taeto

Dear това́рищ Dima, thank you for your praises! Unfortunately I suspect that such a special arc cannot exist. Referring to my diagram in the preceding post, I have tried to look at the case of the two lines defined by \(t = \pi/6\) and \(t=\pi/3.\) I calculate that they intersect on the \(y\)-axis only if \[\frac{\alpha}{\pi/2} = 1 - \frac{4(\sqrt{3} -1)\cos \alpha}{6(2-\sqrt{3})\sin \alpha +3}.\] (If you prefer to calculate in degrees instead of radians, the only change is to replace \(\pi/2\) by \(90.\) ) But when I plug some values of \(\alpha\) into this, it always seems that the expression on the right side is a larger number than the left side value.

Being non-physicist this is puzzling to me. What is a mathematical theory in which lines have radius? By default a line is a certain geometrical object, and in geometry it is a circle that has a radius, whereas a line does not. The notion of stiffness seems to relate to a rigidity notion, though I have not seen it as a defined concept anywhere. Why is 'stiffness' important enough to explicitly exclude as a relevant property? You did not add "without colour". It is superfluous to say that lines have no physical properties. They are abstract objects. To 'fluctuate' only makes sense for systems for which you have a number or other measure to assign to their size. A single object does not fluctuate. But if you speak of any kind of time dependence here, then it contradicts the first part of your description, which speaks of time independent entities. If you say 'randomly', your description to be complete has to provide information about a relevant (type of) probability distribution.

Now that particles no longer exist in the real world, but the term 'particle' continues to be used in physical models to describe certain systems that have mass and volume, etc., then maybe we can simply scrap the old-fashioned use of the word particle, and instead begin to use the word for physical objects that have particle-like properties? 🙂

Sorry to say, this kind of thing always irks me a bit. You are absolutely right to say that d\(S\) is a function of four independent variables, since \(S\) itself is defined as a function with Minkowski space as its domain. But then the OP goes on to say that d\(S\) is defined separately in some special way. That does not make sense. Once \(S\) is defined, then d\(S\) is supposed to be defined as well as the differential of \(S,\) you do not get to make a separate definition, because if you begin to do that, then everything gets left in absolute confusion. I admit that I did not bother to check whether the OP's auxiliary definition of d\(S\) does agree with the usual definition. If it does, he should not state it like it is a new definition. End of rant.

Then let us say that the arc has length \(\pi - 2\alpha\) and is placed symmetrically around the \(y\)-axis as in this diagram. Then the arc itself is the segment of the circle between the points \((\cos \alpha,\sin \alpha)\) and \((-\cos \alpha,\sin \alpha)\) in the upper half of \(\mathbb{R}^2.\) And its chord is the horizontal line segment connecting its endpoints as shown. If we take a random point \(p=(\sin t,\cos t)\) in the upper right quadrant, i.e. the point for which its distance to the \(y\)-axis along the arc is \(t,\) with \(0 < t < \pi/2,\) then the line defined by this point is the line \(L\) through \(p\) and the point \(q=(s,\sin \alpha)\) on the chord, such that the ratio \(s/\cos \alpha\) of the distance from the \(y\)-axis to \(q\) equals the ratio \(t/(\pi/2-\alpha)\) of the distance along the arc between the \(y\)-axis and \(p\) itself to the entire piece of arc in the upper right quadrant. So \(s = \frac{\cos \alpha}{\pi/2-\alpha}t\) follows. The equation for \(L\) becomes \(L : y = \frac{\cos t - \sin \alpha}{\sin t -s}(x - s) + \sin \alpha,\) which is checked by setting \(x = s,\sin t.\) Clearly the \(y\)-intercept of \(L\) is the point \((0,-\frac{\cos t - \sin \alpha}{\sin t - s}s+\sin \alpha)\) by setting \(x=0\) in the equation for \(L.\) If this is correct, then the question seems to be whether there exists a value of \(\alpha\) with \(0 < \alpha < \pi/2\) for which this intersection point does not depend on \(t.\)

Your diagram shows a sensible interpretation of Dima's construction. But it does not seem to say anywhere, nor is it made clear from the diagram, whether the lines are supposed to intersect in a common point. That part may have been something that I inadvertently invented.

If my interpretation is correct, then it is not possible that all the lines will intersect in a single point. If they did, the common intersection must lie on the line \(L\) defined by the midpoint \(m\) of the arc and the midpoint of its chord, which is the symmetry axis of the whole setup. If you compute how far down along \(L\) you find the intersection with another line \(\ell\), then you get an expression that contains nonzero terms with \(\cos t\) and \(\sin t,\) where \(t\) is the distance along the arc from \(m\) to the point on the arc that defines \(\ell.\) As well as it contains other rational nonzero terms that contain terms like \(t\) in their denominator. It is not difficult to see that if you differentiate w.r.t. \(t,\) then the derivative still contains such terms. There is no way that they may cancel each other out, hence the derivative cannot vanish. This means that the intersection points move around on \(L\) as \(t\) varies, in particular they are different for different pairs of lines. For the argument to be completely convincing, it would be better to calculate the precise expressions. But they get pretty complicated.

