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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.