Elementary+Short+Useful - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T05:21:14Z http://mathoverflow.net/feeds/question/60457 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60457/elementaryshortuseful Elementary+Short+Useful Anton Petrunin 2011-04-03T17:34:34Z 2011-12-04T10:56:13Z <p>Imagine your-self in front of a class with very good undergraduates who plan to do mathematics (professionally) in the future. You have 30 minutes after that you do not see these students again. You need to present a <strong>theorem which will be 100% useful</strong> for them.</p> <blockquote> <p>What would you do? </p> </blockquote> <p><strong>One theorem per answer please.</strong> Try to be realistic.</p> <p>For example: 30 min is more than enough to introduce metric spaces, prove existence of partition of unity, and explain how it can be used later.</p> <p><strong>P.S.</strong> Many of you criticized the vague formulation of the question. I agree. I was trying to make it short --- I do not read the questions if they are longer than half a page. Still I think it is a good approximation to what I really wanted to ask. Here is <strong>an other formulation</strong> of the same question, but it might be even more vague.</p> <p>Before I liked <em>jewellery-type</em> theorems; those I can put in my pocket and look at it when I want to. Now I like <em>tool-type</em> theorems those which can be used to dig a hole or build a wall. It turnes out that there are jewellery-type and tool-type theorems at the same time. I know few and I want to know more.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60459#60459 Answer by Andrey Rekalo for Elementary+Short+Useful Andrey Rekalo 2011-04-03T17:44:18Z 2011-04-03T17:44:18Z <p><a href="http://en.wikipedia.org/wiki/Banach_fixed_point_theorem" rel="nofollow">The Banach fixed point theorem</a>.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60463#60463 Answer by Stefan Geschke for Elementary+Short+Useful Stefan Geschke 2011-04-03T18:03:38Z 2011-04-03T18:03:38Z <p>Completeness theorem for first order logic.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60469#60469 Answer by Willie Wong for Elementary+Short+Useful Willie Wong 2011-04-03T18:34:14Z 2011-11-09T11:38:49Z <p><a href="http://en.wikipedia.org/wiki/Picard-Lindelof_theorem" rel="nofollow">Picard–Lindelöf theorem</a> on existence and uniqueness of solutions to ordinary differential equations, introducing Picard iteration along the way. </p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60475#60475 Answer by Dirk for Elementary+Short+Useful Dirk 2011-04-03T19:06:38Z 2011-04-05T09:08:48Z <p><a href="http://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem" rel="nofollow">The Arzelà-Ascoli theorem</a>.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60476#60476 Answer by Grant Olney Passmore for Elementary+Short+Useful Grant Olney Passmore 2011-04-03T19:11:47Z 2011-04-03T19:11:47Z <p>Compactness of First Order Logic (using ultraproducts, not as a corollary of completeness; they get Łoś's Theorem for ultraproducts as a freebie.)</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60479#60479 Answer by Igor Khavkine for Elementary+Short+Useful Igor Khavkine 2011-04-03T19:50:13Z 2011-04-03T19:50:13Z <p><a href="http://en.wikipedia.org/wiki/Stokes%27_theorem" rel="nofollow">Stokes' Theorem</a></p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60482#60482 Answer by Ed Dean for Elementary+Short+Useful Ed Dean 2011-04-03T20:09:12Z 2011-04-03T20:09:12Z <p><a href="http://en.wikipedia.org/wiki/Stone%27s_representation_theorem_for_Boolean_algebras" rel="nofollow">Stone's representation theorem</a>.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60483#60483 Answer by Willie Wong for Elementary+Short+Useful Willie Wong 2011-04-03T20:25:30Z 2011-04-03T20:25:30Z <p>Maybe (a suitably weak version of) <a href="http://en.wikipedia.org/wiki/Brouwer_fixed_point_theorem#A_proof_using_Stokes.27s_theorem" rel="nofollow">Brouwer fixed point theorem</a>? For example you can prove the version for smooth maps, or the topological version in low dimensions. And there are so many generalizations of the theorem that it seems the students are bound to run into some version of topological fixed points in the future. </p> <p>You can even mention, as an application of topological fixed points, Littlewood's proof that there always exists a way to put a rod standing on one end in a train travelling between Kings Cross and Cambridge such that it would not fall over. (In fact, isn't that entire chapter of the <em>Miscellany</em> [Chapter 1, Mathematics with minimum raw material] consisting of answers to your question?)