Obelix

I would like to ask a couple of questions regarding the vestigial strucure known as the Coccyx. Some people deny its vestigiality arguing that it serves as an anchor point of nine - 9 - muscles. My questions are: 1) How do these 9 muscles and their functions compare to the muscles attached on the tailbone of caudate animals? For example (please correct me if I'm wrong!): One of the 9 muscles attached on the human Coccyx is the Levator Ani muscle, which in caudate animals serves to the motion of the tail, whereas in humans it plays a role in defecation. It stroke me though noticing more than one of the cats that live in my garden moving their tail like the handle of a pump while defecating, which means (my guess!) that the levator's role in that function exists in caudate anumals as well. 2) Is it possible that some of those 9 attachments are themselves vestigial? For example: One of the 9 muscles attached on the Coccyx is Gluteus Maximus, the largest and most massive muscle of the human body. To the best of my knowledge (once again: Please correct me if I am wrong!) this muscle is mainly attached to the Saccrum, whereas there is only a minute attachment to the Coccyx, which I wonder whether it is of any function at all or not. Indeed, the Gluteus Maximus is not only a very large and massive muscle, but it also serves in stabilizing the torso in the upright position. How could such a function of such a muscle receive any service at all from its minute attachment to such a frail and (if I am not mistaken!) inappropriate structure as the Coccyx (4 tiny vertebrae instead of a rigid formation)? Could anyone suggest any references that deal with the above topics?

NO MORE THAN two - 2 - atoms.

Do you confound your reasoning within Schwarzschild spacetime?

I first encountered him in a magazine called "Periscope of Science", published in Greece since 1977 (the "Scientific Greek" you might call it). Last year it was struck by the crisis and switched publisher. I have also been writting in it, from 2007 onwards. As regards Tsolkas, he had then published one of his uproarious "proofs" that "Galileo and Einstein were both wrong in claiming that bodies accelerate equally in a gravitational field, regardless of mass". All he "discovered" then (as well as innumerable times, afterwards!) was the simple fact that the relativity principle essentially has to do with a one - body problem (i.e., an IDEALIZED MODEL) whereas, in the actual universe, one needs to deal with two body problems, where the mass of both bodies has to be taken into account. The said magazine is a serious one, and always mantained a high quality. Yet they made the mistake to take that article seriously ("We have been unable to spot any mistakes in it..." was their foreward). And I can ensure you, after chatting many times with the editor about it, they have CURSED the moment they chose to publish him, with all their heart! After they realized what they had done, whenever they were suspicious about anybody whose theories they were about to publish, they expressed their suspicion by saying: "What if he is another Tsolkas?" You're very much right! But I happen to be a college tutor and students ask for details about the veracity of what they're taught. One can't just tell them: "It is a solved problem and you have to stick to it!" Now my topic is Mathematics and Relativity, whereas this falls into Celestial Mechanics. So I'm not experienced with literature. Besides, sadly enough, there are people influnced by charlatans like Tsolkas. If not entirely, they may at least wonder: "What if he is right, after all?" I don't want to "revisit a solved problem". I just want to show anybody asking that Sun's motion round the barycentre HAS been taken itno account. If one wants to work out details, one is welcome to do so oneself!

I know what Tsolkas is, only too well... I've been watching him since March 1986. What does "BTW" stand for? Do you know of any papers on the topic (perihelion precession) that explicitly mention consideration of sun's motion relative to the barycentre?

I appologize in advance if this topic has been discussed before in this forum. If it has, I have not encountered it. A "dedicated enemy of Einstein and Relativity Theory" - whom I believe some of you already know, his name is Tsolkas - has recently claimed to have explained the precission of Mercury's perihelion by the fact that the sun itself orbits around the barycentre of the solar system. He claims that nobody else, from Leverrier to nowdays, did ever take into account this fact, and that "all astronomers before...him ("Tsolkas Magnus") always considered the solar system's barycentre as essentially identical with that of the sun"! It would be a waste of time to trouble you with the rediculous reasoning he uses in his "derivation" of the precission. My question is this: I can't possibly believe that such a fact realy escaped Leverrier's notice - let alone ALL astronomers to the present day. I believe that it was somehow taken into account in pre - relativity astronomy, still leaving the well known gap of 43" / century. So, does anyone know of any books/papers where the consideraton of the sun's motion around the barycentre is taken into account? Is this motion actually dealt with in the theory of perturbations caused by the planets in Mercury's motion?

It's an old book, classical I'd say. I have a copy in english my father bought back in 1971 (McGraw & Hill) and a greek translation I bought in 2004. I'll look the matter up and try to compare it with the same topic in other books. It would be of help if bibhu gave a few indicative details as to what he finds inappropriate in Beiser's derivation.

