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George Baloglou
science forum beginner

Joined: 09 May 2005
Posts: 7

Posted: Thu Sep 22, 2005 9:56 am    Post subject: Re: This Week's Finds in Mathematical Physics (Week 221)

In article <dgrghh$63i$1@dizzy.math.ohio-state.edu>,
John Baez <baez@math.removethis.ucr.andthis.edu> wrote:

[snip]

 Quote: Archimedes did more work on calculus than previously believed! We know this now because a manuscript of his that had been erased and written over has recently been read with the help of a synchrotron X-ray beam! http://www.mlahanas.de/Greeks/ArchimedesPal.htm http://news-service.stanford.edu/news/2005/may25/archimedes-052505.html This manuscript also reveals for the first time that he did work on combinatorics: http://www.mlahanas.de/Greeks/ArchimedesComb.htm

For a modern/combinatorial understanding of Archimedes' work on the puzzle
STOMACH(ION), one should visit http://www.math.ucsd.edu/~fan/stomach/ ;
that was also front page news on the Sunday New York Times of 12/14/03!

George Baloglou
anz3
science forum beginner

Joined: 21 Sep 2005
Posts: 12

 Posted: Thu Sep 22, 2005 8:00 pm    Post subject: Re: extension of Minkowski's inequality Thank you very much for your answer. Since I am still searching for a prove of the validity of Minkowski's inequality with parameter \infty < p < 0, I would appreciate very much if someone could tell me where in the literature that can be found. Again, it would be great if the result were formulated in the terminology of random variables.
martin cohen
science forum Guru Wannabe

Joined: 18 May 2005
Posts: 104

Posted: Fri Sep 23, 2005 1:00 pm    Post subject: Re: extension of Minkowski's inequality

anz3 wrote:

 Quote: Thank you very much for your answer. Since I am still searching for a prove of the validity of Minkowski's inequality with parameter \infty p < 0, I would appreciate very much if someone could tell me where in the literature that can be found. Again, it would be great if the result were formulated in the terminology of random variables. Both Hardy, Littlewood, and Polya's "Inequalities"

and Beckenbach and Bellman's "Inequalities" have the theorems
Ali Taghavi

Joined: 14 May 2005
Posts: 73

Posted: Sat Sep 24, 2005 1:30 pm    Post subject: Re: Which TVS is compatible with Lie bracket?

 Quote: In fact the Example of Robert Israel shows that the following question 1 has negative answer,since the spectrum is unbounded,But

what about the following second question 2(search for TVS structur
compatible to the data?):

in the following V is a linear space and T is a linear map from V to V

Question 1:Is there a norm on V such that T is a bounded operator

question 2:is there a topological vector space structure on V such
that T is continious linear map?

Ali Taghavi
 Quote: According to my example, there's no possibility of such a norm on any subspace that includes 1 and exp(kx) for arbitrarily large k. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
Anton Deitmar
science forum beginner

Joined: 25 Jul 2005
Posts: 24

Posted: Mon Sep 26, 2005 6:30 pm    Post subject: Re: Which TVS is compatible with Lie bracket?

 Quote: in the following V is a linear space and T is a linear map from V to V Question 1:Is there a norm on V such that T is a bounded operator

Counterexample: let V have the basis e_1, e_2, e_3, ...
Define T by

T(e_j) = j e_j

Then for any norm N:

N(T(e_j)) = j N(e_j)

so T is not bounded.

 Quote: question 2:is there a topological vector space structure on V such that T is continious linear map?

Yes, let N be a norm on V and set

s_k(v) = N(T^k(v))

Then the family s_k, k=0,1,2,...
of seminorms defines a topology on V and one has

s_k(T(v)) = s_{k+1}(v),

hence T is continuous wrt this topology.

Cheers,
Anton
M.A.Fajjal
science forum Guru Wannabe

Joined: 13 Sep 2005
Posts: 115

Posted: Tue Oct 11, 2005 1:20 am    Post subject: Re: Solving solvable QUINTICS using two sextics

 Quote: Hello all, For those interested in the quintic, this paper might be of some help: "An Easy Way To Solve Solvable Quintics Using Two Sextics" ABSTRACT: Using a method initially developed by George Young (1819- 1889), Arthur Cayley (1821-1895), and later by George Watson (1886- 1965), an explicit quartic is constructed to enable the solution in radicals of a quintic when it is a solvable equation. Not one, but two sextic resolvents are derived which are important to forming the coefficients of this quartic. Certain difficulties and their solutions as well as a novel consequence to the method are also addressed in this paper. Mathematics Subject Classification. Primary: 12E12; Secondary: 12F10. http://www.geocities.com/titus_piezas/ Just click on the link in the website to the pdf file. Sincerely, Titus (tpiezasIII@uap.edu.ph -> remove III for email)

