Index of uploads 2012 11

SECTION 5 Index of Expressions The numbers refer to the unit in which the idiom is taught. about It'll be all right on the night. 50 You've got to keep your wits about you. 47 It's all hands on deck. 6 accounts It's all in your mind. 39 By all accounts he's pretty good. 86 It's all over now, so go home. 86 acquainted It's all over your face. 26 I'm not very well acquainted with it. 60 It's all up in the air. 51 action It's all yours. 86 Actions speak louder than words. 113 Ifs been all go in the office today. 86 add It's been difficult all along the line. 44 It just doesn't add up. 63 It's open all year round. 86 That added more fuel to the fire. 28 N0t a^ ajj gg To add insult to injury, they didn't even say she.s on the g0 all day 83 thank you. 49, 109 There were flve of us all told 86 advantage They stopped aU 0f a sudden. 86 He's trying to take advantage of you. 54, 68 Wre all in the same boat 6 afford When all's said and done. 86 ^1 can't afford more than a week off. 1 you can>t wln them aU 50 You mustn't put all your eggs in one basket. 74 He s a man after my own heart. 34 . TT,n , I knew it all along. 86 We meet up now and again. 82 ., . . „ „ ambition You can say that again. 48 , . , , Her burning ambition was to be an actress. 28, age The golden age of drama. 38 It's unusual in this day and age. 82 ancient . " '"' That's ancient history now. 61 He's a breath of fresh air here. 52 and .(see Pa£es 188 ~ 191^ I felt as if I was walking on air. 66 angling It's all up in the air. 51 He's ang11^ for something. 29 We need to clear the air. 72 another aU It's been one thing after another. 83 By all accounts he's pretty good. 86 Tomorrow's another day. 50, 82 By all means help yourself. 86 You've got another think coming. 81 He was drunk, and to cap it all, he'd been ants drinking my wine. 109 He's got ants in his pants. 14 I knew it all along. 86 anything I want to get away from it all. 59 Don't take anything for granted. 47 I won't, if it's all the same to you. 86 He'll do anything for a quiet life. 70 I'll tell you once and for all. 40 apart I'm all at sea without her. 45 They're poles apart in sport. 75 I'm all fingers and thumbs. 27 arm I'm all for doing it now. 86 I'd give my right arm for that. 16 It was a good day all in all. 86 OK, twist my arm. 68 It wasn't all it's cracked up to be. 57 Private education costs an arm and a leg. 16, 64 245 arms Don't take your eye off the ball. 62 They are up in arms about it. 66 He's on the ball. 41 around I want to start the ball rolling. 78 He's always throwing his weight around. 68 It's a whole new ball game. 41, 75 arrive The ball's in their court. 41 He thinks he's really arrived. 4 They won't play ball. 41 aside They won't run with the ball. 41 I try to put a bit of money aside each month. 64 balloon asleep The joke went down like a lead balloon. 38 He's fast asleep. 76, 111 bang Sorry, I was half asleep. 76 You're banging your head against a brick wall. The baby's sound asleep. Ill 18,49 awake baptism It's late but I'm wide awake. Ill It was a baptism of fire. 28, 55 away bargain I want to get away from it all. 59 j picked up a bargain yesterday. 64 When the cat's away, the mice will play. 19 It was harder than I had bargained for. 81 awful bark I can't tell you - it's too awful for words. 113 His bark ls worse than his bite. 50 baby You're barking up the wrong tree. 62 Don't throw the baby out with the bath water. 47 barrel She's the baby of the family. 56 You're scraping the bottom of the barrel. 57 bachelor base Paul's a confirmed bachelor. 67 Fm going to touch base Ge ng the most out of ABA Therapy – a Workshop for Families and Therapists ] What every family who has a child with Au sm needs to know Hear the ps and strategies used by those who have been through ABA programs and succeeded! ABA Therapy is a very intensive and expensive therapy Ensure you get the most value you can from your child’s ABA program and therapists Vicky Sal s has been working as Psychologist specialising in ABA Therapy and as a Senior ABA Therapist and trainer for the past 10 years In this workshop, Vicky will present the keys to running a successful ABA Program for your child with Au sm Find out what has worked for children who have excelled through their ABA Therapy program and take home many prac cal strategies that will benefit your child Saturday the 2nd of March, 2013, 10:30 am through until 2:00pm (Gourmet Lunch Included) Wheelers Hill Library Conference & Meeting Room: 