecture of Ein and Lazarsfeld [EL15]. Tautological bundles on symmetric products of curves have been studied since the 1960s [Sch61, Sch64, Mat65], but there are still new results about these bundles discovered nowadays; see, for example, [Wan16, MOP17, BD18]. A natural problem is to decide when a tautological bundle is stable. Here, stability means (n−1) (n) slope stability with respect to the ample class Hn that is represented by C +x ⊂ C for any x ∈ C; see Subsection 1.3 for details.
    [Show full text]
  • Characteristic Classes and K-Theory Oscar Randal-Williams
    Characteristic classes and K-theory Oscar Randal-Williams https://www.dpmms.cam.ac.uk/∼or257/teaching/notes/Kthy.pdf 1 Vector bundles 1 1.1 Vector bundles . 1 1.2 Inner products . 5 1.3 Embedding into trivial bundles . 6 1.4 Classification and concordance . 7 1.5 Clutching . 8 2 Characteristic classes 10 2.1 Recollections on Thom and Euler classes . 10 2.2 The projective bundle formula . 12 2.3 Chern classes . 14 2.4 Stiefel–Whitney classes . 16 2.5 Pontrjagin classes . 17 2.6 The splitting principle . 17 2.7 The Euler class revisited . 18 2.8 Examples . 18 2.9 Some tangent bundles . 20 2.10 Nonimmersions . 21 3 K-theory 23 3.1 The functor K ................................. 23 3.2 The fundamental product theorem . 26 3.3 Bott periodicity and the cohomological structure of K-theory . 28 3.4 The Mayer–Vietoris sequence . 36 3.5 The Fundamental Product Theorem for K−1 . 36 3.6 K-theory and degree . 38 4 Further structure of K-theory 39 4.1 The yoga of symmetric polynomials . 39 4.2 The Chern character . 41 n 4.3 K-theory of CP and the projective bundle formula . 44 4.4 K-theory Chern classes and exterior powers . 46 4.5 The K-theory Thom isomorphism, Euler class, and Gysin sequence . 47 n 4.6 K-theory of RP ................................ 49 4.7 Adams operations . 51 4.8 The Hopf invariant . 53 4.9 Correction classes . 55 4.10 Gysin maps and topological Grothendieck–Riemann–Roch . 58 Last updated May 22, 2018.
    [Show full text]
  • 2019 NFCA Texas High School Leadoff Classic Main Bracket Results
    2019 NFCA Texas High School Leadoff Classic Main Bracket Results Bryan College Station BHS CSHS 1 Bryan College Station 17 Thu 3pm Thu 3pm Cedar Creek BHS CSHS EP Eastlake SA Brandeis 49 Brandeis College Station 57 Cy Woods 1 BHS Thu 5pm Thu 5pm CSHS 2 18 Thu 1pm Brandeis Robinson Thu 1pm EP Montwood BHS CSHS Robinson 11 Huntsville 3 89 Klein Splendora 93 Splendora 9 BHS Fri 2pm Fri 2pm CSHS 3 Fredericksburg Splendora 19 Thu 9am Thu 9am Fredericksburg 6 VET1 VET2 Belton 6 Grapevine 50 58 Richmond Foster 11 BHS Thu 5pm Klein Splendora Thu 5pm CSHS 4 20 Thu 11am Klein Richmond Foster Thu 11am Klein 11 BHS CSHS Rockwall 2 SA Southwest 9 97 SA Southwest Cedar Ridge 99 Cy Ranch 7 VET1 Fri 4pm Fri 4pm CP3 5 Southwest Cy Ranch 21 Thu 11am Thu 1pm Temple 5 VET1 CP3 Plano East 5 Clear Springs 2 51 Southwest Alvin 59 Alvin VET1 Thu 3pm Thu 5pm CP3 6 22 Thu 9am Vandegrift Alvin Thu 3pm Vandegrift 5 BHS CSHS Lufkin San Marcos 5 90 94 Flower Mound 0 VET2 Fri 12pm Southwest Cedar Ridge Fri 12pm CP4 7 San Marcos Cedar Ridge 23 Thu 9am Thu 1pm Magnolia West 3 VET2 CHAMPIONSHIP CP4 RR Cedar Ridge 12 McKinney Boyd 3 52 Game 192 60 SA Johnson VET2 Thu 3pm San Marcos BHS Cedar Ridge Thu 5 pm CP4 8 4:00 PM 24 Thu 11am M. Boyd BHS 5 10 BHS Johnson Thu 3pm Manvel 0 129 Southwest vs. Cedar Ridge 130 Kingwood Park Bellaire 3 Sat 10am Sat 8am Deer Park VET4 BRAC-BB 9 Cedar Park Deer Park 25 Thu 11am Thu 1pm Cedar Park 5 VET4 BRAC-BB Leander Clements 13 53 Cedar Park MacArthur 61 SA MacArthur VET4 Thu 3pm Thu 5pm BRAC-BB 10 26 Thu 9am Clements MacArthur Thu 3pm Waco University 0 BHS CSHS Tomball Memorial Cy Fair 4 91 Friendswood Woodlands 95 Woodlands 2 VET5 Fri 8am Fri 8am BRAC-YS 11 San Benito Woodlands 27 Thu 11am Thu 1pm San Benito 10 VET5 BRAC-YS SA Holmes 1 Friendswood 10 54 62 Santa Fe VET5 Thu 3pm Friendswood Woodlands Thu 5pm BRAC-YS 12 28 Thu 9am Friendswood Santa Fe Thu 3pm Henderson 0 VET1 VET2 Lake Travis Ridge Point 16 98 100 B.