</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60485#60485 Answer by Willie Wong for Elementary+Short+Useful Willie Wong 2011-04-03T20:52:40Z 2011-04-03T20:52:40Z <p>Okay, last one from me tonight. </p> <p><a href="http://en.wikipedia.org/wiki/Separating_hyperplane_theorem" rel="nofollow">Separating hyperplane theorem</a> and/or the <a href="http://en.wikipedia.org/wiki/M._Riesz_extension_theorem" rel="nofollow">Riesz extension theorem</a>. The finite (or 2) dimensional version is fairly easy to illustrate and not too hard to prove. And of course as an example application you can assume the infinite dimensional version and derive Hahn-Banach Theorem (the version about extending linear functionals). Consider its use in convex and functional analysis, at least some of the students will run into something like this in the future. </p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60487#60487 Answer by Daniel Moskovich for Elementary+Short+Useful Daniel Moskovich 2011-04-03T20:54:59Z 2011-04-03T21:54:16Z <p>Simplicity of the alternating group A<sub>n</sub> for $n\geq 5$, contrasted with its non-simplicity for $n\leq 4$.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60488#60488 Answer by Anton Petrunin for Elementary+Short+Useful Anton Petrunin 2011-04-03T20:57:10Z 2011-04-03T20:57:10Z <p><a href="http://en.wikipedia.org/wiki/Euler_characteristic" rel="nofollow">Euler formula</a> $V - E + F = 2$.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60490#60490 Answer by Sebastian Scholtes for Elementary+Short+Useful Sebastian Scholtes 2011-04-03T20:59:10Z 2011-04-03T20:59:10Z <p><a href="http://en.wikipedia.org/wiki/Hilbert_projection_theorem" rel="nofollow">Hilbert projection theorem</a></p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60491#60491 Answer by Anton Petrunin for Elementary+Short+Useful Anton Petrunin 2011-04-03T21:02:16Z 2011-04-03T21:02:16Z <p><a href="http://en.wikipedia.org/wiki/Sperner%27s_lemma" rel="nofollow">Sperner's lemma</a> (Two-dimensional case)</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60492#60492 Answer by Anton Petrunin for Elementary+Short+Useful Anton Petrunin 2011-04-03T21:05:50Z 2011-04-03T21:05:50Z <p>Introduce <a href="http://en.wikipedia.org/wiki/Generating_function" rel="nofollow">generating functions</a> and give couple of applications.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60494#60494 Answer by ght for Elementary+Short+Useful ght 2011-04-03T21:26:49Z 2011-04-03T21:26:49Z <p>The <a href="http://en.wikipedia.org/wiki/Central_limit_theorem" rel="nofollow">Central Limit Theorem</a>.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60495#60495 Answer by Ed Dean for Elementary+Short+Useful Ed Dean 2011-04-03T21:29:13Z 2011-04-03T21:29:13Z <p><a href="http://en.wikipedia.org/wiki/Law_of_large_numbers#Strong_law" rel="nofollow">Strong law of large numbers</a></p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60497#60497 Answer by Todd Trimble for Elementary+Short+Useful Todd Trimble 2011-04-03T21:46:48Z 2011-04-04T00:26:04Z <p>The <a href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem" rel="nofollow">Chinese Remainder Theorem</a>. This is ripe for giving some nice applications, some of which are given in this <a href="http://mathoverflow.net/questions/10014/applications-of-the-chinese-remainder-theorem" rel="nofollow">MO thread</a> (hat tip to Pete Clark; I presume this is the one he meant). </p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60504#60504 Answer by Alex R. for Elementary+Short+Useful Alex R. 2011-04-03T23:16:40Z 2011-04-04T17:27:23Z <p>Singular Value Decomposition, probably one of the most useful and ubiquitous concepts out there. Half the time can be devoted to listing all the synonyms it goes by in various fields such as statistics and finance. </p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60526#60526 Answer by Roland Bacher for Elementary+Short+Useful Roland Bacher 2011-04-04T07:20:37Z 2011-04-04T15:14:02Z <p>Series representations for functions and the fact that $\mathbb C$ is "rigid" in contrast to $\mathbb R$ when discussing differentiability and series developements.