303rd Bombardment Group by Brian D O'Neill, Mark Styling

By Brian D O'Neill, Mark Styling

The 1st name within the Elite devices sequence to accommodate an American bombardment team, this identify makes a speciality of the 303rd BG, dubbed the 'Hells Angels.' one of many first actual B-17 devices assigned to the newly created 8th Air strength in England in September 1942, the 303rd used to be within the leading edge of the sunlight bombing crusade via to VE-Day. offered a distinctive Unit quotation in January 1944, the 303rd additionally had of its aircrewmen awarded with the Medal of Honor, Americas final army ornament. Brian O Neill brings the group's vibrant strive against historical past to lifestyles with a mixture of first-hand debts, uncooked information and concise project narrative.

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Let V be a complex vector space with a linear circle action that fixes only the origin. For any invariant Hermitian inner product, ·, · , the function Ψ(v) = v, v is an abstract moment map. It is proper if and only if ·, · is positive or negative definite. mj |zj |2 for the Hamiltonian circle action In particular, the moment map 12 on Cn with weights −mj ∈ Z is proper if and only if all the mj ’s are positive or all are negative. More generally, for a torus action on Cn with weights −αj ∈ Z∗G , the moment map 21 |zj |2 αj is proper if and only if there exists η ∈ g such that αj (η) > 0 for all j.

Similarly, the Duistermaat–Heckman integral I(ξ) can be replaced (up to sign) by the Fourier transform DHM . Concretely, the Liouville distribution and the Duistermaat–Heckman distribution are the continuous linear functionals f→ f ω n /n! and M ϕ→ g∗ ϕ DHM = M (ϕ ◦ Φ) ω n /n! on the spaces Cc∞ (M ) and Cc∞ (g∗ ) of compactly supported smooth functions. The Fourier transform DHM associates to a function S on g the value g∗ S(α) DHM = S(ξ)e−i g∗ g α,ξ S(ξ)I(−ξ)dξ”. dξ DHM = “ g This is a distribution (in fact, a tempered distribution) if DHM is a tempered distribution (see [Rudi]).

We may fix a convex open subset U of g∗ and assume that the moment image is contained in U and that the moment map is proper as a map to U . The convexity, connectedness, and stability theorems continue to hold; see [LMTW]. We can also work with proper Hamiltonian cobordisms over U . An interesting observation is that cobordism is a local notion: if two Hamiltonian spaces are properly cobordant over U and over V , then they are properly cobordant over U ∪ V (possibly through an orbifold). 5. HAMILTONIAN COMPLEX COBORDISMS 29 The idea of the proof is to “cut” (a la Lerman [Ler1]) each manifold into pieces, each of which maps properly to U or V , and note that the result of the cutting is cobordant to the original manifold.

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