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17 Jan 2016: C/Λ2 are conformally equivalent if and only if thelattices are related by Λ2 = λΛ1 for some λ C. ... 5. Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if and only if τ2 =. (aτ1 b)/(cτ1 d), for some matrix(a bc d

21 Nov 2016: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... 7. Let S be an nn real symmetric matrix with Sk = I for some k 1.

7 Nov 2016: a) All words w {0, 1} consisting of either the string 01 repeated some number oftimes (possibly none), or the string 010 repeated some number of times (possiblynone). ... b) All words w {a,b,c} consisting of some number of a’s (possibly none), followed

15 Feb 2016: Could there be some other Euclidean function φ making Z[3] into a Euclidean domain? ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written as rasbfor some r,s R.

11 Nov 2016: u(0,x) =1. 1 x2. 1. 5. (Cauchy-Kovalevskaya Theorem for systems of ODEs) Suppose b > 0 and F : u0 (b,b)d Rd bereal analytic in a neighbourhood of ... u′(t) = F(t,u(t)), u(0) = u0 Rd,. on (a,a) for some a > 0 with u((a,a)) u0 (b,b)d.

17 Jan 2016: 8. By considering the singularity at or otherwise, show that any injective analytic map f : C Chas the form f(z) = az b, for some a C and b C. ... conformally equivalent). 10. Define an equivalence relation on C by z w iff z = 2sw for some s Z.

27 Oct 2016: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... vn} C such that{B(v1),. ,B(vn)} cover C. (b) Show that there exists some > 0 such that for every v C, B(v) Uα for some α.Deduce that

9 Feb 2016: 5. Let f be a function on a set A and let B A. ... some c (a,b) the vectors(f(b),g(b). )(f(a),g(a). )and. (f ′(c),g′(c). )in R2.

18 Feb 2016: f(x) = limx. g(x) = 0. Assume that for some c > 0, we have g′(x) 6= 0 for all x > c and that f′(x)g′(x). ... Show thatif f ′(a) < y < f ′(b) for some a < b in I and y R, then there exists x I witha < x < b and f ′(x) = y.

18 Jan 2016: C/Λ2 are conformally equivalent if and only if thelattices are related by Λ2 = λΛ1 for some λ C. ... 5. Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if and only if τ2 =. (aτ1 b)/(cτ1 d), for some matrix(a bc d

12 Dec 2016: Bn(θ) =a. n+. b. n2+ O(n3). for some real numbers a,b. ... 5. Let f,g : R [0,) be bounded probability density functions such that f(x) Mg(x)for all x R and some constant M > 0.

27 Oct 2016: 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 ... 1.9. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.

6 Jan 2016: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... irreducible. 3 Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible.

18 Jan 2016: 8. By considering the singularity at or otherwise, show that any injective analytic map f : C Chas the form f(z) = az b, for some a C and b C. ... conformally equivalent). 10. Define an equivalence relation on C by z w iff z = 2sw for some s Z.

16 Nov 2016: This result is called Burnside’s Lemma, or Cauchy-Frobenius Lemma, or just the Count-ing Lemma.)Deduce that if G acts transitively and |X| > 1 then there must exist some ... Canyou also find some analogues considering subgroups of D2n? 2.

28 Oct 2016: Suppose for some θ0 in the interior of Θ and every ε > 0 small enough we haveS(θ0 ε) < 0 < S(θ0 ε), and that Sn has exactly one zero θ̂n ... model {f(,φ) : φ = Φ(θ) for some θ Θ} equals Φ(θ̂MLE).b) Now consider a mapping Φ that is not

1 Feb 2016: This strengthens Liouville’s theorem.)(ii) If for some a C and > 0, f never takes values in B(a; ), show that f is constant.(iii) If f = u iv and ... Show that f(1/n) 6= 1/(n 1) for some n Z>0. 9.

15 Feb 2016: Could there be some other Euclidean function φ making Z[3] into a Euclidean domain? ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written as rasbfor some r,s R.

29 Feb 2016: α for some α.(iv) Any limit ordinal can be written in the form α. ... Given a subset b Pα with more thanone element, consider the least β belonging to some but not all members of b.].

9 Mar 2016: 71). Find an element of OK with norm 2a 3b 5 for some. ... Hint: Let x1,. ,xm generate OL as an OK -module. If pOL = OLthen we can write xi =. aijxj for some aij p.] Deduce that if a and b are ideals.

9 Nov 2016: What happensif Θ = [a,b] for some 0 < a < b <? 9. Let X be multivariate normal N(θ,I) where θ Θ = Rp,p 3, and where I is the ppidentity ... a) Show that the James-Stein estimator θ̃(1) dominates all estimators θ̃(c),c 6= 1.b) Let θ̂ be the

21 Nov 2016: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... 7. Let S be an nn real symmetric matrix with Sk = I for some k 1.

16 May 2016: 7. (a) If f : [0, 1] [0, 1] is continuous, show there is some x [0, 1] withf(x) = x. ... For eachinteger n > 1, show that there is some x [0, 1] with f(x) = f(x 1.

25 Apr 2016: x > 0,Ax 6 b} if and only iffor some z Rm,. ... b) What happens if we try to solve this problem with the transportation algorithm?

23 Feb 2016: If f has poles on the boundary we can find some b C such that no poles lieon the translate b P, and integrate around this curve instead.(iii/iv) Same ... More generally, let F : L C be a function on lattices which satisfies F(αΛ) =αkF(Λ) for some k Z.

23 Mar 2016: Given elements e l ,. , e, of A which induce the operations on some E-algebra B = ( b ,. , b,, e ,. , era) in 3 4 , we define the E-algebra O ( e l ,. , ... We have achieved our purpose in proving the theorem above. However, some con- cluding remarks

23 Mar 2016: Given elements e l ,. , e, of A which induce the operations on some E-algebra B = ( b ,. , b,, e ,. , era) in 3 4 , we define the E-algebra O ( e l ,. , ... We have achieved our purpose in proving the theorem above. However, some con- cluding remarks

27 Apr 2016: Automorphy of some residually S5 Galois representations. Chandrashekhar B. Khare and Jack A. ... 5 3000)x4 (. 5 5)x/2 (1440. 5 3600). 4 Some group theory.

28 Nov 2016: This paper is about the arithmeticstructures which are (in some cases conjecturally) associated to π. ... If we writeπp = n-Ind. G(Qp)B(Qp) χ, for some unramified character χ : T(Qp) C. ,

18 Aug 2016: Therefore. nj=1. jF1xj. = ξF2 andnj=1. mjF1xj. = ξF1. for some ξ K. ... Firstwe need some linear algebra. If B is an nn matrix and S {1,.

25 Apr 2016: 16 4 Conditions for Strong Duality. for some b > 0. ... 1B AN)i > 0 for some i, then we can increase the value of.

16 Mar 2016: We now want to define some global cohomology groups HiS,T which can be used to. ... We continue with the notation of 2.1, and define some local deformation problems.

25 May 2016: Thepaper closes with some general remarks on the mathematical significance of theeffective topos. ... We collect some further folkloric facts about these notions in the next theorem.

25 May 2016: Thepaper closes with some general remarks on the mathematical significance of theeffective topos. ... We collect some further folkloric facts about these notions in the next theorem.

30 Apr 2016: Some algebraic topology 598.3. Proving V is transversal to E˙ 608.4. ... Consider. P.x;y;z/ WD. 2kXiD0. aix2kizi;. for some constants ai 2 R.