Induction, absolutely. But even that will not work unless the statement is true. And in a case like \(a_0=a_1=a_2=k=l=m=0\) the inverse of the determinant on the right does not exist. I am asking about the condition to ensure that the equation in the OP makes sense at all.

The inverse of the matrix on the right is a known "constant" matrix, depending on the six parameters. Is it known what it is? You appear to assume that it is invertible, how you do know it?

Thanks Dima, that is helpful; Спасибо! I stand to be corrected. But is the question anything like this: There is a special measure \(\alpha\) of a circular arc, with \(\alpha < \pi\), for which if you divide the arc of measure \(\alpha\) into \(N\) equidistant points, and you divide the chord between the two ends of the arc into \(N\) equidistant points, for any \(N > 0,\) and the line \(\ell_n\) is defined as the line that passes through the \(n\)'th point on the arc and the \(n\)'th point on the chord, for each \(n=1,2,\ldots,N-1,\) then the lines \(\ell_n\) all intersect in a common point? Maybe I just misunderstand the figure that got posted by studiot a couple of times now.

That is impossible, since there is no nearest point. It would be like asking for the smallest positive number.

His matrices are not given explicitly though.

Then for every \(a \neq 0\) you can calculate an \(x\) and a \(y.\) Is there anything wrong with that?

It answers one question in the original post, is that not right? A little more formally one would state that an angle is a congruence class (instead of "shape") of arcs of the unit circle. Does the original question about definitions of trigonometric functions still need an answer?

If somebody caused and created the big bang, were they inside the universe already, or outside of it? I suppose we would have to look the answers up in the holy bible. But I seem to have an outdated edition.

Dima has been courteous enough to explain some of his use of language. He said earlier that he does consider an arc to be a (connected) piece of the unit circle, and that he identifies the essential property of an arc to be its length. He considers arcs "modulo rotation" so to speak. Thus the use of language such as "the arc \(\pi\)" is unambiguous. We know by definition that the entire unit circle is the arc \(2\pi\). Therefore it makes no difference to speak of an arc which is \(1/n\)'th of the whole circle, and an arc of length \(\frac{2\pi}{n},\) for any \(n > 0.\) What gets a bit murky is how then it can make a difference how to "define an arc", when it is just the same as a real number between \(0\) and \(2\pi?\) Is the question really about how to define the various trigonometric functions? What we would answer then is that if \(\alpha \) is a real number in this range, then \(\cos \alpha\) is the \(x\)-coordinate of the left endpoint of the arc \(\alpha\) when it is positioned so that its right endpoint is \( (1,0). \) And so on, to define the functions \(\sin,\) \(\tan,\) \(\cot,\) etc. Or is it about something else entirely?

Mars has a very low population density compared to Earth, and appears to have none or very little biomass. Logically another useful thing on your list might be to actively kill as many people and as many other living organisms as possible to get prepared for your venture.

Anyway, when you are just typing in some math, you are not usually copy-pasting anyway, but you might likely be editing, and it is by editing from "=" to ":=" that this occurs.

Very well: "Then I type \( x := 0 \), but I edit it to become like in the line above by adding the ":"." and indeed it now renders correctly. But if I paste it once again, but I now delete the : and reinsert it I get "Then I type \( x := 0 \), but I edit it to become like in the line above by adding the ":"." When I copy/paste the previous line without any editing I get "Then I type \( x := 0 \), but I edit it to become like in the line above by adding the ":"." Interesting.

The : character seems to come in two versions, and one of them breaks the math display. I will have to be careful about it in the future.

OK, I try two things now. I type \( x := 0 \) just like that. Then I type \( x := 0 \), but I edit it to become like in the line above by adding the ":". Clearly the newly edited expression with added ":" character works differently than when the : is just typed in normally.

When you type in \( f(x) := 0 \) does it render correctly with TeX?

Every real number has a "non ending" representation in its decimal expansion. E.g. the standard representation of 1 is \(.999\ldots\) in base 10. You want to refer to the property that \(\pi\) has as opposed to rational numbers, that its representation is "non repeating".

Okay, then I try here: [math] f(x) := 0 [/math]. And with parentheses: \( f(x) := 0 \), which should be the same. I see. I cannot trust the \ ( shorthand in this particular case. If I do \ ( f(x) = 0 \ ) and omit the blanks between the backslashes and the brackets, I get \(f(x) = 0, \) which works fine. Curious. Apparently the point is that I use \ ( and \ ) instead of math and /math. Is that deprecated?

Typing f(x) = 0 within the LaTeX brackets produces \( f(x) = 0\) whereas typing f(x) := 0 leads to \( f(x) := 0.\) Is that something fixable, or one just has to be aware of it?