User kale - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:53:16Z http://mathoverflow.net/feeds/user/22146 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100292/hypergeometric-method Hypergeometric method Kale 2012-06-21T21:05:12Z 2012-06-21T21:05:12Z <p>Suppose I have an approximation $p_0/q_0$ to an irrational number $\alpha$. The hypergeometric method uses the fact that if $|Q_n\log-P_n| &lt; C*b^{-n}$ and $|Q_n| &lt; a^n$ then $log a/log b$ is the exponent of diophantine approximation. I'm wondering how I can determine when this is an immprovement on liouville's inequality. I know it clearly depends on the an and b, so for my question I'm looking for a and b to be given explicitly and then a description of how I might be able to determine when loga/logb is an improvement on liouville's inequality. </p> http://mathoverflow.net/questions/99739/how-do-you-bound-exponent-of-x21yp how do you bound exponent of x^2+1=y^p Kale 2012-06-15T19:22:20Z 2012-06-15T19:22:20Z <p>for p a prime exponent using linear forms in logs?</p> <p>So far I have (x-i)(x+i)=y^p which are coprime and hence x+i=(a+ib)^p , now how do I get a linear form in logs so that I can find an upper bound on p?</p> http://mathoverflow.net/questions/98446/folklore-lemma-p-adic-case Folklore Lemma (p-adic case) Kale 2012-05-31T01:24:00Z 2012-05-31T01:24:00Z <p>Is there a p-adic version of the so called 'folklore lemma' which relates a sequence of diophnatine approximations to an algebraic number to the exponential of irrationality.</p> http://mathoverflow.net/questions/98425/sharpenings-of-liouvilles-inequality Sharpenings of Liouville's inequality Kale 2012-05-30T22:12:51Z 2012-05-30T22:29:10Z <p>The norm of an algebraic number $\alpha$ is the product of its conjugates, $N(\alpha)$.</p> <p>Suppose that I have an inequality of the form $|x-\alpha*y| > c X^{n-\gamma}$ where $X=max{|x|,|y|}$ and c is some constnat, can this be used to find an upper bound on the norm of $x-\alpha*y$ of the form $|N(x-\alpha*y)| &lt; C *X^{n-\gamma}$ $|x-\alpha*y|$ where C is some constant and $\gamma \geq 1$</p> <p>In the case of $\gamma=1$ it can since: $|N(x-\alpha*y)|=\prod (x-\alpha_i*y)|=|x-\alpha*y|X^{n-1} \prod (x/X-\alpha_i*y/X) \leq |x-\alpha*y| \prod(1|+|\alpha_{i}|)X^{n-1}$ </p> <p>It's unclear to me how it might be possible to find a sharper upper bound on the norm of $x-\alpha*y$ by using a sharper exponent in liouiville-type inequalities.</p> <p>I should add that the constant C should be effective, as it is in the case of $\gamma=1$.</p> http://mathoverflow.net/questions/97247/p-adic-approximations-to-hypergeometric-functions p-adic approximations to hypergeometric functions Kale 2012-05-17T18:26:48Z 2012-05-17T18:26:48Z <p>I'm trying to research this question and I need links to any of the following papers.</p> <p>"Baker A (1964) Rational Approximation to certain algebraic numbers. Proc. London Math. Soc 4: 385-308"</p> <p>"Alladi K and Robinson ML (1980) Legendre polynomials and irrationality. K. Reine Angew. Math 318: 137-155"</p> <p>I can't find any of them online though.</p> http://mathoverflow.net/questions/95453/computing-the-measure-of-a-generator-in-a-fixed-number-field-k computing the measure of a generator in a fixed number field K Kale 2012-04-28T19:53:19Z 2012-05-12T21:22:00Z <p>Suppose $K=\mathbb{Q}(\alpha)$ is a fixed number field with $[K:\mathbb{Q}]=d$ and fixed basis $b_1,b_2,..,b_d$. Define <code>$$m(\alpha) = max \{ | p_i| , |q_i| : 1 \leq i \leq d \},$$</code> where the max is taken over all representatives of $\alpha$ of the form <code>$$\alpha = \frac{p_1b_1+p_2b_2+\ldots+p_db_d}{q_1b_1+q_2b_2+\ldots+q_db_d}.$$</code></p> <p>Are there any papers which describe techniques to compute this number? </p> http://mathoverflow.net/questions/96156/the-p-adic-subspace-theorem The p-adic subspace Theorem Kale 2012-05-06T21:02:45Z 2012-05-06T21:02:45Z <p>Could someone explain how the subspace theorem could be used to transfer results from archimedian valuatons to nonarchimedian ones?</p> http://mathoverflow.net/questions/96048/p-adic-version-of-liouvilles-approximation-theorem p-adic version of Liouville's approximation theorem Kale 2012-05-05T05:57:56Z 2012-05-05T08:31:42Z <p>Does anyone know of a p-adic analogue of Liouville's approximation theorem <a href="http://mathworld.wolfram.com/LiouvillesApproximationTheorem.html" rel="nofollow">http://mathworld.wolfram.com/LiouvillesApproximationTheorem.html</a> with proof?</p> <p>I'm aware of Roth's theorem and subspace theorems which can be generalized to p-adic numbers, these are different in that the constants are not effective. </p> http://mathoverflow.net/questions/95957/p-adic-diophantine-approximation p-adic diophantine approximation Kale 2012-05-04T05:31:27Z 2012-05-04T13:38:04Z <p>Suppose you have a sequence of rational numbers that gives a diophantine approximaion an irrational, what can be said p-adically about this sequence?</p> <p>I'm interested in the p-adic analoges of these theorems (such as Thue-siegel-roth), but can't find any straightforward resources on the subject. I can't even find what a good diophantine approximation would mean over a p-adic field.