[The following essay appeared in the November,1992 issue of SIAM News and the March, 1993 issue of the Bulletin of the Institute for Mathematics and Applications.] THE DEFINITION OF NUMERICAL ANALYSIS Lloyd N.Trefethen Dept.of Co nputer Science 1992 What is numerical analysis?I believe that this is more than a philosophical question.A certain wrong answer has taken hold among both outsiders to the field and insiders,distorting the image of a subject at the heart of the mathematical sciences. Here is the wrong answer: Numerical analysis is the study of rounding errors. (D1) that it v ould be hard to devis but the fundamental.If(D1)is a common perception,it is hardly surprising that numerical analysis is widely regarded as an unglamorous subject.In fact,mathematicians,physicists,and computer scient I tended to hold numerical analysis in low esteem for many years-a most asserts (DI)quite as But consider the following om some Isaacson&Keller(1966):1.Norms,arithmetic,and well-posed computations. Hamming (1971):1.Roundoff and function evaluation. Dahlquist&Bjorck (1974):1.Some general principles of numerical calculation. 2.How to obtain and estimate accuracy... Stor&Bulirsch(980：1.Error analysis, Conte&de Boor (1980):1.Number systems and errors. Atkinson (1987):1.Error:its sources,propagation,and analysis Kahaner,Moler Nash(1989):1.Introduction 2.Computer arithmetic and computational errors comp netic on opening sucn b
❘✴✟ ❍✄✞ ✞✄❈✕✗✼ ✟ ✜ ✜ ✍✓ ✍ ✟ ✍✁✟✩ ✕✗ ✑✴✟ ☎✄✭✟✡✆ ✟✁ ✪ ⑩ ❶ ❶ ✂ ✕ ✜ ✜✺✟ ✄❍ ✁✄➼➴ ➳➧ ✲➭ ✍✗✩ ✑✴✟ ✸✍✁✏✴ ✪ ⑩ ❶ ❶ ➤ ✕ ✜ ✜✺✟ ✄❍ ✑✴✟ ➵➦ ➦➧ ➜➺➫ ➯➲ ➜➙ ➧ ✄➫➭ ➜➺ ➜➵➜ ➧ ➲➯ ➸ ➴➛➜➙ ➧➩➛➜➺ ➻ ➭ ➛➫ ★ ➼✥✥ ➦➺ ➻➛ ➜➺ ➯➫➭ ✷ ✝ ✞✟✠ ☛✠✍✏➝✏✞✏✔➝ ✔✍ ➝✗✘✠✚✏✜✣✤ ✣➝✣✤✦★✏ ★ ②✞✄✓✩ ☎ ✷ ❘✁✟❍✟✑✴✟✗ ➐✟ ✑ ✷ ✄❍ ✄✡ ✺✑ ✟✁ ❱ ✏✕ ✟✗✏ ✟ ✄✁✗✟✞✞ ✣✗✭✕ ✟✁✜ ✕✑✓ ②☎❘✫✏ ✜ ✷ ✏ ✄✁✗✟✞ ✞ ✷ ✟✩✺ ⑩ ❶ ❶ ✂ ✟✴✍✑ ✕ ✜ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ❐ ❦ ✆ ✟✞ ✕ ✟✭✟ ✑✴✍✑ ✑✴✕ ✜ ✕ ✜ ✡✄✁✟ ✑✴✍✗ ✍ ✴✕ ✞✄ ✜ ✄ ✴✕ ✏ ✍✞ ➅✺✟ ✜ ✑ ✕ ✄✗ ✷ ➔ ✏ ✟✁✑ ✍✕✗ ❈✁✄✗✼ ✍✗✜❈✟✁ ✴✍✜ ✑ ✍❙✟✗ ✴✄✞✩ ✍✡✄✗✼ ✆ ✄✑✴ ✄✺✑ ✜ ✕✩✟✁ ✜ ✑ ✄ ✑✴✟ ♥✟✞✩ ✍✗✩ ✕✗✜ ✕✩✟✁✜ ✪ ✩✕ ✜ ✑ ✄✁✑ ✕✗✼ ✑✴✟ ✕✡✍✼✟ ✄❍ ✍ ✜✺✆ ✟ ✏✑ ✍✑ ✑✴✟ ✴✟ ✍✁✑ ✄❍ ✑✴✟ ✡✍✑✴✟✡✍✑ ✕ ✏ ✍✞ ✜ ✏✕ ✟✗✏ ✟ ✜ ✷ ➶✟✁✟ ✕ ✜ ✑✴✟ ❈✁✄✗✼ ✍✗✜❈✟✁ ⑦ ➳➵➩➧ ➸➺ ➻➛➦ ➛➫ ➛ ➦➾➭ ➺ ➭ ➺ ➭ ➜➙ ➧ ➭ ➜➵ ★➾ ➯➲ ➸➯ ➵➫ ★➺➫✪ ➧ ➸➸➯ ➸➭ ✭ ➌➐ ⑩ ➎ ❘✴✟ ✁✟ ✍✩✟✁ ❈✕ ✞ ✞ ✍✼✁✟ ✟ ✑✴✍✑ ✕✑ ❈✄✺✞✩ ✆ ✟ ✴✍✁✩ ✑ ✄ ✩✟✭✕ ✜ ✟ ✍ ✡✄✁✟ ✺✗✕✗✭✕ ✑ ✕✗✼ ✩✟ ✜ ✏✁✕ ✑ ✕ ✄✗ ✄❍ ✍ ♥✟✞✩ ✷ ❿✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✍✁✟ ✕✗✟✭✕ ✑ ✍✆✞ ✟ ✪ ✓✟ ✜ ✪ ✆✺✑ ✑✴✟✓ ✍✁✟ ✏ ✄✡ ✞✕ ✏ ✍✑ ✟✩ ✍✗✩ ✑ ✟✩✕ ✄✺✜ ✍✗✩ ❜➫ ➯ ➜ ➲➵➫ ★ ➛➩➧ ➫➜ ➛ ➦✭ ❦❍ ➌➐ ⑩ ➎ ✕ ✜ ✍ ✏ ✄✡✡✄✗ ✟✁✏ ✟ ✑ ✕✄✗ ✪ ✕ ✑ ✕ ✜ ✴✍✁✩✞✓ ✜✺✁ ✁✕ ✜ ✕✗✼ ✑✴✍✑ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✕ ✜ ❈✕✩✟✞✓ ✁ ✟✼✍✁✩✟✩ ✍✜ ✍✗ ✺✗✼✞ ✍✡✄✁✄✺✜ ✜✺✆ ✟ ✏✑ ✷ ❦✗ ❍✍✏✑ ✪ ✡✍✑✴✟✡✍✑ ✕ ✏✕ ✍✗✜ ✪ ✴✓✜ ✕ ✏✕ ✜ ✑ ✜ ✪ ✍✗✩ ✏✄✡ ✺✑ ✟✁ ✜ ✏✕ ✟✗✑ ✕ ✜ ✑ ✜ ✴✍✭✟ ✍✞ ✞ ✑ ✟✗✩✟✩ ✑ ✄ ✴✄✞✩ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✕✗ ✞✄❈ ✟ ✜ ✑ ✟ ✟✡ ❍✄✁ ✡✍✗✓ ✓✟ ✍✁✜❜✍ ✡✄ ✜ ✑ ✺✗✺✜✺✍✞ ✏ ✄✗✜ ✟✗✜✺✜ ✷ ❍ ✏ ✄✺✁✜ ✟ ✗✄✆ ✄ ✩✓ ✆ ✟✞✕ ✟✭✟ ✜ ✄✁ ✍✜ ✜ ✟✁✑ ✜ ➌➐ ⑩ ➎ ➅✺✕✑ ✟ ✍✜ ✆ ✍✞✩✞✓ ✍✜ ❈✁✕ ✑ ✑ ✟✗ ✷ ➄✺✑ ✏ ✄✗✜ ✕✩✟✁ ✑✴✟ ❍✄✞ ✞✄❈✕✗✼ ✄ ✟✗✕✗✼ ✏✴✍ ✑ ✟✁ ✴✟ ✍✩✕✗✼✜ ❍✁✄✡ ✜ ✄✡✟ ✜ ✑ ✍✗✩✍✁✩ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✑ ✟✌✑ ✜ ⑦ ✭✆✜✜✦✆✓★ ✮ ✰✁✕✕ ✁✡ ✫✭ ✯ ✱ ✱✴ ✵ ⑩ ✷ ☎✄✁✡✜ ✪ ✍✁✕✑✴✡✟✑ ✕ ✏ ✪ ✍✗✩ ❈✟✞ ✞ ✐ ✄ ✜ ✟✩ ✏ ✄✡ ✺✑ ✍✑ ✕✄✗✜ ✷ ❦✜❑❑✛★✙ ✫✭ ✯ ✰✭ ✴ ✵ ⑩ ✷ ❿✄✺✗✩✄◆ ✍✗✩ ❍✺✗✏✑ ✕ ✄✗ ✟✭✍✞✺✍✑ ✕ ✄✗ ✷ ✣✜✻✕❄ ✿✛✆ ✞ ✮ ✵✷✓✹✡✦♦ ✫✭ ✯ ✰✻✴ ✵ ⑩ ✷ ❱ ✄✡✟ ✼✟✗✟✁✍✞ ✁✕✗✏✕ ✞ ✟ ✜ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✏ ✍✞ ✏✺✞ ✍✑ ✕✄✗ ✷ ✂ ✷ ➶✄❈ ✑ ✄ ✄✆✑ ✍✕✗ ✍✗✩ ✟ ✜ ✑ ✕✡✍✑ ✟ ✍✏ ✏✺✁ ✍✏✓ 2 2 2 ✷ ✽✞ ✓ ✁✡ ✮ ✵✿✕✛✡✆✦✻ ✫✭ ✯❞❛✴ ✵ ⑩ ✷ ❏✁✁✄✁ ✍✗✍✞✓✜ ✕ ✜ ✷ ✑✓★✞ ✁ ✮ ❁✁ ✵✓ ✓✡ ✫✭ ✯❞❛✴ ✵ ⑩ ✷ ☎✺✡✆ ✟✁ ✜✓✜ ✑ ✟✡✜ ✍✗✩ ✟✁✁✄✁ ✜ ✷ ✐✞♦✛★✆✓★ ✫✭ ✯❞ ✰✴ ✵ ⑩ ✷ ❏✁✁✄✁ ⑦ ✕✑ ✜ ✜ ✄✺✁✏ ✟ ✜ ✪ ✁✄ ✍✼✍✑ ✕ ✄✗ ✪ ✍✗✩ ✍✗✍✞✓✜ ✕ ✜ ✷ ✰✜✻ ✜★ ✁✡❤ ✾✓✕✁✡ ✮ ✍✜✆✻ ✫✭ ✯❞ ✯✴ ✵ ⑩ ✷ ❦✗✑ ✁✄ ✩✺✏✑ ✕ ✄✗ ✷ ✂ ✷ ✄✡ ✺✑ ✟✁ ✍✁✕✑✴✡✟✑ ✕ ✏ ✍✗✩ ✏ ✄✡ ✺✑ ✍✑ ✕✄✗✍✞ ✟✁✁✄✁✜ ✷ ♦❏✁✁✄✁ ♣ 2 2 2 ♦✁✄✺✗✩✄◆♣ 2 2 2 ♦ ✏✄✡ ✺✑ ✟✁ ✍✁✕ ✑✴✡✟✑ ✕ ✏ ♣ ❜ ✑✴✟ ✜ ✟ ✍✁✟ ✑✴✟ ❈✄✁✩✜ ✑✴✍✑ ❙✟ ✟ ✁ ✟ ✍ ✟ ✍✁✕✗✼ ✷ ✟✴✍✑ ✕✡ ✁✟ ✜ ✜ ✕ ✄✗ ✩✄ ✟ ✜ ✍✗ ✕✗➅✺✕ ✜ ✕ ✑ ✕✭✟ ✏ ✄✞✞ ✟✼✟ ✜ ✑✺✩✟✗✑ ✼✟✑ ✺ ✄✗ ✄ ✟✗✕✗✼ ✜✺✏✴ ✆ ✄ ✄❙✜ ❐ ✁ ✏✄✗✜ ✕✩✟✁ ✑✴✟ ✩✟♥✗✕✑ ✕ ✄✗✜ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✕✗ ✜ ✄✡✟ ✩✕ ✏✑ ✕✄✗✍✁✕ ✟ ✜ ⑦ ⑩
Webster's New Collegiate Dictionary (1973):"The study of quantitative approxi matotheofhcoemofth errors and bounds to the errors involved. Chambers20th Century Dictionary(1983):"The study of methods of approximation and their accuracy,etc. Themer Dct()or oximate solutions to mathematical problems,taking into account the extent of p An T+ ne that these definitions would serve most effectively to deter the curious from investigating further The singular value decomposition (SVD)affords another example of the perception of nu- merical analysis as the science of rounding errors.Although the roots of the SVD go back more than 100 years,it is mainly since the 1960s,through the work of Gene Golub and other ental an ide t has ax e e Its pre of pro D is a ll kinds of c nd av Yet today.thirty vears later.most mathematical scientists and even many applied mathe maticians do not have a working knowledge of the SVD.Most of them have heard of it,but the impress on se ems to be widespread that the SVD is just a too for combating rounding othTs a I am convinced that consciously or unconsciously,many people think that(D1)is at least half true.In actuality,it is a very small part of the truth.And although there are historical explanations for the influence of(D1)in the past,it is a less appropriate definition today and is destined to become still less appropriate in the future. I propose the following alternative definition with which to enter the new century: Num rical analysis is the study of algorithms for the problems of continuous mathematics (D2 na nition an be perfect.But it s to me tha(characterization as you could come upwith for most disciplines The pivotal word is algorithms.Where was this word in those chapter headings and dictionary definitions?Hidden between the lines,at best,and yet surely this is the center of numerical analysis:devising and analyzing algorithms to solve a certain class of problems. at real or complex aretheamanofotherenptlemoiproblems,whiealg rithms for disc te proble Let us consider the implications of (D2).First of all it is clear that since real and complex numbers cannot be represented exactly on computers,(D2)implies that part of the business umerical analysis must be t o approximate them.This is where the rounding errors come 2
