Re: Mice problem

ValeryOchkov · ‎Oct 10, 2018

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!

ValeryOchkov · ‎Oct 13, 2018

3D variant - not mice but bats!

ValeryOchkov · ‎Oct 14, 2018

3D animation without three 2D animations is not an animation!

ValeryOchkov · ‎Nov 27, 2018

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

Werner_E · ‎Nov 27, 2018

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

...

AlvaroDíaz · ‎Nov 28, 2018

Hi Valery. Try this, using complex numbers.

Best regards.

Alvaro.

ValeryOchkov · ‎Nov 28, 2018

@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?

AlvaroDíaz · ‎Nov 28, 2018

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.

ValeryOchkov · ‎Nov 29, 2018

@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

AlvaroDíaz · ‎Nov 29, 2018

@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.

AlvaroDíaz · ‎Nov 29, 2018

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".

ValeryOchkov · ‎Nov 30, 2018

@AlvaroDíaz wrote:

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".

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...

AlvaroDíaz · ‎Nov 30, 2018

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.

ValeryOchkov · ‎Nov 28, 2018

@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