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Oct 10, 2018
01:59 AM

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Oct 10, 2018
01:59 AM

Mice problem

See please here Mice problem about this task!

You can see in attach a Mathcad 15 file with solution of this problem.

But I think that will be better to use not a X-Y system but a Polar system of coordinate - for the calculation and for the plotting - not only for 4 animals but for 3, 5, 6 etc. Help me please!

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13 REPLIES 13

Oct 13, 2018
03:33 PM

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Oct 14, 2018
03:10 AM

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Nov 27, 2018
01:37 PM

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Nov 27, 2018
01:37 PM

Sorry - it is not **bats** but **bugs** problem! But for me bugs are not 3D but 2D animals!

- Arnold M., Zharnitsky V. Cyclic evasion in the three bugs problem //American Mathematical Monthly. 2015. Vol. 12, issue 4. Pp. 377-380. URL: https://doi.org/10.4169/amer.math.monthly.122.04.377
- Chapman S., Lottes J., Trefethen L. Four bugs on a rectangle // Proc. of The Royal Society A. 2011. Vol. 467(2127). Pp. 881–896. DOI: 10.1098/rspa.2010.0506

Nov 27, 2018
02:58 PM

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Nov 27, 2018
02:58 PM

http://www.utdallas.edu/~maxim.arnold/pub/triangle.pdf

http://eprints.maths.ox.ac.uk/1012/1/finalOR43.pdf

http://www.mikeraugh.org/MiscPapers/FourBugs.pdf

https://math.stackexchange.com/questions/77652/4-bugs-chasing-each-other-differential-equation

http://faculty.missouri.edu/~casazzap/pdf/teach/bug.pdf

...

Nov 28, 2018
02:34 AM

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Nov 28, 2018
02:34 AM

Hi Valery. Try this, using complex numbers.

Best regards.

Alvaro.

Nov 28, 2018
10:49 AM

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Nov 28, 2018
10:49 AM

@AlvaroDíaz wrote:

Hi Valery. Try this, using complex numbers.

Best regards.

Alvaro.

Thanks Alvaro - its very interesting!

And what about a 3D problem - a bugs/bats problem with complex numbers?

Nov 28, 2018
11:41 PM

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Nov 28, 2018
11:41 PM

Hi Valery. Sorry, but one of my old complex variable books have a lot of examples of applications in 2D of complexes: fluids, mechanical systems, thermal systems ... but nothing about 3D, so I assume that there are not way to cover with complexes that. I loose that book and can't remember the author. So, maybe only with quaternions you can improve the speed and get more elegant equations, but it imply a lot of work.

Best regards.

Alvaro.

Nov 29, 2018
12:04 AM

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Nov 29, 2018
12:04 AM

@AlvaroDíaz wrote:

Hi Valery. Sorry, but one of my old complex variable books have a lot of examples of applications in 2D of complexes: fluids, mechanical systems, thermal systems ... but nothing about 3D, so I assume that there are not way to cover with complexes that. I loose that book and can't remember the author. So, maybe only with quaternions you can improve the speed and get more elegant equations, but it imply a lot of work.

Best regards.

Alvaro.

In my students time I have had a dream about 3D "complex numbers": CN := 1 + 1i + 1j

Nov 29, 2018
12:41 AM

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Nov 29, 2018
12:41 AM

@ValeryOchkov wrote:

In my students time I have had a dream about 3D "complex numbers": CN := 1 + 1i + 1j

Hi Valery. You can't, because you lost then usual algebra properties, the soul of the "number" concept: those things that can be added, multiply, divided, etc.

From the wikipedia article: https://en.wikipedia.org/wiki/Hypercomplex_number

"The Cayley–Dickson construction used involutions to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition algebras are the reals ℝ, the complexes ℂ, the quaternions ℍ, and the octonions 𝕆, and the Frobenius theorem says the only real associative division algebras are ℝ, ℂ, and ℍ. In 1958 J. Frank Adams published a further generalization in terms of Hopf invariants on *H*-spaces which still limits the dimension to 1, 2, 4, or 8.^{[2]"}

^{Best regards.}

^{Alvaro.}

Nov 29, 2018
01:56 AM

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Nov 29, 2018
01:56 AM

Hi. Maybe some complicate citation. What those theorems says is that if you define "numbers" like a+bi, a+bi+cj, a+bi+cj+dk, etc, you have algebra properties only for the cases 1,2,4 and sometimes 8. That's reals, complexes, quaternions and octonions. So, those are the only "true" numbers, because they have something like an algebra (for instance, a finite-dimensional real composition algebras) For the case 3 you can define z = a+bi+cj but can't call that thing as "number".

Nov 30, 2018
02:46 AM

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Nov 30, 2018
02:46 AM

@AlvaroDíaz wrote:

I think about the using complex number Mathcad and Math in general functions for the solving nor only 2D problems (your for example) bot 2D problems too...

Nov 30, 2018
05:08 AM

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Nov 30, 2018
05:08 AM

Oh, I see. But I can't get an algorithm which take some visible less time than your original vector-component solution, like in the two dimensions case.

Best regards.

Alvaro.

Nov 28, 2018
02:32 PM

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Nov 28, 2018
02:32 PM

@AlvaroDíaz wrote:

Hi Valery. Try this, using complex numbers.

Best regards.

Alvaro.

We can see someone about this method here https://math.stackexchange.com/questions/77652/4-bugs-chasing-each-other-differential-equation