Recent content by SrEstroncio

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    Eigenvalues of perturbed matrix. Rouché's theorem.

    Homework Statement Let \lambda_0 \in \mathbb{C} be an eingenvalue of the n \times n matrix A with algebraic multiplicity m , that is, is an m-nth zero of \det{A-\lambda I} . Consider the perturbed matrix A+ \epsilon B , where |\epsilon | \ll 1 and B is any n \times n matrix...
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    Proof sin(10) is irrational.

    I should suppose sin(10) is rational, if i am to contradict the statement, shouldnt i?
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    Proof sin(10) is irrational.

    Homework Statement Prove \sin{10} , in degrees, is irrational. Homework Equations None, got the problem as is. The Attempt at a Solution Im kinda lost.
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    Anomalous integral

    @Hernaner28 Floor's not even a continuous function, much less differentiable that cant possibly be a primitive for \sqrt{1-\sin{x}} .
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    Quasilinear PDEs in industry, finance or economics.

    Homework Statement I've been reading some introductory PDE books and they always seem to motivate the search for solutions of the first order quasilinear PDE by the method of characteristics, by introducing a flow model; i was thinking, could someone give an example of a phenomenon modeled by...
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    Boundary of closed sets (Spivak's C. on M.)

    Sorry for the inactivity, my computer decided to self-destruct under the heat. Well, that (a_1,a_2,...,a_n) does not lie in our rectangle centered about the point x is not much of a problem, since said rectangle was constructed inside our original and arbitrary rectangle, not necessarily...
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    Boundary of closed sets (Spivak's C. on M.)

    Let R be an open rectangle such that x \in R , R=(a_1,b_1)\times ... \times (a_n,b_n) . If x=(x_1,...,x_n) , we construct an open rectangle R' with sides smaller than 2\min{(b_i - x_i, x_i-a_i)} for 1\leq i \leq n , and centered about the point x . By construction R' \subset R and...
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    Boundary of closed sets (Spivak's C. on M.)

    The points x\in R^n for which any open rectangle A with x\in A contains points in both U and R^n - U are said to be the boundary of U.
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    Boundary of closed sets (Spivak's C. on M.)

    Homework Statement I have been self studying Spivak's Calculus on Manifolds, and in chapter 1, section 2 (Subsets of Euclidean Space) there's a problem in which you have to find the interior, exterior and boundary points of the set U=\{x\in R^n : |x|\leq 1\}. While it is evident that...
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    Compactness of point and compact set product

    I get it now, thanks. Now, at the risk of seeming kind of stubborn, imagine you've just been given the definition of compact sets and you were immediately asked to prove this (which is the case with Spivak's book), how would you do it without constructing the functions \phi and \psi , which...
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    Compactness of point and compact set product

    While I don't doubt there's nothing wrong with your argument, I am not familiar with homeomorphisms and your proof seems a little out of my grasp right now. I am trying to prove it by means of covers, I suppose A is a cover of \{x\}\times B , and I want to prove there is a finite subcollection...
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    Compactness of point and compact set product

    I was reading Spivak's Calculus on Manifolds and in chapter 1, section 2, dealing with compactness of sets he mentions that it is "easy to see" that if B \subset R^m and x \in R^n then \{x\}\times B \subset R^{n+M} is compact. While it is certainly plausible, I can't quite get how to handle...
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    Self studying little Spivak's, stuck on problem 1-6

    ok ok ok I get it now, thank you very much ;D
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    Self studying little Spivak's, stuck on Schwartz ineq. for integrals

    We do not know if f and g are continuous, we only assume them to be integrable, so it is not necessarily true that \int (f-\lambda g)^{2}=0 implies (f-\lambda g)^{2}=0, since f-\lambda g could be zero except at an isolated number of points (it's integral would still be zero but the...
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    Self studying little Spivak's, stuck on Schwartz ineq. for integrals

    Homework Statement In an effort to keep me from spending all summer lying on the couch, I recently started reading Michael Spivak's Calculus on Manifolds; while working on problem 1-6 I got stuck on a technical detail and I was wondering if anyone could provide a little insight. Problem 1-6...
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