KJW

Is this connected to Fermat's little theorem?

Because deceleration is an acceleration in the opposite direction. To think of acceleration as increasing speed and deceleration as decreasing speed indicates that you are thinking in terms of some absolute velocity, whereas by changing to some other inertial frame of reference, it becomes clear that acceleration and deceleration are actually the same notion.

This seems confused or at least confusing to me. The only way I can make any sense of it at all is if it is describing a multiverse. Is it describing a multiverse?

There is also the uncanny valley to consider.

What does a typical five-year-old know about politics? However, it wasn't until 1962 that Aboriginal and Torres Strait Islander people were granted the right to vote in Australian federal elections (voting became compulsory for them in 1984).

Yes it does. Thank you. I wasn't aware of the technique of "partitioning". My own exploration of the case of 4 x 4 matrices being partitioned into 2 x 2 matrices of 2 x 2 matrices indicated that multiplication produces the same result. This exploration appeared to indicate that partitioning mn x mn matrices into m x m matrices of n x n matrices would also work although I didn't formulate a proof. However, the partitioning of matrices into differently sized matrices given in the textbook was unanticipated.

One question that has interested me is whether a 4 x 4 matrix of scalars can be treated as a 2 x 2 matrix of 2 x 2 matrices with multiplication producing the same result in both cases: [math]\begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24}\\ a_{31} & a_{32} & a_{33} & a_{34}\\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix} \equiv \begin{pmatrix} \begin{pmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{pmatrix} & \begin{pmatrix} a_{13} & a_{14}\\ a_{23} & a_{24} \end{pmatrix} \\ \begin{pmatrix} a_{31} & a_{32}\\ a_{41} & a_{42} \end{pmatrix} & \begin{pmatrix} a_{33} & a_{34}\\ a_{43} & a_{44} \end{pmatrix} \end{pmatrix}[/math]

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Who is "we"? For Australian citizens aged 18 and over, voting is compulsory. And those who didn't vote in an election will receive a "why didn't you vote" form in the mail which must be filled out with an acceptable reason and returned in order to avoid a fine.

About 30 years ago, I experienced something quite extraordinary: I got glasses to correct my short-sightedness. The new clarity of vision was like being reborn. But everywhere I looked, it was like I was in a depression, with the ground and other horizontal surfaces sloping upward away from me. Then after about six months, I noticed that I was no longer in a depression, the ground and other surfaces were flat again. I didn't notice my sight correcting itself over the time. I had simply forgotten about the distorted vision until I suddenly noticed that the distortion was gone. It's extraordinary because it raises the question of how the brain managed to correct the distorted perception.

I don't know if it's mainstream, but it does imply that the laws of physics are arbitrary as if given to us from above. On the other hand, if the laws of physics are purely logical (or mathematical), then there is only one set of laws across all universes. When one considers a multiverse of all possible universes, it's important to note that it's all possible universes. It doesn't include impossible universes. Therefore, one can't simply say that the multiverse contains representatives from each side of every dichotomy as if everything is a possibility. The notion that life is improbable is based on mathematical logic, and therefore would be uniform across all universes. The anthropic principle only applies to our existence, it says nothing about the existence of other lifeforms elsewhere in the universe. There may be universes with many inhibited planets, beyond probabilistic likelihood, but these are unlikely, and it is more likely that our universe is not one of them. Or it could be that your premise is wrong, and that life really is rare in the universe, with our existence and only our existence guaranteed by the anthropic principle in a multiverse.

No, it's not the accelerations at the start and finish of the journey that's responsible, but the acceleration in the middle corresponding to the travelling twin's turnaround. An interesting approach to the twin paradox is to consider not time dilation but Doppler shifts. While each twin is seen to be moving farther from the other, they appear redshifted, but while each twin is seen to be moving closer to the other, they appear blueshifted. However, the two twins do not see each other symmetrically. For the stay-at-home twin observing the travelling twin, the change from redshift to blueshift occurs after the light from the travelling twin's turnaround has reached the stay-at-home twin. But for the travelling twin observing the stay-at-home twin, the change from redshift to blueshift occurs immediately at the travelling twin's turnaround. Thus, the travelling twin sees the redshift and blueshift of the stay-at-home twin for equal times, whereas the stay-at-home twin sees the redshift of the travelling twin for a longer time than the blueshift.

