J. Hubbard, Proc.R.Soc. London Ser. A, 240 (1957) ibid 243, 244(1958)  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Smoluchowski (kinetic) equation for f(m; t) is given by (e.g. 6, 2, 32] kT fRVMS ) R [m mf ] 1) x = r and R = m are the spatial and the rotational gradient operator, respectively; m m a[D m D : mmm] 2) is the Je ery orbit of axisymmetric ellipsoidal molecules [12], D are the rate of strain and vorticity tensors, respectively de ned by D = 1 2 (rv rv ) 1 2 (rv rv ) 3) 1 a 1 is the molecular shape parameter related to the molecule aspect ratio r by a = 1 : 4) The coecient D r is an averaged rotary di usivity, taken to be ....

Jeery, G. B., Proc. Roy. Soc. London Ser. A 102, 161 (1922).

.... [31, 14, 27] Df = R [D r (m; a) Rf kT fRVMS ) R [m mf ] 1) denotes the material derivative t ( v r( x = r and R = m m are the spatial and the rotational gradient operator, respectively; m m a[D m D : mmm] 2) is the Je rey orbit of ellipsoids [63], D are the rate of strain and vorticity tensors, de ned by (with the convention (rv) ij = v i x j D = 1 2 (rv rv ) 1 2 (rv rv ) 3) 1 a 1 is the molecular shape parameter related to the molecular aspect ratio r by a = 1 : 4) Note that a 1 corresponds ....

....i the orientation tensor is in plane, giving easy recognition of in plane versus out of plane solutions either by monitoring these entries or by visualizing the directors relative to the shearing plane. 4. 3 Tensorial analog of the Leslie director alignment criterion Classical studies of Je rey [63], Ericksen [35, 36] Leslie [77] Jenkins [64] Hinch Leal [60] Larson [72] and many others in the past decade have aimed toward criteria for steady versus unsteady motion of liquid crystals and nematic polymers. We refer to two recent articles [118, 127] where L E theory is applied to model ....

Jeery, G. B., Proc. Roy. Soc. London Ser. A 102, 161-179 (1922).

....operations are justified: d(k) 1 1 2i dk A : E.3) Here the prime denotes the derivative with respect to the argument of the Hankel functions. Let us introduce the abbreviation = Following Ref. 50] we apply the Watson contour method [51] to (E.3) m = 1 : E.5) Here the contour C encircles in a counter clock wise manner a small semi infinite strip D which completely covers the real axis, but which only has a small finite extend into the positive and negative imaginary direction. As in Ref. 16] the contour C ....

G.N. Watson, Proc. Roy. Soc. London Ser. A 95 (1918) 83.

....round, cusped, or pointed with 0t = 120 at the tip (see [4] There is a general consensus that these bubbles exist as solutions of this problem. The numerical solutions of Vanden Broeck [9] show a pointed bubble rising at a speed F = 0.3577. This is consistent with a conjecture of Garabedian [5]. There are still some open questions about the pointed bubble. The equations of this problem contain F as a free parameter. Vanden Broeck [9] uses a Fourier collocation method and determines F numerically by treating F as a free parameter. He finds F = 0.3577. He resolves this problem but with ....

P. R. Garabedian, Proc. R. Soc. London Ser. A 241, 423 (1957).

....onto a nite dimensional subspace. The idea of approximating path integrals using a nite subset of basis functions has been suggested before in the literature. Davison was one of the rst to consider the use of orthogonal function expansions in the representation of Feynman path integrals [15], although he did not explore truncating the expansion. In a related article on Wiener integration of a di erent class of functionals, Cameron proposed using a nite set of orthogonal basis functions, and investigated the convergence of Fourier appear in the Journal of Computational (spectral) ....

.... for the approximate density matrix: p) b; a) J exp 2 (b i a i ) d exp i K i ; 12) where x ( a (b a) y ( For the Fourier case, one typically calculates J by requiring that the discretization be exact when applied to an ideal gas (i.e. V 0) [4, 5, 15]. We can apply this same technique to a general subspace method. Assuming that K is symmetric positive de nite, each integral over i can be evaluated by diagonalizing K and applying a linear change of variables; e.g. expf K g d = 1=2 : 13) Applying this formula to ....