Thank you for your simple method for solving solvable quintics by using two sextics. However, I have faced some problems when using your method for solving quintics. The following example shows these problems.

x^5-10102/5*x^3+1368821/25*x^2-67067178/125*x+5658741371/3125=0

c=-10102/50
d=1368821/250
e=-67067178/625
f=5658741371/3125

p=0
The resolvent quartic

z^4+5658741371/3125*z^3+14030242707268383381/9765625*z^2+18604015291701468087887634121/30517578125*z +10808705528596191952313957779475163001/95367431640625=0

ans =

[ -5658741371/12500+5051/100*5523005^(1/2)+5051/2500*(-28816410650-11203210*5523005^(1/2))^(1/2)]
[ -5658741371/12500+5051/100*5523005^(1/2)-5051/2500*(-28816410650-11203210*5523005^(1/2))^(1/2)]
[ -5658741371/12500-5051/100*5523005^(1/2)+5051/2500*(-28816410650+11203210*5523005^(1/2))^(1/2)]
[ -5658741371/12500-5051/100*5523005^(1/2)-5051/2500*(-28816410650+11203210*5523005^(1/2))^(1/2)]

Which root shall be assigned for z1, z2, z3, and z4? Why?

As all z1, z2, z3 and z4 are complex numbers

each of z1^(1/5), z2^(1/5), z3^(1/5) and z4^(1/5) shall have 5 possible values. Which value for each z_i shall be taken? Why?

I think, it should be some assigned formulae to determine each z_i and each z_i^(1/5)

Best regards
Titus Piezas III
science forum Guru Wannabe

Joined: 10 Mar 2005
Posts: 102

Posted: Fri Oct 14, 2005 11:18 am    Post subject: Re: Solving solvable QUINTICS using two sextics

M.A.Fajjal wrote:
 Quote: Hello all, For those interested in the quintic, this paper might be of some help: "An Easy Way To Solve Solvable Quintics Using Two Sextics" ABSTRACT: Using a method initially developed by George Young (1819- 1889), Arthur Cayley (1821-1895), and later by George Watson (1886- 1965), an explicit quartic is constructed to enable the solution in radicals of a quintic when it is a solvable equation. Not one, but two sextic resolvents are derived which are important to forming the coefficients of this quartic. Certain difficulties and their solutions as well as a novel consequence to the method are also addressed in this paper. Mathematics Subject Classification. Primary: 12E12; Secondary: 12F10. http://www.geocities.com/titus_piezas/ Just click on the link in the website to the pdf file. Sincerely, Titus (tpiezasIII@uap.edu.ph -> remove III for email) Thank you for your simple method for solving solvable quintics by using two sextics. However, I have faced some problems when using your method for solving quintics. The following example shows these problems. x^5-10102/5*x^3+1368821/25*x^2-67067178/125*x+5658741371/3125=0 c=-10102/50 d=1368821/250 e=-67067178/625 f=5658741371/3125 p=0 The resolvent quartic z^4+5658741371/3125*z^3+14030242707268383381/9765625*z^2+18604015291701468087887634121/30517578125*z +10808705528596191952313957779475163001/95367431640625=0 ans = [ -5658741371/12500+5051/100*5523005^(1/2)+5051/2500*(-28816410650-11203210*5523005^(1/2))^(1/2)] [ -5658741371/12500+5051/100*5523005^(1/2)-5051/2500*(-28816410650-11203210*5523005^(1/2))^(1/2)] [ -5658741371/12500-5051/100*5523005^(1/2)+5051/2500*(-28816410650+11203210*5523005^(1/2))^(1/2)] [ -5658741371/12500-5051/100*5523005^(1/2)-5051/2500*(-28816410650+11203210*5523005^(1/2))^(1/2)] Which root shall be assigned for z1, z2, z3, and z4? Why? As all z1, z2, z3 and z4 are complex numbers each of z1^(1/5), z2^(1/5), z3^(1/5) and z4^(1/5) shall have 5 possible values. Which value for each z_i shall be taken? Why? I think, it should be some assigned formulae to determine each z_i and each z_i^(1/5) Best regards

Fajjal,

Wow, I didn't think anyone would reply to this post after a year.
Anyway, as Dummit pointed out in his paper, this is the quintic "casus
irreducibilis" analogous to the cubic version, when all roots are real.

