Atheist Nexus2015-07-04T17:50:56ZKeith Brian Johnsonhttp://atheistnexus.org/profile/KeithBrianJohnsonhttp://api.ning.com/files/UR2cIyul0Vx6LWTUQciYExH5llWU*C15XVTVY0guVH6tsUdpCTaH652EIfLZ6FhkCyCjStkUt9UmN-ezqDhRZk-2Rr6WuErg/009Cropped.jpg?width=48&height=48&crop=1%3A1http://atheistnexus.org/group/mathematics/forum/topic/listForContributor?user=0k51gbtdxyik5&feed=yes&xn_auth=noResearcher shows that black holes do not existtag:atheistnexus.org,2015-06-17:2182797:Topic:26123622015-06-17T11:20:56.135ZKeith Brian Johnsonhttp://atheistnexus.org/profile/KeithBrianJohnson
<p><span> I have yet to actually read her papers on this, but it's something to think about for sure. Black holes have always been a mathematical concept, we've never directly observed them, though the same could be said of the electron, and before that the atom. With evolving technology who knows where we'll be in 50 years. <br></br><br></br>Hydrogen atom through an electron microscope (false coloring obviously)…<br></br></span></p>
<p><span> I have yet to actually read her papers on this, but it's something to think about for sure. Black holes have always been a mathematical concept, we've never directly observed them, though the same could be said of the electron, and before that the atom. With evolving technology who knows where we'll be in 50 years. <br/><br/>Hydrogen atom through an electron microscope (false coloring obviously)<br/><a href="http://api.ning.com:80/files/bAnmdOW7Naht38Ye-ekkQVGrl7MKoRnkbaHXQpq4wFnVPENigeocIPeNnohJsauwYorueEJzdFOjtlYQ0miXdEvthHFinfvZ/hydrogenatom.jpg" target="_self"><img src="http://api.ning.com:80/files/bAnmdOW7Naht38Ye-ekkQVGrl7MKoRnkbaHXQpq4wFnVPENigeocIPeNnohJsauwYorueEJzdFOjtlYQ0miXdEvthHFinfvZ/hydrogenatom.jpg" width="208" class="align-full"/></a><a href="http://phys.org/news/2014-09-black-holes.html#jCp"><br/><br/>http://phys.org/news/2014-09-black-holes.html#jCp</a><br/><br/><br/>"I'm still not over the shock," said Mersini-Houghton. "We've been studying this problem for a more than 50 years and this solution gives us a lot to think about."</span><br/><br/><a href="http://phys.org/news/2014-09-black-holes.html#jCp"></a></p>
<p>Mersini-Houghton describes an entirely new scenario. She and Hawking both agree that as a star collapses under its own gravity, it produces Hawking radiation. However, in her new work, Mersini-Houghton shows that by giving off this radiation, the star also sheds mass. So much so that as it shrinks it no longer has the density to become a black hole.</p>
<p><a href="http://phys.org/news/2014-09-black-holes.html#jCp"></a></p>
<p>The paper, which was recently submitted to ArXiv, an online repository of physics papers that is not peer-reviewed, offers exact numerical solutions to this problem and was done in collaboration with Harald Peiffer, an expert on numerical relativity at the University of Toronto. An earlier paper, by Mersini-Houghton, originally submitted to ArXiv in June, was published in the journal Physics Letters B, and offers approximate solutions to the problem.</p>
<p></p>
<p><a href="http://phys.org/news/2014-09-black-holes.html#jCp"><br/><br/></a><a href="http://phys.org/news/2014-09-black-holes.html#jCp"><br/></a></p> Fractal Fourier Analysistag:atheistnexus.org,2014-12-03:2182797:Topic:25148972014-12-03T19:32:33.112ZKeith Brian Johnsonhttp://atheistnexus.org/profile/KeithBrianJohnson
<p><a href="http://www.sciencedaily.com/releases/2014/12/141203111214.htm" target="_blank">Fractal geometry: Finding the simple patterns in a complex world</a></p>
<p>Professor Michael Barnsley invented a new way to uncover the simple patterns hidden in complexity, fractal Fourier analysis. We "break loose from the thrall of continuity and differentiability".</p>
<blockquote><p>An ANU mathematician has developed a new way to uncover simple patterns that might underlie apparently complex…</p>
</blockquote>
<p><a href="http://www.sciencedaily.com/releases/2014/12/141203111214.htm" target="_blank">Fractal geometry: Finding the simple patterns in a complex world</a></p>
