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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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    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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    Vectors in R^4

    As other people pointed out, do the dot product for all three vectors and you'll get a system of 4 unknowns and 3 equations whose answer is most likely a 4-3=1-dimensional subspace of R^4. The answer should be a line.
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    Bernoulli's (differential) equation.

    I have been doing some self study on differential equations using Tom Apostol's Calculus Vol. 1. and I got stuck on a problem (problem 12, section 8.5, vol. 1). Homework Statement Let K be a non zero constant. Suppose P and Q are continuous in an open interval I. Let a\in I and b a real...
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    To prove right inverse implies left inverse for square matrices.

    Homework Statement Let A be a square matrix with right inverse B. To prove A has a left inverse C and that B = C. Homework Equations Matrix multiplication is asociative (AB)C=A(BC). A has a right inverse B such that AB = I The Attempt at a Solution I dont really know where to...
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    Linear function F continuous somewhere, to prove continuous everywhere

    I don't seem to be catching the drift. I can't figure out how to use the linearity property in order to get to where I want to be.
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    Linear function F continuous somewhere, to prove continuous everywhere

    Homework Statement Let f:A\subset{\mathbb{R}}^{n}\mapsto \mathbb{R} be a linear function continuous a \vec{0} . To prove that f is continuous everywhere. Homework Equations If f is continuous at zero, then \forall \epsilon>0 \exists\delta>0 such that if \|\vec{x}\|<\delta then...
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    Distance from a 3 space line

    There is a way to do this using vector projection, does anyone remember it?