User ivane - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:10:59Z http://mathoverflow.net/feeds/user/2196 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergraduate-mathematics-thats-currently-not-or-isn/7293#7293 Answer by ivane for What should be offered in undergraduate mathematics that's currently not (or isn't usually)? ivane 2009-11-30T16:55:47Z 2012-01-08T07:32:00Z <p>Once students have been exposed to linear algebra and vector calculus, build calculus on manifolds using many examples; i.e. go from real $\mathbb{R}$-abstract multilinear to the de Rham complex, illustrating in $\mathbb{R}^3$. All that easen differential geometry, differential topology Riemannian geometry, etc. </p> http://mathoverflow.net/questions/68090/finite-sums-with-binomial-and-catalan-inverses Finite sums with Binomial and Catalan inverses ivane 2011-06-17T20:22:14Z 2011-06-18T01:21:39Z <p>In a recent failed-post about some partial sums with respect to the Central Binomial and Catalan number the formulas $$\sum_{k=0}^n\frac{4^k}{B_k}=\frac{4^{n+1}(2n+1)}{3 B_{n+1}}+\frac{1}{3}$$ $$\sum_{k=0}^n\frac{4^k(k+1)}{B_k}=\frac{4^{n+1}(2n^2+5n+2)}{5 B_{n+1}}+\frac{1}{5}$$ were mention, here in MO, and I forgot to ask, so let me do it now: </p> <p>Is This just one instance of some broader well known pattern?</p> <p>Here, consider $B_m={2m \choose m}$ and $C_m=\frac{B_m}{m+1}$ for those set of numbers.</p> http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Finite sum of integers inverses ivane 2011-06-14T17:53:22Z 2011-06-15T17:37:01Z <p>Recently, we learned from Renzo Sprugnoli that $\sum_{k=0}^n \frac{4^k}{B_k} =\frac{2n+1}{3}\frac{4^{n+1}}{B_{n+1}}+\frac{1}{3}$, where $B_n$ are the famous <strong>central binomial coefficients</strong>, $B_n={2n \choose n}$.</p> <p>Nowadays it is posible to find; using W|A, <em>The Wolframalpha Calculator</em>, that:</p> <p>$$\sum_{k=0}^n\frac{4^k(k+1)}{B_k}=\frac{2n^2+5n+2}{5}\frac{4^{n+1}}{B_{n+1}}+\frac{1}{5}.$$</p> <p>My "begs" are for someone to help me to pursuit some interesant generalizations or applications about the involved <strong>Catalan numbers</strong>. </p> <p><strong>Ref:</strong> R. Sprugnoli, "Sum of reciprocals of the central binomial coefficients", INTEGERS: Electronic journal of combinatorial number theory 6 (2006), #A27. </p> <p><strong>Update:</strong></p> <p>Is the formula $\sum_{k=0}^n\frac{4^k(k+1)}{B_k}=\frac{2n^2+5n+2}{5}\frac{4^{n+1}}{B_{n+1}}+\frac{1}{5}$ well known?</p> http://mathoverflow.net/questions/2147/most-helpful-math-resources-on-the-web/22560#22560 Answer by ivane for Most helpful math resources on the web ivane 2010-04-26T05:04:27Z 2010-04-26T17:22:54Z <p>People: consider <a href="http://www.digizeitschriften.de/" rel="nofollow">http://www.digizeitschriften.de/</a> tons of classical papers in english...</p> <p>I think it is worth to check the 39 journals collection on world class referee-ed mathwork.</p> <p>One paper on Mathematische Annalen (which is the very amusing): "On the <em>holymorphic</em> flow with an isolated singularity", is the famous GSV, gives you an index formula... </p> http://mathoverflow.net/questions/11307/what-do-you-call-the-product-of-a-circle-and-an-annulus/14786#14786 Answer by ivane for What do you call the product of a circle and an annulus? ivane 2010-02-09T17:29:05Z 2010-02-09T17:50:50Z <p>that corresponds to the complement of a trivial (but essencial) torus knot in a open solid torus. For those -Fico had mention- they are called cable spaces and have nice foliation into circles. Its name is <strong>CS(1,0)</strong>. Can you see what is CS(2,1)?