A Closed Form Solution To The Universe

[Doctordick]

Ok Anssi, here we go! I will try to be careful but I will probably make a few errors as the algebra is not trivial. The fundamental equation (squared) was displayed in the “Geometric Proof” thread:
 

 

[math]
\left\{\nabla^2+2g^2(r)\right\}\vec{\Psi}=-2k^2m^2\vec{\Psi}=-K\vec{\Psi}
[/math]

 

The n+1 dimensional expansion of [math]\nabla^2[/math] into hyper spherical coordinates is also shown there:
 

 

[math]
\nabla^2=\sum_{i=0}^n\frac{\partial^2}{\partial x_i^2}= \left(\frac{1}{r}\right)^n\frac{\partial}{\partial r}r^n\frac{\partial}{\partial r}+\frac{1}{r^2}\sum_{i=1}^n\left(\prod_{l=1}^{i-1}csc^2\theta_l \right)(csc\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}(sin\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}
[/math]

 

I just spotted an error in the post I just quoted. The first term of the hyper-spherical representation should have a factor of [math]\left(\frac{1}{r}\right)^n[/math] instead of the "2" in the original post. I have fixed it but Anssi's quote still shows the "2".

I will tell you a little story on that expansion. When I first thought of looking at that representation, the expansion of [math]\nabla^2[/math] into hyper spherical coordinates was obvious to me. (It is actually a very complex expansion but the cross terms drop out.) Back then, the fact that most of the terms dropped out and the reason they dropped out was absolutely clear to me. In 1982, the first time I tried to get the thing published, I wanted to make sure that expansion was correct. It took me three days to prove it was right.

In 2002, when I put my work on a web site, I again decided to prove that expansion was correct. That time I wrote the expansion out completely, identifying every term. The stupid result took almost ten pages of terms, each of which took at least an entire line some, two lines. I think it took me almost a month to find all my errors and figure out which terms canceled which terms. I am not going to do it again; I just don't have the mental wherewithal to accomplish such a thing. When I was posting on the “scienceforum” someone commented that it was a well known expansion so it must be out there somewhere if someone wants to check it. At any rate, I am pretty sure what I have written out there is correct now.

Please note that [math]\vec{\Psi}[/math] is nothing more than an abstract vector of solutions so each component must itself be a solution. For that reason, in searching for solutions, I will use [math]\Psi[/math] for the general solution. Substituting my supposed solution for [math]\Psi[/math],
 

 

[math]
\Psi=\frac{1}{\sqrt{r^n}}U(r)\prod_{i=1}^n\Phi_i(\theta_i),
[/math]

 

into my n+1 dimensional equation; I obtain the following
 

[math]
\nabla^2\frac{1}{\sqrt{r^n}}U(r)\prod_{i=1}^n\Phi_i(\theta_i)=\left\{-2g^2(r)-K\right\}\frac{1}{\sqrt{r^n}}U(r)\prod_{i=1}^n\Phi_i(\theta_i)
[/math]

 

Where I have moved everything except [math]\nabla^2[/math] to the right side of the equation. Now, in order to keep things simple, I will point out an important fact about the expression of [math]\nabla^2[/math] in hyper-spherical coordinates. It consists of a sum of terms where each term contains differential operators of but one of those indicated coordinates. In addition, the first term contains a factor [math]r^{-n}[/math] which is outside the differentials with respect to “r”. Secondly, internally to each of the terms of that sum, there are no sums: i.e., every term consists of a product of simple trigonometric functions with a power of “r” or one over “r” to some power. What this means is that each term may be analyzed independently.

Let us first look at the only term containing the differentials with respect to “r”. Since there are no differentials with respect to [math]\prod_{i=1}^n\Phi_i(\theta_i)[/math], in looking for the differential, we may divide by that term. In addition, because the opening term of [math]\nabla^2[/math] contains the factor [math]r^{-n}[/math], we may also multiply through by [math]r^{n}[/math]. In essence, the differentiable part of the first term may be written:

 

[math]
\frac{\partial}{\partial r}r^n\frac{\partial}{\partial r}\frac{1}{\sqrt{r^n}}U(r)
[/math]

 

Note that, when we finish simplifying this differential, in order to recover the original equation, me must multiply by the factor [math]r^{-n}\prod_{i=1}^n\Phi_i(\theta_i)[/math]

Meanwhile, this differentiation can be simplified into a sum of two terms. You should be familiar with the chain rule of differentiation. Starting from the above expression, follow the following algebra carefully

