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    Continuous functions

    I'm having trouble with the third part of a three part problem (part of the problem is that I don't even see how what I'm trying to prove can be true). The problem is: Let X and Y be topological spaces with X=E u F. We have two functions: f: from E to Y, and g: from F to Y, with f=g on the...
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    Int(A) + ext(A) not dense

    Can someone help me find an example of how the union of int(A) and ext(A) doesn't have to be dense in some space X? Thanks.
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    Rational points on a circle

    Does anyone have an idea how to prove the following (or prove that it is not true): For any positive integer k, you can find k points on a circle such that each point is a rational distance from every other point.
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    Inverses for units

    Okay, if we are looking at a typical extended number field Q(w), and it's corresponding ring of integers, we know that for any given element in this field, it is not necessary that all of it's conjugates are in the same field. A typical example being: Q(\theta), \theta=\root 3\of{3}, \theta...
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    Roots of unity proof

    I'm having trouble following one step in a proof I'm studying. I'm sure I'm missing something obvious, but I just can't get it to work out (it supposed to be "obvious" which is why they left out the details). Anyway, it's part of a proof showing that if you have a monic polynomial with all...
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    Free abelian group proof help

    I'm working on a proof for subgroups of free abelian groups and am having trouble with a step (I know other methods, but would like to try and make this one work if possible). The basic idea is let G be a free abelian group with generators (g_1...g_n) and let H be a subgroup of G. Assuming a...
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    Fintely generated ideals

    I would appreciate some help with developing a simple proof that the ideals in the ring of integers for a number field have the same rank as the ring of integers itself. In other words, assuming from the start that all the ideals are finitely generated, all ideals require the same number of...
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    Parametric representation

    Does anyone have any ideas on how to even start this problem? I am supposed to find a general solution in rational numbers for (aside from the trivial ones): x^3+y^3=u^3+v^3 Actually, I'm given the answer (which is really messy) and am supposed to show how to derive it. The book gives the...
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    Another Euclidean ring

    While I'm on the topic, here is another ring I need to show Euclidean. I'll show more of the work this time too. The ring is Z[{\sqrt 2 \over 2}(1+i)] So, using the standard norm difference approach, we pick any element alpha in the field and try and show we can always find an element beta in...
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    Euclidean rings

    Let \displaystyle{\zeta = e^{{2\pi i} \over 5}} I need to show that Z[\zeta] is a Euclidean ring. The only useful technique I know about is showing that given an element \epsilon \in Q(\zeta) we can always find \beta \in Z[\zeta] such that N(\epsilon - \beta) < 1 (using the standard norm for...
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    Question related to Pell's Equation

    For \alpha and \beta odd integers and X,Y integers, we have the following (by collecting terms): (\alpha + \beta \sqrt{5})^n = x + y \sqrt{5} My question is how do we know that x and y are both divisible by exactly 2^{n-1}? (no more and no less 2's in each) I can show this with...
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    Coset of ideals question

    Okay, I have this problem in my book, an I'm pretty sure I solved it, but I there is something that is confuses me in the way the problem was asked. Assume all polynomials are in the ring {F_2[x] \over x^n-1} where n=2^m-1 and m>2 Let g(x) be the minimal polynomial of a primitive...
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    Min number dependent columns in a matrix

    Is there some easy way to determine the minimum number of dependent columns in a matrix? Assume the matrix entries are binary for convenience. Let's say I have an m x n, m>n matrix in the form (I | A) So that I is nxn, and A is m-n x n. This obviously depends on A. So is there some easy...
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    Finding units in polynomial quotient rings

    Is there a simple method for finding all the units in a polynomial quotient ring over a finite field? For example: {F_2[x] \over x^7-1} I can see the easy ones like 1, and all power of x, but I wanted a general rule or method for finding all of them if it exists (besides testing each...
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    Shortening LInear Codes

