The transition from simple to fractal basin boundary structure displayed by unimodal maps

One-dimensional unimodal maps of the form X_{n + 1} = F (r, x_n) exhibit periodic orbits, or p-cycles, confined within specific values of the control parameter r. For any p-cycle almost all initial conditions x_O will iterate to one of attracting fixed points of the map under iteration of the pth iterate of the map F^(p). The basin boundaries which confine points attracted to the fixed points are simple or fractal depending upon whether the cycle is a member of the period doubling cascade or is a cycle in the aperiodic region beyond the period doubling accumulation point. The topology of these basin structures and the transition from simple to fractal behaviour is analyzed as a function of control parameter in terms of a scenario which parallels that of the progression from period doubling to the aperiodic regime.     

The one-site distribution of Gibbs states on Bethe lattice are probability vectors of period⩽2 for a nonlinear transformation

We prove that the one-site distribution of Gibbs states (for any finite spin setS) on the Bethe lattice is given by the points satisfying the equation π=T ²π, whereT=h·A·?, with?(x)=x ^(q?1/q,h(x)=(x∥x∥ _q ) ^q ,A=(a(r, s)∶r, s∈S), and $$a(r,s) = \exp (K[r,s] + (1/q)[N,r + s])$$ We also show that forA a symmetric, irreducible operator the nonlinear evolution on probability vectorsx(n+1)=Ax(n) ^p ∥Ax(n) ^p ∥₁ withp>0 has limit pointsξ of period?2. We show thatA positive definite implies limit points are fixed points that satisfy the equationAξ ^p=λξ. The main tool is the construction of a Liapunov functional by means of convex analysis techniques.     

Glauber Dynamics for the Mean-Field Potts Model

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q??3 states and show that it undergoes a critical slowdown at an inverse-temperature ?? _s (q) strictly lower than the critical ?? _c (q) for uniqueness of the thermodynamic limit. The dynamical critical ?? _s (q) is the spinodal point marking the onset of metastability. We prove that when ??<?? _s (q) the mixing time is asymptotically C(??,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At ??=?? _s (q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n ^4/3. For ??>?? _s (q) the mixing time is exponentially large in n. Furthermore, as ?????? _s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n ^?2/3) around ?? _s . These results form the first complete analysis of mixing around the critical dynamical temperature??including the critical power law??for a model with a first order phase transition.     

Lyapunov number for a noisy 2 n cycle

A stochastic one-dimensional map which produces a sequence of period doubling bifurcations is theoretically studied. We obtain analytic expressions, to a second-order approximation, of the local distribution function of fluctuating orbital points and the Lyapunov number for a noisy 2 ⁿ cycle. The expressions satisfy scaling laws and well agree with the results of numerical experiments when the external noise is weak. A scaling factor for the noise level is formulated in terms of the derivatives of a deterministic map. From it, the scaling factor is refined to be 6.6190 .... The Lyapunov number shows that, when the external noise is weaker than some extent, the noisy orbit is more stable rather than the deterministic one.     

Level structure of 68 149 Er81 and high-spin isomerism in proton-richN=81, 82, 83 nuclei

The level structure of theN=81 nucleus¹⁴⁹Er has been studied by γ-ray spectroscopy following the reaction⁹²Mo+255 MeV⁶⁰Ni. Yrast levels in¹⁴⁹Er are established up to ~3.3 MeV, including 0.61 and 4.8 μs isomeric states. Most of the observed levels are interpreted as seniority-three states arising from the coupling of s_1/2, d_3/2 and h_11/2 neutron holes with πh ₁₁ ^2/n . Isomers identified in the reaction⁹⁶Ru+255 MeV⁵⁸Ni are tentatively assigned to¹⁵¹Yb. TheB (E2) values of high-spin isomers inZ=66?70,N=81?83 nuclei are surveyed.     

Mass distribution in n—polymer stochastic aggregation and large—mass behaviour

The exact solutions of the rate equations of the n-polymer stochastic aggregation involving two types of clusters, active and passive for the kernel \dprⁿ_k=1s_(ik)(s_(ik)=i_k) and \dsumⁿ_k=1s_(ik)(s_(ik)=i_k), are obtained. The large-mass behaviours of the final mass distribution of the active and passive clusters have scaling-like forms, although the models exhibit different properties. Respectively, they have different decay exponents γ=\dfrac{2n+1}{2(n-1)} and γ=q+\dfrac{2n+1}{2(n-1)} for \dprⁿ_{k=1}s(ik)(s(ik)=ik) and γ=\dfrac 3{2(n-1)} and γ=q+\dfrac 3{2(n-1)} for \dsumⁿk=1}s(ik)(s(ik)=ik), which include exponents of two-polymer stochastic aggregation. We also find that gelation is suppressed for kernel \dprⁿk=1s(ik)(s(ik)=ik) which is different from the deterministic aggregation.}     

Role of metastable atoms and molecules of helium in excitation transfer to hydrogen atoms