<p>Professor Michael Barnsley invented a new way to uncover the simple patterns hidden in complexity, fractal Fourier analysis. We "break loose from the thrall of continuity and differentiability".</p>
<blockquote><p>An ANU mathematician has developed a new way to uncover simple patterns that might underlie apparently complex systems, such as clouds, cracks in materials or the movement of the stockmarket.</p>
<p>The method, named fractal Fourier analysis, is based on new branch of mathematics called fractal geometry.</p>
<p>"Fractal Geometry is a new branch of mathematics that describes the world as it is, rather than acting as though it's made of straight lines and spheres. There are very few straight lines and circles in nature. The shapes you find in nature are rough."</p>
<p>The new analysis method is closely related to conventional Fourier analysis, which is integral to modern image handling and audio signal processing.</p>
<p>"Fractal Fourier analysis provides a method to break complicated signals up into a set of well understood building blocks, in a similar way to how conventional Fourier analysis breaks signals up into a set of smooth sine waves," Professor Barnsley said.</p>
<p>"There are terrific advances to be made by breaking loose from the thrall of continuity and differentiability," Professor Barnsley said.</p>
<p>"The body is full of repeating branch structures -- the breathing system, the blood supply system, the arrangement of skin cells, even cancer is a fractal."</p>
<p style="text-align: center;"><a target="_blank" href="http://images.sciencedaily.com/2014/12/141203111214-large.jpg"><img class="align-center" src="http://images.sciencedaily.com/2014/12/141203111214-large.jpg"/></a>Professor Michael Barnsley with a fractally transformed teacup and pot.</p>
</blockquote> January and Exponential Ebolatag:atheistnexus.org,2014-09-25:2182797:Topic:24806522014-09-25T19:39:59.300ZKeith Brian Johnsonhttp://atheistnexus.org/profile/KeithBrianJohnson
<p style="text-align: center;"><span class="font-size-3" style="color: #ff00ff;">The greatest shortcoming of the human race is the inability to understand the exponential function.</span> Dr. Albert Bartlett</p>
<p>Will this shortcoming be our "<a href="http://rationalwiki.org/wiki/The_Great_Filter" target="_blank">Great Filter</a>"?</p>
<blockquote><p>... the US is sending 1,500 - 1,700 beds. Those hospitals will take 50 - 100 days to be up and functional.</p>
<ul>
<li><p>So <strong>75…</strong></p>
</li>
</ul>
</blockquote>
<p style="text-align: center;"><span style="color: #ff00ff;" class="font-size-3">The greatest shortcoming of the human race is the inability to understand the exponential function.</span> Dr. Albert Bartlett</p>
<p>Will this shortcoming be our "<a href="http://rationalwiki.org/wiki/The_Great_Filter" target="_blank">Great Filter</a>"?</p>
<blockquote><p>... the US is sending 1,500 - 1,700 beds. Those hospitals will take 50 - 100 days to be up and functional.</p>
<ul>
<li><p>So <strong>75 days</strong> (between 50 - 100 to be fair)</p>
</li>
<li><p><strong>Divided 21</strong> day doubling period (if we don't assume it will shrink, which it very well could)</p>
</li>
<li><p>Gives us <strong>3.4 doubling periods</strong> before the hospitals are open and ready.</p>
</li>
<li><p>We're looking at between <strong>40,000 - 50,000</strong> people infected <strong>before</strong> the opening day of those hospitals.</p>
</li>
</ul>
<p>So you can see why 1,500 beds being deployed NOW for use in 50 - 100 days is too little too late. I am glad that 1,500 more beds will be open for a few thousand lucky people who won't die in the dirt, but it's not much.</p>
<p><a class="imgScanned" href="http://www.nytimes.com/2014/09/26/world/africa/obama-warns-of-slow-response-to-ebola-crisis.html?_r=0" rel="nofollow">Even Obama is criticizing the response times</a></p>
<p>“But I want us to be clear: We are not moving fast enough. We are not doing enough,” the president said. “<strong>There is still a significant gap between where we are and where we need to be.</strong>”</p>
