User alfonso gracia-saz - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T03:59:09Z http://mathoverflow.net/feeds/user/3065 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14950/on-using-field-extensions-to-prove-the-impossiblity-of-a-straightedge-and-compass On using field extensions to prove the impossiblity of a straightedge and compass construction Alfonso Gracia-Saz 2010-02-10T21:43:36Z 2013-04-25T20:38:31Z <p>Let $z \in \mathbb{C}$. Consider the following statements: <OL><LI>The point $z$ can be constructed with straightedge and compass starting from the points ${ 0,1}$. <LI>There is a field extension $K / \mathbb{Q}$ which has a tower of subextensions, each one of degree 2 over the next, and such that $z \in K$ <LI>The field extension $\mathbb{Q}(z) / \mathbb{Q}$ has a tower of subextensions, each one of degree 2 over the next. </OL></p> <p>The usual way to prove that a geometric construction is impossible is to use that 1 and 2 are equivalent. My question is: are 2 and 3 equivalent? At first sight this looked like it was going to be true and elementary, but I could not prove it or find a counterexample.</p> http://mathoverflow.net/questions/83254/algebra-decomposition-of-a-matrix-polynomial/83259#83259 Answer by Alfonso Gracia-Saz for Algebra - Decomposition of a matrix polynomial Alfonso Gracia-Saz 2011-12-12T17:16:58Z 2011-12-12T17:16:58Z <p>If you only need to show that $E$ exists, that is easy. $A$ is a domain, so it has a field of quotients $K$. Let $E$ be the algebraic closure of $K$. Then $E$ satisfies the condition you want. This is non-constructive, however.</p> http://mathoverflow.net/questions/83243/breaking-down-an-impartial-game-into-nim-equivalent/83257#83257 Answer by Alfonso Gracia-Saz for Breaking down an impartial game into Nim equivalent Alfonso Gracia-Saz 2011-12-12T17:05:08Z 2011-12-12T17:05:08Z <p>On step 1, you forget that a single heap of size 1 is not the only terminal position in this game. Any increasing sequence is a terminal position.</p> <p>On step 2, your decomposition does not work. For example, the game [3,4,1] is not the sum of the games [3] and [4,1], because when you put those numbers together, they do not form two independent games. To be more precise, the game [3] has nim-value 0; the game [4,1] has nim-value 1; but the game [3,4,1] has nim-value 2. The difficulty of this game (and its interest) lies in the fact that you cannot (at least not at first sight) decompose it as sum of various smaller games.</p> <p>Finally, on step 3, how did you calculate that "Grundy sequence"? If you have a game whose positions are defined by a single integer, then the n-th term of the Grundy sequence is the nim equivalent for position n. This is not the case here, so I have no idea what that sequence means or how you calculated it.</p> http://mathoverflow.net/questions/42395/surprising-behaviour-of-polynomial-that-generates-the-series-1-2-4-8-2k-1/42400#42400 Answer by Alfonso Gracia-Saz for Surprising behaviour of polynomial that generates the series 1,2,4,8,...2^(k-1) Alfonso Gracia-Saz 2010-10-16T18:58:55Z 2010-10-16T18:58:55Z <p><I> Many (all?) integer series f(k), where k = 1,2,3,..K-1,K can be generated by a polynomial of order K-1.</I></p> <p>Well, yes, all of them. Given distinct numbers $a_1, \ldots, a_n$, the polynomial $$p_i(X)= \prod_{j\neq i}\frac{X-a_j}{a_i - a_j}$$ takes value $1$ at $a_i$ and $0$ at $a_j$ for all $j \neq i$. A linear combination of those gives you your desired outcome.</p> http://mathoverflow.net/questions/11105/an-example-of-two-elements-without-a-greatest-common-divisor An example of two elements without a greatest common divisor Alfonso Gracia-Saz 2010-01-08T05:17:19Z 2010-07-12T16:41:46Z <p>Is there an easy example of an integral domain and two elements on it which do not have a greatest common divisor? It will have to be a non-UFD, obviously.</p> <p>"Easy" means that I can explain it to my undergrad students, although I will be happy with any example.</p> http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/29752#29752 Answer by Alfonso Gracia-Saz for Do names given to math concepts have a role in common mistakes by students? Alfonso Gracia-Saz 2010-06-28T07:38:57Z 2010-06-28T07:38:57Z <p>Sequence vs series? Particularly if the two notions are introduced one right after the other in a calculus course, students are doomed to mix them up.</p> http://mathoverflow.net/questions/27344/what-are-your-experiences-of-handouts-in-mathematics-lectures/27416#27416 Answer by Alfonso Gracia-Saz for What are your experiences of handouts in mathematics lectures? Alfonso Gracia-Saz 2010-06-08T01:23:26Z 2010-06-08T01:23:26Z <p>My short answer is to give students the notes in advance, and if that discourages them from coming to class and making the most of it, then I am a poor lecturer.