Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp

[bo198214]

No, for all real points .

Such a function can not exist. If then there must exist a such that for all , because otherwise - if in each left neighborhood of there is an with - there is a sequence with which means that does not exist or is 1.
But if in whole left neighborhood of then for .

[Ivars]

So we have to skip reals. At least formally it seems possible to define such function on hyperreals or superreals (second is more likely as they are discontinued by definition, with gaps) , but as I do not know enough to make such definition I will study a little from both Conway orginal book and Hyperreals, and Cantors ordinals/cardinals because these things does not work without them.

It seems that definition of functions on these numbers is not a very popular topic, most works try to link them to real functions as soon as possible, but I have seen brief mentioning of transcendental functions of surreals in the net, as well as surcomplex numbers so far made by simple adding of I to surreals.

Here is excerpt of one link I have started to read:

Nicolau C Saldanha on surreal functions

Unfortunately, I know of no written reference to this material.
I will do my best to reproduce here what I learned from Conway.

But first some warnings:

- There will be NO talk about continuity or differentiability of
surreal functions. In fact, this approach seems inadequate to
me, due to the existence of gaps in the ``surreal line''.

- Notice that series can NOT be added using epsilon-delta deffinitions
since that would make all non-trivial series diverge by falling
into a gap. For instance, we would not have 1/2 + 1/4 + 1/8 +... = 1,
since 1 - 1/omega is still larger than any partial sum.
When we write such things as 1 + omega^(-1) + omega^(-2) + ...
as the Cantor-Conway normal form of a surreal number, this ``series''
is NOT to be understood in the epsilon-delta sense, lest we get a
divergent series.

( By the way, 0.999... = 1 for the surreals ( since it holds for the
reals ) in the ``only'' meaningful sense the expressions have;
again, epsilon-delta definitions are out. )

- This does not use explicitly any form of summation like Cesaro's;
nor implicitly, as far as I can tell ( I could be wrong ).

- This DOES use the technique of defining things inductively, defining
x by means of x_L and x_R. This follows the same spirit as the
definitions of x + y, x * y, 1/x and the like.

And now a most crucial warning:

- The ( usual ) extentional notion of a function is NOT adequate here.
Two functions assuming the same values at each and every surreal number
must be considered DIFFERENT if the left and right options are not the
same. As an example, the functions:

f(x) = {|};

and

g(x) = { g(x_L) - 1 | g(x_R) + 1 };

are both constant equal zero. Usually, we would say that they are the
same function --- this is the extentional notion of what a function
is: a function is known if its extention, i.e., the values it assumes,
are known. Here, however, we have to adopt a radically different
point of view: in order to know a function, we have to know how it is
defined, and different definitions give different functions even if
the values coincide. In this sense, f and g are DIFFERENT.

An other way of looking at the situation, less radical but less
satisfactory, is to think of functions as having ``good'' and ``bad''
definitions. The above definition of f is probably ``good'', but
g is almost certanly ``bad''.

This is means we have to be careful about certain things we usually
take for granted. For instance, in a minute we will be defining log
as the integral of 1/x. In order to do this, the following definition
of 1/x is not satisfactory:

1/x = y iff xy = 1;

We now need a recursive, ``constructive'' definition of the form:

1/x = { ( stuff depending on x, x_L and x_R ) |
( other stuff ) };

such a definition is possible ( but not very easy ) to obtain.

Another danger: once we have log, we can't really invert it to get
exp: that would give us the values of exp, but not a definition we
could use later if we want to use exp to get new functions.

Now for the real action. We will adopt the convention:

f(x) = { f_L(x,x_L,x_R) | f_R(x,x_L,x_R) };

What follows is a definition of integration.

$
\int_a^b f(t) dt =

{
\int_a^{b_L} f(t) dt + \intd_{b_L}^b {f_L}(t) dt ,
\int_a^{b_R} f(t) dt + \intd_{b_R}^b {f_R}(t) dt ,
\int_{a_R}^b f(t) dt + \intd_a^{a_R} {f_L}(t) dt ,
\int_{a_L}^b f(t) dt + \intd_a^{a_L} {f_R}(t) dt
|
\int_a^{b_L} f(t) dt + \intd_{b_L}^b {f_R}(t) dt ,
\int_a^{b_R} f(t) dt + \intd_{b_R}^b {f_L}(t) dt ,
\int_{a_R}^b f(t) dt + \intd_a^{a_R} {f_R}(t) dt ,
\int_{a_L}^b f(t) dt + \intd_a^{a_L} {f_L}(t) dt
}
$

I used \TEX notation for integrals, subscripts and superscripts.
``\intd'' should be written as an integral sign with a capital `D'
over it, in the middle. It means direct integration, which means
do not chop the domain into pieces. Notice that some integrations
in the above definition will go from right to left, which means you
have to do the usual change of signs.

