Search

12 Oct 2023: a, b], and suppose that fn converges pointwise to a continuous function f. ... Show that there is some subinterval [a, b] of [0, 1] with a < b on which (fn) is uniformly.

23 Mar 2009: As R is a PID we have (a, b) = d for some d R. ... So. i=1. (ai) = (b). 1. for some b R. So b (ak) for some k, and then.

23 May 2024: Now, consider theset. B′(Op) = {b B(Op) | vp(pdi (b)) < di for some i}. ... Write ξ = aξ′ for some a A(k). Writing a = b2 for some b A(k), we haveξ = b b ξ′ = b ξ′ b1.

17 May 2024: C93. “Combining information from different sources: Some surprising phenomena,” with B. ... C95. “Mutual information, synergy and some curious phenomena for simple channels,” with B.

29 Jan 2010: Proof. We can assume that c = [Z′/k] for some integral closed subscheme Z′/k. ... 2.1.1. We recall some basic facts about modular curves (see [4] or [9]).

27 Oct 2016: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... vn} C such that{B(v1),. ,B(vn)} cover C. (b) Show that there exists some > 0 such that for every v C, B(v) Uα for some α.Deduce that

22 Jul 2007: There is some pointto this question, as the following example will make clear. ... We can alsoprove TI 6 TJ 6 TI as follows. Since I 6 J there is some axiom B in I which is not in J.

21 Nov 2023: 4. Let K be a field containing a primitive mth root of unity for some m > 1. ... Showthat f and g have the same splitting field if and only if b = cmar for some c Kand r N with gcd(r,m) = 1.

19 Jan 2021: a) Show that there is a surjection π1Sg G, for some g.(b) Show that G is a subgroup of Mod(Sh), for some h. ... a) Prove that TαTβ(α) = β. (b) Prove the braid relation: TαTβTα = TβTαTβ.

17 May 2024: C93. “Combining information from different sources: Some surprising phenomena,” with B. ... C95. “Mutual information, synergy and some curious phenomena for simple channels,” with B.

11 Mar 2024: f(B) then y = f(a) for some a A andy = f(b) for some b B but then f(a) = y = f(b) so a = b A Band y ... If y f(A) f(B) then y = f(a) for some a A and y = f(b) for someb B.

29 Jan 2010: Show that f (x) = 0 for allx. 8. Let f : [a, b] R be bounded. ... 13. Let f : R R be a function which has the intermediate value property: If f (a) < c < f (b),then f (x) = c for some x between a and b.

15 Oct 2009: k=1 uk(x0) converges at some point x0 [a, b] and. (ii) the series of derivatives. ... Then for x [a, b] we can write g(x) g(x0) = g′(ξ)(x x0) with some ξ [x0, x] [a, b] (or ξ [x, x0] [a, b]), using the

19 Jan 2022: Prove that f is continuous on (a,b). Must it be continuous at a andb too? ... 13. Let f : R R be a function which has the intermediate value property: Iff(a) < c < f(b), then f(x) = c for some x between a and b.

8 Aug 2021: Replace‘Similarly, we can can take a = sin φ, b = cos φ for some real φ.’ by. ... Similarly, we can can take c = sin φ, d = cos φ for some real φ.’.

19 Dec 2003: k}. Show that equality holds for some. fixed r if and only if B(p,r) is isometric to the ball of radius r in the constant curvature space formof curvature ... limr. V (p,r)wnrn. = 1. for some p M, must be isometric to the Euclidean space.

22 Nov 2022: 11. Let K be a field containing a primitive mth root of unity for some m > 1. ... Show that f and g have the same splittingfield if and only if b = cmar for some c K and r N with gcd(r,m) = 1.

6 Feb 2021: Prove that f is continuous on (a,b). Must it be continuous at a andb too? ... 13. Let f : R R be a function which has the intermediate value property: Iff(a) < c < f(b), then f(x) = c for some x between a and b.

6 Oct 2019: Why cohomology? A moduli space M is anything which classifies some kind of families:. ... It always depends on the specifics of the groups Gn, though thereare some general principles.

21 Apr 2013: 4. We can get round this in the usual way(cf. part (b) proof of [11, 5.2]): take X = X′ = X(3) for some auxiliary integerN 3, and define ... If p 1 mod 3, then for some αp Zp (ordp(αp) = 2).