Symmetery of a finite sequence of numbers

[Paul Mackenzie]

Hi All;

I attach a pdf file on something I have been working on for some time.
Any feedback would be appreciated.

Regards

Garbagebin

 

[SteveL27]

Hi All;

I attach a pdf file on something I have been working on for some time.
Any feedback would be appreciated.

Regards

Garbagebin

In your Intro, it would be incredibly helpful to say what you are planning to discuss or prove. Instead, you dive right into your presentation, leaving the reader disoriented.

Your definition of f[x] is nonstandard function notation and is incoherent besides. You really need to clean it up so that it makes sense.

At the end of the Intro you say, "zero is not even number" which is of course completely false and shows a lack of understanding of the most basic facts of arithmetic.

Then you go on with some equations for a while, and then you mention Goldbach's conjecture, and then you have some pretty multi-colored graphs. But there doesn't seem to be a conclusion to any of this.

I'd call it cranky, but it's not even that. Overall, incoherent.

I would suggest starting with the Intro and making your notation and intentions much more clear.

 

[Norwegian]

Hi, I found the pictures at the end very nice and interesting. I have never seen this Goldbach comet before, and your scaled version following it shows even clearer the "good" values of n, and the structural bands of the comet.

(The goldbach comet plots the number of prime pairs adding to n, while the 2nd picture plots the ratio of primes belonging to such a pair.)

I presume those n having many small different prime factors will give a high valued point, so the "local max" around 30k must be the product of primes up to 13. Then I guess that second floor of points are for those N divisible by 6, and the bottom floor those who are not. Looking at this comet of course begs the question, how come nobody has yet proven Goldbach?

Back to the paper: should be more structured, with an introduction including a summary. Your mathematical language is not very precise, and you will lose many readers there. Then, the statement that zero is not even, was surprising, but I continued hoping for a "proof" of goldbach or something. That was not to come, but instead a (de)tour into Fourier analysis, where a statement about the symmetric properties of a sequence were translated into a statement about real/imaginary parts of a transformed sequence. No indication as to why this would be useful in relation to the rest of your paper was given. Instead we jump to a different and imho more fun subject, prime numbers and goldbach partitions, and you give us your nice pictures.

 

[Paul Mackenzie]

Hi, I found the pictures at the end very nice and interesting. I have never seen this Goldbach comet before, and your scaled version following it shows even clearer the "good" values of n, and the structural bands of the comet.

(The goldbach comet plots the number of prime pairs adding to n, while the 2nd picture plots the ratio of primes belonging to such a pair.)

I presume those n having many small different prime factors will give a high valued point, so the "local max" around 30k must be the product of primes up to 13. Then I guess that second floor of points are for those N divisible by 6, and the bottom floor those who are not. Looking at this comet of course begs the question, how come nobody has yet proven Goldbach?

Back to the paper: should be more structured, with an introduction including a summary. Your mathematical language is not very precise, and you will lose many readers there. Then, the statement that zero is not even, was surprising, but I continued hoping for a "proof" of goldbach or something. That was not to come, but instead a (de)tour into Fourier analysis, where a statement about the symmetric properties of a sequence were translated into a statement about real/imaginary parts of a transformed sequence. No indication as to why this would be useful in relation to the rest of your paper was given. Instead we jump to a different and imho more fun subject, prime numbers and goldbach partitions, and you give us your nice pictures.

Thank you and steve for your comments.

I made a big error concerning zero and even numbers, and have corrected that.

As you said the notation concerning f[x] needs some clarification which I will attend.

I will add a more thorough introduction and conclusion.

I was not really trying to prove anyhing, just discussing symmetrical properties of sequences.

In particular how the number of goldbach partitions is related to the symmetry of a prime number sequence, and how it's symmetry, viz the ratio g[2N]/π[2N] approaches zero as 2N approaches infinity.

And how the number of goldbach partitions g[2N] is related to the Fourier transform of the prime number sequence f(x), where f[x] =1 when x is prime and f[x] =0 when x is not prime

In particular how the number of goldbach partitions g[2N] equals
g[2N] = 1/2N * Ʃ(Re[F[l])^2 - (Im[F[l])^2

where F[l] is the Fourier transform of f(x) the prime number sequence.