</p> <p>Edit at: utc-6 = 11:50 approx</p> <p>you could also say <strong>the trivial I-bundle over the torus</strong></p> http://mathoverflow.net/questions/9147/branched-coverings-over-orbifolds-with-reflector-lines Branched coverings over orbifolds with reflector lines ivane 2009-12-17T02:15:57Z 2009-12-24T23:16:55Z <p>It is well known that if $F\to B$ is a $n$-finite branched covering over an orbifold with cone-points then the orbifold Euler's characteristics are related via $\chi(F)=n(\chi(B)-\sum_i^r\frac{a_i-1}{a_i})$, where $r$ is the number of cones with stabilizer of orders $a_1,...,a_r$ respectively. </p> <p>Now, I don't know if everyone would like to know what is the corresponding relation when $B$ has <em>reflector intervals</em> or <em>reflector circles</em>, as I'd rather... so I dare to question: </p> </p> <p>Is there a generalization in this direction? </p> http://mathoverflow.net/questions/7746/periodic-mapping-classes-of-the-genus-two-orientable-surface Periodic mapping classes of the genus two orientable surface ivane 2009-12-04T04:14:19Z 2009-12-17T20:31:26Z <p>Please, any information on the periodic mapping classes of the genus two orientable surface, $O_2$, will be greatly thanked. We had been studying the topological structure of 3d surface bundles and reintrepreting them as a circle bundles over orbifolds. </p> <p>In the <a href="http://web.archive.org/web/20070316045651/http://www.smm.org.mx/SMMP/html/modules/Publicaciones/AM/Cm/35/artExp08.pdf" rel="nofollow">http://web.archive.org/web/20070316045651/http://www.smm.org.mx/SMMP/html/modules/Publicaciones/AM/Cm/35/artExp08.pdf</a> -work you would like see the cases $O_1$, among $N_1$ and $N_2$, solved. Any feedback on the results and conjectures, some of them obviously false, will bring a lot of happiness :) </p> http://mathoverflow.net/questions/8223/reducible-3d-torus-bundles Reducible 3d torus bundles ivane 2009-12-08T18:11:45Z 2009-12-17T20:30:33Z <p>Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So,</p></p> <p>could anyone give me a hint to classify them?</p></p> <p>In contrast, do you agree that -in the sense of connected sum- all these bundles are irreducible?</p> http://mathoverflow.net/questions/7968/two-solid-n-3-glued-by-its-boundary Two solid N_3 glued by its boundary ivane 2009-12-06T05:31:59Z 2009-12-17T19:56:32Z <p>Let $N_3$ be the genus three non orientable surface. Do we have an analogous 3d manifold as the solid torus and the solid Klein bottle for $N_3$? I don't see how to extend the ideas related to the 3d lens spaces. Any feedback would be super-welcome</p> http://mathoverflow.net/questions/8445/learning-topology/8471#8471 Answer by ivane for Learning Topology ivane 2009-12-10T18:40:52Z 2009-12-12T03:42:43Z <p>Books on <strong><em>geometric group theory</em></strong>, nowaday are talking and applying algebra and topology <em>technologies</em> to algorithms, solvability of word problems, membership problems, context free languages, etc...</p></p> <p>All these together with the notes of Peter May refered above, will almost show you the state-of-the-art.</p></p> <p>Further, check Graham NIblo work at <a href="http://www.personal.soton.ac.uk/gan/Welcome.html" rel="nofollow">http://www.personal.soton.ac.uk/gan/Welcome.html</a></p></p> <p><strong>UPDATE</strong>: Follow <a href="http://mathoverflow.net/questions/3858/introductive-text-on-geometric-group-theory" rel="nofollow">books on geometric-group theory</a> to a more especific titles mentioned here in MO.