User dan - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T02:14:17Z http://mathoverflow.net/feeds/user/8381 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/132943/the-paradox-with-the-first-uncountable-ordinal The paradox with the first uncountable ordinal Dan 2013-06-06T13:31:34Z 2013-06-07T00:05:15Z <p>Suppose we have a set $M = (0,1) \subset R$ of reals well-ordered as the first uncountable ordinal.</p> <p>Let $M(a) = \lbrace x \in M : x &lt; a \rbrace$. For every $a \in M$ set $M(a)$ is countable. That's why every increasing sequence is bounded:<br> $$(*) ~~~~~~~~~~ \forall \lbrace a_1,...,a_n,...\rbrace \subset M ~~\exists b \in M : a_i &lt; b ~~\forall i \in \mathbb{N}.$$</p> <p>Now suppose that we can pick elements from $M$ at <b>random</b>. And let's try to build an increasing random sequence by the following algorithm. Let we have an increasing sequence of elements ${a_1,...,a_n}$. Pick some random number $b$. If $b > a_n$ set $a_{n+1} = b$. Otherwise pick other random number instead of $b$ and check $b > a_n$ condition. Continue this till success. Since we pick numbers at random it shouldn't be a problem to construct an infinite sequence. But every infinite sequence is bounded! Which means that in our infinite process we will never be able to pick random number which is greater than some number $c \in M$. This is even more astonishing since $M(c)$ is countable and $M \setminus M(c)$ is uncountable!</p> <p>Any thoughts how to "solve" this paradox?</p> http://mathoverflow.net/questions/54820/physics-and-churchturing-thesis Physics and Churchâ€“Turing Thesis Dan 2011-02-08T22:09:44Z 2012-12-16T12:24:06Z <p>Is there constructed some set of physical laws from which we can logically obtain that any function that can be implemented in some device is Turing computable?</p> <p>EDIT</p> <p>I believe that if we restrict ourselves to classical mechanics (I mean if we suppose that any device obey just classical mechanics laws) then CT thesis can be proved.</p> http://mathoverflow.net/questions/105514/presentations-of-infinite-index-subgroups Presentations of infinite index subgroups Dan 2012-08-26T10:47:53Z 2012-08-26T19:15:03Z <p>Suppose we have a finitely presented group $G$ with a concrete presentation and a subgroup $H$, generated by a finite set of elements from $G$. How to find the presentation for $H$?</p> <p>If $H$ has finite index we can use Reidemeister-Schreier procedure. But what if $H$ has infinite index? What methods exist? Are there some methods if $H$ is normal?</p> http://mathoverflow.net/questions/102879/bijection-between-irreducible-representations-and-conjugacy-classes-of-finite-gro Bijection between irreducible representations and conjugacy classes of finite groups Dan 2012-07-22T16:52:05Z 2012-07-24T10:28:05Z <p>Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of $S_n$)?</p> http://mathoverflow.net/questions/98198/representations-induced-from-factors Representations "induced" from factors Dan 2012-05-28T15:46:41Z 2012-05-28T15:46:41Z <p>There is a well known concept of <a href="http://en.wikipedia.org/wiki/Induced_representation" rel="nofollow">induced representation</a>. That is, for groups for example, we take representation of a subgroup and construct representation of the whole group.</p> <p>Is there way to construct representations of the whole group from reps of its quotient group?</p> http://mathoverflow.net/questions/95794/ways-to-prove-an-inequality-in-groups Ways to prove an inequality in groups Dan 2012-05-02T18:06:05Z 2012-05-03T20:40:29Z <p>Suppose we have a (possibly infinite) group given by generators and relations. One way to prove some inequality is to construct the representation of the group and show inequality in the representation. Is there some other methods?