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About Manifold

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  1. Hi! I've got a question concerning neighbourhoods of points in 2- and 3-dimensional space. How can we explicitly show, using only the definition of a given metric, that the according neighbourhood is some figure? For example the three metrics are given: 1) [math]d(x,y)=\sqrt{\sum_{i=1}^{n}{(x_i-y_i)^2}}[/math]; 2) [math]d(x,y)=\displaystyle\max_i|x_i-y_i|[/math]; 3) [math]d(x,y)=\sum_{i=1}^{n}|x_i-y_i|[/math] If [math]a\in{\mathbb{R}}^2[/math] and the metric is 1), then it's clear: [math]d(a,x)=\sqrt{(a_1-x_1)^2+(a_2-x_2)^2}<\epsilon[/math] and it's clear that it's a circle...because one can rewrite:[math](a_1-x_1)^2+(a_2-x_2)^2<{\epsilon}^2[/math] But I have problems to see a picture when looking at the other metrics: [math]d(a,x)=\displaystyle\max_2\{|a_1-x_1|,|a_2-x_2|\}<\epsilon[/math]; [math]d(a,x)=|a_1-x_1|+|a_2-x_2|<\epsilon[/math] tell me nothing at the moment about what the according neighbourhood might look like. Could you please enlighten me on this case? :smile:
  2. Manifold


    Ok, thanks Matt Grime! Wow, Jesus, this was our first class on induction at university and our exercise group all solved the task the way I did, so it appears that we all have a wrong solution! Very funny! From your post I could not get whether a) was alright... but anyway... First I find it now confusing that "Assume that A(n) is true" is a wrong sentence to start the induction step, since our prof says it exactly in the same way. Second, didn't I show that (n+1)^2<=2n^2 with the string of inequalities?... In my first attempt for b) for example I used A(n): n^2<=2^n is equivalent n^2+2n+1<=2^n+2n+1, which is true according to one of the order axioms. I actually use them throughout,... in the secon case in b) I used the fact that n^2<=n^2 and 2n+1<=3n-1 and again the order axiom: from a<=b and c<=d follows a+c<=b+d. The same thing with a). Here if we assume A(n) i.e. 2n+1<=2^n we can write 2n+1+2<=2^n+2 (again order axioms) and 2^n+2<=2^n+2^n since 2<=2^n. I can't see why this argumentation should be wrong.
  3. Manifold


    Hello! We've got the following to prove by induction: a) [math]2n+1\le{2^n}[/math] b) [math]n^2\le{2^n}[/math] (It is assumed that 0 is a natural number) a) This inequality is not valid for [math]n=1,2[/math], so to prove the inequality one has to show its validity for all [math]n\ge{3}[/math]: 1) [math]A(3):7\le{8}[/math] (true) 2) Assume that [latex]A(n)[/latex] is true. [math]A(n+1): 2(n+1)+1\le{2^{n+1}}[/math] [math]2(n+1)+1=(2n+1)+2\le{2^n+2}\le{2^{n+1}}[/math], since for [math]2^n+2\le{2^{n+1}}[/math] we have [math]n\ge{1}[/math] Is there a difference if I make the induction step this way?: 2)[math]A(n+1): 2(2n+1)\le{2^{n+1}}[/math], so to prove A(n) we must show that [math]2n+3\le{4n+2}[/math], which is also true for [math]n\ge{1}[/math] The thing that disturbs me is that if we assume 0 to be a natural number the last inequalities in 2) in both cases do not hold...but I think this case doesn't play a role since we are to prove it for n greater 3...does it? b) This inequality is invalid for all [math]n=3[/math], so to show that the inequality is valid we must show that it is valid for all [math]n\ge{4}[/math] 1) [math]A(4)=16\le{16}[/math] (true) 2) Assume that [math]A(n)[/math] is true. [math]A(n+1): (n+1)^2\le{2^{n+1}}[/math] [math](n+1)^2=n^2+2n+1\le{2^n+2n+1}\le{2^{n+1}}[/math]. The last inequality in this string is equivalent to [math]2n+1\le{2^n}[/math], which was proved in a). Here, equally, the same question...Is there a difference if I do it this way?: 2) [math]A(n+1): 2n^2\le{2^{n+1}}[/math] [math](n+1)^2=n^2+2n+1\le{n^2+3n-3}\le{n^2+n^2-3}\le{2n^2}[/math]
  4. Manifold


    OK, forget the last thing you don't understand ...I now see myself that it's a bit of nonsense... I got what you meant and have a clear picture of the matter now. Thanks a lot!
  5. Manifold


    Well, I understood why we took the 11 smallest numbers, but I don't quite understood what is actually meant by this. The 11 smallest of what...of which sets/subsets? Or do you simply mean that in the product of 100 numbers one can pick 11 whose product is fewer than the the product of the rest?...which follows from: if ab>1 then either a>1 and b>1 or a>1 and b<1 or a<1 and b>1...? Aren't the expressions in terms of this and in terms of "smallest" self-excluding? (Although if so, this doesn't make the solution wrong!)
  6. Manifold


