taeto

I could not figure out what \( [n] \) means. So maybe you are way ahead of me.

It is an interesting remark. In some cases I was asked to evaluate articles by eastern European authors, where it became immediately clear that they were not comfortable with writing in English. However, just around when there was significant issues of mathematical clarity being important, their language straightened up remarkably. Whereas in parts of the paper that seemed more motivational and explanatory, their grammar got horrific. In such cases it just made it obvious to check for previously published results that were similar or identical, and I never had a miss, and it turned out always plagiarized. In the end I assume that this is just something which would be harder for me to detect if the plagiarizer happens to be quite comfortable with writing in English.

Okay then: primes. Formally a natural number \(p\) is called "prime" if \(p > 1\) and it is true that if you take any integer numbers \(a\) and \(b,\) and \(p\) divides the product \(ab\), then \(p\) divides \(a\) or \(b.\) The first primes are \(2,3,5,7,11,13,\) and then there are some more larger ones as well. The "primorial" \(p\#\) of a prime \(p\) is the product \(2\times 3 \times 5 \times 7 \times 11 \times \cdots \times p\) of all the primes \(2,3,5,7,11,...,p.\) The "Euler totient function" \(\varphi\) is a more complicated function, but all you have to worry about is its value at primorial \(p\#,\) which is simply the product \( \varphi(p\#) = 1 \times 2 \times 4 \times 6 \times 10 \times \cdots \times (p-1)\) of all numbers less by 1 than a prime up to \(p.\) This is enough that you have to know, before you can meaningfully consider the Riemann Hypothesis. The Riemann Hypothesis is logically equivalent to the following statement: \[ \frac{p\#}{\varphi(p\#) \ln \ln p\#} > e^\gamma, \] for all primes \(p,\) where \(\gamma\) is known as the Euler-Mascheroni constant, with \[ \gamma := \int_1^\infty (\frac{1}{\lfloor x \rfloor} - \frac{1}{x}) dx \approx 0.577\ldots, \] where the value of the floor function \( \lfloor x \rfloor \) is the largest integer that is not greater than \(x.\) Reference: J. Nicolas, Petites valeurs de la fonction d’Euler, J. Number Theory 17, 3, 1983, pp. 375–388.

We can explain what a second is to you. If you are holding in your hand, while managing not to shake it very much, a caesium 133 atom which you observe being in its ground state, and you count 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of this ground state, then you have just experienced one second of time passed. Moreover, anybody or anything that stayed fixed in space with respect to you during the entire counting process has experienced exactly one second as well. Now just as you can measure the distance between different things using a meter stick, you can measure the time passed between different events using this device, which is commonly called a "clock".

It depends on the X. Maybe it was a very brilliant X. I admit I did not bother to read far enough into the OP to discover what it means. I trust that you investigated and determined that the X is actually kind of stupid. Note taken. Maybe he just has some questions about logarithms or RH. I will attempt to elucidate. The "natural logarithm" function \( \ln, \) or nowadays mostly written \( \log,\) can be conveniently defined by \[ \ln(x) := \int_1^x \frac{1}{t} dt \] for any positive real number \(x.\) Then \( \ln(1) = 0\) is obvious by definition, and using standard substitution of variables in the Riemann integral, you can easily deduce basic properties, such as \( \ln(ab) = \ln(a) + \ln(b) \) for all \( a,b > 0, \) as well as \( \ln(a^b) = b \ln(a), \) for all \(a > 0\) and all \(b.\) To formulate the Riemann Hypothesis in terms of logarithms it also helps to know about primes. I will wait for the OP to indicate whether this is the intention before I go on to explain about primes.

He said he wants to talk about it, not necessarily solve it, wtf.

What is that?

Yes. And somewhat unremarkably \( dy/dt = dy/dx \times dx/dt \) holds, under reasonable assumptions, so that in this case \[ da/dt = d( \sqrt[3]{345/n^2} )/dt = d( \sqrt[3]{345} \times n^{-2/3} )/dt = \] \[ \sqrt[3]{345}\times d( n^{-2/3} )/dn\times dn/dt = \sqrt[3]{345}\times -2n^{-5/3}/3\times 123 = -82\sqrt[3]{345}/\sqrt[3]{n^5}. \]

Mildly OT, but: what would be wrong with baseball without the catcher, now we are brainstorming on?

It becomes tough going to paraphrase: "The correct method in mathematics would be to say nothing except what can be said (...) -- he would not have the feeling that we were teaching him mathematics". It sounds interesting. But not exactly in the great ballpark of what I am asking for.

I like to engage mathematical cranks, occasionally, and mathematical cranks like to throw Wittgenstein back at you. But when I hear what things he gets quoted for, and besides which I am not an expert who knows every quote in advance, then I just think that, well, this guy is famous and supposed to be knowledgeable about, uh, some things, in particular having done a lot of thinking about mathematics. But how is it possible that he could have misunderstood so badly how mathematics functions and what mathematicians try to do? So I just want to ask in this forum, what is your personal favorite quote, opinion or observation about mathematics, which was astute, on the point and insightful, and made by Wittgenstein?

I tried various substitutions, and checking with tables of known integrals. Unfortunately I see nothing to simplify things further, everything I try just seems to make it more ugly. It is not unlike my past history with potted plants. I hope you can do something with it. Is it unfair to ask what kind of research this is related to? It looks a tiny bit cosmological, with the k and the r and the angles.

