User jcollins - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T22:05:52Z http://mathoverflow.net/feeds/user/3280 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13682/which-mathematical-ideas-have-done-most-to-change-history Which mathematical ideas have done most to change history? JCollins 2010-02-01T14:43:02Z 2012-08-14T18:58:31Z <p>I'm planning a course for the general public with the general theme of "Mathematical ideas that have changed history" and I would welcome people's opinions on this topic. What do you think have been the most influential mathematical ideas in terms of what has influenced science/history or changed the way humans think, and why?</p> <p>I won't expect my audience to have any mathematical background other than high-school.</p> <p>My thoughts so far are: non-Euclidean geometry, Cantor's ideas on uncountability, undecidability, chaos theory and fractals, the invention of new number systems (i.e. negative numbers, zero, irrational, imaginary numbers), calculus, graphs and networks, probability theory, Bayesian statistics.</p> <p>My apologies if this has already been discussed in another post.</p> http://mathoverflow.net/questions/50538/good-ways-to-engage-in-mathematics-outreach/50946#50946 Answer by JCollins for Good ways to engage in mathematics outreach? JCollins 2011-01-02T19:37:52Z 2011-01-02T19:37:52Z <p>In the UK there are a lot of good programmes for engaging the public with mathematics, some of which have been touched on in other answers. Here is a list of the ones I think have worked the best or been most innovative:</p> <p>1) Royal Institution <a href="http://www.rigb.org/contentControl?action=displayContent&amp;id=00000001857" rel="nofollow">Masterclasses</a>: a UK-wide series of masterclasses aimed at age 13 school pupils. Classes are on Saturday mornings for between 5 and 10 weeks. Each class has a different presenter and each class lasts for 2.5 hours with time for lectures, exercises and refreshments. I've been involved in giving the Edinburgh and Glasgow masterclasses and they are a lot of fun for all involved. The kids really appreciate getting to see some maths that they'd never encounter through school, and especially enjoy seeing the diversity and usefulness of the subject. (In addition, <a href="http://www.rigb.org/contentControl?action=displayContent&amp;id=00000001882" rel="nofollow">a set of masterclasses</a> gets broadcast on the BBC every Christmas!)</p> <p>2) <a href="http://www.maths.liv.ac.uk/lms/funmaths/" rel="nofollow">FunMaths Roadshow</a>: a collection of 350 maths activities, each suitable for a particular age group between 5 and 20. The resources are cheap and easy to make, as well as being easy to transport, so that they can be taken all around the local area and used with different groups of people.</p> <p>3) <a href="http://www.mathsbusking.com/" rel="nofollow">Maths busking</a>: a new phenomenon which has been surprisingly successful! 'Buskers' go out onto the streets and perform games and tricks which appear to be magic but which can be explained by mathematics. It's a great way to engage with people who would otherwise never attend a maths lecture or think of going to a science museum.</p> <p>4) Mathematical walking tours: do a tour that opens people's eyes to the mathematics around your city. This can be historical curiosities, geometrical constructions in the architecture or optimization in the town planning. Such tours are currently being designed for Oxford and London, or get examples from <a href="http://www.newyorker.com/talk/2009/08/03/090803ta_talk_paumgarten" rel="nofollow">Manhattan</a> or this <a href="http://mathourism.blogspot.com/" rel="nofollow">maths tourism blog</a>.</p> <p>Whatever you decide to do, just go ahead and do it with plenty of enthusiasm! Good luck!</p> http://mathoverflow.net/questions/11880/what-primes-divide-the-discriminant-of-a-polynomial What primes divide the discriminant of a polynomial? JCollins 2010-01-15T16:32:52Z 2010-01-16T00:57:19Z <p>Given a monic polynomial $p(t) = t^n + ... + c_1 t + c_0$ with integer (or rational) coefficients and with roots $a_1, \dots a_n$, we can compute its discriminant, which is defined to be $\prod_{i&lt; j}(a_i - a_j)^2$.</p> <p>In my case, I have a polynomial which is the characteristic polynomial of some invertible matrix $T$. It is palindromic -- i.e., $c_{n-i} = c_i$ for all $0 \leq i \leq n$ -- so the roots come in inverse pairs $a$ and $\frac{1}{a}$. There are no repeated roots, so the discriminant is non-zero.</p> <p>My question is: is there any way of knowing which primes divide this discriminant, i.e. from the coefficients of the polynomial or from the matrix $T$?</p> http://mathoverflow.net/questions/11880/what-primes-divide-the-discriminant-of-a-polynomial/11892#11892 Answer by JCollins for What primes divide the discriminant of a polynomial? JCollins 2010-01-15T18:19:22Z 2010-01-15T18:19:22Z <p>(I haven't worked out how to leave comments on here rather than answers - help!)