</blockquote>
<p><a href="http://www.reddit.com/r/ebola/comments/2hg498/a_great_story_to_illustrate_exponential_growth/ckseq16" target="_blank">source</a></p>
<blockquote><p>... one of the key features of compound growth…the one thing I want you take away from all this. With exponential functions, the action really only heats up in the last few moments.</p>
</blockquote>
<p><a href="http://www.reddit.com/r/ebola/comments/2hg498/a_great_story_to_illustrate_exponential_growth/" target="_blank">source</a></p>
<p>Will the public still be oblivious at year's end, on the brink of the last few doubling times?</p>
<p><a target="_blank" href="http://img.reitingi.lv/news/0104170415_growth-small.jpg"><img class="align-center" src="http://img.reitingi.lv/news/0104170415_growth-small.jpg?width=400" width="400"/></a><a href="http://www.reitingi.lv/lv/news/ekonomika/92118-apstrades-rupniecibas-apgrozijums-julija-pieauga-par-0-7.html" target="_blank"></a>A "fast tracked" vaccine is alleged to be ready for <em>limited</em> distribution sometime in the first half of 2015. That'll be a big help, right?</p> Very cute math problemtag:atheistnexus.org,2014-07-31:2182797:Topic:24533882014-07-31T21:20:17.067ZKeith Brian Johnsonhttp://atheistnexus.org/profile/KeithBrianJohnson
<p>Suppose a_0, a_1, ..., a_n, ... are positive integers, and a_0 < a_1 < ... < a_n < ... </p>
<p>Prove there's a unique n >= 1, with </p>
<p>a_n < (a_0+...+a_n) / n <= a_(n+1). </p>
<p></p>
<p>>= means "greater than or equal"</p>
<p><= means "less than or equal".</p>
<p>_ denotes a subscript. </p>
<p></p>
<p></p>
<p></p>
<p>Suppose a_0, a_1, ..., a_n, ... are positive integers, and a_0 < a_1 < ... < a_n < ... </p>
<p>Prove there's a unique n >= 1, with </p>
<p>a_n < (a_0+...+a_n) / n <= a_(n+1). </p>
<p></p>
<p>>= means "greater than or equal"</p>
<p><= means "less than or equal".</p>
<p>_ denotes a subscript. </p>
<p></p>
<p></p>
<p></p> Puzzle in real analysistag:atheistnexus.org,2014-04-19:2182797:Topic:24111182014-04-19T14:53:18.100ZKeith Brian Johnsonhttp://atheistnexus.org/profile/KeithBrianJohnson
<p>Suppose f is a continuous function on some open subset S of the real numbers, to the positive real numbers. Suppose the integral of f over S is finite. <br></br>An open subset of the real line is a union of disjoint open intervals (which could have endpoints at infinity.) Define the length of the open subset as the sum of the length of the intervals.<br></br>So the question is, show that for any epsilon > 0, there's a delta such that for any open subset T of S with length(T) < delta,…</p>
<p>Suppose f is a continuous function on some open subset S of the real numbers, to the positive real numbers. Suppose the integral of f over S is finite. <br/>An open subset of the real line is a union of disjoint open intervals (which could have endpoints at infinity.) Define the length of the open subset as the sum of the length of the intervals.<br/>So the question is, show that for any epsilon > 0, there's a delta such that for any open subset T of S with length(T) < delta, the integral of f over T is less than epsilon. </p> 1+2+3+4+5+... = -1/12tag:atheistnexus.org,2014-02-09:2182797:Topic:23778462014-02-09T01:00:14.146ZKeith Brian Johnsonhttp://atheistnexus.org/profile/KeithBrianJohnson
<p>A strange derivation:<iframe frameborder="0" height="360" src="//www.youtube.com/embed/w-I6XTVZXww?feature=player_embedded&wmode=opaque" width="640"></iframe>
</p>
<p>This sum 1+2+3+... = -1/12 is used in physics a lot believe it or not.</p>
<p>This can be rationalized sort of:</p>
<p>The Riemann zeta function is zeta(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ...</p>
<p>where x^s means "x to the s'th power", and s is a complex number. </p>
<p>This series converges when the real part of …</p>
<p>A strange derivation:<iframe width="640" height="360" src="//www.youtube.com/embed/w-I6XTVZXww?feature=player_embedded&wmode=opaque" frameborder="0"></iframe>
</p>
<p>This sum 1+2+3+... = -1/12 is used in physics a lot believe it or not.</p>
