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     

Random Matrix Theory and L-Functions at s= 1/2

Recent results of Katz and Sarnak [8, 9] suggest that the low-lying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U(N), O(N) or USp(2N). We here explore the link between the value distributions of the L-functions within these families at the central point s= 1/2 and those of the characteristic polynomials Z(U,θ) of matrices U with respect to averages over SO(2N) and USp(2N) at the corresponding point θ= 0, using techniques previously developed for U(N) in [10]. For any matrix size N we find exact expressions for the moments of Z(U,0) for each ensemble, and hence calculate the asymptotic (large N) value distributions for Z(U,0) and log Z(U,0). The asymptotic results for the integer moments agree precisely with the few corresponding values known for L-functions. The value distributions suggest consequences for the non-vanishing of L-functions at the central point. Received: 1 February 2000 / Accepted: 24 March 2000     

Orientational order parameter studies on 3.Om and 3O.Om liquid crystals

Orientational order parameter S is evaluated in the nematic phase of six liquid crystal compounds, N-(p-n-propyl benzylidene)-p-n-alkoxy anilines, 3.Om and N-(p-n-propyloxy benzylidene)-p-n-alkoxy anilines, 3O.Om compounds with m = 6, 7 and 8, using different methods. The techniques employed are S from birefringence δn, Haller's approximation from (1?T/T_c) ^β, effective geometry parameter α_g and Vuks’ scaling factor S_C. The values of S obtained using the above methods are compared with one another and with the results on a number of liquid crystals; the liquid crystals favor isotropic Vuks’ method.     

Zeros of the Jones Polynomial for Multiple Crossing-Twisted Links

Let D be a general connected reduced alternating link diagram, C be the set of crossings of D and C′ be the nonempty subset of C. In this paper we first define a multiple crossing-twisted link family {D ⁿ (C′)|n=1,2,…} based on D and C′, which produces (2,2n+1)-torus knot family, the link family A _n defined in Chang and Shrock (Physica A 301:196–218, 2001) and the pretzel link family P(n,n,n) as special cases. Then by applying Beraha-Kahane-Weiss’s Theorem we prove that limits of zeros of Jones polynomials of {D ⁿ (C′)|n=1,2,…} are the unit circle |z|=1 (It is independent of the selections of D and C′) and several isolated limits, which can be determined by computing flow polynomials of subgraphs of G corresponding to D. Furthermore, we use the method of Brown and Hickman (Discrete Math. 242:17–30, 2002) to prove that, for any ε>0, all zeros of Jones polynomial of the link D ⁿ (C) lie inside the circle |z|=1+ε, provided that n is large enough. Our results extend results of F.Y. Wu, J. Wang, S.-C. Chang, R. Shrock and the present authors and refine partial result of A. Champanerkar and L. Kofman.     

Moments and Distributions of Trajectories in Slow Random Monads

Given a finite set B (basin) with n>1 elements, which we call points, and a map M:B→B, we call such pairs (B,M) monads. Here we study a class of random monads, where the values of M(⋅) are independently distributed in B as follows: for all a,b∈B the probability of M(a)=a is s and the probability of M(a)=b, where a≠b, is (1−s)/(n−1). Here s is a parameter, 0≤s≤1. We fix a point ⊙∈B and consider the sequence M ^t (⊙), t=0,1,2,… . A point is called visited if it coincides with at least one term of this sequence. A visited point is called recurrent if it appears in this sequence at least twice; if a visited point appears in this sequence only once, it is called transient. We denote by Vis _n , Rec _n and Tra _n the numbers of visited, recurrent and transient points respectively. We prove that, when n tends to infinity, Vis _n and Tra _n converge in law to geometric distributions and Rec _n converges in law to a distribution concentrated at its lowest value, which is one. Now about moments. The case s=1 is trivial, so let 0≤s<1. For any natural number k there is a number such that the k-th moments of Vis _n , Rec _n and Tra _n do not exceed this number for all n. About Vis _n : for any natural k the k-th moment of Vis _n is an increasing function of n. So it has a limit when n→∞ and for all n it is less than this limit. About Rec _n : for any k the k-th moment of Rec _n tends to one when n tends to infinity. About Tra _n : for any k the k-th moment of Tra _n has a limit when n tends to infinity.     

