%# &% 39 85,507-0 44 3 1:3.9 43, 06:,9 438,9 81 0/- # 02,33 09,1:3.9 43,.9:,,.44507,9 ;0,204150710.9 31472,9 43-09 003 98.4389 9:03990728 070 / 11070392,9 02,9., 1:3.9 43, 8 2-4 8 3. 9 0-089897,90,/4590/- 0,. 5, 0794 4.,90 0748432,9 02,9., 1 0 / 0,/894/ 8.4;07 3 9 0248989,- 0,77,3 02039415 8., 4.,9 4334397 ;, 074841# 02,33 09,1:3.9 43. 3 9:73 0,/894%#&% & $$ #! % $ $ 8; 8:, 0/-,; / -079/;,3.0/2,9 02,9.8 8,.9:,,,20-09 003 / 11070392,9 02,9., 8 2-4 8 070/ 11070398 2-4 814 4.079, 3/01 30/7: 08,2 4 3 940 903/ 8; 0 89,9 3 9,9039 702,9 02,9.8 3:2-079 047 3 5,79.:,7 980 1 8,.9:,,,20 070/ 11070395, 0785,,.44507,9 ;0,2094 70,.,99 0248989,- 006: -7 :289, 0 The mathematical theory of games was invented by John von Neumann and Oskar Morgenstern (1944).Game theory is the study of the ways in which strategic interactions among agents produce outcomes with respect to the preferences (or utilities) of those agents, where the outcomes in question might have been intended by none of the agents.. All situations in which at least one agent can only act to maximize his utility through anticipating (either consciously, or just implicitly in his behavior) the responses to his actions by one or more other agents is called a game. Agents involved in games are referred to as players. If all agents have optimal actions regardless of what the others do, as in purely parametric situations or conditions of monopoly or perfect competition we can model this without appeal to game theory; otherwise, we need it. Each player in a game faces a choice among two or more possible strategies. A strategy is a predetermined programme of play that tells her what actions to take in response to every possible strategy other players might use. The significance of the italicized phrase here will become clear when we take up some sample games below.
The simplest games (from the perspective of logical structure) are those in which agents have perfect information, meaning that at every point where each agent's strategy tells her to take an action, she knows everything that has happened in the game up to that point. This is so because in such games (as long as the games are finite, that is, terminate after a known number of actions) players and analysts can use a straightforward procedure for predicting outcomes. A player in such a game chooses her first action by considering each series of responses and counter-responses that will result from each action open to her. She then asks herself which of the available final outcomes brings her the highest utility, and chooses the action that starts the chain leading to this outcome. This process is called backward induction (because the reasoning works backwards from eventual outcomes to present choice problems). Nash equilibrium (NE) stages are the set of strategies so that no player can maximize payoff by unilateral deviations. John Forbes Nash devised a theorem for it which are known as Nash equilibrium stages. SNE(strong Nash equilibrium stages are the refinements of Nash equilibrium such that no player is going to benefit by unilateral or bilateral defimitions. 5742 3039 :809 0944 841,209 047 941 3/4:9/ 1107039,8 06: -7 :2 89, 08 39 81:3.9 43,,205, 0/-09 0032,9 02,9., 8 2-4 8,3// 8.4;07 3 9 0$9743,8 6: -7 :2 $ 89, 0941 3/4:99 0248906: -7 :289,90,3/9 :8 9 057010770/,/4590/80941897,90-09 003/ 11070395, 078,9 02,9.8 070 3:2-078,70 -,8.,,3 :, 0 94 705708039 8420 5 8.,,850.9 33,9:70:2-078,702,907, 89.705708039,9 4384184205 8.,,850.98 413,9:70 % 0805 8.,,850.98,70 3907.4330.90/ 9 0,. 49 07 33,9:70 9,9 8