<p>This can be rationalized sort of:</p>
<p>The Riemann zeta function is zeta(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ...</p>
<p>where x^s means "x to the s'th power", and s is a complex number. </p>
<p>This series converges when the real part of s is >1. This function can be analytically continued over the whole complex plane, except for s=1, and this analytic continuation is called the zeta function. </p>
<p>Define a function yeta in TWO complex variables by</p>
<p>yeta(s,z) = 1/1^s + z/2^s + z^2/3^s + z^3/4^s + ...</p>
<p>The yeta function converges for |z|<1, for all s. </p>
<p>Then we have the functional equation</p>
<p>yeta(s,z)- 2z/2^s yeta (s,z^2) = yeta (s, -z), for |z| < 1. </p>
<p>When Re(s) > 1, we can take the limit as z -> 1, to get the equation<br/>zeta(s)(1 - 2/2^s) = lim (z -> 1) yeta (s, -z)<br/>Since the zeta function can be analytically continued to the complex plane except for a pole at s=1, the RH side can be analytically continued as well.</p>
<p>We have yeta (-1, -z) = 1 - 2z + 3z^2 - 4z^3 + 5z^4 - ... = 1/(1+z)^2 </p>
<p>Taking the limit as z -> 1, we get</p>
<p>lim (z -> 1) yeta(-1,-z)=1/4.</p>
<p>So if the analytic continuation of lim (z -> 1) yeta (s, -z) = lim (z -> 1) yeta (-1, -z) for s=1, we have</p>
<p>(-3)zeta(-1) = 1/4, or zeta(-1) = -1/12.</p>
<p> </p> p-adics for undergraduates!tag:atheistnexus.org,2014-01-12:2182797:Topic:23650212014-01-12T20:53:59.675ZKeith Brian Johnsonhttp://atheistnexus.org/profile/KeithBrianJohnson
<p>I just finished reading the course in p-adics for undergraduates that I found online, making various annotations. </p>
<p>It turns out this was a preliminary version of a book <a href="http://www.amazon.com/Analysis-Compared-Student-Mathematical-Library/dp/082184220X" target="_blank">P-adic Analysis Compared With Real</a> that the author didn't realize was online! She said most of the errors/typos were fixed in the published version. </p>
<p>P-adic analysis isn't usually an undergraduate…</p>
<p>I just finished reading the course in p-adics for undergraduates that I found online, making various annotations. </p>
<p>It turns out this was a preliminary version of a book <a href="http://www.amazon.com/Analysis-Compared-Student-Mathematical-Library/dp/082184220X" target="_blank">P-adic Analysis Compared With Real</a> that the author didn't realize was online! She said most of the errors/typos were fixed in the published version. </p>
<p>P-adic analysis isn't usually an undergraduate topic, but this simple introduction is nicely done. </p>
<p></p> Speaking of math errors ...tag:atheistnexus.org,2013-12-22:2182797:Topic:23545902013-12-22T21:38:16.522ZKeith Brian Johnsonhttp://atheistnexus.org/profile/KeithBrianJohnson
<p>I found this false lemma in a math text online:</p>
<p>Suppose f: A --> B is a <a href="http://en.wikipedia.org/wiki/Monotonic_function" target="_blank">monotone</a> bijection between two subsets of R (the real numbers). Then f is a <a href="http://en.wikipedia.org/wiki/Homeomorphism" target="_blank">homeomorphism</a>. </p>
<p> </p>
<p>whew! Somebody needs some coffee!</p>
<p>This was in a text that was written by a math professor for an undergraduate course.</p>
<p></p>
<p>I found this false lemma in a math text online:</p>
<p>Suppose f: A --> B is a <a href="http://en.wikipedia.org/wiki/Monotonic_function" target="_blank">monotone</a> bijection between two subsets of R (the real numbers). Then f is a <a href="http://en.wikipedia.org/wiki/Homeomorphism" target="_blank">homeomorphism</a>. </p>
<p> </p>
<p>whew! Somebody needs some coffee!</p>
<p>This was in a text that was written by a math professor for an undergraduate course.</p>
<p></p> Progress on twin primes conjecture by a non-experttag:atheistnexus.org,2013-09-22:2182797:Topic:23058922013-09-22T16:51:54.039ZKeith Brian Johnsonhttp://atheistnexus.org/profile/KeithBrianJohnson
<p>The twin primes conjecture says there are infinitely many primes p such that p+2 is also prime. </p>
