Comments - Extreme Events Modeling Using Continued Fractions - Data Science Central2020-02-22T01:55:31Zhttps://www.datasciencecentral.com/profiles/comment/feed?attachedTo=6448529%3ABlogPost%3A879706&xn_auth=noInteresting comment by Jean F…tag:www.datasciencecentral.com,2019-09-04:6448529:Comment:8826102019-09-04T00:32:36.362ZVincent Granvillehttps://www.datasciencecentral.com/profile/VincentGranville
<p>Interesting comment by <a href="https://www.linkedin.com/in/jean-fivaz/" rel="noopener" target="_blank">Jean Fivaz</a>:</p>
<p><em>I have encountered this in my efforts to automate 3 phase transformer design. The spec often requires a voltage ratio which is a multiple of sqrt(3), but must be accomplished using whole number winding ratios. The optimum core steel and copper mass will determine the starting point, but then the algorithm has to find these sweet spots where the turns ratio is…</em></p>
<p>Interesting comment by <a href="https://www.linkedin.com/in/jean-fivaz/" target="_blank" rel="noopener">Jean Fivaz</a>:</p>
<p><em>I have encountered this in my efforts to automate 3 phase transformer design. The spec often requires a voltage ratio which is a multiple of sqrt(3), but must be accomplished using whole number winding ratios. The optimum core steel and copper mass will determine the starting point, but then the algorithm has to find these sweet spots where the turns ratio is near the ideal voltage ratio, yet still close to the optimum starting point. So I end up with these odd number adjustments which I couldn't really see a pattern to. I haven't given it further thought really, the algorithm just does its work, but this article explains what I saw, pretty clearly. Unexpectedly insightful.</em></p>
<p>It is amazing to see the connection with "harmonious music", and other applications such as this one. I have no doubt, one day, I will also prove the connection with harmonious numbers such as Pi, SQRT(2), log 2 and so on, breaking open one of the deepest secrets of mother nature (are their digits randomly distributed?) -- something more difficult than sending a man on Mars. I have been working for many years, spending a good part of my life working on this. Millions of great mathematicians tried over hundreds of years. I just have to finish it, I can't afford not to do it.</p> Also, the most fascinating pa…tag:www.datasciencecentral.com,2019-09-01:6448529:Comment:8805402019-09-01T14:50:27.113ZVincent Granvillehttps://www.datasciencecentral.com/profile/VincentGranville
<p>Also, the most fascinating part here, in my opinion and from a theoretical perspective, is about the last part of the exercise mentioned in the article. </p>
<p>This problem can be re-phrased as follows: we are trying to find the best approximation of the form <em>Aq </em>− <em>B</em> for an number <em>s</em> (<em>s</em> = 1/<em>e</em> in the exercise), with <em>A</em>, <em>B</em> being strictly positive integers, using the irrational number <em>q</em>. The <em>n</em>-th best approximant for…</p>
<p>Also, the most fascinating part here, in my opinion and from a theoretical perspective, is about the last part of the exercise mentioned in the article. </p>
<p>This problem can be re-phrased as follows: we are trying to find the best approximation of the form <em>Aq </em>− <em>B</em> for an number <em>s</em> (<em>s</em> = 1/<em>e</em> in the exercise), with <em>A</em>, <em>B</em> being strictly positive integers, using the irrational number <em>q</em>. The <em>n</em>-th best approximant for <em>s</em> is { <em>t</em>(<em>n</em>) <em>q</em> } = <em>t</em>(<em>n</em>) <em>q</em> - INT(<em>t</em>(<em>n</em>) <em>q</em>) and of course { <em>t</em>(<em>n</em>) <em>q</em> } → <em>s</em> as <em>n</em> tends to infinity. See details <a href="https://math.stackexchange.com/questions/3339843/strange-property-of-irrational-numbers-linked-to-continued-fractions" target="_blank" rel="noopener">here</a>. This generalizes the concept of using continued fractions to approximate irrational numbers by rational numbers, to the idea of approximating irrational numbers (<em>q</em> = log(3) / log(2) here) by other pre-specified irrational numbers such as <em>s</em> = 1/<em>e</em>.</p> Hi LZ,
Perhaps the easiest wa…tag:www.datasciencecentral.com,2019-09-01:6448529:Comment:8805372019-09-01T14:19:44.439ZVincent Granvillehttps://www.datasciencecentral.com/profile/VincentGranville
<p>Hi LZ,</p>
<p>Perhaps the easiest way to explain it is to show my source code (Perl) also available in text format, <a href="https://storage.ning.com/topology/rest/1.0/file/get/3500824330?profile=original" rel="noopener" target="_blank">here</a>. Here <em>q</em> = log(3) / log(2).</p>
<p><a href="https://storage.ning.com/topology/rest/1.0/file/get/3500820031?profile=original" rel="noopener" target="_blank"><img class="align-full" src="https://storage.ning.com/topology/rest/1.0/file/get/3500820031?profile=RESIZE_710x"></img></a></p>
<p>And below is the output when you run this…</p>
<p>Hi LZ,</p>
<p>Perhaps the easiest way to explain it is to show my source code (Perl) also available in text format, <a href="https://storage.ning.com/topology/rest/1.0/file/get/3500824330?profile=original" target="_blank" rel="noopener">here</a>. Here <em>q</em> = log(3) / log(2).</p>
<p><a href="https://storage.ning.com/topology/rest/1.0/file/get/3500820031?profile=original" target="_blank" rel="noopener"><img src="https://storage.ning.com/topology/rest/1.0/file/get/3500820031?profile=RESIZE_710x" class="align-full"/></a></p>
<p>And below is the output when you run this program:</p>
<p></p>
<p><a href="https://storage.ning.com/topology/rest/1.0/file/get/3500815572?profile=original" target="_blank" rel="noopener"><img src="https://storage.ning.com/topology/rest/1.0/file/get/3500815572?profile=RESIZE_710x" class="align-full"/></a></p>
<p></p> How is the sequence of t_n de…tag:www.datasciencecentral.com,2019-09-01:6448529:Comment:8805332019-09-01T13:17:48.738ZLZ Zhanghttps://www.datasciencecentral.com/profile/LZZhang
<p>How is the sequence of t_n defined?</p>
<p>How is the sequence of t_n defined?</p>