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4 Jun 2024: A sequence an iscalled subadditive if amn am an for all m,n. ... d(α(t),β(t)) 2δ d(α(L),β(L)). for all t [0,L]. 7. Let X be a proper metric space (i.e.

26 Feb 2024: a) Let L be a transverse collection of n lines in the hyperbolic plane, with n > 1.Then, there exists a number R = R(L) > 0 such any line intersecting all ... b) For each k > 0, there are finitely many π1Sg-orbits of transverse collections of klines in L

11 Jan 2024: Set. A = {ω : ω An infinitely often}, B = {ω : ω An for all sufficiently large n}. ... Calculate the probability thatm given people will all be on the committee (a) directly, (b) using the inclusion-exclusion formula.Deduce that (. n mr m. )=. mj=0.

10 Apr 2024: The argument. All spaces of embeddings and so on are given by (realisation of)simplicial sets. ... Embt(D2, D4) Immt(D. 2, D4). is cartesian by [Las76, (a) and (b)] (Lashof supposes in this paper that the ambientdimension is 5, but for this particular

4 Jun 2024: homology spheres discussed in lectures. (a) Show that z is central in Gn, for all n. ... F({a1,. ,an}). Prove that F2 contains subgroupsisomorphic to Fn for all n 2.

24 Jan 2024: Which of (1)–(4)must be true? (1) If f is increasing then f ′(x) > 0 for all x (a,b).(2) If f ′(x) > 0 for all x (a,b) ... then f is increasing.(3) If f is strictly increasing then f ′(x) > 0 for all x (a,b).(4) If f ′(x) > 0 for all x (a,b)

11 Jan 2024: a) Show that, for all p (0, ) and all x (0, ),. P(|X| x) E(|X|p)xp. (b) Show that, for all β 0,P(X x) E(eβX)eβx. ... 3. Let X be a Poisson random variable of parameter λ (0, ). (a) By optimizing the estimate of Question 2(b) over β, show that, for

8 Feb 2024: Show that f is a polynomial, of degree k, if and only if there is aconstant M for which |f(z)| < M(1 |z|)k for all z. ... Show that if thereexists k such that |f(z)| |z|k for all z with |z| sufficiently large, then f is a rational function (i.e.

8 Feb 2024: Show that the sum of the residuesof f at all its poles equals zero. ... 6. Let p(z) = z5 z. Find all z such that |z| = 1 and Im p(z) = 0.

20 Feb 2024: 𝑓 𝑗 (𝑥)𝑑𝑥 𝐸. 𝑓 (𝑥)𝑑𝑥. for all bounded measurable sets 𝐸 R𝑛 of positive measure. ... i) Show that 𝑥𝑖 𝑥 if and only if (𝑦,𝑥𝑖) (𝑦,𝑥) for all 𝑦 𝐻.

5 Feb 2024: 8 Prove/verify the following statements.(i) If K is a field containing F2, then (a b)2 = a2 b2 for all a,b K.(ii) If P F2[X] and ... K is a field containing F2, then P(a)2 = P(a2) for all a K.(iii) Let K be a field containing F2 in which X7 1 factorises

11 Mar 2024: Both are proofs of the samestatement. Proposition. A function f : X Y is injective if and only if f(A B) =f(A) f(B) for all subsets A, B X. ... After all, there are no other symbolsyet introduced, so there is nothing on which ϵ can depend!

28 Jan 2024: N′I(α) =. {NI(α) if all elements of Iα are orthogonal,. 2NI(α) otherwise. ... FGi,F) = B(Gi,Fi) for all i.

23 May 2024: squarefree as an element of Op.For b B(F ), we say that (b) is squarefree if (b) is squarefree at p for all finite primes p. ... Tho15, Lemma 2.8]):. Lemma 2.11. There exists a non-zero N0 OF such that for all primes p and for all b B(Op) wehave N0 κb V

8 Mar 2024: In the field Q(α), all elements have a unique representation of theform. ... 6 PÉTER P. VARJÚ. For all i = 1,. ,n, there are integers mi,1,.

16 Jan 2024: Nevertheless, even in the case of a non-zero winding number, not all is lost. ... Proposition 2 holds for almost all w W whenever F, f and f̃ depend on additional parametersw, by virtue of proposition 1.

31 Jan 2024: i) For all a,b,c G, we have (a b) c = a (b c). ... 12 H. Definition (Abelian group). A group G is abelian if a b = b a for all a,b G.

31 Jan 2024: are continuous. Almost all of these situations will be dealt with usingthe following convenient lemma. ... Thus it requires all the technical hypotheses of Theorem 3.3.1. Proof.

29 May 2024: b and a 6= b is not trichotomous.Note that a b means: (x a) (x b) for all x X, which includes thecase a = b. ... consists of all countable ordinals. Set ω1 = sup B. If ω1 is countable, then so isω1 , and hence ω.

5 Jun 2024: X01 := X0 X1 and Y01 := Y0 Y1 are all finite CW-complexes, and. ... All throughout let Md denote a closed smooth manifold of dimension d 5.