Conditional Entropy and the Rokhlin Metric on an Orthomodular Lattice with Bayessian State

The present paper deals with the study of conditional entropy and its properties in a quantum space (L,s), where L is an orthomodular lattice and s is a Bayessian state on L. First, we obtained a pseudo-metric on the family of all partitions of the couple (B,s), where B is a Boolean algebra and s is a state on B. This pseudo-metric turns out to be a metric (called the Rokhlin metric) by using a new notion of s-refinement and by identifying those partitions of (B,s) which are s-equivalent. The present theory has then been extended to the quantum space (L,s), where L is an orthomodular lattice and s is a Bayessian state on L. Applying the theory of commutators and Bell inequalities, it is shown that the couple (L,s) can be equivalently replaced by a couple (B,s ₀), where B is a Boolean algebra and s ₀ is a state on B.     

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.     

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.     

Nonlinear optical effects for plasma diagnostics

Experimental investigations show that the 1/f noise intensityC and the contact resistanceR can be used to analyse contacts. The simply prepared contacts are fritted by discharging a capacitor, resulting in a multi-spot contact. A model relatesC andR to a number of contact spotsk with radiusa. More impulse-frittings at increasing energies decreaseC andR, thus enhancing the values ofk anda. From experimentalC vsR plots two regions are determined for GaAs: A fritting (a=constant) and A+B fritting (a∝k). Calculated values ofk are in good agreement with the number of peaks or pits formed by etching the semiconductor surface. From experimentalC vsW orR vsW curves, withW the cumulative impulse-fritting energy, the conclusion can be made thatka ³ is proportional toW.     

Minimum Polynomial for Single-Mode Parabose Parasupercharge and Hamiltonian

We calculate the minimum polynomial φ(x,y) of parasupercharge Q and Hamiltonian H for single-mode parabose parasupersymmetry (P-PSUSY). Suppose that φ(x,y) satisfies the homogeneity ∀ λ∈ℝ,φ(λ x,λ ² y)=λ ⁿ φ(x,y), then the parafermionic order p _f is restricted to either 1, 2, or 4. Under the P-PSUSY, the homogeneity of the φ(x,y) is equivalent to the parasuperconformality of Q and H. The physical meaning of the parasuperconformality is discussed, in connection with the spin of the elementary particle.     

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     

Computational study of collisional energy transfer and quasi-continuous population inversions in laser-pumped Ca vapor

A computational study of population inversion between several pairs of excited states viz 3d4p ³ F-4s3d ³ D, 4s5s ³ S-4s4p ³ P and 4s3d ³ D-4s4p ³ P in Ca vapor pumped on the 4s ^{2 1} S ₀-4s4p ³ P ₁ transition is presented. The main aim is to investigate the influence of various atomic processes in creating and sustaining the population inversion for long times after the excitation pulse. The delicate interplay between superelastic energy transfer to free electrons, energy pooling collisions and cascaded recombination is particulary examined. It is noted that quasi-continuous population inversion can be readily excited on the 4s3d ³ D-4s4p ³ P transitions; and under some conditions, also on the 4s5s ³ S-4s4p ³ P transitions. Furthermore, inversion on the 3d4p ³ F-4s3d ³ D transitions can also be excited for a considerable length of time. The results may be useful in designing and developing quasi-cw metal vapor lasers.     

Dual properties of orthogonal polynomials of discrete variables associated with the quantum algebra <Emphasis Type="Italic">U</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(<Emphasis Type="Italic">su</Emphasis>(2))

We show that for every set of discrete polynomials y _n (x(s)) on the lattice x(s), defined on a finite interval (a, b), it is possible to construct two sets of dual polynomials z _k (ξ(t)) of degrees k = s-a and k = b-s-1. Here we do this for the classical and alternative Hahn and Racah polynomials as well as for their q-analogs. Also we establish the connection between classical and alternative families. This allows us to obtain new expressions for the Clerbsch-Gordan and Racah coefficients of the quantum algebra U _q (su(2)) in terms of various Hahn and Racah q-polynomials. Dedicated to the memory of our teacher and friend Arnold F. Nikiforov (18.11.1930–27.12.2005).     

On the connectivity index for lattices of nonintegral dimensionality

We define the connectivity indexc for an infinite graph by the requirement that to disconnect a subset of at leastV points from the rest of the graph requires the deletion of a minimum ofS(V) bonds whereS(V) ∼V ^(c−1)/c for largeV. For ad-dimensional hypercubical lattice withd integral,c=d. We construct explicit examples of lattices with nonintegral connectivity indexc, 1<c<∞. It is argued that the connectivity index is an important parameter determining the critical behaviour of Hamiltonians on these lattices.     

R-matrix formulation of the quantum inhomogeneous groups ISO q,r (N) and ISp q,r (N)

The quantum commutationsRTT=TTR and the orthogonal (symplectic) conditions for the inhomogeneous multiparametricq-groups of theB _n ,C _n ,D _n type are found in terms of theR-matrix ofB _n+1 ,C _n+1 ,D _n+1 .A consistent Hopf structure on these inhomogeneousq-groups is constructed by means of a projection fromB _n+1 ,C _n+1 ,D _n+1 .Real forms are discussed; in particular, we obtain theq-groups ISO _q,r (n+1,n–1), including the quantum Poincaré group. The inhomogeneusq-groups do not contain dilatations when the parameters satisfy certain conditions. For example, we find a dilatation-freeq-Poincaré group depending on one real parameterq.     

