CRITICAL TORUS !!!

tommy1729 · science forum beginner Joined: 23 Feb 2006 Posts: 33

Dear mathfriends (?)

In observing math , ive noticed , and im not alone, that many (proof)problems in math (especially the harder ones ) can be transformed into some kind of " critical line " problem.

The most famous example is of course Riemann's Hypothese.

But also the tanc conjecture can be rewritten as a critical line problem.

Representions theorems can be rewritten as a critical line problem.

For all clarity , i consider "critical line problems" as follows

1) a subgroup of zeros all lie on the critical line

2) none of the zeros of a subgroep lie on the critical line

3) no 3 zeros lie on a line (probably reducable to 1) or 2) so less important i guess )

Every upperbound or lowerbound conjecture is equivalent to some critical line problem

You can just literally bend functions to get a critical line , if ya already have a simple critical "function/figure/path"

SO if ya have a "critical monotonic increasing polynoom " like x^a , you can easily transform it to a line , bye transforming ( bending ) the function

And don't forget that e.g. circels are the product of 2 complex lines.

So critical problems in 2D are often easily transformable into "critical line problems in 2D"

So the "2D equivalent problems " are mainly and often simply reducable to critical lines

A bit like belonging to an NP class to make a comparison.( but for proofs instead of computations )

However !!!

some problems might not be expressible in 2D criticals , but in 3D criticals

with the concept 3D being most general !

so for instance allowing "polysigned" expressions (enter in google if you dont know what polysigned is , there is a whole website explaining the basics )

i mentioned before transforming polynoom to a line

but in 3D we have topology joining the party;

we could have a " critical torus "

or " polysigned critical torus "

or even more complicated stuff , not easily , or not at all , "transformable to a line , or even 2D disk"

we all know topology is strongly related to number theory
and metamathematics is getting popular

i hope ive made clear why i believe so strongly in this "critical theory " importance.
consider it a bit like NP=P , extended Riemann Hypothese and metamathematics.

so i am deeply intrested in the "critical torus" since it is relating so much math concepts.

and it relates them in only 2 words, nice :-)

i do not claim to be a genius or anything , but as far as i know , this has never been studied.

and as an intuitionist and constructivist , this is very appealing to me.

i could continue with more than 3 dimensions , but i think that we should start with the simplest extension ( 2D --> 3D ; holes 0 --> +1 )

so does anybody know functions that have or might have a critical torus ?

also this appears to me as a nice puzzle

would be nice to see this online , e.g. at websites and forums

this mail is not complete in the sence that the critical torus can be considered in different ways;

do the zeros lie on the surface of the torus , or in the torus ?

the most logical and intresting is according to me the critical surface of a torus , since other wise the condition is very loose and its more of a generalization to a strip , not a line.

there is also probably a connection with eigenvalues and cellular automata , but thats confusing me at the moment.

hope you think about it.

i find it a logical question anyway..

greetz

tommy1729

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