taeto

You are married too much to notation. You have to look at the form of the equation that you want to solve, which is \( x + \alpha \cos (\beta x + \gamma ) = c,\) where you have known values for \(\alpha,\beta,\gamma, c.\) You get that form by adding \(F(a)\) on both sides of your last expression, and dividing through by \(D\) on both sides. Or in other words, you want a solution to the equation \(h(x)=0,\) where \(h(x) = x + \alpha \cos (\beta x + \gamma ) - c.\) This cannot be solved as a closed expression, all you can do is to find an approximate solution. But so far as I can determine, your constants have values that are results of measurements, so they are themselves approximate? First try to make a drawing of the graph \(y = h(x)\) to check if it is reasonable that there exists a solution to \(h(x) = 0\) or even several solutions. If some value \(x_0\) looks to be close to a solution, then use a software that is able to perform Newton's method for finding a zero of a function \(h(x)\) close to a starting value \(x = x_0.\) You may have to supply an expression for the derivative \(h'(x).\) But since it is essentially a basic trigonometric function, this is an easy task.

Let \(F\) be an antiderivative of your \(f;\) \(F = \int f(x) dx.\) That is easy, since \(f\) is a simple trigonometric function. Then solve \(F(b)-F(a) = 1200\) to determine \(a\). Dunno what Excel can do. If it cannot do it, then use something else which can, like Maple or Mathematica, or code Newton-Raphson yourself. Once you have the solution, use Excel to document the correct solution to your boss.

Maybe use \(\cos(2\pi - \varphi) = \cos \varphi\) to change the integral from \(0\) to \(2\pi\) to two times an integral from \(0\) to \(\pi.\) Then use \(\cos(\pi - \varphi) = -\cos \varphi\) to change this integral to an integral from \(0\) to \(\pi/2.\) Finally substitute \(s = \sin \varphi\) to get an integral from \(0\) to \(1\) without any trigonometric functions (but a \(\log s\) ). I will try to look if that can be done.

en.wikipedia.org/wiki/List_of_integrals_of_rational_functions

Good, that helps. Now the integral over \(r\) is of standard form \(\int \frac{r}{a+br}dr,\) with \(b=\cos \varphi.\) An antiderivative is \( \frac{r}{b} - \frac{a}{b^2}\log | a+br|.\) So we get \(\int_0^a \frac{r}{a+br}dr = \frac{a}{b} (1-\frac{\log (1+b)}{b} ) \) when \(b \not\in \{0,-1\}.\) Agree?

Indeed. I misread the integrand, sorry about that. But re-reading it now, I wonder if you copied it correctly from its source. The differentials \(dr\) and \(d \varphi\) are in the wrong order with respect to the order of the integration symbols. You have to know which integral is the "outer" and which is the "inner", since in this special case you are not free to swap them around as you please. The integrand is not defined at \(r = a\) and \(\varphi = \pi,\) and that makes it harder than usual.

Great. After substituting \(r\) do you see \(1 + \cos \varphi\) somewhere, and does that remind you of something, like as if it could be another expression squared?

You could begin by substituting \(r = a\cdot s,\) that surely helps. What is the upper integration bound for \(\varphi,\) it looks like \(\gamma_\pi?\)

Of all entities, you in particular should appreciate the fact that it is only when we face a full-fledged AI and crack open its scull to see what is inside, that we will truly know the meaning of "intelligence". Wouldn't work with you, apparently.

Darn. You cut away my following sentence which explained how we will figure it out.

It seems that "intelligence" is a key notion. Used for the name of certain concepts, and also used in the definition of those same concepts. We will only be equipped to recognize an AI/AGI when we have learned the exact meaning of "intelligence". We may hope that once an AGI has formed, we will be able to get into its head.

Now come on, a real number does not have "a neighbouring real number", right? It clearly has both a smaller and a larger neighbouring number 😆. Just kidding. But the total order is not a problem, since the lexicographic order is total, which is why you can fairly easily look up entries in a dictionary or telephone directory. In particular if \(dx\) is a fixed positive infinitesimal, then \(a+b\cdot dx\) is to the left of \(a'+b'\cdot dx\) if \(a < a'\) or ( \(a=a'\) and \(b< b'\) ). A well-ordering is not required anyway.