Initially I started out working on goldbach's conjecture in the Fourier domain, without much success. Then I realized goldbach conjecture forms part of a more general topic, viz symmetry of sequences.

As to your comments concerning the factors, I did try to introduce this into the Fourier domain but without any success. But as aside, the zero and nth harmonic of F[l] equal π[2N] and -π(2N) repectivley and as such their contribution to the goldbach partitions is then 1/2N*(2*π[2N]^2) = 1/N*π^2(2N). I ploted this function on my goldbach comet graph and it goes right down the middle. It seems the contribution to the number of goldbach partitions made by the AC harmonics are either negative or postitve depending on the factors of the even number.

I have some source code which computes the number of goldbach partitions from the Fourier transform of the prime number sequence. If anyone wants a copy please do ask.

Kind Regards

 

[SteveL27]

I have some source code which computes the number of goldbach partitions from the Fourier transform of the prime number sequence. If anyone wants a copy please do ask.

Can you back up to the beginning and help me to understand your notation?

What's the sequence for the even numbers? What's the sequence for the primes? I think if you can simply show a couple of examples, we can help sort out the notational issues.

It's not possible for anyone (well, for me, anyway) to follow your argument since the definition of f[x] was so garbled. I get that to each term of a sequence you assign a pair of numbers, but I'm unclear on what those numbers are. Then you start talking about 2N, sort of out of the blue, without defining N. And if you're not sure whether 0 is an even number, let's talk about that.

In other words, let's nail down the Intro before going forward.

 

[Paul Mackenzie]

Can you back up to the beginning and help me to understand your notation?

What's the sequence for the even numbers? What's the sequence for the primes? I think if you can simply show a couple of examples, we can help sort out the notational issues.

It's not possible for anyone (well, for me, anyway) to follow your argument since the definition of f[x] was so garbled. I get that to each term of a sequence you assign a pair of numbers, but I'm unclear on what those numbers are. Then you start talking about 2N, sort of out of the blue, without defining N. And if you're not sure whether 0 is an even number, let's talk about that.

In other words, let's nail down the Intro before going forward.

Hi Steve and all:

Thanks for your help regarding the notation.



To understand where I coming from, I will describe f[x]
with reference to the prime number sequence {2,3,5,7,11,13}.

In this example the sequence is limited to all primes less
than 16 (viz 2N = 16).

I attach a pdf file which shows the functions f(x), f(2N-x),
and f(x).f(2N-x) where 2N = 16 and
where f(x) = 1 when x is prime
f(x) = 0 when x is otherwise;
and similarly f(2N-x) = 1 when 2N-x is prime
f(2N-x) = 0 when 2N-x is otherwise

The function f(x).f(2N-x) is one only when both x and 2N-x are
both prime so this function illustrates the goldbach partitions
of the even number 16. viz (3,13)(5,11)(11,5)(13,3).

The number of Goldbach partitions for 2N = 16
is then the sum of f(x).f(2N-x) from x = 0 through to x = 15 (2N-1),
which in this case four.

Looking at this problem you can see that figs. 1 and 2
are mirror images of each other about N = 8.

Furthermore Fig. 3 shows where these functions shown in
Fig. 1 and 2 are symmetrical.

The value 2N comes about so as to limit the prime number sequence.
So if 2N = 24 then the prime number sequence includes all primes
upto 24. I choose an even 2N because we are concerned with mirror
symmetry. Furthermore the sum of f(x).f(2N-x) from x = 0 through to 2N-1
will give you the number of goldbach partitions for that even number 2N.


