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15 Oct 2021: Let a, b K such that thepolynomials f = xm a, g = xm b are irreducible. ... Show that G is solvable if and only if there is an i such that.

14 Oct 2021: y2= 0 (3). Construct a sequence of real analytic functions (uk)k=1 with uk : R2 R solving (4) such. ... u1. 1We write A B to mean that there exists a universal constant C such that A CB.

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

22 Nov 2021: S aSbb | T, T bTaa | S | ǫ. (3) Give a CFG for each of the following CFL’s, and then transform each such CFG intoCNF (giving a justification for the transformation). ... b) Show that the language L := {anbncn | n 1} is not a CFL.

29 Nov 2021: Before we begin, let us define the notion of base. A base for a topology on a setX is a family B of open subsets of X such that every open ... such that x B U.Examples The family of open balls in a metric space is a base for the metrictopology.

14 Oct 2021: Show thatS1M = 0 if and only if there exists an s S such that sM = 0. ... ii) There exists nonzero elements e1,e2 A such that e1e2 = 0, e21 = e1, e22 = e2 and e1 e2 = 1.iii) A is isomorphic to A1 A2 for some rings

8 Aug 2021: As a corollary we obtain a result promised at the beginning of this section.Lemma 3.4.13 If A and B are n n matrices such that AB = I, then ... find elementary matrices L1, L2,. , Lp and M1, M2,. , Mq together witha diagonal matrix D such that.

28 Oct 2021: A = {x M : (xn) in A such that xn x} and A = {x M : r > 0 Dr(x) A}. ... Some more questions. 13. (a) A topological space is said to be second countable if there is a countable family Bof open sets such that every open set is a union

15 Oct 2021: Incomponents we write B. j,γ,δ= ((bklj,γ,δ)) and c. γ,δ = (c. ... j, c such that the corresponding. series for u converges near x = 0.

4 Mar 2021: AC B,ρA ρB) D.5 For an inverse system (Ai,ρi) D such that limAi CO then. ... Example (Determinant). Fix a continuous character ψ : Γ O such that ψ mod λ det ρ̄.

27 Oct 2021: if any), and why? (b) By replacing A in the construction of B with a suitably chosen set, construct a setC N such that neither C nor N C are ... and such that Wn is recursive foreach n I but. nI Wn is not r.e.

19 Feb 2021: Regarding T as a map from C into C by identifyingR2 with C in the usual way, show that there exist unique complex numbers A, B such thatfor every z C, ... thatthere need not exist holomorphic f : U C such that u = Re (f) on all of U.

3 Feb 2021: b) We say that a measurable f : Rn! C is in weak-Lp(Rn), written f 2 Lp,w(Rn) ifthere exists a constant C such that. ... Deduce. N[. i=1. Bi. 6 3n. kX. j=1. Bij. 3. b) () Suppose {Bj : j 2 J} is an arbitrary collection of balls in Rn such that each

26 Feb 2021: 6. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elementsa and b in R is an element d in R such that d divides both ... a and b, and if c dividesboth a and b then c divides d.

24 Feb 2021: there exists 2 C such that:. u[] =. Z. R(x)dx, for all 2 D(R). () ... Rm(x, y)dy. b) Show that there exists a sequence of smooth functions fn 2 C1c (Rn) such that Tfn!

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 ... fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj.

8 Oct 2021: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 13. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

16 Mar 2021: 2u u = f. b) Show further that there exists C > 0 such that kukHs4. ... q|u[ ] =. 1. |q|u[ 0] =: M(u). b) Define the dual lattice by := {x 2 Rn : g x 2 2Z, 8g 2 }. Show that thereexists a basis. = {1,. n} such

15 Apr 2021: form χ det.Such representations do not occur as constituents of automorphic representations. ... Two such representations areisomorphic iff they come from the same characters, possibly after reordering.

10 Mar 2021: There is a distinguished uniformizer t B. dR. such that F kBdR = tkBdR and GQ acts on t through the -adic cyclotomic character. ... Such a functor is known to exist on the subcategory of smooth quasi-projective varieties [Fon92, 6.7].