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10 Mar 2020: ȧ. a= ḃ. b= S sin θ cos θ, θ̇ =. S(b2 cos2 θ a2 sin2 θ)a2 b2. Combine these effects to obtain. ... ȧ. a= ḃ. b= S sin θ cos θ,. θ̇ =S(b2 cos2 θ a2 sin2 θ).

27 Feb 2020: 2 S)zByBz. Bx. Bz. K cos(Kz). sin(Kz+) 1. z+. ,. vy = v2az2. ... BxBz. K cos(Kz). sin(Kz+). The meridional (x and z) components of B are called the poloidal magnetic field, while the azimuthal(y) component is called the toroidal magnetic field.

26 Oct 2020: Substituting y,a,b and f by theirFourier series and using (3.3)-(3.5) we obtain an infinite dimensional system of linear equationsfor the Fourier coefficients ŷn:. ... For instance: choosinga(x) = f(x) = cos πx, b(x) = sin 2πx (which incidentally

23 Dec 2020: vr = rν cos φ[c(ν 1)Pν1 b(ν 1) cos θ Pν. ], ... 2.12a). vθ = rν cos φ[cP′ν1 2a csc θ Pν b.

15 Oct 2020: S =. α β. β α. β. β α.  ,are given by λk = α 2 iβ cos kx , and wk = (i. ... λk (A) =λk (C ). λk (B )=. 1 12 iµ cos kx.

4 Feb 2020: in Ow(l i)]B 1 •- ,. cos cos cos k2• sin 20.• -k} cos 20.,)A• iT.,•, 2k•, cos. ... d{ - 3a./.} cos X sin Or,( 1 i) - 2iR•vki, cos Ov] B•.

12 Oct 2020: Thus, the linear system Au = b can be written explicitlyin the block form. ... By Exercise 4, its eigenvalues and orthonormal eigenvectors aregiven as. B q = λq, λ = 4 2 cos πm1,.

28 Sep 2020: z = r cos θ. Then compute the Christoffel symbols by directly applying (1.8). ... l2(1 e cos φ) (4.16). where e is the eccentricity, a free parameter.

12 Feb 2020: Hence show that, for large Reynolds number, an energy amplification of(cos θ. ... ȧ. a= ḃ. b= S sin θ cos θ ,. θ̇ =S(b2 cos2 θ a2 sin2 θ).

7 Oct 2020: Thus, the linear system Au = b can be written explicitlyin the block form. ... By Exercise 4, its eigenvalues and orthonormal eigenvectors aregiven as. B q = λq, λ = 4 2 cos πm1,.

13 May 2020: 220. (|A|2 3. 4x20. ),. εv =1. 220(Br sin 0t Bi cos 0t)2. ... 1. 220(Br cos 0t Bi sin 0t). 2. =1. 220|B|2,. as required.

22 Jan 2020: Consider the matrix representation of the Euclidean group E(2) in twodimensions  cos θ sin θ asin θ cos θ b. ... a cos ψ sin θ. b+ sin ψ. 1. l. θ.

12 Oct 2020: S =. α β. β α. ββ α.  ,are given by λk = α 2 i β cos kx, and wk = (im sin kmx)Mm=1, where x = πh =. πM1. So, all ... λk(A) =λk(C). λk(B)=. 1 12 i µ cos kx. 1 12 i µ cos kx |λk(A)| = 1, k = 1.M.

18 Sep 2020: φ = U(r a2/r) cos θ,. with associated streamfunction. ψ = U(r a2/r) sin θ,. ... As before. u. t= ν. 2u. y2in y > 0. u = U cos ωt on y = 0.

26 Oct 2020: y ′′ a(x )y ′ b(x )y = f (x ), y (1) = y (1), (6). ... For instance: choosinga(x ) = f (x ) = cos πx , b(x ) = sin 2πx (which incidentally evenleads to a sparse matrix) we get.

9 Oct 2020: S =. α β. β α. β. β α.  ,are given by λk = α 2 i β cos kx , and wk = (i. ... λk (A) =λk (C ). λk (B )=. 1 12 i µ cos kx.

21 Oct 2020: By Exercise 4, its eigenvalues and orthonormal eigenvectors aregiven as. B q = λq, λ = 4 2 cos πm1,. ... 25 / 28. Splitting of inhomogeneous systems. Note that our analysis should accommodate b = b(t ), sinceboundary conditions might vary in time!

28 Oct 2020: For instance: choosinga(x ) = f (x ) = cos πx , b(x ) = sin 2πx (which incidentally evenleads to a sparse matrix) we get. ... Tn(x ) := cos n arccos x, x [1, 1],. or, in a more instructive form,.

19 Oct 2020: 12µ 1. . The matrix A is normal, with the eigenvalues λ = 1 iµ cos πh(see Example 2.15), so that ‖A‖2 = 1 µ2, hence instability for anyµ. ... By Exercise 4, its eigenvalues and orthonormal eigenvectors aregiven as. B q = λq

11 Nov 2020: x(t) = X R sin(!B(t t0)) and y(t) = Y R cos(!B(t t0)). ... AN=. g. 4r. 1 cos. sin(1.25). The resulting magnetic field is.

5 Feb 2020: Sébastien Michelin, a Simon Game,b Eric Lauga, c Eric Keaveny b andDemetrios Papageorgiou b. ... usingsinh(iz) = i sin z and cosh(iz) = cos iz.

5 Feb 2020: PnmdyðfC0. sin f. cos f. 0. 0BBB@. 1CCCAeimfdf:. (29). The components of the angular velocity are. ... 1). Fig. 4 Concentration distributions: quantitative comparison of theory(solid curves) and simulations (symbols) for the asymmetric (black circles,k0

8 Sep 2020: 2(B) and2(C)] and starts propagating from head (tapered end) to tail at speed v0 = vk cos(φ ) = vk sin(θ /2),which is the projected speed on the axis ... K̃13 = d3β sin θ , (40)K̃23 = 3 (d cos θ 3/2), (41).

14 Dec 2020: 2. (14). Finally we define the vector t = ( cos θ , 0, sin θ ). An illustrative sketch is provided in Fig. 3(b).The flow component of interest for direct interactions, which ... 2, 0, cos θ. 2. ), (28). and 1 < δ < 0 for R > 1.

22 Sep 2020: φp. 0–π π. x2. x1x3. ds/dt. –0.4 0.40. (a). (b). (c). ... x1. 6. 4. 2. 0–1–2 21. x3. x1. (a). (c). (b).

5 Oct 2020: φ = U(r a2/r) cos θ,. with associated streamfunction. ψ = U(r a2/r) sin θ,. ... As before. u. t= ν. 2u. y2in y > 0. u = U cos ωt on y = 0.

24 Nov 2020: 12k. {. [. (b l sin θ1 l sin θ2)2 (l cos θ2 l cos θ1)2]1/2. ... the directions (1, 0), ( cos 60, sin 60), ( cos 60, sin 60), i.e.