Solving the constrained modified KP hierarchy by gauge transformations

In this paper, we mainly investigate two kinds of gauge transformations for the constrained modified KP hierarchy in Kupershmidt-Kiso version. The corresponding gauge transformations are required to keep not only the Lax equation but also the Lax operator. For this, by selecting the special generating eigenfunction and adjoint eigenfunction, the elementary gauge transformation operators of modified KP hierarchy T_D(Φ) = (Φ^?1)^?_x¹? Φ^?1 and T_I (Ψ) = Ψ^?1? ^?1Ψ_x, become the ones in the constrained case. Finally, the corresponding successive applications of T_D and T_I on the eigenfunction Φ and the adjoint eigenfunction Ψ are discussed.     

Bilinear Identities and Hirota’s Bilinear Forms for the (γn, σk)-KP Hierarchy

In this paper, we discuss how to construct the bilinear identities for the wave functions of the (γ_n, σ_k)-KP hierarchy and its Hirota’s bilinear forms. First, based on the corresponding squared eigenfunction symmetry of the KP hierarchy, we prove that the wave functions of the (γ_n, σ_k)-KP hierarchy are equal to the bilinear identities given in Sec.3 by introducing N auxiliary parameters z_i, i = 1, 2,?…?, N. Next, we derived the bilinear equations for the tau-function of the (γ_n, σ_k)-KP hierarchy. Then, we obtain the bilinear equations for the taufunction of the mixed type of KP equation with self-consistent sources (KPESCS), which includes both the first and the second type of KPESCS as special cases by setting n = 2 and k = 3. Finally, using the relation between the Hirota bilinear derivatives and the usual partial derivatives, we show the procedure of translating the Hirota’s bilinear equations into the mixed type of KPESCS.     

The gauge transformation of the q-deformed modified KP hierarchy

In this paper, we mainly study three types of gauge transformation operators for the q-mKP hierarchy. The successive applications of these gauge transformation operators are derived. And the corresponding communities between them are also investigated.     

Investigation of the mass dependence of self-diffusion coefficients by molecular dynamics calculations: binary and ternary isotopic mixtures of atoms

Extensive molecular dynamics calculations have been used to study for the first time systematically the dependence of the self-diffusion coefficients D_i (i = 1, 2, 3) in binary and ternary atomic isotopic mixtures on the particle mass ratios m*₂ = m ₂/m ₁ and m*₃ = m ₃/m ₁ at different reduced temperatures T* and reduced particle number densities n*, using a Lennard-Jones 12-6 potential and a hard-soft-spheres potential. In addition, the dependence of D_i values of binary mixtures on the mole fraction x ₁ = 1—x ₂ was also investigated for some thermodynamic states. The calculated values of D_i can be represented quantitatively by exponential estimates of the form D_i = D* _i (m*₂)^ex _i in the case of binary mixtures and D_i = D ⁰ _i (m*₂)^{ex _i} (m ₃)^{ext _i} in the case of ternary mixtures. D ⁰ _i are the self-diffusion coefficients in reference mixtures of mass ratios m*₂ = m*₃ = 1. The dependence of the exponents ex _i (m*₂, T*, n*, x ₁) of binary mixtures on m*₂, T*, n*, and x ₁ and the dependence of the exponents ext _i (m*₂, m*₃, n*) of equimolar ternary mixtures at T* = 1.8 on the exponents ex _i of the constituent binary mixtures and on m*₂, m*₃, and n* are discussed.     

丙烯酰胺基光致聚合物全息光栅的动力学研究

研究了光致聚合物的光化学反应和单体扩散反应的理论模型及动力学参数. 合成了以聚乙烯醇为粘结剂,三乙醇胺为引发剂,藻红B敏化的丙烯酰胺基光致聚合物全息存储材料,通过对测定的透过率和衍射效率进行曲线拟合,分析了染料浓度对光致聚合物的摩尔吸收系数ε、量子产率Φ和光化漂白速率常数k等光化动力学参数的影响规律,以及曝光强度对聚合反应速率k₀、扩散时间常数τ_D和最大折射率调制度Δn等扩散动力 关键词： 光致聚合物 透过率 衍射效率 动力学参数     

Simplified closed form expressions for irreducible tensor operators

A general closed form expression is derived which enables one to write a spherical tensor of rank L and projection k, T_uk^L, in terms of the familiar angular momentum operators, S_z and S². The simplicity of this expression results from an intriguing relationship between these spherical tensor operators (diagonal operator equivalents) and successive derivatives of Tchebichef orthogonal polynomials.     