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.     

On the Residual Entropy of the Blume-Emery-Griffiths Model

By means of a transfer matrix method, we show that the residual entropy S of the two dimensional square lattice Blume-Emery-Griffiths model on the antiquadrupolar-disordered and ferromagnetic-antiquadrupolar-disordered phase boundaries satisfies the inequalities (ln λ _1,n )/(n+1)≤S≤(ln λ _1,n )/n, where λ _1,n is the largest eigenvalue of a transfer matrix F _n on a strip of width n. These bounds imply the existence of a O(1/n) correction in the approximation of S by (ln λ _1,n )/n. Using these bounds, we calculate numerically the value of S, with precise estimates on the errors.     

一种新型的两态叠加多模Schrödinger猫态光场的等阶N次方Y压缩

本文根据量子力学的线性叠加原理,构造了由多模(即q模)相干态的相反态|{-Z_j}〉q及多模虚相干态|{Z_j}〉q这两者的线性叠加所组成的一种新型的多模Schrdinger猫态光场|Ψ⁽²⁾〉q,利用新近建立的多模压缩态理论,研究了态|Ψ⁽²⁾〉q的N次方Y压缩效应。结果发现:①当压缩阶数N=2p且p=2m(m=1,2,3,…,…)时,态|Ψ⁽²⁾〉q总是恒处于N-Y最小测不准态;②当压缩阶数N=2p且p=2m+1(m=0,1,2,3,…,…)时,如果各模的初始相位φ_j,态间的初始相位差θ_pq^(I)-θ_nq^(R)以及各单模相干态光场的平均光子数之总和 等满足一定的量子化条件,则态|Ψ⁽²⁾〉q可呈现出周期性变化的、任意阶的N次方Y压缩效应;③当压缩阶数N=2p′+1时,无论p′=2m(m=0,1,2,…,…)还是p′=2m+1(m=0,1,2,3,…,…),只要各模的初始相位φ_j满足一定的量子条件,则当两态叠加几率幅满足r_pq^(I)=r_nq^(R)时,态|Ψ⁽²⁾〉q就恒处于N-Y测不准态,始终不呈现N-Y最小测不准态和N次方Y压缩;而当r_pq^(I)≠r_nq^(R)时,态|Ψ⁽²⁾〉q始终不呈现N-Y测不准态、N-Y最小测不准态和N次方Y压缩效应。     

Spin-wave theory for finite classical magnets and superparamagnetic relation

Analytical calculations based on finite-size spin-wave theory and Monte Carlo (MC) simulations are performed to investigate the validity of the well-known relation m(H, T) = M(H, T)B _D[M(H, T) H/T] between the induced magnetization m of the magnetic particle and its intrinsic magnetization M for the Ising and isotropic classical models (B _D(x) is the Langevin function, D is the number of spin components, is the number of atoms in the particle). It follows from general arguments and from our analytical results for the Heisenberg model at T≪T _c that this relation is not exact for any finite D and nonzero temperature. Nevertheless, corrections to this formula remain very small practically in the whole range T < T _c if ≫ 1, as confirmed by our Monte Carlo calculations. At T T _c/4 there is a good agreement between the MC and finite-size spin-wave calculations for the field dependence of m and M for the Heisenberg model with free boundary conditions. Received 1st December 2000     

C M ? C N D -computable state

The notions of D-computable state and D-concurrence are generalized to the C ^M ⊗ C ^N system. A class of D-computable state on C ^M ⊗ C ^N is given and the calculating method of the lower bound of D-concurrence is provided. The obvious expression of the lower bound of D-concurrence for the state mixed by two D-pure states is derived. Supported by the National Natural Science Foundation of China (Grant No. 60433050) and the Key Project of the Science Foundation of Xuzhou Normal University (Grant No. 06XLA05)     

New explicit nonperiodic solutions of the homogeneous wave equation

We exhibit a newansatz for the solution of the homogeneous three-dimensional time-dependent wave equation in spherical coordinates of the form Φ(r,t)=Y(θ, φ)(I(r)+G(g)), whereg ≡ct/r. FunctionG(g) has explicit solution in terms of three independent nonperiodic functionss _ℓ,t _ℓ,u _ℓ (s _ℓ andt _ℓ are related to the associated Legendre functions of the first and second kinds).G(g) is nonperiodic and may be cast as a superposition of incoming and outgoing waves. To obtainG(g), we solved a nonhomogeneous associated Legendre equation (this solution, to our knowledge, is also new).G(g) may prove useful in many microscopic and macroscopic problems, representable by homogeneous wave equations.     