[math]
\frac{\partial}{\partial r}r^n\frac{\partial}{\partial r}r^{-\frac{1}{2}n}U(r)=\frac{\partial}{\partial r}r^n\left\{-\left(\frac{n}{2}\right) r^{-\frac{1}{2}(n+2)}U(r)+r^{-\frac{1}{2}n}U'(r)\right\}
[/math]


[math]
=\frac{\partial}{\partial r}\left\{-\left(\frac{n}{2}\right) r^{\frac{1}{2}(n-2)}U(r)+r^{\frac{1}{2}n}U'(r)\right\}
[/math]

[math]
=-\left(\frac{n}{2}\right)\left(\frac{n-2}{2}\right)r^{\frac{1}{2}(n-4)}U(r)-\left(\frac{n}{2}\right) r^{\frac{1}{2}(n-2)}U'(r)+\left(\frac{n}{2}\right)r^{\frac{1}{2}(n-2)}U'(r)+r^{\frac{1}{2}n}U''(r)
[/math]


[math]
=-\left(\frac{n(n-2)}{4}\right)r^{\frac{1}{2}(n-4)}U(r)+r^{\frac{1}{2}n}U''(r)
[/math]

 

Multiplying by that term I factored out earlier, the first term of [math]\nabla^2[/math] becomes

 

[math]
r^{-n}\prod_{i=1}^n\Phi_i(\theta_i)\left\{-\left(\frac{n(n-2)}{4}\right)r^{\frac{1}{2}(n-4)}U(r)+r^{\frac{1}{2}n}U''(r)\right\}
[/math]

[math]
=\left\{-\left(\frac{n(n-2)}{4}\right)r^{-\frac{1}{2}(n+4)}+r^{-\frac{1}{2}n}\frac{U''(r)}{U(r)}\right\}U(r)\prod_{i=1}^n\Phi_i(\theta_i)
[/math]

[math]
=\left\{-\frac{n(n-2)}{4r^2}+\frac{U''(r)}{U(r)}\right\}\frac{1}{\sqrt{r^n}}U(r)\prod_{i=1}^n\Phi_i(\theta_i)
[/math]

 

and my n+1 dimensional equation becomes

[math]
\left\{-\frac{n(n-2)}{4r^2}+\frac{U''(r)}{U(r)}+\frac{1}{r^2}\sum_{i=1}^n\left(\prod_{l=1}^{i-1}csc^2\theta_l \right)(csc\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}(sin\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}\right\}\frac{1}{\sqrt{r^n}}U(r)\prod_{i=1}^n\Phi_i(\theta_i)
[/math]

[math]
=\left\{-2g^2(r)-K\right\}\frac{1}{\sqrt{r^n}}U(r)\prod_{i=1}^n\Phi_i(\theta_i)
[/math]

 

or, moving the "n(n-2)" term to the right, one has:

[math]
\left\{\frac{U''(r)}{U(r)}+\frac{1}{r^2}\sum_{i=1}^n\left(\prod_{l=1}^{i-1}csc^2\theta_l \right)(csc\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}(sin\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}\right\}\frac{1}{\sqrt{r^n}}U(r)\prod_{i=1}^n\Phi_i(\theta_i)
[/math]

[math]
=\left\{\frac{n(n-2)}{4r^2}-2g^2(r)-K\right\}\frac{1}{\sqrt{r^n}}U(r)\prod_{i=1}^n\Phi_i(\theta_i)
[/math]

 

Division of this equation by [math]\Psi=\frac{1}{\sqrt{r^n}}U(r)\prod_{i=1}^n\Phi_i(\theta_i)[/math] yields a rather interesting differential equation”

[math]
\left\{\frac{1}{U(r)}\frac{d^2}{d r^2}U(r)+\frac{1}{r^2\prod_{i=1}^n\Phi_i}\sum_{i=1}^n\left(\prod_{l=1}^{i-1}csc^2\theta_l \right)(csc\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}(sin\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}\prod_{i=1}^n\Phi_i\right\}
[/math]

[math]
=\left\{\frac{n(n-2)}{4r^2}-2g^2(r)-K\right\}
[/math]

 

(Note that the derivative with respect to r is no longer a partial as the function it operates on has only one variable.) This equation may be solved by simple separation of variables. I am tired at the moment as I just wasted most of Saturday correcting all the accidental errors I made in the LaTex in this post.

Anssi, if you were able to follow it this far and want to go on, let me know and I will lay out the correct solutions to that equation. Meanwhile, you might have a look at the above reference I gave for the "separation of variables" method.