B. Davison, Proc. Roy. Soc. London Ser. A 225, 252 (1954).

....average directly from the study of the general solution of the inverse Laplace transform of (1.3) Its potential nonuniqueness is lifted, in our context, by imposing compatibility with hyperasymptotics an important improvement in asymptotic calculations proposed by M. Berry ( 22] 23] 24] [25]) 1.1.2 Nonresonance (1) i ; i = 1; n 1 are assumed Z linearly independent for any d. 2) Let 2 [0; 2) and = i 1 ; i p ) where fi fi arg i j Gamma fi fi 2 ( Gamma=2; 2) those eigenvalues contained in the open half plane H centered along e i ) We require that ....

M.V. Berry Proc. Roy. Soc. London Ser. A 434 no. 1891, 465--472. (1991)

....balanced average directly from the study of the general solution of the inverse Laplace transform of (1.3) Its potential nonuniqueness is lifted, in our context, by imposing compatibility with hyperasymptotics an important improvement in asymptotic calculations proposed by M. Berry ( 22] 23] [24], 25] 1.1.2 Nonresonance (1) i ; i = 1; n 1 are assumed Z linearly independent for any d. 2) Let 2 [0; 2) and = i 1 ; i p ) where fi fi arg i j Gamma fi fi 2 ( Gamma=2; 2) those eigenvalues contained in the open half plane H centered along e i ) We require ....

M.V. Berry, C.J. Howls Proc. Roy. Soc. London Ser. A 443 no.

....three times repeated periodic orbit that is spanned by the two disks, including all prefactors, Maslov indices, and symmetry reductions. Note that the two disk system has only one classical periodic orbit. In the mean time, one of us has shown that, with the help of Watson resummation techniques [25, 26] and by complete induction, the semiclassical reduction of the quantum mechanical traces of any non overlapping 2 n 1 disk system (where in addition grazing or penumbra orbits [27, 28] are avoided in order to guarantee unique isolated saddle point contributions) reads as follows [24] ....

Watson G N 1918, Proc. Roy. Soc. London Ser. A 95 83

....with respect to the argument of the Hankel functions. Let us introduce the abbreviation = H (2) 0 (ka j ) H (2) ka j ) Gamma H (1) 0 (ka j ) H (1) ka j ) D.4) Following refs. 11, 12] which are based on W. Franz seminal work [47] we apply the Watson contour method [48] to (D.3) d(k) a j 2i 1 X m= Gamma1 m = a j 2i 1 2i I C d e Gammai sin( D.5) where the contour C encircles counter clock wise a small semi infinite strip D which completely covers the real axis but which only has a small finite extend into the positive and negative ....

G.N. Watson, Proc. Roy. Soc. London Ser. A 95 (1918) 83.

....Page 1 2 Hilbert J. Kappen and Wim J. Wiegerinck The Boltzmann Gibbs distributions is widely used in physics, and mean field theory has been known for these distributions for a long time. For instance, for the Ising model on a square lattice, it is known as the Bragg Williams approximation [3] and it is generalized to other models in the Landau theory [4] One can show that the above lower bound corresponds to the first term in a Taylor series expansion of the free energy around a factorized model. This Taylor series can be continued and the second order term is known as the Thouless ....

W.L. Bragg and E.J. Williams. Proc. Soc. London Ser. A, 145:699, 1934.

....2 B n x Delta y = x 1 y 1 Phi Delta Delta Delta Phi x n y n ) 2 B (the operations on the RHS s being addition and multiplication mod 2 in B. Note that x Delta y is the parity of the number of places where x and y both have a bit value of 1. Early Days The earliest quantum algorithms [1, 2] were concerned with a situation in which we are given a black box or oracle that computes a function f : B n B and we are required to decide whether a certain global property (i.e. a joint property of all the function values) holds of f . For quantum computation the black box is given as ....