So now you know what log is! You define:

log(x) = \int_1^x 1/t dt;

This definition is good for any positive surreal number and satisfies
all the usual properties.

A similar definition can be found for the ``solution of the differential
equation''

dy/dt = g(t,y);

( Notice we are not *really* talking about derivatives )

from which definitions of exp , sin and cos can be obtained.

I will write down the definition later if someone wants me to, but I
think it is obvious if you understand the above definition of
integration.

I am not really sure of this, but I think gamma, bessel, zeta and
other functions can be done about as easily.

Exercise: Prove the Riemann Hypothesis for the surcomplex numbers.
( :-] )

Intuitively, however, it is relatively easy to see how to define
trigonometric functions for all surreals. Just say that TWOPI
is a period in the following surreal sense:

sin( x + TWOPI*n ) = sin(x);

for any surreal x and any n in the class Oz of surreal integers.

( Definition:

n is in Oz iff n = { n-1 | n+1 } )

The above definitions have this property. Notice that omega/TWOPI
is an integer, and therefore omega is a period.

Very roughly, the reason this works is the following: up to day omega
( exclusive ) you have been working only with finite numbers. When
the time comes to define cos(omega) and sin(omega) you still have NO
information about their values. What you do is of course pick the
simplest coherent answer ( this is what you do all the time with
surreal numbers ), and this is of course
cos(omega) = 1 and sin(omega) = 0.

By the same reasoning,

cos(omega^r) = 1,
sin(omega^r) = 0;

for any positive surreal number r.
( This is Cantor's exponentiation, not the analytic one )

By the way, there seems to be someone in Rutgers who is very interested
in this stuff. This someone may want to talk about this more directly.
Send e-mail or we might talk by telephone sometime.

--
Nicolau Corcao Saldanha

I have not any opinion about this, yet.


Ivars

[bo198214]

Sorry, but I dont see any meaning for hyper operations in defining a fucked up function that is so crude that you can not define them on the reals.

That may be an interestic topic for itself but then rather goes in the direction of Cantor sets or general topology. Good luck.

[Ivars]

Sorry, but I dont see any meaning for hyper operations in defining a fucked up function that is so crude that you can not define them on the reals.

Well I found a popular citation that expresses my thoughts rather well and perhaps is more authoritative. From my point of view, given the fast divergent nature of hyperoperations, there can not be a better place to use smaller infinitesimals than used in normal calculus than hyperoperations.

According to Kruskal, these problems could disappear if
theorists use infinitesimals, numbers smaller than any imaginable
positive real numbers. A series that involves such numbers can
be prevented from diverging essentially because infinitesimals
are so small that they "mop up" any tendency a series might
have to zoom off to infinity. "The surreals give us a way of
working with infinitesimals, and thus perhaps of working with
divergent series," says Kruskal. Divergent integrals, another
common bugbear in theoretical physics, may also bow to the
surreal approach.

Ivars

[andydude]

According to Kruskal, these problems could disappear if
theorists use infinitesimals, numbers smaller than any imaginable
positive real numbers.

You need to learn the difference between divergent and false.

Andrew Robbins

[Ivars]

Sorry, but I dont see any meaning for hyper operations in defining a fucked up function that is so crude that you can not define them on the reals.

You need to learn the difference between divergent and false.

The other point is who cares about functions on reals if they are so fucked up in principle that they can not deal with simple reduction of tetration to simpler analysis. Disctinction between true and false in tetration and above can not be determined by the properties of function on reals.

But of course this is just uneducated intuitive opinion. Though I do not see a problem to define a function like f(xL,x)= 0, f(x, xR)=1. xL,
Where xL, xR are Left and Right surreal numbers between any 2 real values of x.Function is just a unique relationship between 2 sets, be it reals and surreals or what ever. Once defined, it exists. Next step is to study if it has any reasonable properties and does it fit the purpose of understanding hyperoperations.


Ivars

[bo198214]

The other point is who cares about functions on reals if they are so fucked up in principle that they can not deal with simple reduction of tetration to simpler analysis.

I dont know what you mean, the tetrational functions we considered so far are quite well behaved, continuous or even analytic. There is no complicated analysis needed, as far as I can see.

For your function you need complicated analysis, and I still dont see the connection/meaning that it could have for hyperoperations.