With closed or half-open intervals, it is more awkward as to which real numbers to assign to the endpoints if you explicitly demand a 1-1 function. Mainly by specifying open intervals I would avoid the closed intervals of the form \([0,0]\) ...

It seems an interesting perspective. The problem is the huge gap between the original Church-Post-Turing (hypo)thesis about AI and what you describe. They suggested that whatever computational problem can be solved can in particular be solved by mechanical means. It does not necessarily include the problem of "to come alive" or "how to think". We know a lot about what we can do with computers, and yet we have no answer to CPT, and it appears unlikely that it is scientifically possible to discover or describe one. Maybe we suspect that "being alive" or "thinking" represent solutions to computational challenges. But that seems purely philosophical and by far not in the scope of science.

That seems to beg the question. To be "connected in geometric sense" ought to mean that any two points are contained in a common line. So if we declare that the set of all hyperreal numbers forms a single line, then this condition will be satisfied trivially.

Which should probably have said "all uncountable subsets", and even that is assuming the Continuum Hypothesis. Or simpler, all open intervals.

Maybe if we compare the set of all whole numbers from \(0\) to \(10\) with the set of all half numbers \(0, 1/2, 1, 3/2,\ldots, 19/2,10,\) then we see that the second set contains more numbers than the first. And yet each set has finite size, so in that sense, none is bigger than the other? It seems that you are missing that to say that two sets both have finite size, or both infinite size, does not imply that they have the exact same size.

The notion of a "number line" does not really come up in any serious mathematical context at all. It seems to be a teaching aid for illustrating the ordering and arithmetical properties of real numbers in particular. As such the properties which it demonstrates are no different from the properties of rational or algebraic numbers, that is, the total order and the Archimedean properties. Wikipedia distinguishes between "basic mathematics" in which a number line represents the real numbers, and "advanced mathematics" in which only the "real number line" represents the real numbers, but others are possible. The "hyperreal number line" is the number line which includes the infinitesimals.

We could challenge that reasoning by saying that there is also a "number line" every point of which corresponds to a rational number, and another "number line" of algebraic numbers, and so on. So why not a "number line" with all infinitesimals? After all, a number line is meant to represent the total order of some kinds of numbers, and if you look at the sums of real numbers with infinitesimal numbers, then they are totally ordered too, so why not?

Sure, segment is fine. I do not want to start an argument either. Just make the point clear whether we are going into real numbers or not. In basic geometry it takes a few steps before you get to Archimedean and even Cantor-Dedekind properties. It seems to me that Dima wants to discuss at the level of measures of angles and real-valued trigonometric functions, which is many levels away from talking about arcs and angles.

Not really. If we want to be painfully clear, then we should distinguish between what is an "arc" and what is an "angle", and your site does not seem willing to do just that. The page certainly does not like to consider that an arc is just a bunch, each with a particular property, of points taken from the circle, which is the view that I intended. Angles define some special kinds of arcs. If you have two straight lines through the origin, you have angles that partition the circle into arcs, usually four of them. Each angle defines a separate arc. There is no involvement whatsoever of real numbers yet. So you should explain the need to introduce real numbers in the first place. They seem largely irrelevant to the introduction of these terms.

Except that such a species is assumed to be created by humans, as opposed to by evolution? Now if an AI is created which reproduces itself, then it must have been constructed so that it has this property, is that not true? By a human, you think? Or by another AI, which was itself constructed by human?

I do not understand why you say that an arc is "very complex". If we already agree about what is the unit circle, what is difficult about saying that an arc is a certain piece of the circle? Are you aware of the difference between an "arc" and an "angle"?

But even without knowing the trigonometric functions and their inverses, is it not possible to still understand that an arc is a connected subset of points of the unit circle?

Must be due to the elevated shipping costs for sending to isolated tribes out there.