So in that case you really only need the definite integral \( \int_0^t \) where \(0 \leq t < \pi/2. \) Because \( \cos x \) is symmetric around \(x =0,\) and the only way you can avoid an imaginary root is to stop some time before \( \pi/2, \) depending on the constants. By the trick of moving \(\sqrt{A}\) outside the integration, as in the "easy" special case, it seems enough to know the value of \( F_a(t) := \int_0^t \sqrt{1 - a\sec^2 x} dx \) for \(a \geq 0 \) and \(0 \leq t < \pi/2.\) It certainly seems to open it up nicely for a numerical approach. Whether it helps to get an analytical solution, I don't see that yet.

Do you assume anything about the values of A and B? E.g. if B is positive you have an imaginary integrand, so the question is if you are happy with a final answer that is not real? We can look at the very special case \( A = B\frac{k^2}{r^2}: \) \[ \int \sqrt{ A - B(\frac{k}{r\cos x})^2 } dx = \int \sqrt{ A - A(1/\cos x)^2 } dx = \sqrt{A} \int \sqrt{1 - (1/\cos x)^2 } dx = \] \[ \sqrt{A} \int \sqrt{\frac{\cos^2x - 1}{\cos^2x}} dx = \sqrt{A} \int \sqrt{- \frac{\sin^2x}{\cos^2x}} dx = \sqrt{-A} \int \sqrt{\tan^2x} dx = \sqrt{-A} \int | \tan x | dx. \] The last indefinite integral does not appear standard. But its definite versions can calculated easily by splitting up into intervals with negative and positive values of the tangent function. So if I didn't make a silly mistake, this could be the most benign of the results you should expect.

Nice. But you should use \(\sqrt{x^2} = |x|\) instead of \(\sqrt{x^2} = x.\)

The engineering way to do it is to say: if you have your number x = 1/2 and your other number y = 1/32, and you wonder how to compute a number a for which x^a = y, then you just have to grab your calculator and punch in (log y)/(log x) to get the answer (approximately). That skips some amount of mathematics which is involved. Especially if you are not familiar with how logarithms work. Logarithms have the property that if you want to compute a product xz of two numbers x and z, but the multiplication looks hard, then you can instead compute log(x) and log(z), and then the sum y = log(x) + log(z) will have the property log(xz) = y, so you can deduce the answer by looking for a number w such that log(w) = y. It seems a roundabout method, but some people can do this very routinely by the use of slide rules. A slide rule will allow you to find log(x) and log(z) quickly, and to add their values, and it allows you to quickly read off the value of w as well. More generally, logarithms have the property log(x^n) = n log(x), i.e, if you multiply x by itself n times, then the logarithm of the answer has to be what you get by adding the log(x) to itself n times. So if x^n = y, then n log(x) = log (y), and you immediately get n = log(y)/log(x).

If you have not already done so, you can first go and read retractionwatch.com/2013/11/25/want-to-report-a-case-of-plagiarism-heres-how/

You beat me to it. It just occurred to me how physicists like to say "neutrinoes have finite mass" to mean that they have positive rest mass. Some particle has "finite radius" meaning positive radius. And so on. It is almost like mass ought to be measured in 1/kg and sizes of things in 1/m. Similarly to how the charge of an electron really should be positive instead of negative, given the direction in which electricity actually flows. But as opposed to mathematics, where things intrinsically are allowed to become infinite, that is not usually so in physics. In mathematics, "finite" rightly means the opposite of "infinite", and the dichotomy does not apply to physics, because then it is just a matter of choice of units.

It is hard to come up with any generally applicable definition by which zero should not be finite. Dedekind: 0 (understood as the empty set, so that the definition applies) does not allow an injection into a proper subset. Hence 0 is finite. Russell: 0 has a bijection to a set {1,2,...,n} for a natural number n, namely for n = 0 and empty bijection. Hence 0 is finite. All well-orderings of 0 are isomorphic. This implies 0 is finite. Every non-empty family of subsets of 0 has an inclusionwise minimal element (Tarski). Etc. There are strange sets that are Dedekind-finite but Kuratowski-infinite. 0 is not one of them, since 0 is Kuratowski-finite by definition. I prefer Russell myself, probably because I come from Combinatorics. For us it is important not only to know whether some things exists or not, but also to count the number of them. An expression like n^m for natural numbers n and m means, by definition, the number of different functions from a set of m elements to a set of n elements. In particular you are quite aware that 0^0 = 1 means that there is the unique function \( \emptyset \) from the empty set to the empty set, so Russell's definition applies nicely.

Or it crunches back, depending on the value of lambda, no?

I think that I wrote it wrong to you, by missing that you said that isotropy is your assumption. Clearly the torus is anisotropic, just from the description of what happens if you observe different directions. My idea is better explained by suggesting that the Universe is flat and finite, but very large, so much that we cannot detect anisotropy by looking at the observable part. It seems quite reasonable to say that an absolutely flat and isotropic Universe has to be infinite. Interesting.

That does not follow. If the universe is a 3D torus, then you might be able to look in some direction and see yourself from the back, and if you look to your straight left, you might see the right side of your body. Or look straight up, and probably get dizzy. The same things repeat in all directions, and in total the universe has only finite content.

Compare with boxing, and here I suppose that I am thinking of the classical sport, where you are not allowed to kick our opponent in the face, but in principle it does not matter a lot. You cannot deny that it has excitement, just look at the crowds that the big fights gather, and the interest from the media. And yet, the decision of the fight typically comes down to just one single deciding hit. The excitement is all about that anything could happen in the next second to decide the outcome.

A forum for "Theoretical Computer Science" would be great for me personally. The current "Computer Science" forum is more like a "Computer Repairs" (not wanting to be facetious, though maybe a little bit) item.

Volume?