I know Ivars you are great in speculations, but perhaps then mathematics is not the right discipline for you. In mathematics there is the possibility (and the necessity) to verify things, to verify ideas. If you dont like verifying (and this impression is quite strong) then go to philosphy where nobody can disprove you for sure.
The gap between what you know and what you want to be true is just too huge.

[Ivars]

I know Ivars you are great in speculations, but perhaps then mathematics is not the right discipline for you. In mathematics there is the possibility (and the necessity) to verify things, to verify ideas. If you dont like verifying (and this impression is quite strong) then go to philosphy where nobody can disprove you for sure.
The gap between what you know and what you want to be true is just too huge.

Unfortunately I absolutely am sure that they both converge ultimately ( mathematics and philosophy). So there is no chance to prefer one over another, though the gap you mention is present in both, and my aim is to reduce it as much and as fast as possible using these speculations as motivators.

Excuse me for heating up much above the level of clarity of the problem statement I presented. Hyperoperations is the place where new things will be discovered, because they are required by nature to operate, and I want to be part of it, even if only as active observer, if not able to be a contributor really.

Ivars

[Ivars]

Hi,

After some pause, I have found a solution why generalized logarithms will work, despite the proofs by bo, Andy that real number line does not support such extension. I can give only a general confirmation, without details yet.

Briefly, because they will take values from INSIDE real number line , where the organization of Real number line will be changed by each level generalized logarithm. From the representation of numbers as ordered lengths of Euclidean straight lines to much more complex ways to organize the Numbers.

Each of these INTEGER level number organization ways will be linked to each other via transformation groups of respective projective spaces to spinor spaces (e.g. h(CP1)=Spin(2)). NON-INTEGER levels will be presented by a non-integer iteration point in process a continuous transformation ( like tetration) that maps Real number line with one organization to Real number line with different ordering.

Thus, also non-integer operations and dimensions, iterations will obtain their geometric meaning and intuitive understanding will be possible.

Here is a link to more detailed first ideas about how tetration transforms +RP1, -RP1, CP1 etc. with comments from Tony Smith about group structure of such transformations.

http://math.eretrandre.org/tetrationforu...hp?tid=216

Plus some ideas from Tony Smith about this transformation ( still working on it):

CP1 = SU(2) / U(1) = S3 / S1 = S2 by the Hopf fibration S1 -> S3 -> S2 = 2-sphere

Since SU(2) = S3 = Spin(3) = 3-sphere
and U(1) = S1 = Spin(2) = circle

I think that maybe when you write Spin(1) you should be writing U(1) = Spin(2) = circle

Since we are talking about a complex projective space CP1
the 2 in Spin(2) may refer to the 2-real-dim nature of 1-complex-dim space.

What the Hopf fibration S1 -> S3 -> S2 = CP1 means geometrically
is
that the 3-sphere S3 looks like a 2-sphere S2 = CP1
with a little circle = S1 = U(1) = Spin(2) attached to each of its points.

Tony

Ivars

[BenStandeven]

Sorry, but I dont see any meaning for hyper operations in defining a fucked up function that is so crude that you can not define them on the reals.

You need to learn the difference between divergent and false.

The other point is who cares about functions on reals if they are so fucked up in principle that they can not deal with simple reduction of tetration to simpler analysis. Disctinction between true and false in tetration and above can not be determined by the properties of function on reals.

But of course this is just uneducated intuitive opinion. Though I do not see a problem to define a function like f(xL,x)= 0, f(x, xR)=1. xL,
Where xL, xR are Left and Right surreal numbers between any 2 real values of x.Function is just a unique relationship between 2 sets, be it reals and surreals or what ever. Once defined, it exists. Next step is to study if it has any reasonable properties and does it fit the purpose of understanding hyperoperations.


Ivars

Let 0 = {|}, 1 = {0|}, -1 = {|0}, as usual.
Then {-1 | 1} = 0.
So 1 = f({|0}, 0) = f(-1, 0) = f(-1, {-1 | 1}) = 0.

The function is not well-defined. (It would be well-defined on combinatorial games, except that it is of course not defined on all of them.)

Also, if f(x^y) = f(x) f(y) on the surreals, then either f is identically 1, f is identically zero, or there is some y with 1^y not equal to 1, or some x with x^1 not equal to x. This follows by bo198214's proof. I daresay all the standard definitions of exponentiation on the surreals have the latter two properties, so if you still want this setup, you'll need a new definition of exponentiation. I would recommend making it commutative, since surreal multiplication is.