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17 May 2024: On the average number of 2-Selmer elements of elliptic curves over F. ... Elliptic curves over Q. are modular. J. Eur. Math. Soc. 21 (2019), no.

5 May 2004: Finally, suppose that the average of f is significantly higher over A than it is over B. ... This probability increases with t, and the increase is very considerable when t is a couple of standard deviations higher than the average of f over B.

11 Jan 2024: 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 all x λ,. ... yet appeared? (b) What is the probability that the entire culture dies out eventually?

29 Jan 2002: It says that the dimension of Q(b) over Q equals the dimension of Q(a) over Q times the dimension of Q(b) over Q(a). ... But what is the dimension of Q(b) over Q(a)? It means the dimension of the vector space of all linear combinations of powers of b,

12 Oct 2023: Write down its negation insymbolic form. (Here m, n, a, b should be understood as ranging over all natural numbers.). ... m n a b (n m) [(a = 1) (b = 1) (ab 6= n)].

24 Nov 2023: 1. The square matrices A and B over the field F are congruent if B = PT AP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

25 Apr 2002: 2. =b? To answer that, we need to put it back in the original form, by multiplying out the bracket and taking b over to the left hand side. ... If we wish to carry this approach over to the cubic, then we should have clear answers to the following two

30 Oct 2023: 5. Let A and B be n n matrices over a field F. ... Which over R? 8. Find the eigenvalues and give bases for the eigenspaces of the following complex matrices: 1 1 00 3 20 1 0.

10 Oct 2023: Suppose that α has minimal polynomial f over Q, and let β = α4.Find a, b, c Q such that β = a bα cα2. ... The elements α andβ are then said to be algebraically dependent over K.].

26 Jun 2024: 2019). 148,. 103. (doi: 10.1090/proc/14749). Cut-off phenomenon for the axb Markov chain over a finite field. ... M Gross, B Siebert. (2019). Explicit moduli spaces for congruences of elliptic curves.

1 Nov 2021: DIFFERENTIAL GEOMETRY (PART III) MICHAELMAS 2021. EXAMPLE SHEET 2. 1. Let T , U, and V be finite-dimensional vector spaces over a field K. ... Show that M M is trivial.(b) Show that for all n we have TSn R = Rn1 over Sn.

23 Oct 2023: Prof. Dr. B. Löwe. Example Sheet #2. (14) Work over the alphabet Σ = {0, 1} and for n 1 consider. ... 16) Let L and M be languages over an alphabet Σ and consider the set equation X = LX M.

21 Nov 2018: form over R is equivalent to a direct sum of copies of the form(. ... b) Prove that if E and F are oriented vector bundles over a space X, their Euler classessatisfy eEF = eE eF.

11 Nov 2018: bundle of realrank one) over the circle, and prove that bundle is non-trivial. ... 7. Let p : E B be a fibre bundle over a path-connected space B with fibre F = p1(b).Suppose that H(F) is finitely generated and free, and that

29 Jan 2010: More generally, for any compact open subgroup K of Gf , there is a modular curve MK defined over Qwith. ... Thus over Z[1/q], uq is nonvanishing, and so in particular uq O(Y0(q)Q) O.

27 Feb 2015: Nj=1 ‖γ(tj) γ(tj1)‖ where the sup is taken over all. finite partitions 0 = t0 < t1 <. < tN = 1. (i) Give an example for which (γ) =. If γ is continuously ... b) = inf (γ), wherethe inf is taken over all continuous γ: [0, 1] E with γ(0) = a,

19 May 2003: n then. B = P1AP. Corollary 3.3 Let V be an n dimensional vector space over F. ... 15. Theorem 3.10 If A and B are n n matrices over F then detAdetB =det(AB).

29 Mar 2011: affordsus no variables (as it might be ‘F’ and ‘G’) to range over the second orderobjects captured by the branching quantifiers which would enable us to say‘F = G’. ... 8). Let us instantiate ‘u’ for the lower-case variables in (8)

29 Jan 2010: The vectors {ug | g G} span Sp over k, since φ is a generator forIndGB δ. ... We work in the category MQ Q of Chow motives over Qwith coefficients in Q.

19 Dec 2014: A filter F over κ is normal if for every {Xα F : α < κ}, the diagonal. ... ii) Prove Ulam’s theorem: there is no κ+ -saturated κ+ -complete ideal over κ+.