    [Show full text]
  • Classification Made Easy Class 1
    Classification Made Easy Class 1 (CP1) The most severely disabled athletes belong to this classification. These athletes are dependent on a power wheelchair or assistance for mobility. They have severe limitation in both the arms and the legs and have very poor trunk control. Sports Available: • Race Runner (RR1) – using the Race Runner frame to run, track events include 100m, 200m and 400m. • Boccia o Boccia Class 1 (BC1) – players who fit into this category can throw the ball onto the court or a CP2 Lower who chooses to push the ball with the foot. Each BC1 athlete has a sport assistant on court with them. o Boccia Class 3 (BC3) – players who fit into this category cannot throw the ball onto the court and have no sustained grasp or release action. They will use a “chute” or “ramp” with the help from their sport assistant to propel the ball. They may use head or arm pointers to hold and release the ball. Players with a impairment of a non cerebral origin, severely affecting all four limbs, are included in this class. Class 2 (CP2) These athletes have poor strength or control all limbs but are able to propel a wheelchair. Some Class 2 athletes can walk but can never run functionally. The class 2 athletes can throw a ball but demonstrates poor grasp and release. Sports Available: • Race Runner (RR2) - using the Race Runner frame to run, track events include 100m, 200m and 400m. • Boccia o Boccia Class 2 (BC2) – players can throw the ball into the court consistently and do not need on court assistance.
    [Show full text]
  • On Positivity and Base Loci of Vector Bundles
    ON POSITIVITY AND BASE LOCI OF VECTOR BUNDLES THOMAS BAUER, SÁNDOR J KOVÁCS, ALEX KÜRONYA, ERNESTO CARLO MISTRETTA, TOMASZ SZEMBERG, STEFANO URBINATI 1. INTRODUCTION The aim of this note is to shed some light on the relationships among some notions of posi- tivity for vector bundles that arose in recent decades. Positivity properties of line bundles have long played a major role in projective geometry; they have once again become a center of attention recently, mainly in relation with advances in birational geometry, especially in the framework of the Minimal Model Program. Positivity of line bundles has often been studied in conjunction with numerical invariants and various kinds of asymptotic base loci (see for example [ELMNP06] and [BDPP13]). At the same time, many positivity notions have been introduced for vector bundles of higher rank, generalizing some of the properties that hold for line bundles. While the situation in rank one is well-understood, at least as far as the interdepencies between the various positivity concepts is concerned, we are quite far from an analogous state of affairs for vector bundles in general. In an attempt to generalize bigness for the higher rank case, some positivity properties have been put forward by Viehweg (in the study of fibrations in curves, [Vie83]), and Miyaoka (in the context of surfaces, [Miy83]), and are known to be different from the generalization given by using the tautological line bundle on the projectivization of the considered vector bundle (cf. [Laz04]). The differences between the various definitions of bigness are already present in the works of Lang concerning the Green-Griffiths conjecture (see [Lan86]).
    [Show full text]
  • The US Livestock Industry
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1983 The SU livestock industry: an evaluation of the adequacy and relevance of three models of consumer and producer behavior Stephen Stanley Steyn Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Agricultural and Resource Economics Commons, and the Agricultural Economics Commons Recommended Citation Steyn, Stephen Stanley, "The SU livestock industry: an evaluation of the adequacy and relevance of three models of consumer and producer behavior " (1983). Retrospective Theses and Dissertations. 7653. https://lib.dr.iastate.edu/rtd/7653 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This reproduction was made from a copy of a document sent to us for microfilming. While the most advanced technology has been used to photograph and reproduce this document, the quality of the reproduction is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help clarify markings or notations which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure complete continuity.
