Search

11 Oct 2023: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... an,bn] such that the Lebesgue measure of A4B(= (Ac B) (ABc)) is less than ε.

17 Jul 2023: a) Show that the unique minimizer is x = 0. (b) Find constants 0 < m < M and L > 0 such that m f ′′(x) M and |f ′′′(x)| L for all x. ... Given constants b1, b2 such that b1 eb2 0 use the Lagrangian method to.

8 Mar 2023: Does 10f exist? 3. Give an example of a continuous function f : [0,) [0,), such that0f. ... Must f vanishidentically?[We say g vanishes near a and b if there exists > 0 such that g(x) = 0 for x 6(a ,b ).]. 1. 2 ANALYSIS I EXAMPLES 4.

24 Nov 2023: If A is the matrix of the first of these (say), find a non-singular matrix P such that PT AP is diagonalwith entries 1. ... an be real numbers such that a1 an = 0 and a21 a2n = 1.

8 Nov 2023: 12. (a) Show that if x ̸= y are vectors in the finite dimensional vector space V , then there is a linear func-tional θ V such that θ(x) ̸= ... Let A, B V. Prove that A B if and only if Ao Bo.Show that A = V if and only if Ao = {0}.

6 Feb 2023: a) Construct a continuous surjection p : T2 T2 from the torus to itself such that for every x T2, p1(x)consists of exactly two points, and such that p is ... Such a map is calleda covering map.]. (b) Show that T2 also admits a covering map p̂ : T2 K to

9 Nov 2023: SC =2π. κ. C. [Q[ξ] ξ BH]. However, it is not possible for θ to be of the form of a totalvariation δBH on H in general because such a ... M =. [Q[t] t B]. and there is no difficulty in obtaining an M that satisfies thedesired relation.

23 Oct 2023: Show that:. (a) There is a nondeterministic automaton N with n 1 states such that L(N) = Ln and(b) if D is a deterministic automaton with fewer than 2n states, ... Prove that there is a proper excerpt e of v such that xey L.

24 Jan 2023: 6) Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements aand b in R is an element d in R such that d divides both ... a and b, and if c divides both aand b then c divides d.

19 Jan 2023: i) is a set;. (ii) F is a σ-field on , i.e., F P() such that. ... 2 Outer measures. 2.1 Given a set , an outer measure on is a function µ : P() [0,]such that.

19 Jan 2023: A sesquilinear form θ is bounded if there is aconstant C > 0 such that. | ... 1.6 Adjoints For T B(H) there is a unique map T : H H such that〈Tx,y〉 = 〈x,Ty〉 for all x,y H.

28 Nov 2023: Operator-valued measures. A resolution of the identity of H over K is amap P : B B(H) such that. ... over K. Then there is an isometric, unital -isomorphism Φ of L(P) onto acommutative, unital C-subalgebra of B(H) such that.

29 Sep 2023: The discriminant = (a, b)=16(4a327b2)is non-zero. Conversely, for any pair (a, b) O2K such that (a, b) , 0, the equation(2.1) defines an elliptic curve. ... ap bp = cp. to the Fermat equation in exponent p; thus a, b, c Z are coprime non-zero

10 Oct 2023: Suppose that α has minimal polynomial f over Q, and let β = α4.Find a, b, c Q such that β = a bα cα2. ... 9. Let K and L be subfields of a field M such that M/K is finite.

14 Sep 2023: If χ: A B is an algebra homomorphismthen there is b B such that χ(x) = bxb1 for all x A. ... Evidently such a pair is also an element of AutA(M/X/G). (b) Given a group homomorphism σ : G AutA(M/X/G) such that pr1 σ =1G, the restriction

28 Nov 2023: ii) If A is compact and B is closed, then there exists f X such that supA f <infB f. ... An algebra norm on A is a norm ‖‖ on A such that ‖ab‖ 6 ‖a‖‖b‖ for alla,b A.

