The modified Korteweg-de Vries equation on the quarter plane with t-periodic data

We study the modified Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic Dirichlet boundary datum u(0,t) in the sense that u(0,t) tends to a periodic function g?₀ (t) with period τ as t → ∞. We consider the perturbative expansion of the solution in a small ε > 0. Here we show that if the unknown boundary data u_x(0,t) and u_xx(0,t) are asymptotically t-periodic with period τ which tend to the functions g?₁ (t) and g?₂ (t) as t → ∞, respectively, then the periodic functions g?₁ (t) and g?₂ (t) can be uniquely determined in terms of the function g?₀ (t). Furthermore, we characterize the Fourier coefficients of g?₁ (t) and g?₂ (t) to all orders in the perturbative expansion by solving an infinite system of algebraic equations. As an illustrative example, we consider the case of a sine-wave as Dirichlet datum and we explicitly determine the coefficients for large t up to the third order in the perturbative expansion.     

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     

Measurement of K β K /α values using proton beamvalues using proton beam

The Measurement of K _β K _/α intensity ratios are measured in some 3d shell elements by using a 2 MeV proton beam along with a high resolution Si(Li) detector. The present Measurement of K _β K _/α intensity ratios are in good agreement with Scofield modified theoretical values, thus supporting the basic assumptions in that theory. From the present Measurement of K _β K _/α intensity ratios, it is evident that due to chemical effects, the experimental Measurement of K _β K _/α intensity ratios will be increased while they will be decreased due to the presence of simultaneous M-shell vacancies which are produced due to proton excitation.     

Connection Considerations of Gravitational Field in Finsler Spaces

Some alternative connection structures of the Finslerian gravitational field are considered by modifying the independent variables (x,y) (x: point and y: vector) in various ways. For example, (x ^k ,y ⁱ ) (k,i = 1,2,3,4) are changed to (x ^k ,y ⁰) (y ⁰: scalar) or (x ⁰,y ⁱ ) (x ⁰: time axis); (x ^k ,y ⁱ ) are generalized to (x ^k ,y ⁱ ,p _i ) (p _i : covector dual to y ⁱ ) or (x ^k ,y ⁱ ,q _a ) (q _a : covector different from p _i ); (x ^k ,y ⁱ ) are further generalized to (x ^k ,y ^(a)i ) (a = 1,2,…,m), (y ^(a): (a)th vector), etc.     

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.     

Calculation of the Hyperfinestructure and gJ-Values of the 3d 4s 4p-Configuration of Scandium

Wave functions for the 3d 4s 4p, 3d² 4p and 4s² 4p configurations of ScI are calculated, taking into account departures from SL-coupling and configuration interaction and on fitting the radial integrals to the experimental fine structure energies. Using these wave functions g J -values are derived. The intermediate coupled hfs matrix elements of the 3d 4s 4p configuration are reduced to the unknown electron coupling constants a_s, a_p and a_d and calculated on estimating these constants from the spin-orbit coupling constants and fitting them to some experimental A-values. By this way the absolute phases of the experimental A((¹P)²D)- and A((¹P)²F)-values are obtained. Good agreement between calculated and most experimental data has been achieved.     

Superconductivity in amorphous thin films of tin–copper

The properties of the superconducting state in the amorphous Sn_1?xCu_x thin films were characterized. The concentration of copper changes in the range from 0.08 to 0.41. The calculations were conducted in the framework of the strong-coupling formalism, wherein the Eliashberg functions determined in the tunnel experiment were used. The value of the Coulomb pseudopotential equal to 0.1 was adopted. It will be shown that the critical temperature (T_C) decreases from 7 to 3.9 K. The ratio, R_Δ = 2Δ(0)/k_BT_C, differs from the BCS value: R_Δ ∈ <4.4, 3.95>, where Δ(0) represents the order parameter. Similarly behave the ratios: R_C = ΔC(T_C)/C^N(T_C) ∈ <2.2, 1.75> and R_H = T_CC^N(T_C)/H²_C(0) ∈ <0.141, 0.154>. The parameter ΔC(T_C) is the specific heat jump, C^N(T_C) denotes the specific heat of the normal state and H_C(0) is the thermodynamic critical field.     