Anisotropy of rotation of nitroxide probes in nematogenic liquid crystals

An electron paramagnetic resonance (EPR) line shape simulation for nitroxide spin probes in the motional narrowing region was carried out assuming axially symmetricg andA tensors and with different anisotropies of rotationN (=R _∥/R _⊥) whereR _∥ andR _⊥ are, respectively, elements of the diffusion tensor along and perpendicular to its principal axisz′. In addition, it was assumed that the principal axes of the diffusion tensor coincide with the molecular axes. Each of three casesz′=x,z′=y andz′=z, which result from cyclic permutations of the molecular axesx, y andz with thez′,y′ andx′ axes of the diffusion tensor, yields its typical EPR spectrum characterized by the relative intensities of the low-, center- and high-field lines. The parameter δ defined by and calculable from the intensities of the three lines was found to vary linearly withN for thez′=x andz′=y cases and, as anticipated, to be practically constant at a value of 1 for thez′=z case. This suggested a method for estimatingN for a probe from its EPR spectrum. Experimental spectra over a narrow temperature range (1°C) in the vicinity of the nematic-to-isotropic transition (about 34.6°C) ofN-(4-n-butylbenzilidene)-4-amino-2,2,6,6-tetramethylpiperidine-1-oxide at a mole fraction of 1·10⁻³ in 4-n-pentyl-4′-cyanobiphenyl showed a pattern of peak heights characteristic of thez′=x case with δ values that gave, neglecting effects of the mean field, higher and lowerN values in the nematic and isotropic regions, respectively. Analysis of other similar systems in the literature gave similar results.     

An exact formulation of the Blume-Emery-Griffiths model on a two-fold Cayley tree model

A two-fold Cayley tree graph with fully q-coordinated sites is constructed and the spin-1 Ising Blume-Emery-Griffiths model on the constructed graph is solved exactly using the exact recursion equations for the coordination number q = 3. The exact phase diagrams in (kT/J, K/J ) and (kT/J, D/J) planes are obtained for various values of constants D/J and K/J, respectively, and the tricritical behavior is found. It is observed that when the negative biquadratic exchange (K) and the positive crystal-field (D) interactions are large enough, the tricritical point disappears in the (kT/J, K/J) plane. On the other hand, the system always exhibits a tricritical behavior in the phase diagram of (kT/J, D/J) plane. Received 8 June 2001 and Received in final form 28 September 2001     

Optical-Oscillator strengths for Ca X,Sc XI,Ti XII,Cu XIX,Zn XX,Br XXV and Kr XXVI

Summary We have applied the Tiwary approach to calculate the opticaloscillator strengths, of both the length and velocity forms, for the inner-shell excitation1s ² 2s ² 2p ⁶ 3s ² S ^e →1s ² 2s ² 2p ⁵ 3s ^{2 2} P ⁰ transition in Ca X, Sc XI, Ti XII, Cu XIX, Zn XX, Br XXV and Kr XXVI ions of the sodium isoelectronic sequence employing Hartree-Fock (HF) and configuration interaction (CI) wave functions of both the initial and final states involved in the transition. Our present calculated results demonstrate that the Tiwary approach is capable of yielding encouraging results. A part of this work was done while the author was Research Director and Professor, CNRS Laboratory, University of Paris-Sud and Observatoire de Paris, Meudon, Paris, France.     