How are functions represented? Suppose one has an equation of the form: [math]x_{n} = f(x_{1}, x_{2}, ..., x_{n-2}, x_{n-1})[/math] In this case, one has [math]n[/math] variables parameterizing an [math]n[/math]-dimensional space in which the equation is describing an embedded [math](n-1)[/math]-dimensional space. Such an equation could also be expressed in implicit form: [math]g(x_{1}, x_{2}, ..., x_{n-1}, x_{n}) = 0[/math] However, suppose one has [math]m[/math] independent simultaneous equations: [math]g_{1}(x_{1}, x_{2}, ..., x_{n-1}, x_{n}) = 0[/math] [math]g_{2}(x_{1}, x_{2}, ..., x_{n-1}, x_{n}) = 0[/math] [math]...[/math] [math]g_{m-1}(x_{1}, x_{2}, ..., x_{n-1}, x_{n}) = 0[/math] [math]g_{m}(x_{1}, x_{2}, ..., x_{n-1}, x_{n}) = 0[/math] Then in this case, the [math]n[/math]-dimensional space contains an embedded [math](n-m)[/math]-dimensional space. That is, the number of dimensions of the embedded space described by the set of equations is equal to the total number of variables minus the total number of equations (assumed to be independent). Each equation is a constraint on the values of the variables that satisfy the equations, and each constraint reduces by one the dimension of the space formed by the values of the variables that satisfy the equations. Another way to describe a [math]d[/math]-dimensional space embedded in an [math]n[/math]-dimensional space is to introduce [math]d[/math] additional variables that parameterize the [math]d[/math]-dimensional space. Then the [math]d[/math]-dimensional space is described by the [math]n[/math] equations: [math]x_{1} = h_{1}(t_{1}, t_{2}, ..., t_{d-1}, t_{d})[/math] [math]x_{2} = h_{2}(t_{1}, t_{2}, ..., t_{d-1}, t_{d})[/math] [math]...[/math] [math]x_{n-1} = h_{n-1}(t_{1}, t_{2}, ..., t_{d-1}, t_{d})[/math] [math]x_{n} = h_{n}(t_{1}, t_{2}, ..., t_{d-1}, t_{d})[/math] Note that there are [math]n+d[/math] variables and [math]n[/math] equations describing a [math]d[/math]-dimensional space. In the case of a one-dimensional curve in an [math]n[/math]-dimensional space, one has: [math]x_{1} = h_{1}(t)[/math] [math]x_{2} = h_{2}(t)[/math] [math]...[/math] [math]x_{n-1} = h_{n-1}(t)[/math] [math]x_{n} = h_{n}(t)[/math] Therefore: [math]t = h^{-1}_{n}(x_{n})[/math] And eliminating [math]t[/math] from the remaining [math]n-1[/math] equations: [math]x_{1} = h_{1}(h^{-1}_{n}(x_{n}))[/math] [math]x_{2} = h_{2}(h^{-1}_{n}(x_{n}))[/math] [math]...[/math] [math]x_{n-2} = h_{n-2}(h^{-1}_{n}(x_{n}))[/math] [math]x_{n-1} = h_{n-1}(h^{-1}_{n}(x_{n}))[/math] Note also that: [math]h^{-1}_{1}(x_{1}) = t[/math] [math]h^{-1}_{2}(x_{2}) = t[/math] [math]...[/math] [math]h^{-1}_{n-1}(x_{n-1}) = t[/math] [math]h^{-1}_{n}(x_{n}) = t[/math] And therefore: [math]h^{-1}_{1}(x_{1}) = h^{-1}_{2}(x_{2}) =\ ...\ = h^{-1}_{n-1}(x_{n-1}) = h^{-1}_{n}(x_{n})[/math]

I didn't ask for 45 pages of explanation. I asked specifically about the fine structure constant. This is a number, and explaining it would amount to calculating it from first principles, not just in terms of the dimensioned physical constants that define it. Note that the fine structure constant is the square of the elementary charge expressed in dimensionless form, so explaining it is equivalent to explaining why the elementary charge has the value it does. BTW, I didn't suggest that you broke any rules. You did supply enough text about your theory to get the ball rolling. But now you need to supply text about requested specifics to continue the discussion.

The forum rules require that one be able to discuss your theory without downloading a file or visiting an external website. So, I will ask you to elaborate on your claim about successfully explaining the fine structure constant.

And Jehovah's Witnesses do not vote, not even in countries where voting is compulsory.

It should be remarked that a rotating superfluid behaves differently to other rotating fluids, exhibiting quantized vortices.

I don't agree. For example, it was a conservative Supreme Court that recently overturned Roe v. Wade. Yes, there is that. But what it means is that Christianity doesn't align itself properly along left/right political lines. For one thing, there is this whole "obedience to God" aspect of Christianity that does not sit well with the left. For example, the members of science forums tend to be leftist, but science forums are rather unfriendly places for anyone promoting Christianity.

No. However, I shall defer to Wikipedia on how neurons function.

As I see it, the left tends to be atheistic or at least secular in attitude. Also, leftist liberalism tends to be viewed as decadent and against the word of God by Christians. Furthermore, communist regimes tend to ban religions, regarding them as the "opium of the people".

As swansont said, not really an earth-shattering observation here. Indeed, there is a term that covers the notion you are describing: solipsism.

I think one needs to remember that Newtonian gravitation only contributes half of any gravitational lensing effects.

I think, you don't understand the numbers... I think you don't understand what I said.

I think billionaires is the new millionaires.

I very recently watched a documentary in which this was mentioned. I was not aware of it prior to this. However, it seems to me that the notion of something preventing supermassive black holes from merging is not consistent with general relativity. According to general relativity, if one has a system containing an arbitrary arrangement of black holes, an increase of the masses of all the black holes by a constant factor, as well as an increase of all the distances and times by the same factor will result in the system's behaviour being unchanged. For example, if one has two black holes of mass equal to ten million solar masses in mutual circular orbit ten million astronomical units apart, then the amount of time required for the orbit to decay by emission of gravitational radiation to five million astronomical units apart will be one million times as long as two black holes of ten solar masses in mutual circular orbit ten astronomical units apart decaying by emission of gravitational radiation to five astronomical units apart. Yes, the supermassive black holes will take a lot longer to merge, but there isn't anything other than time itself preventing the merge. However, the above does assume a background spacetime that is flat. In a de Sitter or anti-de Sitter background spacetime, the scaling of the cosmological constant is different and will lead to larger black holes behaving differently to smaller black holes.