User miguel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:49:54Z http://mathoverflow.net/feeds/user/5316 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22247/geometrical-meaning-of-grassmann-algebra/22275#22275 Answer by Miguel for Geometrical meaning of Grassmann Algebra Miguel 2010-04-22T23:00:29Z 2010-04-22T23:00:29Z <p>You could start by considering the vector product in 3 dimensions...</p> http://mathoverflow.net/questions/21486/good-economical-textbook-for-undergraduate-intro-to-diff-eq-for-engineers/21493#21493 Answer by Miguel for Good/Economical textbook for undergraduate intro to diff.eq. for engineers? Miguel 2010-04-15T19:02:33Z 2010-04-15T19:02:33Z <p>I could do worse than to direct you to Dover Publications. They have at least half a dozen introductory differential equation books, all for less than $50.</p> <p>This one has good reviews and it's only$25: Ordinary Differential Equations by Morris Tenenbaum &amp; Harry Pollard <a href="http://store.doverpublications.com/0486649407.html" rel="nofollow">http://store.doverpublications.com/0486649407.html</a></p> http://mathoverflow.net/questions/21471/can-you-have-a-spherical-plane/21478#21478 Answer by Miguel for Can you have a spherical plane? Miguel 2010-04-15T17:02:25Z 2010-04-15T17:02:25Z <p>Intrinsically, no, because one way to understand positive curvature is as a "force" that pulls diverging geodesics closer together. If the curvature is bounded from below away from zero, then any two geodesic arcs leaving a given point will intersect at a finite distance as Ulrich comments above.</p> <p>But extrinsically, the situation is a lot more interesting and fun. You can embed an intrinsically flat plane in a hyperbolic 3-space and, from within the hyperbolic 3-space, it will look like it has constant positive curvature with respect to geodesic planes (intrinsically, hyperbolic planes) of the hyperbolic 3-space. There are models of the hyperbolic 3-space where the 3-space is represented by the interior of a ball of radius 1, and then one of this embedded flat planes is represented by a 2-sphere internally tangent to the surface of the 3-ball.</p> <p>See <a href="http://en.wikipedia.org/wiki/Horoball" rel="nofollow">http://en.wikipedia.org/wiki/Horoball</a>, and (for a nice picture) <a href="http://en.wikipedia.org/wiki/Horocycle" rel="nofollow">http://en.wikipedia.org/wiki/Horocycle</a></p> http://mathoverflow.net/questions/21327/is-it-still-impossible-to-partition-the-plane-into-jordan-curves-without-choice/21342#21342 Answer by Miguel for Is it still impossible to partition the plane into Jordan curves without choice? Miguel 2010-04-14T14:40:05Z 2010-04-14T14:40:05Z <p>Isn't the "proper" 3D analogue of filling the plane with closed curves to fill space with surfaces?</p> <p>I mean, the decomposition of the 3 sphere into two tori (Hopf fibration) is nice and all, but you're using a result involving link theory as the analogue of a result from measure theory...</p> <p>What I would consider the proper analogue of partitioning the plane with Jordan curves is to partition ${\mathbb R}^3$ with closed, orientable surfaces. And what makes that question superficially a lot more interesting than the ${\mathbb R}^2$ case is that all Jordan curves are homeomorphic to a circle, but closed orientable surfaces can have any genus. However, the proof you give still goes through in that case because a closed orientable surface has nonempty interior...</p> http://mathoverflow.net/questions/21090/smooth-gelfand-duality/21191#21191 Answer by Miguel for smooth Gelfand-duality Miguel 2010-04-13T10:58:13Z 2010-04-14T14:24:09Z <p>If I am not mistaken the algebraic data distinguishing $C(M)$ from $C^\infty(M)$ is that $C^\infty(M)$ is equipped with a space of derivations which is a module over the algebra $C^\infty(M)$.</p> <p>A derivation in this case is an $\mathbb R$-linear map $D$ of $C^\infty(M)$ to itself satisfying Leibniz's product rule: $D(fg) = D(f)g + fD(g)$ for all $f,g\in C^\infty(M)$.</p> <p>I don't think the Gelfand duality itself is different from what you'd expect. In fact, the point of the Gelfand duality in this case would be to prove that $C(M)$ is the closure of $C^\infty(M)$ under the compact-open topology. The differentiable manifold structure is given by the derivations.</p> http://mathoverflow.net/questions/21168/how-to-classify-the-algebras-cm/21193#21193 Answer by Miguel for How to classify the algebras C^∞(M)? Miguel 2010-04-13T11:03:45Z 2010-04-13T11:03:45Z <p>I think the appropriate category here is not that of algebras but algebras with derivations (linear maps satisfying Leibniz's product rule). If you don't look at the derivations you're forgetting the differentiable structure of the manifold and all the manifolds homeomorphic (possibly not diffeomorphic) to your manifold support your algebra of functions. I posted an answer along the same lines in your previous questions.</p> http://mathoverflow.net/questions/21182/making-an-intuition-precise/21190#21190 Answer by Miguel for Making an intuition precise Miguel 2010-04-13T10:43:45Z 2010-04-13T10:43:45Z <p>I would say the "correct conditions" for a statement are discovered by trying to prove it and seeing what you need to be true in order for the statement to follow.