    [Show full text]
  • Manifolds of Low Dimension with Trivial Canonical Bundle in Grassmannians Vladimiro Benedetti
    Manifolds of low dimension with trivial canonical bundle in Grassmannians Vladimiro Benedetti To cite this version: Vladimiro Benedetti. Manifolds of low dimension with trivial canonical bundle in Grassmannians. Mathematische Zeitschrift, Springer, 2018. hal-01362172v3 HAL Id: hal-01362172 https://hal.archives-ouvertes.fr/hal-01362172v3 Submitted on 26 May 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Manifolds of low dimension with trivial canonical bundle in Grassmannians Vladimiro Benedetti∗ May 27, 2017 Abstract We study fourfolds with trivial canonical bundle which are zero loci of sections of homogeneous, completely reducible bundles over ordinary and classical complex Grassmannians. We prove that the only hyper-K¨ahler fourfolds among them are the example of Beauville and Donagi, and the example of Debarre and Voisin. In doing so, we give a complete classifi- cation of those varieties. We include also the analogous classification for surfaces and threefolds. Contents 1 Introduction 1 2 Preliminaries 3 3 FourfoldsinordinaryGrassmannians 5 4 Fourfolds in classical Grassmannians 13 5 Thecasesofdimensionstwoandthree 23 A Euler characteristic 25 B Tables 30 1 Introduction In Complex Geometry there are interesting connections between special vari- eties and homogeneous spaces.
    [Show full text]
  • An Advanced System of the Mitochondrial Processing Peptidase and Core Protein Family in Trypanosoma Brucei and Multiple Origins of the Core I Subunit in Eukaryotes
    GBE An Advanced System of the Mitochondrial Processing Peptidase and Core Protein Family in Trypanosoma brucei and Multiple Origins of the Core I Subunit in Eukaryotes Jan Mach1, Pavel Poliak2,3,AnnaMatusˇkova´ 4,Vojteˇch Zˇa´rsky´1,Jirˇı´ Janata4, Julius Lukesˇ2,3,and Jan Tachezy1,* 1Department of Parasitology, Faculty of Science, Charles University, Prague, Czech Republic 2Institute of Parasitology, Biology Centre, Czech Academy of Sciences, Prague, Czech Republic 3Faculty of Science, University of South Bohemia, Cˇ eske´ Budeˇjovice, Budweis, Czech Republic 4Institute of Microbiology, Czech Academy of Sciences, Prague, Czech Republic *Corresponding author: E-mail: [email protected]. Accepted: April 2, 2013 Abstract Mitochondrial processing peptidase (MPP) consists of a and b subunits that catalyze the cleavage of N-terminal mitochondrial- targeting sequences (N-MTSs) and deliver preproteins to the mitochondria. In plants, both MPP subunits are associated with the respiratory complex bc1, which has been proposed to represent an ancestral form. Subsequent duplication of MPP subunits resulted in separate sets of genes encoding soluble MPP in the matrix and core proteins (cp1 and cp2) of the membrane-embedded bc1 complex. As only a-MPP was duplicated in Neurospora,itssingleb–MPP functions in both MPP and bc1 complexes. Herein, we investigated the MPP/core protein family and N-MTSs in the kinetoplastid Trypanosoma brucei, which is often considered one of the most ancient eukaryotes. Analysis of N-MTSs predicted in 336 mitochondrial proteins showed that trypanosomal N-MTSs were comparable with N-MTSs from other organisms. N-MTS cleavage is mediated by a standard heterodimeric MPP, which is present in the matrix of procyclic and bloodstream trypanosomes, and its expression is essential for the parasite.