Have fun -- Dick

Edited May 21, 2014 by Doctordick

[AnssiH]

The fundamental equation (squared) was displayed in the “Geometric Proof” thread:

 

[math]

\left\{\nabla^2+2g^2®\right\}\vec{\Psi}=-2k^2m^2\vec{\Psi}=-K\vec{\Psi}

[/math]

 

Okay, I'm going to take the validity of that on faith for time being.

 

I already have questions that need to be sorted out.

 

First, I had to Google "closed form solution", and found out it means it's an expression consisting only of elementary functions. Not sure what the significance of that is. Does that have something to do with this being a one-body equation? (If so, I don't really get it because the number of variables is still very large)

 

Second, just to be sure I've interpreted this right, those variables are essentially equivalent to angular positions, not to angular momentum? The fact that one of the variables is "r" leads me to believe it is angular positions. Is that correct?

 

[imath]g®[/imath] stands for an integration over all the variables except "r"?

 

EDIT: Fixed [imath]G®[/imath] in quote, something substituted an ® with "registered" symbol...

 

-Anssi

[Doctordick]

Okay, I'm going to take the validity of that on faith for time being.

Ok, but it is only some rather simple algebra compared to what we are about to do.

 

First, I had to Google "closed form solution", and found out it means it's an expression consisting only of elementary functions.

That is exactly what it is. Actually it is a rather unusual event. Most serious differential equations can not be solved in “closed form”. Most of them, in the end, can only be solved with series type solutions which do not terminate: i.e., the series goes on to an infinite number of terms. There is a whole field of mathematics/physics devoted to that problem; it is called “perturbation theory”. Perturbation theory starts with an approximate solution and then comes up with corrections, yielding a more accurate solution which is, nevertheless, still an approximation. In most cases, this process goes on forever.

 

Perturbation theory exists because proving that the series actually converges when it goes to infinity can be difficult. Nuclear structures for example have never been solved in closed form (at least they hadn't been forty years ago when I studied physics).

 

Does that have something to do with this being a one-body equation?

Yes, the fact that it can be done is closely related to the fact that it is a rather simple one-body equation. The issue is that there is a serious difference between “very large” and “infinite”. Closed form does not mean that there is enough paper in the universe to write the answer down; it is the fact that a finite answer exists.

 

Second, just to be sure I've interpreted this right, those variables are essentially equivalent to angular positions, not to angular momentum? The fact that one of the variables is "r" leads me to believe it is angular positions. Is that correct?

That is correct. Since human beings can not mentally “picture” a space of more than three dimensions, it is best to look at a three dimensional Euclidean space to get a picture of what is going on here. If you set up such a space, you can reference a specific point via the coordinates (x,y,z) laid out from three orthogonal axes. You can also reference that same point via what are called “spherical coordinates”, [math](r,\phi,\theta)[/math]. I prefer to use what they call the (U,N,E) convention. The angle theta is defined to be an angle in the (x,y) plane which points in the direction of interest (in that plane). This means that [math]tan\theta=\frac{y}{x}.[/math] The angle phi is defined to be an angle in the plane defined by the line just defined and the z axis. Since the length of the vector pointing in the (x,y) plane is given by [math]\sqrt{x^2+y^2},[/math] the angle is defined [math]tan\phi=\frac{z}{\sqrt{x^2+y^2}}.[/math] And “r” is the distance from the origin to the point of interest.

 

[imath]g®[/imath] stands for an integration over all the variables except "r"?

Not really, nothing has been integrated over. Symmetry issues remove the possibility of any dependence on the angles (we are talking about the entire universe here). Any real dependence on the magnitude of r also vanishes but, at this point, there seems to be no simple way to argue away some relationship there. That is why I left it in.

 

EDIT: Fixed [imath]G®[/imath] in quote, something substituted an ® with "registered" symbol...

So, that bug is still there! I complained about it right after they got the new system and thought they had fixed it. Ah, it appears to be a font issue; take a look at this -> ®.

 

 

Since I don't want to get too far out ahead of you, I thought I would just post the algebra I have done so that you can get an idea as to where this thing goes. Might as well get into the questions as soon as possible.

 

Have fun -- Dick

[AnssiH]

Just a quick response for now...

 

Ok, but it is only some rather simple algebra compared to what we are about to do.

 

Yeah, I think I could follow it through if I just refresh my memory of when we did something similar earlier, but I'll probably do that later.

