User rogelio fern&#225;ndez-alonso - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T19:45:04Z http://mathoverflow.net/feeds/user/20947 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/122687/dual-concept-for-the-p-primary-component Dual concept for the p-primary component Rogelio Fernández-Alonso 2013-02-23T02:51:38Z 2013-02-23T11:43:35Z <p>Is there a dual concept for the p-primary component of an abelian group? Please name some books/papers where it is studied.</p> http://mathoverflow.net/questions/87276/an-equivalence-relation-on-the-power-set-of-the-plane An equivalence relation on the power set of the plane. Rogelio Fernández-Alonso 2012-02-01T21:16:54Z 2012-02-01T22:41:03Z <p>Let $R\subseteq\mathbb{R}^2$. Consider the set of all "horizontal sections" $H_R =$ {$Rb|b\in\mathbb{R}$} where $Rb=${$a\in\mathbb{R} | (a,b)\in R$}. Similarly consider the set of "vertical sections" of $R$, $V_R =${$aR|a\in\mathbb{R}$} where $aR=${$b\in\mathbb{R} | (a,b)\in R$}. Now define the equivalence relation on $\wp (\mathbb{R^2})$ such that $R \sim S$ if, and only if, $H_R=H_S$ and $V_R=V_S$.</p> <ol> <li>Do you have any reference to this equivalence relation or a similar one?</li> <li>What connections does it have to topology?</li> <li>As an example, ¿can you describe the equivalence class of a disk?</li> </ol> <p>Of course this can be generalized to any set of binary relations, but I want to understand it in the case of the plane.</p> http://mathoverflow.net/questions/87047/tangent-lines-to-2-circles-tangent-planes-to-3-spheres-and-so-on Tangent lines to 2 circles, tangent planes to 3 spheres, and so on. Rogelio Fernández-Alonso 2012-01-30T17:46:07Z 2012-01-31T06:33:53Z <p>Although it is known the solution to the first two questions, somebody may have different nice answers, so I include them:</p> <ol> <li><p>Given two circles in the plane, there is (at least) a line which is tangent to both of them.</p></li> <li><p>Given three spheres in the space, there is a plane which is tangent to all of them.</p></li> <li><p>In general, given $n$ n-spheres in the n-dimensional space, is there a hyperplane which is tangent to all of them?</p></li> <li><p>What other generalizations does this problem admit?</p></li> </ol> <p>EDIT: As @Noam kindly remarked below, the existence of the tangent objects is not always true. I think that in #2 the hypothesis must be: one of the spheres not in the cone determined by the other two. In #1 the "cone" determined by one circle is the circle itself. So in #3 we need a suitable definition for the "cone" determined by $n-1$ n-spheres.</p> http://mathoverflow.net/questions/86930/an-algorithm-for-constructing-the-ar-quiver-of-a-path-algebra-corresponding-to-a An algorithm for constructing the AR-quiver of a path algebra corresponding to a change in the orientation. Rogelio Fernández-Alonso 2012-01-28T22:27:41Z 2012-01-29T05:00:16Z <p>Considering the path algebra of the quiver $\mathbb{A}_n$, it is well known its Auslander-Reiten quiver with the canonical orientation of $\mathbb{A}_n$, that is, with all the arrows from, say, left to right. I can find as examples in several texts the AR-quivers of $\mathbb{A}_n$ with other orientations.</p> <p>QUESTION: Is there an algorithm to construct the AR-quiver of $\textit{any}$ orientation of $\mathbb{A}_n$ ?</p> <p>Clearly it will suffice to describe the effect in the AR-quiver if it is changed the orientation of one arrow. Observing the examples I think there is some pattern, but I can't figure out the algorithm.</p> <p>Of course this algorithm could be applied to quivers of other type than $\mathbb{A}_n$, but I think it is better to understand this in the simplest case.</p> http://mathoverflow.net/questions/86871/the-functor-category-functr-mod-s-mod The functor category Funct(R-Mod, S-Mod) Rogelio Fernández-Alonso 2012-01-28T00:13:59Z 2012-01-28T14:16:33Z <p>Can you name some properties of the functor category Funct(R-Mod,S-Mod), where R,S are associative rings with unit?</p> <p>EDIT: I am sorry for the lack of precision of my (first in MO) question. I was thinking in a sort of Yoneda's Lemma for functors $F\colon R\mbox{-Mod}\to S\mbox{-Mod}$, maybe using tensorizing functors $h_U$ to describe $Nat(h_U,F)$.</p> http://mathoverflow.net/questions/8846/proofs-without-words/19341#19341 Comment by Rogelio Fernández-Alonso Rogelio Fernández-Alonso 2012-01-29T16:51:36Z 2012-01-29T16:51:36Z In this pretty solution there is another pretty geometric problem: Given three spheres there is a plane which is tangent to all three. http://mathoverflow.net/questions/86930/an-algorithm-for-constructing-the-ar-quiver-of-a-path-algebra-corresponding-to-a/86940#86940 Comment by Rogelio Fernández-Alonso Rogelio Fernández-Alonso 2012-01-29T04:06:44Z 2012-01-29T04:06:44Z Thank you Steve, this is certainly an algorithm to construct the AR-quiver directly from the oriented $A_n$-quiver. I was wondering if there is anyone that in each step constructs the AR-quiver corresponding to one-arrow change from the AR-quiver that we had before that change. http://mathoverflow.net/questions/86871/the-functor-category-functr-mod-s-mod Comment by Rogelio Fernández-Alonso Rogelio Fernández-Alonso 2012-01-28T04:44:06Z 2012-01-28T04:44:06Z Thank you Benjamin, these are the adjoint functors, and I think they form an interesting subclass of the functor category. http://mathoverflow.net/questions/86871/the-functor-category-functr-mod-s-mod Comment by Rogelio Fernández-Alonso Rogelio Fernández-Alonso 2012-01-28T04:16:29Z 2012-01-28T04:16:29Z Thank you Spice-the-Bird for this point of view. I think that it may be connected with Morita theory (as Yosemite-Sam mentioned), in the sense that equivalence functors are special ones in Funct(R-Mod,S-Mod), and I believe they may not be in the image of this map, for example two rings not connected by an homomorphism, may be Morita equivalent.