<p>A <a href="https://www.simonsfoundation.org/quanta/20130519-unheralded-mathematician-bridges-the-prime-gap/" target="_blank">new proof</a> by a mathematician who isn't one of the known experts in the field, has made a big contribution towards proving the conjecture.</p>
<p>Apparently he was very dogged in his pursuit of the proof. </p>
<p>It reminds me of myself - I've proved things…</p>
<p>The twin primes conjecture says there are infinitely many primes p such that p+2 is also prime. </p>
<p>A <a href="https://www.simonsfoundation.org/quanta/20130519-unheralded-mathematician-bridges-the-prime-gap/" target="_blank">new proof</a> by a mathematician who isn't one of the known experts in the field, has made a big contribution towards proving the conjecture.</p>
<p>Apparently he was very dogged in his pursuit of the proof. </p>
<p>It reminds me of myself - I've proved things and noticed things in math that other people don't because I get my teeth into a math problem and don't let go. <end of dog metaphors></p> Math and physics popularizations with substancetag:atheistnexus.org,2013-06-05:2182797:Topic:22456092013-06-05T16:13:23.743ZKeith Brian Johnsonhttp://atheistnexus.org/profile/KeithBrianJohnson
<p>There's a style of "popular" books about math and physics that involves actually learning some of the math and physics. They don't just talk ABOUT the subject, they try to convey some OF it.</p>
<p>In the math books, many theorems are stated without proof. So you have to learn enough to understand the statement of the theorem, but you don't have to wade through the proof. Some of the theorems are illustrated by simple examples.</p>
<p>A couple of such math books are "…</p>
<p>There's a style of "popular" books about math and physics that involves actually learning some of the math and physics. They don't just talk ABOUT the subject, they try to convey some OF it.</p>
<p>In the math books, many theorems are stated without proof. So you have to learn enough to understand the statement of the theorem, but you don't have to wade through the proof. Some of the theorems are illustrated by simple examples.</p>
<p>A couple of such math books are "<a href="http://www.amazon.com/Elliptic-Tales-Curves-Counting-Number/dp/0691151199" target="_blank">Elliptic Tales</a>" and "<a href="http://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-Numbers/dp/0691138710" target="_blank">Fearless Symmetry</a>" both by Ash and Gross. <span style="text-decoration: underline;">Elliptic Tales</span> is about elliptic curves, an important but difficult part of mathematics. <span style="text-decoration: underline;">Fearless Symmetry</span> is their "popularized" introduction to algebraic number theory, including the Wiles-Taylor proof of Fermat's Last Theorem. I wrote an <a href="http://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-Numbers/product-reviews/0691138710/ref=cm_cr_dp_see_all_btm?ie=UTF8&showViewpoints=1&sortBy=bySubmissionDateDescending" target="_blank">Amazon review</a> of <span style="text-decoration: underline;">Fearless Symmetry</span> on 6/4/13.</p>
<p>A physics book like that is <a href="http://www.amazon.com/Road-Reality-Complete-Guide-Universe/dp/0679776311" target="_blank">Road to Reality</a> by Roger Penrose. I was <em>obsessed</em> with this book for months, I did solved all the exercises that looked challenging except for one or two, and I learned a LOT from it. It's about mathematical physics.</p>
<p>I put "popular" in quotes because most people would find such books too difficult and challenging, and it requires thought and commitment to understand them. <span style="text-decoration: underline;">Road to Reality</span> could be <em>more</em> challenging than a physics textbook, just because he doesn't explain the concepts and equations rigorously - so you have to figure out the rigorous meaning, if you want to fully understand.</p>
<p>But I find them interesting. I'm too stressed and my mind has been too fuzzy with allergies to read a "real" math book, containing the proofs and all the technicalities. These are math lite. Anyone enjoyed other books like this?</p>