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18 Oct 2012: y =. . . . . . 0 , x < 0 ,. 1 cos x , 0 < x < ǫ ,. cos(x ǫ) cos x , x > ǫ. ... 1 cos x. x. )2. dx = π. (b) Use Parseval’s theorem and the result of question 7b to evaluate the integral.

20 Apr 2012: ex2. sin 2x). ,. (ii)d12. dx12(x cos x) using (a) the Leibniz rule and (b) repeated application of the product rule,. ... i) (x2 a)3/2,. (ii) ln(cos x),. iii) exp. {. 1(x a)2. }. ,.

7 Feb 2012: a magnetic moment m aligned along its axis. Let there be a magnetic field B, so that. ... each molecule has a potential energy mB cos θ (θ being the angle between the axis.

31 Oct 2012: x = ax′ , y = by′ , z = cz′ ,. x′ = r′ sin θ′ cos φ′ , y′ = r′ sin θ′ sin φ′ , z′ = r′ cos θ′ ,. where a, b and c are positive constants. Calculate ... If an extra transverse force F sin(πx/L) cos(πct/L) per unit length

17 Apr 2012: 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.

3 Sep 2012: 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.

2 Sep 2012: sider the circular trajectory [see Fig. 1(b)]. R ðÞ bðcos; sin; 0Þ: (10)For this trajectory, we have jR0ðÞj b and tðÞ ð sin; cos; 0Þ. ... Whenthey are not parallel, the phase dependence is proportionalto RðÞ. For example, for perpendicular

2 Sep 2012: B. Liverpool, Phys. Rev. E 74, 061913 (2006).[10] D. Saintillan and M. ... E. Paullet and G. B.Ermentrout, SIAM J. Appl. Math. 54, 1720 (1994).

2 Sep 2012: In a coordinate system moving at the speed of the colony, theradial and polar velocities are ur U[(c x3)cos A(x, )]and u U[(d x3)sin B(x, )], ... B 254, 107–113.12. Berg, H. C. & Purcell, E. M. (1977) Biophys.

28 Feb 2012: 3.4) wx =DW(z). exp(ζt) cos( x),. where D denotes a differentiation with respect to z. ... cos. ((m 1)κz. )]}= 0,(3.22). where A is given by (3.11) and B, a modified inertial parameter, is defined as.

8 Mar 2012: Here. b = R sin α , θ = π 2α =. b = R sin. (π. 2 θ. 2. )= R cos. ... This gives. Aκ cos κa. A sin κa=. B k cos (ka δ0).

2 Sep 2012: Fig. 4 Snapshots of turbulent spiral waves in a 2D array of hydrodynamically coupled rotors, with the greyscale representing cos[f(r) f].100 (a) dp/4.Dynamical steady state. ... b) dp/3. Initial developments of spirals from a pair of topological defects.

20 Oct 2012: 10-6. 0.01. 100. 106. 1010. (a) (b) (c). Figure 1: (a): cos(θn,n) against n. ... As is evident, the quantity cos (θn,n) is exponentially small in n.

6 Jun 2012: σ = 0. Then, from the above, cos(ωαT) = 0 and ω = sin(ωαT). ... If σ = 0 then. A = B cos ωT. ω = B sin ωT.

27 Aug 2012: fprop = (ξ‖ ξ)u sin θ cos θ ex. (13)In addition, in order to generate a net propulsion from a. ... u(x, h(x, t)) = ht. ey = bω cos(kx ωt)ey , (21).

2 Sep 2012: However, for all but the first, cos((v/2)Al(1-z)) changes sign in the range 0 < z < l , so the logarithm in equation (29) becomes singular. ... f r o m w h i c h we find t h a t :. COS -? A3(z --z) (47). T h e cell d i s t r i b

22 Aug 2012: 1 p0a cos q> 0q> 2• sin {b p0G dz (7) Thus in the steady state limit the nonlocal control of the mass flux expressed by (5) has become a purely ... atfi if ar• atfi •• •=(a cos q>)G (10). at a a• az.