</p> <p>This "explains" for example how pocket calculators compute trigonometric functions, logarithms and exponentials.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60529#60529 Answer by Stefan Waldmann for Elementary+Short+Useful Stefan Waldmann 2011-04-04T07:39:30Z 2011-04-04T07:39:30Z <p>The definition of the tensor product and existence/uniqueness/associativity properties.</p> <p>I know, this is perhaps not a single theorem but in my eyes one of the most useful "elemetary" concepts. Personally, I had two semesters of linear algebra without mentioning the tensor product. And from this I suffered for a long time during my further studies. Now it is my first homework/exercise for students in my lectures (e.g. diff geo).</p> <p>If the student is really clever, one can even do something like the tensor algebra in these 30 min.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60530#60530 Answer by Johannes Ebert for Elementary+Short+Useful Johannes Ebert 2011-04-04T08:51:49Z 2011-04-04T08:51:49Z <p>Let $G$ be a finite group and $V_i$, $i=1,...,r$ be the irreducible representations, $d_i:=dim(V_i)$. Then $|G|=\sum_i d_{i}^{2}$.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60532#60532 Answer by Zsbán Ambrus for Elementary+Short+Useful Zsbán Ambrus 2011-04-04T09:14:21Z 2011-04-04T09:14:21Z <p>Sanov's theorem of large deviations. </p> <p>I don't have to prove anything, right? If they want a proof, they'll look it up in a book later. </p> <p>Assume the students already know about the central limit theorem. Explain how the two theorems talk about limits in different direction: let $S_n$ be the sum of $n$ independent variables of identical distributions (real valued, with zero mean and finite variance), the central limit theorem gives a limit of the unscaled probability $P(S_n/\sqrt{n} &lt; c)$, this limit is strictly between 0 and 1; whereas large deviation theorems give the rate of decrease of a probability like $P(S_n/n &lt; c)$.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60551#60551 Answer by Chuck for Elementary+Short+Useful Chuck 2011-04-04T13:34:33Z 2011-04-04T17:50:02Z <p>I would say something far far more elementary than all the other suggestions here (perhaps assuming the audience is in their first semester as undergraduates)</p> <p>I would define an equivalence relation and an equivalence class and prove that equivalence classes on $X$ define a partition of $X$. (And then spend the remaining 29 minutes talking about their philosophical significance :) )</p> <p>Its usefulness is of course immense but that doesn't mean we should attribute it solely to its obviousness. In my mind it also encodes so many very deep intuitions that separate high-school from college-level mathematics. To name a few: </p> <ul> <li>The fact that there is nothing metaphysically 'special' about the relation of equality, which foreshadows the algebraic paradigm-shift towards isomorphisms</li> <li>The fact that information about certain properties is better captured when we look at classes of objects satisfying a relation</li> <li>That the foundations of analysis are a lot more conceptually flexible (and amenable to reinterpretation or even reinvention) than 'functions and derivatives'.</li> <li>The information encoded by the definition of an equivalence relation is absolutely minimal and trivial to understand (which is why most undergraduates, I've found, almost scoff when a lecturer spends time defining it) and yet responsible for profoundly deep intuitions - think of the Grothendieck group.</li> <li>It brings out the significance of structuralist thinking at a very early, pre-algebraic stage (this is more personal, but still)</li> </ul> http://mathoverflow.net/questions/60457/elementaryshortuseful/60570#60570 Answer by Alexander Duncan for Elementary+Short+Useful Alexander Duncan 2011-04-04T16:22:20Z 2011-04-04T16:22:20Z <p>Using <a href="http://en.wikipedia.org/wiki/Grobner_bases" rel="nofollow">Groebner Bases</a> to solve equations. Just use the lexicographic ordering without disucssing theory. Mash generalized polynomial long division and Buchberger's algorithm into one mechanical procedure. 