Search

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.

5 Jun 2020: nlog Qn(B(X. n1 , D)) some finite R bits/symbol, w.p.1. – This suggests a natural lossy analog for the Shannon code-lengths:. ... with either: (a) n(θ) =. dim(Qθ)2 log n. (b) n(θ) =. (θ) for θ in some countable Γ Θ;. otherwiseConsistency? Does

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.

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.

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

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.

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.

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),

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,.

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.

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.

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.

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

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.

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.

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.

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.

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.

16 Oct 2019: 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,.

6 Mar 2023: Hence deduce that two analytic toriC/Λ1 and C/Λ2 are conformally equivalent if and only if the lattices are relatedby Λ2 = λΛ1 for some λ C. ... τ2 = aτ1 b. cτ1 d. for some matrix (a bc d. )

6 Mar 2023: Hence deduce that two analytic toriC/Λ1 and C/Λ2 are conformally equivalent if and only if the lattices are relatedby Λ2 = λΛ1 for some λ C. ... τ2 = aτ1 b. cτ1 d. for some matrix (a bc d. )

4 Oct 2017: 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

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.

11 Nov 2018: 4. Let K be a field containing a primitive mth root of unity for some m > 1. ... Show that f andg have the same splitting field if and only if b = cmar for some c K and r N withgcd(r,m) = 1.

28 Mar 2010: Deduce that every finiteabelian group is the Galois group of some Galois extension K/Q. ... only if b = cnar for some c Kand r N with gcd(r, n) = 1.

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 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

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.

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.

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 Oct 2021: ii) M R n N x S |fn(x)| 6 M. (b) In this case assume that S is a metric space. ... Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f isbounded.

21 May 2005: for some x,y Z with (x,y) = 1, then there exist a,b Z such that p = a2 b2. ... subgroup of Q/Q2 generated by a,b. Determine whether the field Q(.

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.

6 Mar 2023: By considering the singularity at or otherwise, show that any injective an-alytic map f C C has the form f(z) = az b, for some a C and b ... 9. Define an equivalence relation on C by z w iff z = 2sw for some s Z.Show that the quotient space R = C/ has

4 Oct 2017: 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: 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.

7 Nov 2016: 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), followed

15 Oct 2021: b. a) Show that the sequences (f(xj. )). 1j=0. and (f(yj. )). ... sup. x2(a,b). f. 0i. (x).  M. for some M independent of i.

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 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.

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.

28 Feb 2022: some a,b 0. Hence prove that the class group of K is cyclic and find itsorder.

16 Jan 2024: γABA′B′ = φABεA′B′ ψA′B′εAB. for some symmetric spinors φ an ψ. ... for some complex Zα, α = 0, 1, 2, 3, where Z2 6= 0.

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.

28 Feb 2019: b) For some n > 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T, and. ... 4. For some n 2 let X1,. ,Xniid U[θ, 2θ], for some θ > 0.

6 Aug 2021: i) fn f pointwise ae. ii) |fn|6 g ae for some integrable g. ... Exercise 1.7. Suppose that f Lr(Rn) for some 1 6 r <. Show that‖f‖L = limp‖f‖Lp.[Hint: you may find the estimates in Exercises 1.2 b),

15 Nov 2019: 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),

6 Mar 2023: By considering the singularity at or otherwise, show that any injective an-alytic map f C C has the form f(z) = az b, for some a C and b ... 9. Define an equivalence relation on C by z w iff z = 2sw for some s Z.Show that the quotient space R = C/ has