....jyi Gamma jx 1 i jx 2 i : jx n i jy Phi f(x 1 ; x n )i (3) 2 (We will often abbreviate jx 1 i jx 2 i : jx n i as jxi for x 2 B n . Thus if y is initially set to 0 the value of f may be read from the last qubit. For our first problem, referred to as Deutsch s XOR problem [1], we have n = 1 so that f is one of the four possible functions f : B B. We are to decide whether f(0) Phi f(1) is 0 or 1. Equivalently we wish to decide whether f is a constant function or a balanced function (where balanced means that f takes one value 0 and one value 1) Clearly any ....

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Deutsch, D. (1985) Proc. Roy. Soc. London Ser. A 400, 97.

....variable cell MD, pionereed by Andersen [78] and by Parrinello and Rahman [16] to a slab with variable curvature. In our formulation the curvature is a single, global Lagrangian degree of freedom. Surface stress can be extracted as a direct result of the calculation through elasticity equations [79], which will be described below. The outline of the chapter is as follows: x7.2 will present the theory and geometric considerations underlying our simulation. In x7.3 the phenomenology of a bent plate is reviewed in order to extract the pertinent equations. In section x7.4 we present some ....

.... Delta (s) Gamma Y 6 (1 Gamma 2 ) k M t 2 (7.32) where Y is the Young s modulus, the Poisson s number, k M the curvature and t the thickness of the sample. This relation is known as Stoney s equation, and was first derived in 1909 (except for the biaxial nature of the stress [79, 66]) We stress that this formula is valid for a uniaxial bending, whereas for an isotropic bending in two directions the factor Gamma 1 Gamma 2 Delta should be replaced by (1 Gamma ) Another limit of validity is that the plate must be infinitely large, relative to its thickness. In other ....

G. G. Stoney, Proc. R. Soc. London Ser. A 82(1909)172.

....is viewed as consisting of two main parts: quantum gates and quantum wires [29] 14 By basic quantum gates we mean any set of quantum gates which can perform any desired quantum computation. A universal quantum gate is the one whose combination can be used to simulate any other quantum gate [29 31]. A quantum wire is used as a representation of that part of computation of any qubit where the evolution is a simple identity operation (i.e. no gate operates on the qubit) as well as the time qubit spends during the gate operation. B. Errors in Computation There are two main types of errors ....

A. Barenco, Proc. R. Soc. London Ser. A 444, 678 (1995).

....of quantum error correction codes when applied in real atoms. III. ACCURACY THRESHOLDS TO QUANTUM COMPUTATION A. Quantum Computer An input to a quantum computer is a string of quantum bits qubits. A quantum computer is viewed as consisting of two main parts: quantum gates and quantum wires [29]. 14 By basic quantum gates we mean any set of quantum gates which can perform any desired quantum computation. A universal quantum gate is the one whose combination can be used to simulate any other quantum gate [29 31] A quantum wire is used as a representation of that part of computation of ....

....is viewed as consisting of two main parts: quantum gates and quantum wires [29] 14 By basic quantum gates we mean any set of quantum gates which can perform any desired quantum computation. A universal quantum gate is the one whose combination can be used to simulate any other quantum gate [29 31]. A quantum wire is used as a representation of that part of computation of any qubit where the evolution is a simple identity operation (i.e. no gate operates on the qubit) as well as the time qubit spends during the gate operation. B. Errors in Computation There are two main types of errors ....

D. Deutsch, Proc. R. Soc. London Ser. A 425, 73 (1989).

.... question how evolutionary branching and speciation take place is for example approached by developing formal models which demonstrate the formation of groups [2 6] These models are individual based in contrast to macroevolution models which assume a species or group as a given elementary unit [1,7,8]. The diffusion and separation of individuals in genotype or trait space is either achieved by drift in a neutral fitness landscape [4] or by introducing an explicit fitness function [2,3,9] which causes disruptive selection. Sometimes additional explicit functions are introduced to model strength ....

