@JM: now that you mention it, I remember. I even unconsciously copied the same expression... I had forgotten and probably I was hoping for a successful "Eeeek! My profile page looks like a nuclear dashboard..." intervention on meta.SO.
Hi, everybody, what is a formal way of finding the number of real roots with regard of multiplicity. Also, does the way of Ferdinand von Lindemann (1852-1939) expresses the roots of an arbitrary polynomial in terms of theta functions help with my question?
Again, this is for the general quintic. Doesn't preclude us for being able to use easy methods for things like x^5-1 or x^5-15 x^4+85 x^3-225 x^2+274 x-120 ...
For things like x^5-x-1 ... then you need the heavy machinery.
the OP might have meant there's more than just *zeta*-regularization, but in addition you can discard the function aspect and just use straight-up summability methods, which don't all give the same answer
To be frank, I am in Didier's boat really. I didn't follow the hint completely; then again I didn't pay much attention because the question is simple. =)
@Srivatsan Like me here in the chat room droning on and on about something that you're not interested in and perhaps going too fast and no chance for questions until the end...
Well, video lectures have the huge deficit that they are not interactive. Nevertheless I like to watch talks from MSRI or the ENS or Cambridge or wherever else they have good collections.
If I didn't have time to prepare, I understood little from the lectures; if I prepared well, some lectures would be a waste of time. I agree with @robjohn, I prefer asking questions after reading.
I really think I had some outstanding lectures, containing stuff presented in a way that you won't find in books (some of them are books now, though). Still, I don't think the books alone could replace the lectures and my notes but rather they complement each other excellently.
Also books could be too dense; I feel a lecture cannot be as fast or pack as much material. Also, the lecturer is forced to break down the contents into smaller units than usually seen in a book.
I wouldn't describe a book as unromantic. Losing attention makes it unromantic either way. And @Sri: No, but it makes going back and forth between it and other things (on a computer) more efficient, mentally.
Me too. Since I started earning, I used to buy books at the rate of one per month (or couple months). I stopped temporarily because of office space constraints
I think what makes a good (mathematical) lecturer, at least those I witnessed, is that they actually think in front of the crowd. If they are good at tuning their pace to the audience, the audience can sort of witness some of the thought processes that a more advanced person has. That kind of thing is often missing in books.
I often lost the big picture. But I seldom blamed the lecturers. If I had the time to, I would rather work hard by myself. After all, learning is one's own business.
@Srivatsan There were several things. You nested math environments and MathJaX doesn't like it. Don't use \text{for all $x \in X$ satisfying} but rather \text{for all } x \in X \text{ satisfying} (notice the blanks that take care of proper spacing). Then doubling up backslashes is sometimes necessary because backslashes are also used for escaping things, like * to get an asterisk. I'll link to a better explanation in a moment.
@JM Now that you just edited the post, can you fix the tags as well for the math.stackexchange.com/questions/71851/…? Elementary-number-theory will do... // Thanks @J.M.
hi everybody, is there any ways to find the exact number of real roots for arbitary polynomial with regards to multiplicity by hand. in the discussion above, jm have given a long yet effective way to find the roots for quintic
@JM - is that the reason that most of the math competition provide the information , for example, the polymial have a double real root and triple real root in the question?
@jm - i mean every people have to study something, from my philsophy, if you get down a water well, but you don't know how to tie a knot, even somebody have reached you, they can't save you...
@Victor I'm new here but what do you mean by "no way to know the multiplcity of the real roots"? One can check the derivative of the polynomial to decide whether there is a double root at a certain point, would this suffice?
Oh t.b. ... would you happen to remember offhand a reference for constructing continuous but nowhere differentiable functions like Riemann's or Weierstrass's?
Apparently it's not as simple as \sum a_n\cos(c_n x) ... for some rapidly increasing c_n .
Oh, yeah, batrachions. The reason I was asking is this question. It looks to me that one only has to show the fractal behavior is inherent in the components themselves...
@JM Well, it looks to me as if the main point was mentioned by Henning: these functions look like very close to having the unit circle as a natural boundary (even though the usual lacunary series theorem doesn't apply), so you want the boundary to be mapped to a Jordan curve that is nowhere smooth, so in that sense yes I agree. I'm waiting for our new complex analyst mathstribble to clarify the issue...
@JM No, but I've seen enough of the (excellent) answers to believe that it's the strongest complex analyst we have here along with Hans and maybe Zarrax.