860 Ferntree Gully Road, Wheelers Hill $120 per person (GST inclusive) To Register please contact Vicky Saltis (Psychologist) at Play Psychology: 0421 865 410 - vicky@playpsychology.com.au Or to download an application form please visit: www.playpsychology.com.au  6 THE END OF ANGEVIN BRITTANY, 1186±1203 The death of Duke Geoffrey brought yet another transformation to the Angevin regime in Brittany, introducing its ®nal phase. The new situation was largely a return to that prevailing between 1156 and 1166; a native ruler was allowed to govern with minimal interference provided his (now her) loyalty to the Angevin lord was assured. This chapter is divided into two parts. The ®rst will discuss the government of Brittany under the last dukes to be subject to Angevin rule, Duchess Constance and her son, Duke Arthur. The second part will proceed by way of a narrative account of political relations between the Angevin kings and the province of Brittany to 1203. As a general principle, after 1186, the Angevin kings permitted the dukes to rule Brittany in their own right. Angevin sovereignty did not extend to direct government, as it had between 1166 and 1181. On the other hand, Angevin sovereignty was vigorously asserted in speci®c acts of royal intervention. In 1187, Henry II entered Brittany, led a military campaign in the far western barony of Le  on and, after this show of force, according to one source took oaths of allegiance from the Breton magnates. In 1196, Richard I sought the custody of Arthur, the young heir to Brittany, and when the Bretons refused, invaded the duchy while Constance was held captive. Apart from these episodes, Henry II and Richard I in turn were content to allow Duchess Constance to rule Brittany without interference. King John seems to have followed the same policy after making peace with Constance and Arthur in September 1199. As his father had exercised his right to give Constance in marriage, so did John, marrying her to the loyal Guy de Thouars. From then until 1203, John allowed ®rst Constance, then Arthur, to rule without inter- ference. Some change is indicated, though, by the fact that in June 1200 John issued orders directly to vicecomites in Guingamp, Lamballe 146 and Dinan. 1 This may have been justi®ed under the terms of the peace settlement, which are unfortunately unknown. the seneschal of brittany With the exception of Ralph de Fouge Á res, the seneschal of Brittany (with or without this title) had been Henry II's deputy in Brittany at various times since 1158. 2 For this reason, I have included this discussion of the institution in the period after 1186 in the context of the role of the Angevin kings, rather than of the dukes' internal government. Roger of Howden's account of the rebellion of Guihomar and Harvey de Le  on in the autumn of 1186 includes the detail that the custodians of the castles seized had been appointed by Ralph de Fouge Á res on the orders of Henry II. 3 From this it can be inferred that, in the immediate aftermath of Geoffrey's death, the king recognised Ralph's position as `seneschal of Brittany' and issued royal writs to him, but this state of affairs was not to last. Two seneschals of Brittany are recorded for the period 1187±1203: Maurice de Craon and Alan de Dinan, the lord of Becherel, although it is impossible to determine when each held the of®ce. 4 What is signi®cant is that neither was a `foreigner' to Brittany. Alan de Dinan was a native, but Maurice de Craon also had strong Breton connections. Jean-Claude Meuret has demonstrated how, in the eleventh and The Invariance of the Index of Elliptic Operators Constantine Caramanis ∗ Harvard University April 5, 1999 Abstract In 1963 Atiyah and Singer proved the famous Atiyah-Singer Index Theorem, which states, among other things, that the space of elliptic pseudodifferential operators is such that the collection of operators with any given index forms a connected subset. Contained in this statement is the somewhat more specialized claim that the index of an elliptic operator must be invariant under sufficiently small perturbations. By developing the machinery of distributions and in particular Sobolev spaces, this paper addresses this more specific part of the famous Theorem from a completely analytic approach. We first prove the regularity of elliptic operators, then the finite dimensionality of the kernel and cokernel, and finally the invariance of the index under small perturbations. ∗ cmcaram@fas.harvard.edu 1 