Integrability ofSU(N) spin chains with elliptic exchange

For some one-parameter setH _N of linear combinations ofN(N−1)/2 elementary transpositions {P _jk} (1≤j<k≤N) at arbitrary naturalN≥3 one can construct a variety {I _m} (3≤m≤N) of operators which commute withH _N. Being applied toSU(n) spin representations of the permutation group, this proves the integrability of 1D periodic spin chains with elliptic short-range interaction. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Desorption kinetics and surface diffusion of potassium,rubidium and cesium on a silicon(111)7 × 7-surface

Using pulsed atomic beam technique and a surface ionization ion microscope, the desorption kinetics and the surface diffusion of the alkalis potassium, rubidium and cesium were investigated on a Si(111)7 × 7-surface at extremely low alkali coverages. In the temperature range 1120 … 800 K, the mean adsorption lifetime τ(T) = τ₀ · exp(E_desⁱ/kT) and the mean diffusion length x(T) - defined in the equilibrium between adsorption, diffusion and desorption - were measured. From these data the diffusion constant D(T) = D₀ · exp(-E_diff/kT) was obtained as D = x^?2/τ. For temperatures T ? 750 K, the diffusion constant was calculated from nonstationary alkali concentration profiles using the Boltzmann-Matano method. From the temperature dependence of these quantities the parameters of desorption (E_des,ⁱ τ₀) and surface diffusion (E_diff, D₀) for K, Rb and Cs on Si(111) were obtained. The values of E_diff and D₀ are comparably high and may be interpreted by non-localized diffusion according to a model proposed by Bonzel (Surf. Sci. 21 (1970) 45).     

D-dimensional massless particle with extended gauge invariance

We propose the model ofD-dimensional massless particle whose Lagrangian is given by theN-th extrinsic curvature of world-line. The system hasN+1 gauge degrees of freedom constitutingW-like algebra; the classical trajectories of the model are space-like curves which obey the conditionsk _N+a=k_N−a, k_2N =0,a=1, ...,N−1,N≤[(D−2)/2], while the firstN curvaturesk _i remain arbitrary. We show that the model admits consistent formulation on the anti-DeSitter space. The solutions of the system are the massless irreducible representations of Poincaré group withN nonzero helicities, which are equal to each other. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

A new theorem of information theory

Consider a random experiment whose possible outcomes arez ₁,z ₂,...,z _n. Let the prior probabilities be p₁ ⁰, ...,p_n ⁰, and let the posterior probabilities bep ₁,...,p _n. It is shown that, subject to certain prescribed and intuitively reasonable conditions, the expressionI =k p _i In (p _i/p _i ⁰), wherek is a positive constant, is the unique expression for the information contained in a message which alters the probabilities from thep _i ⁰ to thep _i.     

Clarifying the nature of the Johari-Goldstein β-relaxation and emphasising its fundamental importance

ABSTRACT

Since 1998 the primitive relaxation time τ ₀(T,P) of the Coupling Model (CM) and the Johari-Goldstein (JG) β-relaxation time τ_JG (T,P) are shown approximately equal in many glass-formers. The CM relation between τ ₀(T,P) and τ_α (T,P) at any T and P is exact. Additionally from the CM relation τ_α (T,P)/τ ₀(T,P) is exactly invariant to variations of T and P while τ_α (T,P) is kept constant, and τ ₀ is exactly a function of ρ ^γ/T like τ_α . However, since τ_JG (T,P) ≈ τ ₀(T,P), the exact invariance of τ_α (T,P)/τ ₀(T,P) leads to approximate invariance of τ_α (T,P)/τ_JG (T,P), and τ_JG is approximately a function of ρ ^γ/T. Notwithstanding, the CM prediction of the approximate relations between τ_β and τ_α were mistaken as exact relations by some researchers. In this paper, we remove this misunderstanding by demonstrating via simulations and experiments that the JG β-relaxation is comprised of processes with different length-scales and degrees of cooperativity, and the process is heterogeneous. The distribution of processes makes τ_JG (T,P) equivocal, because it is just a single relaxation time used to represent the different processes within the distribution, which may change on varying T and P, at constant τ_α (T,P). The problem is compounded if the β-relaxation is not resolved, and fitting procedure used to extract τ_JG (T,P) and τ_α (T,P). Despite the relations of τ_JG (T,P) to τ_α (T,P) are approximate, we show these properties of τ_JG (T,P) are truly remarkable, fundamental, general, and important.     