Ok, we've warmed up enough! Now for the real thing: Let [math](G,\cdot,T)[/math] and [math](H,\ast,S)[/math] be topological groups, [math]H[/math] being connected as a topological space, and let [math]f : G \rightarrow H: g \rightarrow f(g)[/math] be a homomorphism of topological groups, i.e.: [math]f : G \rightarrow H :[/math] Continuous, and: [math]f(g_1\cdot g_2)= f(g_1) \ast f(g_2)[/math] (or [math]f(g_1 g_2) = f(g_1) f(g_2)[/math], in simplified notation) [math]\forall g_1, g_2 \in G[/math]. Suppose moreover that [math]int_{S}f(G) \equiv f(G)^{\circ} \neq \emptyset[/math], i.e.: [math]\exists \, g \in G, U \in S[/math] such that: [math]f(g) \in U \subseteq f(G)[/math]. Then [math]f(G) = H[/math], i.e.: [math]f : G \rightarrow H[/math] is onto. Can anyone suggest a proof? Then I will explain why I picked this problem up.

Ok, let's make it more interesting! Let [math]G[/math] be a group, and suppose there exist three consecutive integers: [math]n-1, n, n+1[/math] such that: [math](ab)^{n-1} = a^{n-1} b^{n-1}, (ab)^n = a^n b^n, (ab)^{n+1} = a^{n+1} b^{n+1}, \forall a,b \in G[/math]. Show that [math]G[/math] is abelian. Also show that such a conclusion needs not hold if the condition is assumed for only two consecutive integers.

I call it chatting! Well then: Any integer [math]\pm a_n a_{n-1} \dots a_1 a_0,[/math] where: [math]0 \leq a_i \leq 9, \ \ \forall i = 0, 1, 2, ..., n, n:[/math] Natural, is actually of the form: [math] \pm (a_n 10^n + a_{n-1} 10^{n-1} + \dots + a_10 + a_0)[/math]. We can proceed by induction on [math]n[/math]: For [math]n = 0[/math] the integer is a one - digit number [math]\large 0 \leq a \leq 9[/math], whence the sum of its digits is [math]a[/math], whereas [math]a - a = 0[/math] clearly divisible by 9. Suppose the statement holds for [math]n = m[/math], i.e. for any number of the form: [math]a_m 10^m + \dots + a_1 10 + a_0[/math] we have: [math]a_m 10^m + \dots + a_1 10 + a_0 - (a_m + a_{m-1} + \dots + a_1 +a_0 = 9k), k:[/math] integer. Then, for [math]n = m + 1: a_{m+1} 10^{m+1} + a_m 10^m + \dots + a_1 10 + a_0 = [/math] [math]a_{m+1} 10^{m+1} + (a_m 10^m + \dots + a_1 10 + a_0)[/math]. The quantity in parenthsis fullfils the required property, according to the assumption of the induction, whence: [math]a_{m+1} 10^{m+1} + a_m 10^m + \dots + a_1 10 + a_0 - (a_{m+1} + a_m + \dots + a_1[/math] [math]+ \, a_0) = (a_{m+1} 10^{m+1} - a_{m+1})+[/math] [math][a_m 10^m + \dots + a_1 10 + a_0 - (a_{m+1} + a_m + \dots + a_1 + a_0)][/math] [math]= a_{m+1} (10^{m+1} - 1) + 9k = [/math] [math]a_{m+1} (10 - 1) (10^m + 10^{m-1} + \dots + 10 + 1) + 9k =[/math] [math]9a_{m+1} (10^m + 10^{m-1} + \dots + 10 + 1) + 9k [/math] [math]= 9[a_{m+1} (10^m + 10^{m-1} + \dots + 10 + 1) + k][/math], Q.E.D. Set ANY base [math]b \geq 0[/math] in the place of 10, replace 9 by [math]b-1[/math], consider [math]0 \leq a_i \leq b, \ \ \forall i = 0, 1, 2, ..., n[/math] and the proof generalizes directly! For your information, I am a tutor of Mathematics. Why don't you take a slight trouble to check my profile? And yes, it was an assignment - for YOU.

The following holds: If from any integer we subtract the sum of its digits, then the result is divisible by 9. E.g. take 1456. The sum of its digits is: 1 + 4 + 5 + 6 = 16. Then: 1456 - 16 = 1440 = 160 x 9. Give a rigorous proof of the above statement! How does it generalize?

So: "Let's believe for safety!"...Is that what you mean? If one believes in religion A, then one is still threatened by hell fire by a number of other religions. So much for "safety"... Does one "choose" what to believe? If yes, I'd like to believe I have $ 10,000,000 in my bank account!

1) Kindly define "good" and "bad form". 2) You too do multiple (and a bit impolite!) posting: The Science forum.

18 Kindly justify your answer. On two - 2 - I found it here: PHORUM. Is anything wrong with multiple posts? I'm collecting different solutions.

How many numbers lie between 11 and 1111 which when divided by 9 leave a remainder of 6 and when divided by 21 leave a remainder of 12?