introduction to complexity theory lecture notes - oded goldreich

Ngày tải lên: 12/05/2014, 02:42

... d 2 . u v - 3 - 1 3–8 WWLChen : Introduction to Complex Analysis It is easy to check that the function exp : R 0 → C \{0}, defined for every z ∈R 0 by exp(z)=e z , is one -to- one and onto. Remark. The region R 0 is ... Cauchy-Riemann equations (3) hold. Furthermore, we have (4) f  (z)= ∂u ∂x +i ∂v ∂x and f  (z)= ∂v ∂y − i ∂u ∂y . A natural question to ask is whether the Cauc...

Ngày tải lên: 31/03/2014, 15:17

... the strategic form game correspondin g to this gam e is HH HT TH TT Head -1 ,1 -1 ,1 1 ,-1 1 ,-1 Tail 1 ,-1 -1 ,1 1 ,-1 -1 ,1 Inform ation sets are very important! To see this, consider the follo w in ... Information 1 Head 2 Head Tail Tail 2 Head Tail O O (-1 , 1) (1, -1 ) (1, -1 ) (-1 , 1) The tree co nsists of 7 nodes. The first one is allocated t o play...

Ngày tải lên: 08/04/2014, 12:16

... Chapter 1 Introduction to Probability Theory 1.1 The Binomial Asset Pricing Model The binomial asset pricing model provides a powerful tool to understand arbitrage pricing theory and probability theory. ... removed in the Cantor set construction of Example 3.2.  CHAPTER 1. Introduction to Probability Theory 35 In addition to tossing a coin, another common random experiment...

Ngày tải lên: 18/10/2013, 03:20

Ngày tải lên: 17/12/2013, 15:19

... z)=x 2 + z 2 (6) which is illustrated in Figure xxx. 15 -2 -1 .75 -1 .5 -1 .25 -1 -0 .75 -0 .5 -0 .25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 -2 -1 .25 -0 .5 0.25 1 1.75 0 1 2 3 4 5 6 7 8 y x z y = x^2 + z^2 Figure ... 2001 1 Introduction to Portfolio Theory Consider the following investment problem. We can invest in two non-dividend paying stocks A and B over...

Ngày tải lên: 17/12/2013, 15:21

Ngày tải lên: 01/01/2014, 17:43

... boundaries mid-pt. freq. xf x 2 f xf 1 5. 0-9 . 0- 7 3 21 147 2 9. 0-1 3. 0- 11 5 55 605 3 13. 0-1 7. 0- 15 7 105 1,575 4 17. 0-2 1. 0- 19 6 114 2,166 5 21. 0-2 5. 0- 23 3 69 1,587 6 25. 0-2 9.0 27 1 27 729 Totals 25 ... previous MAT test are summarized below. class frequency, f xf x 2 f 40 -5 0- 4 50 -6 0- 6 6 0-7 0- 10 7 0-8 0- 15 8 0-9 0-...

Ngày tải lên: 17/03/2014, 03:20

... (3.2.44) 18 arXiv:hep-th/9709062 v2 30 Mar 1998 CERN-TH/9 7-2 18 hep-th/9709062 INTRODUCTION TO SUPERSTRING THEORY Elias Kiritsis ∗ Theory Division, CERN, CH-1211, Geneva 23, SWITZERLAND Abstract In these lecture notes, ... In the quant um theory, coordinates and momenta are non-commuting operators. A specific ordering prescription has to be made in order to define them as wel...

Ngày tải lên: 17/03/2014, 14:48