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I'm a graduate student at UC Berkeley, studying mathematical logic. I'm especially interested in reverse mathematics and abstract model theory.
11h

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Appropriately choosing the parameter so that one function maximizes while other minimizes
Well, it need not exist, e.g. if $f=g$. And if we know a priori that such a simultaneous extremum does exist, is there some reason we can't find the maxima of $f$, find the minima of $g$, and look for points in both sets? 
17h

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The relation on the set of functions
But what if 17 is not constant? (An actually serious comment: what is the motivation for this? Without knowing that, it's not clear how to alter the problem to avoid Joel's examples and still yield something interesting.) 
20h

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Who know about Rumek proof
One further point: it appears there is no "Professor Rumek" in mathematics, at all! Friedman was very careful to make this recognizably false. Friedman's point was to pose a thought experiment, to drive home the point of just how revolutionary an inconsistency in ZFC would be. Keep in mind that his position is that the consistency of ZFC  and much, much more  is perfectly justifiable by philosophical arguments. This email is meant (I believe) as tongueincheek "evidence" for that consistency. 
20h

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Who know about Rumek proof
Ooh, I like that point! 
20h

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Who know about Rumek proof
As I said above, there is no such thing  this is a hypothetical scenario developed by Harvey Friedman. Google "professor Rumek set theory" if you doubt me. (Also, "middlegate" and "westgate" universities don't exist, except in the case of westgate as an online degree program. Seriously, do some research.) 
20h

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Who know about Rumek proof
That's not a thing that actually happened  that's Harvey Friedman writing a hypothetical newspaper article to illustrate (what he believes) would be the impact of a proof of inconsistency of ZFC. If nothing else, the "Harvey Friedman, reporting" at the bottom should be a dead giveaway. 
2d

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reference on aperiodicity and cluster
What result is this? (For that matter, what is the "this" on which you are doing research?) If you have such a result, then doesn't that answer your own question? 
2d

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reference on aperiodicity and cluster
Wait, why do you believe there is such a message? 
Aug 29 
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When are two algorithms essentially the same?
If you believe they are all equivalent, then why did you ask this question? 
Aug 29 
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Are there any relevance between coefficients of simple continued fraction of quadradic algebraic number and algebraic number with degree $2^n$
That's an extremely stringent requirement. Given that the question wavers from the trivially general to that level of specificity, I'm skeptical that there is a good question here. Also, it is not clear that you have put much work into trying to figure out this question yourself. In particular, have you tried looking at any specific nontrivial examples? 
Aug 29 
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impossibility and mode of convergence
I'm a little confused by the use of "convergence" here, but is certainly true that "events with probability zero can happen" (more precisely, sets with measure zero need not be empty). 
Aug 29 
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Any result, example or conjecture about the computational complexity of transcendental number that can not be computed by linear time Turing Machine
It is still the case that there are trivially transcendental numbers which are computable but arbitrarily complicated (e.g., not computable by a Turing machine running in exponential time, or Ackermann function time, or . . .). And I still have no idea what you are looking for in an answer  do you want a numbertheoretic property which has complexitytheoretic implications? (E.g., "irrational but algebraic" is conjectured to imply "not lineartime computable".) At present, this seems more like a fishing expedition than a real question. 
Aug 29 
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Any result, example or conjecture about the computational complexity of transcendental number that can not be computed by linear time Turing Machine
Of course there are transcendental numbers with different computational complexities, since there are uncountably many transcendental numbers. 
Aug 29 
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Any result, example or conjecture about the computational complexity of transcendental number that can not be computed by linear time Turing Machine
I did not vote to close, but this doesn't seem like a real question: it seems "not computable by a linear time Turing machine" is a pretty straightforward characterization of the computational complexity of a transcendental number that cannot be computed by a linear time Turing machine! It's unclear what you're asking for. 
Aug 28 
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When are two algorithms essentially the same?
I don't think this is a real question yet. You seem to be begging the question: why do you suspect there is a definition of equivalence/similarity which one might want to apply in these special cases? Given that in general no such definition seems to exist, and that there are lots of potentially contradictory properties we might want such a definition to satisfy, I think you need to explain at the very least why you think these particular classes of machines are likely to admit such definitions. 
Aug 28 
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Maximum length of a chain of topologies on $\Bbb R$
One small remark: Call a collection of sets $\mathcal{X}\subseteq\mathcal{P}(\mathbb{R})$ independent if for each $A\in\mathcal{X}$, $A$ is not in the topology generated by $\mathcal{X}\{A\}$. Obviously any independent set yields a chain of topologies; unfortunately, there are "small" maximal independent sets  for example, $\{\{r\}: r\in\mathbb{R}\}$  and I don't see an obvious way to build large independent sets. But maybe there is one? 
Aug 28 
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Maximum length of a chain of topologies on $\Bbb R$
Note, though, that we can bound the cofinality of $\mathfrak{I}$: pick a wellordered cofinal sequence through $\mathfrak{I}$. Each successor topology in that sequence must contain some set of reals not in any previous topology in that sequence, and so the length of the sequence is at most $\vert 2^{\vert\mathbb{R}\vert}\vert$. 
Aug 28 
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$\infty$Borel Determinacy?
If I understand that answer: start with a model $V$ of ZF+AD, in which every set of reals is Ramsey; now force with $\mathcal{P}(\omega)/fin$ to get a generic extension $V[G]$. Every set in the ground model remains determined, since the forcing adds no new reals, and every infinityBorel set lives in the ground model, so is determined. So $V[G]\models (*)$. I don't see, though, how to show that $V[G]\models \neg AD$, which would be necessary to answer question 1. Is it clear that $V[G]\models\neg AD$? 
Aug 27 
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What does a Turing machine compute?
@Francois, you're quite right; I should have been more specific. What I meant, was that there is probably not a body of research specifically around varying the i/o interpretations of a given model. (That was what I meant by the last sentence of my first comment.) I should probably clarify that, too, though: some i/o interpretation changes of course have dramatic effects (e.g., unary vs. binary), but Hans' variations fixed some method of representing numbers and varied where on the tape these numbers are to be located, and it was this specific approach that I was saying didn't yield a topic. 
Aug 26 
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What does a Turing machine compute?
Basically, anything that looks reasonably like a Turing machine (and some things that don't: murecursion, lambda calculus, MRDP theorem, . . .) results in either the same notion of computability, or a strictly weaker notion. I'm not sure of anywhere where it's specifically focused on, but see section 3.2 of Sipser's book "Introduction to the theory of computation." It's obviously hard to make this rigorous, but it's the sort of thing that becomes fairly clear after working with Turing machines and their ilk for a while. 