The afterglow of a discharge in helium with a small admixture of hydrogen is studied spectroscopically (p=40 Torr, [e]≤10¹¹ cm^?3). The time-resolved measurements of intensities of the first four lines of the Balmer series are performed. The concentrations of metastable helium atoms and molecules are evaluated from the relative intensity of the absorption lines. The ratios of excitation transfer rates from atoms He(2 ³ S ₁) k ₁(n) and molecules of helium He₂(a 2sσ ³Σ _u ⁺ ) k ₂(n) to atomic hydrogen H*(n) are measured to be k ₁(n=3)/k ₂(n=3)=0.04±0.02 and k ₁(n=4)/k ₂(n=4)=0.01±0.02. The ratios of excitation rate constants k ₂(n) corresponding to different states H(n) are measured to be k ₁(n=4)/k ₂(n=3)=0.023±0.01; k ₁(n=5)/k ₂(n=3)≤0.013; and k ₁(n=6)/k ₂(n=3)≤0.007.     

Moments of the Percus-Yevick hard-sphere correlation function

A simple recursive relation is derived for the momentsM _n ,n=1, 2,..., of the Percus-Yevick correlation functionh(r) for identical hard spheres. TheM _n are rational functions of the volume fractionw occupied by the spheres; the first ten are given explicitly, and a single-term asymptotic form is obtained to suffice for the rest. Applications of theM _n(w) include testing different approximations forh by numerical integration ofh(r) r ⁿ . We compare exact moments with shell approximationsM _n [h ^s ] corresponding to integration fromr=0 tos+1 fors=3–8, and with hybrid approximationsM _n [h ^s +h ^a ] which supplement the shell approximations with integrals of an asymptotic tail froms+1 to . For a givens, the hybrid approximation is better forw increasing than the shell approximation, andM _n [h ³+h ^a ] is even better thanM _n [h ⁸]     

一维单峰映象的拓扑熵

在对Logistic映象数值计算的基础上,我们分析了一维单峰映象的逆轨道结构,证明了不同参数处逆轨道总数N(n)随求逆次数n而变化的递推公式。借此解析地求得了在倍周期区中h(f)≡0;在U序列RLR²¹的m=3+2l周期点上h(f)=logα_mp,其中α_mp为方程α^m-2α^m-2-1=0的最大实根;在2^j-1常和2j带交界处h_j(f)=(1/2)^jlog2,由此可得聚点μ_∞处拓扑熵的标度指数t=0.449806…。在此基础上,我们还求得了混沌区的周期窗口,U序列RL^aR^b所对应的各点处的拓扑熵,以及h_R*Q(f)=(1/2)h_Q(f)的关系。证明是在M.S.S.规则和“*”乘法则的基础上进行的。所以本文的结果对一维单峰映象是普适的。 关键词：     

Inelastic collisions between excitons in semiconductors—I: Collisions with excited final states

The exciton-exciton (Exc-Exc) inelastic collision cross sections for the different semiconductors CdS, ZnO, CdSe, Si and Cu₂O have been calculated in the Born approximation. The simple transitions Exc (1s/B) + Exc (1s/A)→ Exc (1s/B) + Exc (nl /A) where nl = 2s, 2p, 3s, 3p and 3d have been The calculations are carried out as functions of the different available values of σ = m¹_e/m¹_h where m¹ and m¹_h are, respectively, the electron and the hole effective masses for the corresponding semiconductors.     

Dissociative excitation of even states of tin atoms in e-SnCl2 collisions

Dissociative excitation of even levels of tin atoms in collisions of slow electrons with tin dichloride molecules is studied experimentally. The levels studied belong to the configurations 5s ²5pnp ^{1, 3} L (n=6, 7; L=S, P, D) and 5s ²5p 4f. Absolute values of cross sections for dissociative excitation at the electron energy of 100 eV are 1.05–2.56 times greater than cross sections for excitation in electron-atom collisions at the energy of 50 eV.     

Galvanomagnetic properties of air-stable and highly conductive potassium-intercalated graphite sheet

Magnetoresistance and Hall coefficient of air-stable potassium-intercalated graphite sheets (hereafter abbreviated as K-PGS) were determined at room temperature. The magnitude of the magnetoresistance and the absolute value of Hall coefficient of K-PGS decreased with increasing potassium content of K-PGS, n_K/n_C. Two-carrier model was used for calculating carrier density and mobility. The electron density increased with increasing n_K/n_C: 3.07×10²⁰ cm⁻³ (n_K/n_C=0.005), 5.67×10²⁰ cm⁻³ (n_K/n_C=0.008) and 6.40×10²⁰ cm⁻³ (n_K/n_C=0.011). The value of the electron density of K-PGS with n_K/n_C=0.011 (nominal composition KC₉₁) was about 80% of the reported value, 7.8×10²⁰ cm⁻³, for KC₄₈ (n_K/n_C=0.021) prepared from HOPG (highly oriented pyrolytic graphite). The mobility decreased with increasing n_K/n_C: 2.11×10³ cm² V⁻¹ s⁻¹ (n_K/n_C=0.005), 1.42×10³ cm² V⁻¹ s⁻¹ (n_K/n_C=0.008) and 1.34×10³ cm² V⁻¹ s⁻¹ (n_K/n_C=0.011). The value of the mobility of K-PGS with n_K/n_C=0.011 was about 60% of the reported value (2300 cm² V⁻¹ s⁻¹) for KC₄₈ prepared from HOPG.     

