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Int. J. of Mathematical Sciences and Applications, Vol. 1, No. 3, September 2011 Copyright Mind Reader Publications www.journalshub.com QUANTUM – TYPE AND CONTINUOUS COMPRESSIONS I. Szalay University of Szeged, HUNGARY [email protected] Abstract The compression of the world is easier imaginable. In the first step we consider a compressor ; x ; R f 1,1 , where is called the first compressed of the real number x R and the open interval th x function, for example x 1 e x ex f x th x x x e e R 1 1,1 is called the first compressed real axis. In the second step, having a point X x, y, z R 3 we say that its first compressed is the point X 1 x 1 , y 1 , z 1 which is situated inside the open cube R 3 1 x, y, z R 3 : 1 x 1;1 y 1,1 z 1 . This 3 open cube is very similar to the euclidean space R , it is a so - called sub – euclidean space. (See [2]) Next step yields x 2 R 2 th1, th1 , X 2 x 2 , y , z 2 and 2 R 3 2 x, y , z R 3 : x, y , z R 2 . Following the compressions we obtain a sequence of open cubes R n ; n 1,2,3,... These kinds of compressions are called quantum – type 3 compressions. Adding a positive parameter c we have the c –compressed x c c th x ;xR. c The open interval R c c, c is called the c - compressed real axis. Moreover, having a point X x, y, z R 3 we say that its c - compressed is the point X c x c , y c , z c which is situated inside the open cube R c x, y, z R : c x c;c y c,c z c . As c may change continuously, these kinds of compression are called continuous compressions. Our aim is to compare the quantum – type and continuous compressions. 3 3 AMS Subj. Classification: 83F05 Keywords: compression and explosion of numbers, compressed space, sub – function, sub – operations. 1. Quantum – type compressions In [1] we detailed that the algebraic properties of R 1 1,1 are based on the isomorphism between R , , R,, created by the compressor function 1 (1.1) with (1.2) and (1.3) 1 1 e x ex x 1 th x x x e e ; x R x 1 1 y 1 x y 1 ; x, y R x 1 1 y 1 x y 1 , x, y R . 1303 I. Szalay Operations under (1.2) and (1.3) are called sub – addition and sub – multiplication, respectively. As R , , an ordered field, we can say that 1 1 1 R,, is is the first compressed of the field of real numbers. Geometrical, it is the first compressed of the real number line. If SR then the set S 1 x 1 R 1 : x S is called the first compressed set of S. The inverse of our compressor function is called exploder function and the number x (1.4) 1 1 x areath x ln ; x R 1 2 1 x 1 is called the first exploded of x R 1 . If S R 1 , then the set S 1 1 x R : x S is called the first exploded of S. The following inversion identity x x; x R , 1 (1.5) 1 x and 1 (1.6) 1 x; x R 1 are valid. Let f be a real function with D f R ; R f R . We define the function sub f .The number x R 1 belongs to domain D subf if 1 x D f , moreover subf x (1.7) ;xD f x 1 1 subf For example, in the case of function subth we have D subth R 1 moreover (1.8) subth x th x ; x 1,1 . (See [3], (4.28).) The latter means, that the graph of the function subth is situated in the domain T x, y R 2 : 1 x 1;th1 y th1 . Hence, (1.9) Rsubth th1, th1 . The second compression is happened in the following way. Instead of field consider R , , the field 1 1 1 R,, and function th we and the function subth, respectively. Now, we get the field R , , , where R th1, th1 , and by 2 2 2 2 x 2 subth x 1 th x 1 th th x th2 x ; x R , (1.10) is obtained. Moreover, similarly to (1.2) and (1.3) we have x 2 2 y x 1 1 y ; x 1 , y (1.11) 2 1 1 1 R 1 and (1.12) x 2 2 y 2 x 1 1 y 1 ; x 1 , y 1 R 1 . 