Radiotracer Studies on the Extraction of Metal Ions Part 10: Extraction of Calcium,Iron, Nickel,Copper, and Zinc with Some Aromatic Schiff Bases)

Using the radioactive indicators Ca ⁴⁵, Fe ⁵⁹, Ni ⁶³, Cu ⁶⁴, and Zn ⁶⁵ the yield of extraction (%E) of Ca(II), Fe(III), Ni(II), Cu(II), and Zn(II) with quadridentate Schiff bases, which are products of condensation of salicylaldehyde and its derivatives with ethylenediamine, 1,3-propylenediamine and o-phenylenediamine has been investigated as a function of the pH of the electrolyte.     

Clusters of cycles

A cluster of cycles (or (r,q)-polycycle) is a simple planar 2-connected finite or countable graph G of girth r and maximal vertex-degree q, which admits an (r,q)-polycyclic realization P(G) on the plane. An (r,q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q; (ii) all interior faces (denote their number by p_r) are combinatorial r-gons; (iii) all vertices, edges and interior faces form a cell-complex.An example of (r,q)-polycycle is the skeleton of (r^q), i.e. of the q-valent partition of the sphere, Euclidean plane or hyperbolic plane by regular r-gons. Call spheric pairs (r,q)=(3,3),(4,3),(3,4),(5,3),(3,5). Only for those five pairs, P((r^q)) is (r^q) without exterior face; otherwise, P((r^q))=(r^q).Here we give a compact survey of results on (r,q)-polycycles. We start with the following general results for any (r,q)-polycycle G: (i) P(G) is unique, except of (easy) case when G is the skeleton of one of the five Platonic polyhedra; (ii) P(G) admits a cell-homomorphism f into (r^q); (iii) a polynomial criterion to decide if given finite graph is a polycycle, is presented.Call a polycycle proper if it is a partial subgraph of (r^q) and a helicene, otherwise. In [ARS Comb. A 29 (1990) 5], all proper spheric polycycles are given. An (r,q)-helicene exists if and only if p_r>(q−2)(r−1) and (r,q)≠(3,3). We list the (4,3)-, (3,4)-helicenes and the number of (5,3)-, (3,5)-helicenes for first interesting p_r. Any outerplanar (r,q)-polycycle G is a proper (r,2q−2)-polycycle and its projection f(P(G)) into (r^2q−2) is convex. Any outerplanar (3,q)-polycycle G is a proper (3,q+2)-polycycle.The symmetry group Aut(G) (equal to Aut(P(G)), except of Platonic case) of an (r,q)-polycycle G is a subgroup of Aut((r^q)) if it is proper and an extension of Aut(f(P(G))), otherwise. Aut(G) consists only of rotations and mirrors if G is finite, so its order divides one of the numbers 2r, 4 or 2q. Almost all polycycles G have trivial AutG.Call a polycycle G isotoxal (or isogonal, or isohedral) if AutG is transitive on edges (or vertices, or interior faces); use notation IT (or IG, or IH), for short. Only r-gons and non-spheric (r^q) are isotoxal. Let T^*(l,m,n) denote Coxeter’s triangle group of a triangle on S², E² or H² with angles π/l, π/m, π/n and let T(l,m,n) denote its subgroup of index 2, excluding motions of 2nd kind. We list all IG- or IH-polycycles for spheric (r,q) and construct many examples of IH-polycycles for general case (with AutG being above two groups for some parameters, including strip and modular groups). Any IG-, but not IT-polycycle is infinite, outerplanar and with same vertex-degree, we present two IG-, but not IH-polycycles with (r,q)=(3,5),(4,4) and AutG=T(2,3,∞)PSL(2,Z), T^*(2,4,∞). Any IH-polycycle has the same number of boundary edges for each its r-gon. For any r≥5, there exists a continuum of quasi-IH-polycycles, i.e. not isohedral, but all r-gons have the same 1-corona.On two notions of extremal polycycles:

1. We found for the spheric (r,q) the maximal number n_int of interior points for an (r,q)-polycycle with given p_r; in general case, (p_r/q)≤n_int<(rp_r/q) if any r-gon contains an interior point.

2. All non-extendible (r,q)-polycycles (i.e. not a proper subgraph of another (r,q)-polycycle) are (r^q), four special ones, (possibly, but we conjecture their non-existence) some other finite (3,5)-polycycles, and, for any (r,q)≠(3,3),(3,4),(4,3), a continuum of infinite ones.

On isometric embedding of polycycles into hypercubes Q_m, half-hypercubes and, if infinite, into cubic lattices Z_m, : for (r,q)≠(5,3),(3,5), there are exactly three non-embeddable polycycles (including (4³)−e, (3⁴)−e); all non-embeddable (5,3)-polycycles are characterized by two forbidden sub-polycycles with p₅=6.     

BD-FRW models in the framework of Israel-Stewart-Hiscock theory

BD-FRW universe filled with imperfect fluid having bulk viscosity is investigated under the framework of Israel-Stewart-Hiscock causal theory. The field equations have been solved by using the relationφ=KR ^α whereK andα are constants, between the Brans-Dicke scalar fieldϕ and the scale factorR. This relation, in fact, leads to a constant deceleration parameterq. It is shown that the constancy of the deceleration parameter permits only two possibilities i.e. eitherH=constant withm=1 orm=(1+b −α)/(2(1+b) −α), irrespective of the value ofɛ.