A very strong difference property for semisimple compact connected lie groups

Let G be a topological group. For a function f: G → ℝ and h ∈ G, the difference function Δ _h f is defined by the rule Δ _h f(x) = f(xh) − f(x) (x ∈ G). A function H: G → ℝ is said to be additive if it satisfies the Cauchy functional equation H(x + y) = H(x) + H(y) for every x, y ∈ G. A class F of real-valued functions defined on G is said to have the difference property if, for every function f: G → ℝ satisfying Δ _h f ∈ F for each h ∈ G, there is an additive function H such that f − H ∈ F. Erdős’ conjecture claiming that the class of continuous functions on ℝ has the difference property was proved by N. G. de Bruijn; later on, F. W. Carroll and F. S. Koehl obtained a similar result for compact Abelian groups and, under the additional assumption that the other one-sided difference function ∇ _h f defined by ∇ _h f(x) = f(xh) − f(x) (x ∈ G, h ∈ G) is measurable for any h ∈ G, also for noncommutative compact metric groups. In the present paper, we consider a narrower class of groups, namely, the family of semisimple compact connected Lie groups. It turns out that these groups admit a significantly stronger difference property. Namely, if a function f: G → ℝ on a semisimple compact connected Lie group has continuous difference functions Δ _h f for any h ∈ G (without the additional assumption concerning the measurability of the functions of the form ∇ _h f), then f is automatically continuous, and no nontrivial additive function of the form H is needed. Some applications are indicated, including difference theorems for homogeneous spaces of compact connected Lie groups.     

(GL(<Emphasis Type="Italic">n</Emphasis> + 1, ℝ), GL(<Emphasis Type="Italic">n</Emphasis>, ℝ)) is a generalized Gelfand pair

Denote by G = GL(n + 1, ℝ) the group of invertible (n + 1) × (n + 1) matrices with real entries, acting on ℝ ⁿ⁺¹ in the usual way, and let H ₁ = GL(n, ℝ) be the stabilizer of the first unit vector e ₀. Let H ₀ = GL(1, ℝ) and set H = H ₀ × H ₁. It is known that the pair (G,H) is a generalized Gelfand pair. Define a character χ of H by χ(h) = χ(h ₀ h ₁) = χ0(h ₀) where χ0 is a unitary character of H ₀ (h ₀ ∈ H ₀, h ₁ ∈ H ₁). Let σ be the anti-involution on G given by σ(g) = ^t g. In this note, we show that any distribution T on G satisfying T(h ₁ gh ₂) = χ(h ₁ h ₂) T(g) (g ∈ G; h ₁, h ₂ ∈ H) is invariant under the anti-involution σ. This result implies that (G,H ₁) is a generalized Gelfand pair.     

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)).     

Glassy behaviour of random field Ising spins on Bethe lattice in external magnetic field

The thermodynamics and the phase diagram of random field Ising model (RFIM) on Bethe lattice are studied by using a replica trick. This lattice is placed in an external magnetic field (B). A Gaussian distribution of random field (hi) with zero mean and variance hi2 = HR2F is considered. The free-energy (F ), the magnetization (M) and the order parameter (q) are investigated for several values of coordination number (z). The phase diagram shows several interesting behaviours and presents tricritical point at critical temperature TC = J/k and when HRF = 0 for finite z. The free-energy (F) values increase as T increases for different intensities of random field (HRF) and finite z. The internal energy (U) has a similar behaviour to that obtained from the Monte Carlo simulations. The ground state of magnetization decreases as the intensity of random field HRF increases. The ferromagnetic (FM)-paramagnetic (PM) phase boundary is clearly observed only when z →∞. While FM-PM-spin glass (SG) phase boundaries are present for finite z. The magnetic susceptibility (χ) shows a sharp cusp at TC in a small random field for finite z and rounded different peaks on increasing HRF.     