1 Of course, we have the isomorphisms (second) inversion identity R , , R , , R,, . For the shake of 2 2 2 1 1 1 x x; x R , 2 2 we define the second exploded of x. (1.13) x 2 areathareath x areath 2 x ; x R 2 . By (1.13) the inversion identity x 2 2 x; x R 2 is valid. Using (1.11) and (1.13) we define the concept of the function sub2 f , where f is a real function with D f R ; R f R . The number x R 2 belongs to domain D sub 2 f if x 1304 2 D f , moreover QUANTUM – TYPE AND CONTINUOUS COMPRESSIONS sub2 f x f x (1.14) We remark, that sub2 f = sub(subf). 2 2 ; x D sub 2 f Going on we construct fields R n , n , n ; n 1,2,3,... ,where x n th thn 1 x thn x; x R R 0 ; th0 x x ; th1 x th x , (1.15) x n n y n x n1 n1 y n 1 ; x n 1 , y n 1 R n 1 ; 0 (1.16) 1 and x n n y n x n 1 n 1 y n 1 ; x n 1 , y n 1 R n 1 ; 0 . (1.17) 1 Similarly to (1.14) , by x n areathareathn 1 x areathn x ; x R n ; areath0 x x; areath1 x areath x we can define the iterated function sub n f . All of fields R n , n , n ; n 1,2,3,... are isomorphic with the field R,, . 2. Continuous compressions Let c an arbitrary positive number. The algebraic properties of the open interval the isomorphism between (2.1) R c c, c are based on R c , c , c R,, created by the compressor function x x c c th ; c 0; x R c with (2.2) and x c c y c x y c ; x, y R (2.3) x c c y c x y c , x, y R . x c is called the c –compressed of x and R c is called the c- compressed of the real number line. If S R then the set S c x c R c : x S is called the c – compressed of S. The inverse of compressor The number function (2.1) is called c - exploder function and the number c x x c areath ; x R c c is called the first exploded of x R c .Clearly, the inversion identities (2.4) x c c (2.5) and x; x R , x x; x R c (2.6) c c are valid. Let f be a real function with D f R ; R f R . We define the function. c x R c belongs to domain D subc f if x D f , moreover (2.7) subc f x ;xD f x c c subc f . By (2.5),(2.4),(2.2),(2.1) the addition – formula of the function th gives (2.8) c ; , R c 1 2 c and by (2.5),(2.4),(2.3) and (2.1) 1305 subc f .The number I. Szalay c c th c areath areath ; , R c c c (2.9) is obtained. 3. Similar properties of quantum – type and continuous compressions Property of extension. The sets R n ; n 1,2,3,...; R c ; c 0 are proper subsets of the set of real numbers R such that the fields R n , n , n ; n 1,2,3,... and R c , c , c ; c 0 are isomorphic with the field R,, . In the cases of quantum – type compressions the isomorphism is created by the map (1.15). Using (1.15) and (1.1) by induction we are able to prove, that (1.16) and (1.17) have the forms x n n y n x y n ; x, y R; n 1,2,3,... and x n n y n x y n ; x, y R; n 1,2,3,... , respectively. In the case of continuous compression the isomorphic was already characterised by (2.1),(2.2) and (2.3). Property of monotonity. Le tus denote d n ; d c the lenghts of the sets R n ; n 1,2,3,...; R c ; c 0 measured in R. So, d n 2thn 11; n 1,2,3,...; d c 2c . The sequence d n n 1 is strictly monotonic decreasing, d c is strictly monotonic increasing in the interval 0, . „Big Bang” property. lim n d n 0 ; lim c 0 0 d c 0 . We have to prove the firs one, only. As while d n n 1 is strictly monotonic decreasing sequence it is convergent, such that d . By (1.15) the continuity of the function th 2 d d yields that lim n thn 1 lim n thth n11 thlim n th n11 th . So, d = 0. 2 2 2 3 The compressions of the spaces R and R . Using the notations x ; x R; x R ; R,, for lim n d n inf d n d 0 . Hence, lim n thn 11 compressions mentioned under (1.1),(1.10),(1.15) and (2.1), in general, the sets R 2 , R 2 : , R and R 3 , , R 3 : , , R 2 3 3 2 are the compresseds of R and R , respectively. R (and similarly R , too) is a linear space by the operations 1 2 1 2 ,1 2 , 1 2 ; 1 , 2 R 3 and Let X x, y, z R 3 and , , ; R; R 3 . , , R 3 be the compressed and exploded of the points X x, y, z R 3 and , , R 3 , respectively. It is valid, that the linear space R 3 is normed, with the norm metrical, with the distance R3 R3 ; R3 d R 3 1 , 2 d R 3 1 , 2 ; 1 , 2 R 3 and eucledian with the inner product 1 2 1 2 ; 1 , 2 R 3 . 