Would anybody remember the precise mathematical statement of the problem written by John Nash (interpreted by Russell Crowe) on the black board during his math lesson and that Alicia, his future wife, tried to solve it ? Thanks.
On Feb 14, 6:52 am, giorgio.garzi...@tin.it wrote:
> Would anybody remember the precise mathematical statement of the > problem written by John Nash (interpreted by Russell Crowe) on the > black board during his math lesson and that Alicia, his future wife, > tried to solve it ? > Thanks.
V = { F : R^3 \ X -> R^3 so Del x F = 0 } W = { F = Del g } dim( V / W ) = ?
On Feb 13, 8:05 pm, achille <achille_...@yahoo.com.hk> wrote:
> On Feb 14, 6:52 am, giorgio.garzi...@tin.it wrote:
> > Would anybody remember the precise mathematical statement of the > > problem written by John Nash (interpreted by Russell Crowe) on the > > black board during his math lesson and that Alicia, his future wife, > > tried to solve it ? > > Thanks.
> V = { F : R^3 \ X -> R^3 so Del x F = 0 } > W = { F = Del g } > dim( V / W ) = ?
In article <e1b04544-37e6-4a17-b782-e0fa8d238...@f8g2000vba.googlegroups.com>, Joe
Copper <giorgio.garzi...@tin.it> wrote: > On Feb 13, 8:05 pm, achille <achille_...@yahoo.com.hk> wrote: > > On Feb 14, 6:52 am, giorgio.garzi...@tin.it wrote:
> > > Would anybody remember the precise mathematical statement of the > > > problem written by John Nash (interpreted by Russell Crowe) on the > > > black board during his math lesson and that Alicia, his future wife, > > > tried to solve it ? > > > Thanks.
> > V = { F : R^3 \ X -> R^3 so Del x F = 0 } > > W = { F = Del g } > > dim( V / W ) = ?
> > where 'Del' is the downward pointing triangle.
On Feb 15, 10:06 am, A N Niel <ann...@nym.alias.net.invalid> wrote:
> In article > <e1b04544-37e6-4a17-b782-e0fa8d238...@f8g2000vba.googlegroups.com>, Joe
> Copper <giorgio.garzi...@tin.it> wrote: > > On Feb 13, 8:05 pm, achille <achille_...@yahoo.com.hk> wrote: > > > On Feb 14, 6:52 am, giorgio.garzi...@tin.it wrote:
> > > > Would anybody remember the precise mathematical statement of the > > > > problem written by John Nash (interpreted by Russell Crowe) on the > > > > black board during his math lesson and that Alicia, his future wife, > > > > tried to solve it ? > > > > Thanks.
> > > V = { F : R^3 \ X -> R^3 so Del x F = 0 } > > > W = { F = Del g } > > > dim( V / W ) = ?
> > > where 'Del' is the downward pointing triangle.
On Feb 16, 2:06 pm, Joe Copper <giorgio.garzi...@tin.it> wrote:
> On Feb 15, 10:06 am, A N Niel <ann...@nym.alias.net.invalid> wrote:
> > In article > > <e1b04544-37e6-4a17-b782-e0fa8d238...@f8g2000vba.googlegroups.com>, Joe
> > Copper <giorgio.garzi...@tin.it> wrote: > > > On Feb 13, 8:05 pm, achille <achille_...@yahoo.com.hk> wrote: > > > > On Feb 14, 6:52 am, giorgio.garzi...@tin.it wrote:
> > > > > Would anybody remember the precise mathematical statement of the > > > > > problem written by John Nash (interpreted by Russell Crowe) on the > > > > > black board during his math lesson and that Alicia, his future wife, > > > > > tried to solve it ? > > > > > Thanks.
> > > > V = { F : R^3 \ X -> R^3 so Del x F = 0 } > > > > W = { F = Del g } > > > > dim( V / W ) = ?
> > > > where 'Del' is the downward pointing triangle.
Joe Copper wrote: > On Feb 15, 10:06 am, A N Niel<ann...@nym.alias.net.invalid> wrote: >> In article >> <e1b04544-37e6-4a17-b782-e0fa8d238...@f8g2000vba.googlegroups.com>, >> Joe
>> Copper<giorgio.garzi...@tin.it> wrote: >>> On Feb 13, 8:05 pm, achille<achille_...@yahoo.com.hk> wrote: >>>> On Feb 14, 6:52 am, giorgio.garzi...@tin.it wrote:
>>>>> Would anybody remember the precise mathematical statement of >>>>> the problem written by John Nash (interpreted by Russell >>>>> Crowe) on the black board during his math lesson and that >>>>> Alicia, his future wife, tried to solve it ? Thanks.
