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27 Feb 2020: B. t+ u B = B u η2B. Supposing that the velocity retains purely the form of a linear shear flow, u = Sx ey,show that the induction equation has solutions in ... Verify that B B = 0 for this solution, and confirm that the magnetic field has noinfluence

20 Feb 2020: Now use the vector identity. (A B) = B A A B A( B) B( A).

13 May 2020: where x0 and y0 are real constants and A and B are complex constants. ... u. t+ u u = Φ 1. ρΠ. 1. µ0ρB B ,. B. t+ u B = B u ,. u = B = 0 ,. where Π = p |B|22µ0. ]. Part III, Paper 321 [TURN OVER].

13 Apr 2020: One can verifythat the block matrix. [A A#B. A#B B. ]is positive semidefinite since[. ... A A#BA#B B. ]=. [A1/2. A1/2(A1/2BA1/2)1/2. ][A1/2. A1/2(A1/2BA1/2)1/2. ] 0. Other changes The wrong expression for

10 Mar 2020: If x is a solution then it minimises. f(x) := ‖Axb‖2 = 〈Axb, Axb〉 = x>A>Ax 2x>A>b b>b. ... Hence f(x) = 0. But 12f(x) = A>AxA>b, hence A>(Axb) = 0.

12 May 2020: Then for any x wehave. (f1 f2)(x) = f1(x) f2(x). (The right-hand side is the Minkowski sum of sets A B = {a b : a A,b B}).

5 May 2020: Also one cancheck that (F1 F2)(u) = F1(u) F2(u) where the latter is the Minkowski addition of sets (i.e.,A B = {a b : a A,b B}). ... minxRn. ‖Ax b‖22 λ‖x‖1. (8). where A Rmn and b Rm.

2 Mar 2020: Such a matrix is called sparse and efficient solution ofAx = b should exploit sparsity. ... Since thecolumns of A are assumed linearly independent, b 6= 0 and we set q2 = b/‖b‖, hence q1 and q2 are orthonormal.Moreover,.

21 Oct 2020: Thus, the linear system Au = b can be written explicitlyin the block form. ... 25 / 28. Splitting of inhomogeneous systems. Note that our analysis should accommodate b = b(t ), sinceboundary conditions might vary in time!

4 Mar 2020: Since thecolumns of A are assumed linearly independent, b 6= 0 and we set q2 = b/‖b‖, hence q1 and q2 are orthonormal.Moreover,. ... This can be accomplished by multiplying the givenvector, e.g., the right-hand side b if we are solving a linear system