Investigation of the mass dependence of self-diffusion coefficients by molecular dynamics calculations: binary non-isotopic mixtures of atoms

Extensive molecular dynamics calculations have been used to study systematically for the first time the dependence of the self-diffusion coefficients D_i (i = 1,2) in binary equimolar non-isotopic atomic mixtures (mole fractions x ₁ = x ₂ = 0.5, reduced temperature T? = 1.6) on the particle mass ratio m?₂ = m ₂/m ₁ (1 ≤ m?₂ ≤ 16.5) in the ranges of reduced particle number density 0.15 ≤ n? ≤ 0.85 and reduced length parameter 0.5 ≤ σ?₂₂ ≤ 2.0 for a Lennard-Jones 12-6 potential. The calculated D_i values can be represented quantitatively by an exponential estimate of the form: D_i = D ⁰ _i (m?₂) ^ex _i . D ⁰ _i are the self-diffusion coefficients in reference mixtures with m?₂ = 1. The observed dependence of the exponents ex _i (m?₂, n?, σ?₂₂) on m?₂, n? and σ?₂₂ is discussed in this paper.     

Capacity of Multivariate Channels with Multiplicative Noise: Random Matrix Techniques and Large-N Expansions (2)

We study memoryless, discrete time, matrix channels with additive white Gaussian noise and input power constraints of the form Y i = ∑ _j H _ij X _j + Z _i , where Y _i , X _j and Z _i are complex, i = 1… m, j = 1… n, and H is a complex m× n matrix with some degree of randomness in its entries. The additive Gaussian noise vector is assumed to have uncorrelated entries. Let H be a full matrix (non-sparse) with pairwise correlations between matrix entries of the form E[H ik H ^* jl] = 1/n C ij D kl, where C, D are positive definite Hermitian matrices. Simplicities arise in the limit of large matrix sizes (the so called large-n limit) which allow us to obtain several exact expressions relating to the channel capacity. We study the probability distribution of the quantity f(H) = log (1+PH ^† SH) . S is non-negative definite and hermitian, with TrS = n and P being the signal power per input channel. Note that the expectation E[f(H)], maximised over S, gives the capacity of the above channel with an input power constraint in the case H is known at the receiver but not at the transmitter. For arbitrary C, D exact expressions are obtained for the expectation and variance of f(H) in the large matrix size limit. For C = D = I, where I is the identity matrix, expressions are in addition obtained for the full moment generating function for arbitrary (finite) matrix size in the large signal to noise limit. Finally, we obtain the channel capacity where the channel matrix is partly known and partly unknown and of the form α; I+ β H, α,β being known constants and entries of H i.i.d. Gaussian with variance 1/n. Channels of the form described above are of interest for wireless transmission with multiple antennae and receivers.     

Low frequency asymptotics for the spin-weighted spheroidal equation in the case of s=1/2

The spin-weighted spheroidal equation in the case of s=1/2 is thoroughly studied by using the perturbation method from the supersymmetric quantum mechanics.The first-five terms of the superpotential in the series of parameter β are given.The general form for the n-th term of the superpotential is also obtained,which could also be derived from the previous terms W k,k < n.From these results,it is easy to obtain the ground eigenfunction of the equation.Furthermore,the shape-invariance property in the series of parameter β is investigated and is proven to be kept.This nice property guarantees that the excited eigenfunctions in the series form can be obtained from the ground eigenfunction by using the method from the supersymmetric quantum mechanics.We show the perturbation method in supersymmetric quantum mechanics could completely solve the spin-weight spheroidal wave equations in the series form of the small parameter β.     