30 minutes is pretty tight, but doable.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60573#60573 Answer by fedja for Elementary+Short+Useful fedja 2011-04-04T16:50:45Z 2011-04-04T16:50:45Z <p>Newton's method for solving the non-linear (systems of) equations. How to make the presentation depends on the level and interests of the students. It can range from a fast algorithm for finding the square root with high precision to some advanced topics in dynamics. </p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60580#60580 Answer by Rbega for Elementary+Short+Useful Rbega 2011-04-04T17:30:07Z 2011-04-04T17:30:07Z <p>A short presentation on the <a href="http://en.wikipedia.org/wiki/Hopf_fibration" rel="nofollow"> Hopf fibration </a> could be very useful as it is such a central example. The idea to make it elementary would be to take a concrete point of view and include lots of pictures. </p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60586#60586 Answer by anonymous for Elementary+Short+Useful anonymous 2011-04-04T18:38:29Z 2011-04-04T18:38:29Z <p>Lagrange's theorem (order of a sugroup divides the order of the group).</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60587#60587 Answer by Gil Kalai for Elementary+Short+Useful Gil Kalai 2011-04-04T18:44:11Z 2011-04-10T16:33:40Z <p><a href="http://en.wikipedia.org/wiki/Robinson-Schensted_algorithm" rel="nofollow">Robinston-Schensted-Knuth algorithm</a></p> <p>This is a map between permutations to pairs of standard tableaux. So it immediately gives various wonderful facts. It is elementary, short and useful.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60588#60588 Answer by Gil Kalai for Elementary+Short+Useful Gil Kalai 2011-04-04T18:46:26Z 2011-04-10T16:36:48Z <p><a href="http://en.wikipedia.org/wiki/Borsuk-Ulam_theorem" rel="nofollow">Borsuk-Ulam theorem</a>. A very useful topological theorem. It is very easy to state and to describe some applications, or alternatively to describe what is involved in a proof. </p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60589#60589 Answer by Gil Kalai for Elementary+Short+Useful Gil Kalai 2011-04-04T18:47:51Z 2011-04-11T06:13:36Z <p><a href="http://en.wikipedia.org/wiki/Helly%27s_theorem" rel="nofollow">Helly theorem</a>. It is easy to motivate state and prove in 30 minutes. It is very useful in terms of application as a fundamental example of a result in combinatorial geometry.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60603#60603 Answer by Jeff Schenker for Elementary+Short+Useful Jeff Schenker 2011-04-04T19:55:42Z 2011-04-04T19:55:42Z <p>Min-max principle and spectral theorem as a corollary for real symmetric matrices. I often teach this quickly in my vector analysis course as an example of finding extrema of functions in $\mathbb{R}^n$. </p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60605#60605 Answer by James for Elementary+Short+Useful James 2011-04-04T20:41:52Z 2011-04-04T20:41:52Z <p>The <a href="http://en.wikipedia.org/wiki/Pigeonhole_principle" rel="nofollow">Pigeonhole Principle</a></p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60646#60646 Answer by Phillip Williams for Elementary+Short+Useful Phillip Williams 2011-04-05T04:48:27Z 2011-04-05T04:48:27Z <p>I've always been thrilled by the fact that the coefficients of a (monic) polynomial are obtained by taking the elementary symmetric functions in (minus) the roots of that polynomial:</p> <p>$$\prod_{i=1}^n (X+\alpha_i) = \sum_{k=0}^n (\sum_{i_1 &lt; \cdots &lt; i_k} \alpha_{i_1}\cdots \alpha_{i_k})X^{n-k}$$ A lot is built on this, I think. I'd like to explain the connection to automorphisms and fixed fields and how the roots of a polynomial are permuted by an automorphism that fixes the coefficient field of that polynomial. Then maybe mention the beginnings of Galois theory.