User nekochan - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T07:16:10Z http://mathoverflow.net/feeds/user/14552 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96434/particular-subset-of-integers-generating-rational-numbers particular subset of integers generating rational numbers Nekochan 2012-05-09T11:19:16Z 2012-05-10T09:36:57Z <p>Hello, maybe this is a naive question, but so far I did not found anything related to the subject.</p> <p>I would like to consider a subset of integers, say E, such that the set ${ \frac{x}{y}, x \in E, y \in E, y \neq 0 }$ is $\mathbb{Q}$.</p> <p>Do such sets have a particular name? Is anyone known for having studied them? And is it possible to define such a set for which any (positive or non-zero) rational is uniquely represented as a ratio of elements in $E$? </p> <p>Thanks by advance for your comments!</p> http://mathoverflow.net/questions/91174/on-unitary-fractions On unitary fractions Nekochan 2012-03-14T13:24:43Z 2012-03-14T15:55:16Z <p>My apologies if the question has already been discussed somewhere else, I did not found anything related to unitary fractions with the search tool...</p> <p>It is a nice exercise for high-school students to prove that any positive rational number less than 1 can be written as a sum of unitary fractions with distinct denominators. One possible proof is to consider the lesser integer n such that p/q-1/n is positive; we obtain a fraction whose numerator is np-q which can be at most p-1.</p> <p>So I was considering the following questions : </p> <ul> <li><p>in some cases, since nq-p can be equal to p-1, it seems possible that one cannot write p/q as a sum of less than p unitary fractions. Is it true that one can find such a fraction for any integer p ?</p></li> <li><p>finding a sum of unitary fractions which is equal to a fraction p/q is not difficult, but is it easy to find the sum with a minimum number of fractions ? And the sum which minimize the sum of denominators ?</p></li> </ul> <p>Thanks by advance for any hint or reference concerning those questions! (Also, I would be very interested in questions related to this topic that could be solved at high-school level)</p> http://mathoverflow.net/questions/82622/numbers-of-intersection-points-and-lines Numbers of intersection points and lines Nekochan 2011-12-04T13:58:15Z 2011-12-08T13:46:31Z <p>Hello, </p> <p>I don't know if this question has already been posted, I have made a little search with keywords and did not found it, sorry if I missed anything.</p> <p>Is it possible to characterize the set of pairs of integers ($l$,$i$) such that one can draw $l$ lines on the euclidean plane with exactly $i$ intersection points?</p> <p>It is quite trivial to see is that given $l$, an upper bound for $i$ is $l(l+1)/2$. More generally, given $l$, any additive decomposition of $l$ of the form $\underset{j=1}{\overset{k}{\sum}} l_i$ provides a value for $i$ which is $\underset{i=1}{\overset{k}{\sum}}$ $\underset{j>i}{\overset{k}{\sum}} l_i l_j$ if we fix for any $i \in [1,k]$ exactly $l_i$ parallel lines such that there is no intersection of three or more lines at the same point. </p> <p>It is not difficult to see that there are pairs ($l$,$i$) that are not of this form. For instance, if you try all decompositions of 6, you may draw 6 lines with 5, 8, 9, 11, 12, 13, 14 or 15 intersection points with this method, but 7 and 10 are missing (they can be obtained with intersection points of three lines).</p> <p>Here is a link containing some observations (http://www.ics.uci.edu/~eppstein/junkyard/how-many-intersects.html). As far as I know, this is the only place where this problem has been seriously considered, but it is quite old and maybe lacks of results. So any additional comment will be welcomed :)</p> <p>Just a final remark, thanks to some projective properties, this question is the same as finding $c$ circles sharing a common point with exactly $i+1$ intersection points. Don't know if this can help.</p> http://mathoverflow.net/questions/67351/a-game-on-numbers a game on numbers Nekochan 2011-06-09T15:39:16Z 2011-07-08T08:55:49Z <p>Hello, here is a little two-players game.</p> <p>Players A and B choose three numbers : a, b and c for A, a', b' and c' for B. The values are numbers between 0 and 1, their sum is 1, and they are ordered: $a \geq b \geq c$ and $a' \geq b' \geq c'$.</p> <p>Then, the players A and B compare their choices : $a$ vs $a'$, $b$ vs $b'$ and $c$ vs $c'$. If a player has 2 values bigger than the other, he wins (otherwise it's a tie).</p> <p>I would like to study whether there is a good strategy in this game but I don't know how to start. Do you have an idea on the general way of studying this kind of game? Any reference of book/article is welcomed :)</p> http://mathoverflow.net/questions/62394/asymptotic-behaviour-of-a-sequence Asymptotic behaviour of a sequence Nekochan 2011-04-20T11:17:37Z 2011-04-26T08:03:05Z <p>Hello, I am interested in some kind of sequence that are "not finitely recurrent".