Wikipedia:Reference desk/Archives/Mathematics/2014 October 7

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October 7

Visualizing Klein Bottle

Suppose a bright glowing orange mobius strip appeared in space for just an instant and then disappeared, except for its glowing orange edge, which remains suspended motionless in space for a moment, and then the mobius strip reappears again, just for an instant, in the same location. Questions: 1. Have I successfully visualized a klein bottle, embedded in R4? 2. Could it be fiddled with to make it parameretized by a vector with 4 components, each component a function of u and v? 3. Can it be made i"sotropic"--every point on surface looking like every other point, so a there would be transitive group of rigid motions of R4 on the surface? Thanks, Rich Peterson155.97.8.213 (talk) 04:17, 7 October 2014 (UTC)

If you take a Klein bottle like this: File:Klein bottle.svg and cut it with its symmetry plane and then deform slightly both halves so each of them does not self-intersect, then you'll have two Möbius strips with opposite chirality (one right-twisted, the other one left-twisted). So I suppose your visualisation would need some movement of the 'orange edge' curve so it would easily accomodate a plane-reflected strip. Or the initial and final strips would have to be differently suspended on the same edge. But I can't imagine how it would look like in details, so can't also explain that. --CiaPan (talk) 06:30, 7 October 2014 (UTC)

Thanks again.2601:7:6580:5E3:8D53:DA72:5AB7:2AC5 (talk) 03:52, 10 October 2014 (UTC)

Here's a nice video of the embedding you seek [1]. Hard to tell from your description, but I think you had it not-quite right. 3: I'm fairly confident we can present the klein bottle as a surface of Constant_curvature, which is I think what you're getting at with your "isotropic" sentence. SemanticMantis (talk) 14:38, 7 October 2014 (UTC)

It's a surface withouut a boundary, i think, so if it's not a klein bottle, as I think you are saying, then what would it be? Simply a torus? thanks again2601:7:6580:5E3:BDB2:413B:BE6B:D8D4 (talk) 15:13, 8 October 2014 (UTC)

Since the Mõbius strip, which is only part of the surface envisioned in the given construction, is already a non-orientable surface, the final manifold is non-orientable. A torus is orientable, so this is immediately ruled out, as is the sphere. A consideration of the fundamental polygons of the Mõbius strip and of the Klein bottle would lead me to think that this is indeed a Klein bottle. I do not think that CiaPan's consideration of chirality applies: every shape that can be embedded in 3-space can be reflected by applying a rigid rotation in 4-space. And as to the question of isotropy, my guess that the shape would not be isotropic (as in that every direction is the same), but this does not rule out homogeneity (every point being the same). —Quondum 13:40, 10 October 2014 (UTC)

Perhaps it would need to be embedded in 5 dimensions to allow isotropy, just as a torus needs 4 dimensions to allow isotropy, although it can of coursebe embedded in 3 dimensions? Thanks.2601:7:6580:5E3:8C0D:AAA4:C74C:9388 (talk) 14:39, 11 October 2014 (UTC)

You might be interested in the parametrization of a flat embedding of a Klein bottle in R⁵ given in p.115 and [2]. —Quondum 23:42, 11 October 2014 (UTC)

I shall check it out.2601:7:6580:5E3:6CDB:A3CD:121E:88B5 (talk) 04:41, 12 October 2014 (UTC)

Demographic mathematics

The Philippines has a population of roughly 100 million people as of 2014 with a fertility rate of 3.1 children per woman in childbearing years. My question is, how high must be the fertility rate (given the death rate of 5 per 1,000 people remains stable for the time being) from now on that the Philippines will have a population of 250 million in 2064? How to calculate it? --112.198.82.2 (talk) 15:05, 7 October 2014 (UTC)

For convenience additional data: roughly 33% of the population are under 15, just 5% over 60. Birth rate is 25 per 1,000 people. How can the question be solved? --112.198.82.2 (talk) 15:09, 7 October 2014 (UTC)

Population is probably a bit more complicated to model than you're indicating. First, you're ignoring immigration/emigration which can have a significant effect. Even ignoring that you have take into account changes in life expectancy. Fertility rate is related but what you really want is birth rate for the total population, not just women of child bearing age. I would think there are other factors involved that experts in population dynamics would know about, a subject for which math is required but not all that is required. --RDBury (talk) 16:10, 7 October 2014 (UTC)

Yes, it is indeed more complicated. No model can capture everything, so it becomes a matter of what sort of assumptions are good enough for the question at hand. I've done this sort of think for plants and non-human animals, but humans are generally a bit harder to model. For starters, OP might want to read up on the Leslie matrix and links therein. It is a linear model, but the linear approximation is useful for most phases of logistic growth, it just can't capture the transitions. SemanticMantis (talk) 18:37, 7 October 2014 (UTC)

Oh, haven't considered life expectancy and migration. I add additional parameters: life expectancy remains same at about 71 years (regardless the gender). Immigration - emigration is set at zero sum, no loss or gains by immigration/emigration. --112.198.82.61 (talk) 22:30, 7 October 2014 (UTC)

You can ignore immigration, but you have to allow for death, because otherwise the population will grow way too fast. If this were just for fun, I would only follow females, and assume the population is half male and half female. Then I would divide the population into three age groups, one for 'sexually immature', one for 'reproductive', and one for 'too old to reproduce'. Then you can use some of your info to fill out the transition rates in the Leslie matrix, and do some simple linear algebra to get a projection for 2064. If this isn't just for fun, then you'll have to do a lot more work, and probably get better data and read some research papers on modern demographic forecasting. SemanticMantis (talk) 15:08, 8 October 2014 (UTC)

maths

I want to learn maths form begining what should I do please tell me.. I had practised so much for my university test preparation and did lits of harswork but still I cannot place in my mind.. — Preceding unsigned comment added by 65.255.37.207 (talk) 23:15, 7 October 2014 (UTC)

To have any hope of helping you, we need to know whether you know

• arithmetic?
• algebra?
• analytic geometry?
• basic calculus?

—Tamfang (talk) 02:50, 8 October 2014 (UTC)

To forget trigonometry would be a sin. StuRat (talk) 03:34, 8 October 2014 (UTC)

At least one would have plenty of co-sinners. —Tamfang (talk) 06:53, 9 October 2014 (UTC)

I remember a little humorous pamphlet in Italian, called Dizionario balneare or some such, that pretended to be a phrasebook for words you might need at the beach. On two pages, there were drawings of a man and a woman, with body parts labeled. One of the woman's breasts was labeled seno. The other, naturally — coseno. --Trovatore (talk) 07:11, 9 October 2014 (UTC)