The 2‐matrix of the spin‐polarized electron gas: contraction sum rules and spectral resolutions

The spin‐polarized homogeneous electron gas with densities ρ_↑ and ρ_↓ for electrons with spin ‘up’ (↑) and spin ‘down’ (↓), respectively, is systematically analyzed with respect to its lowest‐order reduced densities and density matrices and their mutual relations. The three 2‐body reduced density matrices γ_↑↑, γ_↓↓, γ_a are 4‐point functions for electron pairs with spins ↑↑, ↓↓, and antiparallel, respectively. From them, three functions G_↑↑(x,y), G_↓↓(x,y), G_a(x,y), depending on only two variables, are derived. These functions contain not only the pair densities according to g_↑↑(r) = G_↑uarr;(0,r), g_↓↓(r) = G_↓↓(0,r), g_a(r) = G_a(0,r) with r = | r ₁ ‐ r ₂|, but also the 1‐body reduced density matrices γ_↑ and γ_↓ being 2‐point functions according to γ_s = ρ_sf_s and f_s(r) = G_ss(r, ∞) with s = ↑,↓ and r = | r ₁ ‐ r ^′₁|. The contraction properties of the 2‐body reduced density matrices lead to three sum rules to be obeyed by the three key functions G_ss, G_a. These contraction sum rules contain corresponding normalization sum rules as special cases. The momentum distributions n_↑(k) and n_↓(k), following from f_↑(r) and f_↓(r) by Fourier transform, are correctly normalized through f_s(0) = 1. In addition to the non‐negativity conditions n_s(k),g_ss(r),g_a(r) ≥ 0 [these quantities are probabilities], it holds n_s(k) ≤ 1 and g_ss(0) = 0 due to the Pauli principle and g_a(0) ≤ 1 due to the Coulomb repulsion. Recent parametrizations of the pair densities of the spin‐unpolarized homogeneous electron gas in terms of 2‐body wave functions (geminals) and corresponding occupancies are generalized (i) to the spin‐polarized case and (ii) to the 2‐body reduced density matrix giving thus its spectral resolutions.     

非对称量子点中磁极化子性质的磁场和温度依赖性

采用Tokuda线性组合算符法和Lee-Low-Pines变换法,研究了温度和磁场对非对称抛物量子点中强耦合磁极化子性质的影响,简捷地得到了作为量子点的横向受限强度ω₁、纵向受限强度ω₂、电子-声子耦合强度α、外磁场的回旋频率ω_c和温度参数γ的函数的磁极化子的振动频率λ、基态能量E₀和有效质量m 关键词： 非对称量子点 强耦合磁极化子 磁场和温度依赖性     

Heavy quark currents in ultra-high energy neutrino interactions

We discuss heavy quark contributions to the neutrino-nucleon total cross section at very high energies, well above the real top production threshold. The top-bottom weak current is found to generate strong left-right asymmetry of neutrino-nucleon interactions. We separate contributions of different helicity states and make use of the κ-factorization to derive simple and practically useful formulas for the left-handed (F _L ) and right-handed (F _R ) components of the conventional structure function 2xF ₃ = F _L − F _R in terms of the integrated gluon density. We show that F _L ≫ F _R and, consequently, xF ₃ ≈ F _T , where F _T is the transverse structure function. The conventional structure function F ₂ = F _S + F _T at Q ² ≪ m _t ² appears to be dominated by its scalar (also known as longitudinal) component F _S and the hierarchy F _S ≫ F _L ≫ F _R arises naturally. We evaluate the total neutrino-nucleon cross section at ultra-high energies within the color dipole BFKL-formalism.     

Phase transition studies in polymesomorphic compounds

Summary The structural parameters of theS _A andS _B mesophases and the phase transitionS _G -S _B andS _B -S _A of the compound N-(p-n-octyloxybenzylidene)p-n-butyl aniline were investigated by X-ray diffraction. This work is complementary to the previous structural determination in the crystalline solid phase andS _G phase of the same compound. TheS _B hexagonal lattice parameters atT=66°C area=b=(5.03±0.01) ? andc=(27.6±0.2) ?. TheS _A structural data atT=80°C are layer thicknessd=(26.9±0.2) ? and intermolecular distanceD=(5.04±0.05) ?. The interlayer and the in-plane intermolecular distances were obtained as a function of temperature through all the mesophases. Pretransitional effects were observed at theS _G -S _B phase transition, which corresponds to a progressive reduction to zero of the molecule inclination angle with respect to the normal to the smectic layers. The layer thickness in theS _A phase appeared smaller than that in theS _B due to molecular axial fluctuations. To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.     

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.     

