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  1. R

    Roots of higher derivatives

    Could you please be a bit more specific about your second line? The k here means the k-th derivatives. The power of (x-1)(x+1) is a fix n, and I don't think I'm supposed to do an induction on that. f^(r)(x) has to be a pretty messy function, is there a clear way to take derivative out of that?
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    Roots of higher derivatives

    I tried to prove by induction on r. But I'm not sure how to express the k-th derivative of f(x). r=0 or r=n are special cases, they clearly holds. My problem is how to generalize it.
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    Roots of higher derivatives

    Let f (x) = (x^2 − 1)^n . Prove (by induction on r) that for r = 0, 1, 2, · · · , n, f^ (r) (x)(the r-th derivative of f(x)) is a polynomial whose value is 0 at no fewer than r distinct points of (−1, 1). I'm thinking about expanding f(x) as the sum of the (n+1) terms, then it's easier to...
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    Uniform convergence and continuity

    Thanks for your reply. I think clearly y=x is not uniformly convergent, so I guess kn(x) isn't either? About the second one, I tried to work with the epsilon-delta definition, but the result seemed still depend on n. Could you please be a bit more specific how you would do it?
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    Uniform convergence and continuity

    1.kn (x) = 0 for x ≤ n x − n, x ≥ n, Is kn(x) uniformly convergent on R? I can show that it is uniformly convergent on any closed bounded interval [a,b], but I don't think it is on R. Could anyone please give me some hints how to prove it? 2.Fix 0 < η < 1. Suppose now...
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    Compactness theorem proof

    An expression of the compactness theorem for sets of sentences is that: let T be a set of sentences in L. Then T has a model iff every finite subset of T has a model. Could anyone give me some hints how to prove this? The first direction is straightforward: every model of T is a model of...
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    Uniform continuity

    Homework Statement Show that if h is continuous on [0, ∞) and uniformly continuous on [a, ∞), for some positive constant a, then h is uniformly continuous on [0, ∞). Homework Equations The Attempt at a Solution I'm thinking of using the epsilon-delta definition of continuity...
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    First order language in logic

    Thanks. What if I change Q into all non-positive rational numbers, then it has a maximum. Would that work? Also, that's only one l-structure. Could you give me some hints about the other two possible l-structure?
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    First order language in logic

    Thanks a lot for your help. I can only think of <Q, >>, which would make 1,2 true and 3 false. And I'm not sure that I've interpreted 3 correctly. Could you explain it a little bit more please?
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    First order language in logic

    1. Homework Statement Let L = {P } be a first-order language with a binary relation symbol P as only non-logical symbol. By exhibiting three suitable L-structures prove (informally) that no two of the following sentences logically implies the other (i) ∀x∀y∀z(P (x, y) → (P (y, z) → P (x...
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    Discontinuity at certain points

    Homework Statement 1.Find a function f : R → R which is discontinuous at the points of the set {1/n : n a positive integer} ∪ {0} but is continuous everywhere else. 2. Find a function g : R → R which is discontinuous at the points of the set {1/n : n a positive integer} but is continuous...
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    Convergent series

    Homework Statement Find a sequence (an) of positive real numbers such the sum of an from 1 to infinity is convergent but the number of k such that a(k+1)>ak divided by n tends to 1 as n tends to infinity. Homework Equations The Attempt at a Solution I don't have a clue how to find...
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    Complex number and power series

    I can write it as the sum of (z^n)*(1+w^n+w^2n)/n!, n from 0 to infinity. But I'm still not sure how to simplify 1+w^n+w^2n from 1+w+w^2=0. Could you explain it in a bit more details? Thanks a lot!
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    Complex number and power series

    Homework Statement Let ω be the complex number e^(2πi/3), Find the power series for e^z + e^(ωz) + e^((ω^2) z). Homework Equations The Attempt at a Solution I can show that 1+w+w^2=0, don't know if it would help. Could anyone please give me some hints? Any input is appreciated!
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    Convergence test

    I've got them. Thanks a lot!
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    Convergence test

    There are five statements: (a) If n^2 an → 0 as n → ∞ then ∑ an converges. (b) If n an→ 0 as n → ∞ then ∑an converges. (c) If ∑an converges, then ∑((an )^2)converges. (d) If ∑ an converges absolutely, then ∑((an )^2) converges. (e) If ∑an converges absolutely, then |an | < 1/n for all...
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    Homeomorphism and project space