</p> http://mathoverflow.net/questions/8361/hnn-extensions-which-are-free-products HNN extensions which are free products ivane 2009-12-09T16:03:21Z 2009-12-11T18:46:08Z <p>which HNN-extensions are free products? this question is related with another still unsolved about Nielsen-Thruston-reducibility and connected-sum-irreducibility of 3d-torus- bundles... </p> http://mathoverflow.net/questions/3858/introductory-text-on-geometric-group-theory/8588#8588 Answer by ivane for Introductory text on geometric group theory? ivane 2009-12-11T18:35:22Z 2009-12-11T18:35:22Z <p>What about this wonder:</p></p> <p>Peter Scott, Terry Wall, <strong>Topological methods in group theory</strong>, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press (1979) 137-203.</p> </p> <p>simply beauty and useful</p> http://mathoverflow.net/questions/8521/nice-proof-of-the-jordan-curve-theorem/8532#8532 Answer by ivane for Nice proof of the Jordan curve theorem? ivane 2009-12-11T05:01:00Z 2009-12-11T05:01:00Z <p>You should compare with: "Geometric Topology in Dimensions 2 and 3", Moise, Edwin E. (1977). Springer-Verlag and tell</p> http://mathoverflow.net/questions/8361/hnn-extensions-which-are-free-products/8519#8519 Answer by ivane for HNN extensions which are free products ivane 2009-12-11T02:07:31Z 2009-12-11T02:16:15Z <p>Let me add an explicit partial solution to the question above: for a torus bundle $E$ over the circle, the fundamental group of $E$ can't be a free product of groups, because if it were, the fundamental subgroup of the fiber would be a free product (by the Kurosch's theorem) which is impossible for the torus, then $E$ isn't a connected sum, hence irreducible.</p></p> <p>At least the HNN extensions which are free products can't be torus bundles, and in fact, no other surface bundles unless the surface be the 2-sphere </p> http://mathoverflow.net/questions/7957/books-well-motivated-with-explicit-examples/8466#8466 Answer by ivane for books well-motivated with explicit examples ivane 2009-12-10T17:48:43Z 2009-12-10T18:10:04Z <p><strong>Characteristic classes</strong> by Milnor-Stasheff, 1974. This book from Princeton marks (i think) the synthesis of several years of maturation for the real beginnings of modern topology, the next years that came...</p></p> <p>In their 20 chapters, preface, 3 appendices, bibliograph and index, anyone gonna see a jewel master piece of math</p> http://mathoverflow.net/questions/8216/less-elementary-group-theory/8239#8239 Answer by ivane for less elementary group theory ivane 2009-12-08T19:57:14Z 2009-12-08T19:57:14Z <p>re-develop your course employing <strong>category theory</strong> vision: Once you construct objects construct arrows. Example, if you have defined groups inmediately define homomorphisms, if you define subgroups and quotients, define kernels...etc, arrive to the homomorphism's fundamental theorems... </p> http://mathoverflow.net/questions/5499/which-mathematicians-have-influenced-you-the-most/8062#8062 Answer by ivane for Which mathematicians have influenced you the most? ivane 2009-12-07T01:49:10Z 2009-12-07T01:54:37Z <p>Consider looking forward: <strong>Grisha Perelman</strong>, his strange history <a href="http://en.wikipedia.org/wiki/Grigori_Perelman" rel="nofollow"> wiki G.Perelman</a>. However the first millenium prize awarded. Even note this: Terence Tao said... "well, it's amazing"</p> http://mathoverflow.net/questions/7957/books-well-motivated-with-explicit-examples/7967#7967 Answer by ivane for books well-motivated with explicit examples ivane 2009-12-06T05:05:58Z 2009-12-07T00:53:59Z <p>"Differential Topology", Guillemin-Pollack, 1974. </p> http://mathoverflow.net/questions/7957/books-well-motivated-with-explicit-examples/7958#7958 