</p> <p>I used to think that I was giving a good lecture if I was more useful to the students than an hour reading the book. Now I think that I am giving a good lecture if I am more useful to the students than an hour reading <I>my own set of notes</I>. This is to say, if I can be entirely replaced by my notes, and if giving students my notes in advance means that they will not come to class, then I am not doing things right.</p> <p>Yes, I understand that I am setting a very high standard, one by which most lectures by most professors (and me included) are not that good, but I think that this is the standard to aim for nevertheless. I like many of the things that Tao says in his teaching statement, in particular about how lectures are to complement, not reproduce or replace, the book or notes. And yes, I admit that this is easier said than done.</p> http://mathoverflow.net/questions/16829/what-are-your-favorite-instructional-counterexamples/16878#16878 Answer by Alfonso Gracia-Saz for What are your favorite instructional counterexamples? Alfonso Gracia-Saz 2010-03-02T17:11:45Z 2010-03-04T01:08:29Z <p>My favourite counterexample is purely academic: it does not have any applications, but I think it is pretty.</p> <p>Let $X = \mathbb{N} \times \mathbb{N}$. Define a non-empty set $U \subseteq X$ to be open if for cofinitely many $x \in \mathbb{N}$ the set <code>$\{ y \in \mathbb{N} \vert (x,y) \in U\}$</code> is cofinite.</p> <p>Construct a sequence in $X$ that hits every point in $X$ exactly once. In other words, take a bijection $\mathbb{N} \rightarrow X$. Then: <UL> <LI> $X$ is countable; <LI> every point in $X$ is an accumulation point of this sequence, but <LI> the sequence has no convergent subsequences. </UL></p> <p>In particular, this is an example in a countable set that accumulation point of a sequence does not have to be a limit of a subsequence. I call this the Herreshoff topology for the (high-school) student of mine who came up with it. (I could not find it anywhere else, although I do not discard that I did not look hard enough.)</p> http://mathoverflow.net/questions/16899/an-example-where-gcd-depends-on-the-domain An example where GCD depends on the domain Alfonso Gracia-Saz 2010-03-02T19:18:27Z 2010-03-03T02:35:42Z <p>First some notation. Given a domain $R$ and $x,a,b \in R$, I write $x=gcd(a,b)_R$ to mean that $x$ is <I>one</I> gcd of $a$ and $b$ in $R$.</p> <p>I want to find an example of an GCD-domain $R$, a subdomain $S \subseteq R$, and two elements $a, b \in S$ such that there isn't any $x \in S$ such that $x=gcd(a,b)_R$ and $x=gcd(a,b)_S$. Notice that it is not enough to find one element $x \in S$ such that $x=gcd(a,b)_R$ but $x \neq gcd(a,b)_S$.</p> <p>I can prove that this is impossible in as little as a Bezout domain, but I cannot prove that this is impossible in a mere GCD-domain. I do not know that many examples of GCD-domains which are not Bezout domains in the first place.</p> <p>ETA: As suggested below, I also wanted $S$ to be a GCD-domain.</p> http://mathoverflow.net/questions/16587/specializing-early/16825#16825 Answer by Alfonso Gracia-Saz for Specializing early Alfonso Gracia-Saz 2010-03-02T03:55:45Z 2010-03-02T03:55:45Z <p>I agree with José's comment above: I do not think early specialisation is a good idea. Did I understand correctly that you want to give a one week to a 15-year old to decide on which area of mathematics to specialize? </p> <p>I want to add something different, however. I fail to see how "some undergraduate topics currently taught compulsorily are a bit of a burden". Mathematics is not a set of disconnected areas. They are all highly related. Most research problems, while staying in one area, may be related to another, motivated by another, applicable in another, or steal ideas or techniques from another. One general course in, say, real analysis, complex analysis, abstract algebra, differential geometry, discrete mathematics, or topology is not a burden, but I dare say an actual necessity for anybody wanting to do research on <I>any</I> topic in pure math. To use your own example, someone doing research in Lie theory will benefit from, rather than be burdened by, a solid understanding of basic differential geometry. Or to use my own case, I am a Poisson geometer, but I have used ideas or results from all the above topics in my research.