</p> <p>Proofs are generally discovered in the opposite direction than they are written because proofs are written for elegance and conciseness, not for teaching purposes. The last thing you discover is the first step of your elegantly-presented proof. Which is why textbooks can be very confusing to students. They understand the statement of the theorem and then the proof starts with "Let blah blah be " and the student is left wondering where that came from. Well, it came from spending a long time working on the proof... and the process of discovery is almost never shown.</p> <p>Sometimes the very statement of a theorem is confusing because it is not clear where they got the maddeningly detailed conditions of the theorem. They got them by starting with the conclusion and working out the proof. </p> <p>And sometimes the basic definitions of an entire theory are obscure and baroque, and only much later one realises that the definition is just what one needs for a certain big theorem to be true.</p> http://mathoverflow.net/questions/21142/maximal-ideal-of-codimension-1 Maximal ideal of codimension >1 Miguel 2010-04-12T18:59:14Z 2010-04-12T19:04:12Z <p>To assuage my conscience over an unsourced statement in a paper I'm writing:</p> <p>I am looking for an example of a commutative algebra over the complex numbers having a maximal ideal of codimension >1, or a statement of inexistence.</p> http://mathoverflow.net/questions/22635/can-we-prove-set-theory-is-consistent Comment by Miguel Miguel 2010-04-27T20:22:35Z 2010-04-27T20:22:35Z What theory could you use to prove this statement?&lt;blockquote&gt;if Set1 is consistent, then it cannot prove the consistence of Set2.&lt;/blockquote&gt;What does it mean to say &quot;Set1 is consistent&quot;? In what level is that statement? Like there is a naive set theory Set1 used to define a formal logic Logic1 within which Set2 is a theory, is there a naive logic (Logic0?) used to reason about Set1? Is the statement &quot;Set1 is consistent&quot; a statement in Logic0? Is there hope of proving &quot;If Set1 is consistent it cannot prove the consistency of Set2&quot; within Logic0 which, being naive, isn't too powerful? http://mathoverflow.net/questions/12638/taking-lecture-notes-in-lectures/12673#12673 Comment by Miguel Miguel 2010-04-27T08:03:06Z 2010-04-27T08:03:06Z Live TeXing is definitely the way to go. http://mathoverflow.net/questions/12638/taking-lecture-notes-in-lectures/12666#12666 Comment by Miguel Miguel 2010-04-27T08:01:52Z 2010-04-27T08:01:52Z I agree on the Abe Lincoln effect. It helps me that I can hear the lecture, see what's written on the board or presented in slides, in addition to writing the notes (often I write my own version of what I think the lecturer is saying - so some processing is going on) and then reading my own notes as I write them. I think this helps me remember the lecture better. In fact, though I sometimes take nice notes, I hadly ever revisit my own notes, which I take as further evidence that the purpose of my note-taking is not reference but an aid to acquiring the information by involving many channels... http://mathoverflow.net/questions/22549/how-often-are-irrational-numbers-well-approximated-by-rationals/22566#22566 Comment by Miguel Miguel 2010-04-26T07:00:16Z 2010-04-26T07:00:16Z I knew continued fractions had to appear somewhere... http://mathoverflow.net/questions/21486/good-economical-textbook-for-undergraduate-intro-to-diff-eq-for-engineers/21493#21493 Comment by Miguel Miguel 2010-04-21T23:25:28Z 2010-04-21T23:25:28Z Another classic book is Hirsh and Smale: <a href="http://www.amazon.com/Differential-Equations-Dynamical-Introduction-Mathematics/dp/0123497035" rel="nofollow">amazon.com/&hellip;</a> http://mathoverflow.net/questions/21327/is-it-still-impossible-to-partition-the-plane-into-jordan-curves-without-choice/21342#21342 Comment by Miguel Miguel 2010-04-14T18:15:49Z 2010-04-14T18:15:49Z That's analogous to considering closed space-filling curves in ${\mathbb R}^2$. http://mathoverflow.net/questions/21182/making-an-intuition-precise/21190#21190 Comment by Miguel Miguel 2010-04-13T11:07:32Z 2010-04-13T11:07:32Z As an aftethought: you might want to get ahold of Polya's book &quot;How to Solve it&quot;. Googling about, I have also found a book called &quot;How to Prove it&quot; by Daniel J. Velleman. The former is a classic. Does anyone have an opinion of the latter? http://mathoverflow.net/questions/21142/maximal-ideal-of-codimension-1/21144#21144 Comment by Miguel Miguel 2010-04-12T19:10:26Z 2010-04-12T19:10:26Z I like the bit about the codimension being necessarily infinite. Where could I find a proof of that? What kinds of techniques are involved? http://mathoverflow.net/questions/21142/maximal-ideal-of-codimension-1/21145#21145 Comment by Miguel Miguel 2010-04-12T19:08:18Z 2010-04-12T19:08:18Z Thanks, Pete - I knew about the Nullstellensatz but I am indeed interested in infinitely generated algebras.