7 Jan 2023: Suppose that b′ > b. Then we know by the rules for N+ that wecan find a c N+ such that b′ = b c. ... If c C then, since g is surjective, we can find b B such that g(b) = cand, since f is surjective we can find a A such that f (a) =

29 Nov 2023: 10. )〉. So. R2 =〈(. 10. )〉〈(. 01. )〉is the only such decomposition in this case. ... Ws such that W = sj=1Wj. Thus. V =ri=1. Ui sj=1.

25 Oct 2023: 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.

20 Dec 2023: Proposition 2.1.1. ([25, Exercise 2F]) For every ring homomorphism φ: A Band element b B such that bφ(a)φ(a)b = φ(δ(a)) for all a A, ... there is a uniquering homomorphism ψ : A[T; δ] B such that ψ|A = φ and ψ(T) = b.

25 Oct 2023: 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.

12 Jan 2023: 20. are constants K,C, such that for all a,b MCG(S),. dMCG(S)(a,b) K,CYS. ... Let (X,S) be a HHS. Thereis a constant T0 > 0 such that for all T T0, there are constants K,C, such that for all a,b X,.

1 Feb 2023: STP that if0 6= b Qn then Tb 6= 0, or equivalently there exists c Qn such that tcTb 6= 0.But if β =. bjαj, γ =. cjαj then. tcTb = TrK/Q( ... There is an invertiblelinear transformation u: Rn Rn such that u(Λ) = Zn.

6 Mar 2023: Logarithmic Gromov-Witten theory defines the notion of a logarithmic stable mapto such targets X B; more generally, this morphism just needs to be toroidal ratherthan normal crossings, i.e., log ... The type of such a family consistsof the data τ := (G,

24 Oct 2023: Vn such that V ' V1 Vn. Theorem 4.12. Suppose that G is finite and |G| 6= 0 in F. ... central character, such that z v = λ(z) v forall z Z(G) and v V.

23 Jan 2023: Suppose there exists a class [b] B(G,{µ}) such that ν[b] = µ(such a class is necessarily unique if it exists by the above remark). ... When G is quasi-split,this class is just [b]µ. However if G is not quasi-split, there may be no [b] B(G,{µ})such

24 Nov 2023: Let a,b K such that the polynomials f = Xm a, g = Xm b are irreducible. ... Use asuitable Lagrange resolvent to find a Q(ζ3) such that K = Q(ζ3, 3.

7 Feb 2023: b) We say that a measurable 𝑓 : R𝑛 C is in weak-𝐿𝑝(R𝑛), written 𝑓 𝐿𝑝,𝑤 (R𝑛) if there exists aconstant 𝐶 such that. |{ ... Deduce 𝑁𝑖=1. 𝐵𝑖. 6 3𝑛 𝑘︁𝑗=1. 𝐵𝑖𝑗. b) () Suppose {𝐵 𝑗 : 𝑗

5 Oct 2023: Theleast such n is called the order of g.). (b) Show from first principles that there is a positive integer N such that gN = e for all g G. ... 9. Let H and G be groups and let X G such that 〈X〉 = G.

27 Nov 2023: X̃1, that is, β̃1 = (X̃T1 X̃1). 1X̃T1 Ỹ. (c) Find a condition on X0 and X1 such that β̃0 = β̂0. ... b) The model is to be reparametrised in such a way that it can be specialised to (i) two parallellines for the two groups, (ii) two lines with

10 Feb 2023: g :(D2,S1. ) (M,B). such that[g|S1. ]6 N. We first explain why the Loop Theorem implies Dehn’s Lemma:. ... g :(D2,S1. ) (M,B). such that[g|S1. ]6= 1. Set c =. [g|S1. ] π1B, and write c = ka b, with. k, Z.

25 Nov 2023: Let θ : A B be a -homomorphism from a Banach -algebra A to a C-algebraB. ... H) such thatz(T) = T, where z(λ) = λ for all λ K.

13 Mar 2023: E,E′) such that Hom(π,π′)s = IdX gives for each x X. ... Choose vectors en1,. ,eN FN such that. s1(x),. ,sn(x),en1,. ,eN FN.

1 Feb 2023: problem. To describe such a problem, let (M,g) be a compact Rieman-. ... function on R2 such that |x|kf L(R2) for any k 0.