 

That is correct. Since human beings can not mentally “picture” a space of more than three dimensions, it is best to look at a three dimensional Euclidean space to get a picture of what is going on here. If you set up such a space, you can reference a specific point via the coordinates (x,y,z) laid out from three orthogonal axes. You can also reference that same point via what are called “spherical coordinates”, [math](r,\phi,\theta)[/math]. I prefer to use what they call the (U,N,E) convention. The angle theta is defined to be an angle in the (x,y) plane which points in the direction of interest (in that plane). This means that [math]tan\theta=\frac{y}{x}.[/math] The angle phi is defined to be an angle in the plane defined by the line just defined and the z axis. Since the length of the vector pointing in the (x,y) plane is given by [math]\sqrt{x^2+y^2},[/math] the angle is defined [math]tan\phi=\frac{z}{\sqrt{x^2+y^2}}.[/math] And “r” is the distance from the origin to the point of interest.

 

Yup, okay. And in this case the actual value of r is a function of the number of variables.

 

Not really, nothing has been integrated over. Symmetry issues remove the possibility of any dependence on the angles (we are talking about the entire universe here). Any real dependence on the magnitude of r also vanishes but, at this point, there seems to be no simple way to argue away some relationship there. That is why I left it in.

 

Okay, hmmm... I got the idea that it represents an integration from looking at the steps at the Schrödinger's derivation, but you are saying that here it simply represents whatever impact the "r" would have? I feel a bit shaky about this but I'll just move on for now.

 

Since I don't want to get too far out ahead of you, I thought I would just post the algebra I have done so that you can get an idea as to where this thing goes. Might as well get into the questions as soon as possible.

 

Yes I think that's a better way to do it.

 

So the next step;

 

[math]

\nabla^2=\sum_{i=0}^n\frac{\partial^2}{\partial x^2}= \left(\frac{1}{r}\right)^n\frac{\partial}{\partial r}r^n\frac{\partial}{\partial r}+\frac{1}{r^2}\sum_{i=1}^n\left(\prod_{l=1}^{i-1}csc^2\theta_l \right)(csc\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}(sin\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}

[/math]

 

So the nabla stands for a sum of partial derivatives of all the variables. Considering your story about the expansion, I guess I should not stop and try to check it right now... One thing though, what does the symbol [imath]\prod_{l=1}^{i-1}[/imath] stand for? Also, I'm not sure what the "l" stands for in there.

 

Also;

 

[math]

\Psi=\frac{1}{\sqrt{r^n}}U®\prod_{i=1}^n\Phi_i(\theta_i)

[/math]

 

what does the "U" stand for here; exactly the same as what "g" did earlier?

 

I'll try to get around to continue from here soon...

 

-Anssi

[Doctordick]

So the next step;

 

[math]

\nabla^2=\sum_{i=0}^n\frac{\partial^2}{\partial x^2}= \left(\frac{1}{r}\right)^n\frac{\partial}{\partial r}r^n\frac{\partial}{\partial r}+\frac{1}{r^2}\sum_{i=1}^n\left(\prod_{l=1}^{i-1}csc^2\theta_l \right)(csc\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}(sin\theta_i)^{n-i}\frac{\partial}{\partial \theta_i}

[/math]

 

So the first thing I noticed was an error I made in my “LaTex” expression. That first part of the above equations should be

[math]

\nabla^2=\sum_{i=0}^n\frac{\partial^2}{\partial x_i^2}=

[/math]

 

So the nabla stands for a sum of partial derivatives of all the variables. Considering your story about the expansion, I guess I should not stop and try to check it right now...

 

Oh, the algebra is not so bad if you know how to do differentials but it very quickly gets out of hand.

 

 

[math]

\theta_{i+1}= arccos\left(\frac{x_i}{\sqrt{\sum_{j=i}^n x_j^2}}\right)

[/math]

 

and

 

[math]r=\sqrt{\sum_{i=0}^n x_i^2}[/math]

 

 

 

But since [math]\nabla^2[/math] is defined by a sum over second partials with respect to [math]x_i[/math], we need to do some algebra if our geometry is expressed in these angular coordinates. Essentially, we have to convert partials with respect to [math]x_i[/math] into partials with respect to our spherical coordinates. One uses the following relationship to uncover the coefficients which must be attached to each of those partials with respect to our spherical coordinates.

[math]

\frac{\partial}{\partial x_i}F(spherical \; coordinates)= \left(\frac{\partial}{\partial r}F\right) \frac{\partial r}{\partial x_i}+\sum_j^{n-1}\left( \frac{\partial}{\partial \theta_j}F\right)\frac{\partial \theta}{\partial x_i}

[/math]

 

The terms we need are clearly the entire collection [math]\left(\frac{\partial r}{\partial x_i}\right)[/math] and [math]\left(\frac{\partial \theta_j}{\partial x_i}\right)[/math] which we can obtain from the above equations connecting the two coordinate systems.

