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27 Oct 2021: if any), and why? (b) By replacing A in the construction of B with a suitably chosen set, construct a setC N such that neither C nor N C are ... and such that Wn is recursive foreach n I but. nI Wn is not r.e.

19 Feb 2021: Regarding T as a map from C into C by identifyingR2 with C in the usual way, show that there exist unique complex numbers A, B such thatfor every z C, ... thatthere need not exist holomorphic f : U C such that u = Re (f) on all of U.

3 Feb 2021: b) We say that a measurable f : Rn! C is in weak-Lp(Rn), written f 2 Lp,w(Rn) ifthere exists a constant C such that. ... Deduce. N[. i=1. Bi. 6 3n. kX. j=1. Bij. 3. b) () Suppose {Bj : j 2 J} is an arbitrary collection of balls in Rn such that each

26 Feb 2021: 6. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elementsa and b in R is an element d in R such that d divides both ... a and b, and if c dividesboth a and b then c divides d.

24 Feb 2021: there exists 2 C such that:. u[] =. Z. R(x)dx, for all 2 D(R). () ... Rm(x, y)dy. b) Show that there exists a sequence of smooth functions fn 2 C1c (Rn) such that Tfn!

10 Nov 2021: 5. (a) Suppose that f Matn,n(F) is such that f(AB) = f(BA) for all A,B Matn,n(F) and f(I) = n.Show that f is the ... fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj.

8 Oct 2021: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 13. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

16 Mar 2021: 2u u = f. b) Show further that there exists C > 0 such that kukHs4. ... q|u[ ] =. 1. |q|u[ 0] =: M(u). b) Define the dual lattice by := {x 2 Rn : g x 2 2Z, 8g 2 }. Show that thereexists a basis. = {1,. n} such

15 Apr 2021: form χ det.Such representations do not occur as constituents of automorphic representations. ... Two such representations areisomorphic iff they come from the same characters, possibly after reordering.

10 Mar 2021: There is a distinguished uniformizer t B. dR. such that F kBdR = tkBdR and GQ acts on t through the -adic cyclotomic character. ... Such a functor is known to exist on the subcategory of smooth quasi-projective varieties [Fon92, 6.7].

11 Feb 2021: in R3 defined by rotatingthe circle {(x b)2 z2 = a2} about the z-axis, where 0 < a < b. ... b) Construct a continuous surjection p : T2 K such that for every x K, p1(x) consists of exactly two points.(c) Draw inside the identification square for the

18 May 2021: such that ρ̄π = ρ̄m. McDonald Automorphic forms on definite quaternion algebras: Integral theory 17 / 20. ... and tr(ρ(GF )) B for B A a closedsubring, then there exists a ker(GLn(A) GLn(F)) such that aρa1 factorsthrough GLn(B).

15 Oct 2021: supxA||fi(x)f(x)|| <. b) For all x A, for all i N there exists > 0 such that:. ... xK} [a,b] Q such that [a,b] Km=1(xm δ,xm δ). Since gi converges ateach rational point, for each m = 1,.

15 Oct 2021: Suppose further that f(a) = f(b). Then there existsc (a,b) such that f ′(c) = 0. ... Then there exists c (a,b) such that:. f ′(c) =f(b)f(a). ba.

15 Oct 2021: Such a φ exists, for example we can take φ(x) = C exp(. ... zj}j=1 Rn be a sequence ofpoints such that zj 0 as j.

15 Oct 2021: a(x,y,u)ux b(x,y,u)uy = c(x,y,u). such that for s (α,β) we have [uγ](s) = u(f(s),g(s)) = h(s) ... interval t (s,s), and that moreover the solution depends continuously on s.Later we’ll state more precisely some sufficient conditions on a,b,c such

15 Oct 2021: Proof. Since is open, there exists ,x such that the ball B(x) = {y Rn : |y x| < }is contained in. ... Chapter 1 Test functions 3. a) Show that χ C(R). b) Show that there exists a function ψ C0 (Rn) such that.

25 Feb 2021: CO and tr ρ(Γ) B then there exists an a ker(GLn(A) GLn(k ))such that aρa1 has image in GLn(B). ... The universal property of R induces ahomomorphism O[[X ]]/J O[[X ]]/Jf such that the composition.