</p> http://mathoverflow.net/questions/35217/ackermann-function-in-the-primitive-recursive-arithmetic Ackermann function in the Primitive recursive arithmetic Dan 2010-08-11T12:37:25Z 2012-04-18T05:12:18Z <p>Hello.</p> <p>I study <a href="http://en.wikipedia.org/wiki/Primitive_recursive_arithmetic" rel="nofollow">primitive recursive arithmetic</a> and have the following questions.</p> <p>1) Is it possible to express in the PRA that Ackermann function is total?</p> <p>2) If yes, is such expression decidable in the PRA ?</p> <p>Can u suggest some literature on this topic?</p> <p>Thank you.</p> http://mathoverflow.net/questions/70990/torsion-in-triangle-groups/87729#87729 Answer by Dan for Torsion in triangle groups Dan 2012-02-06T22:38:37Z 2012-02-06T22:38:37Z <p>Also you can reference to theorem 2.10 in W.Magnus "Noneuclidean Tesselations and Their Groups" ACADEMIC PRESS New York 1974.</p> http://mathoverflow.net/questions/105514/presentations-of-infinite-index-subgroups/105568#105568 Comment by Dan Dan 2012-08-27T10:54:23Z 2012-08-27T10:54:23Z Oh I see. The set of relators for $H$ is just an $\bf intersection$ of 2 r.e. sets - words in $S$ and words in $\langle\langle R\rangle\rangle$. And yes, terminology is a bit confusing. http://mathoverflow.net/questions/105514/presentations-of-infinite-index-subgroups/105568#105568 Comment by Dan Dan 2012-08-26T20:42:47Z 2012-08-26T20:42:47Z Thanks for interesting examples. But can you clarify the part about recursive presentation and that naive algorithm? Do you suppose that $G$ have solvable w.p.? How can we decide whether or not word in $S$ lie in $\langle\langle R \rangle\rangle$ without knowing the w.p. solution for $G$? http://mathoverflow.net/questions/105514/presentations-of-infinite-index-subgroups Comment by Dan Dan 2012-08-26T14:35:39Z 2012-08-26T14:35:39Z @Benjamin Steinberg: Yes, but we choose subgroup to be finitely generated, so I think it's reasonable to hope to find at least recursive presentations (under certain conditions). http://mathoverflow.net/questions/98198/representations-induced-from-factors Comment by Dan Dan 2012-05-28T17:14:17Z 2012-05-28T17:14:17Z Well, what MTS said is really elementary. Actually I interested in constructing reps from quotient reps AND subgroup reps. But i'd better think by myself before asking again =) http://mathoverflow.net/questions/98198/representations-induced-from-factors Comment by Dan Dan 2012-05-28T16:34:01Z 2012-05-28T16:34:01Z I cant find anything searching the word &quot;precomposition&quot;. Can u give at least 1 reference where it constructed? http://mathoverflow.net/questions/54820/physics-and-churchturing-thesis/54886#54886 Comment by Dan Dan 2011-02-09T22:27:57Z 2011-02-09T22:27:57Z Of course everyone should regard their text with some criticism. This answer flaged as accepted because they are trying to do exactly what I ask. Though other answers are also very good. http://mathoverflow.net/questions/54820/physics-and-churchturing-thesis/54886#54886 Comment by Dan Dan 2011-02-09T15:41:07Z 2011-02-09T15:41:07Z Cool, they've made an article in 1 day after my question! just kidding =) http://mathoverflow.net/questions/54820/physics-and-churchturing-thesis/54858#54858 Comment by Dan Dan 2011-02-09T11:19:16Z 2011-02-09T11:19:16Z In order to check less-than-one-million-bit programs, you must wait for their halt (or for first fault in trying generate S). Generally this cannot be completed in finite time. http://mathoverflow.net/questions/54820/physics-and-churchturing-thesis Comment by Dan Dan 2011-02-08T22:51:33Z 2011-02-08T22:51:33Z &quot;Is there some set of axioms modeling physics from which you can derive a proof of the Church-Turing thesis.&quot; --- This is exactly what I mean. I thought that if we state some physics law then it is anyway defined in some set of axioms (maybe not so formally defined). Though I am not a physician and can be wrong about this.