 

Just start trying to write that out and you will quickly see what the problem is. Note, in order to get what you want you have to do that again on the first answer you get. It is just an algebra problem but it gets pretty hairy very quickly. Thank god most of the terms end up cancelling each other out.

 

One thing though, what does the symbol [imath]\prod_{l=1}^{i-1}[/imath] stand for? Also, I'm not sure what the "l" stands for in there.

 

The [math]\prod_{l=1}^{i-1}[/math] is the symbolic notation for a “product” just like [math]\sum_{i=1}^n[/math] is the symbolic notation for a sum. The collection of terms starts with l=1 and ends with l=i-1. “l” is used (supposed to be a small letter “L”) for the index on the product because the product is embedded in a term defined by “i” (the sum). Since “i” has already been specified, we need a new index for the product.

[math]

\prod_{l=1}^{i-1}csc^2\theta_l=(csc^2\theta_1)(csc^2\theta_2)(csc^2\theta_3)\cdots(csc^2\theta_{i-1})

[/math]

 

clearly, the number of terms in the product depends on exactly what i-1 happens to be.

 

Also;

 

[math]

\Psi=\frac{1}{\sqrt{r^n}}U(r)\prod_{i=1}^n\Phi_i(\theta_i)

[/math]

 

what does the "U" stand for here; exactly the same as what "g" did earlier?

 

Not at all. “U” is what I am using for the function of r which goes to make up [math]\Psi[/math]. The issue is that, because of the symmetries embedded in the spherical coordinate system, a general solution can be written as a sum over solutions where [math]\Psi[/math] can be written as a product of functions, each depending upon a different coordinate. That is a procedure called "separation of variables". It can't be done with general differential equations. It can be done here because all arguments are totally independent of one another (we are describing the position of a single point: i.e., it is a "one body" problem).

 

Have fun -- Dick

Edited May 29, 2014 by Doctordick

[AnssiH]

Sorry I've been so slow, just have had way too many things going on to be able to concentrate...

 

So the first thing I noticed was an error I made in my “LaTex” expression. That first part of the above equations should be

 

[math]

\nabla^2=\sum_{i=0}^n\frac{\partial^2}{\partial x_i^2}=

[/math]

 

 

Apparently I'm starting to become blind to some of those things. I noticed you did not fix it in the OP yet though.

 

Oh, the algebra is not so bad if you know how to do differentials but it very quickly gets out of hand.

 

...

 

Just start trying to write that out and you will quickly see what the problem is. Note, in order to get what you want you have to do that again on the first answer you get. It is just an algebra problem but it gets pretty hairy very quickly. Thank god most of the terms end up cancelling each other out.

 

Okay I think I see what you mean. And yeah, I would have to be much more fluent in algebra to make it feasible to actually verify that expansion... I'll take it on faith :)

 

But, more importantly, I am little fuzzy about the [imath]x_i[/imath] variables, let's make sure I'm thinking of the correct thing.

 

So for every point on that polyhedron, there exists an axis whose rotation yields no changes to the projected distribution of the points, except for moving that one single point through the entire distribution. And the [imath]x_i[/imath] is basically the projected position of that point?

 

I realize it would be possible to verify this from the relationships you just laid down, but I'm afraid my trigonometry and algebra is too weak for me to really do that..

 

-Anssi

[Doctordick]

Sorry I've been so slow, just have had way too many things going on to be able to concentrate...

 

Don't worry about it. We are taking off to Chicago after next week and will be gone for a while visiting relatives. My sister has gotten into family history so I am restoring some old photographs (from the middle 1800's). It is a time consuming job and I wanted to finish a few for the trip.

 

Apparently I'm starting to become blind to some of those things. I noticed you did not fix it in the OP yet though.

 

Thanks, I thought I had fixed it. I guess there were two, one I had fixed the other I missed.

 

Okay I think I see what you mean. And yeah, I would have to be much more fluent in algebra to make it feasible to actually verify that expansion... I'll take it on faith

 

I am pretty sure it is correct. I wouldn't worry about it if I were you. I only brought up the issue because I thought someone who had actually worked in the field would find what I had done interesting. No one has and I doubt anyone will. Even if you could follow the algebra, you could easily fail to see an error so, in the end you would be taking it on faith anyway. I think a quote from my original paper (written circa 1982) puts the issue quite clearly.

