Atheist Nexus2014-08-22T12:37:52ZKeith Brian Johnsonhttp://www.atheistnexus.org/profile/KeithBrianJohnsonhttp://api.ning.com/files/UR2cIyul0Vx6LWTUQciYExH5llWU*C15XVTVY0guVH6tsUdpCTaH652EIfLZ6FhkCyCjStkUt9UmN-ezqDhRZk-2Rr6WuErg/009Cropped.jpg?width=48&height=48&crop=1%3A1http://www.atheistnexus.org/group/mathematics/forum/topic/listForContributor?user=0k51gbtdxyik5&feed=yes&xn_auth=noVery cute math problemtag:www.atheistnexus.org,2014-07-31:2182797:Topic:24533882014-07-31T21:20:17.067ZKeith Brian Johnsonhttp://www.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:www.atheistnexus.org,2014-04-19:2182797:Topic:24111182014-04-19T14:53:18.100ZKeith Brian Johnsonhttp://www.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:www.atheistnexus.org,2014-02-09:2182797:Topic:23778462014-02-09T01:00:14.146ZKeith Brian Johnsonhttp://www.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:www.atheistnexus.org,2014-01-12:2182797:Topic:23650212014-01-12T20:53:59.675ZKeith Brian Johnsonhttp://www.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:www.atheistnexus.org,2013-12-22:2182797:Topic:23545902013-12-22T21:38:16.522ZKeith Brian Johnsonhttp://www.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:www.atheistnexus.org,2013-09-22:2182797:Topic:23058922013-09-22T16:51:54.039ZKeith Brian Johnsonhttp://www.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:www.atheistnexus.org,2013-06-05:2182797:Topic:22456092013-06-05T16:13:23.743ZKeith Brian Johnsonhttp://www.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> Math for Amateurs?tag:www.atheistnexus.org,2013-02-28:2182797:Topic:21720512013-02-28T05:13:04.214ZKeith Brian Johnsonhttp://www.atheistnexus.org/profile/KeithBrianJohnson
<p>Is there such a thing as an amateur mathematician? I mean I am slowly becoming infatuated with math and numbers, especially their abstract / finite juxtaposition but I'm really slow at the comprehension and a little crippled by using a calculator. But the whole reason I want to learn more math is because I like the theory, I like the higher stuff, the abstract fuzzy stuff that you can use for metaphors in a story. (I'm a writer) so I want to devote a larger chunk of my life toward studying…</p>
<p>Is there such a thing as an amateur mathematician? I mean I am slowly becoming infatuated with math and numbers, especially their abstract / finite juxtaposition but I'm really slow at the comprehension and a little crippled by using a calculator. But the whole reason I want to learn more math is because I like the theory, I like the higher stuff, the abstract fuzzy stuff that you can use for metaphors in a story. (I'm a writer) so I want to devote a larger chunk of my life toward studying math, as it is the universal language ...as it were, like music and music is math in a way; but I know I am not good enough to truly go to school and learn it in an academic setting. I took an Honors Algebra 2 course in HS and failed it almost, so .... yeah. I'm just curious about this and thought I'd toss the question out into the aether to all the math whizzes out there and see what they all think....</p>
<p></p>
<p>:)</p> Fun math puzzletag:www.atheistnexus.org,2013-02-15:2182797:Topic:21641142013-02-15T12:38:59.074ZKeith Brian Johnsonhttp://www.atheistnexus.org/profile/KeithBrianJohnson
<p>Here's a math puzzle for you:</p>
<p>Suppose f is a function from the natural numbers (1,2,3...) to the natural numbers, and (f(m)+n)(f(n)+m) is a square for all m,n. What functions f have this property?</p>
<p>It's easy to guess what f has to be, but the puzzle is to <em>prove</em> that the possibilities for f are limited to your guess - or show what other possibilities there are.</p>
<p>If you happen to see the solution online somewhere, please don't post it! I'm curious about what people…</p>
<p>Here's a math puzzle for you:</p>
<p>Suppose f is a function from the natural numbers (1,2,3...) to the natural numbers, and (f(m)+n)(f(n)+m) is a square for all m,n. What functions f have this property?</p>
<p>It's easy to guess what f has to be, but the puzzle is to <em>prove</em> that the possibilities for f are limited to your guess - or show what other possibilities there are.</p>
<p>If you happen to see the solution online somewhere, please don't post it! I'm curious about what people can come up with on their own. I solved it (I think!). </p>
<p>It's a great puzzle because there are so many possible ways to tackle the problem. And likely, various possible proofs. </p> I need help!tag:www.atheistnexus.org,2012-09-03:2182797:Topic:20442342012-09-03T19:33:13.959ZKeith Brian Johnsonhttp://www.atheistnexus.org/profile/KeithBrianJohnson
<p>I have a problem that seems really basic, but for whatever reason I keep messing up...</p>
<p></p>
<p>f(r) = 7/(1+r)^2</p>
<p></p>
<p>Find the average value of the function on the interval [1,5]</p>
<p></p>
<p></p>
<p>I let u = 1+r</p>
<p>so du = dx</p>
<p></p>
<p>and integral 7u^-2 du</p>
<p></p>
<p>integrated I get -7u^(-1) and the new interval is [2,6]</p>
<p></p>
<p>-7(6)^-1+7(2)^-1 = 7/3</p>
<p></p>
<p>So the answer I keep coming up with is 7/3 which is wrong evidently........ Can…</p>
<p>I have a problem that seems really basic, but for whatever reason I keep messing up...</p>
<p></p>
<p>f(r) = 7/(1+r)^2</p>
<p></p>
<p>Find the average value of the function on the interval [1,5]</p>
<p></p>
<p></p>
<p>I let u = 1+r</p>
<p>so du = dx</p>
<p></p>
<p>and integral 7u^-2 du</p>
<p></p>
<p>integrated I get -7u^(-1) and the new interval is [2,6]</p>
<p></p>
<p>-7(6)^-1+7(2)^-1 = 7/3</p>
<p></p>
<p>So the answer I keep coming up with is 7/3 which is wrong evidently........ Can anyone help?</p>
<p></p>