Santilli s Isomathematical theory for changing modern mathematics

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1 Santilli s Isomathematical theory for changing modern mathematics Chun-Xuan. Jiang P. O. Box 3924, Beijing P. R. China jiangchunxuan@vip.sohu.com Dedicated to the 30-th anniversary of China reform and opening Astract We estalish the Santilli s isomathematics ased on the generalization of the modern mathematics. Isomultiplication a ˆ a atˆ a, isodivision Iˆ, where Iˆ 1 is called an isounit, TI ˆˆ 1, T ˆ inverse of isounit. Keeping unchanged addition and sutraction, (,, ˆ, ˆ) are four arithmetic operations in Santilli s isomathematics. Isoaddition a 0ˆ, isosutraction a ˆ0 where ˆ0 0 is called isozero, ( ˆ, ˆ, ˆ, ˆ) are four arithmetic operations in Santilli s new isomathematics. We give an example to illustrate the Santilli s isomathematics. Santilli [1] suggests the isomathematics ased on the generalization of the multiplication division and multiplicative unit 1 in modern mathematics. It 1

2 is epoch-making discovery. From modern mathematics we estalish the foundations of Santilli s isomathematics and Santilli s new isomathematics. We estalish Santilli s isoprime theory of oth first and second kind and isoprime theory in Santilli s new isomathematics. (1) Division and multiplican in modern mathematics. Suppose that a a a 0, 1 (1) where 1 is called multiplicative unit, 0 exponential zero. From (1) we define division and multiplication a a, 0, a, a (2) 0 a a ( a a) a a a (3) We study multiplicative unit 1 a 1 a, a 1 a,1 a 1/ a (4) n a/ n n / a / a ( 1) 1, ( 1) 1, ( 1) ( 1), ( 1) ( 1) (5) The addition +, sutraction -, multiplication and division are four arithmetic operations in modern mathematics which are foundations of modern mathematics. We generalize modern mathematics to estalish the foundations of Santilli s isomathematics. (2) Isodivision and isomultiplication in Santilli s isomathematics. We define the isodivision ˆ and isomultiplication ˆ [1-5] which are generalization of division and multiplication in modern mathematics. a ˆ a a I1, 0, 0 (6) 0 ˆ where Î is called isounit which is generalization of multiplicative unit 1, 0 exponential isozero which is generalization of exponential zero. We have 2

3 Suppose that From (8) we have ˆa a ˆ I, 0, ˆa ˆaT, (7) 0 a a ˆ ( a) a ˆ a ati ˆˆ a. (8) where ˆ T is called inverse of isounit Î. TI ˆˆ 1 (9) We conjectured [1-5] that (9) is true. Now we prove (9). We study isounit Î ˆ ˆ ˆ ˆ I ˆ2 a ˆ I a, aˆ I, aˆ I a a /, I a (10) aˆ ˆa a nˆ ˆn n ( Iˆ ) ˆI, ( ˆI ) ˆ I,( ˆ I ) ( ˆ 1) Iˆ,( I ) ˆ ( I1) (11) Keeping unchanged addition and sutraction, (,, ˆ, ˆ) are four arithmetic operations in Santilli s isomathematics, which are foundations of isomathematics. When Iˆ 1, it is the operations of modern mathematics. (3)Addition and sutraction in modern mathematics. We define addition and sutraction x a, y a (12) a a a a (13) a a 0 (14) Using aove results we estalish isoaddition and isosutraction (4)Isoaddition and isosutraction in Santilli s new isomathematics. We define isoaddition ˆ and isosutraction ˆ. From (16) we have a c, a c (15) 1 2 a a a c c a (16) c c (17) 3

