Search

6 Feb 2021: Which of(1)–(4) must be true? (1) If f is increasing then f ′(x) 0 for all x (a,b).(2) If f ′(x) 0 for all x (a,b) ... then f is increasing.(3) If f is strictly increasing then f ′(x) > 0 for all x (a,b).(4) If f ′(x) > 0 for all x (a,b) then f

9 Nov 2021: a) Compute dim SnV and dim ΛnV for all n. (b) Let g G and λ1,. , ... even. 7. Find all the characters of S5 obtained by inducing irreducible representations of S4.

19 Jan 2021: tβn. are joined by an edge if, after renumbering:. (a) αi is isotopic to βi for all i > 1;. ... b) if Sα2,.,αn is a one-holed torus then i(α1,β1) = 1;.

6 Feb 2021: Which of(1)–(4) must be true? (1) If f is increasing then f ′(x) 0 for all x (a,b).(2) If f ′(x) 0 for all x (a,b) ... then f is increasing.(3) If f is strictly increasing then f ′(x) > 0 for all x (a,b).(4) If f ′(x) > 0 for all x (a,b) then f

15 Oct 2021: b. a. f(t)dt = 0, 0  f(x), for all x 2 [a, b]. ... R iscontinuous on [a, b], differentiable on (a, b) and satisfies F 0(x) = f(x) for all x 2 (a, b).

15 Oct 2021: Define τ by:. U τ for all x U, there exists B β such that x B and B U. ... xn xm B, for all n,m N.

20 Jan 2021: Set. A = {ω : ω An infinitely often}, B = {ω : ω An for all sufficiently large n}. ... P(Xn = 0) 12π. h. (b) Show further that, for all x R,.

15 Oct 2021: c) Show that if 0  f(x) for all x 2 [a, b] then:. ... 0 Z. b. a. f(x)dx. [Hint: Use part a)]. d) Show that if f(x)  g(x) for all x 2 [a, b] then:Z.

20 Jan 2021: a) Show that, for all p (0, ) and all x (0, ),. P(|X| x) E(|X|p)xp. (b) Show that, for all β 0,P(X x) E(eβX)eβx. ... 3. Let X be a Poisson random variable of parameter λ (0, ).(a) By optimizing the estimate of Question 2(b) over β, show that, for

9 Nov 2021: a) Compute dim SnV and dim ΛnV for all n. (b) Let g G and λ1,. , ... even. 7. Find all the characters of S5 obtained by inducing irreducible representations of S4.

15 Mar 2021: 1. |B|vB. d(v) > (α ε)|A|. Show there is B′ B with |B′| > ε|B| so that d(v) > α|A|, for all v B′. ... Is this bound sharp? (11) Prove that the matrix J (all of whose entries are 1) is a polynomial in the adjacency matrix of a.

29 Jan 2021: Show that G can be decomposedinto cycles if and only if all degrees of G are even. ... Show that there is a partition V = AB so all the vertices inthe graphs G[A] and G[B] are of even degree.

12 Feb 2021: Ck begreat circles on S that don’t all meet at a common point. ... edge) is a straight line. (14) () Let C R2 be a polygon with all vertices in Z2.

10 Nov 2021: b) All words w {a, b, c} consisting of some number of a’s (possibly none), followed bysome number of b’s (at least one), followed by some number of ... If |w| < 4,then w must contain a 1 somewhere. (d) All words w {a,. ,

8 Oct 2021: ψ(m,σ) = E[U(m σZ)].Show that. (a) if U is increasing, then ψ(, t) is increasing for all σ(b) if U is concave, then ψ concave, that ... is. ψ(pm0 qm1,pσ0 qσ1) pψ(m0,σ0) qψ(m1,σ1). for all m0,m1,σ0,σ1 and 0 p = 1 q 1.(c) if U is concave,

21 Nov 2021: 9. Let M be a non-empty compact metric space and f : M M be a function.(a) Show that if d(f(x),f(y)) < d(x,y) for all ... x 6= y in M, then f has a unique fixed point.(b) Show that if f is isometric, i.e., d(f(x),f(y)) = d(x,y) for all

2 Nov 2021: 5) An independent set in a graph G = (V,E) is a subset I V so that x 6 y for all x,y I.Let G = (V,E) be a ... 12) A graph is outer-planar if it can be drawn in the plane so that all of its vertices are on the infinite.

23 Nov 2021: 4. Find all bases for which 39 is an Euler pseudoprime. ... More generally, showthat if p and 2p1 are both prime numbers, then N = p(2p1) is a pseudoprimefor precisely half of all bases.

21 Nov 2021: 9. Let M be a non-empty compact metric space and f : M M be a function.(a) Show that if d(f(x),f(y)) < d(x,y) for all ... x 6= y in M, then f has a unique fixed point.(b) Show that if f is isometric, i.e., d(f(x),f(y)) = d(x,y) for all

15 Oct 2021: loc.(U). Suppose u 2 H1(U). satisfies:. B[u, v] = (f, v)L2(U), for all v 2 H10(U). ... 3 R,show that solutions to ([) arising from sufficiently small5 data exist for all time.