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11 Nov 2020: We now let µ X(T) and let b G(L) such that [b] B(G,µ). ... Thenumber N (µ,b) depends on b only via its σ-conjugacy class [b] B(G).

20 Apr 2007: ii) We say that f : A B is surjective3 if given b B there exists ana A such that f (a) = b. ... leastone solution for each b B; f is bijective if the equation f (a) = b has exactlyone solution for each b B.

25 Jun 2015: Then we may choose smooth functions ϕi : M [0, 1] such that supp(ϕi) int(Vi) and p(ν K) (. ϕi)1(1) and choose B =. B(A, ϕ1,. , ϕr) A ... Our choice of B guarantees that the linearcombination s =. ϕisi has s A.

14 Mar 2005: Proof. Define B [N]2 to be the set of all (x,y) such that xy A. ... labelling vertices of a cube (as they would if a, b and c denoted three orthogonal vectors.

10 Feb 2023: Ṽ = AtC, Ẽ = B.6. • The endpoints of an edge b B are φ(b) A and ψ(b) C.The graph Γ̃Y is bipartite, and therefore has a canonical structure ... Then f inducesmaps A′ A, B′ B, C′ C which we also denote by f.

5 Jun 2020: d sup. jP B A P BP AjP BP A. A B B Bd. ... d supjP B A P BP Aj A B B Bd.

25 Oct 2011: 4. Let AB be a segment of straight line in the plane with endpoints A and B and let be afixed number strictly greater than the length of AB. ... ϕ(s,v) = α(s) r(n(s) cos v b(s) sin v. ),. where n = n(s) and b = b(s) are the normal and binormal vectors

5 Jun 2020: Theorem 2.2:Let Φ = {Φi : i 0} be anarbitraryMarkov chain with values onX, let B B be fixedmeasurable set, and letF1,F2,. ... n 1)H(γ‖γ1 P), (32). where γ is a bivariate probability measure on(X X,B B), whose marginalsγ1 andγ2 satisfy,.

6 Mar 2008: 1. Which of the following propositions are tautologies?(a) (A B) (B A). ... b) A A.(c) (A B) (B A). (d) (A B) ((A B) A).

11 Feb 2014: Now let γ : [a,b] B(p) be a smooth path. If Lg(γ) and L(γ) are the lengthsof γ with respect to g and the Euclidean metric, then. ... theorem implies theremust be some point t [a,b] with d(γ(t)) = /2.

11 Nov 2019: Define an equivalencerelation between elements (b,h) UαG, (b′,h′) UβG, so that (b,h) (b′,h′) preciselyif b′ = b and h′ = ψβα(b)h. ... point b B is determined by the value s(b) at that point.

19 Jan 2021: 11. Let T B(B) compact and λ 6 (σp(T) {0}). Show that T λ is bounded below. ... 15. Given T B(B) define r1(T) := supλσ(T) |λ|. Show that the limit r2(T) :=limn |||Tn|||.

27 Nov 2012: Let (M,g) be a Riemannian manifold. A curve γ : (a,b) Mis said to be a geodesic if it locally minimises arc length, i.e. ... integral curve of V with γ(0) =, then T a < b T+ andγ|(a,b) = γ̃.

16 Mar 2021: ii) Show that iA iB : H̃(A) H̃(B) H̃(M {}) is an isomorphism. ... by pulling backE F B B along the diagonal; relate Th(E F) to Th(E) and Th(F)].

9 Jun 2008: f (z) = 1 ab. cz. a(a 1) b(b 1)1 2 c(c 1). ... b) (1 z)a = F (a, b, b; z).(c) sin1 z = zF ( 12 ,. 12 ,. 32 ; z. 2).6. Consider the matrix form of the Riemann hypergeometric differential equation:.

16 Mar 2021: F i //. Ep //. p. B. {b0} // B B. ... for some monic polynomial p(x) H(B; Z)[x]. Homotopy Theory Example sheet 3.

22 Oct 2017: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... a) f(x) = sin x2 (b) f(x) = inf{|xn2| |n Z} (c) (sin x3)/(x 1).

6 Mar 2023: 2.2) Σ(C) = Γ(G,). // Σ(X). Σ(W) = ω = QR // Σ(B). ... stacks. We remind the reader that allproducts in this square are taken over B.

22 Oct 2020: and for any [b] B(G,{µ}),there exists a straight representative in the admissible set of µ (see [16]). ... Fornonbasic [b] B(G,{µ}), there is another obstruction, coming from the non-uniqueness of thestraight representatives of [b].

5 Oct 2022: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... 1.8. Let B be a Borel subset of the interval [0, 1].