>>>> V = { F : R^3 \ X -> R^3 so Del x F = 0 } W = { F = Del g } >>>> dim( V / W ) = ?
>>>> where 'Del' is the downward pointing triangle.
>>> Thanks. Should we assume that g domain is the sub-space X ?
>> Looks like it should be the same as the domain of F, the complement >> of X.
> However, if that is the case, what is the role played by the > sub-space X ?
This is a computation of 1st Betti number of R^3 \ X via its deRham cohomology in dimension 1.
V comprises vector fields having zero curl. if you replace a vector field : a <x> + b <y> + c <z> by a differential form : a dx + b dy + c dz
The resulting differential forms are the closed 1-forms.
W comprises vector fields that are gradients of real-valued functions on R^3 \ X. The similar vector-field --> differential forms mapping used above yields the so-called "exact" differentials.
It is illustrative to compare this result to the connectivity of X.
On Feb 13, 10:05 pm, achille <achille_...@yahoo.com.hk> wrote:
> V = { F : R^3 \ X -> R^3 so Del x F = 0 } > W = { F = Del g } > dim( V / W ) = ?
> where 'Del' is the downward pointing triangle.
Since I don't know the context of those letters and symbols, I'm just curious, what branch of mathematics is that in? What math course (graduate level would it be?) would it typically appear in?
On 16 helmi, 18:54, MoeBlee <jazzm...@hotmail.com> wrote:
> On Feb 13, 10:05 pm, achille <achille_...@yahoo.com.hk> wrote:
> > V = { F : R^3 \ X -> R^3 so Del x F = 0 } > > W = { F = Del g } > > dim( V / W ) = ?
> > where 'Del' is the downward pointing triangle.
> Since I don't know the context of those letters and symbols, I'm just > curious, what branch of mathematics is that in? What math course > (graduate level would it be?) would it typically appear in?
I don`t know much about this, but since nobody else hasn`t answered this yet, I would say this looks like differential geometry. Take this with a grain of salt.
MoeBlee wrote: > On Feb 13, 10:05 pm, achille<achille_...@yahoo.com.hk> wrote:
>> V = { F : R^3 \ X -> R^3 so Del x F = 0 } W = { F = Del g } dim( V >> / W ) = ?
>> where 'Del' is the downward pointing triangle.
> Since I don't know the context of those letters and symbols, I'm > just curious, what branch of mathematics is that in? What math > course (graduate level would it be?) would it typically appear in?
> MoeBlee
Here's a wee li'l vocabulary for you
R^3 = 3-dimensional vector space over the reals R X some subset
R^3 \ X = the set-theoretic complement of X in R^3
{... : ...} stands for the set of all (left hand side ...) satisfying the condition (right hand side ...)
Del = partial differential (gradient) operator <D_x, D_y, D_z>
F is a vector-valued function
x : vector cross product in R^3
<a,b,c> x <p,q,r> = <br - cq, cp - ar, aq - bp>
Del x F = curl F, formed formally by applying the vector cross product above to Del and F = <F1, F2, F3>
Del g = <D_x g , D_y g, D_z g> = vector of 1st-order partial derivatives of the (real-valued) function g. also called grad g.
And a few facts:
It is frequently proven in Calculus classes that curl(grad(g)) = 0 That curl(F) = 0 doesn't always guarantee F = grad(g) if the domain of F is not contractible is easily seen by noting that the vector field
F = <-y/(x^2 + y^2) , x/(x^2 + y^2), 0>
satisfies curl(F) = 0 on its domain {(x,y,z) in R^3 | (x,y) ~= (0,0)} but there is no function defined on all of the domain for which F is the gradient.
I mentioned in another article in this thread that the problem is really a computation using deRham cohomology (cohomology of differential forms), of the first Betti number of R^3 \ X; by Alexander duality, this will turn out to be the same as the first Betti number of X itself.
Where would you see this? Well, my advanced calculus course went through Spivak's little text "Calculus on Manifolds", so I saw differential forms but didn't see deRham cohomology, and surely didn't hit Alexander duality. You'd want a course in algebraic topology for that.
On the way, you might care to peruse Bott & Tu : Differential forms in Algebraic Topology" or Madsen & Tornehave : From Calculus to Characteristic Classses).
W. Dale Hall wrote: > MoeBlee wrote: >> On Feb 13, 10:05 pm, achille<achille_...@yahoo.com.hk> wrote:
>>> V = { F : R^3 \ X -> R^3 so Del x F = 0 } W = { F = Del g } dim( V >>> / W ) = ?
