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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.).

31 Mar 2024: Show further that if f : X Y is a continuous function between Polish spacesand f is injective on a Borel set B X, then f(B) is Borel. ... 14. Prove that a set A (in some Polish space) is analytic if and only if there exist closedsets An1,.,nk (indexed by

4 Dec 2014: 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.

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

23 Mar 2009: b = uac for some unit u. Then g̃ = uf k̃ and sog = ag̃ = f uak̃ and so f|g in R[X]. ... f (X) = af̃ (X) for some a R and primitive f̃ R[X], and so.

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

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

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 Dec 2013: 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.

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.

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’.

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.

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.

5 Jun 2020: for some appropriate rate function R(P,Q,D), except perhaps for a single value of Ddenoted Dmin. ... 1n. log Q̂nn(B(Xn1 ,D)). a.s. infQΘ1n. log Qn(B(Xn1 ,D)) n,. for some sequence (n)n1 with n 0 as n.

15 Oct 2021: i) If x S, then there exists some B β with x B.ii) If B1, B2 β, then for every x B1 B2 there exists B β with:. ... x B B B1 B2. b) Conversely, suppose that one is given a set S and a collection β of subsets of Ssatisfying i), ii) above.

12 Nov 2012: B) A Theorem of Mertens states that. p6x. log p. p= log x O(1). ... where b|a. At leastcompute some examples and find out what generally is the real represented.

5 Jun 2020: Corollary 2.2: Fix some b > 0 consider the rates RN :=. 1The result of [24] is actually for source coding with side information,which subsumes the fixed-length lossless source setting. ... fix some b R+ and define τN :=. σ2(p)N b and RN :=. H(p) τN.

21 Nov 2016: 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.

7 May 2008: x =V for some n, and of course this cannot happen in ZF. ). ... If w is a molecular term of rank n then, in full generality, it is w′g forsome constant g and some boolean combination w′ of B(zi) for various i.

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.

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),

15 Oct 2021: some function which maps an interval (a,b) with a < b to thereal line R. ... Thenf(x) = C for some C R. Proof. Pick c,d with a < c < d < b.

15 Oct 2021: Show that ifeb|k|. f̂(k) 2 L2(R) for some b > 0 then f is real analytic at each point in R. ... B to mean that there exists a universal constant C such that A  CB.

11 Feb 2021: Motivation. Let F : (Schemes)opp (Sets) be a functor represented by some(noetherian) scheme M. ... A ĈO, and representable if F (A) = Hom(R,A) for some R ĈO.

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

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.

18 Mar 2021: 1 Unramified up to twist: τ = (ψ ψ, 0) for some characterψ : IK C. ... 4 Irreducible: τ = (r0, 0) where r0 is the restriction of some irreducibleWeil-Deligne representation.

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 <

22 Jan 2015: Xn be independent Poisson random variables, with Xi having meaniθ, for some θ > 0. ... b) For some n > 2, let X1,. ,Xn be iid with Xi Exponential(θ).

15 Oct 2021: supx(a,b). f ′i(x) Mfor some M independent of i. Show that F is uniformly equicontinuous. ... supx[a,b]. |fi(x)| K. for some K independent of i. Then F has a subsequence which converges uniformly.

12 Sep 2023: If r = 3 then ker H = 〈e12,e13,ae23 be14〉 for some a,b k. ... If r = 3 then ker H = 〈e12,e23 ae13,e14 be13〉 for some a,b k.

12 Mar 2008: If g is another such function then f(z) = g(z) exp h(z)for some entire function h. ... If g were another such function then g/f would be an entire function with no zeros and thereforeequal to exp h for some entire function h.

17 Jan 2016: 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.

27 Apr 2016: Automorphy of some residually S5 Galois representations. Chandrashekhar B. Khare and Jack A. ... 5 3000)x4 (. 5 5)x/2 (1440. 5 3600). 4 Some group theory.

15 Jan 2021: some a,b 0. Hence prove that the class group of K is cyclic and find itsorder.

3 Feb 2006: 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.

21 May 2005: χ(z w) χ(z) χ(w). Remark for readers of Ahlfors or Jones & Singerman: their σ and ζ are not quite the same asψ and χ here, but for some ... f(z) =a(z z0) bc(z z0) d. ,. for some a,b,c,d,z0 C.

21 Oct 2006: 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.

20 Jan 2021: 6. Let (An : n N) be a sequence of events in some probability space (, F, P). ... P(Xn = 0) 12π. h. (b) Show further that, for all x R,.

23 Jan 2019: As described above, by a theorem of Borel[Bor74] the maps (3.1) are isomorphisms, so ch4kd(ξH) is equal to some poly-nomial p(σ1,σ2,. , ... Hi(B; Q) 0.Now s(τc)|E = 0 so (τc)|E = u (E)x for some x H|c|d(B; Q).

10 Nov 2021: 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),

15 Oct 2021: g2Zngv. Show that if 2 D(Rn) with supp K for some compact K Rn then. ... u[] =X. g2Agv[],. for some finite set A Zn which depends only on K.

18 Sep 2014: components. Consider a map. : 1()! Z=pZ. for some prime p that maps all the boundary. ... If G F and H G then H is. measure-equivalent to some free group.

5 Jun 2020: This makes a direct comparison possible and enables us topresent some theoretical results. ... There exists some number K such that k(n) < K for all n.

15 Feb 2016: 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.

12 Aug 2008: vary in difficulty but cover some instructive points. 1. For what values of a and b does the system of simultaneous linear equations. ... What are the eigenvalues and eigenvectors?Show that if an endomorphism β of V commutes with α then β = p(α) for

6 Mar 2023: The results of4 then apply to give us some information about the natural map φ′ : M (X/B, τ ) M sch(X/B, τ ). Unfortunately, in general it is difficult to ... 1We recall that H2(X) represents some choice of group of curve classes.

22 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 Oct 2006: The proof has three stages. First, we briefly review some facts about quasirandom. ... As promised, let us briefly review some of the standard theory of quasirandomness,.

4 Feb 2008: 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.