Acknowledgements I would like to express my thanks to a number of individuals for their con- tributions to this thesis, and to my development as a student of mathematics. First, I would like to thank Professor Clifford Taubes for advising my thesis, and for the many hours he spent providing both guidance and encouragement. I am also indebted to him for helping me realize that there is no analysis without geometry. I would also like to thank Spiro Karigiannis for his very helpful criti- cal reading of the manuscript, and Samuel Grushevsky and Greg Landweber for insightful guidance along the way. I would also like to thank Professor Kamal Khuri-Makdisi who instilled in me a love for mathematics. Studying with him has had a lasting influence on my thinking. If not for his guidance, I can hardly guess where in the Harvard world I would be today. Along those lines, I owe both Professor Dimitri Bertsekas and Professor Roger Brockett thanks for all their advice over the past 4 years. Finally, but certainly not least of all, I would like to thank Nikhil Wagle, Alli- son Rumsey, Sanjay Menon, Michael Emanuel, Thomas Knox, Demian Ordway, and Benjamin Stephens for the help and support, mathematical or other, that they have provided during my tenure at Harvard in general, and during the re- searching and writing of this thesis in particular. April 5 th , 1999 Lowell House, I-31 Constantine Caramanis 2 Contents 1 Introduction 4 2 Euclidean Space 6 2.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Definition of Sobolev Spaces . . . . . . . . . . . . . . . . . 7 2.1.2 The Rellich Lemma . . . . . . . . . . . . . . . . . . . . . 11 2.1.3 Basic Sobolev Elliptic Estimate . . . . . . . . . . . . . . . 12 2.2 Elliptic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Local Regularity of Elliptic Operators . . . . . . . . . . . 16 2.2.2 Kernel and Cokernel of Elliptic Operators . . . . . . . . . 19 3 Compact Manifolds 23 3.1 Patching Up the Local Constructions . . . . . . . . . . . . . . . . 23 3.2 Differences from Euclidean Space . . . . . . . . . . . . . . . . . . 24 3.2.1 Connections and the Covariant Derivative . . . . . . . . . 25 3.2.2 The Riemannian Metric and Inner Products . . . . . . . . 27 3.3 Proof of the Invariance of the Index . . . . . . . . . . . . . . . . 32 4 Example: The Torus 36 A Elliptic Operators and Riemann-Roch 38 B An Alternate Proof of Elliptic Regularity 39 3 1 Introduction This paper defines, and then examines some properties of a certain 5.10 Polynomial Approximation from Chebyshev Coefficients 197 Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). 5.10 Polynomial Approximation from Chebyshev Coefficients You may well ask after reading the preceding two sections, “Must I store and evaluate my Chebyshev approximation as an array of Chebyshev coefficients for a transformed variable y? Can’t I convert the c k ’s into actual polynomial coefficients in the original variable x and have an approximation of the following form?” f(x) ≈ m−1  k=0 g k x k (5.10.1) Yes, you can do this (and we will give you the algorithm to do it), but we caution you against it: Evaluating equation (5.10.1), where the coefficient g’s reflect an underlying Chebyshev approximation, usually requires more significant figures than evaluation of the Chebyshev sum directly (as by chebev). This is because the Chebyshev polynomials themselves exhibit a rather delicate cancellation: The leading coefficient of T n (x), for example, is 2 n−1 ; other coefficients of T n (x) are even bigger; yet they all manage to combine into a polynomial that lies between ±1. Only when m is no larger than 7 or 8 should you contemplate writing a Chebyshev fit as a direct polynomial, and even in those cases you should be willing to tolerate two or so significant figures less accuracy than the roundoff limit of your machine. You get the g’s in equation (5.10.1) from the c’s output from chebft (suitably truncated atamodest value of m)bycallinginsequencethe followingtwoprocedures: #include "nrutil.h" void chebpc(float c[], float d[], int n) Chebyshev polynomial coefficients. Given a coefficient array c[0 n-1] , this routine generates a coefficient array d[0 n-1] such that  n-1 k=0 d k y k =  n-1 k=0 c k T k (y) − c 0 /2.Themethod is Clenshaw’s recurrence (5.8.11), but now applied algebraically rather than arithmetically. { int k,j; float sv,*dd; dd=vector(0,n-1); for (j=0;j<n;j++) d[j]=dd[j]=0.0; d[0]=c[n-1]; for (j=n-2;j>=1;j--) { for (k=n-j;k>=1;k--) { sv=d[k]; d[k]=2.0*d[k-1]-dd[k]; dd[k]=sv; } sv=d[0]; d[0] = -dd[0]+c[j]; dd[0]=sv; } for (j=n-1;j>=1;j--) d[j]=d[j-1]-dd[j]; d[0] = -dd[0]+0.5*c[0]; free_vector(dd,0,n-1); } 198 Chapter 5. Evaluation of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). void pcshft(float a, float b, float d[], int n) Polynomial coefficient shift. Given a coefficient array d[0 n-1] , this routine generates a coefficient array g [0 n-1] such that  n-1 k=0 d k y k =  n-1 k=0 g k x k ,wherexand y are related by (5.8.10), i.e., the interval −1 <y<1is mapped to the interval a <x< b . The array g is returned in d . { int k,j; float fac,cnst; cnst=2.0/(b-a); fac=cnst; for (j=1;j<n;j++) { First we rescale by the factor const . CHAPTER 5 ELASTICITY AND ITS APPLICATION 105 responds substantially to changes in the price. Supply is said to be inelastic if the quantity supplied responds only slightly to changes in the price. The price elasticity of supply depends on the flexibility of sellers to change the amount of the good they produce. For example, beachfront land has an inelastic supply because it is almost impossible to produce more of it. By contrast, manu- factured goods, such as books, cars, and televisions, have elastic supplies because the firms that produce them can run their factories longer in response to a higher price. In most markets, a key determinant of the price elasticity of supply is the time period being considered. Supply is usually more elastic in the long run than in the short run. Over short periods of time, firms cannot easily change the size of their factories to make more or less of a good. Thus, in the short run, the quantity sup- plied is not very responsive to the price. By contrast, over longer periods, firms can build new factories or close old ones. In addition, new firms can enter a market, and old firms can shut down. Thus, in the long run, the quantity supplied can re- spond substantially to the price. COMPUTING THE PRICE ELASTICITY OF SUPPLY Now that we have some idea about what the price elasticity of supply is, let’s be more precise. Economists compute the price elasticity of supply as the percentage change in the quantity supplied divided by the percentage change in the price. That is, Price elasticity of supply ϭ . For example, suppose that an increase in the price of milk from $2.85 to $3.15 a gal- lon raises the amount that dairy farmers produce from 9,000 to 11,000 gallons per month. Using the midpoint method, we calculate the percentage change in price as Percentage change in price ϭ (3.15 Ϫ 2.85)/3.00 ϫ 100 ϭ 10 percent. Similarly, we calculate the percentage change in quantity supplied as Percentage change in quantity supplied ϭ (11,000 Ϫ 9,000)/10,000 ϫ 100 ϭ 20 percent. In this case, the price elasticity of supply is Price elasticity of supply ϭϭ2.0. In this example, the elasticity of 2 reflects the fact that the quantity supplied moves proportionately twice as much as the price. THE VARIETY OF SUPPLY CURVES Because the price elasticity of supply measures the responsiveness of quantity sup- plied to the price, it is reflected in the appearance of the supply curve. Figure 5-6 shows five cases. In the extreme case of a zero elasticity, supply is perfectly inelastic, 20 percent 10 percent Percentage change in quantity supplied Percentage change in price 100 110 100 125 (a) Perfectly Inelastic Supply: Elasticity Equals 0 $5 4 Supply Quantity 1000 (b) Inelastic Supply: Elasticity Is Less Than 1 $5 4 Quantity 0 (c) Unit Elastic Supply: Elasticity Equals 1 $5 4 Quantity 0 Price 1. An increase in price . . . 2. . . . leaves the quantity supplied unchanged. 2. . . . leads to a 22% increase in quantity supplied. 1. A 22% increase in price . . . Price Price 2. . . . leads to a 10% increase in quantity supplied. 1. A 22% increase in price . . . (d) Elastic Supply: Elasticity Is Greater Than 1 $5 4 Quantity 0 Price (e) Perfectly Elastic Supply: Elasticity Equals Infinity $4 Quantity 0 Price Supply 1. A 22% increase in price . . . 2. At exactly $4, producers will supply any quantity. 1. At any price above $4, quantity supplied is infinite. 2. . . . leads to a 67% increase in quantity supplied. 3. At a price below $4, quantity supplied is zero. Supply Supply 100 200 Supply Figure 5-6 T HE P RICE E LASTICITY OF S UPPLY . The price elasticity of supply determines whether the supply curve is steep or flat. Note that all percentage changes are calculated using the midpoint method. CHAPTER 5 ELASTICITY AND ITS APPLICATION 107 and the supply curve is