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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.

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.

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

19 Jan 2021: Show that E has a subgroup isomorphic toE/M, and show that all such subgroups are conjugate. ... for all n. 3. 8. Let G be a group and let K be a subgroup of G.

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.

1 Nov 2021: Show that M M is trivial.(b) Show that for all n we have TSn R = Rn1 over Sn. ... and Q7. compute HndR(RPn) for all n. 9. Let Σ be a compact 2-manifold-with-boundary, let F : Σ R2 be a smooth map, and let α bethe

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;.

8 Oct 2021: b) Let X Bin(n,θ), where θ Θ = [0, 1]. Find all prior distributions π on Θ for whichthe maximum likelihood estimator is a Bayes rule for estimating θ in ... a) Show that the James-Stein estimator θ̃(1) dominates all estimators θ̃(c),c 6= 1.(b)

13 Oct 2021: foreach γ C there exist elements α(γ) A and β(γ) B, such that for all γ we have:. ... multiplication. An R-linear derivation OX,p R is an R-linearmap d : OX,p R such that d(fg) = d(f)g(p) f(p)d(g) for all f

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 ... f(A) = trA for all A Matn,n(F).(b) Now let V be a non-zero finite dimensional real vector space.

15 Oct 2021: uα =Dαu(x′,0). α!= 0, for all multi-indices α with αn = 0. ... We will proceed inductively tocompute all uα by making use of the equation.

10 Jun 2021: All the other edges of the network are of the form (ai, bj) for some i, j = 1,. , ... A is connected to all ai and B is connected to all ci with capacity 1 edges.The capacities of all other edges is infinite.

14 Oct 2021: Y := {v V : (v,u) = 0 for all u Y}.(a) Show that dim Y dim Y = dim V. ... y,λη) for all (x,ξ) M and λ [0,).

18 Jan 2021: Assume all normalized eigenforms f S2(N) have rational coefficients. Fix a prime p 6= N. ... Define the p-Eisenstein ideal as the ideal a of T generated by p andT ( 1) for all.

8 Oct 2021: 3. (i) Show that the function ψ(A,B) = tr(ABT ) is a symmetric positive definite bilinear form on the spaceMatn(R) of all nn real matrices. ... sn forms a basis for Pn.(iii) For all 1 k n, sk spans the orthogonal complement of Pk1 in Pk.(iv) sk is an

29 Nov 2021: set is a union of somemembers of B; equivalently, for U X, we have that U is open in X if and only iffor all x U there exists B B ... Show that if fn Y for all n N andfn f in X, then f Y.

12 Feb 2021: every z C, |f(z) b| > ε.(iii) f = u iv and |u(z)| > |v(z)| for all z C. ... z|)k for all z.(ii) Show that an entire function f is a polynomial of positive degree if and only if |f(z)|as |z|.

8 Aug 2021: Correction from Greg Price Last line of Exercise 11.5.25 should read:-Show that T̂ s′ and T̂ 6= 0, but T̂b = 0 for all b c00. ... least n 1 diagonalelements having value 1 and all diagonal elements non-zero.’.

11 Feb 2021: thetransition functions for the atlas with charts {π+,π,φ} are all smooth. ... b) Prove that for any decomposition of S (where all vertices have valence at least 3), we have 2E/F 6 (1 χ(S)/F),where χ(S) is the Euler

15 Oct 2021: a) f L1(Rn), g Ck(Rn) with supRn |Dαg| < for all |α| k. ... 3. for all j J. Thuslimj. τzjg gLp(Rn) = 0.4This is another point at which p 6= is crucial.

20 Apr 2021: The étale homotopy type of Spec k is "-isomorphicto the pro-object {B(Gal(K/k))} where K/k runs over the system of all finite Galois extensions K/k. ... So Et! (Spec k) is isomorphic to the pro-object {B(Gal(K/k))}.This implies that all the higher

16 Mar 2021: Show that there exists aconstant Cn, > 0 such that for all x, y 2 Rn:. ... u |x|2u = f (†). if. (u, v)H = (f, v)L2 for all v 2 H. ().

15 Oct 2021: ii) For all f C0([a,b]), if a R, then ||af||C0 = |a| ||f||C0. ... fi(x)fi(y)| < ,for all i N and all x,y [a,b] with |xy| < δ.