The formula (just like Cardano's) remains true generically though
numerically, when the quintic has all roots real one has to properly
determine which 5th roots of the 4 complex roots of its resolvent go
together, as you pointed out.

Dummit had a similar complication with his method though he got around
that by using an "ordering condition" and letting the 5th roots satisfy
a certain equation, similar to what you did in another post.

To make the method as simple as possible I ignored this aspect though,
with some tweaking, it can address this as well.

P.S. You really should write some formal paper on your work on quintics
in some easy-to-read font and post it on a website. Others will
appreciate it. (For some reason, there's always someone wanting to know

--Titus
Ali Taghavi

Joined: 14 May 2005
Posts: 73

Posted: Thu Oct 20, 2005 1:45 am    Post subject: Re: IS quantum mechanics a limit cycle theory??

hello
let we have a classical hamiltonian H,and we quantize it.
I want to learn some relations between the behavior of solutions of X_H(the classic hamiltonian vector field) and some operetor theoretic invariants of the quantum operatores.In particular a nice interprewtation of the number of closed orbits of X_H in quantum language!
Thank you for your suggestions for some deep references
Ali Taghavi

 Quote: Hi I Am Intersted in the Hilbert 16th problem which main object is "Limit Cycle"! Last Year I Found in the web a paper by Cetto and De La Penna :with The Title "Is Quantum Mechanics A Limit Cycle Theory?" this paper is available in Mathscinet. I Invite you to review this paper and discuss on a possible relation between Hilbert 16th Problem And Mathematical Aspect Of QM, for begining:Let We Have A Planner Vector field L(Lienard Equation ) as follow: x'=y-F(x) y'=-x where F is A non even_polynomial ,we are intersted in the number of Limit Cycles Of L,consider the following two questions:(Assume F'(0) is not zero) 1)Does there exist a correspondence between {closed orbites} of L and {closed orbits} of 4 dimensional (classical) Hamiltonian (y-F(x))z-xw? please see also the similar question in : http://www.arxiv.org/abs/math.CA/0409594 2)Let's Quantize the above 4 dimensional hamiltonian:x,y stand for operators multiplication by x and y,while z,w stand for partial derivative with respect to x ,y resp. What Is The Quantum interpretations for (the number of) closed orbits of classical Hamiltonian (y-F(x))z-xw? Thank you Ali Taghavi Iran
Anton Deitmar
science forum beginner

Joined: 25 Jul 2005
Posts: 24

 Posted: Thu Oct 20, 2005 1:35 pm    Post subject: Re: Presheaf as functor Being a presheaf, it is a functor from the category of opens in X to the category of abelian groups. The O(U)-module structure on F(U) is an add-on structure. This is why people rather speak of O_X-modules.
Ali Taghavi

Joined: 14 May 2005
Posts: 73

Posted: Mon Oct 24, 2005 12:30 pm    Post subject: Re: IS quantum mechanics a limit cycle theory??

Due to material of my question in "Is quantum mechanics a limit cycle
theory" , it seems that I should ask, in priori, for an example of a
classical hamiltonian H with a finite (but non zero) number of closed
orbits (of course it is impossible in 2 dim.) If it is impossible to
present an example of hamiltonian with a finite number of closed
orbits, so the quantum interpretation for this "number" would be
meaningless!

 Quote: hello let we have a classical hamiltonian H,and we quantize it. I want to learn some relations between the behavior of solutions of X_H(the classic hamiltonian vector field) and some operetor theoretic invariants of the quantum operatores.In particular a nice interprewtation of the number of closed orbits of X_H in quantum language! Thank you for your suggestions for some deep references Ali Taghavi Hi I Am Intersted in the Hilbert 16th problem which main object is "Limit Cycle"! Last Year I Found in the web a paper by Cetto and De La Penna :with The Title "Is Quantum Mechanics A Limit Cycle Theory?" this paper is available in Mathscinet. I Invite you to review this paper and discuss on a possible relation between Hilbert 16th Problem And Mathematical Aspect Of QM, for begining:Let We Have A Planner Vector field L(Lienard Equation ) as follow: x'=y-F(x) y'=-x where F is A non even_polynomial ,we are intersted in the number of Limit Cycles Of L,consider the following two questions:(Assume F'(0) is not zero) 1)Does there exist a correspondence between {closed orbites} of L and {closed orbits} of 4 dimensional (classical) Hamiltonian (y-F(x))z-xw? please see also the similar question in : http://www.arxiv.org/abs/math.CA/0409594 2)Let's Quantize the above 4 dimensional hamiltonian:x,y stand for operators multiplication by x and y,while z,w stand for partial derivative with respect to x ,y resp. What Is The Quantum interpretations for (the number of) closed orbits of classical Hamiltonian (y-F(x))z-xw? Thank you Ali Taghavi Iran
Han de Bruijn
science forum Guru

Joined: 18 May 2005
Posts: 1285

Posted: Mon Oct 24, 2005 2:30 pm    Post subject: Re: minimum volume ellipse

John wrote:

 Quote: I am looking for a solution that works in high dimensions. Most of these solutions are exponential in the dimension.