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60657#60657 Answer by Henno Brandsma for Elementary+Short+Useful Henno Brandsma 2011-04-05T07:52:26Z 2011-04-05T07:52:26Z <p>The well-ordering theorem and an application (that uses transfinite recursion, after well-ordering a set). Many interesting sets and examples can be built that way. Or maybe Axiom of Choice/Zorn's lemma (show one from the other) and then show the well-ordering theorem.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60670#60670 Answer by Zen Harper for Elementary+Short+Useful Zen Harper 2011-04-05T10:56:43Z 2011-04-06T08:02:27Z <p>Uniform convergence of the averages of the partial sums of the Fourier series, for any continuous function $f$ on $[0, 2 \pi]$ with $f(0)=f(2\pi)$:</p> <p>$$\sigma_N(f, \theta) = \sum_{n = -N}^N \left(1-\frac{|n|}{N+1} \right) \widehat{f}(n)e^{in \theta} \to f(\theta)$$</p> <p>And the Weierstrauss Polynomial Approximation Theorem: the polynomials are uniformly dense in $C[a,b]$. This is a corollary of the Fourier series result, or it can be proved similarly. Finally, if time permits, the Stone-Weierstrauss Theorem.</p> <p>Of course, it would be nice to talk about approximations to the Dirac Delta, convolutions, fundamental solutions to PDEs, e.g. the Heat Equation, etc. etc. but I suppose only a REALLY good class could absorb all this in half an hour...</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60671#60671 Answer by Daniel Miller for Elementary+Short+Useful Daniel Miller 2011-04-05T11:20:40Z 2011-04-05T11:20:40Z <p>At the risk of incurring the wrath of some here, I would propose the <a href="http://en.wikipedia.org/wiki/Yoneda_lemma" rel="nofollow">Yoneda Lemma</a>, along with the minimum of necessary category theory. Like it or not, category theory is hugely useful to algebraists, and early exposure can be very helpful. (It was to me!)</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60677#60677 Answer by ght for Elementary+Short+Useful ght 2011-04-05T11:56:41Z 2011-04-05T11:56:41Z <p>The <a href="http://en.wikipedia.org/wiki/Spectral_theorem" rel="nofollow">spectral theorem</a> for normal operators.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60697#60697 Answer by Evan Jenkins for Elementary+Short+Useful Evan Jenkins 2011-04-05T14:55:03Z 2011-04-05T14:55:03Z <p>How about the <a href="http://en.wikipedia.org/wiki/Probabilistic_method" rel="nofollow">probabilistic method</a>?</p> <p>There are plenty of elementary, self-contained examples to choose from, and it has a pithy slogan that's memorable enough even for non-combinatorialists. (Can't construct something explicitly? Then construct it randomly!) Best of all, it has a nice wow factor: While many undergraduates may be familiar with nonconstructive phenomena in mathematics, the fact that we need to resort to such to say things about <i>finite</i> graphs is rather surprising.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60713#60713 Answer by Paul Siegel for Elementary+Short+Useful Paul Siegel 2011-04-05T16:56:55Z 2011-04-05T16:56:55Z <p>The Gelfand-Naimark theorem: every commutative C* algebra is $C_0(X)$ for some locally compact Hausdorff space $X$. </p> <ul> <li>The spectral theorem is a corollary.</li> <li>The theorem introduces students to the idea that a ring is a geometric object</li> <li>Certain constructions in topology, e.g. the Stone-Cech compactification, become more transparent.</li> </ul> http://mathoverflow.net/questions/60457/elementaryshortuseful/60714#60714 Answer by Paul Siegel for Elementary+Short+Useful Paul Siegel 2011-04-05T17:03:52Z 2011-04-05T17:03:52Z <p>Maybe a stretch, but...</p> <p>Finiteness of the class number via Minkowski's theorem.</p> <ul> <li>Everyone should at least have a rough idea what the class number is.</li> <li>Minkowski's theorem has other amusing and useful applications (e.g. well-definedness of the signature?)</li> <li>One of the first (of <em>many</em>) interesting theorems involving the geometry of lattices.</li> </ul> http://mathoverflow.net/questions/60457/elementaryshortuseful/60718#60718 Answer by Paul Siegel for Elementary+Short+Useful Paul Siegel 2011-04-05T17:08:22Z 2011-04-05T17:08:22Z <p>Heisenberg's uncertainty principle.</p> <ul> <li>Everyone should be exposed to quantum mechanics.