</p> http://mathoverflow.net/questions/1114/whats-a-groupoid-whats-a-good-example-of-a-groupoid/13156#13156 Answer by Alfonso Gracia-Saz for What's a groupoid? What's a good example of a groupoid? Alfonso Gracia-Saz 2010-01-27T18:19:52Z 2010-01-27T18:19:52Z <p>Contra dance (or square dance) gives us a nice example of a groupoid. The objects are the formations (i.e. the positions of the dancers) and the morphisms are the calls up to homotopy. A choreography (or a dance, if you wish) if a set of composable calls whose product is a morphism between two specific objects.</p> http://mathoverflow.net/questions/9402/pedagogical-question-about-linear-algebra/12749#12749 Answer by Alfonso Gracia-Saz for Pedagogical question about linear algebra Alfonso Gracia-Saz 2010-01-23T15:47:31Z 2010-01-23T15:47:31Z <p>I can share what I did having a similar concern in mind, but it was for point-set topology, not linear algebra. I am not sure how much of this can be translated to linear algebra, since student's minds are already full of preconceptions about what a vector space, but not about what a topological space is.</p> <p>After many years of tutoring point-set topology, I observed that students systematically thought of all topological spaces as $\mathbb{R}^n$, and that they always wanted to use balls, even if the topology was non-metrizable. Hence, when I got to teach my own point-set topology course, I tried something a bit radical: I did not talk about metric spaces at all until later in the course.</p> <p>I started with motivation. On the second day, I defined the notions of topology, homeomorphism (but not continuous function), and convergence of a sequence. Then I did only small finite examples first. I gave the students the following exercise: 1) How many topologies can you define in {0,1,2}; 2) How many of them produce homeomorphic topological spaces?; and 3) In how many of them does the sequence $0,1,0,1,0,1, \ldots$ converge to $2$? Then I made sure to give students enough time (and guidance) to solve this exercise before moving to anything else. </p> <p>I wanted to force the students to accept the abstract notion of topology and to not be scared by it (and to realize that everything we do in point-set topology is logical). Also, in this example, there is no way a student is going to attempt to use balls (particularly when I have not talked about balls). I think it worked well.</p> http://mathoverflow.net/questions/12070/what-are-the-advantages-and-disadvantages-of-the-moore-method/12744#12744 Answer by Alfonso Gracia-Saz for What are the advantages and disadvantages of the Moore method? Alfonso Gracia-Saz 2010-01-23T14:28:03Z 2010-01-23T15:00:25Z <p>I taught a course on Galois theory at Canada/USA Mathcamp. The context was certainly different from a regular university or college: there were no exams or evaluations of any kind, and students choose their own courses and drop casually out of them if they wanted. I started with 28 students (many of whom did not belong there), and ended with only 14. I think that 10 out of those 14 had learned a significant amount of material rather than being over their heads, and 4 out of those 10 understood everything we had done perfectly. When you compare this with the outcome of teaching a course on Galois theory with traditional method, it is not bad at all.</p> <p>It is true that the material is covered slower than we traditional lecture, but I believe that students learn more. The students who finished the course were certainly enthusiastic and I had the same experience that Kevin tells in his last paragraph.</p> <p>Finally, I will add that a competitive atmosphere is not necessary for the Moore method to work. (Yes, Moore's original course required aggressive and competitive students, but there are many things we call Moore method now.) In my case, it was more of a community working together. I recall how, after proving the fundamental theorem of Galois theory, a student was attempting to compute $\cos \frac{2\pi}{17}$ explicitly (in an afternoon, preparing for the class for the next day), and he was surrounded by a group of other students, eager following the process and cheering him on.</p> <p>In short, after my limited experience, I am a big supported of the Moore Method and other variations. Particularly for students who want to go on to become mathematicians, it gives them a more realistic taste of what math is than a traditional course.