Search

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.

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.

22 Oct 2020: For some examples of cases when connectedparahorics arise in the study of Shimura varieties we refer to [PR09, 1.b.3] for the case of ramifiedunitary groups and [PR05] for cases ... We assume G isof the form Gx for some x B(G,Qp).

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: 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.

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.

30 Mar 2022: 42. 7.3 Some stacks. 44. 7.4 Orbits of square-free discriminant. 46. ... We then recall and extend some of its basic properties in 3.2–3.6.In 3.7 we introduce the corresponding family of curves C B and in 3.8

19 Dec 2022: We finish in Section 4 by covering some technical points and variants of our results. ... Instead, webegin in Section 2.3 by explaining (iii) and (v), after a broader discussion to provide some motivationand context for the construction of ks.

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 <

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.

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.

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.

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

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,.

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.

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.

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.

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.

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.

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.

8 Oct 2021: We may think of SK(G) as a moduli space of abelian varieties with multiplication by some max-imal order OB in B. ... G′S̃(Q) := {g GLDS (W) | ψ(xg,yg) = c(g)ψ(x,y), for some c(g) Q}.

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.

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.

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

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.

25 Nov 2023: Some more questions. 15. Show that the direct sum A B of C-algebras A and B is a C-algebra withcoordinate-wise operations and with ‖(x,y)‖ = max. ... norm ‖‖. Let ‖‖1 be some (possibly incomplete) algebra-normon A. Show that ‖f‖6 ‖f‖1

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.

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,.

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.

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.

19 Mar 2008: only if U = Y V for some V open in (X, dX ).(ii Show that A is closed in (Y, dY ) if and only if A = Y B for some B ... a contraction? Deduce that the differential equation. dy. dx= 2xy , with y = 1 when x = 0 ,. has a unique solution in some interval

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),

23 Jan 2023: Suppose then that [b′] = [b′]µ′. It follows that ν[b] =. µ α for some α X(ker(G G′))I. But [b] B(G,{µ}) and hence µν[b] isa ... Proof. By [HR17, Theorem 6.1 (b)], there exists h G(OF̆ ) such that h1bσ(h) I(OF̆ )ẇI(OF̆ ) for some w JW

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.

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.