15 Oct 2021: φ(y)dσy. Now, for any δ > 0, since φ is continuous, there exists > 0 such that |φ(y) φ(0)| < δfor all y B(0). ... such that Ki Ki1 and = iKi (see Lemma B.6). Then since (2.10) does not hold for any K,N,C, in particularit does not hold for K = Kj,

15 Oct 2021: Fix > 0. Then thereexist partitions PL of [a,c] and PR of [c,b] such that:. ... b) R such that fi f uniformly, then f : (a,b) R is uniformlycontinuous.

15 Oct 2021: Let > 0. Then there exists N such that for i N we have:. ... Let > 0. Since g is continuous at f(p), we know that there exists δ1 such thatfor any y B with ||y f(p)|| < δ1, we have ||g(y)

15 Oct 2021: b) Find a sequence fi R([a,b]) such that (fi)iN is Cauchy with respect tothe L2 norm, but for which there is no f R([a,b]) such that ... Proof. Since is open, there exists ,x such that the ball B(x) = {y Rn : |y x| < }is contained in.

15 Oct 2021: First, suppose that p B. Then there exists δ > 0 such that Bδ(p) B, and thusχB(q) = 1 for any q with ||q p|| < δ. ... We say that a domain A R2 is of Type I if there exist a,b R and continuous functionsα,β : [a,b] R such that:.

15 Oct 2021: Then there is an open interval Acontaining a and an open interval B containing b with the property that for each x Athere is a unique y B such that ... theexistence of open sets U,V Rn Rm with (a,b) U, (a, 0) V such that F : U Vhas a continuously

15 Oct 2021: φ(y)dσy. Now, for any δ > 0, since φ is continuous, there exists > 0 such that |φ(y) φ(0)| < δfor all y B(0). ... Exercise(). a) Suppose that {φj}j=1 E () is a sequence such that φj φin E (), and χ D (). Show that. χφj χφ in D (). b) Show

15 Oct 2021: a b = b a, a b = b a. iv) There exist unique, distinct, additive and multiplicative identity elements: there exist0 Φ and 1 Φ with 0 6= 1 such that for ... By the definition of τ,there exists B such that x B W and B has the form.

15 Oct 2021: Then there exists a subsequence (Λjk)k=1 and Λ B. ′ such that Λj Λ. ... b) If A is compact, B is closed then there exist Λ X′ and γ1,γ2 R such that. <(

15 Oct 2021: b) Fix x U and suppose that V ,W TxU are two vectors such that for any f C1(U)we have. ... iv) For each a G there exists b G such that a b = b a = e [Existence of an inverse].

15 Oct 2021: Dαφ. b) There exist constants N N, K > 0 depending on u and φ such that:. ... b) Show that f is smooth and periodic. c) Show that there exists a constant cn depending only on n such that:.

25 Aug 2021: Moreover, forthe given b there exists µ1 X(T) such that |Jb(F)Σtop(Xµ1 (b))| = 1. ... It is a reductivegroup over F such that. Jb(R) = {g G(F̆ F R) | g1bσ(g) = b}.

15 Oct 2021: a b = b a, a b = b a. iv) There exist unique, distinct, additive and multiplicative identity elements: there exist0 Φ and 1 Φ with 0 6= 1 such that for ... By the definition of τ,there exists B such that x B W and B has the form.

12 Jan 2021: Then CA = CB implies that there exists a smooth U : M U(n)such that U|M = Id and B = U1dU U1AU. ... This is the simplest non-trivialcase. Suppose there is a smooth map b : SM SU(2) such that f := b1V (b) solvesthe PDE:.

6 Dec 2021: the unique V τ such that V A and, if W τ and. ... Proof. The proof, which follows the standard pattern for such things, is givenon page 69.

15 Oct 2021: µ(B) := infn=1. µ(An),. where the infimum is taken over all sequences (An)n=1 of sets such that An A andB. ... By Tonelli applied tof we find h, g such that. EFfd(µν) =. Ehdµ =. Fgdν. B.4 The Lp-spaces 191. The first integral is ,.