</p> http://mathoverflow.net/questions/83254/algebra-decomposition-of-a-matrix-polynomial/83263#83263 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2011-12-13T05:06:56Z 2011-12-13T05:06:56Z It is not. Ignore my reply, as it was completely wrong. http://mathoverflow.net/questions/83254/algebra-decomposition-of-a-matrix-polynomial/83259#83259 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2011-12-13T05:06:29Z 2011-12-13T05:06:29Z Oops. Silly me. http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/40091#40091 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-10-19T06:26:31Z 2010-10-19T06:26:31Z Correcting that misunderstanding is crucial to prove that an accumulation point of a net is always the limit of some subnet, which does not hold for sequences. http://mathoverflow.net/questions/35600/what-are-your-favorite-puzzles-toys-for-introducing-new-mathematical-concepts-to/35609#35609 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-08-17T03:25:43Z 2010-08-17T03:25:43Z There is also projective set, a version designed by a graduate student at Waterloo, which requires recognizing lines in five-dimensional projective space over the field with 2 elements. http://mathoverflow.net/questions/35600/what-are-your-favorite-puzzles-toys-for-introducing-new-mathematical-concepts-to/35603#35603 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-08-17T03:23:04Z 2010-08-17T03:23:04Z The 15 puzzle is a good way of introducing groupoids. http://mathoverflow.net/questions/27344/what-are-your-experiences-of-handouts-in-mathematics-lectures/27416#27416 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-06-08T02:27:20Z 2010-06-08T02:27:20Z Victor, I agree. What I actually mean is that I want my one hour lecture plus $n-1$ hours reading the book/notes to be more useful for the student than $n$ hours reading the book/notes. http://mathoverflow.net/questions/26518/do-lie-algebroids-pull-back-along-submersions/26521#26521 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-06-08T02:09:20Z 2010-06-08T02:09:20Z In particular, since in Theo's case $Y \to X$ was a vector bundle, then $phi^{**} A \to Y$ is the total space'' of a VB-algebroid $(\phi^{**}A, Y, A, X)$. See section 6.1 on arXiv:0810.0066. Not sure whether this is relevant, but since you mention &quot;in trivialized case comes in pieces&quot;, it may. http://mathoverflow.net/questions/27344/what-are-your-experiences-of-handouts-in-mathematics-lectures/27397#27397 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-06-08T01:22:26Z 2010-06-08T01:22:26Z That is what &lt;a href=&quot;<a href="http://www.youtube.com/watch?v=WwslBPj8GgI&quot;&gt;Eric" rel="nofollow">youtube.com/watch?v=WwslBPj8GgI&quot;&gt;Eric</a> Mazour&lt;/a&gt; does in physics (but it is equally applicable in mathematics). http://mathoverflow.net/questions/27344/what-are-your-experiences-of-handouts-in-mathematics-lectures/27389#27389 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-06-08T01:10:52Z 2010-06-08T01:10:52Z If this had been anybody but Ole Hald, I would be very skeptical. http://mathoverflow.net/questions/27076/why-does-undergraduate-discrete-math-require-calculus/27132#27132 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-06-07T00:55:38Z 2010-06-07T00:55:38Z To back up Noah's claim, I have taught topology and abstract algebra at Mathcamp without calculus as a prerequisite with no problem. http://mathoverflow.net/questions/25059/examples-of-undergraduate-mathematics-separation-from-what-mathematicians-should/25081#25081 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-05-19T07:38:06Z 2010-05-19T07:38:06Z I agree with Mike Skirvin about the Jordan/rational canonical forms: they do appear in problems in many areas. Even if they did not appear, I still consider them important just for illustrating the more general concept of &quot;canonical form&quot; (i.e., the choice of a particular representative of each equivalence class in an equivalence relation). http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/23497#23497 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-05-07T21:21:25Z 2010-05-07T21:21:25Z That there are two different notions that are called &quot;locally compact&quot; does not help. http://mathoverflow.net/questions/22299/what-are-some-examples-of-colorful-language-in-serious-mathematics-papers/22471#22471 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-05-06T05:02:28Z 2010-05-06T05:02:28Z I was a TA on a course taught by Sarason himself following this book. I had two students &quot;solve&quot; that Exercise during one of my office hours. http://mathoverflow.net/questions/19987/math-paper-authors-order/20114#20114 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-04-08T18:15:06Z 2010-04-08T18:15:06Z Yes, it is apocryphal. I heard David Cox explain during a recent dinner that upon meeting Steven Zucker as grad students, they immediately decided they had to co-author a paper together (which would happen several years later). There was never any intention of not using alphabetical order. http://mathoverflow.net/questions/16829/what-are-your-favorite-instructional-counterexamples/16878#16878 Comment by Alfonso Gracia-Saz Alfonso Gracia-Saz 2010-03-10T05:28:04Z 2010-03-10T05:28:04Z Yes, you are right. Thanks.