✁ ✄✆ ✞ ✁✡ ☞✆ ✍✁✏ ✑✓✕✕✁✙✛✜ ✞ ✁ ✣✛✦✞✛✓★ ✜✡✪ ✫✭ ✯ ✰✲✴ ✵ ✷✹✻ ✁ ✆ ✞ ✿ ❁✪ ✓❃ ❄ ✿ ✜★✞✛ ✞ ✜ ✞✛ ❊✁ ✜❋❋✡✓❍✛❏ ❑✜ ✞✛✓★✆ ✞ ✓ ✞✻ ✁ ✆✓✕ ✿ ✞✛✓★✆ ✓❃ ❑✜ ✞✻ ✁❑✜ ✞✛✦✜✕ ❋✡✓ ✄✕✁❑✆ ✛★ ✦✕✿ ❁✛★✙ ✦✓★✆✛❁✁✡✜ ✞✛✓★ ✓❃ ✞✻ ✁ ✁✡✡✓✡✆ ✜★ ❁ ✄ ✓ ✿★ ❁✆ ✞ ✓ ✞✻ ✁ ✁✡✡✓✡✆ ✛★❊✓✕❊✁❁❭ ❪ ✑✻ ✜❑✄ ✁✡✆ ❴❛✞✻ ✑✁★✞ ✿✡✪ ✣✛✦✞✛✓★ ✜✡✪ ✫✭ ✯❞✲✴ ✵ ✷✹✻ ✁ ✆ ✞ ✿ ❁✪ ✓❃❑✁ ✞✻ ✓ ❁✆ ✓❃ ✜❋❋✡✓❍✛❑✜ ✞✛✓★ ✜★ ❁ ✞✻ ✁✛✡ ✜✦✦✿✡✜✦✪❤ ✁ ✞ ✦❭ ❪ ✹✻ ✁ ✐❑✁✡✛✦✜★ ❦✁✡✛ ✞ ✜✙✁ ✣✛✦✞✛✓★ ✜✡✪ ✫✭ ✯ ✯❴✴ ✵ ✷✹✻ ✁ ✆ ✞ ✿ ❁✪ ✓❃ ✜❋❋✡✓❍✛❑✜ ✞ ✁ ✆✓✕ ✿ ✞✛✓★✆ ✞ ✓ ❑✜ ✞✻ ✁❑✜ ✞✛✦✜✕ ❋✡✓ ✄✕✁❑✆❤ ✞ ✜♦✛★✙ ✛★✞ ✓ ✜✦✦✓ ✿★✞ ✞✻ ✁ ✁❍✞ ✁★✞ ✓❃ ❋ ✓✆✆✛ ✄✕✁ ✁✡✡✓✡✆ ❭ ❪ ♦➔✁✄✌✕✡✍✑ ✕ ✄✗✜ ♣ s s s ♦ ✍✏ ✏✺✁ ✍✏✓♣ s s s ♦ ✟✁✁✄✁✜ ♣ ✍✼✍✕✗ ✷ ❦ ✑ ✜ ✟ ✟✡✜ ✑ ✄ ✡✟ ✑✴✍✑ ✑✴✟ ✜ ✟ ✩✟♥✗✕ ✑ ✕ ✄✗✜ ❈✄✺✞✩ ✜ ✟✁✭✟ ✡✄ ✜ ✑ ✟◆✟ ✏✑ ✕✭✟✞✓ ✑ ✄ ✩✟✑ ✟✁ ✑✴✟ ✏✺✁✕✄✺✜ ❍✁✄✡ ✕✗✭✟ ✜ ✑ ✕ ✼✍✑ ✕✗✼ ❍✺✁✑✴✟✁ ✷ ❘✴✟ ✜ ✕✗✼✺✞ ✍✁ ✭✍✞✺✟ ✩✟ ✏✄✡ ✄ ✜ ✕ ✑ ✕✄✗ ➌ ❱✡➐➎ ✍◆✄✁✩✜ ✍✗✄✑✴✟✁ ✟✌✍✡✞ ✟ ✄❍ ✑✴✟ ✟✁✏ ✟✑ ✕✄✗ ✄❍ ✗✺✐ ✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✍✜ ✑✴✟ ✜ ✏✕ ✟✗✏ ✟ ✄❍ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✷ ➔✞✑✴✄✺✼✴ ✑✴✟ ✁✄ ✄✑ ✜ ✄❍ ✑✴✟ ❱✡➐ ✼✄ ✆ ✍✏❙ ✡✄✁✟ ✑✴✍✗ ⑩ → → ✓✟ ✍✁✜ ✪ ✕ ✑ ✕ ✜ ✡✍✕✗✞✓ ✜ ✕✗✏ ✟ ✑✴✟ ⑩ ❶ ✝ → ✜ ✪ ✑✴✁✄✺✼✴ ✑✴✟ ❈✄✁❙ ✄❍ ❢✟✗✟ ❢✄✞✺✆ ✍✗✩ ✄✑✴✟✁ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✑ ✜ ✪ ✑✴✍✑ ✕ ✑ ✴✍✜ ✍✏✴✕ ✟✭✟✩ ✕ ✑ ✜ ✁ ✟ ✜ ✟✗✑ ✩✟ ✼✁✟ ✟ ✄❍ ✁✄✡✕✗✟✗✏ ✟ ✷ ❘✴✟ ❱✡➐ ✕ ✜ ✍✜ ❍✺✗✩✍✡✟✗✑ ✍✞ ✍✗ ✕✩✟ ✍ ✍✜ ✑✴✟ ✟✕✼✟✗✭✍✞✺✟ ✩✟ ✏✄✡ ✄ ✜ ✕ ✑ ✕ ✄✗ ✢ ✕ ✑ ✕ ✜ ✑✴✟ ✗✍✑✺✁✍✞ ✞ ✍✗✼✺✍✼✟ ❍✄✁ ✩✕ ✜ ✏✺✜ ✜ ✕✗✼ ✍✞ ✞ ❙✕✗✩✜ ✄❍ ➅✺✟ ✜ ✑ ✕✄✗✜ ✄❍ ✗✄✁✡✜ ✍✗✩ ✟✌✑ ✁✟✡✍ ✕✗✭✄✞✭✕✗✼ ✗✄✗✜✓✡✡✟✑ ✁✕ ✏ ✡✍✑ ✁✕ ✏ ✟ ✜ ✄✁ ✄ ✟✁✍✑ ✄✁ ✜ ✷ ✍✟✑ ✑ ✄ ✩✍✓✪ ✑✴✕✁✑✓ ✓✟ ✍✁ ✜ ✞✍✑ ✟✁ ✪ ✡✄ ✜ ✑ ✡✍✑✴✟✡✍✑ ✕ ✏ ✍✞ ✜ ✏✕ ✟✗✑ ✕ ✜ ✑ ✜ ✍✗✩ ✟✭✟✗ ✡✍✗✓ ✍✞✕ ✟✩ ✡✍✑✴✟✐ ✡✍✑ ✕ ✏✕ ✍✗✜ ✩✄ ✗✄✑ ✴✍✭✟ ✍ ❈✄✁❙✕✗✼ ❙✗✄❈✞✟✩✼✟ ✄❍ ✑✴✟ ❱✡➐ ✷ ✸✄ ✜ ✑ ✄❍ ✑✴✟✡ ✴✍✭✟ ✴✟ ✍✁✩ ✄❍ ✕✑ ✪ ✆✺✑ ✑✴✟ ✕✡✁ ✟ ✜ ✜ ✕✄✗ ✜ ✟ ✟✡✜ ✑ ✄ ✆ ✟ ❈✕✩✟ ✜✁ ✟ ✍✩ ✑✴✍✑ ✑✴✟ ❱✡➐ ✕ ✜ z ✺✜ ✑ ✍ ✑ ✄ ✄✞ ❍✄✁ ✏ ✄✡✆ ✍✑ ✕✗✼ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✷ ➔ ✼✞ ✍✗✏ ✟ ✍✑ ✍ ❍✟❈ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✑ ✟✌✑✆ ✄ ✄❙✜ ✜✺✼✼✟ ✜ ✑ ✜ ❈✴✓✷ ❦✗ ✄✗✟ ✏ ✍✜ ✟ ✍❍✑ ✟✁ ✍✗✐ ✄✑✴✟✁ ✪ ✑✴✟ ❱✡➐ ✕ ✜ ✆✺✁✕ ✟✩ ✩✟ ✟ ✕✗ ✑✴✟ ✆ ✄ ✄❙ ✪ ✑✓✕ ✏ ✍✞ ✞✓ ✕✗ ✍✗ ✍✩✭✍✗✏ ✟✩ ✜ ✟ ✏✑ ✕ ✄✗ ✄✗ ✁✍✗❙✐ ✩✟♥✏✕ ✟✗✑ ✞ ✟ ✍✜ ✑ ✐ ✜➅✺✍✁✟ ✜ ✁✄✆✞✟✡✜ ✪ ✍✗✩ ✁✟ ✏✄✡✡✟✗✩✟✩ ✡✍✕✗✞✓ ❍✄✁ ✕✑ ✜ ✜ ✑ ✍✆✕ ✞ ✕✑✓ ✁✄ ✟✁✑ ✕ ✟ ✜ ✷ ❦ ✍✡ ✏✄✗✭✕✗✏ ✟✩ ✑✴✍✑ ✏✄✗✜ ✏✕ ✄✺✜ ✞✓ ✄✁ ✺✗✏ ✄✗✜ ✏✕✄✺✜ ✞✓✪ ✡✍✗✓ ✟✄✞ ✟ ✑✴✕✗❙ ✑✴✍✑ ➌➐ ⑩ ➎ ✕ ✜ ✍✑ ✞ ✟ ✍✜ ✑ ✴✍✞❍ ✑ ✁✺✟ ✷ ❦✗ ✍✏✑✺✍✞✕ ✑✓✪ ✕✑ ✕ ✜ ✍ ✭✟✁✓ ✜✡✍✞ ✞ ✍✁✑ ✄❍ ✑✴✟ ✑ ✁✺✑✴ ✷ ➔✗✩ ✍✞✑✴✄✺✼✴ ✑✴✟✁✟ ✍✁✟ ✴✕ ✜ ✑ ✄✁✕ ✏ ✍✞ ✟✌✞✍✗✍✑ ✕✄✗✜ ❍✄✁ ✑✴✟ ✕✗✜✺✟✗✏ ✟ ✄❍ ➌➐ ⑩ ➎ ✕✗ ✑✴✟ ✍✜ ✑ ✪ ✕ ✑ ✕ ✜ ✍ ✞ ✟ ✜ ✜ ✍✁✄✁✕ ✍✑ ✟ ✩✟♥✗✕ ✑ ✕ ✄✗ ✑ ✄ ✩✍✓ ✍✗✩ ✕ ✜ ✩✟ ✜ ✑ ✕✗✟✩ ✑ ✄ ✆ ✟ ✏✄✡✟ ✜ ✑ ✕ ✞ ✞ ✞ ✟ ✜ ✜ ✍✁✄✁✕ ✍✑ ✟ ✕✗ ✑✴✟ ❍✺✑✺✁ ✟ ✷ ❦ ✁✄ ✄ ✜ ✟ ✑✴✟ ❍✄✞ ✞✄❈✕✗✼ ✍✞✑ ✟✁✗✍✑ ✕✭✟ ✩✟♥✗✕ ✑ ✕ ✄✗ ❈✕✑✴ ❈✴✕ ✏✴ ✑ ✄ ✟✗✑ ✟✁ ✑✴✟ ✗✟❈ ✏ ✟✗✑✺✁✓ ⑦ ➳➵➩➧ ➸➺ ➻➛ ➦ ➛➫ ➛ ➦➾➭ ➺ ➭ ➺ ➭ ➜➙ ➧ ➭ ➜➵ ★➾ ➯➲ ➛➦✪ ➯ ➸➺ ➜➙➩➭ ➲➯ ➸ ➜➙ ➧ ✥➸➯ ✁ ➦➧➩➭ ➯➲ ➻➯➫➜➺➫➵ ➯ ➵➭ ➩➛ ➜➙ ➧➩➛ ➜➺ ➻ ➭ ✭ ➌➐✂ ➎ ➄✄✺✗✩✍✁✕ ✟ ✜ ✆ ✟✑❈✟ ✟✗ ♥✟✞✩✜ ✍✁✟ ✍✞❈✍✓✜ ❍✺✘ ✘✓ ✢ ✗✄ ✩✟♥✗✕✑ ✕ ✄✗ ✏ ✍✗ ✆ ✟ ✟✁❍✟ ✏✑ ✷ ➄✺✑ ✕✑ ✜ ✟ ✟✡✜ ✑ ✄ ✡✟ ✑✴✍✑ ➌➐✂ ➎ ✕ ✜ ✍✜ ✜✴✍✁ ✍ ✏✴✍✁ ✍✏✑ ✟✁✕ ✘ ✍✑ ✕ ✄✗ ✍✜ ✓✄✺ ✏✄✺✞✩ ✏ ✄✡✟ ✺ ❈✕ ✑✴ ❍✄✁ ✡✄ ✜ ✑ ✩✕ ✜ ✏✕✞ ✕✗✟ ✜ ✷ ❘✴✟ ✕✭✄✑ ✍✞ ❈✄✁✩ ✕ ✜ ➛ ➦✪ ➯ ➸➺ ➜➙➩➭ ✭ ✟✴✟✁✟ ❈✍✜ ✑✴✕ ✜ ❈✄✁✩ ✕✗ ✑✴✄ ✜ ✟ ✏✴✍✑ ✟✁ ✴✟ ✍✩✕✗✼✜ ✍✗✩ ✩✕ ✏✑ ✕✄✗✍✁✓ ✩✟♥✗✕ ✑ ✕ ✄✗✜ ❐ ➶✕✩✩✟✗ ✆ ✟✑❈✟ ✟✗ ✑✴✟ ✞ ✕✗✟ ✜ ✪ ✍✑ ✆ ✟ ✜ ✑ ✪ ✍✗✩ ✓✟✑ ✜✺✁✟✞✓ ✑✴✕ ✜ ✕ ✜ ✑✴✟ ✏ ✟✗✑ ✟✁ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ⑦ ✩✟✭✕ ✜ ✕✗✼ ✍✗✩ ✍✗✍✞✓✘ ✕✗✼ ✍✞✼✄✁✕ ✑✴✡✜ ✑ ✄ ✜ ✄✞✭✟ ✍ ✏ ✟✁✑ ✍✕✗ ✏✞✍✜ ✜ ✄❍ ✁✄✆✞✟✡✜ ✷ ❘✴✟ ✜ ✟ ✍✁✟ ✑✴✟ ✁✄✆✞ ✟✡✜ ✄❍ ➻➯➫➜➺➫➵ ➯ ➵➭ ➩➛ ➜➙ ➧➩➛ ➜➺ ➻ ➭ ✭ ♦t✄✗✑ ✕✗✺✄✺✜ ♣ ✡✟ ✍✗✜ ✑✴✍✑ ✁✟ ✍✞ ✄✁ ✏ ✄✡✞✟✌ ✭✍✁✕ ✍✆✞ ✟ ✜ ✍✁✟ ✕✗✭✄✞✭✟✩ ✢ ✕✑ ✜ ✄ ✄ ✜ ✕ ✑ ✟ ✕ ✜ ♦✩✕ ✜ ✏✁✟✑ ✟ ✷ ♣ ➔ ✩✄ ✘ ✟✗ ➅✺✍✞✕♥✏ ✍✑ ✕ ✄✗✜ ✍✜ ✕✩✟ ✪ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✐ ✞✓✜ ✑ ✜ ✍✁✟ ✆✁✄ ✍✩✞✓ ✏ ✄✗✏ ✟✁✗✟✩ ❈✕ ✑✴ ✏✄✗✑ ✕✗✺✄✺✜ ✁✄✆✞ ✟✡✜ ✪ ❈✴✕ ✞✟ ✍✞✼✄✁✕ ✑✴✡✜ ❍✄✁ ✩✕ ✜ ✏✁✟✑ ✟ ✁✄✆✞ ✟✡✜ ✍✁✟ ✑✴✟ ✏ ✄✗✏ ✟✁✗ ✄❍ ✄✑✴✟✁ ✏ ✄✡✺✑ ✟✁ ✜ ✏✕ ✟✗✑ ✕ ✜ ✑ ✜ ✷ ② ✟✑ ✺✜ ✏ ✄✗✜ ✕✩✟✁ ✑✴✟ ✕✡✞ ✕ ✏ ✍✑ ✕ ✄✗✜ ✄❍ ➌➐✂ ➎ ✷ ➇✕✁ ✜ ✑ ✄❍ ✍✞ ✞ ✕ ✑ ✕ ✜ ✏✞ ✟ ✍✁ ✑✴✍✑ ✜ ✕✗✏ ✟ ✁ ✟ ✍✞ ✍✗✩ ✏ ✄✡✞✟✌ ✗✺✡✆ ✟✁✜ ✏ ✍✗✗✄✑ ✆ ✟ ✁✟✁✟ ✜ ✟✗✑ ✟✩ ✟✌✍✏✑ ✞✓ ✄✗ ✏ ✄✡✺✑ ✟✁✜ ✪ ➌➐✂ ➎ ✕✡✞ ✕ ✟ ✜ ✑✴✍✑ ✍✁✑ ✄❍ ✑✴✟ ✆✺✜ ✕✗✟ ✜ ✜ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✡✺✜ ✑ ✆ ✟ ✑ ✄ ✍✁✄✌✕✡✍✑ ✟ ✑✴✟✡✷ ❘✴✕ ✜ ✕ ✜ ❈✴✟✁✟ ✑✴✟ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✏✄✡✟ ✕✗ ✷ ☎✄❈ ❍✄✁ ✍ ✏ ✟✁✑ ✍✕✗ ✜ ✟✑ ✄❍ ✁✄✆✞ ✟✡✜ ✪ ✗✍✡✟✞✓ ✑✴✟ ✄✗✟ ✜ ✑✴✍✑ ✍✁✟ ✜ ✄✞✭✟✩ ✆✓ ✍✞✼✄✁✕✑✴✡✜ ✑✴✍✑ ✑ ✍❙✟ ✍ ♥✗✕ ✑ ✟ ✗✺✡✆ ✟✁ ✄❍ ✜ ✑ ✟✜ ✪ ✑✴✍✑ ✕ ✜ ✍✞ ✞ ✑✴✟✁ ✟ ✕ ✜ ✑ ✄ ✕✑ ✷ ❘✴✟ ✁✟✡✕ ✟✁ ✟✌✍✡✞ ✟ ✕ ✜ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✂
wh setvi'Nati eab sysiem evednajie's(a b73en'.ebsia.Ganssia etimi aile'den fave ie n'.ebsia' cempnieb scie'ce issnes sncf as epetajie'cen is a:macfie atffijecintesd yen ave ere py iWsemebe:y ctaims ifai 0Is Jnsi aI But most p.ob ms of continaous mathematies cannot be soded bu finite ato.ithms!o'tike icat a atysis cent: ,ei be setve:eacity eve iywe cent:wetk i eaci a atysisk ew ifisd:me jie'ij ate'Nwiif a ww webr s abeni Abet a:Gateis wfe ifey jeacf atNbiifms Wo cempnii NmaibAeiNe vat ifey ie me jie e Even if.oundinr e..o.s vanished,nume.icaCanasis woud.emaina AppbeAimaji nmbetedifek eea Neiaifmejicdsee aireat iepic maybe eve eepeb bnsi ess ew nmebicat a atysis is appbeAimaji Nn'k ew sd'e e'veb He' eWenb fier is ifaid atlebifms ifai ce' tevmbetic cempnii'me peciauy 防8 mea epebajle's?o N setve: ave h e eveby siep ate'Nife way ewa catcntajie'baifeb ifa i asi Ne aci ai ife e way e'e pbecee:sdi Hleaii'pei i eb symbeticattydife mai'piebtem ewi':i Na hapi.ty Ce'vebNe i at bifm is ife same7 ame in'cajise escbibe ifese Bni ece wel s beN ie be:e:ii is fa iekew wrere ie siepdwb"D20 atse wits ie me'je se me eifeb impebia i majjebs:ifai ifege atNebifms abe impteme a mesi impebia ic aiey a. a, ewappticajie's e mitie's eWcempniebs aten':ife welt:7"3fe pbebtems eWce'nens maifemajics"ate ife pbebtemsifai scie ce a:eN eeti'Nate bnit meife:sdsc ce a:e N eeb!'Nas pbacjice;ie:ay ent eSitemnne ty ie a fati73fey a ie ife Neaj ita. went:i be pbevi,NeAsje ce ifeetems7 3
❍✄✁ ✜ ✄✞✭✕✗✼ ✍ ✞ ✕✗✟ ✍✁ ✜✓✜ ✑ ✟✡ ✄❍ ✟➅✺✍✑ ✕ ✄✗✜ '✉ ✈ ✇ ✷ ❘✄ ✺✗✩✟✁ ✜ ✑ ✍✗✩ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✪ ✓✄✺ ✴✍✭✟ ✑ ✄ ✺✗✩✟✁✜ ✑ ✍✗✩ ✏✄✡B✺✑ ✟✁ ✜ ✏✕ ✟✗✏ ✟ ✕ ✜ ✜✺✟ ✜ ✜✺✏✴ ✍✜ ✄B ✟✁ ✍✑ ✕✄✗ ✏ ✄✺✗✑ ✜ ✍✗✩ ✡✍✏✴✕✗✟ ✍✁✏✴✕ ✑ ✟ ✏✑✺✁✟ ✜ ✪ ✍✗✩ ✓✄✺ ✴✍✭✟ ✑ ✄ ✺✗✩✟✁ ✜ ✑ ✍✗✩ ✑✴✟ B✁✄B ✍✼ ✍✑ ✕ ✄✗ ✄❍ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁ ✜❜✜ ✑ ✍✆✕ ✞✕ ✑✓✷ ❘✴✍✑ ➉ ✜ ✍✞✞ ✓✄✺ ✴✍✭✟ ✑ ✄ ✺✗✩✟✁ ✜ ✑ ✍✗✩ ✪ ✍✗✩ ✕❍ ✜ ✄✡✟✆ ✄ ✩✓ ✏✞ ✍✕✡✜ ✑✴✍✑ ➌➐✂ ➎ ✕ ✜ c ✺✜ ✑ ✍ ✡✄✁✟ B ✄✞ ✕✑ ✟ ✁✟ ✜ ✑ ✍✑ ✟✡✟✗✑ ✄❍ ➌➐ ⑩ ➎ ✪ ✓✄✺ ✏ ✍✗ ➉ ✑ B✁✄✭✟ ✴✕✡ ✄✁ ✴✟✁ ❈✁✄✗✼ ❈✕ ✑✴ ✑✴✟ ✟✌✍✡B✞✟ ✄❍ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✷ ➵➜ ➩➯ ➭ ➜ ✥➸➯ ✁ ➦➧➩➭ ➯➲ ➻➯➫➜➺➫➵ ➯ ➵➭ ➩➛ ➜➙ ➧➩➛ ➜➺ ➻ ➭ ➻➛➫➫ ➯ ➜ ✁➧ ➭ ➯ ➦☎ ➧ ★ ✁ ➾ ✞➫➺ ➜ ➧ ➛ ➦✪ ➯ ➸➺ ➜➙➩➭ ✠ ✣✗✞✕❙✟ '✉ ✈ ✇ ✪ ✍✗✩ ✺✗✞✕❙✟ ✑✴✟ ✩✕ ✜ ✏✁✟✑ ✟ B✁✄✆✞✟✡✜ ✄❍ ✏ ✄✡B✺✑ ✟✁ ✜ ✏✕ ✟✗✏ ✟ ✪ ✡✄ ✜ ✑ ✄❍ ✑✴✟ B✁✄✆✞ ✟✡✜ ✄❍ ✗✺✡✟✁✐ ✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✏✄✺✞✩ ✗✄✑ ✆ ✟ ✜ ✄✞✭✟✩ ✟✌✍✏✑ ✞✓ ✟✭✟✗ ✕❍ ❈✟ ✏ ✄✺✞✩ ❈✄✁❙ ✕✗ ✟✌✍✏✑ ✍✁✕ ✑✴✡✟✑ ✕ ✏ ✷ ☎✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✑ ✜ ❙✗✄❈ ✑✴✕ ✜ ✪ ✍✗✩ ✡✟✗✑ ✕ ✄✗ ✕ ✑ ✍✞✄✗✼ ❈✕✑✴ ✍ ❍✟❈ ❈✄✁✩✜ ✍✆ ✄✺✑ ➔✆ ✟✞ ✍✗✩ ❢✍✞✄✕ ✜ ❈✴✟✗ ✑✴✟✓ ✑ ✟ ✍✏✴ ✍✞✼✄✁✕ ✑✴✡✜ ❍✄✁ ✏ ✄✡B✺✑ ✕✗✼ ✡✍✑ ✁✕✌ ✟✕✼✟✗✭✍✞✺✟ ✜ ✷ ❘✄ ✄ ✄❍✑ ✟✗ ✑✴✟✓ ❍✄✁✼✟✑ ✑ ✄ ✡✟✗✑ ✕✄✗ ✑✴✍✑ ✑✴✟ ✜ ✍✡✟ ✏ ✄✗✏✞✺✜ ✕✄✗ ✟✌✑ ✟✗✩✜ ✑ ✄ ✭✕✁✑✺✍✞ ✞✓ ✍✗✓ B✁✄✆✞✟✡ ❈✕✑✴ ✍ ✗✄✗✞✕✗✟ ✍✁ ✑ ✟✁✡ ✄✁ ✍ ✩✟✁✕✭✍✑ ✕✭✟ ✕✗ ✕✑❜✘ ✟✁✄♥✗✩✕✗✼ ✪ ➅✺✍✩✁✍✑✺✁✟ ✪ ✩✕◆✟✁✟✗✑ ✕ ✍✞ ✟➅✺✍✑ ✕✄✗✜ ✪ ✕✗✑ ✟✼✁✍✞ ✟➅✺✍✑ ✕ ✄✗✜ ✪ ✄B✑ ✕✡✕ ✘ ✍✑ ✕ ✄✗ ✪ ✓✄✺ ✗✍✡✟ ✕ ✑ ✷ ➥☎ ➧ ➫ ➺➲ ➸➯➵➫ ★➺➫✪ ➧ ➸➸➯ ➸➭ ☎ ➛➫➺ ➭ ➙ ➧★➪ ➫➵➩➧ ➸➺ ➻➛ ➦ ➛➫ ➛ ➦➾➭ ➺ ➭ ✲➯➵➦★ ➸➧➩➛➺➫ ✭ ➔BB✁✄✌✕✡✍✑ ✕✗✼ ✡✟✁✟ ✗✺✡✆ ✟✁✜ ✪ ✑✴✟ ✑ ✍✜❙ ✄❍ ✜✄ ✍✑ ✕✗✼✐B ✄✕✗✑ ✍✁✕✑✴✡✟✑ ✕ ✏ ✪ ✕ ✜ ✕✗✩✟ ✟✩ ✍ ✁✍✑✴✟✁ ✜✡✍✞ ✞ ✑ ✄B✕ ✏ ✍✗✩ ✡✍✓✆ ✟ ✟✭✟✗ ✍ ✑ ✟✩✕ ✄✺✜ ✄✗✟ ✷ ❘✴✟ ✩✟ ✟B ✟✁ ✆✺✜ ✕✗✟ ✜ ✜ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✕ ✜ ✍BB✁✄✌✕✡✍✑ ✕✗✼ ✺✗❙✗✄❈✗✜ ✪ ✗✄✑ ❙✗✄❈✗✜ ✷ ❿✍B✕✩ ✏ ✄✗✭✟✁✼✟✗✏ ✟ ✄❍ ✍BB✁✄✌✕✡✍✑ ✕ ✄✗✜ ✕ ✜ ✑✴✟ ✍✕✡✪ ✍✗✩ ✑✴✟ B✁✕✩✟ ✄❍ ✄✺✁ ♥✟✞✩ ✕ ✜ ✑✴✍✑ ✪ ❍✄✁ ✡✍✗✓ B✁✄✆✞ ✟✡✜ ✪ ❈✟ ✴✍✭✟ ✕✗✭✟✗✑ ✟✩ ✍✞✼✄✁✕✑✴✡✜ ✑✴✍✑ ✏✄✗✭✟✁✼✟ ✟✌✏ ✟ ✟✩✕✗✼✞✓ ❍✍✜ ✑ ✷ ❘✴✟ ✜ ✟ B ✄✕✗✑ ✜ ✍✁✟ ✜ ✄✡✟✑ ✕✡✟ ✜ ✄✭✟✁✞✄ ✄❙✟✩ ✆✓ ✟✗✑✴✺✜ ✕ ✍✜ ✑ ✜ ✄❍ ✜✓✡✆ ✄✞ ✕ ✏ ✏ ✄✡B✺✑ ✕✗✼ ✪ ✟ ✜B ✟ ✏✕ ✍✞✞✓ ✁✟ ✏ ✟✗✑ ✏ ✄✗✭✟✁✑ ✜ ✪ ❈✴✄ ✍✁✟ ✍B✑ ✑ ✄ ✑✴✕✗❙ ✑✴✍✑ ✑✴✟ ✟✌✕ ✜ ✑ ✟✗✏ ✟ ✄❍ ✸✍B✞✟ ✄✁ ✸✍✑✴✟✡✍✑ ✕ ✏ ✍ ✁✟✗✩✟✁✜ ✸✍✑ ✞ ✍✆ ✍✗✩ ➇✄✁✑ ✁✍✗ ✄✆✜ ✄✞ ✟✑ ✟ ✷ ❦ ✑ ✕ ✜ ✑ ✁✺✟ ✑✴✍✑ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✏ ✍✗ ✆ ✟ ✡✍✩✟ ✑ ✄ ✭✍✗✕ ✜✴ ✕✗ ✑✴✟ ✜ ✟✗✜ ✟ ✑✴✍✑ ✕✗ B✁✕✗✏✕B✞✟ ✪ ✍✗✓ ♥✗✕ ✑ ✟ ✜ ✟➅✺✟✗✏ ✟ ✄❍ ✍✞✼✟✆✁✍✕ ✏ ✄B ✟✁ ✍✑ ✕ ✄✗✜ ✏ ✍✗ ✆ ✟ ✁✟B✁✟ ✜ ✟✗✑ ✟✩ ✟✌✍✏✑ ✞✓ ✄✗ ✍ ✏✄✡B✺✑ ✟✁ ✆✓ ✡✟ ✍✗✜ ✄❍ ✍BB✁✄B✁✕ ✍✑ ✟ ✜✓✡✆ ✄✞✕ ✏ ✄B ✟✁✍✑ ✕ ✄✗✜ ✷ ✣✗✞ ✟ ✜ ✜ ✑✴✟ B✁✄✆✞ ✟✡ ✆ ✟✕✗✼ ✜ ✄✞✭✟✩ ✕ ✜ ✍ ♥✗✕ ✑ ✟ ✄✗✟ ✪ ✴✄❈✟✭✟✁ ✪ ✑✴✕ ✜ ✄✗✞✓ ✩✟❍✟✁✜ ✑✴✟ ✕✗✟✭✕ ✑ ✍✆✞ ✟ ✍BB✁✄✌✕✡✍✑ ✕✄✗✜ ✑ ✄ ✑✴✟ ✟✗✩ ✄❍ ✑✴✟ ✏ ✍✞✏✺✞ ✍✑ ✕✄✗ ✪ ✆✓ ❈✴✕ ✏✴ B ✄✕✗✑ ✑✴✟ ➅✺✍✗✑ ✕ ✑ ✕ ✟ ✜ ✄✗✟ ✕ ✜ ❈✄✁❙✕✗✼ ❈✕✑✴ ✡✍✓ ✴✍✭✟ ✆ ✟ ✏✄✡✟ ✟✌✑ ✁ ✍✄✁✩✕✗✍✁✕ ✞✓ ✏✺✡✆ ✟✁✜ ✄✡✟ ✷ ➇✞✄ ✍✑ ✕✗✼✐B ✄✕✗✑ ✍✁✕✑✴✡✟✑ ✕ ✏ ✕ ✜ ✍ ✗✍✡✟ ❍✄✁ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✑ ✜ ➉ ✴✍✆✕ ✑ ✄❍ ✩✄✕✗✼ ✑✴✟✕✁ B✁✺✗✕✗✼ ✍✑ ✟✭✟✁✓ ✜ ✑ ✟B ✍✞✄✗✼ ✑✴✟ ❈✍✓ ✄❍ ✍ ✏ ✍✞✏✺✞ ✍✑ ✕✄✗ ✁ ✍✑✴✟✁ ✑✴✍✗ ✕✗ ✍ ✜ ✕✗✼✞✟ ✍✏✑ ✍✑ ✑✴✟ ✟✗✩ ✷ ✟✴✕ ✏✴✟✭✟✁ ❈✍✓ ✄✗✟ B✁✄ ✏ ✟ ✟✩✜ ✪ ✕✗ ✜✄ ✍✑ ✕✗✼✐B ✄✕✗✑ ✄✁ ✜✓✡✆ ✄✞ ✕ ✏ ✍✞ ✞✓✪ ✑✴✟ ✡✍✕✗ B✁✄✆✞ ✟✡ ✄❍ ♥✗✩✕✗✼ ✍ ✁✍B✕✩✞✓ ✏ ✄✗✭✟✁✼✟✗✑ ✍✞✼✄✁✕ ✑✴✡ ✕ ✜ ✑✴✟ ✜ ✍✡✟ ✷ ❦✗ ✜✺✡✡✍✁✓✪ ✕ ✑ ✕ ✜ ✍ ✏ ✄✁✄✞ ✞✍✁✓ ✄❍ ➌➐✂ ➎ ✑✴✍✑ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✕ ✜ ✏ ✄✗✏ ✟✁✗✟✩ ❈✕ ✑✴ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✍✗✩ ✍✞ ✜ ✄ ❈✕ ✑✴ ✑✴✟ ✩✟ ✟B ✟✁ ❙✕✗✩✜ ✄❍ ✟✁✁✄✁✜ ✍✜ ✜ ✄ ✏✕ ✍✑ ✟✩ ❈✕✑✴ ✏✄✗✭✟✁✼✟✗✏ ✟ ✄❍ ✍BB✁✄✌✕✡✍✑ ✕✄✗✜ ✪ ❈✴✕ ✏✴ ✼✄ ✆✓ ✭✍✁✕ ✄✺✜ ✗✍✡✟ ✜ ➌ ✑ ✁✺✗✏ ✍✑ ✕ ✄✗ ✪ ✩✕ ✜ ✏✁✟✑ ✕ ✘ ✍✑ ✕ ✄✗ ✪ ✕ ✑ ✟✁ ✍✑ ✕✄✗➎ ✷ ❍ ✏ ✄✺✁✜ ✟ ✄✗✟ ✏ ✄✺✞✩ ✏✴✄ ✄ ✜ ✟ ✑ ✄ ✡✍❙✟ ➌➐✂ ➎ ✡✄✁✟ ✟✌B✞ ✕ ✏✕ ✑ ✆✓ ✍✩✩✕✗✼ ❈✄✁✩✜ ✑ ✄ ✩✟ ✜ ✏✁✕✆ ✟ ✑✴✟ ✜ ✟ ✍BB✁✄✌✕✡✍✑ ✕ ✄✗✜ ✍✗✩ ✟✁✁✄✁✜ ✷ ➄✺✑ ✄✗✏ ✟ ❈✄✁✩✜ ✆ ✟✼✕✗ ✑ ✄ ✆ ✟ ✍✩✩✟✩ ✕✑ ✕ ✜ ✴✍✁✩ ✑ ✄ ❙✗✄❈ ❈✴✟✁✟ ✑ ✄ ✜ ✑ ✄B ✪ ❍✄✁ ➌➐✂ ➎ ✍✞ ✜ ✄ ❍✍✕ ✞ ✜ ✑ ✄ ✡✟✗✑ ✕✄✗ ✜ ✄✡✟ ✄✑✴✟✁ ✕✡B ✄✁✑ ✍✗✑ ✡✍✑ ✑ ✟✁✜ ⑦ ✑✴✍✑ ✑✴✟ ✜ ✟ ✍✞✼✄✁✕ ✑✴✡✜ ✍✁✟ ✕✡B✞ ✟✡✟✗✑ ✟✩ ✄✗ ✏✄✡B✺✑ ✟✁✜ ✪ ❈✴✄ ✜ ✟ ✍✁✏✴✕✑ ✟ ✏✑✺✁✟ ✡✍✓ ✆ ✟ ✍✗ ✕✡B ✄✁✑ ✍✗✑ B ✍✁✑ ✄❍ ✑✴✟ B✁✄✆✞✟✡✢ ✑✴✍✑ ✁✟✞ ✕ ✍✆✕ ✞ ✕✑✓ ✍✗✩ ✟✌✏✕ ✟✗✏✓ ✍✁✟ B ✍✁ ✍✡✄✺✗✑ ✼✄ ✍✞ ✜ ✢ ✑✴✍✑ ✜ ✄✡✟ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✑ ✜ ❈✁✕✑ ✟ B✁✄✼✁ ✍✡✜ ✍✗✩ ✄✑✴✟✁ ✜ B✁✄✭✟ ✑✴✟ ✄✁✟✡✜ ✢ ✍✗✩ ✡✄ ✜ ✑ ✕✡B ✄✁✑ ✍✗✑ ✪ ✑✴✍✑ ✍✞ ✞ ✄❍ ✑✴✕ ✜ ❈✄✁❙ ✕ ✜ ➛✥✥ ➦➺ ➧★➪ ✍BB✞✕ ✟✩ ✩✍✕ ✞✓ ✍✗✩ ✜✺✏ ✏ ✟ ✜ ✜ ❍✺✞ ✞✓ ✑ ✄ ✑✴✄✺✜ ✍✗✩✜ ✄❍ ✍BB✞ ✕ ✏ ✍✑ ✕ ✄✗✜ ✄✗ ✡✕ ✞ ✞✕ ✄✗✜ ✄❍ ✏✄✡B✺✑ ✟✁✜ ✍✁✄✺✗✩ ✑✴✟ ❈✄✁✞✩ ✷ ♦❘✴✟ B✁✄✆✞✟✡✜ ✄❍ ✏ ✄✗✑ ✕✗✺✄✺✜ ✡✍✑✴✟✡✍✑ ✕ ✏ ✜ ♣ ✍✁✟ ✑✴✟ B✁✄✆✞✟✡✜ ✑✴✍✑ ✜ ✏✕ ✟✗✏ ✟ ✍✗✩ ✟✗✼✕✗✟ ✟✁✕✗✼ ✍✁✟ ✆✺✕ ✞✑ ✺B ✄✗ ✢ ❈✕ ✑✴✄✺✑ ✗✺✡✟✁✕ ✏ ✍✞ ✡✟✑✴✄ ✩✜ ✪ ✜ ✏✕ ✟✗✏ ✟ ✍✗✩ ✟✗✼✕✗✟ ✟✁✕✗✼ ✍✜ B✁✍✏✑ ✕ ✏ ✟✩ ✑ ✄ ✩✍✓ ❈✄✺✞✩ ✏ ✄✡✟ ➅✺✕ ✏❙✞✓ ✑ ✄ ✍ ✴✍✞✑ ✷ ❘✴✟✓ ✍✁✟ ✍✞ ✜ ✄ ✑✴✟ B✁✄✆✞ ✟✡✜ ✑✴✍✑ B✁✟ ✄ ✏ ✏✺B✕ ✟✩ ✡✄ ✜ ✑ ✡✍✑✴✟✡✍✑ ✕ ✏✕ ✍✗✜ ❍✁✄✡ ✑✴✟ ✑ ✕✡✟ ✄❍ ☎✟❈✑ ✄✗ ✑ ✄ ✑✴✟ ✑❈✟✗✑ ✕ ✟✑✴ ✏ ✟✗✑✺✁✓✷ ➔✜ ✡✺✏✴ ✍✜ ✍✗✓ B✺✁✟ ✡✍✑✴✟✡✍✑ ✕ ✏✕ ✍✗✜ ✪ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✑ ✜ ✍✁✟ ✑✴✟ ✴✟✕✁✜ ✑ ✄ ✑✴✟ ✼✁✟ ✍✑ ✑ ✁✍✩✕✑ ✕ ✄✗ ✄❍ ❏✺✞✟✁ ✪ ② ✍✼✁ ✍✗✼✟ ✪ ❢✍✺✜ ✜ ✍✗✩ ✑✴✟ ✁✟ ✜ ✑ ✷ ❦❍ ❏✺✞ ✟✁ ❈✟✁✟ ✍✞✕✭✟ ✑ ✄ ✩✍✓✪ ✴✟ ❈✄✺✞✩✗ ➉ ✑ ✆ ✟ B✁✄✭✕✗✼ ✟✌✕ ✜ ✑ ✟✗✏ ✟ ✑✴✟✄✁✟✡✜ ✷ ➤
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❘✟✗ ✓✟ ✍✁✜ ✍✼✄ ✪ ❦ ❈✄✺✞✩ ✴✍✭✟ ✜ ✑ ✄%% ✟✩ ✍✑ ✑✴✕ ✜ % ✄✕✗✑ ✷ ➄✺✑ ✑✴✟ ✟✭✄✞✺✑ ✕ ✄✗ ✄❍ ✏ ✄✡%✺✑ ✕✗✼ ✕✗ ✑✴✟ % ✍✜ ✑ ✩✟ ✏ ✍✩✟ ✴✍✜ ✼✕✭✟✗ ✑✴✟ ✩✕◆✟✁ ✟✗✏ ✟ ✆ ✟✑❈✟ ✟✗ ➌➐ ⑩ ➎ ✍✗✩ ➌➐✂ ➎ ✍ ✗✟❈ ✑ ✄%✕ ✏ ✍✞ ✕ ✑✓✷ ② ✟✑ ✺✜ ✁✟✑✺✁✗ ✑ ✄ Z✉ ✈ ✇ ✷ ✸✺✏✴ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✏✄✡%✺✑ ✍✑ ✕✄✗ ✩✟% ✟✗✩✜ ✄✗ ✞ ✕✗✟ ✍✁ ✍✞✼✟✆✁ ✍ ✪ ✍✗✩ ✑✴✕ ✜ ✴✕ ✼✴✞✓ ✩✟✭✟✞✄% ✟✩ ✜✺✆d ✟ ✏✑ ✴✍✜ ✆ ✟ ✟✗ ✑✴✟ ✏✄✁ ✟ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✜ ✕✗✏ ✟ ✑✴✟ ✆ ✟✼✕✗✗✕✗✼ ✷ ☎✺✐ ✡✟✁✕ ✏ ✍✞ ✞ ✕✗✟ ✍✁ ✍✞✼✟✆✁✍ ✜ ✟✁✭✟✩ ✍✜ ✑✴✟ ✜✺✆d ✟ ✏✑ ❈✕✑✴ ✁✟ ✜% ✟ ✏✑ ✑ ✄ ❈✴✕ ✏✴ ✑✴✟ ✗✄❈ ✜ ✑ ✍✗✩✍✁✩ ✏✄✗✏ ✟%✑ ✜ ✄❍ ✜ ✑ ✍✆✕ ✞ ✕✑✓✪ ✏✄✗✩✕✑ ✕✄✗✕✗✼ ✪ ✍✗✩ ✆ ✍✏❙❈✍✁✩ ✟✁✁✄✁ ✍✗✍✞✓✜ ✕ ✜ ❈✟✁ ✟ ✩✟♥✗✟✩ ✍✗✩ ✜✴✍✁% ✟✗✟✩ ✪ ✍✗✩ ✑✴✟ ✏ ✟✗✑ ✁ ✍✞ ♥✼✺✁✟ ✕✗ ✑✴✟ ✜ ✟ ✩✟✭✟✞✄%✡✟✗✑ ✜ ✪ ❍✁✄✡ ✑✴✟ ⑩ ❶ ❒ → ✜ ✑ ✄ ✴✕ ✜ ✩✟ ✍✑✴ ✕✗ ⑩ ❶ ❷ ✝ ✪ ❈✍✜ ➹✕✡ ✟✕ ✞❙✕✗✜ ✄✗ ✷ ❦ ✴✍✭✟ ✡✟✗✑ ✕ ✄✗✟✩ ✑✴✍✑ Z✉ ✈ ✇ ✴✍✜ ✑✴✟ ✺✗✺✜✺✍✞ ❍✟ ✍✑✺✁✟ ✑✴✍✑ ✕ ✑ ✏ ✍✗ ✆ ✟ ✜ ✄✞✭✟✩ ✕✗ ✍ ♥✗✕ ✑ ✟ ✜ ✟➅✺✟✗✏ ✟ ✄❍ ✄% ✟✁ ✍✑ ✕ ✄✗✜ ✷ ❦✗ ❍✍✏✑ ✪ Z✉ ✈ ✇ ✕ ✜ ✡✄✁✟ ✺✗✺✜✺✍✞ ✑✴✍✗ ✑✴✍✑ ✪ ❍✄✁ ✑✴✟ ✜ ✑ ✍✗✩✍✁✩ ✍✞✼✄✁✕✑✴✡ ❍✄✁ ✜ ✄✞✭✕✗✼ ✕✑ ✪ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞ ✕✡✕✗✍✑ ✕ ✄✗ ✪ ✑✺✁✗✜ ✄✺✑ ✑ ✄ ✴✍✭✟ ✟✌✑ ✁✍✄✁✩✕✗✍✁✕ ✞✓ ✏✄✡%✞ ✕ ✏ ✍✑ ✟✩ ✜ ✑ ✍✆✕ ✞ ✕✑✓ %✁✄% ✟✁✑ ✕ ✟ ✜ ✷ ✡✄✗ ☎✟✺✡✍✗✗ ❈✁✄✑ ✟ ⑩ ❷ → % ✍✼✟ ✜ ✄❍ ✡✍✑✴✟✡✍✑ ✕ ✏ ✜ ✄✗ ✑✴✕ ✜ ✑ ✄%✕ ✏ ✢ ❘✺✁✕✗✼ ❈✁✄✑ ✟ ✄✗✟ ✄❍ ✴✕ ✜ ✡✍d ✄✁ % ✍% ✟✁✜ ✢ ✟✕ ✞❙✕✗✜ ✄✗ ✩✟✭✟✞✄% ✟✩ ✍ ✑✴✟✄✁✓ ✑✴✍✑ ✼✁✟❈ ✕✗✑ ✄ ✑❈✄ ✆ ✄ ✄❙✜ ✍✗✩ ✍ ✏ ✍✁ ✟ ✟✁ ✷ ✍✟✑ ✑✴✟ ❍✍✏✑ ✁✟✡✍✕✗✜ ✑✴✍✑ ❍✄✁ ✏ ✟✁✑ ✍✕✗ ✏ ✒ ✏ ✡✍✑ ✁✕ ✏ ✟ ✜ ✪ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞ ✕✡✕✗✍✑ ✕ ✄✗ ❈✕ ✑✴ % ✍✁✑ ✕ ✍✞ %✕✭✄✑ ✕✗✼ ✍✡%✞ ✕♥✟ ✜ ✁✄✺✗✩✕✗✼ ✟✁✁✄✁✜ ✆✓ ✍ ❍✍✏✑ ✄✁ ✄❍ ✄✁✩✟✁ ✂ ✖ ✪ ✡✍❙✕✗✼ ✕✑ ✍ ✺✜ ✟✞✟ ✜ ✜ ✍✞✼✄✁✕ ✑✴✡ ✕✗ ✑✴✟ ❈✄✁✜ ✑ ✏ ✍✜ ✟ ✷ ❦ ✑ ✜ ✟ ✟✡✜ ✑✴✍✑ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞ ✕✡✕✗✍✑ ✕ ✄✗ ❈✄✁❙✜ ✕✗ %✁✍✏✑ ✕ ✏ ✟ ✆ ✟ ✏ ✍✺✜ ✟ ✑✴✟ ✜ ✟✑ ✄❍ ✡✍✑ ✁✕ ✏ ✟ ✜ ❈✕ ✑✴ ✜✺✏✴ ✆ ✟✴✍✭✕ ✄✁ ✕ ✜ ✭✍✗✕ ✜✴✕✗✼✞✓ ✜✡✍✞ ✞ ✪ ✆✺✑ ✑ ✄ ✑✴✕ ✜ ✩✍✓✪ ✗✄✆ ✄ ✩✓ ✴✍✜ ✍ ✏ ✄✗✭✕✗✏✕✗✼ ✟✌%✞ ✍✗✍✑ ✕ ✄✗ ✄❍ ❈✴✓ ✑✴✕ ✜ ✜✴✄✺✞✩ ✆ ✟ ✜ ✄ ✷ ❦✗ ✡✍✗✕❍✄✞✩ ❈✍✓✜ ✪ ✑✴✟✗ ✪ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞ ✕✡✕✗✍✑ ✕ ✄✗ ✕ ✜ ✍✑✓%✕ ✏ ✍✞ ✷ ➇✟❈ ✗✺✡✟✁✕ ✏ ✍✞ ✍✞✼✄✁✕✑✴✡✜ ✴✍✭✟ ✜✺✏✴ ✜✺✆✑ ✞✟ ✜ ✑ ✍✆✕ ✞ ✕✑✓ %✁✄% ✟✁✑ ✕ ✟ ✜ ✪ ✍✗✩ ✏ ✟✁✑ ✍✕✗✞✓ ✗✄ ✄✑✴✟✁ ❈✍✜ ✜ ✏✁✺✑ ✕✗✕ ✘ ✟✩ ✕✗ ✜✺✏✴ ✩✟%✑✴ ✆✓ ✭✄✗ ☎✟✺✡✍✗✗ ✪ ❘✺✁✕✗✼ ✪ ✍✗✩ ✟✕ ✞❙✕✗✜ ✄✗ ✷ ❘✴✟ ✟◆✟ ✏✑ ❐ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✪ ❈✴✕ ✏✴ ✜✴✄✺✞✩ ✴✍✭✟ ✆ ✟ ✟✗ ✍ ✜ ✕✩✟ ✜✴✄❈✪ ✞ ✕✗✼✟✁✟✩ ✕✗ ✑✴✟ ✜% ✄✑ ✞ ✕ ✼✴✑ ❈✴✕ ✞ ✟ ✄✺✁ ♥✟✞✩ ❈✍✜ ✓✄✺✗✼ ✍✗✩ ✼✁✟❈ ✕✗✑ ✄ ✑✴✟ ✏ ✍✗✄✗✕ ✏ ✍✞ ✍✞✼✄✁✕ ✑✴✡ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✷ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✜ ✟✑ ✑✴✟ ✍✼✟✗✩✍ ✪ ✟✕ ✞❙✕✗✜ ✄✗ ✜ ✟✑ ✑✴✟ ✑ ✄✗✟ ✪ ✍✗✩ ✑✴✟ ✩✕ ✜ ✑ ✁✟ ✜ ✜ ✕✗✼ ✁✟ ✜✺✞✑ ✴✍✜ ✆ ✟ ✟✗ ➌➐ ⑩ ➎ ✷ ❍ ✏ ✄✺✁✜ ✟ ✑✴✟✁✟ ✕ ✜ ✡✄✁✟ ✑✴✍✗ ✑✴✕ ✜ ✑ ✄ ✑✴✟ ✴✕ ✜ ✑ ✄✁✓ ✄❍ ✴✄❈ ➌➐ ⑩ ➎ ✍✏➅✺✕✁✟✩ ✏✺✁✁✟✗✏✓✷ ❦✗ ✑✴✟ ✟ ✍✁✞✓ ✓✟ ✍✁✜ ✄❍ ✏ ✄✡%✺✑ ✟✁✜ ✪ ✕ ✑ ❈✍✜ ✕✗✟✭✕✑ ✍✆✞✟ ✑✴✍✑ ✍✁✕✑✴✡✟✑ ✕ ✏ ✕ ✜ ✜✺✟ ✜ ❈✄✺✞✩ ✁✟ ✏ ✟✕✭✟ ✏ ✄✗✏ ✟✁✑ ✟✩ ✍✑ ✑ ✟✗✐ ✑ ✕ ✄✗ ✷ ➇✕✌✟✩✐% ✄✕✗✑ ✏✄✡%✺✑ ✍✑ ✕ ✄✗ ✁✟➅✺✕✁ ✟✩ ✏ ✍✁ ✟❍✺✞ ✑✴✄✺✼✴✑ ✍✗✩ ✗✄✭✟✞ ✴✍✁✩❈✍✁✟ ✢ ✜✄ ✍✑ ✕✗✼✐% ✄✕✗✑ ✏ ✄✡%✺✑ ✍✑ ✕ ✄✗ ✍✁✁✕✭✟✩ ✍✜ ✍ ✜ ✟ ✏✄✗✩ ✁ ✟✭✄✞✺✑ ✕ ✄✗ ✍ ❍✟❈ ✓✟ ✍✁✜ ✞ ✍✑ ✟✁ ✷ ✣✗✑ ✕ ✞ ✑✴✟ ✜ ✟ ✡✍✑ ✑ ✟✁ ✜ ❈✟✁✟ ❈✟✞ ✞ ✺✗✩✟✁ ✜ ✑ ✄ ✄ ✩ ✕✑ ❈✍✜ ✗✍✑✺✁✍✞ ✑✴✍✑ ✍✁✕ ✑✴✡✟✑ ✕ ✏ ✕ ✜ ✜✺✟ ✜ ✜✴✄✺✞✩ ✆ ✟ ✍ ✏ ✟✗✑ ✁ ✍✞ ✑ ✄%✕ ✏ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞ ✐ ✓✜ ✕ ✜ ✪ ✍✗✩ ✪ ✆ ✟ ✜ ✕✩✟ ✜ ✑✴✕ ✜ ✪ ✍✗✄✑✴✟✁ ❍✄✁✏ ✟ ❈✍✜ ✍✑ ❈✄✁❙ ✷ ❘✴✟✁✟ ✕ ✜ ✍ ✼✟✗✟✁ ✍✞ %✁✕✗✏✕%✞ ✟ ✄❍ ✏✄✡%✺✑ ✕✗✼ ✑✴✍✑ ✜ ✟ ✟✡✜ ✑ ✄ ✴✍✭✟ ✗✄ ✗✍✡✟ ⑦ ➜➙ ➧ ➲➛➭ ➜ ➧ ➸ ➜➙ ➧ ➻➯➩✥➵➜ ➧ ➸➪ ➜➙ ➧ ➩➯ ➸➧ ➺➩✥➯ ➸➜ ➛➫➜ ➜➙ ➧ ➭✥➧ ➧ ★ ➯➲ ➛ ➦✪ ➯ ✬ ➸➺ ➜➙➩➭ ✭ ❦✗ ✑✴✟ ✟ ✍✁✞✓ ✓✟ ✍✁ ✜ ✪ ❈✕✑✴ ✑✴✟ ✟ ✍✁✞✓ ✏ ✄✡%✺✑ ✟✁✜ ✪ ✑✴✟ ✩✍✗✼✟✁✜ ✄❍ ✕✗✜ ✑ ✍✆✕ ✞✕ ✑✓ ❈✟✁✟ ✗✟ ✍✁✞✓ ✍✜ ✼✁ ✟ ✍✑ ✍✜ ✑✴✟✓ ✍✁✟ ✑ ✄ ✩✍✓✪ ✍✗✩ ❍✍✁ ✞✟ ✜ ✜ ❍✍✡✕ ✞ ✕ ✍✁ ✷ ❘✴✟ ✼✍%✜ ✆ ✟✑❈✟ ✟✗ ❍✍✜ ✑ ✍✗✩ ✜ ✞✄❈ ✍✞✼✄✁✕✑✴✡✜ ✪ ✴✄❈✟✭✟✁ ✪ ❈✟✁✟ ✗✍✁✁✄❈✟✁ ✷ ➔ ✩✟✭✟✞✄%✡✟✗✑ ✴✍✜ ✄ ✏ ✏✺✁✁✟✩ ✕✗ ✁✟ ✏ ✟✗✑ ✓✟ ✍✁✜ ✑✴✍✑ ✁✟✜✟ ✏✑ ✜ ✴✄❈ ❍✍✁ ❈✟ ✴✍✭✟ ✏✄✡✟ ❍✁✄✡ ✑✴✍✑ ✑ ✕✡✟ ✷ ❦✗✜ ✑ ✍✗✏ ✟ ✜ ✴✍✭✟ ✆ ✟ ✟✗ ✍✏ ✏✺✡✺✞✍✑ ✕✗✼ ✕✗ ❈✴✕ ✏✴ ✪ ✟✭✟✗ ✑✴✄✺✼✴ ✍ ♥✗✕ ✑ ✟ ✍✞✼✄✁✕ ✑✴✡ ✟✌✕ ✜ ✑ ✜ ❍✄✁ ✍ %✁✄✆✞ ✟✡✪ ✍✗ ✕✗♥✗✕ ✑ ✟ ✍✞✼✄✁✕ ✑✴✡ ✡✍✓ ✆ ✟ ✆ ✟✑ ✑ ✟✁ ✷ ❘✴✟ ✩✕ ✜ ✑ ✕✗✏✑ ✕ ✄✗ ✑✴✍✑ ✜ ✟ ✟✡✜ ✍✆✜ ✄✞✺✑ ✟ ❍✁✄✡ ✍ ✞✄✼✕ ✏ ✍✞ % ✄✕✗✑ ✄❍ ✭✕ ✟❈ ✑✺✁✗✜ ✄✺✑ ✑ ✄ ✴✍✭✟ ✞ ✕✑ ✑ ✞✟ ✕✡% ✄✁✑ ✍✗✏ ✟ ✕✗ %✁ ✍✏✑ ✕ ✏ ✟❜✍✗✩ ✕✗ ❍✍✏✑ ✪ ➔✆ ✟✞ ✍✗✩ ❢✍✞✄✕ ✜ ✗✄✑❈✕ ✑✴✜ ✑ ✍✗✩✕✗✼ ✪ ✞✍✁✼✟✐ ✜ ✏ ✍✞✟ ✡✍✑ ✁✕✌ ✟✕✼✟✗✭✍✞✺✟ %✁✄✆✞✟✡✜ ✍✁✟ ✍✆ ✄✺✑ ✍✜ ✟ ✍✜✓ ✑ ✄ ✜ ✄✞✭✟ ✕✗ %✁ ✍✏✑ ✕ ✏ ✟ ✍✜ ✞✕✗✟ ✍✁ ✜✓✜ ✑ ✟✡✜ ✄❍ ✟➅✺✍✑ ✕✄✗✜ ✷ ➇✄✁ Z✉ ✈ ✇ ✪ ✕ ✑ ✟✁ ✍✑ ✕✭✟ ✡✟✑✴✄ ✩✜ ✍✁✟ ✆ ✟ ✏ ✄✡✕✗✼ ✡✄✁✟ ✍✗✩ ✡✄✁✟ ✄❍✑ ✟✗ ✑✴✟ ✡✟✑✴✄ ✩✜ ✄❍ ✏✴✄✕ ✏ ✟ ✍✜ ✏ ✄✡%✺✑ ✟✁✜ ✼✁✄❈ ❍✍✜ ✑ ✟✁ ✪ ✡✍✑ ✁✕ ✏ ✟ ✜ ✼✁✄❈ ✞ ✍✁✼✟✁ ✍✗✩ ✞ ✟ ✜ ✜ ✜% ✍✁✜ ✟ ➌✆ ✟ ✏ ✍✺✜ ✟ ✄❍ ✑✴✟ ✍✩✭✍✗✏ ✟ ❍✁✄✡ ✂➐ ✑ ✄ ➤➐ ✜ ✕✡✺✞ ✍✑ ✕ ✄✗✜ ➎ ✪ ✍✗✩ ✑✴✟ ➋ ➌➍✱ ➎ ✄% ✟✁ ✍✑ ✕✄✗ ✏✄✺✗✑ ✜ ✄❍ ✑✴✟ ✺✜✺✍✞ ✩✕✁ ✟ ✏✑ ➌✈ ♥✗✕ ✑ ✟ ➎ ✍✞✼✄✁✕ ✑✴✡✜ ✆ ✟ ✏ ✄✡✟ ✟✭✟✁ ✡✄✁✟ % ✍✕✗❍✺✞ ✷ ❘✴✟ ✗✍✡✟ ✄❍ ✑✴✟ ✗✟❈ ✼ ✍✡✟ ✕ ✜ ➺ ➜ ➧ ➸➛ ➜➺ ➯ ➫ ✲➺ ➜➙ ✥➸➧ ➻➯ ➫ ★➺ ➜➺ ➯ ➫➺➫✪ ✭ ❦✗✏✁✟ ✍✜ ✕✗✼✞✓ ✄❍✑ ✟✗ ✕ ✑ ✕ ✜ ✗✄✑ ✄%✑ ✕✡✍✞ ✑ ✄ ✑ ✁✓ ✑ ✄ ✜ ✄✞✭✟ ✍ ❸
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✁✄✆✞✟✡ ✟✌✍✏✑ ✞✓ ✕✗ ✄✗✟ ✍✜ ✜ ✢ ✕✗✜ ✑ ✟ ✍✩ ✪ ✜ ✄✞✭✟ ✕✑ ✍✁✄✌✕✡✍✑ ✟✞✓✪ ✑✴✟✗ ✕ ✑ ✟✁ ✍✑ ✟ ✷ ✸✺✞✑ ✕✼✁✕✩ ✡✟✑✴✄ ✩✜ ✪ ✟✁✴✍✜ ✑✴✟ ✡✄ ✜ ✑ ✕✡ ✄✁✑ ✍✗✑ ✩✟✭✟✞✄✡✟✗✑ ✕✗ ✗✺✡✟✁✕ ✏ ✍✞ ✏✄✡✺✑ ✍✑ ✕✄✗ ✕✗ ✑✴✟ ✍✜ ✑ ✑❈✟✗✑✓ ✓✟ ✍✁✜ ✪ ✍✁✟ ✆ ✍✜ ✟✩ ✄✗ ✍ ✁ ✟ ✏✺✁✜ ✕✭✟ ✍✞ ✕ ✏ ✍✑ ✕ ✄✗ ✄❍ ✑✴✕ ✜ ✕✩✟ ✍ ✷ ❏✭✟✗ ✩✕✁✟ ✏✑ ✍✞✼✄✁✕✑✴✡✜ ✴✍✭✟ ✆ ✟ ✟✗ ✍◆✟ ✏✑ ✟✩ ✆✓ ✑✴✟ ✗✟❈ ✡✍✗✗✟✁ ✄❍ ✏✄✡✺✑ ✕✗✼ ✷ ❘✴✍✗❙✜ ✑ ✄ ✑✴✟ ❈✄✁❙ ✄❍ ❱❙✟ ✟✞ ✍✗✩ ✄✑✴✟✁✜ ✪ ✕✑ ✴✍✜ ✆ ✟ ✟✗ ✗✄✑ ✕ ✏ ✟✩ ✑✴✍✑ ✑✴✟ ✟✌ ✟✗✜ ✟ ✄❍ ✡✍❙✕✗✼ ✍ ✩✕✁✟ ✏✑ ✡✟✑✴✄ ✩ ✜ ✑ ✍✆✞✟❜✜ ✍✓✪ ✄❍ ✕✭✄✑ ✕✗✼ ✕✗ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗❜✡✍✓ ✕✗ ✏ ✟✁✑ ✍✕✗ ✏ ✄✗✑ ✟✌✑ ✜ ✆ ✟ ✏✄ ✜ ✑ ✐ ✕✗✟◆✟ ✏✑ ✕✭✟ ✷ ❦✗✜ ✑ ✟ ✍✩ ✪ ✜❙✕ ✑✴✍✑ ✜ ✑ ✟❜✜ ✄✞✭✟ ✑✴✟ ✁✄✆✞ ✟✡ ✩✕✁ ✟ ✏✑ ✞✓ ✆✺✑ ✺✗✜ ✑ ✍✆✞✓✪ ✑✴✟✗ ✩✄ ✄✗✟ ✄✁ ✑❈✄ ✜ ✑ ✟✜ ✄❍ ✕ ✑ ✟✁✍✑ ✕✭✟ ✁ ✟♥✗✟✡✟✗✑ ✷ ♦❏✌✍✏✑ ♣ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✆ ✟ ✏ ✄✡✟ ✜ q ✺✜ ✑ ✍✗✄✑✴✟✁ ✁✟ ✏ ✄✗✩✕ ✑ ✕✄✗✟✁ r s✑✴✟✁ ✁✄✆✞✟✡✜ ✆ ✟ ✜ ✕✩✟ ✜ t✉ ✈ ✇ ✴✍✭✟ ✺✗✩✟✁✼✄✗✟ ✍✗✍✞✄✼✄✺✜ ✏✴✍✗✼✟ ✜ ✪ ✍✗✩ ✑✴✟ ❍✍✡✄✺✜ ✟✌✍✡✞ ✟ ✕ ✜ ✞ ✕✗✟ ✍✁ ✁✄✼✁ ✍✡✡✕✗✼ ✷ ②✕✗✟ ✍✁ ✁✄✼✁✍✡✡✕✗✼ ✁✄✆✞ ✟✡✜ ✍✁✟ ✡✍✑✴✟✡✍✑ ✕ ✏ ✍✞✞✓ ♥✗✕✑ ✟ ✪ ✍✗✩ ❍✄✁ ✩✟ ✏ ✍✩✟ ✜ ✪ ✟✄✞✟ ✜ ✄✞✭✟✩ ✑✴✟✡ ✆✓ ✍ ♥✗✕✑ ✟ ✍✞✼✄✁✕ ✑✴✡⑦ ✑✴✟ ✜ ✕✡✞ ✟✌ ✡✟✑✴✄ ✩ ✷ ❘✴✟✗ ⑨✍✁✡✍✁❙✍✁ ✍✗✗✄✺✗✏ ✟✩ ✕✗ ⑩ ❶ ❷ ❸ ✑✴✍✑ ✕ ✑ ✟✁ ✍✑ ✕✭✟ ✪ ✕✗♥✗✕ ✑ ✟ ✍✞✼✄✁✕✑✴✡✜ ✍✁✟ ✜ ✄✡✟✑ ✕✡✟ ✜ ✆ ✟✑ ✑ ✟✁ ✷ ❘✴✟ ✁✟ ✜✺✞✑ ✴✍✜ ✆ ✟ ✟✗ ✏ ✄✗✑ ✁✄✭✟✁✜✓✪ ✕✗✑ ✟✞✞ ✟ ✏✑✺✍✞ ✟✌✏✕✑ ✟✡✟✗✑ ✪ ✍✗✩ ✍ ✟✁✏ ✟✑ ✕✆✞ ✟ ✜✴✕❍✑ ✄❍ ✑✴✟ ✟✗✑ ✕✁✟ ♥✟✞✩ ✄❍ ✞ ✕✗✟ ✍✁ ✁✄✼✁✍✡✡✕✗✼ ✍❈✍✓ ❍✁✄✡ ✑✴✟ ✁ ✍✑✴✟✁ ✍✗✄✡✍✞✄✺✜ ✄ ✜ ✕ ✑ ✕✄✗ ✕ ✑ ✴✍✜ ✑ ✁✍✩✕ ✑ ✕✄✗✍✞✞✓ ✄ ✏ ✏✺✕ ✟✩ ✑ ✄❈✍✁✩✜ ✑✴✟ ✡✍✕✗✜ ✑ ✁ ✟ ✍✡ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✏✄✡✺✑ ✍✑ ✕ ✄✗ ✷ ❦ ✆ ✟✞✕ ✟✭✟ ✑✴✍✑ ✑✴✟ ✟✌✕ ✜ ✑ ✟✗✏ ✟ ✄❍ ♥✗✕ ✑ ✟ ✍✞✼✄✁✕✑✴✡✜ ❍✄✁ ✏ ✟✁✑ ✍✕✗ ✁✄✆✞✟✡✜ ✪ ✑ ✄✼✟✑✴✟✁ ❈✕✑✴ ✄✑✴✟✁ ✴✕ ✜ ✑ ✄✁✕ ✏ ✍✞ ❍✄✁✏ ✟ ✜ ✪ ✴✍✜ ✩✕ ✜ ✑ ✁✍✏✑ ✟✩ ✺✜ ❍✄✁ ✩✟ ✏ ✍✩✟ ✜ ❍✁✄✡ ✍ ✆ ✍✞ ✍✗✏ ✟✩ ✭✕ ✟❈ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✷ ❿✄✺✗✩✕✗✼ ✟✁✁✄✁ ✜ ✍✗✩ ✕✗✜ ✑ ✍✆✕ ✞ ✕ ✑✓ ✍✁✟ ✕✡ ✄✁✑ ✍✗✑ ✪ ✍✗✩ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✑ ✜ ❈✕ ✞✞ ✍✞❈✍✓✜ ✆ ✟ ✑✴✟ ✟✌ ✟✁✑ ✜ ✕✗ ✑✴✟ ✜ ✟ ✜✺✆q ✟ ✏✑ ✜ ✍✗✩ ✍✑ ✍✕✗✜ ✑ ✄ ✟✗✜✺✁✟ ✑✴✍✑ ✑✴✟ ✺✗❈✍✁✓ ✍✁✟ ✗✄✑ ✑ ✁✕ ✟✩ ✺ ✆✓ ✑✴✟✡✷ ➄✺✑ ✄✺✁ ✏ ✟✗✑ ✁✍✞ ✡✕ ✜ ✜ ✕ ✄✗ ✕ ✜ ✑ ✄ ✏ ✄✡✺✑ ✟ ➅✺✍✗✑ ✕ ✑ ✕ ✟ ✜ ✑✴✍✑ ✍✁ ✟ ✑✓✕ ✏ ✍✞ ✞✓ ✺✗✏ ✄✡✺✑ ✍✆✞ ✟ ✪ ❍✁✄✡ ✍✗ ✍✗✍✞✓✑ ✕ ✏ ✍✞ ✄✕✗✑ ✄❍ ✭✕ ✟❈✪ ✍✗✩ ✑ ✄ ✩✄ ✕✑ ❈✕ ✑✴ ✞ ✕✼✴✑✗✕✗✼ ✜ ✟ ✟✩ ✷ ➇✄✁ ✼✺✕✩✍✗✏ ✟ ✑ ✄ ✑✴✟ ❍✺✑✺✁✟ ❈✟ ✜✴✄✺✞✩ ✜ ✑✺✩✓ ✗✄✑ ❢✍✺✜ ✜ ✕ ✍✗ ✟✞✕✡✕✗✍✑ ✕✄✗ ✍✗✩ ✕ ✑ ✜ ✆ ✟✼✺✕ ✞ ✕✗✼ ✜ ✑ ✍✆✕ ✞ ✕✑✓ ✁✄ ✟✁✑ ✕ ✟ ✜ ✪ ✆✺✑ ✑✴✟ ✩✕ ✍✆ ✄✞✕ ✏ ✍✞✞✓ ❍✍✜ ✑ ✏✄✗q ✺✼✍✑ ✟ ✼✁ ✍✩✕ ✟✗✑ ✕✑ ✟✁✍✑ ✕ ✄✗❜✄✁ ❢✁✟ ✟✗✼✍✁✩ ✍✗✩ ❿✄❙✴✞✕✗ ➉ ✜ ➋ ➌➍➎ ✡✺✞✑ ✕ ✄✞ ✟ ✍✞✼✄✁✕ ✑✴✡ ❍✄✁ ✍✁✑ ✕ ✏✞ ✟ ✜ ✕✡✺✞ ✍✑ ✕ ✄✗✜❜✄✁ ✑✴✟ ✟✌ ✄✗✟✗✑ ✕ ✍✞ ✏✄✗✭✟✁✼✟✗✏ ✟ ✄❍ ✜ ✟ ✏✑ ✁✍✞ ✡✟✑✴✄ ✩✜ ❍✄✁ ✜ ✄✞✭✕✗✼ ✏ ✟✁✑ ✍✕✗ ➏➐❏✜❜✄✁ ✑✴✟ ✏✄✗✭✟✁✼✟✗✏ ✟ ✕✗ ➋ ➌ ⑩ ➎ ✕ ✑ ✟✁ ✍✑ ✕ ✄✗ ✍✏✴✕ ✟✭✟✩ ✆✓ ✡✺✞✑ ✕✼✁✕✩ ✡✟✑✴✄ ✩✜ ❍✄✁ ✡✍✗✓ ❙✕✗✩✜ ✄❍ ✁✄✆✞✟✡✜❜✄✁ ✟✭✟✗ ➄✄✁❈✟✕✗ ✍✗✩ ➄✄✁❈✟✕✗ ➉ ✜ ✡✍✼✕ ✏ ✍✞ ➔❢✸ ✕ ✑ ✟✁ ✍✑ ✕ ✄✗ ❍✄✁ ✩✟✑ ✟✁✡✕✗✕✗✼ ⑩ ✪ → → → ✪ → → → ✩✕✼✕ ✑ ✜ ✄❍ ➣ ✕✗ ✑✴✟ ✆✞ ✕✗❙ ✄❍ ✍✗ ✟✓✟ ✷ ↔➙ ➛ ➜ ✕ ✜ ✑✴✟ ✴✟ ✍✁✑ ✄❍ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✷ ➝➞➟ ➠➡ ✸✍✗✓ ✟ ✄✞ ✟ ✪ ✑ ✄ ✄ ✗✺✡✟✁✄✺✜ ✑ ✄ ✗✍✡✟ ✪ ✁✄✭✕✩✟✩ ✏ ✄✡✡✟✗✑ ✜ ✄✗ ✩✁✍❍✑ ✜ ✄❍ ✑✴✕ ✜ ✟ ✜ ✜ ✍✓✷ ❘✴✟✕✁ ✜✺✼✼✟ ✜ ✑ ✕✄✗✜ ✞✟✩ ✡✟ ✑ ✄ ✡✍✗✓ ✺✆✞ ✕ ✏ ✍✑ ✕ ✄✗✜ ✑✴✍✑ ❦ ❈✄✺✞✩ ✄✑✴✟✁❈✕ ✜ ✟ ✗✄✑ ✴✍✭✟ ❍✄✺✗✩ ✷ ❦ ✩✄ ✗✄✑ ✏✞ ✍✕✡ ✑✴✍✑ ✍✗✓ ✄❍ ✑✴✟ ✕✩✟ ✍✜ ✟✌✁✟ ✜ ✜ ✟✩ ✴✟✁✟ ✍✁✟ ✟✗✑ ✕✁✟✞✓ ✗✟❈✷ ❦✗ ❍✍✏✑ ✪ ➤ → ✓✟ ✍✁✜ ✍✼✄ ✪ ✕✗ ✴✕ ✜ ➥➦➧➩➧ ➫➜➭ ➯➲ ➳➵➩➧ ➸➺ ➻➛ ➦ ➼➫ ➛ ➦➾➭ ➺ ➭ ➪ ➏✟✑ ✟✁ ➶✟✗✁✕ ✏✕ ✩✟♥✗✟✩ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✍✜ ♦✑✴✟ ✑✴✟ ✄✁✓ ✄❍ ✏ ✄✗✜ ✑ ✁✺✏✑ ✕✭✟ ✡✟✑✴✄ ✩✜ ✕✗ ✡✍✑✴✟✡✍✑ ✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✷ ♣ s✑✴✟✁✜ ✴✍✭✟ ✟✌✁✟ ✜ ✜ ✟✩ ✜ ✕✡✕ ✞ ✍✁ ✭✕ ✟❈✜ ✢ ➹ ✄ ✜ ✟✴ ❘✁ ✍✺✆ ➌ ➘➯➩➩➵➫➺ ➻➛ ➜➺ ➯➫➭ ➯➲ ➜➙ ➧ ➼ ➘➴➪ ➷ ➬ ➮➱➎ ✪ ❍✄✁ ✟✌✍✡✞ ✟ ✪ ✩✟♥✗✟✩ ✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✍✜ ♦✑✴✟ ✍✗✍✞✓✜ ✕ ✜ ✄❍ ✏ ✄✗✑ ✕✗✺✄✺✜ ✍✞✼✄✁✕✑✴✡✜ ✷ ♣ ➇✄✁ ✑✴✍✑ ✡✍✑ ✑ ✟✁ ✪ ✆ ✄✑✴ ✑✴✟ ❿✍✗✩✄✡ ➶✄✺✜ ✟ ✍✗✩ ✑✴✟ s✌❍✄✁✩ ❏✗✼✞ ✕ ✜✴ ✩✕ ✏✑ ✕✄✗✍✁✕ ✟ ✜ ✄◆✟✁ ✆ ✟✑ ✑ ✟✁ ✩✟♥✗✕ ✑ ✕✄✗✜ ✑✴✍✗ ✑✴✟ ✑✴✁ ✟ ✟ ➅✺✄✑ ✟✩ ✴✟✁✟ ✷ ➔✗✩ ✜✴✄✺✞✩ ✑✴✟ ♥✟✞✩ ✆ ✟ ✏ ✍✞✞ ✟✩ ♦✗✺✡✟✁✕ ✏ ✍✞ ✍✗✍✞✓✜ ✕ ✜ ✪ ♣ ♦ ✜ ✏✕ ✟✗✑ ✕♥✏ ✏ ✄✡✺✑ ✕✗✼ ✪ ♣ ✄✁ ✜ ✄✡✟✑✴✕✗✼ ✟✞ ✜ ✟ ✟✗✑ ✕✁✟✞✓❐ ➌ ♦✡✍✑✴✟✡✍✑ ✕ ✏ ✍✞ ✟✗✼✕✗✟ ✟✁✕✗✼❐ ♣ ➎ ✷ ❘✴✍✑ ✕ ✜ ✍✗✄✑✴✟✁ ✟ ✜ ✜ ✍✓✷ ❒