Download QUANTUM – TYPE AND CONTINUOUS COMPRESSIONS Author:I. Szalay

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  ex
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
;xR.
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  ex
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
SR
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)
  ;xD
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   th2  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 
 areathareath 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 sub2  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
 
sub2  f  x   f x
(1.14)
We remark, that
sub2  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 thn 1 x   thn  x; x  R  R 0  ; th0  x  x ; th1 x  th x ,
(1.15)
x n   n  y n   x  n1   n1 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 
 areathareathn 1 x   areathn  x ; x  R n  ; areath0  x  x; areath1 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  
  ;xD
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   2thn 11; 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 thn 1  lim n thth n11  thlim n th n11  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  thn 11 
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   thth 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    ththareath 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 ,
thth1  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
2th12



„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