The Spectral Flow for Dirac Operators on Compact Planar Domains with Local Boundary Conditions

Let D_t, ${0\;\leqslant\;t\;\leqslant\;1}$ be a 1-parameter family of Dirac type operators on a two-dimensional disk with m ? 1 holes. Suppose that all operators D_t have the same symbol, and that D₁ is conjugate to D₀ by a scalar gauge transformation. Suppose that all operators D_t are considered with the same elliptic local boundary condition. Our main result is a computation of the spectral flow for such a family of operators. The answer is obtained up to multiplication by an integer constant depending only on the number of holes in the disk. This constant is calculated explicitly for the case of the annulus (m = 2).     

Concentration and temperature dependence of the refractive index of ethanol-water mixtures: Influence of intermolecular interactions

The temperature and concentration dependence of the refractive index, n _D(x, T) , in ethanol-water mixtures agrees with previous data in the ethanol-rich concentration range. The refractive index versus concentration x determined at 20 ^° C shows the expected maximum at about 41 mol% water (22 mass% water). The temperature derivative of the refractive index, dn _D /dT, shows anomalies at lower water concentrations at about 10 mol% water but no anomaly at 41 mol% water. Both anomalies are related to intermolecular interactions, the one in n_D seems to be due to molecular segregation and cluster formation while the origin of the second one in dn _D /dT is still not clear.     

Non-abelian T-duality,Ramond fields and coset geometries

We extend the recently constructed double field theory formulation of the low-energy theory of the closed bosonic string to the heterotic string. The action can be written in terms of a generalized metric that is a covariant tensor under O(D, D + n), where n denotes the number of gauge vectors, and n additional coordinates are introduced together with a covariant constraint that locally removes these new coordinates. For the abelian subsector, the action takes the same structural form as for the bosonic string, but based on the enlarged generalized metric, thereby featuring a global O(D, D + n) symmetry. After turning on non-abelian gauge couplings, this global symmetry is broken, but the action can still be written in a fully O(D, D + n) covariant fashion, in analogy to similar constructions in gauged supergravities.

     

Perturbation allowed transitions in the infrared spectrum of 14ND3: determination of the K-dependent parameters in the ground state

ABSTRACT

Accurate values of the K-dependent constants ^{( i )} C, ^{( i )} D_K and ^{( i )} H_K in the ground state of ¹⁴ND₃, with i = s, a, have been determined for the first time thanks to the detections of ‘perturbation allowed’ transitions in the ν₁, ν₂, ν₃, ν₄ and 2ν₄ infrared bands. The rotation–inversion and inversion transitions from the literature, together with 7289 ground state combination differences from the infrared vibration–rotation–inversion transitions have been simultaneously analysed. The adopted rotation–inversion Hamiltonian includes distortion constants up to the eighth power and the Δk = ±3 and Δk = ±6 interaction terms. Precise values of the diagonal constants and of the Δk = ±3 interaction coefficients have been obtained. Accurate values of the ground state term values have been calculated for both s and a levels up to J = 21.     

The N-mode squeezed state with enhanced squeezing

It is known that exp [iλ (Q1P1i/2)] is a unitary single-mode squeezing operator,where Q1,P1 are the coordinate and momentum operators,respectively.In this paper we employ Dirac’s coordinate representation to prove that the exponential operator S n ≡ exp [iλ sum((QiPi+1+Qi+1Pi))) from i=1 to n ],(Qn+1=Q1,Pn+1=P1),is an n-mode squeezing operator which enhances the standard squeezing.By virtue of the technique of integration within an ordered product of operators we derive S n ’s normally ordered expansion and obtain new n-mode squeezed vacuum states,its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.     

Superstring on pp-wave orbifold from large-N quiver gauge theory

We extend the proposal of Berenstein, Maldacena and Nastase to the Type IIB superstring propagating on a pp-wave over the R ⁴/Z _k orbifold. We show that first-quantized free string theory is described correctly by the large-N, fixed gauge coupling limit of [U(N)] ^k quiver gauge theory. We propose a precise map between gauge theory operators and string states for both untwisted and twisted sectors. We also compute leading-order perturbative correction to the anomalous dimensions of these operators. The result is in agreement with the value deduced from the string energy spectrum, thus substantiating our proposed operator-state map. Received: 14 March 2002 / Published online: 5 July 2002