User ivane - MathOverflow

Answer by ivane for What should be offered in undergraduate mathematics that's currently not (or isn't usually)?

2009-11-30T16:55:47Z

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.

Finite sums with Binomial and Catalan inverses

2011-06-17T20:22:14Z

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:

Is This just one instance of some broader well known pattern?

Here, consider $B_m={2m \choose m}$ and $C_m=\frac{B_m}{m+1}$ for those set of numbers.

Finite sum of integers inverses

2011-06-14T17:53:22Z

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 central binomial coefficients, $B_n={2n \choose n}$.

Nowadays it is posible to find; using W|A, The Wolframalpha Calculator, that:

$$\sum_{k=0}^n\frac{4^k(k+1)}{B_k}=\frac{2n^2+5n+2}{5}\frac{4^{n+1}}{B_{n+1}}+\frac{1}{5}.$$

My "begs" are for someone to help me to pursuit some interesant generalizations or applications about the involved Catalan numbers.

Ref: R. Sprugnoli, "Sum of reciprocals of the central binomial coefficients", INTEGERS: Electronic journal of combinatorial number theory 6 (2006), #A27.

Update:

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?

Answer by ivane for Most helpful math resources on the web

2010-04-26T05:04:27Z

People: consider http://www.digizeitschriften.de/ tons of classical papers in english...

I think it is worth to check the 39 journals collection on world class referee-ed mathwork.

One paper on Mathematische Annalen (which is the very amusing): "On the holymorphic flow with an isolated singularity", is the famous GSV, gives you an index formula...

Answer by ivane for What do you call the product of a circle and an annulus?

2010-02-09T17:29:05Z

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 CS(1,0). Can you see what is CS(2,1)?

Edit at: utc-6 = 11:50 approx

you could also say the trivial I-bundle over the torus

Branched coverings over orbifolds with reflector lines

2009-12-17T02:15:57Z

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.

Now, I don't know if everyone would like to know what is the corresponding relation when $B$ has reflector intervals or reflector circles, as I'd rather... so I dare to question:

Is there a generalization in this direction?

Periodic mapping classes of the genus two orientable surface

2009-12-04T04:14:19Z

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.

In the http://web.archive.org/web/20070316045651/http://www.smm.org.mx/SMMP/html/modules/Publicaciones/AM/Cm/35/artExp08.pdf -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 :)

Reducible 3d torus bundles

2009-12-08T18:11:45Z

Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So,

could anyone give me a hint to classify them?

In contrast, do you agree that -in the sense of connected sum- all these bundles are irreducible?

Two solid N_3 glued by its boundary

2009-12-06T05:31:59Z

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

Answer by ivane for Learning Topology

2009-12-10T18:40:52Z

Books on geometric group theory, nowaday are talking and applying algebra and topology technologies to algorithms, solvability of word problems, membership problems, context free languages, etc...

All these together with the notes of Peter May refered above, will almost show you the state-of-the-art.

Further, check Graham NIblo work at http://www.personal.soton.ac.uk/gan/Welcome.html

UPDATE: Follow books on geometric-group theory to a more especific titles mentioned here in MO.

HNN extensions which are free products

2009-12-09T16:03:21Z

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...

Answer by ivane for Introductory text on geometric group theory?

2009-12-11T18:35:22Z

What about this wonder:

Peter Scott, Terry Wall, Topological methods in group theory, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press (1979) 137-203.

simply beauty and useful

Answer by ivane for Nice proof of the Jordan curve theorem?

2009-12-11T05:01:00Z

You should compare with: "Geometric Topology in Dimensions 2 and 3", Moise, Edwin E. (1977). Springer-Verlag and tell

Answer by ivane for HNN extensions which are free products

2009-12-11T02:07:31Z

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.

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

Answer by ivane for books well-motivated with explicit examples

2009-12-10T17:48:43Z

Characteristic classes 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...

In their 20 chapters, preface, 3 appendices, bibliograph and index, anyone gonna see a jewel master piece of math

Answer by ivane for less elementary group theory

2009-12-08T19:57:14Z

re-develop your course employing category theory 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...

Answer by ivane for Which mathematicians have influenced you the most?

2009-12-07T01:49:10Z

Consider looking forward: Grisha Perelman, his strange history wiki G.Perelman. However the first millenium prize awarded. Even note this: Terence Tao said... "well, it's amazing"

Answer by ivane for books well-motivated with explicit examples

2009-12-06T05:05:58Z

"Differential Topology", Guillemin-Pollack, 1974.

Answer by ivane for books well-motivated with explicit examples

2009-12-06T04:19:54Z

"Riemannian Geometry", Gallot-Hulin-Lafontaine, 1987, plenty of examples and exercises and the motivation: the own one helps...

Answer by ivane for Undergraduate Differential Geometry Texts

2009-12-06T02:35:22Z

commentaries like without much formalism like those to the Thorpe's book, I think, are really discouraging for everyone, then students don't want to do complex calculations 'cuz they are ugly... Nah!

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: O'Neil is suitable!

Answer by ivane for Memorizing theorems

2009-12-06T01:00:00Z

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...

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!