So as to compare a value which is representative of the
symmetry for different values of 2N, I normalised this
sum to get a symmetry value I. The normalising value is
the sum of f(x).f(x) from x = 0 through to x = 15 (2N-1)

So the symmetry value I is the sum of f(x).f(2N-x)/ sum of f(x).f(x) from x = 0
through to x = 15 (2N-1)

which in this example I = 4/6. [Note the demoninator equals pi[2N=16] = 6]

As mentioned in my post, this symmetry value I for prime number sequences upto 2n
is then equal to g(2n)/pi(2n). I plotted this value for even numbers upto 80,000
and it seemed as though g(2n)/pi(2n) approaches zero as 2n appeoaches infinity

I realized this problem can be generalised for many integer sequences to obtain a symmetry
value [ with some restrictions]. I realized for example you could not apply this technique
to Fibonacci sequence. i.e. {0,1,1,2,3,5,...} because two elements of the sequence are
equal, viz the second and third element "1". It seems that's where the confusion arises,
when I tried to deal with this.

I also gave an example regarding even numbers which probably does not correctly describe
the function f[x]. But I hope my depiction of the functions in these diagrams will assist
in your understanding. You are correct that my statement that "zero is not a even number"
was erroneous. I have removed it.


I think this generalisation could be applied to non-integer values and maybe some
real world applications. For instance you could take a finite seqment of speech, sample it,
reverse it, and multiply the sampled signal and reversed sampled signal together and divide it by the
power of the signal to obtain a value for its symmetry. But this requires a lot more work.

I hope these diagrams help with your understanding

Kind Regards

 

[SteveL27]

Hi Steve and all:

Thanks for your help regarding the notation.
To understand where I coming from, I will describe f[x]
with reference to the prime number sequence {2,3,5,7,11,13}.

In this example the sequence is limited to all primes less
than 16 (viz 2N = 16).

I attach a pdf file which shows the functions f(x), f(2N-x),
and f(x).f(2N-x) where 2N = 16 and
where f(x) = 1 when x is prime

I'm just asking you to define f clearly. It shouldn't take more than two or three lines. Tell me its domain, tell me its range, tell me what it does to a typical input value. I don't know how else to say this. Just tell us what f is. Two or three short lines. All the rest of this exposition is running before you've caught the ball. Just define f. That's step 1. Beyond that I can't offer any more advice.

 

[willem2]

I'm just asking you to define f clearly. It shouldn't take more than two or three lines. Tell me its domain, tell me its range, tell me what it does to a typical input value. I don't know how else to say this. Just tell us what f is. Two or three short lines. All the rest of this exposition is running before you've caught the ball. Just define f. That's step 1. Beyond that I can't offer any more advice.

Right at the start of the pdf:

For the purposes of this post, a sequence of integers is represented in the x-y Cartesian coordinate
system in the following form.
f [x]=1 if x is a member of the sequence, otherwise
f [x]= 0 , for all x = 0,1,…, 2N.

The domain of f is obviously meant to be the integers from 0 to 2N, and it would
be nice to mention what the numbers in the sequence are first.
I also would drop all references to "the x-y cartesian coordinate system" here and in
the rest of the pdf.

 

[Norwegian]

I had no trouble understanding the first page of the write-up as follows:

Let N be a positive integer, let D={0,1,2,...,2N}, and let S=s₀,s₁,...,s_m be a strictly increasing sequence of elements from D. Let f:D--> {0,1} have the values 1 for numbers in S, and 0 otherwise.

In fairness to the author, we really shouldn't ask him to further specify S or f, as that is unambigous from the paper. I did find the use of the f-notation very practical when defining the symmetry measure I(S), and even more when taking the Fourier transforms. We may criticize the immature manner in which the author expresses all of this, and question the usefulness of involving the Fourier transforms. But I hope further comments can be about the contents of the remaining paper, and not just from people pretending not to understand its first page.

 

[SteveL27]

But I hope further comments can be about the contents of the remaining paper, and not just from people pretending not to understand its first page.

I was confused by the author's example:


As an example consider the sequence S’ of all even numbers up to 2N. The function
sequence f [x]=1 if x is a member of this sequence S’, otherwise f [x]= 0 , for all x =
0,1,…, 2N. becomes {(0,0)(1,0),(2,1),(3,0),(4,1)(5,0),(6,1)…..,(2N-1,0), (2N,1)} when
expressed in x-y Cartesian co-ordinates.

In other words he really did mean to map onto pairs of numbers and not just numbers. My feedback on the lack of clarity is valid, even if it is in fact possible to figure out what he really means. Apologies if I'm the only one who considered the exposition unclear.