Generalized holographic and Ricci dark energy models

In this paper, we consider generalized holographic and Ricci dark energy models where the energy densities are given as ρ _R =3c ² M _pl² Rf(H ²/R) and ρ _h =3c ² M _pl² H ² g(R/H ²), respectively; here f(x), g(y) are positive defined functions of the dimensionless variables H ²/R or R/H ². It is interesting that holographic and Ricci dark energy densities are recovered or recovered interchangeably when the function f(x)=g(y)≡1 or f(x)=Id and g(y)=Id are taken, respectively (for example f(x),g(x)=1−ε(1−x), ε=0or1, respectively). Also, when f(x)≡xg(1/x) is taken, the Ricci and holographic dark energy models are equivalent to a generalized one. When the simple forms f(x)=1−ε(1−x) and g(y)=1−η(1−y) are taken as examples, by using current cosmic observational data, generalized dark energy models are considered. As expected, in these cases, the results show that they are equivalent (ε=1−η=1.312), and Ricci-like dark energy is more favored relative to the holographic one where the Hubble horizon was taken as an IR cut-off. And the suggested combination of holographic and Ricci dark energy components would be 1.312R−0.312H ², which is 2.312H²+1.312[(H)\dot]2.312H^{2}+1.312\dot{H} in terms of H ² and [(H)\dot]\dot{H} .     

Effects of impurity on the entanglement of the three-qubit Heisenberg XXX spin chain

We investigate the entanglement of the three-qubit Heisenberg XXX chain in the presence of impurity and obtain the analytical expressions of the concurrence C. It is found that for impurity entanglement, C appears only when J ₁ > J for J > 0, and J ₁ > 0 for J < 0, and in these two regions C increases with the increase of J ₁, so is the critical temperature T _c. When J ₁ ≫ | J |, C reaches its maximum value 0.5 and T _c reaches the asymptotic value T _c = 3.41448J ₁. For entanglement between the normal lattices, C appears only when J > 0 and −2J < J ₁ < J, and initially increases with the increase of J ₁ and arrives at the maximum value C _max = (e^4J/T −3)/(e^4J/T + 3) before it decays to zero gradually, so is the critical temperature T _c with, however, the maximum value T _cmax = 4J/In3. Supported by the Natural Science Research Project of Shaanxi Province (Grant No. 2004A15)     

Coherent inelastic processes on nuclei at ultrarelativistic energies

The coherent inelastic processes of the type a → b, which may take place in the interaction of hadrons and γ quanta with nuclei at very high energies (the nucleus remains the same), are theoretically investigated. For taking into account the influence of the nucleus matter, the optical model, based on the conception of the refraction index, is used. Analytical formulas for the effective cross section σ _coh (a → b) are obtained, taking into account that, at ultrarelativistic energies, the main contribution into σ _coh (a → b) is provided by very small transferred momenta in the vicinity of the minimal longitudinal momentum transferred to the nucleus. It is shown that the cross section σ _coh (a → b) may be expressed through the “forward” amplitudes of inelastic scattering f _a+N+b+N (0) and elastic scattering f _a+N+a+N(0), f _b+N+b+N(0) on a separate nucleon, and it depends on the ratios L _a /R and L _b /R (L _a and L _b are the mean lengths of the free path in the nucleus matter for the particles a and b, respectively, and R is the nucleus radius). In particular, when L _a /R ≫ 1, but L _b /R ≪ 1 (or L _a /R ≪ 1, but L _b /R ≫ 1), σ _coh (a → b) is equal to the ratio of the “forward” cross sections of inelastic scattering a + N → b + N and elastic scattering of the particle b (or a) on a nucleon, multiplied by the cross section of scattering on the “black” nucleus πR ². When both conditions L _a /R ≫ 1 and L _b /R ≫ 1 are satisfied, σ _coh (a → b) is proportional to the factor R ⁴/k ², where k is the initial energy of particle a in the laboratory frame. The text was submitted by the authors in English.     

Self-diffusion in liquid metals

A survey is made of all published data on self-diffusion in liquid metals and an examination is carried out on the temperature-dependence of D for studies covering a wide range of temperatures. Log D versus 1/T and D versus T represent the best data equally well, while D versus T ^1/2 and D versus T ² do not. The proper functional form is taken to be D=CT experimentally, and a simple free diffusion theory is developed from the standpoint of vibrational atom motion. An equation having no adjustable parameters results (D=k ² ? _D T/hk), in which κ is the vibrational force constant. Values calculated for D at the melting point are in good agreement with experimental values for potassium, sodium, copper and silver, but are below the measured values for lead, indium, zinc and tin.     