Answer by ivane for Canonical examples of algebraic structures

2009-11-30T18:06:27Z

study presentations, zum beispiel:

$\langle a,b\mid a^2=b^3=e, ab=b^2a \rangle$

$\langle a,b\mid a^2=b^3=e, ab=ba \rangle$ or $\langle a\mid a^6=e \rangle$

compare nuances...

Other: the $Out\pi_1(F)$ of any surface. Btw this is the famous mapping class group of the surface $\cal{MCG}(F)$.

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...

N_3 and N_4 periodic and pseudo Anosov auto-homeomorphisms

2009-11-30T19:41:44Z

It is well know that the genus three non orientable surface, N₃, has only periodic and reducible auto-homeomorphisms, meanwhile the surface N₄ is the first non orientable surface with pseudo Anosov maps. Also, recently profesor B.Szepietowski gave the MCG presentation of N₄, from where, I calculated that there are seven periodic mapping classes. The question is: are there more?

Curiously, the torus and N₃ show also seven periodic mapping classes each but we would like to understand better why N₃ loses pseudo Anosov maps which contrast with the fact that N₃ is got from the 2-torus via an one-point blow up...

Answer by ivane for Circle bundles over $RP^2$

2009-11-30T16:34:16Z

I think that they have Seifert fiber space presentation as: $(On,1|(1,b))$.

Or $(On,1|(1,b),(a_1,b_1),...,(a_r,b_r))$, if you allow an orbifold with cone points in $RP^2$.

You can look at the cases by decomposing $RP^2=Mo\cup_{\partial}D$, so the orientable 3-manifold will be the

1) orientable $Q=Mo\tilde{\times}S^1$, the twisted circle bundle over the mobius band, very well known being equivalent to the orientable I-bundle over the Klein bottle, with boundary a torus $T$,

2) and a Dehn-filling in the remaining disk $D$, with a whichever fibered solid torus or tori.

We could say that $(On,1\mid (1,b))=Q\cup_T W(1,b)$, for a fibered $(1,b)$ solid torus $W$

Answer by ivane for What is the first interesting theorem in (insert subject here)?

2009-12-01T03:49:09Z

3-manifolds, existence and uniqueness of decomposition into irreducible pieces in the connected sum (Kneser, Milnor-Wallace)

Answer by ivane for Algorithmic Combinatorics resources?

2009-11-30T18:47:52Z

a key ingredient in modern software like mathematica is using hypergeometric functions. Let me recall the Knuth's statement in his Concrete Mathematics... "with hypergeometric functions you are likely to embrace each elementary functions studied in a technological career"... well, sort of

Answer by ivane for de Rham Cohomology of surfaces

2009-11-30T17:45:31Z

check these of J. Harrison and the ivancevics bros

1)http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.4991v3.pdf

and

2)http://arxiv.org/PS_cache/math-ph/pdf/0501/0501001v2.pdf

these skip a little Mayer-Vietoris... (but believe me: it is unwise avoid Mayer-Vietoris :)

Answer by ivane for Great mathematical figures and/or diagrams?

2009-11-30T16:47:19Z

see my starting effort to help see 3-manifolds at http://commons.wikimedia.org/wiki/Category:3-manifolds

Comment by ivane

2012-10-12T01:48:13Z

@Ryan: $M\ddot{o}\times S^1$ is the non-orientable twisted $I$-bundle over the torus.

Comment by ivane

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

Comment by ivane

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 $\sum_{k=0}^n\frac{4^k(k+1)}{B_k}=\frac{4^{n+1}(2n^2+5n+2)}{5B_{n+1}}+\frac{1}{5}$

Comment by ivane

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?

Comment by ivane

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. "Begs" 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

Comment by ivane

2011-06-14T23:16:27Z

does it hurts that someone tells you that your theorems can be reached by W|A?

Comment by ivane

2011-06-14T20:29:54Z

Excellent ref, just what i was looking for, Lot of Thanx

Comment by ivane

2011-06-14T18:23:42Z

a simple one: how W|A did it? he.he

Comment by ivane

2011-06-14T18:09:34Z

Even more complex are the relations inter $\sum_{k=0}^n\frac{1}{{2k\choose k}}$ and $\sum_{k=0}^n\frac{k+1}{{2k\choose k}}$...

Comment by ivane

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...

Comment by ivane

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 $v\in F=\mathbb{R}^n$

Comment by ivane

2010-02-09T16:56:44Z

at commons.wikimedia.org/wiki/Category:3-manifolds you could see some propose. Would be a place to start proposal. Personaly I like "moxi" for $M\ddot{o}\times I$ and twisted-moxi to $M\ddot{o}^\stackrel{\sim}\times I$...

Comment by ivane

2009-12-14T01:31:46Z

i.e. in the complement of the invariant multicurve?

Comment by ivane

2009-12-14T00:58:35Z

@Sam: 10-4, gonna check, super-thanks

Comment by ivane

2009-12-14T00:36:55Z

@Sam: but N_3 don't have pA according to: R.C.Penner. "A construction of pseudo-Anosov homeomorphisms", Trans.Amer.Math. Soc. 310 (1988) No 1, 179-197.