Differential Geometry from Differential Equations

We first show how, from the general 3rd order ODE of the form , one can construct a natural Lorentzian conformal metric on the four-dimensional space . When the function satisfies a special differential condition the conformal metric possesses a conformal Killing field, , which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space ) or equivalently, on the space of solutions of the original differential equation. This construction is then generalized to the pair of differential equations, z _ss =S(z,z _s ,z _t ,z _st ,s,t) and z _tt =T(z,z _s ,z _t ,z _st ,s,t), with z _s and z _t the derivatives of z with respect to s and t. In this case, from S and T, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,z _s ,z _t ,z _st ,s,t). When the S and T satisfy differential conditions analogous to those of the 3^rd order ode, the 6-space then possesses a pair of conformal Killing fields, and which allows, via the mapping to the four-space of (z,z _s ,z _t ,z _st ) and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions on S and T, thus includes all (local) solutions of the Einstein equations. Received: 10 October 2000 / Accepted: 26 June 2001     

ZnO-based heterojunction light-emitting diodes on p-SiC(4H) grown by atomic layer deposition

Atomic layer deposition was used to grow n-type Al-doped ZnO (n-ZnO) and undoped ZnO (i-ZnO) layers on p-type 4H-SiC substrates, to fabricate n-ZnO/p-SiC and n-ZnO/i-ZnO/p-SiC heterojunction light-emitting diodes (LEDs). Electroluminescence (EL) from the n-ZnO/p-SiC LED originated from radiative recombination of donor–acceptor pairs in SiC due to the predominant electron injection from n-ZnO into p-SiC. On the other hand, the n-ZnO/i-ZnO/p-SiC LED exhibited dominant ultraviolet (UV) emission at 393 nm from ZnO. This difference is attributable to the insertion of the undoped i-ZnO layer between n-ZnO and p-SiC, leading to the injection of holes from p-SiC and electrons from n-ZnO into the i-ZnO layer and thus the generation of UV EL from ZnO.     

Solvable models of temporally correlated random walk on a lattice

We seek the conditional probability functionP(m,t) for the position of a particle executing a random walk on a lattice, governed by the distributionW(n, t) specifying the probability ofn jumps or steps occurring in timet. Uncorrelated diffusion occurs whenW is a Poisson distribution. The solutions corresponding to two different families of distributionsW are found and discussed. The Poissonian is a limiting case in each of these families. This permits a quantitative investigation of the effects, on the diffusion process, of varying degrees of temporal correlation in the step sequences. In the first part, the step sequences are regarded as realizations of an ongoing renewal process with a probability densityψ(t) for the time interval between successive jumps.W is constructed in terms ofψ using the continuous-time random walk approach. The theory is then specialized to the case whenψ belongs to the class of special Erlangian density functions. In the second part,W is taken to belong to the family of negative binomial distributions, ranging from the geometric (most correlated) to the Poissonian (uncorrelated). Various aspects such as the continuum limit, the master equation forP, the asymptotic behaviour ofP, etc., are discussed.     

Solutions of Doubly and Higher Order Iterated Equations

Michael Fisher once studied the solution of the equation f(f(x))=x ²–2. We offer solutions to the general equation f(f(x))=h(x) in the form f(x)=g(ag ^–1(x)) where a is in general a complex number. This leads to solving duplication formulas for g(x). For the case h(x)=x ²–2, the solution is readily found, while the h(x)=x ²+2 case is challenging. The solution to these types of equations can be related to differential equations.     

Scaling contributions to the free energy in the 1/n expansion of O(n) nonlinear sigma models in d-dimensions

Within the 1/N expansion of O(N) nonlinear σ models for d≤4 it is possible to separate consistently the spin-wave and the massive-mode contributions to the scaling part of the free energy near criticality, and to evaluate them to O(1/N). For critical dimensions d=2+2/n the Abe-Hikami anomaly is recovered, while for d=2 the removal of the spin-wave term is justified.     

Survival of grand unified monopoles

If a grand unified theory with a compact group G is spontaneously broken to H, magnetic monopoles are created. The fate of such an H-monopole under a subsequent breaking to K⊂H is shown to depend on the behaviour of its non-Abelian charge Q introduced by Goddard, Nuyts, and Olive: if Q belongs to the Lie algebra k of K, the monopole survives: if Q can be H-rotated to k, it can be converted. A necessary condition for an H-monopole to survive is that its Higgs charges satisfy a topological constraint.     