</li> <li>Appears frequently in analysis and probability (not to mention physics). </li> <li>Showcases some of the highlights of Fourier theory.</li> </ul> http://mathoverflow.net/questions/60457/elementaryshortuseful/60719#60719 Answer by Paul Siegel for Elementary+Short+Useful Paul Siegel 2011-04-05T17:14:15Z 2011-04-05T17:14:15Z <p>The isoperimetric inequality.</p> <ul> <li>Ubiquitous in geometry.</li> <li>Among the easier examples of variational problems.</li> <li>Can be used to illustrate why we need rigorous proofs of things that are "obvioius".</li> </ul> http://mathoverflow.net/questions/60457/elementaryshortuseful/60837#60837 Answer by tetrapharmakon for Elementary+Short+Useful tetrapharmakon 2011-04-06T17:22:35Z 2011-04-06T17:22:35Z <p>Yoneda Lemma. :D</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60942#60942 Answer by ABayer for Elementary+Short+Useful ABayer 2011-04-07T14:16:34Z 2011-04-07T14:16:34Z <p>Elementary symmetric polynomials generate the ring of symmetric polynomials.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60954#60954 Answer by Ethan Fetaya for Elementary+Short+Useful Ethan Fetaya 2011-04-07T15:40:18Z 2011-04-07T15:40:18Z <p>Something that I found very interesting and very useful is <a href="http://en.wikipedia.org/wiki/Singular_value_decomposition" rel="nofollow">Singular value decomposition</a>. It shows that every operator is "almost diagnosable", and is skipped in a lot of basic linear algebra courses I have seen.</p> <p>I has many application, for example - solving sum of least squares of example. You can give a 30 minute talk on this in various levels as well.</p> <p>There are prettier theorems (Stokes, Uniformization, and many more) but I think with the 3 constraints (interesting, useful, little background) this is a good topic. </p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60970#60970 Answer by picakhu for Elementary+Short+Useful picakhu 2011-04-07T16:57:41Z 2011-04-07T16:57:41Z <p>The <a href="http://en.wikipedia.org/wiki/Martingale_%28probability_theory%29" rel="nofollow">Martingale stochastic process</a></p> http://mathoverflow.net/questions/60457/elementaryshortuseful/61057#61057 Answer by Ivan Rothstein for Elementary+Short+Useful Ivan Rothstein 2011-04-08T12:48:38Z 2011-04-08T12:48:38Z <p>My first choice was taken, Picard iteration using Fixed point principles. I'll try not to have a repeat. I have been teaching a history of math class this semester so this sort of thing has been on my mind recently.</p> <p>I would definitely consider different choices depending on how advanced the students I expected were.</p> <p>Pre-Calculus but talented: Archimedes method for finding $\pi$. Calculus: Fermat method for finding the integral of $x^n$ Differential Equations: Picard iterations/fixed point principles more advanced. The Brachistichrone.</p> <p>Another topic that I like, specifically for analysis is to take some of the different definitions of continuity and show that they are equivalent.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/61060#61060 Answer by pageman for Elementary+Short+Useful pageman 2011-04-08T13:21:10Z 2011-04-08T13:21:10Z <p>I can just imagine what would have happened if I was introduced to <a href="http://en.wikipedia.org/wiki/Kepler%2527s_conjecture" rel="nofollow">Kepler's Conjecture</a> and Thomas Hales' approach earlier ...</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/61086#61086 Answer by L Spice for Elementary+Short+Useful L Spice 2011-04-08T18:24:35Z 2011-04-08T18:24:35Z <p>Pursuant to <a href="http://mathoverflow.net/questions/60457/elementaryshortuseful/60530#60530" rel="nofollow">Johannes's answer</a>, I would like to give a talk entitled “How to factor $x_0^4 + x_1^4 + x_2^4 + x_3^4 - 2x_0^2 x_1^2 - 2x_0^2 x_2^2 - 2x_0^2 x_3^2 - 2x_1^2 x_2^2 - 2x_1^2 x_3^2 - 2x_2^2 x_3^2 - 8x_0 x_1 x_2 x_3$”.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/61196#61196 Answer by yaoxiao for Elementary+Short+Useful yaoxiao 2011-04-10T08:56:59Z 2011-04-10T08:56:59Z <p>Every matrix can be represented by the linear combinations of four orthogonal matrix</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/61214#61214 Answer by TriThang Tran for Elementary+Short+Useful TriThang Tran 2011-04-10T14:36:32Z 2011-04-10T14:36:32Z <p><a href="http://en.wikipedia.org/wiki/Schur%2527s_lemma" rel="nofollow">Schur's Lemma</a>. After which one can as an application, classify the simple modules for cyclic groups.