User paarshad - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:48:17Z http://mathoverflow.net/feeds/user/695 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3997/are-there-any-interesting-connections-between-game-theory-and-algebraic-topology Are there any interesting connections between Game Theory and Algebraic Topology? paarshad 2009-11-03T21:00:01Z 2013-02-12T17:42:52Z <p>I've been learning game theory on my own and was just curious how it connected with previous things I've learned. So are there any interesting connections between Game Theory and Algebraic Topology? or possibly other branches of topology?</p> http://mathoverflow.net/questions/24958/showing-e-is-transcendental-using-its-continued-fraction-expansion Showing e is transcendental using its continued fraction expansion paarshad 2010-05-17T01:25:03Z 2012-01-13T03:40:28Z <p>Can the transcendence of e be shown using its continued fraction expansion e = [2;1,2,1,1,4,1,1,6,...]?</p> http://mathoverflow.net/questions/4917/on-euclids-proof-of-the-infinitude-of-primes-and-generating-primes On Euclid's proof of the infinitude of primes and generating primes paarshad 2009-11-10T20:27:04Z 2010-09-13T05:05:51Z <p>So looking at Euclid's proof he says 1)take a finite family of primes (F) 2)multiply all the primes and add one 3)this new number has at least 1 new prime factor</p> <p>So I was wondering about what kind of primes you get by recursively feeding this process into it self.</p> <p>Since the number you must factor grows exponentially, it's hard to get a lot of numerical evidence for what happens.<br /> I calculated a few:</p> <p>[2]-> [2,3]-> [2,3,7]->[2,3,7,43]->[2,3,7,43,13,139]->[2,3,7,43,13,139,3263443] ->[2,3,7,43,13,139,3263443,547,607,1033,31051]-> cannot factor 113423713055421844361000443</p> <p>[5] (x5)-> [5,2,3,31,7,19,37,3343,79,193662529] -> cannot factor 234069798025176583891</p> <p>Obviously quite a few primes are missing, 5,11,19,etc from the first list, but could show up later.</p> <p>So my question is does a finite family of primes exist that eventually generates all the primes? I figure this probably doesn't have an easy answer, but any information related to this process would be appreciated, or even why it can't be done.</p> http://mathoverflow.net/questions/31980/2-adic-valuation-of-the-class-number 2-adic valuation of the class number paarshad 2010-07-15T08:57:31Z 2010-07-15T18:08:57Z <p>I came across the following line and was wondering what it meant exactly and how you go about showing it. Let d be a fundamental discriminant. Let P(d) = the divisors of d except for the largest.</p> <p>The cardinality of P(d) is at most the 2-adic valuation of the h (the class number).</p> <p>I only have a vague notation of what is meant by the 2-adic valuation, so any clarification on that as well as how to prove the statement would be helpful. Thank you.</p> http://mathoverflow.net/questions/31769/number-of-integers-sqrtd-2-represented-by-an-indefinite-quadratic-form Number of integers $<\sqrt{d}/2$ represented by an indefinite quadratic form paarshad 2010-07-13T22:30:49Z 2010-07-14T03:17:02Z <p>Given an indefinite integral quadratic form $Q(x,y)=ax^2 +bxy + cy^2$ with $b^2-4ac=d>0$, is there an easy way to count the number of integers in $t \in (-\sqrt{d}/2, \sqrt{d}/2)$ such that there exists $(m,n)\in Z \times N$ with $Q(m,n)=t$, ie. m any integer, n a positive integer? Good upper bounds would also be acceptable.</p> http://mathoverflow.net/questions/29850/primes-as-the-first-coefficient-of-a-reduced-indefinite-quadratic-form Primes as the first coefficient of a reduced indefinite quadratic form paarshad 2010-06-29T00:46:48Z 2010-06-29T18:10:08Z <p>Given a discriminant d>0 (make it fundamental if that is easier), when can a prime p be the the $x^2$ coefficient of a reduced indefinite quadratic form?</p> <p>That is, for what p is there a reduced form $px^2 + bxy + cy^2,$ with $b^2-4pc=d$?</p> http://mathoverflow.net/questions/24842/bsd-conjecture-and-l-functions-with-zeroes-of-order-g BSD conjecture and L functions with zeroes of order g paarshad 2010-05-16T00:01:08Z 2010-05-16T01:16:26Z <p>If the group of rational points of $E$, which is finitely generated by the Mordell-Weil Theorem, has $g$ generators of infinite order, then the Birch-Swinnerton-Dyer conjecture gives</p> <p>$L_E(s)$ has a zero of order $g$ at $s=1$.</p> <p>Assuming the BSD conjecture, is it possible to (and if so how) to construct such $L_E(s)$? Specifically, if we want $g=3$ or $4$?