....f sel represents the advantage which stems from occupying a lowly populated niche. 2) The macroscopic pattern of the evolving population structure is similar to patterns derived from the fossil record. 3) The model can be easily extended to include: bounded genotype space, environmental stress [7], sexual recombination, external fitness pressure, a spatial world, higher dimensional genotypes, etc. 4) Compared to macroevolutionary models which assume a species to be a given entity [1,7] the seceder model can explain group formation and evolution based on local interactions of individuals. ....

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M. E. J. Newman, Proc. R. Soc. London Ser. B 263, 1605 (1996).

....analogy, and like most, should not be taken too literally. Here the complex operators FLIP, AND, and AND y are speci ed in more detail. In operator notation, FLIP = F 0 c2 c1 F 0 c2 x j (n j 1; z pj 6= z p 1j ) 6) Alternatively, FLIP may be represented as a quantum network as in [12]. Fig. 2 shows one such network for a 3 input function. In the gure, a indicates that the F 0 operator is applied only if the value of the bit in question di ers from the value of that bit for the previous example. Also, in operator notation AND = A zn1 xngn 2 gn 1 A z3 ....

D. Deutsch, Proc. Roy. Soc. London Ser. A 425, 73 (1989).

....quantum algorithms, amplitude manipulation. 1 INTRODUCTION The study of computation as a physical process has produced fascinating results in the form of new approaches to problem solving and information processing that exhibit impressive speed up over classical approaches for some problems [1, 2, 3, 4, 5, 6]. All of these algorithms operate on a superposition 1 of all basis states, usually assuming prior knowledge of some function f that can be e ected as a unitary evolution of the system. However, it is often the case in computational problem solving that a function f is not known but rather that ....

D. Deutsch and R. Jozsa, Proc. Roy. Soc. London Ser. A 439, 553 (1992).

.... including optical gate interferometers (Milburn 1989) magnetic bubbles (Chang 1982) Josephson junctions or discrete state systems (single electron parametron) Likharev 1982; Likharev and Korotkov 1996) split level charge recovery devices (Younis and Knight 1993) and quantum devices (Deutsch 1985; Lloyd 1996; Orlov, et al. 1997; Gershenfeld and Chuang 1997; Cory, et al. 1997) It is historically interesting to note that Bennett proposed in his original paper (1973) and again in a review with Landauer (1985) that enzymatic reactions close to equilibrium could represent a chemical ....

Deutsch, D., 1985, Proc. R. Soc. London Ser. A 400, 97-117.

....of turbulent spectra [1, 2] The main idea is to connect small scale quantities such as the energy dissipation rate ffl with the large scale quantities such as the outer length scale L and a rms velocity component, u 1;rms . More precisely, following Richardson s cascade picture of turbulence [3, 2], it is argued that for fully developed turbulence ffl = c ffl u 3 1;rms =L (1) holds, where c ffl is a dimensionless constant in the range of 1. However, it is not clear, for what Reynolds number Re the large Re limit (1) is reached. For small Re the dimensionless number c ffl clearly depends ....

L. F. Richardson, Proc. R. Soc. London Ser. A 110, 709 (1926).

....have, on average, less speciation then an old one. This view, of the origin of species per saltum, i.e. the creation of new genera by extremely rare mutations that trigger an avalanche of speciation, as opposed to the gradual adaptation of species through natural selection, was taken up by Yule [3] in a remarkable paper. He developed a mathematical theory of evolution based on this picture, which matched the species abundance curves obtained by Willis with high accuracy. The theory proposed was simple. For one ancestral genus, he assumes a certain probability for a specific mutation which ....

....This allowed him to fit most observed distributions with a parameter ae 2, which fit the species abundance relations available to him (see Fig. 1) Fig. 1. Number Ng of genera (binned) of Leguminosae with n species, compiled by Willis from Dictionary of Flowering plants, as quoted by Yule [3] (errors are unavailable) The solid line is Yule s model fit (infinite time) for ae = 2:457, i.e. Ds = 1:4. The finite time fit raises the power to Ds = 1:5. Furthermore, he concluded from his model that there is indeed, as postulated by Willis, a relationship between age and size of a genus, ....

