Qualifying Examination Intended to Promote Students′ Interest in Basic Engineering Course
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چکیده
منابع مشابه
Qualifying Examination
so ‖f‖p ≤ ‖f‖q · μ(X) 1 p − 1 q . Hence if f is in Lq, the left-hand side is finite hence so is the right-hand side, so f is in Lp. Also, the inequality shows that if ‖f‖p is small then ‖f‖q is also small, hence the inclusion Lq ↪→ Lp is continuous 2. Let X ⊂ Pn be an irreducible projective variety of dimension k, G(`, n) the Grassmannian of `-planes in Pn for some ` < n− k, and C(X) ⊂ G(`, n) ...
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2. (T) Let CPn be complex projective n-space. (a) Describe the cohomology ring H∗(CPn,Z) and, using the Kunneth formula, the cohomology ring H∗(CPn × CPn,Z). (b) Let ∆ ⊂ CPn×CPn be the diagonal, and δ = i∗[∆] ∈ H2n(CP×CP,Z) the image of the fundamental class of ∆ under the inclusion i : ∆ → CPn × CPn. In terms of your description of H∗(CPn × CPn,Z) above, find the Poincaré dual δ∗ ∈ H2n(CPn × C...
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1. (a) Prove that the Galois group G of the polynomial X6 + 3 over Q is of order 6. (b) Show that in fact G is isomorphic to the symmetric group S3. (c) Is there a prime number p such that X6 + 3 is irreducible over the finite field of order p? Solution. We initially work over any field k in which the polynomial X6 + 3 is irreducible. Clearly k cannot have characteristic 2 or 3. Let α be a root...
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Solution: By the adjunction formula, the canonical divisor class is KC = OC(d− 3), that is, plane curves of degree d− 3 cut out canonical divisors on C. It follows that if d ≥ 4 then any two points p, q ∈ C impose independent conditions on the canonical series |KC |; that is, h(KC(−p − q)) = g − 2, so by Riemann-Roch h(OC(p+ q)) = 1, i.e., C is not hyperelliptic. Similarly, if d ≥ 5 then any th...
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Solution: Since V is locally trivializable and M is compact, one can find a finite open cover Ui, i = 1, . . . , n, of M and trivializations Ti : V |Ui → Rk. Thus, each Ti is a smooth map which restricts to a linear isomorphism on each fiber of V |Ui . Next, choose a smooth partition of unity {fi}i=1,...,n subordinate to the cover {Ui}i=1,...,n. If p : V → M is the projection to the base, then ...
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ژورنال
عنوان ژورنال: Journal of JSEE
سال: 2009
ISSN: 1341-2167,1881-0764
DOI: 10.4307/jsee.57.5_40