    [Show full text]
  • Technology Enhancement and Ethics in the Paralympic Games
    UNIVERSITY OF PELOPONNESE FACULTY OF HUMAN MOVEMENT AND QUALITY OF LIFE SCIENCES DEPARTMENT OF SPORTS ORGANIZATION AND MANAGEMENT MASTER’S THESIS “OLYMPIC STUDIES, OLYMPIC EDUCATION, ORGANIZATION AND MANAGEMENT OF OLYMPIC EVENTS” Technology Enhancement and Ethics In the Paralympic Games Stavroula Bourna Sparta 2016 i TECHNOLOGY ENHANCEMENT AND ETHICS IN THE PARALYMPIC GAMES By Stavroula Bourna MASTER Thesis submitted to the professorial body for the partial fulfillment of obligations for the awarding of a post-graduate title in the Post-graduate Programme, "Organization and Management of Olympic Events" of the University of the Peloponnese, in the branch "Olympic Education" Sparta 2016 Approved by the Professor body: 1st Supervisor: Konstantinos Georgiadis Prof. UNIVERSITY OF PELOPONNESE, GREECE 2nd Supervisor: Konstantinos Mountakis Prof. UNIVERSITY OF PELOPONNESE, GREECE 3rd Supervisor: Paraskevi Lioumpi, Prof., GREECE ii Copyright © Stavroula Bourna, 2016 All rights reserved. The copying, storage and forwarding of the present work, either complete or in part, for commercial profit, is forbidden. The copying, storage and forwarding for non profit-making, educational or research purposes is allowed under the condition that the source of this information must be mentioned and the present stipulations be adhered to. Requests concerning the use of this work for profit-making purposes must be addressed to the author. The views and conclusions expressed in the present work are those of the writer and should not be interpreted as representing the official views of the Department of Sports’ Organization and Management of the University of the Peloponnese. iii ABSTRACT Stavroula Bourna: Technology Enhancement and Ethics in the Paralympic Games (Under the supervision of Konstantinos Georgiadis, Professor) The aim of the present thesis is to present how the new technological advances can affect the performance of the athletes in the Paralympic Games.
    [Show full text]
  • The Chern Characteristic Classes
    The Chern characteristic classes Y. X. Zhao I. FUNDAMENTAL BUNDLE OF QUANTUM MECHANICS Consider a nD Hilbert space H. Qauntum states are normalized vectors in H up to phase factors. Therefore, more exactly a quantum state j i should be described by the density matrix, the projector to the 1D subspace generated by j i, P = j ih j: (I.1) n−1 All quantum states P comprise the projective space P H of H, and pH ≈ CP . If we exclude zero point from H, there is a natural projection from H − f0g to P H, π : H − f0g ! P H: (I.2) On the other hand, there is a tautological line bundle T (P H) over over P H, where over the point P the fiber T is the complex line generated by j i. Therefore, there is the pullback line bundle π∗L(P H) over H − f0g and the line bundle morphism, π~ π∗T (P H) T (P H) π H − f0g P H . Then, a bundle of Hilbert spaces E over a base space B, which may be a parameter space, such as momentum space, of a quantum system, or a phase space. We can repeat the above construction for a single Hilbert space to the vector bundle E with zero section 0B excluded. We obtain the projective bundle PEconsisting of points (P x ; x) (I.3) where j xi 2 Hx; x 2 B: (I.4) 2 Obviously, PE is a bundle over B, p : PE ! B; (I.5) n−1 where each fiber (PE)x ≈ CP .
    [Show full text]
  • WITT GROUPS of GRASSMANN VARIETIES Contents Introduction 1
    WITT GROUPS OF GRASSMANN VARIETIES PAUL BALMER AND BAPTISTE CALMES` Abstract. We compute the total Witt groups of (split) Grassmann varieties, over any regular base X. The answer is a free module over the total Witt ring of X. We provide an explicit basis for this free module, which is indexed by a special class of Young diagrams, that we call even Young diagrams. Contents Introduction 1 1. Combinatorics of Grassmann and flag varieties 4 2. Even Young diagrams 7 3. Generators of the total Witt group 12 4. Cellular decomposition 15 5. Push-forward, pull-back and connecting homomorphism 19 6. Main result 21 Appendix A. Total Witt group 27 References 27 Introduction At first glance, it might be surprising for the non-specialist that more than thirty years after the definition of the Witt group of a scheme, by Knebusch [13], the Witt group of such a classical variety as a Grassmannian has not been computed yet. This is especially striking since analogous results for ordinary cohomologies, for K-theory and for Chow groups have been settled a long time ago. The goal of this article is to explain how Witt groups differ from these sister theories and to prove the following: Main Theorem (See Thm. 6.1). Let X be a regular noetherian and separated 1 scheme over Z[ 2 ], of finite Krull dimension. Let 0 <d<n be integers and let GrX (d, n) be the Grassmannian of d-dimensional subbundles of the trivial n- n dimensional vector bundle V = OX over X. (More generally, we treat any vector bundle V admitting a complete flag of subbundles.) Then the total Witt group of GrX (d, n) is a free graded module over the total Witt group of X with an explicit basis indexed by so-called “even” Young diagrams.
    [Show full text]