On the p,q-binomial distribution and the Ising model

We employ p, q-binomial coefficients, a generalisation of the binomial coefficients, to describe the magnetisation distributions of the Ising model. For the complete graph this distribution corresponds exactly to the limit case p = q. We apply our investigation to the simple d-dimensional lattices for d = 1, 2, 3, 4, 5 and fit p, q-binomial distributions to our data, some of which are exact but most are sampled. For d = 1 and d = 5, the magnetisation distributions are remarkably well-fitted by p,q-binomial distributions. For d = 4 we are only slightly less successful, while for d = 2, 3 we see some deviations (with exceptions!) between the p, q-binomial and the Ising distribution. However, at certain temperatures near T _c the statistical moments of the fitted distribution agree with the moments of the sampled data within the precision of sampling. We begin the paper by giving results of the behaviour of the p, q-distribution and its moment growth exponents given a certain parameterisation of p, q. Since the moment exponents are known for the Ising model (or at least approximately for d = 3) we can predict how p, q should behave and compare this to our measured p, q. The results speak in favour of the p, q-binomial distribution's correctness regarding its general behaviour in comparison to the Ising model. The full extent to which they correctly model the Ising distribution, however, is not settled.     

On the range of validity of a semiclassical relation between the critical exponents at the Anderson localization transition

Using Wegner's result relating critical exponents s and ν for conductivity and localization length, respectively, via dimensionality d and that for ν given by García-García, we derive what we term a semiclassical (sc) relation for ν in terms of s, which is independent of dimensionality. Forming the ratio s/ν versus d from the above relations, s/ν=0 at d=2 is due to a singularity in the sc relation for ν. We argue that, in reality, s/ν=0 results from s being zero at d=2. Finally we conjecture that (i) Wegner's prediction s/ν=1 when d=3 and (ii) ν tends to 1/2 at large s, are both insensitive to interactions.     

BiFeO_3结构性质与相转变的第一性原理研究

采用基于密度泛函理论的第一性原理计算,系统地研究了BiFeO3的7种不同空间群(R3c,R3m,P4mm,Cm,Pm3m,R3m和R3c)结构及其转变关系.结果表明,铁电相R3c结构是基态,不同结构之间也存在着一定的转变关系,其变化主要包括两种形式,在[111]方向上Bi3+相对FeO6八面体存在一定的位移和FeO6八面体绕[111]极化轴的反铁扭曲旋转.此外,还得出BiFeO3的薄膜结构受到衬底结构的作用会导致其从三方相(R3c)向四方相(P4mm)转变.     

Nonlinear Steepest Descent Asymptotics for Semiclassical Limit of Integrable Systems: Continuation in the Parameter Space

The initial value problem for an integrable system, such as the Nonlinear Schrödinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent ([4,5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface ${\mathcal {R} = \mathcal {R}(x,t)}The initial value problem for an integrable system, such as the Nonlinear Schr?dinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent ([4,5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface R = R(x,t){\mathcal {R} = \mathcal {R}(x,t)} in the spectral variable, where the space-time variables (x, t) play the role of external parameters. The curves in the x, t plane, separating regions of different genuses of R(x,t){\mathcal {R}(x,t)}, are called breaking curves or nonlinear caustics. The genus of R(x,t){\mathcal {R}(x,t)} is related to the number of oscillatory phases in the asymptotic solution of the integrable system at the point x, t. The evolution theorem ([10]) guarantees continuous evolution of the asymptotic solution in the space-time away from the breaking curves. In the case of the analytic scattering data f(z; x, t) (in the NLS case, f is a normalized logarithm of the reflection coefficient with time evolution included), the primary role in the breaking mechanism is played by a phase function á h(z;x,t){{\Im\,h(z;x,t)}}, which is closely related to the g function. Namely, a break can be caused ([10]) either through the change of topology of zero level curves of á h(z;x,t){\Im\,h(z;x,t)} (regular break), or through the interaction of zero level curves of á h(z;x,t){{\Im\,h(z;x,t)}} with singularities of f (singular break). Every time a breaking curve in the x, t plane is reached, one has to prove the validity of the nonlinear steepest descent asymptotics in the region across the curve.     

