User campello - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:35:20Z http://mathoverflow.net/feeds/user/21398 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/123384/primitive-orthogonal-vectors-unimodular-matrices Primitive orthogonal vectors/Unimodular matrices Campello 2013-03-01T23:30:01Z 2013-03-02T05:46:55Z <p>Primitive sets of vectors are very important in the theory of point lattices, since they constitute the sets of vectors that are part of a basis for the lattice.</p> <p>A set of integer vectors $v_1,\ldots,v_k$ is primitive in $\mathbb{Z}^n$ if there are $v_{k+1},\ldots,v_{n}$ such that the matrix whose columns are $v_i$ has determinant 1 (i.e., unimodular). Is there an easy way to characterize primitive vectors which are orthogonal? In the end, I want to find families of vectors $v_1(m),\ldots,v_k(m)$ which are primitive, orthogonal and the norm increases with m. Is there any such family, is it known?</p> http://mathoverflow.net/questions/109229/hypergeometric-function-2f-1-n-r12 hypergeometric function $_2F_1(-n;-r;1;2)$ Campello 2012-10-09T14:12:25Z 2012-10-11T21:28:13Z <p>The hypergeometric function $_2F_1(-n;-r;1;2)$ appears in many different situations. For instance, it counts the number of integer points within a sphere in the $l_1$ norm, i.e., </p> <p>$$_2F_1(-n;-r;1;2) = \mbox{#} \lbrace x \in \mathbb{Z}^n : |x_1| + \cdots + |x_n| \leq r \rbrace$$</p> <p>and another formula for $_2F_1(-n;-r;1;2)$ is given by</p> <p>$$_2F_1(-n;-r;1;2) = \sum_{i=1}^{\min\lbrace{n,r}\rbrace}{n \choose i}{r \choose i}2^i$$</p> <p>Does anyone know an asymptotic formula for this function when $n$ is large?</p> <p>I know there are some closed formulas for the Gaussian hypergeometric (i.e. $_2F_1(a,b;2,1)$) and in some other cases, but haven't find any clue in this case.</p> http://mathoverflow.net/questions/123384/primitive-orthogonal-vectors-unimodular-matrices/123393#123393 Comment by Campello Campello 2013-03-04T16:23:29Z 2013-03-04T16:23:29Z Yes, this will do it. Is there any reference where you found this example? Based on this, I can find other ones by guesswork, but it would be nice to find a systematic way. http://mathoverflow.net/questions/109229/hypergeometric-function-2f-1-n-r12 Comment by Campello Campello 2012-10-14T16:59:19Z 2012-10-14T16:59:19Z What do you mean comparable? $r=O(n)$ ? Also it is not clear for me how the $i=n+r−\sqrt{n^2+r^2}$ comes out... http://mathoverflow.net/questions/109229/hypergeometric-function-2f-1-n-r12 Comment by Campello Campello 2012-10-11T21:48:00Z 2012-10-11T21:48:00Z Thank you Noam. The mistake was corrected. Gerald, the approximation $2^r n^r/r!$ (volume of ball in the $l_1$ norm) is good for fixed $r$ and (very) large $n$. However, when $r$ also increases (for example, if r = O(n)), what should be the behavior?