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/61221#61221 Answer by Gil Kalai for Elementary+Short+Useful Gil Kalai 2011-04-10T16:31:09Z 2011-04-10T16:31:09Z <p><a href="http://en.wikipedia.org/wiki/Hall%27s_marriage_theorem" rel="nofollow">Hall Marriage theorem</a></p> <p>This is a very useful theorem in combinatorics, analysis, algebra, computational complexity, and more.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/61249#61249 Answer by Jérôme JEAN-CHARLES for Elementary+Short+Useful Jérôme JEAN-CHARLES 2011-04-11T00:51:21Z 2011-04-11T00:51:21Z <p>I would tell them <strong>"What is real maths"</strong>. To achieve this use Lakatos way about Euler's formula ( $V - E + F = 2$ ).<br> It is a set of successive reformulations (more and more precise) each followed by a counter example justifying the next reformulation. </p> <p>Reference is : I. Lakatos, "Proofs and Refutations: The Logic of Mathematical Discovery</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/61260#61260 Answer by Andy for Elementary+Short+Useful Andy 2011-04-11T04:35:57Z 2011-04-11T04:35:57Z <p>Sperner's Theorem on antichains in subset lattice and the Sunflower Lemma. Two great theorems in combo which require little to no theory to introduce and have extremely beautiful proofs. </p> http://mathoverflow.net/questions/60457/elementaryshortuseful/61261#61261 Answer by Wai Kit for Elementary+Short+Useful Wai Kit 2011-04-11T04:40:32Z 2011-04-11T04:40:32Z <p>Fundamental Theorem of Finitely Generated Abelian Groups.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/61365#61365 Answer by Derek Anderson for Elementary+Short+Useful Derek Anderson 2011-04-12T04:57:41Z 2011-04-12T04:57:41Z <p><a href="http://en.wikipedia.org/wiki/G%25C3%25B6del%2527s_incompleteness_theorems" rel="nofollow">Gödel's incompleteness theorems</a></p> <p>A non-technical overview could be done in a fairly short amount of time, thus allowing for some discussion of its various implications, particularly regarding possible roles of mathematics.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/61381#61381 Answer by Pantsman0 for Elementary+Short+Useful Pantsman0 2011-04-12T10:47:31Z 2011-04-12T10:47:31Z <p><a href="http://en.wikipedia.org/wiki/Integration_by_parts/" rel="nofollow">Integration by Parts</a> It's a powerful analytical tool and it can be used for reduction of order on complex functions.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/71648#71648 Answer by Chandrasekhar for Elementary+Short+Useful Chandrasekhar 2011-07-30T11:24:39Z 2011-07-30T11:24:39Z <ul> <li>I would go for <a href="http://en.wikipedia.org/wiki/Cayley%27s_theorem" rel="nofollow">Cayley's theorem</a> which asserts that every group is isomorphic to a subgroup of $S_{n}$ for some $n$. </li> </ul> <p>One, can even look into this following post:</p> <ul> <li><a href="http://math.stackexchange.com/questions/10029/importance-of-cayleys-theorem" rel="nofollow">http://math.stackexchange.com/questions/10029/importance-of-cayleys-theorem</a></li> </ul> http://mathoverflow.net/questions/60457/elementaryshortuseful/71649#71649 Answer by Chandrasekhar for Elementary+Short+Useful Chandrasekhar 2011-07-30T11:27:50Z 2011-07-30T11:27:50Z <ul> <li>The famous <a href="http://en.wikipedia.org/wiki/Heine%25E2%2580%2593Borel_theorem" rel="nofollow">Heine - Borel theorem</a> which says that a closed a bounded subset of $\mathbb{R}^{n}$ is compact.</li> </ul> http://mathoverflow.net/questions/60457/elementaryshortuseful/71655#71655 Answer by Mark Schwarzmann for Elementary+Short+Useful Mark Schwarzmann 2011-07-30T13:49:16Z 2011-07-30T13:49:16Z <p>Jordan normal form.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/80466#80466 Answer by aglearner for Elementary+Short+Useful aglearner 2011-11-09T11:09:32Z 2011-11-09T11:09:32Z <p>I would introduce Bezout's Theorem (there is an article on wiki). It will be hard to prove this statement in the full generality, but the proof of the weaker statement:</p> <p><em>The system of two polynomials $P(x,y)$ and $Q(x,y)$ without common factors of degrees $m$ and $n$ correspondingly has at most $mn$ solutions.