15 Oct 2021: f ′(x) = g′(x), for all x (a,b). Show that:f(x) = g(x) C. ... f(k)(x) = g(k)(x), for all x (a,b). Show that:f(x) = g(x) pk1(x).

15 Oct 2021: Now we assume that a(f(s),g(s),h(s))g′(s) b(f(s),g(s),h(s)f ′(s) 6= 0 for all s (α,β), so that ... Suppose that a(Γ(s))g′(s)b(Γ(s))f ′(s) 6= 0 for all s (α,β)where Γ(s) = (f(s),g(s),h(s)).

15 Oct 2021: Define τxφ by:. τxφ : C,y 7 φ(y x). (1.2). Then there exists > 0 such that τxφ Ck0 () for all x B(0). ... supxRn. (1 |x|)NDαφ(x) < for all multi-indices α and all N N.

15 Oct 2021: f. 0(x) = g. 0(x), for all x 2 (a, b). ... 3. b) Let g(x) = f(x)exp(x). Show that:. g. 0(x) = 0, for all x 2 R g(0) = 1.

15 Oct 2021: for all x,y,z Rn and a R. b) For t R and x,y Rn, show that:. ... Supposethat the xi satisfy ||xi|| < r for all i and some r > 0.

15 Oct 2021: u[φ]| C supxK. |α|N. |Dαφ(x)| , for all φ C0 (K). ... 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).

15 Oct 2021: b]. a) Show that f(x) = 0, for all x [a,b]. b) Does the result hold if we do not assume f is continuous? ... fi(x) f(x)| <. 2(ba),. for all x [a,b]. Henceforth assume i N.

15 Oct 2021: This states that iff : [a,b] R is continuous and F : [a,b] R satisfies F ′(x) = f(x) for all all x (a,b)then: b. ... and moreover α and β are differentiable at all x (a,b) with α′,β′ : (a,b) R piecewisecontinuous.

15 Oct 2021: Rn Rmthere exists M such that ||λx|| M ||x|| for all x Rn. ... b) Show that Df(p) is invertible for all p. c) Show that f : R2 is not injective.

15 Oct 2021: Define τxφ by:. τxφ : C,y 7 φ(y x). (1.2). Then there exists > 0 such that τxφ Ckc () for all x B(0). ... for all x (a,b).Then: b.

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). ... Thus, we conclude:. lim0. 1. |B(0)|. B(0). φ(y)dσy = φ(0). Putting this all together, we have:.

15 Oct 2021: E A,• B A A for all A,B A with A B,. ... We say that A E is µ-measurableif, for all B E we have:.

15 Oct 2021: Λj(xk) Λ(xk) as j , for all k. Secondly, since the weak- topology on B′ is metric, compactness is equivalent to sequentialcompactness. ... i) a a for all a S. (Reflexivity). ii) If a b and b a, then a = b.

15 Oct 2021: 100 Appendix B Bonus Material. i.e. τ is the set of all unions of elements of β. ... for all x,y,z X. We have the following useful result. Lemma B.13.

26 Mar 2021: 1p 1. if n 2,. andρ(n, 1) = 1 for all n 1,. ... AB AB = {ab | a A, b B}.If Res(a,b) Zp for all a A and b B, then this bijection is measure-preserving.

15 Oct 2021: Λω) [v] = ω [Λv] , for all v V. A.3 Differential geometry 101. ... i) a b G for all a,b G [Closure]. ii) (a b) c = a (b c) for all a,b G [Associativity].

6 Dec 2021: x} = B(x, 1/2). and all subsets of X are open. ... αA Uα τ.(iii) If Uj τ for all 1 j n, then.

12 Jan 2021: dxu(X(x)) dIdRu(x)(B(x)) = 0. for all x N. If G is a matrix group we can write this more succinctly as. ... Thus for every meromorphic funtion we obtain a cohomologically trivial connection.Are these all?

15 Oct 2021: a a1 = 1. vi) The multiplication operation is distributive over addition: the following identity holdsfor all a,b Φ:. ... Define τ by:. U τ for all x U, there exists B β such that x B and B U.

15 Oct 2021: hi φ Diφ in S , as h 0. b) First, we note the following simple inequality which holds for all x,y Rn:. ... for all x K, so that:. 1 (1 R) 1 |x g|1 |g|.