</p> http://mathoverflow.net/questions/23386/lower-bounds-for-split-primes-in-real-quadratic-fields Lower bounds for split primes in Real quadratic fields paarshad 2010-05-03T22:47:35Z 2010-05-06T19:21:32Z <p>Snippet portion: From Iwaniec and Kowalski's Analytic Number Theory:</p> <p>If the class number $h=h(D)$ is small, then there are only few prime ideals $\bf{p}$ of degree one with small norm. Indeed, if $p=\bf{p \bar{p}}$ with $(\bf{p},\bf{\bar{p}})=1$, then $\bf{p}^h$ is a principal ideal generated by $\frac{1}{2}(m+n\sqrt{D})$ with $n \ne 0$, when $p^h = \frac{1}{4}(m^2-n^2 D) \ge \frac{|D|}{4}$.</p> <p>Therefore the least prime $p_1 = p_1(D)$ with $\chi_D(p_1)=1$ satisfies $p_1 \ge {(\frac{|D|}{4})}^{1/h}$.</p> <p>Hence $\chi_D(n)$ agrees with $\mu(n)$ on all squarefree numbers $n \le {(\frac{|D|}{4})}^{1/h}$ with $(n,{(\frac{|D|}{4})}^{1/h})=1$. This property is not likely to hold in long segments (because $\chi_D$ is periodic while $\mu$ is not), therefore this suggests that h is rather large.</p> <p>Question portion: Although the above argument would not work in a Real quadratic field ($D > 0$ so the last inequality in the first paragraph does not hold), it seems that if we replace the class number h with h times the regulator this should work.</p> <p>Any ideas on how to actually show this?</p> http://mathoverflow.net/questions/3820/how-hard-is-it-to-compute-the-number-of-prime-factors-of-a-given-integer/4337#4337 Answer by paarshad for How hard is it to compute the number of prime factors of a given integer? paarshad 2009-11-06T05:23:00Z 2009-11-06T05:23:00Z <p>This paper <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa49/aa4913.pdf" rel="nofollow">On some estimates involving the number of prime divisors of an integer</a> might have something that helps you out.</p> http://mathoverflow.net/questions/31980/2-adic-valuation-of-the-class-number/32048#32048 Comment by paarshad paarshad 2010-07-15T22:44:57Z 2010-07-15T22:44:57Z Yep I picked it up from the library yesterday and have Cox's book laying around somewhere. I think my main confusion was due to the 2-adic part. Once that was cleared up it made sense and the proof is straight forward. Thanks! http://mathoverflow.net/questions/31769/number-of-integers-sqrtd-2-represented-by-an-indefinite-quadratic-form/31776#31776 Comment by paarshad paarshad 2010-07-14T00:57:02Z 2010-07-14T00:57:02Z Also so I don't waste a trip to the library, is the book you're referring to &quot;Binary Quadratic Forms: Classical Theory and Modern Computations&quot; by Duncan Buell? http://mathoverflow.net/questions/31769/number-of-integers-sqrtd-2-represented-by-an-indefinite-quadratic-form/31776#31776 Comment by paarshad paarshad 2010-07-14T00:19:43Z 2010-07-14T00:19:43Z Sorry about the non-square. I would be happy with d fundamental as well. Let's say I modify it so Q is reduced. You mentioned that I was essentially talking about the cycle length. Does that mean that the number of integers &lt;$\sqrt{d}/2$ represented is bounded by the cycle length? http://mathoverflow.net/questions/29850/primes-as-the-first-coefficient-of-a-reduced-indefinite-quadratic-form Comment by paarshad paarshad 2010-06-29T09:54:59Z 2010-06-29T09:54:59Z @Wadim, I'm just summing up coefficients of reduced quadratic forms and want to make sure I'm considering the correct possibilities. http://mathoverflow.net/questions/29850/primes-as-the-first-coefficient-of-a-reduced-indefinite-quadratic-form/29860#29860 Comment by paarshad paarshad 2010-06-29T09:52:54Z 2010-06-29T09:52:54Z Thank you for your answer. Concerning the case where p|d, does this sound like the correct generalization. If (a,b,c) is a quadratic form, and p|a, then (d|p)=1 or 0? http://mathoverflow.net/questions/24958/showing-e-is-transcendental-using-its-continued-fraction-expansion Comment by paarshad paarshad 2010-05-17T02:42:35Z 2010-05-17T02:42:35Z Thank you for the reference. Interesting read. http://mathoverflow.net/questions/24842/bsd-conjecture-and-l-functions-with-zeroes-of-order-g/24846#24846 Comment by paarshad paarshad 2010-05-17T01:21:18Z 2010-05-17T01:21:18Z This is what I was looking for. Thank you. http://mathoverflow.net/questions/24842/bsd-conjecture-and-l-functions-with-zeroes-of-order-g/24846#24846 Comment by paarshad paarshad 2010-05-16T07:32:30Z 2010-05-16T07:32:30Z I'm in case 1. I'm just curious where to find such curves E, and how to turn them in to $L_E(s)$. http://mathoverflow.net/questions/4917/on-euclids-proof-of-the-infinitude-of-primes-and-generating-primes Comment by paarshad paarshad 2009-11-10T20:48:34Z 2009-11-10T20:48:34Z So obviously they can cause different sequences. I am interested in know why adding multiplicity would make the sequence more manageable. The first time I saw a change was for [11] without multiplicity [11] (x5)-&gt; [11,2,3,67,4423,43,454849,7,37,2029,727929913] with multiplicty [11] (x5)-&gt; [11,2,2,3,7,19,97,181,103,139,21529,13,7308166138386889]