</p> <p>The matrix has rational coefficients but is usually not symmetric. It is indeed only the polynomial which is symmetric in its coefficients, as you say. </p> <p>Not every integer/rational polynomial is the characteristic polynomial of a rational matrix (see <a href="http://mathoverflow.net/questions/3939/when-is-a-monic-integer-polynomial-the-characteristic-polynomial-of-a-non-negativ" rel="nofollow">here</a>), so I hope that this extra information may be of some help.</p> <p>I appreciate that I can simply compute the discriminant in particular cases and see what happens, but I was hoping that there would be a general theorem to say something about the prime divisors. So far I can't seem to figure anything out, even using the resultant.</p> http://mathoverflow.net/questions/21424/how-to-draw-knots-with-latex/21456#21456 Comment by JCollins JCollins 2011-03-25T10:44:29Z 2011-03-25T10:44:29Z You can also create .eps files from the .png files by using the 'Trace' tool in programs like Illustrator, Inkscape or Corel Draw. This is a great thing if you want to adjust the knot, scale it or change colours. http://mathoverflow.net/questions/50343/what-would-you-want-to-see-at-the-museum-of-mathematics/50754#50754 Comment by JCollins JCollins 2011-01-02T19:09:19Z 2011-01-02T19:09:19Z There is also the Arithmeum (<a href="http://www.arithmeum.uni-bonn.de/en/home/" rel="nofollow">arithmeum.uni-bonn.de/en/home</a>) in Bonn, which features a history of calculational devices, many of which the visitors can touch and play with. Interestingly, the museum also houses a collection of modern art which has been inspired by mathematics. http://mathoverflow.net/questions/13682/which-mathematical-ideas-have-done-most-to-change-history/13751#13751 Comment by JCollins JCollins 2010-02-02T16:11:46Z 2010-02-02T16:11:46Z Thanks for the great reference! http://mathoverflow.net/questions/13682/which-mathematical-ideas-have-done-most-to-change-history Comment by JCollins JCollins 2010-02-01T20:32:38Z 2010-02-01T20:32:38Z Ok, let's remove Bayesian statistics from the list then. http://mathoverflow.net/questions/13682/which-mathematical-ideas-have-done-most-to-change-history Comment by JCollins JCollins 2010-02-01T15:36:05Z 2010-02-01T15:36:05Z The 'butterfly effect' is a concept very much in the public consciousness, and the general theory of chaos has applications to a wide range of modern-day life (economics, weather-prediction, turbulence in aircraft). http://mathoverflow.net/questions/13682/which-mathematical-ideas-have-done-most-to-change-history Comment by JCollins JCollins 2010-02-01T15:33:16Z 2010-02-01T15:33:16Z Cantor's ideas on there being different sizes of infinity had a big impact on religion, since the notion of infinity was very closely tied to the notion of God. http://mathoverflow.net/questions/13682/which-mathematical-ideas-have-done-most-to-change-history/13689#13689 Comment by JCollins JCollins 2010-02-01T15:12:17Z 2010-02-01T15:12:17Z I'm lumping general relativity in with 'non-euclidean geometry'. Is there a particular idea or theory in maths which helped develop optics? http://mathoverflow.net/questions/13682/which-mathematical-ideas-have-done-most-to-change-history/13688#13688 Comment by JCollins JCollins 2010-02-01T15:10:47Z 2010-02-01T15:10:47Z It is interesting to read about modern-day tribes or communities which still don't have counting words above two. They count their livestock by simply having a name for each creature! http://mathoverflow.net/questions/13682/which-mathematical-ideas-have-done-most-to-change-history/13683#13683 Comment by JCollins JCollins 2010-02-01T15:06:22Z 2010-02-01T15:06:22Z This is why I added the &quot;and why?&quot; to the end of my question! http://mathoverflow.net/questions/13682/which-mathematical-ideas-have-done-most-to-change-history Comment by JCollins JCollins 2010-02-01T14:56:28Z 2010-02-01T14:56:28Z Done! Thanks for letting me know. http://mathoverflow.net/questions/11880/what-primes-divide-the-discriminant-of-a-polynomial/11892#11892 Comment by JCollins JCollins 2010-01-15T18:45:05Z 2010-01-15T18:45:05Z @Pete L.Clark: Thanks for fixing my question, but if you are going to use $a_i$ for the coefficients of my polynomial, please call the roots something else, like $\alpha_i$. http://mathoverflow.net/questions/11880/what-primes-divide-the-discriminant-of-a-polynomial/11892#11892 Comment by JCollins JCollins 2010-01-15T18:37:44Z 2010-01-15T18:37:44Z Sorry, you are right about the companion matrix. Thanks for the link. Thank you also to Ben for the Newton Identities and the determinant formula. Looks tricky to compute but may turn out to help. (I am having issues with Mathoverflow so that the 'me' who asked the question is not the 'me' who is commenting here; thus I have no reputation and cannot comment on my own question! Argh.)