Ultraviolet and visible upconversion luminescence of Tm(0.1)Yb(5):FOV oxyfluoride nanophase vitroceramics

The ultraviolet upconversion luminescence of Tm³⁺ ions sensitized by Yb³⁺ ions in oxyfluoride nanophase vitroceramics when excited by a 975 nm diode laser was studied. An ultraviolet upconversion luminescence line positioned at 363.6 nm was found. It was attributed to the fluorescence transition of ¹D₂→³H₆ of Tm³⁺ ion. Several visible upconversion luminescence lines at 450.7 nm, (477.0 nm, 462.5 nm), 648.5 nm, (680.5 nm, 699.5 nm) and (777.2 nm, 800.7 nm) were also found, which result respectively from the fluorescence transitions of ¹D₂→³F₄, ¹G₄→³H₆, ¹G₄→³F₄, ³F₃→³H₆ and ³H₄→³H₆ of Tm³⁺ ion. The careful measurement and analysis of the variation of upconversion luminescence intensity F as a function of the 975 nm pumping laser power P prove that the upconversion luminescence of ¹D₂ state is partly a five-photon upconversion luminescence, and the upconversion luminescence of ¹G₄ state and ³H₄ state are respectively the three-photon and two-photon upconversion luminescence. The theoretical analysis suggested that the upconversion mechanism of the 363.6 nm ¹D₂→³H₆ upconversion luminescence is partly the cross energy transfer of {³H₄(Tm³⁺), ³F₄(Tm³⁺), ¹G₄(Tm³⁺)→¹D₂(Tm³⁺)} and {¹G₄(Tm³⁺)→³F₄(Tm³⁺), ³H₄(Tm³⁺)→¹D₂(Tm³⁺)} between Tm³⁺ ions. In addition, the upconversion luminescence of ¹G₄ and ³H₄ state results respectively from the sequential energy transfer {²F_5/2(Yb³⁺)→²F_7/2(Yb³⁺), ³H₄(Tm³⁺)→¹G₄(Tm³⁺)} and {²F_5/2(Yb³⁺) →²F_7/2(Yb³⁺), ³F₄(Tm³⁺)→³F₂(Tm³⁺)} from Yb³⁺ ions to Tm³⁺ ions. Supported by the National Natural Science Foundation of China (Grant No. 10674019)     

激光束在漫射表面上的散射(Ⅱ)

首先在前文的基础上,对于用激光根据“光核、光带比”(D₂/D₁)来测定磨削工件表面光洁度的原理加以系统总结,然后按照经验关系R_x=5R_a(对于▽7以上光洁度),确定了表面随机高度的概率密度函数中的衰减系数。对于有限负指数型函数P₁(h)={e^{(-b(|h|/h_m))} 当|h|≤h_m; 0 当|h|>h_m, 定出b=1.23,对于正则型函数P₂(h)=e^{(-a2(h/h_m)2)} 定出a²=2.分别讨论了以上两种函数中h_m的物理意义(皆对应于1/2R_z)将前文中公式加以精确改进后,对P₁(h)和P₂(h)分别计算了D₂/D₁与R_x的关系曲线,即绝对定标曲线。最后还计算了衍射图样半强度宽与R_x的关系曲线。 关键词：     

Asymptotic Heat Kernel Expansion in the Semi-Classical Limit

Let H_(^h/2p) = (^h/_2p)²L +V{H_\hbar = \hbar^{2}L +V}, where L is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and V is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of H_(^h/2p){H_\hbar} as (^h/_2p) \searrow 0{\hbar \searrow 0}. As a consequence we get an asymptotic expansion for the quantum partition function and we see that it is asymptotic to the classical partition function. Moreover, we show how to bound the quantum partition function for positive (^h/_2p){\hbar} by the classical partition function.     

Finite- N effects for ideal polymer chains near a flat impenetrable wall

This paper addresses the statistical mechanics of ideal polymer chains next to a hard wall. The principal quantity of interest, from which all monomer densities can be calculated, is the partition function, G _N(z) , for a chain of N discrete monomers with one end fixed a distance z from the wall. It is well accepted that in the limit of infinite N , G _N(z) satisfies the diffusion equation with the Dirichlet boundary condition, G _N(0) = 0 , unless the wall possesses a sufficient attraction, in which case the Robin boundary condition, G _N(0) = - G _N ^′(0) , applies with a positive coefficient, . Here we investigate the leading N ^-1/2 correction, G _N(z) . Prior to the adsorption threshold, G _N(z) is found to involve two distinct parts: a Gaussian correction (for z aN ^1/2 with a model-dependent amplitude, A , and a proximal-layer correction (for z a described by a model-dependent function, B(z) .     

