Search

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.

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.

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.’.

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]).

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.

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.

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 Feb 2012: 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.

3 Feb 2011: 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.

24 Nov 2003: Deducethat there exists a pair of irrational numbers a, b such that ab is rational. ... c there is some x with a < x < b and f(x) = c.

25 Nov 2005: Show that f and g havethe same splitting field if and only if b = cmar for some c K and r N with gcd(r, m) = 1. ... Deduce that every finite abelian groupis the Galois group of some Galois extension K/Q.

23 Mar 2009: a = qb r. for some q, r R with r = 0 or d(r) < d(b).For example. ... N by. N(a b. 2. )= a2 2b2. (=. a b22) (a, b Z).

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. ... 7. Let S be an n n real symmetric matrix with Sk = I for some k 1.

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.

19 Dec 2014: Note that g M and P(P 2) hascardinality β for some β < κ (by strong inaccessibility). ... Say that A and B arepartially isomorphic, denoted A 'p B if some non-empty family F Part(A, B) of thepartial isomorphisms from A to B is a

19 Jan 2021: for every x X there exists some B B such that x B; and. ... for all B1,B2 B and for all x B1 B2 there exists some B3 B suchthat x B3 B1 B2.

11 Sep 2013: C/Λ2 are conformally equivalent if and only if thelattices are related by Λ2 = λΛ1 for some λ C. ... 5. Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if and only if τ2 =. (aτ1 b)/(cτ1 d), for some matrix(a bc d

26 Apr 2021: I hope you were not expecting to do it by induction on a or b!),However, one needs to take some thought. ... But of course GCD(a b, b) = GCD(a, b) so we get.

27 Nov 2019: Show that f and g have the same splitting field if andonly if b = cmar for some c K and r N with gcd(r,m) = 1. ... Deduce that every finite abelian group is the Galois group of some Galois extensionK/Q.

11 Jan 2024: Could there be some other Euclidean function φ making Z[3] into a Euclidean domain? ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written as rasbfor some r,s R.

8 Nov 2022: Show that H is normal in G if and only if H is a unionof some conjugacy classes of G. ... 1. |G|gG|Fix(g)|. Deduce that if G acts transitively and |X| > 1, then there is some g G with no xed point.How many distinct ways are there to

11 Sep 2013: 8. By considering the singularity at or otherwise, show that any injective analytic map f : C Chas the form f(z) = az b, for some a C and b C. ... conformally equivalent). 10. Define an equivalence relation on C by z w iff z = 2sw for some s Z.

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).

15 Oct 2021: for some b > 0, and invoke. the contraction mapping principle]. ...  CbT(1 T) ||w v||X. for some constant C depending on b but not T.

15 Oct 2021: 0. for some ai. 2 R. Show that there is a unique f : R! ... Show that:f(x) = g(x) p. k1(x). for some polynomial pk1 of order k 1.

15 Oct 2021: V 0[|E|2 |B|2. ] V µTµ0[F ]. 1. 4V 0. [|E|2 |B|2. ... T(X,Y ) = TσµνXµY νeσ. for some Crfunctions Tσµν := eσ [T(eµ,eν)].

17 Mar 2014: There is some overlap between these examples andthe examples in my lecture notes. ... 6. This example arose from some psychiatry data provided by Professor I.Goodyerof Cambridge University.

4 Feb 2021: BBB = {x : (B B)(x B)}. (a) Let B P() and C P(). ... B C) (BB. B) (CC. C). (b) This part seems to have caused problems for some.

8 Nov 2021: Show that H is normal in G if and only if H is a unionof some conjugacy classes of G. ... 1. |G|gG|Fix(g)|. Deduce that if G acts transitively and |X| > 1, then there is some g G with no xed point.How many distinct ways are there to

20 Oct 2007: f(z) =a(z z0) bc(z z0) d. ,. for some a,b,c,d,z0 C. ... 1 < |z| < 2}, {0 < |z| < 1}, {0 < |z| < }. 8. (i) Let R and S be some Riemann surfaces, f : R S a continuous map, and p a pointin R.

14 Jul 2024: 100. 1.59 Some tho’rts about the theory of wellfounded sets in NF. ... 228. 9.5 Some more recent tho’rts on Ehrenfeucht games for duality.