Does anyone know if it's true that, when b, a_1, . . . , a_n are not squares in Q, and \sqrt{b} cannot be expressed with the \sqrt{a_i} using only multiplication and division that
Hm. This is the kind of attitude I don't like. This user asked 16 questions (rather non-trivial ones at that, sometimes) and only cast a single vote...
I hope I didn't phrase my question poorly. The reason I'm curious, is for example, sqrt(5) \notin Q(sqrt(2),sqrt(3)), so it's true in this case.
I'm trying to read some Galois Theory, and for some exercises I'll say something like Q(sqrt(2),sqrt(3),sqrt(5)) has dimension 8 over Q, without actually checking that adjoining each successive root gives an extension of degree 2. So I wonder if there's a theorem that will make me more comfortable about saying things like this.
I would agree here. But I don't know how to get \sqrt{2} from \sqrt{2} + \sqrt{3} (which generates Q(\sqrt{2},\sqrt{3}) = Q(\sqrt{2} + \sqrt{3})) using multiplication and division only
Ah, that would make sense, although I don't think \sqrt{2}+\sqrt{3} has form \sqrt{a} for some integer a. I guess if I allowed elements which are sums of square roots of nonsquares, I'd have to add that \sqrt{b} cannot be gotten through basic arithmetic operations (+,-,x,/) to avoid trivial problems.
@tb Yes, I think that's what I wanted to communicate, in hopes of an easier way to check that it's not in the field extension. In the same example, showing \sqrt{5} is not a rational linear combination of the basis elements 1,\sqrt{2},\sqrt{3},\sqrt{6} is too messy.
As I said, my field theory is very rusty (there is only little rust that can be on the little theory I ever knew). I had a horrible algebra course and I was happy with a few of the classical applications (ruler and compass constructions, testing for integrability in elementary terms), forgot about the rest and lived happily ever after in blissful ignorance...
To cut a long story short: I don't remember any generally applicable trick for that, but it sounds pretty basic and important.
Oh, no, not at all! I was more like talking to myself, observing that it is a bit scary that I still have the same repulsive feelings when it comes to a subject which is certainly beautiful and very important, but whose appeal was messed up by an idiotic teacher. And this after 12 years...
I always liked analysis, yes, but not so much the computational aspects, so functional analysis fits my tastes pretty well. But I'm also very interested in geometric things, metric and differential geometry, algebraic and differential topology and the like.
Is it difficult to get by in algebraic topology with a distaste for algebra? I thought algebraic topology would require quite a bit of abstract algebra.
@yunone I have absolutely nothing against abstract algebra. I don't like number theory very much, and our algebra course focused very much on that. I just couldn't get my head around all those different rings and couldn't motivate myself to look closer. If you understand abelian groups well enough that's plenty enough for most of the basic applications of algebraic topology. Of course, you need to understand the homological algebra bits, but luckily these weren't treated in my algebra course...
@Srivatsan Yeah, I know. Badp thinks I'm the bad guy...
Is there a good way of expressing the exterior algebra of a vector space? I mean: the wedge in \wedge V is too small and in \bigwedge V it is too big, especially in displayed formulas. One could go for the \Lambda V kludge but I'd prefer a sans serif version of the \Lambda.
SE has changed. Again. It's almost like facebook. It works perfectly fine but they have so much money and time that they rearrange the UI without adding new features.
@Matt Well, it's not so bad, I exaggerated a bit. Nevertheless, there are very few persons on this planet to which I can apply such nice words with a good conscience...
Hey! I found something strange. In the famous book L'intégration dans les groupes topologiques et ses applications, the axioms for a group do not assume that the cancellation law holds, is this not strange?
He doesn't require uniqueness. For the others: the first is associativity of the operation and the second axiom states for every pair x,y of elements there exist z and z' such that xz = y and z'x = y
But even though any element can acquire an identity, the identity for two elements can differ, or even there are two identities for an element, is it not?
A comment from here: "I'm sorry isn't there an easier way to do this without over-complicating it? I would like to see something that is elementary and cleaner.", and I would like a six pack of beer waiting for me in the classroom today.
@robjohn I assumed you wanted to fix that so that it uses both processors. I know nothing about Mathematica so I don't know whether there still is a Parallelise command in M8...
Hmm, is there an easy way to tell which subgroups of QD_16 are normal? I know those with index 2 are normal, but I don't really want to go through and compute the left and right cosets for the rest.
