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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. ,