AlAs相变及热动力学性质的第一性原理研究

利用基于密度泛函理论的全势能线性糕模轨函法研究了闪锌矿（B3）,NiAs（B8）和岩盐（B1）结构的AlAs的相变、结构性质以及热动力学性质.对B3-B8和B3-B1结构的能量体积曲线做公切线,得到了B3→B8相变压力为5.44 GPa,并预测到B3→B1相变压力为6.46 GPa.同时计算了高压下B8相的结构性质,结果显示当V/V₀≈0.7—1.05时,c/a基本保持恒定（仅有约 0.2%的波动）;当V/V₀≈0.4—0.7,c/a随着V/V₀的减小而近似线性地增大.通过状态方程拟合,得到了AlAs的相对体积V/V₀与压强P的函数关系,B8相的状态方程与实验结果符合很好.最后利用准谐德拜模型得到了AlAs的体弹模量B随压力P的变化关系以及不同压强下热容C_V与温度T的关系. 关键词： 相变 热力学性质 第一性原理     

Jahn–Teller distortions considered as steady state bifurcations

The steady state bifurcation model can be used to determine the Jahn–Teller distortion modes of a molecule having symmetry point group G to lower symmetry structures having symmetry point groups H _j ???G. In order to find such distortion modes, the trace formula arising from bifurcation theory is applied to all subgroups H _j of G to give a sum of irreducible representations Γ(G?→?H _j ) corresponding to the possible distortion modes of G to H _j . These subgroups H _j correspond to epikernel subgroups of G for any irreducible representation of G found in Γ(G?→?H _j ). This method has been applied to Jahn–Teller distortions of square (D _{4 h} ), octahedral (O _h ), and icosahedral (I _h ) molecules leading to lower symmetry structures.     

K±p elastic scattering in the QCD inspired eikonalized model

Based on our previous study of the QCD inspired eikonalized model for describing vector meson photoproduction, pp, and $ \bar p $ \bar p p elastic scattering at high energies, we apply the mode to high energy K ^± p elastic scattering. The total cross section σ _tot(s), differential cross section dσ/dt, the ratio of the real part to imaginary part of the forward scattering amplitude ρ(s), and nuclear slope parameter function β(s) are calculated in the model. Our results show that the theoretical prediction for σ _tot(s) is in a good agreement with the experimental data within error bars of the data. For the other theoretical predictions there are no data to test the predictive power of the model. We need the corresponding experimental data to examinate the validity of our QCD inspired eikonalized model. However, our calculations clearly show that the Odderon exchange in the process makes a significant contribution to the observable of ρ(s) and β(s). Therefore, we may conclude that there is a good opportunity to find the QCD Odderon in the K ^± p elastic scattering at high energies.     

Numerical study of local and global persistence in directed percolation

The local persistence probability P _l (t) that a site never becomes active up to time t, and the global persistence probability P _g (t) that the deviation of the global density from its mean value does not change its sign up to time t are studied in a (1+1)-dimensional directed percolation process by Monte-Carlo simulations. At criticality, starting from random initial conditions, P _l (t) decays algebraically with the exponent . The value is found to be independent of the initial density and the microscopic details of the dynamics, suggesting is an universal exponent. The global persistence exponent is found to be equal or larger than . This contrasts with previously known cases where . It is shown that in the special case of directed-bond percolation, P _l (t) can be related to a certain return probability of a directed percolation process with an active source (wet wall). Received: 15 December 1997 / Revised: 6 April 1998 / Accepted: 29 May 1998     

The Modified Korteweg-de Vries Equation on the Half-Line with a Sine-Wave as Dirichlet Datum

Boundary value problems for integrable nonlinear evolution PDEs, like the modified KdV equation, formulated on the half-line can be analyzed by the so-called unified transform method. For the modified KdV equation, this method yields the solution in terms of the solution of a matrix Riemann-Hilbert problem uniquely determined in terms of the initial datum q(x,0), as well as of the boundary values {q(0, t),qx(0, t),qxx(0, t)}. For the Dirichlet problem, it is necessary to characterize the unknown boundary values qx(0, t) and qxx(0, t) in terms of the given data q(x, 0) and q(0, t). It is shown here that in the particular case of a vanishing initial datum and of a sine wave as Dirichlet datum, qx(0, t) and qxx(0, t) can be computed explicitly at least up to third order in a perturbative expansion and that at least up to this order, these functions are asymptotically periodic for large t.     