    Hi, I've just picked up the following problem and got stuck...although seems easy. You have 100 positive real numbers. The product of each 11 of them (11 different) is greater than 1. Show that the product of all hundred numbers is greater than 1. Please, give me a hint first. Thanks
  7. I think you could also show it this way: 1. An electron is attracted towards the nucleus by the force: [math]\vec{F}=\frac{q_e{q_p}}{4\pi{\epsilon_0}r^3}\vec{r}[/math] where [math]\vec{F}\cdot\vec{r}=1[/math]; 2. On the other side [math]\frac{q_e{q_p}}{4\pi{\epsilon_0}r^3}\vec{r}=m\vec{a}[/math], since the electric field is non-homogeneous, and [math]\frac{q_e{q_p}}{4\pi{\epsilon_0}mr}=\vec{a}\vec{r}\neq{0}[/math], the electron gets in the electric field of the nucleus an acceleration with the normal component directed towards the nucleus, which is in the extreme case has a zero angle with the radius-vector (radial acceleration); 3. It is now clear that the trajectory of the electron is anyway curvilinear. Since it moves in the non-homogeneous electric field with increasing force gradient directed towards the nucleus, it gets angular acceleration, moment of inertia and hence orbital moment of force [math]L=rmv[/math]. This is that classical model, which allows the electron to fall on the nucleus.
  8. Back to your question again, Swansot. The electric field is mainly generated by the transformator, which works as plasma heater. Because there are always fluctuations of charge in plasma current, the electric field is in places where differences in charge occur. There is another thing I don't understand. It says that spiral magnetic field lines, which are set by both toroidal and poloidal fields, are the best protection from the cross drift. It also says, that they go along on interleaving toroidal surfaces. At the same time, it reads that a particle, traveling along such a magnetic field line, moves through magnetic field of varying intensity, that is through all layers with different magnetic field strength. Now how can that be? The toroidal surfaces have constant cross-section radius, and hence the B-field spirals themselves. A particle moves along only one such spiral, a B-field line with constant intensity value. It cannot change its trajectory can it?... so why should it travel through B-field of different strength?
  9. Thanks Mezarashi! The link is just great! The fact that the Lorentz force causes spiral-like motion of a particle along the magnetic field axis is clear to me. It reads on the page you've given me that there is a gradient of magnetic field - along the inner axis of the torus the magnetic field is at strongest. Ok, but since the field is weaker also to the right and to the left for example, why is there no drift in these directions, but only the cross drift? I find it confusing, that the ions should move upward and electrons downward. My source claims that the converse is true (Soros Educational Journal). I also can't understand, why we are dealing with homogeneous magnetic field here then. How can there be a gradient???...Oh sorry, have just noticed it...In a toroid the field is non-homogeneous!!!...I'll make corrections above. Swansot, you asked exactly the same question I asked myself. I read it in other sources again and again...it's only magnetic field involved, although I can't get it either...plasmas are also (very good) conductors, and as far as I know there is also electric field around any conductor segment. I'm totally confused now!
  10. I've got an understanding problem. It's about the behaviour of charged particles in plasma inside of a tokamak. It says that a particle in the non-homogeneous magnetic field (toroidal field), moving on the spiral trajectory all along the magnetic field lines, tends to drift downward (for ions) or upward (for electrons), together with the axis of the Larmor-spiral, which makes plasma highly unstable and damages the chamber walls. That's the reason why the second field (poloidal field) is necessary, which is provided by plasma current itself. Which force is it, that causes the particles to drift (toroidal drift) downward/upward?
  11. Hello! What I'd like to ask is, whether there is a mathematical theory of consciousness in development or any promising serious research projects in describing conscious process mathematically or physically with sufficient mathematical background
  12. I think it's inappropriate to make such an example in this situation. I read a number of biographies about prominent mathematicians, who invented problems of their own when they were young, in their pre-university years. They had succeeded in a number of olympiads themselves, so they were good problem solvers. But somehow they managed to make a transition from problem-solvers to problem-inventors. You can't put it down to their assumed genius, can you? I really think there is more to this than their mathematical ingenuity.
  13. hm well I think it will at most help training those problem solving strategies...I've got enough books to do that , although you can assume that similar methods would be applicable to few of the problems. I can imagine refining an already solved problem, investigating conditions, constraints, looking for special cases and so on. But the basis for this will be still the problem you already solved. I wonder however if it is the only method of invention in mathematics. What if I go straight on the "body" of mathematics and try to invent a new problem out of the material I'm working on. I can imagine asking various questions about each theorem, lemma, corollary or whatever. So far it has led me to clarification of the whole picture of how things work in a particular situation, but not to new problems in traditional sense.
  14. Hey guys! (I posted the same thread in physicsforums. To grasp a wider audience I decided to post it here too.) They say the best way to develop and train problem solving skills is by creating one's own problems. I'm still a "passive" problem solver but I'd like to learn solving problems in an "active" way. Could you give me some advice, name some guidelines on how to create new, own problems? Thanks in advance.
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