    1. (1) (a) Let X be a topological space. Prove that the set Homeo(X) of home- omorphisms f : X → X becomes a group when endowed with the binary operation f ◦ g. (b) Let G be a subgroup of Homeo(X). Prove that the relation ‘xRG y ⇔ ∃g ∈ G such that g(x) = y ’ is an equivalence relation. (c)...
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    Compactness of closed unit ball

    thanks for the reply, but I'm still not sure about which non-convergent sequence to choose, would something like 1/n work? I just don't know how to use the complete norm here.
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    Real projective plane

    I'm working on a proof to show there exists an embedding of the real projective plane P R2 in R4. The initial setup is as follows: Let S2 denote the unit sphere in R3 given by S2 = {(x, y, z) ∈ R3 : x2 + y 2 + z 2 = 1}, and let f : S2 → R4 be defined by f (x, y, z) = (x2 − y 2 , xy, yz, zx)...
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    Compactness of closed unit ball

    Homework Statement Let l∞ be the space of bounded sequences of real numbers, endowed with the norm ∥x∥∞ = supn∈N |xn | , where x = (xn )n∈N . Prove that the closed unit ball of l∞ , B(0, 1) = {x ∈ l∞ ; ∥x∥∞ ≤ 1} , is not compact. Homework Equations The Attempt at a Solution I'm...
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    Rings problems

    1.Let R be a ring such that Z ⊂ R ⊂ Q. Show that R is a principal ideal domain. We show that Z is a principal ideal domain, so every ideal in Z which is also in R is principal. But I'm not sure how to use that R is contained in Q. 2. Proof that X^4+1 is reducible in Z/pZ [X] for every...
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    Second and third isomorphism theorem

    I've recently encountered some forms of the second and third isomorphism theorem, but I don't quite get them. Could anyone explain in a bit of details please? I guess my thought was not in the right direction or something. (Second isomorphism theorem) Let A be a subring and I an ideal of the...
  23. R

    Chinese remainder theorem

    (a) Let R and S be rings with groups of units R∗ and S ∗ respectively. Prove that (R × S)∗ = R∗ × S ∗ . (b) Prove that the group of units of Zn consists of all cosets of k with k coprime to n. Denote the order of (Zn )∗ by φ(n); this is Euler’s φ-function. (c) Now suppose that m and n are...
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    Connected component

    Thank you very much for your reply. I have a much clearer picture in my mind. Just one more question, to find the connected component for X = {(z, w) ∈ C2 ; z not equal to w} with the topology induced from C2, we still need to check to see if X is disconnected right? But it's hard to find two...
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    Connected component

    I'm not quite clear about this notion. Could anyone explain a little bit for me? Here is the definition: Let a be an arbitrary point in X . Then there exists a largest connected subset of X containing a, i.e. a set Ca such that: • a ∈ Ca and Ca is connected; • for any connected subset S of...
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    Darboux theorem

    Let I be an open interval in R and let f : I → R be a differentiable function. Let g : T → R be the function defined by g(x, y) =(f (x)−f (y))/(x-y) 1.Prove that g(T ) ⊂ f (I) ⊂ g(T ) (The last one should be the closure of g(T), but I can't type it here) 2. Show that f ′ (I) is an interval...
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    Connectedness in topology

    Thanks a lot for your guys. I wasn't that clear about the concept in the first place, but now I know how to handle this type of problems.
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    Connectedness in topology

    Let A, B be two connected subsets of a topological space X such that A intersects the closure of B . Prove that A ∪ B is connected. I can prove that the union of A and the closure of B is connected, but I don't know what to do next. Could anyone give me some hints or is there another way to...
  29. R

    Finding maximum and minimum value

    Thanks a lot! I finished it already:p
  30. R

    Finding maximum and minimum value

    Homework Statement We have x^2+y^2+z^2=1, and we want to find the maximum and minimum value of xz+xy-yz. Homework Equations The Attempt at a Solution I've simplified the original problem to this point, but I'm not sure what to do next. Could anyone give me some hints? Any help is...
  31. R

    Diagonalisability problem and others

    Thanks a lot! I can't believe I missed it in the first place. Could anyone give me some hints about problem 2?
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    Spectral theorem for self-adjoint linear transformations

    Let P,Q be self-adjoint linear transformations from V to V, Q is also positive-definite. Deduce that there exist scalars λ1 , . . . , λn and linearly independent vectors e1 , . . . , en in V such that, for i, j = 1, 2, . . . , n: (i) P ei = λi Qei ; (ii) <P ei , ej > = δi j λi ; (iii) <Qei ...
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    Diagonalisability problem and others

    Homework Statement 1.Prove that if A is a real matrix then At A is diagonalisable. 2. Given a known 3*3 matrix A, Calculate the maximum and minimum values of ||Ax|| on the sphere ||x|| = 1. Homework Equations The Attempt at a Solution For the first problem, I'm thinking of...
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