</p> http://mathoverflow.net/questions/95815/p-adic-analysis-of-hypergeometric-functions p-adic analysis of hypergeometric functions Kale 2012-05-02T22:57:28Z 2012-05-03T04:53:53Z <p>Are there any p-adic techniques that can be applied to the 2F1 hypergeometric function?</p> <p>For e.g. I'm interested in which values this function converges p-adically.</p> http://mathoverflow.net/questions/95617/lower-bounds-on-binary-forms-of-degree-m Lower Bounds on Binary Forms of degree m Kale 2012-04-30T20:43:42Z 2012-04-30T21:08:51Z <p>The following paper <a href="http://www.math.leidenuniv.nl/~evertse/05-discres.pdf" rel="nofollow">http://www.math.leidenuniv.nl/~evertse/05-discres.pdf</a> looks at the binary form $\sum \alpha_ix^iy^{m-i} where i ranges from 0 to m. It then defines the discriminant.</p> <p>It's well known that for binary quadratic forms how to find a lower bound using the discriminant. I was wondering if analogous theorems hold for a form of arbitrary degree.</p> <p>Ultimately I would like to find a lower bound in terms of the variable y for this binary form I'm working on.</p> <p>A little more details:</p> <p>I'm trying to show that the binary form$x^{n-1}+x^{n-2}y\alpha^2+x^{n-3}y^2\alpha^3+...+y^{n-1}\alpha^{n-1}$is bounded below by$c*y^{n-1}$or something similar where c is some explicit constant.</p> http://mathoverflow.net/questions/94141/is-it-true-that-if-x-is-a-separable-banach-space-and-m-a-inear-manifold-in-the-du Is it true that if X is a separable banach space and M a inear manifold in the dual Kale 2012-04-15T19:29:10Z 2012-04-15T19:29:10Z <p>that weak sequential closure M = weak star closure then M is sequentially dense?</p> http://mathoverflow.net/questions/91217/a-point-in-the-weak-closure-but-not-in-the-weak-sequential-closure A point in the weak closure but not in the weak sequential closure Kale 2012-03-14T20:36:46Z 2012-03-15T21:26:01Z <p>I'm trying to find a proof of this counterexample by von Neumann:</p> <p>Let$x_{mn}\in \ell^2$be defined by $$x_{mn}(m)=n \quad,\quad x_{mn}(n)=m \quad,\quad x_{mn}(k)=0 \hbox{ otherwise,}$$ and let <code>$S=\{ x_{mn} : m, n\geq 1\}$</code>. Von Neumann shows that$0$is in the weak closure of this set but no sequence in$S$convergess weakly to$0$.</p> http://mathoverflow.net/questions/99739/how-do-you-bound-exponent-of-x21yp Comment by Kale Kale 2012-06-15T20:34:52Z 2012-06-15T20:34:52Z not what link is supposed to mean, I'm not asking for a full proof of cataln's conjecture but consider this particular e.g. http://mathoverflow.net/questions/98425/sharpenings-of-liouvilles-inequality Comment by Kale Kale 2012-05-30T22:44:04Z 2012-05-30T22:44:04Z Gerry Myserson: I'm not sure what is stronger/weaker here. I would like to use a stronger Liouiville's inequality to imply an upper bound of a certain form on the norm of the algebraic number$x-\alpha*y$(in absolute value). Basically I want$\prod (x-\alpha_i*y) (where i ranges over all conjugates of $\alpha$ other than itself to be $\leq C*X^(\gamma=1)$ given that the stronger Liouville inequality $|x-\alpha*y| \leq c *X^{\gamma-1}$ does hold. http://mathoverflow.net/questions/98425/sharpenings-of-liouvilles-inequality Comment by Kale Kale 2012-05-30T22:23:30Z 2012-05-30T22:23:30Z To make my questions a bit more clear: In the case of $\gamma=1$, the inequality follows immediately, I would like a proof that works for any $\gamma$, given the lower bound on the absolute value of $x-\alpha*y|$ I think there should be some way to use a lower bound on the absolute value of an algebraic number and transform it into an upper bound on the norm as is the case for $\gamma=1$. http://mathoverflow.net/questions/96156/the-p-adic-subspace-theorem Comment by Kale Kale 2012-05-07T00:16:43Z 2012-05-07T00:16:43Z I was just looking for an e.g. of how to apply it http://mathoverflow.net/questions/96048/p-adic-version-of-liouvilles-approximation-theorem Comment by Kale Kale 2012-05-05T08:51:25Z 2012-05-05T08:51:25Z I'm intereste in ways that ordinary Liouville's theorem relates to p-adic Liouville's theorem http://mathoverflow.net/questions/96048/p-adic-version-of-liouvilles-approximation-theorem Comment by Kale Kale 2012-05-05T08:33:20Z 2012-05-05T08:33:20Z Anatoly: Is this the only paper on the subject? It looks like it's generalized over any extension field. http://mathoverflow.net/questions/95957/p-adic-diophantine-approximation Comment by Kale Kale 2012-05-04T06:26:53Z 2012-05-04T06:26:53Z I'm aware of these particular references listed on the wikipedia page, the problem is I can't find them anywhere. http://mathoverflow.net/questions/94141/is-it-true-that-if-x-is-a-separable-banach-space-and-m-a-inear-manifold-in-the-du Comment by Kale Kale 2012-04-15T20:40:26Z 2012-04-15T20:40:26Z Op here. M is just a linear subspace of X^*. Spaces where weak sequential closure is the same as ordinary closure (e.g. first countable spaces) are called Uryshon Fretchet spaces. Ive seen various characterizations of when M is sequentially dense in the dual of a separable space but none of them are very helpful. I'm trying to determine under what conditiosn M is a Uryshon Fretchet space.