 

It should be clear, even to the uninitiated, that errors in the logical deductions only occur when an attack is newborn and are quickly eliminated by careful examination of those deductions. Errors in deduction are the easiest to eliminate and, in fact,seldom persist long enough to pervade the field. Certainly, if any idea survives long enough to be part of the body of knowledge passed from one generation to another, one can expect to find few if any errors in the deductions; too many people will have been led through those deductions to allow anything but extremely subtle errors to stand for long.

 

It should be clear that actually eliminating “errors in deduction” requires a reasonably large number of persons to be led through those deductions. Something which will certainly not occur with my discovery (at least not in my lifetime).

 

But, more importantly, I am little fuzzy about the [imath]x_i[/imath] variables, let's make sure I'm thinking of the correct thing.

 

The numerical label, [math]x_i[/math], is still just that; a numerical label specifying a specific element of the circumstance [math](x_1,x_2,\cdots,x_n,t)[/math]. We are simply viewing it as a value expressed as an argument of the function [math]\vec{\Psi}[/math] but seen as expressed in an n-dimensional representation. All this does is change the mathematical form of the fundamental equation. A form more amenable to solution as it converts a many-body problem into a one body problem. Really, don't worry about it.

 

So for every point on that polyhedron, there exists an axis whose rotation yields no changes to the projected distribution of the points, except for moving that one single point through the entire distribution. And the [imath]x_i[/imath] is basically the projected position of that point?

 

You are confusing the “Simple Geometric Proof” with this solution to the fundamental equation.

 

What I am talking about here is the general solution to the fundamental equations as seen in an n-dimensional representation. This has to do with solving a differential equation. In many ways, trying to convince you of the form of solution is pretty much a waste of time.

 

The only reason I brought it up is because of the form of the solution. It turns out to be essentially equivalent to the angular momentum solution to the three dimensional problem which yields quantized angular momentum. The only real difference here is that it yields an n-dimensional angular momentum which is essentially quantized in every dimension.

 

This means that the mathematical solution of the fundamental equation is equivalent to quantized rotation of an n-dimensional object. If that is indeed so then “Simple Geometric Proof” has to be a valid representation of how those [math]x_i[/math] change with time. That is what led me to attempt that proof. The two assertions are essentially equivalent.

 

If my solution to the fundamental equation is valid, then it must be possible to see possible distribution of elements in a three dimensional universe as the three dimensional projection of the orientation of an n-dimensional simple polyhedron: i.e., every possible distribution must be representable by a specific orientation of that polyhedron. In my head, I found to be a rather astounding fact and extremely hard to believe.

 

It was that very issue which led me to attempt that proof. If I could not prove it, it could be taken as a serious example of a flaw in my fundamental equation: i.e., a serious counter-example. (Don't worry about how I came to the conclusion the problems were equivalent; just believe me that they are.) At any rate, I am not going to bother with completing the solution to that differential equation. My solution will serve no purpose to your understanding and no one else is going to read it anyway.

 

Furthermore, in the final analysis, anyone familiar with solving second order linear differential equations would have no problem finding the solution if he wanted to (it's little more than standard first year graduate mathematics; perhaps even undergraduate now). At any rate, I think laying out the solution for you is pretty well a waste of time which serves no purpose. Both of us have better things to do with our time.

 

One issue I think is worthwhile to point out is the fact that, with regard to linear differential equations, any sum of solutions is also a solution. In the representation I have presented, the positions of all the “non-hypothetical” indivisible elements in a three dimensional observable universe can be seen as projections of the vertices of an n-dimensional simple polyhedron and all future positions can be seen as represented by the future orientations of that same polyhedron. Conservation of angular momentum (a fact derivable from the symmetry principals) yields an exact model of how the universe must evolve from its current motion.

 

However, since any orientation and any angular momentum is a solution to the equation, any sum of such solutions is also a solution. This is how the uncertainty of the current motion of the universe yields uncertainty in the future state. By the way, this also suggests that the dimensionality of that simple polyhedron is on the order of 10³⁰ orthogonal dimensions. That is not a small number.

 

Anyway, that is about the simplest model of the universe I have ever heard of and it apparently matches exactly all the physical experiments we can perform.

 

Have fun -- Dick

Edited May 29, 2014 by Doctordick

[AnssiH]

The numerical label, [math]x_i[/math], is still just that; a numerical label specifying a specific element of the circumstance [math](x_1,x_2,\cdots,x_n,t)[/math]. We are simply viewing it as a value expressed as an argument of the function [math]\vec{\Psi}[/math] but seen as expressed in an n-dimensional representation. All this does is change the mathematical form of the fundamental equation. A form more amenable to solution as it converts a many-body problem into a one body problem. Really, don't worry about it.