Answer by ivane for books well-motivated with explicit examples ivane 2009-12-06T04:19:54Z 2009-12-06T04:19:54Z <p>"Riemannian Geometry", Gallot-Hulin-Lafontaine, 1987, plenty of examples and exercises and the motivation: the own one helps... </p> http://mathoverflow.net/questions/7834/undergraduate-differential-geometry-texts/7948#7948 Answer by ivane for Undergraduate Differential Geometry Texts ivane 2009-12-06T02:35:22Z 2009-12-06T02:35:22Z <p>commentaries like <em>without much formalism</em> like those to the Thorpe's book, I think, are really discouraging for everyone, then students don't want to do complex calculations <em>'cuz they are ugly</em>... Nah! </p> <p>I believe that the std covariant derivative of $\mathbb{R}^3$ and the induced connection to a surface, via the Gauss equation (to quick deduce a formula for the gaussian curvature), paves the way to grasp better thing like the geodesic curvature and the Gauss-Bonnet must-do's, for: <strong>O'Neil</strong> is suitable!</p> http://mathoverflow.net/questions/3951/memorizing-theorems/7937#7937 Answer by ivane for Memorizing theorems ivane 2009-12-06T01:00:00Z 2009-12-06T01:00:00Z <p>I would advise to study the proofs until plainly pure memory, BUT each time you repeat the reasons involved, the understanding switch always in ON position... that allows you to gain real and great intuitions, believe-me...</p> Ah! if the repetitions are in front of someone else -who is stuying the same- there'll be a lot of more FUN... put attention in that Gowers is telling us: ...eventualy you will be able of finding quick proofs of everything you masters!</p> http://mathoverflow.net/questions/3242/canonical-examples-of-algebraic-structures/7301#7301 Answer by ivane for Canonical examples of algebraic structures ivane 2009-11-30T18:06:27Z 2009-12-06T00:25:00Z <p>study presentations, zum beispiel:</p> </p> <p>$\langle a,b\mid a^2=b^3=e, ab=b^2a \rangle$ </p> </p> <p>vs </p> </p> <p>$\langle a,b\mid a^2=b^3=e, ab=ba \rangle$ or $\langle a\mid a^6=e \rangle$</p> </p> <p>compare nuances...</p> <p>Other: the $Out\pi_1(F)$ of any surface. Btw this is the famous <strong>mapping class group of the surface</strong> $\cal{MCG}(F)$.</p></p> <p>To get a real modern grasp on the subjet you should include topological techniques: homotopy, cohomotopy, homology and cohomology, K-theory... you might enter in contact with a types of structures like free groups, free products, amalgamated products, HNN extensions of groups, Grothendieck groups and the like...</p> http://mathoverflow.net/questions/7308/n-3-and-n-4-periodic-and-pseudo-anosov-auto-homeomorphisms N_3 and N_4 periodic and pseudo Anosov auto-homeomorphisms ivane 2009-11-30T19:41:44Z 2009-12-04T06:49:18Z <p>It is well know that the genus three non orientable surface, N<sub>3</sub>, has only periodic and reducible auto-homeomorphisms, meanwhile the surface N<sub>4</sub> is the first non orientable surface with pseudo Anosov maps. Also, recently profesor B.Szepietowski gave the MCG presentation of N<sub>4</sub>, from where, I calculated that there are seven periodic mapping classes. The question is: are there more?</p> </p> <p>Curiously, the torus and N<sub>3</sub> show also seven periodic mapping classes each but we would like to understand better why N<sub>3</sub> loses pseudo Anosov maps which contrast with the fact that N<sub>3</sub> is got from the 2-torus via an one-point blow up...</p> http://mathoverflow.net/questions/6142/circle-bundles-over-rp2/7288#7288 Answer by ivane for Circle bundles over $RP^2$ ivane 2009-11-30T16:34:16Z 2009-12-02T19:48:07Z <p>I think that they have <strong>Seifert fiber space</strong> presentation as: $(On,1|(1,b))$.