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17 Dec 2023: These issues touch on aspects of nonequilibrium statistical mechanics and fluid dynamics, but also have implications for specific biological phenomena such as quorum sensing. ... subtilis, and algae such as Chlamydomonas and Volvox). Theoretical tools

4 Oct 2023: 3. A spin- 12 particle interacts with a time-varying magnetic field such that. ... H = γB(t) S where B(t) = B0ẑ b (x̂ cos ω1t ŷ sin ω1t).

1 Nov 2023: Show that the eigenvalues of such a matrixare its diagonal entries: λi = Aii (no sum over i). ... 3. Find a, b and c such that. . 1/3 0 a.

10 Oct 2023: Show that. u v = v w = w u ,. and hence obtain the sine rule for ABC.(b) Given any three vectors p, q, r such that. ... a b, b c, c a ] = [a, b, c]2.

1 Nov 2023: c) Let A be a square matrix such that Am = 0 for some positive integer m. ... a0tn ). Hint: Use properties of the determinant such as det(A) det(B) = det(AB).

29 Jan 2023: b) Deduce the existence of a g valued function K = K(w,z,w̃, z̃)such that Aw̃ = zK,Az̃ = wK. ... c) Let Da,a = 1,. , dim(B) be linearly independent vector fields onP such that.

30 Oct 2023: rodu. ced. with. out p. erm. issio. n. 5. (Optional, for enthusiasts.) Suppose that space is filled with a uniform, non-dispersivedielectric medium such that the refractive index is n. ... Write the macroscopic Maxwell’sequations in terms of the

1 Nov 2023: a) Let m, u and a be fixed vectors in R3 such that mu = 0 and au 6= 0. ... xn) in Rn (n > 4) such that their components satisfy.

17 Nov 2023: By. considering other functions of state, such as the free energy F = E TS, derive the. ... this is the original Ginzburg-Landau free energy for a superconductor.). Neglecting interactions, what is the critical dimension dc, such that the coupling.

2 Aug 2023: b. Now choose a metric such that the line element on the worldsheet is dτ2 dσ2. ... b. Explain why A(m) = A(m), and show that the Jacobi identity fixes.

6 Nov 2023: Theiteration matrix for this scheme is HS = I SA. Scheme 4.2 (Splitting) We assume A = B C in such a way that solving a linear system with the matrixC is ... so we should seek a splitting such that S = (AB)1 approximates A1.

27 Jan 2023: Assume that A(r) and B(r) are analytic functionsof r such that both have a simple zero at r = r+ > 0 and are positive for r > r+. ... In lectures we introducedcoordinates (u,,θ,φ) such that I+ is = 0 and the ”unphysical” metric satisfies.

8 Nov 2023: Dr H. Fawzi. Mathematical Tripos Part II: Michaelmas Term 2023. Numerical Analysis – Lecture 16Theorem 4.10 (The Householder–John theorem) If A and B are real matrices such that both A ... in the Gauss–Seidel method.) By definition of H we have Bw =

7 Mar 2023: x = ccoshξ cosη , y = csinhξ sinη , ξ > 0 , 0 6 η < 2π ,. and choose c > 0 such that the boundary of the vortex patch at a particular instantcorresponds to the curve ... b) Determine the effect of a uniform shear flow, u = Sxey, on the shape of an

27 Jan 2023: LξF = 0) then locally thereexists a scalar potential Φ such that iξF = dΦ. ... The decay happens such that P2 initially has dr/dτ 0. P2 subsequently falls into theblack hole and P3 escapes to r =. Let Ei miEi denote the energy of Pi (which

11 Oct 2023: saddle. Sketch examples, where possible, of flows with a periodic orbit such that the set of fixed points. ... enclosed by it: (a) is empty; (b) contains just one sink; (c) contains just the saddle; (d) contains one sink.

1 Nov 2023: 3.15). Chebyshev expansion Since (Tn)n=0 forms an orthogonal sequence, a function f such that 11 |f(x)|. ... Theorem 3.17 Let f be a function on [1, 1] such that it can be extended analytically to the Bernstein ellipse inthe complex plane.

31 Oct 2023: b) Deduce that D is completely reducible. 5. (Note that this question uses physics conventions for the generators ti, such that theyare Hermitian.) Three 3 3 matrices t := (t1, t2, t3) ... m, there existssome a such that Vb = Wa. Let α : Wa Vb be an

1 Dec 2023: As such, they are able to predict (a) that the stress will always fall below the yieldstress sufficiently close to the corner, leading to a plug there, and (b) that only ... figure 9(b) in Taylor-West & Hogg2022). As such, the dominant balances in the

3 Oct 2023: 3. If e1 = (1, 1, 0), e2 = (1,a, 1) and e3 = (1,b,2),(a) find the values of a and b such that e1, e2 and e3 form an ... 9. Four points A,B,C,D are such that AD is perpendicular to BC, and BD is perpen-dicular to AC.

26 Jul 2023: A, meaning that. D(A) D(B) and there exist constants. a, b > 0 such that for any. ... f n D(A) D(B) D(A) D(B1/2)such that. f n f and.

1 Dec 2023: 4(b)]. For theunconfined configuration, the flow pattern in such a fluid is expected to become asymmetrical owingto viscoelastic relaxation, which pushes the maximum shear stresses downstream of the top andbottom ... 9(b)]. Note that such interactions

1 Dec 2023: Theextent to which such hysteresis is a particular feature of thisexperimental material is not clear. ... In the specific model outlined in Sec. II B,α = 3 and β = 1 such that α > β, and so the flux tends to aconstant for large H.