 

 

You are confusing the “Simple Geometric Proof” with this solution to the fundamental equation.

 

Ah yeah, I was looking at it as a continuation to the definitions of the Geometric Proof, that was causing some problems. Anyway, I see what you mean, about what this implies, and I'm perfectly willing to leave the verification of the math to better qualified.

 

-Anssi

[Qfwfq]

It should be clear that actually eliminating “errors in deduction” requires a reasonably large number of persons to be led through those deductions. Something which will certainly not occur with my discovery (at least not in my lifetime).

Especially if you place such high restrictions on who's critique counts.

[Doctordick]

Especially if you place such high restrictions on who's critique counts.

 

High restrictions??? All I ask is that they have some inkling of what I am talking about. You clearly don't. I have examined one issue and one issue alone: what are the constraints on an explanation which are implied by the definition of an explanation! Over and over, you keep bringing up constraints which have utterly nothing to do with that definition.

 

No one here has presented a definition of “an explanation” which is sufficiently explicit to work with and yet they all (except Anssi and Bombadil) insist mine is not acceptable. That is an idiotic response (I suspect due to their great desire to avoid thinking). My definition is quite simple!

 

I have defined what I mean by an explanation: it is a defined procedure (a method, an operation, an act of thinking or whatever you wish to call the means of getting from one to the other) which allows one to come to conclusions as to what is and is not expected (given that they believe that explanation is correct).

 

In my mind that is not a difficult concept to comprehend and I find it astonishing that everyone presumes it is invalid without giving it the slightest first thought (again, Anssi and Bombadil excepted).

 

A definition is an explanation of how to determine whether or not something belongs in the category being defined. If one wants to disagree with my definition, they need only give me something which they consider to be an explanation which yields no information on their expectations or something which does yield expectations which can not be categorized as an explanation. No one has ever even made the first attempt to do so. Why? Well I think it is because of “their great desire to avoid thinking”.

 

One thing absolutely required, to consider those implied constraints, is a notation which can represent any explanation without making any constraints whatsoever on exactly what that explanation is. I have done that.

 

My notation, [math](x_1,x_2,\cdots,x_n)[/math] where the x_i are nothing more than simple numerical labels labeling the fundamental elements (the underlying irreducible concepts required to understand the explanation) is clearly capable of representing any conceivable circumstance in any conceivable language. The possibility of order of those elements being significant can be removed via the introduction of another numerical index “t” such that order is of no significance (if order is significant, we can represent that order with different circumstances). Thus I come up with the notation [math](x_1,x_2,\cdots,x_n,t)[/math] which contains no constraints whatsoever. Show me anything you can represent which can not be represented by that notation. No one even begins to consider that issue at all.

 

Well, the final issue is to use that notation to specify an explanation. If the explanation can yield an estimate for the expectation of any conceivable circumstance (which is essentially my definition of an explanation) then any explanation can be represented by a collection of probabilities represented by the notation,

[math]

P(x_1,x_2,\cdots,x_n,t)

[/math]

 

Now that looks like a mathematical function! Well, all the x_i are no more than numerical labels of the significant fundamental elements necessary to represent that circumstance and probability is nothing except a number bounded by zero and one, so, guess what! The stupid thing is a mathematical function.

 

Everyone (accept, it seems, Anssi and Bombadil) are feverishly searching for reasons to avoid thinking along the lines I propose. Continually bringing up what they suppose to be “counter examples” to what they have decided I must be saying; all without putting the slightest thought into the actual issue I am trying to bring up.

 

All one needs to know, in order to understand the explanation being so represented, is the language from which those elements are derived. Anyone with any brains at all can comprehend that “understanding what is meant in a specific language” is the very essence of understanding an explanation of that language. And Qfwfq, in every argument you raise, you inevitably want to presume you already know the language being used to represent the explanation (among millions of other things).

 

It appears to me that every one of you are doing your greatest effort to discover the assumptions I am making, all the while putting billions of billions of assumptions you have made as a ground work basis of your efforts. The assumptions I am referring to here constitute your world-view, which is clearly an explanation of the universe you find yourself in; something you believe to be undoubtedly correct. I assert that I have made no assumptions and your search for those non-existent assumptions is the most useless waste of time imaginable.

 

Anyone with minimal intellectual sophistication knows that languages themselves place constraints on what can be expressed thus what I am talking about cannot be expressed in any language except via my notation. This is a fact because the results must include all possible explanations expressed in all possible languages (some not yet conceived of; and, with that comment, I refer to both languages and explanations).