    I'm not sure if this is the right forum for this question, but it is a form of linear algebra, so I'll give it a shot. It's about coding theory. The problem is given a q-ary [n,k,d] linear code, fix an arbitrary column number and then collect all the code words that have 0 in that column...
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    Limits of integrals

    What are the rules for when you can and can not move the limit of a definite integral inside the integration sign? For example, if I have a definite integral of a function f(z) and the function includes a constant k, and I want to take the limite of the definite integral as k goes somewhere...
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    Branch cut integration

    I need help with a branch cut intgration. The problem is to show the following for 0< \alpha <1: \int_{0}^{\infty}{x^{\alpha - 1} \over x+1}={\pi \over sin\alpha\pi} I used the standard keyhole contour around the real axis (taking that as the branch cut), but using the residue theorem...
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    Reversing Fourier series

    Anyone have any clues what function f(x) satisfies the following integral (for an integer n > 0): \int^{\pi}_{0} f(x)sin(nx)dx=(-1)^{n+1}
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    Conjugate of an integral

    In the complex conjugate of an integral equal to the integral of the complex conjugate? If so, is there an easy way to show this? Thanks.
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    A diophantine equation

    I think I posted this in the wrong forum before. Let's try again. I need to prove that the equation x^3 + y^3 = 3z^3 has no integer solutions. I can do it easily for all cases except where z has a factor of 3, in which case I don't know what to do. I am assuming the 3 in front of the...
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    Cos/Sin as roots of rationals

    This is indirectly related to issues with cyclotomic polynomials and glaois groups. Is there some easy way to know if you are dealing with a cos or sin that is expressable in terms of roots of rationals? Like \pi/3 for example? If so, is there any straightforward way of figuring it out...
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    Normal Field Extensions

    How do I show that the following field extension is normal? Q(\sqrt{2+\sqrt{2}}):Q As far as I could tell from my limited understanding is that I need to show that: \sqrt{2-\sqrt{2}} is also an element of the new field, which is required for the minimum polynomial to split over the field...
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    Irreducible polynomials over finite fields

    Can someone explain to me why the following is true (ie, show me the proof, or at least give me a link to one): Over the field Zq the following polynomial: x^q^n-x is the product of all irreducible polynomials whose degree divides n Thanks.
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    Homomorphism question

    Excuse me if I get the English names wrong here, I hope my question is clear. I would be happy if someone could explain this problem to me ... I'm having trouble with some aspects of homomorphisms. The question is: Is there a homomorphism Phi: Z24 --> Z18 which fullfills Phi(1)=16...
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    Help with a group theory proof

    I need some help with what should be a really simple group theory proof, but for some reason I'm hitting my head against a wall on this one. I seem to be missing something simple to get me to the next step. I'm learning this stuff in Swedish, so I'm not sure about all the English words, but...
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    Question about exploding neutron stars

    I have a question about the time scale for a certain type occurance causing a neturon star to explode, and a related question about the conditions of this occurrance. If you have a binary star system with one of the stars being a neutron star, I read that if the other star sucks off enough...
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    Artificial gravity

    I was wondering what sort of advances it would take for making artificial gravity seem within reach for ships (not based on spin). I was thinking something along the lines of maybe using extremely pressureized gas. I realize the contribution from pessure is generally small compared to...
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    Stargazing Limits on radio telescopes

    More research for scifi stories: What the "theoretical" limits on the size of a radio telescope (assuming great advances in manufacturing techniques and unlimited resources). I know the resolution is dependent on how far apart you place the individual components, so could you in theory have...
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    What does a neutron star look like

    What does a neutron star "look" like Hi, I have some questions for a scifi story I'm working on. First, what would a neutron star look like? What color are they, and how bright do they tend to be? By look like, I mean both seen from a theortical planet surface orbiting one (or mabye it would...
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    GR vs. SR while accelerating away

    A quick question for those fast with the GR and SR math. Assume you get in a spaceship and start accelerating away from Earth, and during the trip you and the people left behind compare clock speeds periodically (not elapsed time, but rather tick rates). At what combination of acceleration...