Random-field induced interface widths in Ising systems

Ad-dimensional Ising system with interactionJ at zero temperature in random fields ±h is considered, where due to boundary conditions an interface between states of positive and negative magnetization is maintained over a cross-sectionL ^d?1. The root mean square interface widthw(L,h) is found to behave asw(L, h)∝(h/J)² L(d=2) andw(L, h)≦L exp [?const(J/h)²] (d=3). Ford=2 the interface energy is negative forL→∞, and hence domains form spontaneously. Nonzero temperatures are discussed qualitatively, and consequences for the interpretation of experiments on random-field systems are outlined. Ford=2 and nonzero temperatures our treatment agrees with recent results of Grinstein and Ma and Villain.     

Two-nucleon excitations in 66 148 Dy82 from Gamow-Teller decay of the 67 148 Ho81 6− isomer

Two-particle excitations in the two-proton nucleus¹⁴⁸Dy populated inβ-decay of 10-s ¹⁴⁸Ho(6^?) were investigated throughγ-ray measurements following on-line mass separation. Theπh _11/2s_1/2 two-proton doublet and the (vh _9/2 s ₁ ^2/?1 )5^? neutron particle hole configuration are firmly identified, and theπh _11/2 d ₅ ^2/?1 and πh_11/2D_3/2 multiplets are also observed. The results suggest dominantπh _11/2vs ₁ ^2/?1 character for the¹⁴⁸Ho 6^?-isomer.     

Wn(n=3—27)原子团簇结构的第一性原理计算

使用密度泛函理论下的第一性原理方法,对W_n原子团簇(n=3—27)的结构特性进行了理论计算. 得到了W_n团簇(n=3—7)的最低能量结构和(n=8—27)的局域能量极小的典型结构. 使用凝胶模型,提出的电子组态1s²1p⁶1d¹⁰2s²1f¹⁴2p⁶3s ^{关键词： W团簇 结构性质 稳定性 从头计算}     

Dielectronic recombination rate coefficients for Fe23+ ion

The dielectronic recombination rate coefficients are explicitly calculated for the Li-like ion Fe²³⁺, which recombines with the continuum electron to form Fe²²⁺. Both 1s²2s and 1s²2p initial states are treated for the temperature range 1～8 keV. The rate coefficients are obtained from a direct evaluation of the Auger and radiative transition probabilities which are calculated from nonrelativistic Hartree-Fock wave functions. The L-shell electron excitation with Δn≠0 is found to be the dominant transition, while the 2s→2p excitation with Δn = 0 contributes approx. 10–25% of the Δn≠0 value. The K-shell excitation effect is about 1～30% in the temperature range considered, and the cascade effect is estimated to be a reduction in the rate of about 14%.     

Isometric immersions of pseudo-Riemannian space forms

In this paper we study local isometric immersions f:M_sⁿ(K)→N_s+q²ⁿ⁻¹(c) of a time-like n-submanifold M_sⁿ(K) with constant sectional curvature K and index s into a pseudo-Riemannian space form N_s+q²ⁿ⁻¹(c) with constant sectional curvature c and index s+q, where q≥0, 1≤s≤n−1 and K≠c. We first prove the existence of Chebyshev coordinates of a time-like submanifold M_sⁿ(K) in certain conditions. Afterwards, we generalize the classical Bäcklund theorem for space-like (or time-like) submanifolds in N_n−1²ⁿ⁻¹(c) and N₁²ⁿ⁻¹(c). Finally as an application, in the Chebyshev coordinates, we use the Bäcklund theorem to give a Bäcklund transformation and a permutability formula between the generalized sine-Laplace equation and the generalized sinh-Laplace equation.     

Dynamic behaviours of 2∞ attractor and q-phase transitions at bifurcations in logistic map

Dynamic behaviours of the 2^∞ attractor at the accumulation of period doubling in the logistic map are studied by the sum of the local expansion rates S_n(x₁) of nearby orbits. The variance 〈[S_n(x)]²〉 and algebraic exponent ß_n(x₁) = S_n(x₁)/ln(n) exhibits self-similar structures. The critical bifurcations such as intermittency, band merging and crisis-sudden widening of the chaotic attractor are studied in terms of a q-weighted average Λ(q), (− ∞ < q < ∞) of the coarse-grained local expansion rates Λ of nearby orbitals.     

Quark model relation for kaon charge radii

We derive the equation r²_E(K⁰)/r²_E(K⁺) = -(m²_s ? m²_n)/(2m²_s + m²_n) relating the kaon electric charge radii and the strange (m_s) and non-strange (m_n) quark masses in the nonrelativistic quark model, and suggest an inequality which we expect to hold in the presence of relativistic corrections. New data for these charge radii are presently being analysed by two experimental groups.