8 Apr 2021: 1 Stuff to Fit in. Second-order logic is the Work Of The Devil as we all know, but there arestill some points worth making. ... So how about in ZF? It’sinjective; is it a permutation? Well, no, since some of its values are complementsof sets.

7 Dec 2009: 5. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in some open ball around(a,b), and D2f = f/y exists at (a,b). ... 11. Show that there is a continuous square-root function on some neighbourhood of I in Mn;that is, show that there is an

24 Nov 2022: 7. Let f: R2 R and (a, b) R2. (a) Suppose that D1f exists and is continuous in some open ball around (a, b), and that. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a, b), and.

15 Oct 2021: Show that f and g have the same splitting field if andonly if b = cmar for some c K and r N with gcd(r,m) = 1. ... Deduce that every finite abelian group is the Galois group of some Galois extensionK/Q.

24 Jan 2015: C/Λ2 are conformally equivalent if and only if thelattices are related by Λ2 = λΛ1 for some λ C. ... 5. Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if and only if τ2 =. (aτ1 b)/(cτ1 d), for some matrix(a bc d

28 Jan 2009: Proposition 1.2 Isometries of E2. An orientation preserving isometry of E2 is:(a) The identity.(b) A translation.(c) A rotation about some point c E2. ... a) The identity.(b) A translation.(c) A rotation about some line. (d) A screw rotation, that is a

2 Oct 2006: Let kL = kK (b) for some b with bq = a kK kpK , and let u oLbe any lift of b. ... Assume thereexists a surjection (K′/K) (oK′/pλ)d1 for some λ 0. Then.

24 Jan 2015: 8. By considering the singularity at or otherwise, show that any injective analytic map f : C Chas the form f(z) = az b, for some a C and b C. ... conformally equivalent). 10. Define an equivalence relation on C by z w iff z = 2sw for some s Z.

9 Nov 2020: a) All words w {0, 1} consisting of either the string 01 repeated some number oftimes (possibly none), or the string 010 repeated some number of times (possiblynone). ... b) All words w {a, b, c} consisting of some number of a’s (possibly none),

5 Apr 2007: First note the following. Lemma 5.1. T is open iff T (B(1)) B(ǫ) for some ǫ > 0.Proof. ... Since by Theorem 5.3, we have. T (B(1)) B(δ),for some δ > 0, it follows that.

11 Nov 2019: Some more terminology: B is called the base and E the total space of this vector bundle. ... The ψβα’s can be taken from some vector bundle E over B, then P will be‘constructed from E’.

28 Nov 2012: F maps some neighbourhood U of x dif-feomorphically onto a neighbourhood V of (y,T(x)). ... for some function τ(s). (Note that 〈b, ḃ〉 = 0 since b has unit norm.).

21 Nov 2007: Questions 7–10 are more challenging than others and some parts certainly go beyond limitsof the examination. ... neous cubic polynomial in the generalized Weierstrass normal form XZ24Y 3AX2Y BX3,for some complex constants A, B, A3 27B2 6= 0.

10 Mar 2017: 3. Prove Cauchy’s mean value theorem: let f,g : [a,b] R be continuous functionswhich are differentiable on the open interval (a,b); show that for some c ... Show thatif f ′(a) < y < f ′(b) for some a < b in I and y R, then there exists x I witha <

20 Nov 2015: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... 7. Let S be an nn real symmetric matrix with Sk = I for some k 1.

15 Oct 2021: for some a < b. ... Lu = u in I, u(a) = u(b) = 0, (?). for some 2 R. Show that the Wronskian:. W(x) = p(x)u0(x)ũ(x) u(x)ũ0(x).

8 Mar 2024: a1α1. adαdq. for some a1,. ,ad Z. Proof. We take. q =(disc(α1,. , ... In what follows, we will exhibit asuitable α, but this requires some preparations.

17 Jan 2016: C/Λ2 are conformally equivalent if and only if thelattices are related by Λ2 = λΛ1 for some λ C. ... 5. Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if and only if τ2 =. (aτ1 b)/(cτ1 d), for some matrix(a bc d

10 Oct 2006: n}.How large can we make a family A of subsets of [n] subject to thecondition that for all A, A′ A, there is some B B with B A A′? ... n}.How large can we make a family A of subsets of [n] subject to thecondition that for all A, A′ A, there is some