Calorimetric fragility of maltitol

The heat capacity of maltitol has been measured with an adiabatic calorimeter for the crystal from 100 K to 425 K (T _m = 420 K), for the glass from 249 K to T _g (around 311 K) and for the liquid from T _g to 400 K. The heat of melting is 55.068 kJ/mol. The calorimetric glass transition occurs at about T _g = 311 K with a sudden jump of the heat capacity ΔC _p (T _g ) of about 243.6 J/(K mol). The excess entropy between the undercooled liquid and the crystal was calculated from the heat capacity data and was used to estimate the Kauzmann temperature T _K which was found 50 K below T _g . ΔC _p (T _g ) and T _K for maltitol were compared to other compounds like sugars, polyol and hydrogen bonded liquids. It has been found that the glass former maltitol is a "fragile" liquid on the thermodynamic point of view.     

Inclusive production of the meson in hadronic decays of the Z boson

The inclusive production rate of the ρ^±(770) vector meson in hadronic Z decays is measured with the ALEPH detector at the LEP collider. A total of 3.2 million hadronic events are selected from data recorded between 1991 and 1995. Decays of ρ^±→π⁰+π^± are reconstructed for x_E>0.05 and x_p>0.05 where x_p=p_ρ/p_beam and x_E=E_ρ/E_beam. The average ρ^± multiplicity per hadronic event is evaluated to be N(ρ^±)=2.59±0.03±0.15±0.04 where the first error is statistical and the second systematic. The third error is from the uncertainty in the extrapolation to x_p=x_E=0. The rates and differential cross-section are compared with Monte Carlo model predictions and OPAL measurements. Residual Bose–Einstein correlations are found to be an important component in the analysis.     

Dielectric relaxation of binary polar liquid mixture measured in benzene at 10 GHz frequency

The dielectric relaxation times τ _jk ’s and dipole moments μ _jk ’s of the binary (jk) polar liquid mixture of N,N-dimethyl acetamide (DMA) and acetone (Ac) dissolved in benzene (i) are estimated from the measured real σ′ _ijk and imaginary σ″ _ijk parts of complex high frequency conductivity σ* _ijk of the solution for different weight fractions w _jk ’s of 0.0, 0.3, 0.5, 0.7 and 1.0 mole fractions x _j of Ac and temperatures (25, 30, 35 and 40°C) respectively under 9.88 GHz electric field. τ _jk ’s are obtained from the ratio of slopes of σ″ _ijk -w _jk and σ′ _ijk -w _jk curves at w _jk → 0 as well as linear slope of σ″ _ijk -σ′ _ijk curves of the existing method (Murthy et al, 1989) in order to eliminate polarpolar interaction in the latter case. The calculated τ’s are in excellent agreement with the reported τ’s due to Gopalakrishna’s method. μ _jk ’s are also estimated from slopes β’s of total conductivity σ _ijk -w _jk curves at w _jk → 0 and the values agree well with the reported μ’s from G.K. method. The variation of τ _jk ’s and μ _jk ’s with x _j of Ac reveals that solute-solute molecular association occurs within 0.0–0.3x _j of Ac beyond which solute-solvent molecular association is predicted. The theoretical dipole moments μ _theo’s are calculated from bond angles and bond moments to have exact μ’s only to show the presence of inductive, mesomeric and electromeric effects in the substituent polar groups. The thermodynamic energy parameters are estimated from ln(τ _jk T) against 1/T linear curve from Eyring’s rate theory to know the molecular dynamics of the system and to establish the fact that the mixture obeys the Debye-Smyth relaxation mechanism.      

The number of independent traces and supertraces on symplectic reflection algebras

It is shown that A:= H₁, _η (G), the sympectic reflection algebra over ?, has T_G independent traces, where T_G is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group G ? Sp(2N) ? End(?^2N) generated by the system of symplectic reflections.

Simultaneously, we show that the algebra A, considered as a superalgebra with a natural parity, has S_G independent supertraces, where S_G is the number of conjugacy classes of elements without eigenvalue -1 belonging to G.

We consider also A as a Lie algebra A^L and as a Lie superalgebra A^S.

It is shown that if A is a simple associative algebra, then the supercommutant [A^S, A^S] is a simple Lie superalgebra having at least S_G independent supersymmetric invariant non-degenerate bilinear forms, and the quotient [A^L, A^L]/([A^L, A^L] ∩ ?) is a simple Lie algebra having at least T_G independent symmetric invariant non-degenerate bilinear forms.     