How about trying to generalize the theory at:

http://huizen.dto.tudelft.nl/deBruijn/grondig/crossing.htm#BE

to multiple dimensions? Involving a quadratic form with the mean values
of the points as a midpoint, and the inverse of a matrix with variances
giving the coefficients.

Must be not too bad. Just my 5 cents worth ...

Han de Bruijn

..
joe.shmoe.joe.shmoe
science forum beginner

Joined: 19 Oct 2005
Posts: 7

 Posted: Mon Oct 24, 2005 5:00 pm    Post subject: Re: Presheaf as functor Thanks for the replies. This sounds like a good way to do it, but it's not exactly what I had in mind. I started thinking about this after reading the definition of a presheaf in many textbooks, along the lines of: "On a topological space X, a presheaf F of sets (or abelian groups, rings, etc) is an assignment for each open subset..." etc. Clearly, it seems beneficial to prove statements about presheaves in the general context of "contravariant functors from opens(X) to C" for some C. Then looking at stalks, for example, works the same for all categories in which direct limits exist. Or another example - constructing the sheaf associated to a presheaf, which in the case of sets, abelian groups etc., gives you a (pre)sheaf with values in the same category you started with. This is the sort of thing I'm trying to achieve with some degree of generality. One could add slightly modify the definition of a presheaf, starting with a fixed "base" functor R: Opens(X) --> B, and a "projection" functor P:C --> B, and then saying a presheaf into C ("relative to R and P") is a functor F:Opens(X)--> C such that P·F=R. However, this does make things a lot more cumbersome, and I have not seen this sort of thing done anywhere. Any thoughts on this?
Robert B. Israel
science forum Guru

Joined: 24 Mar 2005
Posts: 2151

Posted: Mon Oct 24, 2005 5:30 pm    Post subject: Re: IS quantum mechanics a limit cycle theory??

Ali Taghavi wrote:
 Quote: Due to material of my question in "Is quantum mechanics a limit cycle theory" , it seems that I should ask, in priori, for an example of a classical hamiltonian H with a finite (but non zero) number of closed orbits (of course it is impossible in 2 dim.) If it is impossible to present an example of hamiltonian with a finite number of closed orbits, so the quantum interpretation for this "number" would be meaningless!

Consider a particle moving in 3 dimensions in the potential
V(x,y,z) = (1 - x^2 - y^2)^2 z + z^3 + x^2 + y^2

Note that dV/dz > 0 except on the circle x^2 + y^2 = 1, z = 0, so
the only possible closed orbits are on that circle. On that circle
there are closed orbits in both directions.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Han de Bruijn
science forum Guru

Joined: 18 May 2005
Posts: 1285

Posted: Tue Oct 25, 2005 2:30 pm    Post subject: Re: IS quantum mechanics a limit cycle theory??

israel@math.ubc.ca wrote:

 Quote: Consider a particle moving in 3 dimensions in the potential V(x,y,z) = (1 - x^2 - y^2)^2 z + z^3 + x^2 + y^2 Note that dV/dz > 0 except on the circle x^2 + y^2 = 1, z = 0, so the only possible closed orbits are on that circle. On that circle there are closed orbits in both directions.

IMHO this cannot have any physical relevance, because these orbits are
highly _unstable_. Am I wrong?

Han de Bruijn

..
Ali Taghavi

Joined: 14 May 2005
Posts: 73

Posted: Tue Oct 25, 2005 2:30 pm    Post subject: Re: IS quantum mechanics a limit cycle theory??

the domain of hamiltonian is even dimensional space! But can Morse
theory help to my question, namely is dynamic fix if we do not pass a
critical value (However I prefer to not change the main subject of my
question, a possible relation between limit cycle theory and
quantization)

 Quote: Consider a particle moving in 3 dimensions in the potential V(x,y,z) = (1 - x^2 - y^2)^2 z + z^3 + x^2 + y^2 Note that dV/dz > 0 except on the circle x^2 + y^2 = 1, z = 0, so the only possible closed orbits are on that circle. On that circle there are closed orbits in both directions. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada

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