1 Dec 2023: Such a style of motion is, in fact, often observed for real organisms (Gidmarket al. ... This feature can be seen in figure 7(b), where the speeddata for Bs = 50 and 103 approach the limit Ws 1 for such amplitudes, independently ofthe waveform.

1 Dec 2023: As B is increased, these plugs lengthenin the flow direction, to become much longer than the diameter of the obstacle, andwiden such that an increasingly large fraction of the disk lies ... For yet larger B, the channelsbecome thinner and wend their way

1 Dec 2023: As such, we work with this parameterised formof the layers throughout this study (see figure 1). ... Thisreorganisation is more pronounced at larger Ω, as demonstrated by the snapshots infigure 4(b,d) which show the flow before (a,b) and after (c,d)

1 Dec 2023: 2006a), yet theconfounding role that thixotropy can play in such a device is usually ignored. ... back of the current in such a way as to obtain the correct fluidvolume.

1 Dec 2023: As such, no specialtreatment was required in the numerical scheme to describe this moving front. ... The coordinate x measures the radial distancefrom the point of injection, such that r =|x|.

27 Dec 2023: For such flows, one of the main10interests is understanding the relationship between the flow rate 𝑞 and the pressure drop Δ𝑝,11which, to date, is studied primarily using numerical simulations. ... 2013), and in various industrial applications

1 Dec 2023: Such upper bounds can beimproved by posing trial velocity fields guided by the slipline fields. ... Such aconclusion is problematic as it implies that the viscoplastic theory does not convergeto perfect plasticity.

1 Dec 2023: Such instabilities can be driven by a number ofdifferent mechanisms including chemical reactions (Almarcha et al. ... chosen to be sufficiently large such that theseboundaries are far from the fingered region.

1 Dec 2023: Infact, for all the numerical results presented here, we further restrict our attentionto log-permeabilities which vary sinusoidally (De Wit & Homsy 1997a,b; Sajjadi &Azaiez 2013), such that. ... Againwe impose periodic boundary conditions and zero-mean

26 Jan 2023: such remarkable precision – dierent devices dier only by 3 parts in 1010 – the integer. ... There is a unique such state. given by. LLL,m=0 e|z|2/4l2B. We can now construct the higher states by acting with b†.

1 Dec 2023: As such,the equilibrium height, or far-field compression height, is simply. h=(pρigd) b0. ... As such, we model theoverlying ice as an elastic beam of bending stiffness B = Eid3/12(1 ν2), in termsof its Young’s modulus Ei and Poisson ratio ν.

14 Jan 2023: that is free to rotate. It also includes more exotic quantum materials such as Bose-. ... short distance modes, resulting in the critical exponent 6= 0. There are no such.

24 Aug 2023: Proposition 3.1 (Change of conformal scale). Given choices of scale ϵAB, ϵA′B′ definedover U M there are unique connections on S,S′ defined over U such that AA′ϵBC = ... The integral therefore vanishes for such contours.Accordingly, let N

1 Dec 2023: An inclined straight rod. Consider a straight rod falling under the action of a force such as gravity. ... Such configurations were also examined byMadani et al. (2010) in their experimental study.

3 Oct 2023: on the right a b points into the page. to both a and b in such a way that the set a,b,ab forms a right-handed. ... the order of a,b,c. Permutations (aside). • Suppose we have an ordered sequence of elements (such as a,b,c). •

13 Sep 2023: space volume is finite). Take k0 > k such that! k,k0 = Dk Dk0 6=. ... simply i =! it. Whenever such a transformation exists, the system is said to be.

16 Jun 2023: bfitx (m),bfity (m)} Mm=1 such that bfity (m) = F(bfitx (m)), m = 1,. ... The measure scrE prime assigns anorthogonal projector to each Borel measurable subset of BbbT such that.

20 Nov 2023: output. Such convergence becomes incredibly important as it provides an errorcontrol from below. ... In particular,suppose we have non-negative θ1,θ2 such that θ1 θ2 < π.

19 Nov 2023: We’ll see a variety of examples of such quantum computational benefits (including fac-toring) in the second half of the course. ... Such measurementshave the curious physical feature that generally there will be no states |ψ〉 having anyone of the

18 Dec 2023: Let t be a real number such that. σ1(y, A) < t < min. { ... Then there exists an index i {1, 2,. , N} such that|xi| = t.

21 Feb 2023: forward to adapt our methods to other initial domains such as Schwartz space. ... If > 0 is such that Sp.A/B2.z/ D zº, there is at most one point in n2;n1.A/ that also lies in B.z/.

18 Dec 2023: Then, Γ1 fails on a set F such that. • B(b, U, r) F has Lebesgue measure 0,• B(b, U, α1) F. ... If stsp(y, A) > 0 then there existsS BN such that Ξ(y, A) = {S}.

11 Jan 2023: The total number of iterations, T , neededto reach such an xs is bounded by. ... We call such an h a schedule criterion function, or simply a schedule criterion.

24 Aug 2023: pn, q1,. , qn)) withx 2 U such that takes the form 1.1.7. ... such that. lnk1(!) = lnk2(µ). (1.1.26). Accordingly, the map kH! {! a1.ak2 k. H|! b. a1.ak2b= 0} defined by! 7! l(µ) (1.1.27).

10 Feb 2023: on timelike geodesics cannot be expected to behave well near such conformal boundaries. - ... embeds M as a manifold with smooth boundary M in M̃ such that:.

29 Dec 2023: Since we know nothing else about the system, such a democratic approach seems. ... Hamiltonian. But such a term would shift the energy levels of each system.

27 Dec 2023: topics that exist for real word QCD, such as confinement and chiral symmetry breaking,. ... Are there other. theories that also exhibit such symmetry? What properties might they have?