Recent content by zinq

thaiqi: Did you mean to write "doubt" or "believe"?

Partly for difficulties with the English language and possibly for other reasons, I am not able to understand what question is being asked.

If two smooth manifolds are diffeomorphic, that just means that the functions defining a homeomorphism between them can be chosen so as to be differentiable. Differentiability in differential geometry is usually taken to be "smooth" — which means infinitely differentiable. But also, a...

yossell wrote: "But there is a question about such 'infinite' verifications. We can't complete an infinite process and so there's a question whether we can talk about verification here. "For instance, would you argue that the continuum hypothesis is either true or false because you can just run...

In answer to the original question: Cantor's proof is indeed a proof and it is entirely rigorous. In outline, it shows that IF there existed a one-to-one correspondence between the positive integers Z+ and the real numbers, then it is always possible to find a real number that no integer in Z+...

There very well may be true statements (such as, about the integers) that cannot be proved true and cannot be proved unprovable (with a given system of axioms). One candidate for this is the Twin Prime Conjecture (TPC), which states that there exists an infinite number of pairs of prime numbers...

The "2nd law of thermodynamics" is not really a law, in that unlike other laws of physics, it does not always hold. Rather it is a statistical law, which is different. Simple models of entropy show that (on a given experimental trial) it is overwhelmingly likely that entropy will increase, but...

In #27 above, I meant to write B(r) for the closed ball of radius r in CP2, but somehow ended up putting the r as a superscript (resulting in Br instead).

Here's how I like to think of ℂℙ2: Pretend you start out at any point p ∈ ℂℙ2 and consider what happens when you consider the 4-dimensional ball Br of radius r > 0 in ℂℙ2 centered at p as r increases. Its boundary ∂Br = S3 is a 3-dimensional sphere. Geometrically, any point of ℂℙ2 looks like...

"... because one could then add a point to get a projective plane" I don't see that argument. (Where would you add a point to a skinny Möbius band that stays close to its core circle?) The argument against a Möbius band M ⊂ R3 with everywhere positive curvature is just that the positive...

But my favorite Möbius band is the so-called "Sudanese Möbius band", embedded minimally in S3 as follows. In S3 = {(z,w) ∈ ℂ2 | |z|2+|w|2 = 1} consider the family of all great hemispheres that have the same great circle C as their boundary. This family is parametrized by a circle. Each...

Whether the projective plane of constant curvature embeds in R4 is apparently unknown at present, according to Robert Bryant.

In case people are still interested in whether the Möbius band can be smoothly embedded in R^4 with constant positive curvature, a differential geometer has answered my query: "Regarding your specific question about whether there is a Möbius strip of constant Gaussian curvature in Euclidean...

"For instance has a product space structure (e.g. there exist projections on the first and second factor) whereas the (trivial) fiber bundle may not have it." This is not correct. For any trivial fibre bundle π : E → B, there always exists a homeomorphism h : E → F × B of E with the...

"Your goal is to formulate a staking strategy to maximize the geometric growth rate of wealth." What is left unstated here is this: What is the criterion by which the maximum is determined? Is it expectation, i.e., the average value gained under all equally likely outcomes of the coin...