</em></p> <p>takes one page at most and uses only the fact that polynomials of two variables have a unique factorisation in irreducible polynomial. (for example, you can check page 244 in an appendix of the book "Rational Points on Elliptic curves" of Silverman and Tate).</p> <p>The well-known beautiful (or, say, elementary) application of this theorem is Pascal's theorem. </p> http://mathoverflow.net/questions/60457/elementaryshortuseful/82561#82561 Answer by gggg gggg for Elementary+Short+Useful gggg gggg 2011-12-03T16:03:50Z 2011-12-03T16:03:50Z <p>[I would introduce Taylor's theorem and point out that it has many applications for instance in physics but also in differential geometry. On the one hand very elementary proofs can be given, but on the other hand, for practical computations with "nice" functions it is always helpful to have that theorem in full generality at the ready. For instance in Riemannian Geometry, one uses Taylor expansion in combination with Jacobi fields to expand the metric tensor locally. This does show that locally, we can find coordinates s.t. the metric behaves like the standard Euclidean metric, but there have to be some corrections such as one term involving the Riemannian curvature tensor.][http://en.wikipedia.org/wiki/Taylor's_theorem]</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/82575#82575 Answer by Jose Arnaldo Dris for Elementary+Short+Useful Jose Arnaldo Dris 2011-12-03T19:37:44Z 2011-12-03T19:37:44Z <p>The <a href="http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means" rel="nofollow">Arithmetic Mean-Geometric Mean Inequality</a>.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/82596#82596 Answer by Jonathan Ringstad for Elementary+Short+Useful Jonathan Ringstad 2011-12-04T03:38:18Z 2011-12-04T03:38:18Z <p>My suggestion -- assuming they have not yet taken a class on complex analysis -- would be to talk about <a href="http://en.wikipedia.org/wiki/Eulers_formula" rel="nofollow">Eulers formula</a> and <a href="http://en.wikipedia.org/wiki/De_Moivre%2527s_formula" rel="nofollow">De Moivre's formula</a>, along with the complex representations of the most common trigonometric functions. Perhaps, if there is time left, power series and the <a href="http://en.wikipedia.org/wiki/Cauchy_product" rel="nofollow">Cauchy product</a> could be touched upon.</p> <p>This could help the students to understand better how some trigonometric identities can be derived, which is usually not explained in detail until a first course on complex analysis. </p> <p>Each of the topics is simple enough to introduce in a very short amount of time, so there would probably be time left to show some cool applications.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/82610#82610 Answer by One_math_boy for Elementary+Short+Useful One_math_boy 2011-12-04T10:41:59Z 2011-12-04T10:41:59Z <p><a href="http://en.wikipedia.org/wiki/Cauchy%2527s_integral_theorem" rel="nofollow">Cauchy's integral theorem</a> and <a href="http://en.wikipedia.org/wiki/Cauchy%2527s_integral_formula" rel="nofollow">Cauchy's integral formula</a>.</p> <p>It's really an example of a jewellery-type and tool-type theorem at the same time. It can be introduced and proved for students that even don't know about functions of complex variables in 20 minutes. And other 10 minutes can be spend to say how many applications and generalizations these results have in theory of functions and applied mathematics.</p> http://mathoverflow.net/questions/60457/elementaryshortuseful/82611#82611 Answer by Michael Greinecker for Elementary+Short+Useful Michael Greinecker 2011-12-04T10:56:13Z 2011-12-04T10:56:13Z <p><a href="http://ncatlab.org/nlab/show/Moore+closure" rel="nofollow">Moore closures</a>, their relation to collections of Moore-closed sets and a characterization for closure under finitary operations. </p> <p>One can then discuss why Moore-closed sets form a complete lattice and a lot more, if one feels so inclined.</p> <p>This is certainly something students will encounter over, and over, and over again in different guises. Moore-closures are certainly among the most useful trivialities I know.</p>