1306 QUANTUM – TYPE AND CONTINUOUS COMPRESSIONS R 3 (or in R 2 ) has a mutually unambiguous Our istruments guarantee, that each set S which is decribed in image S in the eucledian space R 3 1 . So, the „world” is compressed into a bounded open cube. R 3 1 . For example, the The images of graphs of two variable real functions can be seen in the open cube Da R 2 ; a x, y x y ; x 1 ; y 1 , so by (1.5) we get addition – function, where 1 x ; 1 y and Dsuba R 2 1 . Applying (1.2), by (1.7) we have 1 1 x 1 1 y 1 a x, y 1 a , 1 1 The graph of function suba can be seen in [2], Fig. 3.22. Another example the multiplication – function, where Dm R ; m 2 (1.3), by (1.7) we have 1 1 x 1 1 y 1 m x, y 1 m , 4. 1 1 1 a , 1 suba , . x, y x y; Dsubm R 2 1 . Applying 1 m , 1 subm , . Different properties between quantum – typed compressions and continuous compressions 0 , while this cardinality for the set of contunuous compressions is continuum 1 . It is well known that 0 1 . Cardinality. The cardinal number of the set of quantum – type compressions is the countable infinity R1 ,1 ,1 at (2.1) are not R 1 , 1 , 1 R 1 ,1 ,1 we can write Operations. The first compression R 1 , 1 , 1 at (1.1) and 1 – compression merely isomorphic but totally equals, so, instead of R , , R , , .For example we can see that 1 1 1 1 1 1 1 1 ; , R 1 R1 1,1 1 Of course, R 2 , 2 , 2 R th1 , th1 , th1 , moreover R 2 th1, th1 R th1 . Whether is it true that R 2 , 2 , 2 R th1 , th1 , th1 ? The answer is affirmative. First, we investigate the operation ont he field R 2 , 2 , 2 . Let be x 2 , y 2 ; x, y R . Applying (1.1), (1.10) and (1.4) we have x 1 th x ; x 2 thth x x 2 th x , so, 1 x x 1 2 1 obtained. Now, by (1.11) and (1.4) we can write 2 x 2 2 y 2 x 1 1 y 1 1 and similarly, 1 1 1 1 1 2 1 1 th 1 1 and similarly 1 1 x 2 1 y 2 1 1 y y 1 2 1 areath areath ; , th1, th1 1 areath areath 2 ththareath areath areath areath ; , th1, th1 . (In detail see [3], (5.4).) On the other hand the for the operations th1 th1 and th1 , (2.8) and (2.9) give ; th1 th1 th th1 areath areath ; , th1, th1 . th1 th1 1 2 th 1 We see, that for both cases the additive unit element is 0, because 0 ; 2 while, the multiplicative unit element for 2 is th1 0 ; th1, th1 , thth1 0,642 , because 1307 are I. Szalay 2 th th1 ; th1, th1 th1 is th1 th 1 0,6588 , because th1 1 th1 th1 th ; ( th1, th1 . th1 Despite of that operations 2 and th1 have a common unit element 0, these operations are different. For and for example, 1 1 2 th 2 2 2areath 1 2 1 1 areath 2 0,687900089 , 2 while 1 1 th1 2 2 1 1 0,698804491. 1 2th12 „Container” properties. Clearly, among the fields R n , n , n ; n 1,2,3,... the field R 1 , 1 , 1 is the largest, because R n R 1 ; n 2,3,4,... . As largest field does not exist, but we remark that lim c d c among the fields R c , c , c lim c x c x; x R , lim c x c c y c x y; x, y R and lim c x c c y c x y; x, y R . References [1] I. Szalay: On the compressed Descartes – plane and its applications, AMAPN 17(2001), 37-46. www.emis.de/journals [2] I. Szalay : On the cube – model of three dimensional eucledian space, Acta Acad. Paed. Agriensis, Sectio Mathematica 29(2002), 13-54. [3] I. Szalay: Exploded and compressed numbers , AMAPN 18(2002), 33-51. www.emis.de/journals Acknowledgements Supported by Foundation Domonkos Bonifert, Szeged, Hungary 1308