Hysteresis behavior in Raman intensities in YBa2Cu3O7-y -family superconductors between T c and T o = 234 K

The intensity of the 646 cm^-1 Raman line in YBa₂Cu_3-x V _x O_7-y has been measured for cooling and heating from 77 K to 234 K. The intensity exhibits intensity hysteresis of order 20% in the cooling versus heating values between T _c = 89 K and T _o = 234 K. This effect is unexplained but is extremely similar to that reported by Ledbetter et al. for the bulk modulus and C ₁₁ elastic coefficient. In each case the hysteresis vanishes below T _c , independent of the value of T _c , which may be adjusted from 60 K to 94 K by changing oxygen concentration y. Qualitative interpretation requires that the structural phase transition near 234 K be intimately related to the superconducting transition at T _c presumably through partial ordering of oxygens.     

空间网络上的随机游走

本文以一维均匀环为基础, 通过添加有限数量的长程连接构造出了一维有限能量约束下的空间网络, 环上任意节点i与j之间存在一条长程连接的概率满足p_ijα d_ij^-α (α≥ 0),其中d_ij为节点i与j之间的网格距离, 并且所有长程连接长度总和受到总能量∧=cN(c≥ 0)的约束, N为网络节点总数.通过研究该空间网络上的随机游走过程,存在最优幂指数α₀ 使得陷阱问题的平均首达时间最短.进一步研究发现,平均首达时间与网络规模N之间存在着幂律关系, 随着网络规模N和总能量∧的增加,最优幂指数α₀单调增加,并趋近最优值1.5.     

Fine-structure energy levels, oscillator strengths and lifetimes in Co XV

Excitation energies from ground state for 97 fine-structure levels as well as oscillator strengths and radiative decay rates for all electric-dipole-allowed and intercombination transitions among the fine-structure levels of the terms belonging to the (1s ²2s ²2p ⁶)3s ²3p, 3s ³ p ², 3s ²3d, 3p ³, 3s3p3d, 3p ²3d, 3s3d ², 3s ²4s, 3s ²4p, 3s ²4d, 3s ²4f, and 3s3p4s configurations of Co XV are calculated, using extensive configuration-interaction (CI) wave functions, obtained with the CIV3 computer code of Hibbert. The important relativistic effects in intermediate coupling are included through the Breit-Pauli approximation via spin-orbit, spin-other-orbit, spin-spin, Darwin and mass correction terms. Small adjustments to the diagonal elements of the Hamiltonian matrices have been made. Our calculated excitation energies, including their ordering, are in excellent agreement with the experimental results and the experimentally compiled energy values of the National Institute for Standards and Technology (NIST) wherever available. The mixing among several fine-structure levels is found to be very strong, with most of the strongly mixed levels belonging to the (1s ²2s ²2p ⁶)3p ²3d and 3s3d ² configurations. The strong mixing among several fine-structure levels makes it very difficult to identify them uniquely. Perhaps, that may be the reason for the lack of both experimental and theoretical results for these levels. We believe that our extensive calculated values can guide experimentalists in identifying the fine-structure levels in their future work. From our radiative decay rates we have also calculated radiative lifetimes of some fine-structure levels. In this calculation we also predict new data for several fine-structure levels where no other theoretical and/or experimental results are available.     

The isotope dependence of diatomic Dunham coefficients

The isotope dependence of the Dunham vibration-rotation coefficients Y_kl of a diatomic molecule is studied. Rovibronic interactions between different electronic states are taken into account by transformation to an effective vibration-rotation Hamiltonian for each electronic state. This contains modified vibrational and rotational reduced masses as well as the adiabatic correction to the potential energy. The effects of these contributions on the vibration-rotation energies are expressed in terms of two functions and for each atom i. The resultant formula for Y_kl is Y_kl=μ_c^−(k+2l)/2U_kl{1+m_eΔ_kl^a/M_a+m_eΔ_kl^b/M_b+O(m_e²/M_i²)}, where U_kl, Δ_kl^a, and Δ_kl^b are isotopically invariant, M_a and M_b are the atomic masses, and μ_c = M_aM_b/(M_a + M_b − Cm_e) is the atomic reduced mass, modified by the molecular charge number C for charged species. The U_kl with l ≥ 2 can be calculated from those with l = 0 and 1. The corrections U_klΔ_klⁱ are related to the functions and and to the Dunham corrections. Recent data for the CO molecule are discussed, and it is suggested that some large Δ_klⁱ values are associated with accidentally small U_kl values, since the size of U_klΔ_klⁱ is not directly related to that of U_kl.