>>> where 'Del' is the downward pointing triangle.
>> Since I don't know the context of those letters and symbols, I'm >> just curious, what branch of mathematics is that in? What math >> course (graduate level would it be?) would it typically appear in?
>> MoeBlee
> Here's a wee li'l vocabulary for you
> R^3 = 3-dimensional vector space over the reals R > X some subset
> R^3 \ X = the set-theoretic complement of X in R^3
> {... : ...} > stands for the set of all (left hand side ...) > satisfying the condition (right hand side ...)
> Del = partial differential (gradient) operator <D_x, D_y, D_z>
> F is a vector-valued function
> x : vector cross product in R^3
> <a,b,c> x <p,q,r> = <br - cq, cp - ar, aq - bp>
> Del x F = curl F, formed formally by applying the vector cross > product above to Del and F = <F1, F2, F3>
> Del g = <D_x g , D_y g, D_z g> = vector of 1st-order partial derivatives > of the (real-valued) function g. also called grad g.
> And a few facts:
> It is frequently proven in Calculus classes that curl(grad(g)) = 0 > That curl(F) = 0 doesn't always guarantee F = grad(g) if the domain > of F is not contractible is easily seen by noting that the vector field
> F = <-y/(x^2 + y^2) , x/(x^2 + y^2), 0>
> satisfies curl(F) = 0 on its domain {(x,y,z) in R^3 | (x,y) ~= (0,0)} > but there is no function defined on all of the domain for which F is > the gradient.
> I mentioned in another article in this thread that the problem is > really a computation using deRham cohomology (cohomology of > differential forms), of the first Betti number of R^3 \ X; by > Alexander duality, this will turn out to be the same as the first > Betti number of X itself.
> Where would you see this? Well, my advanced calculus course went through > Spivak's little text "Calculus on Manifolds", so I saw differential > forms but didn't see deRham cohomology, and surely didn't hit Alexander > duality. You'd want a course in algebraic topology for that.
> On the way, you might care to peruse Bott & Tu : Differential forms in > Algebraic Topology" or Madsen & Tornehave : From Calculus to > Characteristic Classses).
[...]
I read recently about the Novikov Conjecture. From what I read, it seems close to the field of geometric topology.
Also, the name Friedrich Hirzebruch comes up often in the pre-history of the Novikov Conjecture.
Then, I see that there's a theorem known as: "the Hirzebruch Riemann Roch theorem".
I understand the concept of the fundamental group and the importance of the Euler characteristic in classifying 2-manifolds.
One thing I don't understand is exterior algebra or the work of Hermann Grassmann.
I had a look at the Wikipedia article on Grassmann, < http://en.wikipedia.org/wiki/Hermann_Grassmann >, and it's now clear to me that I have no intuitive understanding of the exterior algebra of Grassmann.
I'd appreciate knowing of recommended texts or lecture notes or any suggestions on how to get an intuitive feeling for exterior algebra and/or Grassmann's work in mathematics.
In article <DOqdnXa1D6jXfOHWnZ2dnUVZ_hmdn...@giganews.com>, "W. Dale Hall" <wdunderscorehallatpacbelldotnet@last> wrote:
> On the way, you might care to peruse Bott & Tu : Differential forms in > Algebraic Topology" or Madsen & Tornehave : From Calculus to > Characteristic Classses).
i suspect you are referring to Madsen & tornehave's "From calculus to cohomolgy: de rham cohomology and characteristic classes".
and, FWIW, Tu has published "An Introduction to manifolds", which he says is intended to bridge the gap between first year graduate students and the bott & tu text.
hmm. i see that he intended two more volumes. i wonder if either is out.
vale, rip
-- email address is r i p 1 AT c o m c a s t DOT n e t
richard i pelletier wrote: > In article<DOqdnXa1D6jXfOHWnZ2dnUVZ_hmdn...@giganews.com>, > "W. Dale Hall"<wdunderscorehallatpacbelldotnet@last> wrote:
>> On the way, you might care to peruse Bott& Tu : Differential forms in >> Algebraic Topology" or Madsen& Tornehave : From Calculus to >> Characteristic Classses).
> i suspect you are referring to Madsen& tornehave's "From calculus to > cohomolgy: de rham cohomology and characteristic classes".
Oops, yes of course. Thanks for the correction.
> and, FWIW, Tu has published "An Introduction to manifolds", which he > says is intended to bridge the gap between first year graduate students > and the bott& tu text.
> hmm. i see that he intended two more volumes. i wonder if either is out.
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