Ivars Wrote:No, for all real points \( x_0 \).

Quote: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

bo198214 Wrote: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.

Quote: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 Wrote:

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

bo198214 Wrote: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.

andydude Wrote:You need to learn the difference between divergent and false.

Ivars Wrote: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.

bo198214 Wrote: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.

Quote: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

(06/04/2008, 07:20 AM)Ivars Wrote:

bo198214 Wrote: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.

andydude Wrote: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