4 Suppose that c1 c2 ˆ0, where ˆ0 is called isozero which is generalization of addition and sutraction zero We have a ˆ a 0, ˆ ˆa a ˆ0 (18) When ˆ0 0, it is addition and sutraction in modern mathematics. From aove results we otain foundations of santilli s new isomathematics ˆ Tˆ, ˆ 0 ˆ ; ˆ Iˆ, ˆ 0 ˆ ; a ˆ atˆ, a 0; ˆ a Iˆ, a 0; ˆ a a ˆ a a, a a a; 2 a ˆ a a T, a 2a 0; ˆ a Iˆ 1, a 0ˆ 0; TI ˆˆ 1. (19) ( ˆ, ˆ, ˆ, ˆ) are four arithmetic operations in Santilli s new isomathematics. Remark, a ˆ ( ˆ c) a ˆ ( c 0) ˆ, From left side we have a ˆ ( ˆ c) a ˆ a ˆ ˆ a ˆ c) a ˆ ( ˆ c) a ˆ ( 0 ˆ c), where ˆ 0ˆ also is a numer. a ˆ ( ˆ c) a ˆ ( c 0) ˆ. From left side we have a ˆ ( ˆ c) a ˆ a ˆ ˆ a ˆ c) a ˆ ( ˆ c) a ˆ ( 0 ˆ c), where ˆ ˆ0 also is a numer. It is satisfies the distriutive laws. Therefore ˆ, ˆ, ˆ and ˆ also are numers. It is the mathematical prolems in the 21st century and a new mathematical tool for studying and understanding the law of world. (5 )An Example We give an example to illustrate the Santilli s isomathematics. Suppose that algeraic equation y a ( c) a ( c) (20)

5 We consider that (20) may e represented the mathematical system, physical system, iological system, IT system and another system. (20) may e written as the isomathematical equation yˆ ( ˆ c ) ˆ ( ˆ c ) a Tˆ ( c 0) ˆ 0 ˆ a / Tˆ ( c 0) ˆ. (21) If Tˆ 1 and ˆ0 0, then y yˆ. Let Tˆ 2 and ˆ0 3. From (21) we have the isomathematical suequation yˆ 2 a( c 3) 3 a / 2( c. 3) (22) Let Tˆ 5 and ˆ0 6. From (21) we have the isomathematical suequation yˆ 5 a( c 6) 6 a / 5( c. 6) (23) Let Tˆ 8 and ˆ0 10. From (21) we have the isomathematical suequation yˆ 8 a( c 10) 10 a / 8( c. 10) (24) From (21) we have infinitely many isomathematical suequations. Using (21)-(24), ˆ T and ˆ0 we study staility and optimum structures of algeraic equation (20). Acknowledgements The author would like to express his deepest appreciation to A. Connes, R. M. Santilli, L. Schadeck, G. Weiss and Chen I-Wan for their helps and supports. References [1] R. M. Santillis, Isonumers and genonumers of dimension 1, 2, 4, 8, their isoduals and pseudoduals, and hidden numers of dimension 3, 5, 6, 7, 5

6 Algeras, Groups and Geometries 10, (1993). [2] Chun-Xuan Jiang, Foundations of Santilli s isonumer theory, Part I: Isonumer theory of the first kind, Algeras, Groups and Geometries, 15, (1998). [3] Chun-Xuan Jiang, Foundations of Santilli s isonumer theory, Part II: Isonumer theory of the second kind, Algeras Groups and Geometries, 15, (1998). [4] Chun-Xuan Jiang, Foundations of Santilli s isonumer theory. In: Foundamental open prolems in sciences at the end of the millennium, T. Gill, K. Liu and E. Trell (Eds) Hadronic Press, USA, (1999). [5] Chun-Xuan Jiang, Foundations of Santilli s isonumer theory, with applications to new cryptograms, Fermat s theorem and Goldach s conjecture, International Academic Press, America- Europe- Asia (2002) (also availale in the pdf file http: // www. i--r. org/jiang. Pdf) [6] B. Green and T. Tao, The primes contain aritrarily long arithmetic progressions, Ann. Math.,167, (2008). [7] E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith., 27, (1975). [8] H. Furstenerg, Ergodic ehavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31, (1977). [9] W. T. Gowers, A new proof of Szemerédi s theorem, GAFA, 11, (2001). [10] B. Kra, The Green-Tao theorem on arithmetic progressions in the primes: an ergodic point of view, Bull. Amer. Math. Soc., 43, 3-23 (2006). 6

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