</p> <p>Let $a_i$ be a sequence taking values in ${0,1}$. Consider the sequence $(u_i)$ such that $u_0=1$, and for any positive integer $n$, $u_n =\sum_{i=1}^n u_{n-i}a_i$</p> <p>It is easy to prove that there exists some positive term $\lambda$ such that $(u_n)$ growth faster than $(\lambda-\epsilon)^n$ and slower than $(\lambda+\epsilon)^n$ for any $\epsilon>0$. I would like to know if it is possible to get better bounds, even by adding if necessary some hypothesis on the coefficients $(a_i)$ (but of course this should not be a periodic sequence, otherwise this is well known and too easy).</p> <p>Thanks by advance for your comments !</p> http://mathoverflow.net/questions/96434/particular-subset-of-integers-generating-rational-numbers/96457#96457 Comment by Nekochan Nekochan 2012-05-11T18:17:29Z 2012-05-11T18:17:29Z That is a nice question, thanks! http://mathoverflow.net/questions/96434/particular-subset-of-integers-generating-rational-numbers/96473#96473 Comment by Nekochan Nekochan 2012-05-11T18:16:45Z 2012-05-11T18:16:45Z Thanks for your comment! http://mathoverflow.net/questions/96434/particular-subset-of-integers-generating-rational-numbers/96476#96476 Comment by Nekochan Nekochan 2012-05-11T18:16:15Z 2012-05-11T18:16:15Z Thanks for your comment :) http://mathoverflow.net/questions/96434/particular-subset-of-integers-generating-rational-numbers Comment by Nekochan Nekochan 2012-05-09T11:50:01Z 2012-05-09T11:50:01Z (or $\mathbb{Q}_+$, or $\mathbb{Q}_+^*$, depends if you choose $E$ as a subset of positive integers or integers). http://mathoverflow.net/questions/96434/particular-subset-of-integers-generating-rational-numbers Comment by Nekochan Nekochan 2012-05-09T11:48:27Z 2012-05-09T11:48:27Z Right, I forgot to add these facts, thanks! Anyway the question of uniqueness is not my first concern, mainly I would like to know if for a given set $E$, there are known methods to determine whether the set of quotients is $\mathbb{Q}$. http://mathoverflow.net/questions/93650/stability-under-arithmetics Comment by Nekochan Nekochan 2012-04-10T12:34:42Z 2012-04-10T12:34:42Z Sorry, I did not mean the usual stability, but &quot;almost-stability&quot; with respect to a multiplicative factor, for instance 2(x+y) or 2xy in F, or something else using an algebraic factor instead of 2. http://mathoverflow.net/questions/91174/on-unitary-fractions Comment by Nekochan Nekochan 2012-03-14T14:20:50Z 2012-03-14T14:20:50Z Thanks Sean, that was a mistake, it is np-q as you noticed it. Thanks Charles for the references which solve the first question; however I haven't found anything concerning the minimal sum of denominators. http://mathoverflow.net/questions/82622/numbers-of-intersection-points-and-lines/82961#82961 Comment by Nekochan Nekochan 2011-12-11T13:21:09Z 2011-12-11T13:21:09Z That's very nice, thanks a lot for this reference! http://mathoverflow.net/questions/82622/numbers-of-intersection-points-and-lines Comment by Nekochan Nekochan 2011-12-05T12:28:42Z 2011-12-05T12:28:42Z Thanks Joseph for the book however it won't be possible for me to read it :/ You're right Will, that's l(l-1)/2, not l(l+1)/2 Thanks Gerry for this discussion, I will take a look at this, maybe this can help http://mathoverflow.net/questions/67351/a-game-on-numbers Comment by Nekochan Nekochan 2011-06-09T18:37:58Z 2011-06-09T18:37:58Z Thanks a lot for the name ;) http://mathoverflow.net/questions/67351/a-game-on-numbers/67356#67356 Comment by Nekochan Nekochan 2011-06-09T17:08:55Z 2011-06-09T17:08:55Z Thanks! Actually I know that there is no winning strategy in only one move. I would like to find a set of moves (with a probability of choosing each one) such that the expected gain is non negative. http://mathoverflow.net/questions/67351/a-game-on-numbers Comment by Nekochan Nekochan 2011-06-09T17:01:15Z 2011-06-09T17:01:15Z then a good strategy could be to determine whether there exists a set P (finite?) such that, for any point in the triangle, draw winning and losing areas, and you have to check if the weight (adding a probability of choosing a point...) of winning choices is greater than the weight of losing ones. Well, maybe this is totally unclear, I have to make it more understandable for sure :) http://mathoverflow.net/questions/67351/a-game-on-numbers Comment by Nekochan Nekochan 2011-06-09T16:59:33Z 2011-06-09T16:59:33Z I have an idea but I don't know if it is useful. Consider the values $\lambda=a-b$ and $\mu=b-c$. One may draw a triangle of possible values ($\lambda$,$\mu$) which are one-to-one with the choice of ($a$,$b$,$c$). Then for any point ($\lambda$,$\mu$) in the triangle, one may draw three line crossing in this point (whose direction do not depend on the point) which define areas of winning and losing opponent's choice (they should alternate around the point). http://mathoverflow.net/questions/67351/a-game-on-numbers/67354#67354 Comment by Nekochan Nekochan 2011-06-09T16:42:42Z 2011-06-09T16:42:42Z Hmm I'm not sure that I understand well your answer. It seems to me that your strategy consists in playing values of the form (1/2, 1/2-$\varepsilon$, $\varepsilon$). Then a winning counter-strategy is to play values of the form (1/2+2$\varepsilon$, 1/4-$\varepsilon$, 1/4-$\varepsilon$). Please correct if this doesn't match your idea. http://mathoverflow.net/questions/65790/boundary-of-sums-of-sets Comment by Nekochan Nekochan 2011-05-23T19:51:27Z 2011-05-23T19:51:27Z err... no, D(2X) is not included in D(X+X), it's a mistake.