</p></p> <p>Or $(On,1|(1,b),(a_1,b_1),...,(a_r,b_r))$, if you allow an orbifold with cone points in $RP^2$.</p></p> <p>You can look at the cases by decomposing $RP^2=Mo\cup_{\partial}D$, so the orientable 3-manifold will be the</p> 1) orientable $Q=Mo\tilde{\times}S^1$, the <strong>twisted circle bundle over the mobius band</strong>, very well known being equivalent to the <strong>orientable I-bundle over the Klein bottle</strong>, with boundary a torus $T$,</p> 2) and a <strong>Dehn-filling</strong> in the remaining disk $D$, with a whichever fibered solid torus or tori.</p> <p>We could say that $(On,1\mid (1,b))=Q\cup_T W(1,b)$, for a fibered $(1,b)$ solid torus $W$</p> http://mathoverflow.net/questions/2340/what-is-the-first-interesting-theorem-in-insert-subject-here/7362#7362 Answer by ivane for What is the first interesting theorem in (insert subject here)? ivane 2009-12-01T03:49:09Z 2009-12-01T03:49:09Z <p><strong>3-manifolds</strong>, existence and uniqueness of decomposition into irreducible pieces in the connected sum (Kneser, Milnor-Wallace)</p> http://mathoverflow.net/questions/2851/algorithmic-combinatorics-resources/7303#7303 Answer by ivane for Algorithmic Combinatorics resources? ivane 2009-11-30T18:47:52Z 2009-11-30T19:47:00Z <p>a key ingredient in modern software like mathematica is using hypergeometric functions. Let me recall the Knuth's statement in his <strong>Concrete Mathematics</strong>... "with hypergeometric functions you are likely to embrace each elementary functions studied in a technological career"... well, sort of</p> http://mathoverflow.net/questions/2479/de-rham-cohomology-of-surfaces/7298#7298 Answer by ivane for de Rham Cohomology of surfaces ivane 2009-11-30T17:45:31Z 2009-11-30T17:45:31Z <p>check these of J. Harrison and the ivancevics bros</p></p> <p>1)<a href="http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.4991v3.pdf" rel="nofollow">http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.4991v3.pdf</a></p></p> <p>and</p></p> <p>2)<a href="http://arxiv.org/PS_cache/math-ph/pdf/0501/0501001v2.pdf" rel="nofollow">http://arxiv.org/PS_cache/math-ph/pdf/0501/0501001v2.pdf</a></p></p> <p>these skip a little Mayer-Vietoris... (but believe me: it is unwise avoid Mayer-Vietoris :) </p> http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/7289#7289 Answer by ivane for Great mathematical figures and/or diagrams? ivane 2009-11-30T16:47:19Z 2009-11-30T16:47:19Z <p>see my starting effort to help see 3-manifolds at <a href="http://commons.wikimedia.org/wiki/Category:3-manifolds" rel="nofollow">http://commons.wikimedia.org/wiki/Category:3-manifolds</a> </p> http://mathoverflow.net/questions/11307/what-do-you-call-the-product-of-a-circle-and-an-annulus/11315#11315 Comment by ivane ivane 2012-10-12T01:48:13Z 2012-10-12T01:48:13Z @Ryan: $M\ddot{o}\times S^1$ is the non-orientable twisted $I$-bundle over the torus. http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergraduate-mathematics-thats-currently-not-or-isn/7293#7293 Comment by ivane ivane 2012-08-30T18:24:46Z 2012-08-30T18:24:46Z on the beginning missing duality of finite dimensional vector spaces (preferable over the real field) garanties from minor to almost no understanding :P http://mathoverflow.net/questions/68090/finite-sums-with-binomial-and-catalan-inverses/68104#68104 Comment by ivane ivane 2011-06-18T01:04:57Z 2011-06-18T01:04:57Z you are right, i make a mistake because i should write the next formulas: $\sum_{k=0}^n\frac{4^k}{B_k}=\frac{4^{n+1}(2n+1)}{3B_{n+1}}+\frac{1}{3}$ and $\sum_{k=0}^n\frac{4^k(k+1)}{B_k}=\frac{4^{n+1}(2n^2+5n+2)}{5B_{n+1}}+\frac{1}{5}$ http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Comment by ivane ivane 2011-06-15T17:55:51Z 2011-06-15T17:55:51Z ooh!, i found that Witula and Slota have: FINITE SUMS CONNECTED WITH THE INVERSES OF CENTRAL BINOMIAL NUMBERS AND CATALAN NUMBERS in the AEJM. From the abstract, can anyone infer that Sprugnoli's method is different? http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Comment by ivane ivane 2011-06-15T17:26:56Z 2011-06-15T17:26:56Z Yemon, i'm strucked to know that this formulas correspond to someone which already saw them. I would like to see the ref, please. &quot;Begs&quot; or requests are the same, if i use a ridicule tone nothing distorts the math significance, to, connect wisdom. And to the bead about w|a, it is easy to check they and i can do the calculazhons http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Comment by ivane ivane 2011-06-14T23:16:27Z 2011-06-14T23:16:27Z does it hurts that someone tells you that your theorems can be reached by W|A? http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Comment by ivane ivane 2011-06-14T20:29:54Z 2011-06-14T20:29:54Z Excellent ref, just what i was looking for, Lot of Thanx http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Comment by ivane ivane 2011-06-14T18:23:42Z 2011-06-14T18:23:42Z a simple one: how W|A did it? he.he http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Comment by ivane ivane 2011-06-14T18:09:34Z 2011-06-14T18:09:34Z Even more complex are the relations inter $\sum_{k=0}^n\frac{1}{{2k\choose k}}$ and $\sum_{k=0}^n\frac{k+1}{{2k\choose k}}$... http://mathoverflow.net/questions/18570/why-are-tensors-a-generalization-of-scalars-vectors-and-matrices Comment by ivane ivane 2010-05-31T15:47:01Z 2010-05-31T15:47:01Z would like to add that the grasping of the fundamental sense for these objects and properties, are implanted around the generalization of calculus: differential forms and its applications... http://mathoverflow.net/questions/21235/how-to-shown-that-the-tangent-bundle-of-a-vector-space-is-a-vector-bundle Comment by ivane ivane 2010-04-13T19:02:01Z 2010-04-13T19:02:01Z you're going to see better if the normed vector space is taken the fiber $F$ in your definition of vector bundle AND for the tangent bundle you use this type functions: $(x,v)\to(x,J(\varphi_j^{−1}\circ\varphi_i)(x)v)$ where $J$ is for jacobian and $v\in F=\mathbb{R}^n$ http://mathoverflow.net/questions/11307/what-do-you-call-the-product-of-a-circle-and-an-annulus Comment by ivane ivane 2010-02-09T16:56:44Z 2010-02-09T16:56:44Z at <a href="http://commons.wikimedia.org/wiki/Category:3-manifolds" rel="nofollow">commons.wikimedia.org/wiki/Category:3-manifolds</a> you could see some propose. Would be a place to start proposal. Personaly I like &quot;moxi&quot; for $M\ddot{o}\times I$ and twisted-moxi to $M\ddot{o}^\stackrel{\sim}\times I$... http://mathoverflow.net/questions/8717/reducible-3d-n-3-bundles Comment by ivane ivane 2009-12-14T01:31:46Z 2009-12-14T01:31:46Z i.e. in the complement of the invariant multicurve? http://mathoverflow.net/questions/8223/reducible-3d-torus-bundles/8241#8241 Comment by ivane ivane 2009-12-14T00:58:35Z 2009-12-14T00:58:35Z @Sam: 10-4, gonna check, super-thanks http://mathoverflow.net/questions/8717/reducible-3d-n-3-bundles Comment by ivane ivane 2009-12-14T00:36:55Z 2009-12-14T00:36:55Z @Sam: but N_3 don't have pA according to: R.C.Penner. &quot;A construction of pseudo-Anosov homeomorphisms&quot;, Trans.Amer.Math. Soc. 310 (1988) No 1, 179-197.