26 Mar 2021: Theorem 1. Let p be a prime number and n, d any integers such that 0 d n. ... Let A Zp[x]1m, B Zp[x]1n (resp. B Zp[x]n), and AB Zp[x]1mn (resp.AB Zp[x]mn) be measurable subsets such that multiplication induces

21 Sep 2021: αn1for A, such that for each 1 6 i,j 6 n 1 we have. ... 334 = c. 434 = 0. In general we could normalise our basis by choosing a convention such as.

6 Aug 2021: b) Find a sequence fi R([a,b]) such that (fi)iN is Cauchy with respect tothe L2 norm, but for which there is no f R([a,b]) such that ... Proof. Since is open, there exists ,x such that the ball B(x) = {y Rn : |y x| < }is contained in.

8 Oct 2021: Choose an element δ DS such that δ = δ as in [TX16, Lemma 3.8]. ... τ. We define the integer nτ to be the smallest integer. such that σnττ Σ S.

9 Mar 2021: In both cases, there arepoints x H2 such that d(x,f(x)) is arbitrarily small. ... A bigonis an embedded disc D2 S such the interior is disjoint from α β and theboundary D2 decomposes as a union of closed arcs a b, where a α andb

8 Aug 2021: b) If yr = 0 whenever ar = 0 and there exists an i such that ai = 0,then there are an infinity of solutions with xs = a. ... m n matrix such thatAx Bx = Cx. If A is an m n matrices and B is an m p matrix, then AB = Cwhere C is the n p matrix such

21 Nov 2021: Then there is a point y / E such. that B(y,r) E 6= whenever r > 0. ... Since Uj1 isopen we can find a δj1 > 0 with δj1 δj/4 such that B(xj1,δj1) Uj1.

15 Oct 2021: suppose that f,g L1(Rn) L2(Rn). By Theorem 1.13 part b), there exists asequence {fj}j=1 C0 (Rn) S such that:. ... Dαφ. b) There exist constants N N, K > 0 depending on u and φ such that:.

21 Nov 2021: Theorem 4.2. If f : [a,b] [a,b] is a continuous function, then we canfind a w [a,b] such that f(w) = w. ... Supposethat we can find a a0 < a1 < < an b such that, writing σ = ‖fP‖we have either.

15 Feb 2021: A) any element of M is prime to N,. (B) for any two coprime integers a,b there exists an element m M such that m a mod b. ... Then SR = LZ R. Proof. There exists an integer B 1 such that the map.

15 Oct 2021: Proof. Since is open, there exists ,x such that the ball B(x) = {y Rn : |y x| < }is contained in. ... Chapter 1 Test functions 3. a) Show that χ C(R). b) Show that there exists a function ψ C0 (Rn) such that.

15 Oct 2021: supxA||fi(x) f(x)|| <. b) For all x A, for all i N there exists > 0 such that:. ... Repeating theargument that led to (1.3), we know that for any > 0 there exists N N such that forany y [a,b] and i,j N:.

19 Jan 2021: The pushout A qC B was defined to be agroup equipped with maps iA : A AqC B and iB : B AqC B, such thatiA φA = iB φB, and such that for ... A product of two objects A,B Obj(C)is an object P equipped with morphisms pA : P A and pB : P B, such thatfor

22 Jan 2021: Theorem 4.1. If f : [a,b] R is continuous function and f(a) c f(b),then we can find a y [a,b] such that f(y) = c. ... Theorem 4.2. If f : [a,b] [a,b] is a continuous function, then we canfind a w [a,b] such that f(w) = w.

15 Oct 2021: b) Find a sequence fi R([a,b]) such that (fi)iN is Cauchy with respect tothe L2 norm, but for which there is no f R([a,b]) such that ... Proof. Since is open, there exists ,x such that the ball B(x) = {y Rn : |y x| < }is contained in.

30 Apr 2021: Given w [x,y], (i) implies the existence of z α, a [x,w] and b [w,y]such that d(z,a),d(z,b) 6. ... We pick geodesics. [z,a], [a,x], [a,y] such that [x,y] = [a,x]1 [a,y].