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Yule, G.U. (1924), Proc. Roy. Soc. London Ser. B 213, 21.

....formulation of the model than those caused by lack of randomness in the random number generator. In contrast, the first class of applications can require very precise solutions. Increasingly, computers are being used to solve very well defined but hard mathematical problems. For example, as Dirac [1] observed in 1929, the physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are completely known and it is only necessary to find precise methods for solving the equations for complex systems. In the intervening years fast computers and new ....

P. A. M. Dirac. Proc. R. Soc. London Ser A, 123:734, 1929.

....proteins such as myosin, kinesin and dynein moving along molecular tracks are involved in a wide range of processes essential for life, e.g. cell division, muscle contraction, and intracellular transport of organelles. For many decades exclusively data from physiological measurements on muscles [2] provided experimental information for modeling molecular motors [3, 4] In recent years, a variety of in vitro techniques allowed the observation of single motor proteins [5] and gave new insights into the basic principles underlying their operation. Not only new theoretical models for ....

....produce maximum force than for those which have already done their work and now pull backwards. This idea has already been used by A.F. Huxley [3] Such a dependence is needed for an explanation of the approximately hyperbolic force velocity dependence in muscle systems. Physiological data by Hill [2] are perfectly fitted within the above analytic results by choosing ff dm = 0:55 and t d AE t 0 a . Quantized binding sites and thermal fluctuations are found to play only a minor role (their neglect leads to almost the same curve with ff dm = 0:58) A function that can lead to an anomalous F ....

Hill, A., Proc. R. Soc. London Ser. B 126 (1939) 136.

....problem. N phase solutions of soliton equations such as KdV, NLS and general AKNS systems were first constructed by S. P. Novikov and I. M. Krichever [31] Their method is a rediscovery of classical Riemann surface techniques developed by J. L. Burchnall and T. W. Chaundy [7] and by H. E. Baker [3] for classifying commutative algebras of scalar differential operators. A different characterisation of N phase solutions was given by P. D. Lax (in particularly for the KdV equation) 39] N phase solutions are critical points for a linear combination of integrals of motion. This ....

H. E. Baker, Proc. Royal Soc. London Ser. A, 118 (1928), pp. 573--580.

....a few peopley. The story of phase in quantum mechanics went further on with the work of Dirac in his famous paper on the quantization of the electromagnetic field. Dirac got confronted with the problem, how to relate an operator to a quantized field amplitude. In his original article of 1927 Dirac [7] took the existence of a phase operator for granted. Only later he has written down mode operators with bose character. Nowadays the formalism of quantized fields relies on these creation and annihilation operators rather than phase and amplitude. A renaissance of the phase problem, which still ....

....one is to try to represent the phase by a hermitian operator. This operator should be the conjugate variable to the photon number of a field mode or the energy of a harmonic oscillator. This is analogous to the position and momentum description of quantum mechanics. It was this concept which Dirac [7] had in mind when he wrote down his method of field quantization. We have already mentioned the problems connected to this approach in the introduction and many authors [8, 9, 10, 11] have tried to deal with them. Some of them even completely dismissed the existence of such a hermitian phase ....

Dirac P A M 1927 Proc. Roy. Soc. London Ser. A 114 243

....eld, the anticrossings mainly take place. Perhaps the most remarkable feature of the analyzed data is the strong oscillation of the resonance width of the ground WS resonances. These oscillations are superimposed on the general tendency 1 F exp(const=F ) predicted by the Landau Zener theory [11] and show an hierarchical structure of di erent scales. The physical origin of the resonance width oscillations is attributed to the large number of crossings with WSLs associated with the unstable above barrier resonances, as it will be shown in Sec. 3. We proceed with the analysis of the wave ....

C. Zener, Proc. R. Soc. London Ser. A 145, 523 (1934).

....algorithm based on a general formalism of Kitaev and contrast its structure to the ingredients of Shor s algorithm. Introduction The principal quantum algorithms which provide an exponential speedup over any known classical algorithms for the corresponding problems are Deutsch s algorithm [2], Simon s algorithm [4] and Shor s algorithm [5] Each of these rests essentially on the application of a suitable Fourier transform. In this paper we will outline the construction of the Fourier transform over a general (finite) Abelian group and highlight its origin and utility in the quantum ....