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9,9/4349 3;4 ;0 3.70/ - 09 70,98475742 808$0.43/ 0;07 1 3 90,20 41.425 090 31472,9 43,8,9 0,89430$ 3.4390 9 41 1:3.9 43, 06:,9 43,20 5, 0/ - # 02,33 09, 1:3.9 438 3 9 0 0-9 070,709 45, 078 070.47708543/8948 3,3/.47708543/8 948 3 % 84 :9 43.43.059 3,209 047 3,20 9 047,8 06: -7 :2 89,908 701 302039 41,8 06: -7 :2 89, 08.47708543/ 94 9 0 84 :9 43,.,7/ 3/:.9 43 3,,20 41 50710.9 31472,9 43,, 8 0,/ 94,9 0,89 3,8 06: -7 :2 89, 0 Number system as an originally found quantum group:..47708543/8 94 9 0 84 :9 43 070 9 05 8., 4.,9 434134397 ;, 074841# 02,33 09,1:3.9 43 Number system is a physically FOUND commutative GROUP as in quantum world. a) For every a, b there exists a+b on the number line(group) b) There exists an identity element I su ch that a+i=i+a=a This will also be valid. if operator + is changed to *. Any way ^,* are derived out of + So, with any one operator, the number system is DEFINITELY QUANTUM GROUP FNATURE. On numbr line for everyelement 1) a, there exists 1/a, 2) a there exists a. Thus covering each and every point on the number line. Every a can generate another number b on it through operators +,-,*,/,^ 4)Entire negative part of number line is mapped onto the domain 0-1
e.g. 2^-3=1/8,2^-2=1/4,2^-1=1/2 5)and every positive side on number line can be mapped onto the domain after 1 e.g.,2^1=2,2^2=4,2^3=8 etc. 6) 0 is mapped onto 1 to maintain the symmetry of GROUP i.e. 2^0=1 What I am interested here is to so why NUMBER SYSTEM is an INTERCONNECRED GROUP. So, I wd like to say that THIS HAS ALL THE PROPERTY OF A QUANTUM SYMMETRY GROUP FOUND IN NATURE and being a qwuantum group IT S INTERCONCTED and perfectly informed group. Refe. Eugen Merzbacher -Quantum mechanics Chapter 17, Groups and Symmetrical invariancy in ORIGINALLY FOUND GROUP IN nature.!# :3.9 43, 06:,9 438,9 81 0/-!, 078 8 8 39 0039 70.425 0 /42, 3 8 8430,3/43 4309072430,. 8 /0418 3.,3,3/2:89-0 074,8 8 34906:, 94 343 0743:2-07 8 8 3 8 8 <,88 89 0 4.,9 4314754 0
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!, 070 07. 808 3/459 43 $ ( : 147 147 08009,9 # (147 # (147 % 8,8,8 06: -7 :289,908-44 3,99 080941# :98:2-07 30 8,.425 090 74:5 0470;07 # #< 3/ # #< 55 3 : 3 89 0470289,9089 0702:89-0,9 0,89$ 39 8,204150710.9 31472,9 43 $4 09:834 1 3/$ 8: 2, : < 84 & 2, : < 147
% :8$ 89 0$ 39 8,20-,.,7/ 3/:.9 43 % $ $% 84 :9 43 $89, 0 $$&!! #% & $% #,3/ 03.0 4: /-0 57010770/-, 9 05, 078942, 2 09 05, 411 9 4: 9 8,20,8,8 06: -7 :289,908,83430.,3-0301 9,83430.,3-0301 9- :3,907, /0;,9 438 -:99 09 80941897,90 4: /-09 0 $ $%# $ #"& # &89,90,89 8 8 22:3094-49 :3,907, /0;,9 438,3/.4, 9 43/0;,9 4383/ 03.0-49 9 05, 0782:890 07. 809 03/459 43894 2, 2 09 05, 411 % 8,8807989 097:9 1: 3088419 0# 02,33 549 08 89,997 ;, 0748 0439 0 54 3988,3/97 ;, 0748 0439 0# 8 % :8 9 25 089,9 8 147# 8 Г 9147 # 8,3/, 84 8 147# 8 " Reference: 1.GAME THEORY http://plato.stanford.edu/entries/game-theory/#games 2. Extensive form finite game of perfect information http://docs.google.com/viewer?a=v&q=cache:ljfnd1jnbpyj:www.eecs.harv ard.edu/cs286r/archived/fall08/files/lecture3.pdf+finite+extensive+form+ga me&hl=en&gl=in&pid=bl&srcid=adgeesglax41fnxre9yxw1k51fnhuya19bv 8Yf3WmRbHOwAmahc33o2QaSHi2Ns0Q6dk4IUid2avKWibOoXKmC20Xm3Y
8c7NlxzmceD2qWQzeNFz4FsjlWIaJbAN4NDC_bq3tBAutI5F&sig=AHIEtbT22 Mjxlsq1 3) Official description of Riemann Hypothesis problem by Enrico Bombeiri Pw1hpwmsJONtCFAKQhttp://docs.google.com/viewer?a=v&q=cache:IdUCngHnmOoJ:www.clay math.org/millennium/riemann_hypothesis/riemann.pdf+rimeann+hypothesis+ooficial+descript ion+by+bombeiri&hl=en&gl=in&pid=bl&srcid=adgeesg857veuf9atzsyjwn1bisnb25hoha7m3vuc2irjjrsgs4lbklfgvtijxtsinmmiiy8wuicuih54gkkwy1wjbpfz4fdvk9bfjuqwx o2s4btahyq_beislbqg6gi4cdjohoxaq0&sig=ahietbsa-khwurcachkjvad9zl5b5oj4cq 4) Riemann zeta functional equation in the entire complex plane. http://www.google.co.in/imgres?q=riemann+zeta+function&start=60&hl=e n&safe=off&sa=n&gbv=2&ndsp=21&sout=0&tbm=isch&tbnid=5rsllrm5ngh lkm:&imgrefurl=http://dspace.mit.edu/bitstream/handle/1721.1/37301/18-112fall-2005/ocwweb/mathematics/18-112fall- 2005/CourseHome/index.htm&docid=cECkPL8ZuJTS8M&w=419&h=206&ei =QmkuToahEsTRrQetpMWcAw&zoom=1&iact=hc&vpx=340&vpy=382&dur =451&hovh=80&hovw=163&tx=116&ty=113&page=4&tbnh=80&tbnw=163 &ved=1t:429,r:15,s:60&biw=1280&bih=551