Ballistic diffusion induced by non-Gaussian noise

In this letter,we have analyzed the diffusive behavior of a Brownian particle subject to both internal Gaussian thermal and external non-Gaussian noise sources.We discuss two time correlation functions C(t) of the non-Gaussian stochastic process,and find that they depend on the parameter q,indicating the departure of the non-Gaussian noise from Gaussian behavior:for q ≤ 1,C(t) is fitted very well by the first-order exponentially decaying curve and approaches zero in the longtime limit,whereas for q 1,C(t) can be approximated by a second-order exponentially decaying function and converges to a non-zero constant.Due to the properties of C(t),the particle exhibits a normal diffusion for q ≤ 1,while for q 1 the non-Gaussian noise induces a ballistic diffusion,i.e.,the long-time mean square displacement of the free particle reads [x(t)-x(t)]2∝t2.     

Structural and electronic properties of InNxP1-x alloy in full range (0 ≤ x ≤1)

We have performed first-principles method to investigate structural and electronic properties of InN_xP_1?x ternary semiconductor alloy in full range (0 ≤ x ≤ 1) using density functional theory. We have used modified Becke–Johnson potential to obtain accurate band gap results. From the electronic band structure calculation we have found that InN_xP_1?x become metal between 47 and 80% of nitrogen concentration. Additional to our band gap calculations, we have also used the band anticrossing model. The band anticrossing model supplies a simple, analytical expression to calculate the physical properties, such as the electronic and optical properties, of III-N_xV_1?x alloys. The knowledge of the electron density of states is required to understand and clarify some properties of materials such as the band structures, bonding character and dielectric function. In order to have a deeper understanding of these properties of the studied materials, the total and partial density of states has been calculated. Finally, we have calculated the total bowing parameter b of studied alloys, together with three contributions b_VD, b_CE, and b_SR due to volume deformation, different atomic electron negativities and structural relaxation, respectively.     

Boundary Layer Stability¶in Real Vanishing Viscosity Limit

In the previous paper [20], an Evans function machinery for the study of boundary layer stability was developed. There, the analysis was restricted to strongly parabolic perturbations, that is to an approximation of the form u _t +(F(u)) _x =ν(B(u)u _x ) _x $ (ν≪1) with an “elliptic” matrix B. However, real models, like the Navier–Stokes approximation of the Euler equations for a gas flow, involve incompletely parabolic perturbations: B is not invertible in general. We first adapt the Evans function to this realistic framework, assuming that the boundary is not characteristic, neither for the hyperbolic first order system u _t +(F(u)) _x = 0, nor for the perturbed system. We then apply it to the various kinds of boundary layers for a gas flow. We exhibit some examples of unstable boundary layers for a perfect gas, when the viscosity dominates heat conductivity. Received: 27 November 2000/ Accepted: 16 March 2001     

Ageing characteristics of the metastable gamma phase in U–9 wt.% Mo alloy: experimental observations and thermodynamic validation

Ageing characteristics of the metastable bcc γ-phase in U–9 wt.% Mo alloy, a candidate for high uranium density nuclear fuel for research and test reactors, have been investigated in this study. Analyses of the aged microstructures, employing X-ray diffraction and various microscopy techniques, revealed the decomposition mechanism of the metastable γ-phase to the stable α-U and γ′ (U₂Mo) phases. A discontinuous precipitation reaction, leading to the generation of partially transformed cellular colonies with lamellae comprising of either the α-phase or the γ′-phase in γ-phase matrix, was found to be operative. The in situ transformation of γ interlamellar regions to the γ′-phase was noticed occasionally within the γ + α lamellar colonies. Thermodynamic analysis of the U–Mo system, using free energy–composition diagrams, could associate the observed attributes of the decomposition of γ-phase to the extent of Mo segregation and the chemical driving force required for the nucleation of α- and γ′-phases in the γ-matrix.