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4 Aug 2022: Φ(r) b]. Useful expressions (c 6 a 6 b): b. adx. ... π. (a c)(b c),. b. adx. (xa)(bx). x c = π[a b.

4 Aug 2022: 3. 3 Let AB denote the disjoint occurrence of two events A,B G. ... b) Prove, carefully, that on the d-dimensional integer lattice Zd, for d > 2,.

8 Aug 2022: tv v v 2ez v = Φt 1. ρΨ B B. ... 4πρ,. tB v B = B v ( v)B,. where Φt = (3/2)2x2 (1/2)2z2 is the tidal potential and Ψ is the combined gasand magnetic pressure.

4 Aug 2022: b) By considering the dual 1-forms, show that on U we have. ... Let A be a connection on a principal G-bundle π : P B.(b) Define the horizontal distribution H and curvature F of A, and show that H is.

4 Aug 2022: Recall that an ultrafilter on A is a set U PAsuch that (i) B U and B B′ imply B′ U; (ii) B U and B′ U imply B B′ ... Let (f, g) : A B be a reflexive parallel pair in an additive category C.

8 Aug 2022: the disk midplane is B µ/r3, where µ is the dipole moment. ... 12A2 = A A AA,. (AB) = A(B) B(A) B AA B.].

4 Aug 2022: ρ. (u. t+ (u )u. )= ρΦ p 1. 4π(B) B, (3). ... b) Show that j attains a maximum value at some point as u transitions from sub-sonicto super-sonic regimes.

4 Aug 2022: F[φ] =. [a. 2φ2. b. 4φ4. κ̃. 2(φ)2 γ. 2(2φ)2. ]dr. ... Find the strength of nematic ordering,λ0, in terms of a and b.

4 Aug 2022: B)).]. (b) State what is meant by (i) the data-processing inequality for the quantum relativeentropy, (ii) joint convexity of the quantum relative entropy, and (iii) the strongsubadditivity inequality of the ... for anylinear CPTP map. Λ : B(H) B(K)

4 Aug 2022: y b. x= 0,. where b = b(x,y,t), ψ = ψ(x,y,t), 1 is a constant parameter and. ... 10]. (c) State Parseval’s identity and use the results from part (b) to evaluate.