## Monday, November 25, 2013

### Conjugates of Fields, 5 of 6; Logarithmic and Exponential Fields

5. Logarithmic and Exponential Fields

Logarithmic addition and subtraction (respectively, "accretion" and "erosion") are logarithmic conjugates of plus and minus:
X  ln[+] Y    =    X # Y   =   ln( eX  +  eY )  =  "X along Y"
X  ln[-] Y    =     X \$ Y   =   ln( eX   -  eY )  =  "X beyond Y"
X ln[*] Y   =   ln ( eX * eY )   =   X+Y
Therefore logarithmic addition is a field under +, with identity   - :
X + ( Y # Z )  =  ( X + Y ) # ( X + Z )
X + ( Y \$ Z )  =  ( X + Y ) \$ ( X + Z )
-   #  X         =        X
X  \$  X           =       -

Defining logarithmic negation require using Euler's formula eπi  =  -1.
Define \$X  =  - \$ X  =  ln(-eX)  =  "beyond X";  then
\$X  =  X  + \$0
\$0  =  πi
"Beyond zero equals pi times i." This formula refers, in five symbols, to the five major branches of mathematics; analysis, arithmetic, logic, geometry, and algebra.

We can define log addition to other bases. Let
X  #C  Y  =   logC( CX + CY )  = ( (X lnC)#(Y lnC) ) / lnC  =  X  [#/lnC] Y
X  \$C  Y  =   logC( CX - CY )  = ( (X lnC)\$(Y lnC) ) / lnC  =  X  [\$/lnC] Y

If   P1 = A bX   and  P2 = A bY  then   (P1+P2)  =  A b Z  , where Z  =  X #b Y.
Z  =   X [#/r] Y  =  ( rX # rY ) / r   =   X   ln r\/+  Y  ;  log-fermat addition!

In the decibel scale, sound energy is proportional to 10D/10 , where D is decibels; therefore if two sound energies add linearly, then the decibels add logarithmically;
D12   =   D1   #10^0.1  D2

In the Richter scale, seismic energy is proportional to 10R. If energies add, then;
R12   =   R1   #10  R2

In the pH scale, concentration is proportional to  10 - pH ; therefore, if we mix equal volumes of two solutions and no buffering occurs, so that the concentrations average, then;
pH12   =   ( pH1   #0.1  pH2 )  +  ln2

Exponential Multiplication

Exponential multiplication (or "eruption") is * conjugated by exp:
X  e*  Y    =    X  @  Y   =   e ( lnX * lnY )  =  X lnY  =  Y lnX
Its inverse operation is exponential division, or "subduction":
X e¸ Y     =    X (1/lnY)
Unlike exponentiation, eruption is commutative and associative:
X @ Y  =  Y @ X
X @ ( Y @ Z )   =  ( X @ Y ) @ Z
* is a field under @, with identities 1 and e:
X @ ( Y * Z )   =   ( X @ Y ) * ( X @ Z )
X @ 1              =           1
X @ e              =           X
X @ eN            =           XN
X @ YN           =       (X@Y)N

We can exponentiate eruption:    X  e@  Y   =   e(lnX @ ln Y). This is a field over @, with identities e and ee. And so on; we get an "Exponential Tower" of operations:
#  ,  +  ,  *  ,  @  ,  e@ ,  ...
- each one of which is a field over the preceding operation, with identities
-  ,  0  ,  1 ,  e  ,  ee  ,  ...

Conjecture: Continuous Exponential Tower: there exists a continuous, invertible, analytic "super-exponential" operator E on the positive reals such that E(x+1) = eE(x).  Given such an E, we can define the continuous iterate of exponentiation:
expR(X)   =   E( E-1(X) + R )
So exp-1(X) =  lnX  ;   exp0(X)  =  X  ;  exp1(X)   =  eX  ;  exp2(X)  =  e(e^X) ; ...
Therefore we can make the exponential tower continuous, thus:
X  R+  Y   =    expR( exp-R(X) + exp-R(Y) )
=    E (  E-1{ E[E-1(X) - R]  +  E[E-1(Y) - R] }+ R )
So   -1+   =  #  ,    0+   =  +  ,    1+   =  *   ,    2+   =  @   ,   ...
Each  R+ distributes over R-1+ , and distributes under R+1+ .
Open questions remain. For instance, how does  1/2+  relate to  +  and  * ?