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11 Oct 2022: 17. Let L/K be an extension, and x,y L transcendental over K. ... Show that x is algebraicover K(y) iff y is algebraic over K(x).

12 Aug 2008: 2. Let A and B be n n matrices over a field F. ... 3. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

7 May 2013: we can let B bethe local coefficient system which over {0} BΣ corresponds to the module Z[X],where X is a free transitive Σ-set, and over the other path ... There is an isomorphism of Σ-sets SX =. X,. and so (m )B is the local coefficient system which

26 Oct 2015: 3. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the basis{x1,x2,x3,x4} for ... 9. Let A and B be nn matrices over a field F.

5 Nov 2006: 6. Modify the construction of the Hopf bundle given in Lectures, replacing C everywhere by R toobtain a rank one real vector bundle over S1. ... Show that the tangent bundle TS2 induces on Y a structureof a principal S1-bundle over S2.(ii) Show further

6 Oct 2019: map(B,M) = {families of (.) over B}. Then. H(M) = {characteristic classes of such families}. ... B. n0. BGLn(R).  =: K(R), algebraic K-theory13. The Madsen–Weiss theorem.

9 Nov 2023: Integrate. d[δQ[ξ] ξ θ(φ,δφ)] = 0. over a hypersurface extending from the bifurcation surface B tospatial infinity. ... d[δQ[ξ] ξ θ(φ,δφ)] = ξaδCaIntegration over the region of the horizon bounded by.

27 Feb 2008: Moreover, the restriction of a finitely generated A-module is finitelygenerated over B. ... action explained in the proof of Proposition 2.3. The latter module is just a directsum of dim V copies of ResAB(M) and is hence finitely generated over B.

27 Oct 2019: Hence i and j arehomotopy inverses. Corollary 3.2. All fibres of a fibration π : E B over a path-connected base are homotopy equivalent. ... 3.5. B as delooping. Recall that a principal G-bundle π : E B over a CW complex is a fibrationso there is a

5 Jun 2020: n1 obtained byminimizing the ideal Shannon code-lengths over the family {Qθ}θΘ,. ... Integrating over distortion-balls effectively changes a continuous alphabet intoa discrete one and log Q(B(xn1,D)) still makes sense as an idealized code-length.