....2 B n x Delta y = x 1 y 1 Phi Delta Delta Delta Phi x n y n ) 2 B (the operations on the RHS s being addition and multiplication mod 2 in B. Note that x Delta y is the parity of the number of places where x and y both have a bit value of 1. Early Days The earliest quantum algorithms [1, 2] were concerned with a situation in which we are given a black box or oracle that computes a function f : B n B and we are required to decide whether a certain global property (i.e. a joint property of all the function values) holds of f . For quantum computation the black box is given as ....

[Article contains additional citation context not shown here]

Deutsch, D. and Jozsa, R. (1992) Proc. Roy. Soc. London Ser A 439, 553-558.

....for incompressible, immiscible fluids, we consider fully compressible flow that allows for local compression of volumes elements; the interface should in principle however remain sharp, the smoothing depending on the diffusion of the numerical scheme. The instability has been studied extensively [1 10]. The interest stems from the importance of the Rayleigh Taylor (and the related Richtmyer Meshkov) instability in turbulent mixing in Inertial Confinement Fusion [11] and supernovae [5] The direct simulation of three dimensional compressible RayleighTaylor Instability has generally not been made ....

Taylor, G. I., Proc. Roy. Soc. London Ser. A 201, 192 (1950).

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M.V. Berry, Proc. Roy. Soc. London Ser. A392, 45 (1984).

....by resistivity [1] In this Brief Communication we demonstrate that for increasing Reynolds numbers the quiescent ground state of a voltage driven resistive plane sheet pinch becomes first unstable to two dimensional perturbations. This is a generalization of Squire s theorem in hydrodynamics [2,3]. For the special case of a spatially uniform resistivity and no dc magnetic field in the sheetwise or toroidal direction (the direction of the driving electric field) a proof was given in Dahlburg and Karpen [4] If the resistivity is spatially uniform, the equilibrium current is also uniform ....

.... Gamma1 [ik 3 jj 1k Gamma ( jj 3k ) 0 ] b 3k = ik 2 B e 2 v 3k ik 3 B e 3 v 3k GammaS Gamma1 [ jj 2k ) 0 Gamma ik 2 jj 1k ] 22) where a prime denotes differentiation with respect to x 1 and D = x 1 . Generalizing Squire s transformation of ordinary hydrodynamics [2,3] to the magnetohydrodynamic case, we define k = k 2 2 k 2 3 ) 1=2 ; P k = k = P k =k 2 k v 2 k = k 2 v 2k k 3 v 3k ; v 1 k = v 1k ; k R = k 2 R; k b 2 k = k 2 b 2k k 3 b 3k ; b 1 k = b 1k ; k S = k 2 S: 23) We can assume k 2 6= 0 ....

H. B. Squire, Proc. R. Soc. London Ser. A 142, 621 (1933).

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J. Hubbard, Proc.R.Soc. London Ser. A, 240 (1957) ibid 243, 244(1958)

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M. V. Berry, Proc. R. Soc. London Ser. A 413, 183 (1987).

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D. Gabor, Proc. Roy. Soc. London Ser. A A197, 459 (1949).

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M.V. Berry, C.J. Howls, Proc. Roy. Soc. London Ser. A 443,

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J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).

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Freeland, S. J. and Hurst, L. D. (1998) Proc. R. Soc. London Ser. B 265, 2111--2119

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Harris, H. (1966) Proc. Royal Soc. London Ser. B 164, 298--310

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Taylor, G. I., Proc. R. Soc. London Ser. A 164, (1938) 476.

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Dirac P.A.M. (1925), Proc. Roy. Soc. London ser. A, 109, 642-653.

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P. A. M. Dirac, Proc. Roy. Soc. London Ser. A 114:243 (1927).

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Harris, H. (1966) Proc. Royal Soc. London Ser. B 164, 298--310

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