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     

Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings

We present exact results on the partition function of the q-state Potts model on various families of graphs G in a generalized external magnetic field that favors or disfavors spin values in a subset I _s ={1,…,s} of the total set of possible spin values, Z(G,q,s,v,w), where v and w are temperature- and field-dependent Boltzmann variables. We remark on differences in thermodynamic behavior between our model with a generalized external magnetic field and the Potts model with a conventional magnetic field that favors or disfavors a single spin value. Exact results are also given for the interesting special case of the zero-temperature Potts antiferromagnet, corresponding to a set-weighted chromatic polynomial Ph(G,q,s,w) that counts the number of colorings of the vertices of G subject to the condition that colors of adjacent vertices are different, with a weighting w that favors or disfavors colors in the interval I _s . We derive powerful new upper and lower bounds on Z(G,q,s,v,w) for the ferromagnetic case in terms of zero-field Potts partition functions with certain transformed arguments. We also prove general inequalities for Z(G,q,s,v,w) on different families of tree graphs. As part of our analysis, we elucidate how the field-dependent Potts partition function and weighted-set chromatic polynomial distinguish, respectively, between Tutte-equivalent and chromatically equivalent pairs of graphs.     

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.     

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.     

Weak and Partial Symmetries of Nonlinear PDE in Two Independent Variables

Abstract

Nonclassical infinitesimal weak symmetries introduced by Olver and Rosenau and partial symmetries introduced by the author are analyzed. For a family of nonlinear heat equations of the form u _t = (k(u) u _x)_x + q(u), pairs of functions (k(u), q(u)) are pointed out such that the corresponding equations admit nontrivial two-dimensional modules of partial symmetries. These modules yield explicit solutions that look like u(t, x) = F (θ(t) x + φ(t)) or u(t, x) = G(f(x) + g(t)).     

A simple rule of the sequential bond enthalpies in clusters and the possibility of liquid-phase carbon

This letter reports a simple rule on the sequential bond enthalpies D(B(M)_x − M) in clusters B(M)_x + 1, where B is a bonding particle (neutrals, cations, anions, complexes, biochemical particles), M is a neutral material molecule (inorganic and organic) or atom (metal, non-metal and rare gas), x describes the cluster size, and x = 0, 1, 2, 3, …, etc. Thousands of experimental results for 60 kinds of materials M and 310 kinds of clusters B(M)_x + 1 strongly support the simple rule: the sequential bond enthalpies in clusters regularly drop to an intrinsic value, the vaporization enthalpies of the liquid-phase materials M, i.e., D(B(M)_x − M, large x) ~ Δ_vapH(M). Using this simple rule, we have discussed the possibility of liquid-phase carbon.     

Study on Fractal Character of High Impact Polystyrene with Poly(cis‐butadiene) Rubber Partly Miscible Blends by Fractal Measure Relations and Box‐Counting Methods

Abstract

Phase formation and evolution of high‐impact polystyrene/poly(cis‐butadiene) rubber (HIPS/PcBR) blends during melting and mixing processes were investigated by scanning electron micrographs analysis. The diameter, d _p , was used to describe the evolution of the phase morphology of HIPS/PcBR blends during mixing. Scale functions, S _N (r) and S _M (r), were defined to confirm the self‐similarity of the phase morphology. The plots of S _N (r)/S _N (r) _m [the maximum of S _N (r)] vs. r/r _m (the maximum of r) and S _M (r)/S _M (r) _m [the maximum of S _M (r)] vs. r/r _m showed the phase morphology had self‐similarity. Furthermore, the fractal dimension, D, of different HIPS/PcBR blends was calculated by two different methods (fractal measure relations and box‐counting methods). The results showed that the fractal dimension was an effective parameter for study of the phase morphology of polymer blends.