 

It should be clear to everyone than, knowing the language, has absolutely nothing to do with the constraints on an explanation which are implied by the definition of an explanation!

 

I am tired of dealing with people who find thinking to be absolutely beyond their interest.

 

Have fun -- Dick

Edited May 29, 2014 by Doctordick

[Qfwfq]

you keep bringing up constraints which have utterly nothing to do with that definition.

Rubbish!

 

...and yet they all (except Anssi and Bombadil) insist mine is not acceptable.

Does this include me? What do you mean by "not acceptable"? Do you mean it isn't how most people put it? Where did I say any more than this about it?

 

(I suspect due to their great desire to avoid thinking).

Of course, you have a great excuse for all and any objections to your claims and, of course, it's never the reason for your disagreement with others...:rolleyes:

[Rade]

Claim of Doctordick.....I have defined what I mean by an explanation: it [an explanation]is a defined procedure (a method' date=' an operation, an act of thinking or whatever you wish to call the means of getting from one to the other) which allows one to come to conclusions as to what is and is not expected (given that they believe that explanation is correct)[/color']

I see a number of assumptions in this definition:

(1) it is assumed 100% of explanations begin with a procedure, a means of getting from one to the other,

(2) it is assumed 100% of explanations result in conclusions concerning expectations by both the person doing the explaining, and the person to which something is being explained,

(3) it is assumed 100% of explanations are believed to be 100% correct by both the person explaining and the person to which something is being explained.

 

Thus, this definition of explanation fails for any class of circumstances where any of the above (1,2,3) do not hold to be true either for some person explaining something or for some person to which something is being explained. One example where the definition logically fails is that the definition provided is not held to be 100% correct by 100% of people that read it, given that Doctordick can only provide the names of two people that so hold it to be 100% correct.

 

Second example. Mother to Son..explain to me your actions. Son....NO ! The definition of explanation given by Doctordick fails for the response. While there is a type of weak procedure, a shout by the Son, there is no expectation of a conclusion of understanding by either Mother or Son, nor a believe by the Mother that the explanation provided is correct for the circumstance that initiated the request for explanation.

[Doctordick]

I only write this because I noticed the absolute and total lack of intelligent response to this thread by anyone other than Anssi. Perhaps some of you might read it and think about it a bit. Regarding Rade's response, I never answered it before because it lacks thought of any kind. If anyone disagrees with me, I would like to hear your take on the issue.
 

I see a number of assumptions in this definition:
(1) it is assumed 100% of explanations begin with a procedure, a means of getting from one to the other,

 

Apparently Rade believes explanations stand on nothing more than outbursts of unsupportable rhetoric.
 

(2) it is assumed 100% of explanations result in conclusions concerning expectations by both the person doing the explaining, and the person to which something is being explained,

 

What? The person explaining something intends no conclusions to be embedded in that explanation! Then why does he feel any compulsion to explain it. What purpose does he have in mind. Is it just rhetoric with no purpose?
 

(3) it is assumed 100% of explanations are believed to be 100% correct by both the person explaining and the person to which something is being explained.

 

I never made any assertion as to what people believed and that is never an assumption behind my presentation. But, as an aside, I suspect that anyone who doesn't believe his own explanations must have some other motive regarding his act! A motive I myself would not hold in high respect.
 

One example where the definition logically fails is that the definition provided is not held to be 100% correct by 100% of people that read it, given that Doctordick can only provide the names of two people that so hold it to be 100% correct.

 

Where did I ever set forth that two people hold my presentation to be 100% correct. What I said is that, as far as I know, only two people seem to comprehend what I am talking about; a rather different perspective.
 

Second example. Mother to Son..explain to me your actions. Son....NO !

 

Now, exactly how many people (outside of Rade of course) would hold that the above “NO!” qualifies as an explanation. It sounds more like a refusal to explain to me.
 

The definition of explanation given by Doctordick fails for the response. While there is a type of weak procedure, a shout by the Son, there is no expectation of a conclusion of understanding by either Mother or Son, nor a believe by the Mother that the explanation provided is correct for the circumstance that initiated the request for explanation.

 

And well it should fail for such a response!!

Is the entire world populated by complete idiots?

Edited April 12, 2015 by Doctordick

[Doctordick]

I have been fixing the problems with the "LaTex" issues introduced by the new system used on the forum and felt this post was worth fixing.

 

Haver fun -- Dick

[Dumbass]

I admit I didn't read all the posts but why don't you compare your closed solution to a numerical one?

Also you might wanna look into p-adic analysis.