Er3+，Tm3+共掺的NaY(WO4)2晶体的光谱分析和上转换发光

在室温下，测量了Er:Tm:NaY(WO₄)₂晶体的吸收光谱、激发光谱、发射光谱以及上转换发光，并运用J-O理论对测量的结果进行了计算，得出了Er:Tm:NaY(WO₄)₂晶体的强度参数.报道了Tm，Er离子间特殊的能量传递和相关上转换，解释了离子间的能级跃迁过程.同时，对于Er增强Tm离子近红外发光的特性也作了充分研究. 关键词： 4)₂晶体')" href="#">Er:Tm:NaY(WO₄)₂晶体 吸收光谱 发射光谱 激发光谱 上转换     

Acoustic scattering by an impenetrable spheroid

The scattering of a plane acoustic wave from an impenetrable, soft or hard, prolate or oblate spheroid is considered. Two different methods are used for the evaluation. In the first, the pressure field is expressed in terms of spheroidal wave functions. In the second, a shape perturbation method, the field is expressed in terms of spherical wave functions only, while the equation of the spheroidal boundary is given in spherical coordinates. Analytical expressions are obtained for the scattered pressure field and the various scattering cross-sections, when the solution is specialized to small values of the eccentricity h = d/(2a) , where d is the interfocal distance of the spheroid and 2a is the length of its rotation axis. In this case, exact, closed-form expressions are obtained for the expansion coefficients g ⁽²⁾ and g ⁽⁴⁾ in the relation S(h) = S(0)[1 + g ⁽²⁾ h ² + g ⁽⁴⁾ h ⁴ + O(h ⁶)] expressing the scattered field and the scattering cross-sections. Numerical results are given for various values of the parameters. Published in Russian in Akusticheskiĭ Zhurnal, 2007, Vol. 53, No. 4, pp. 500–513. The text was submitted by the authors in English.     

Universal relation between Green functions in random matrix theory

We prove that in random matrix theory there exists a universal relation between the onepoint Green function G and the connected two-point Green function G_c given by N²G_c(z, w) = (∂²/∂z ∂w) log[(G(z) − G(w))/(z − w) + irrelevant factorized terms]. This relation is universal in the sense that it does not depend on the probability distribution of the random matrices for a broad class of distributions, even though G is known to depend on the probability distribution in detail. The universality discussed here represents a different statement than the universality we discovered some time ago, which states that a²G_c (az, aw) is independent of the probability distribution, where a denotes the width of the spectrum and depends sensitively on the probability distribution. It is shown that the universality proved here also holds for the more general problem of a hamiltonian consisting of the sum of a deterministic term and a random term analyzed perturbatively by Brézin, Hikami, and Zee.     

Mixtures are not derivable

A binary operation ø on probability distribution functions is derivable from a binary operation on random variables if there exists a two-place functionV such that, for any distribution functionsF andG, there exist random variablesX andY, defined on a common probability space, such thatF andG are the distribution functions ofX andY , respectively, and ø(F, G) is the distribution function ofV (X, Y). We show that if ø(F, G) =cF + (1 -c)G, 0 <c < 1, then ø is not derivable; similarly, is not derivable.1. Dedicated to the memory of Charles H. Randall, colleague and friend.     

Nonstandard GLh(n) quantum groups and contraction of covariant q-bosonic algebras

GL_h(n) × GL_h(m)-covariant h-bosonic algebras are built by contracting the GL_q(n) × GL_q(m)-covariant q-bosonic algebras considered by the present author some years ago. Their defining relations are written in terms of the corresponding R _h-matrices. Whenever n = 2, and m = 1 or 2, it is proved by using U_h(sl(2)) Clebsch-Gordan coefficients that they can also be expressed in terms of coupled commutators in a way entirely similar to the classical case. Some U_h(sl(2)) rank-(1/2) irreducible tensor operators, recently constructed by Aizawa in terms of standard bosonic operators, are shown to provide a realization of the h-bosonic algebra corresponding to n = 2 and m = 1.     

BeH,H2和BeH2的分子结构和势能函数

用二次组态相关(QCISD)和密度泛函(B3LYP)方法, 选用6-311++g(d,p), 6-311++g(3df,3pd)和D95(3df,3pd)基组对H₂, BeH和BeH₂分子的结构进行优化. 得到它们的基态电子态分别为H₂(¹Σ_g), BeH(²Σ)和BeH₂(¹Σ_{g 关键词： BeH 2}')" href="#">BeH₂ 2')" href="#">H₂ 二次组态相关(QCISD) 势能函数     

Multiphoton ultraviolet and visible upconversion luminescence of TmYb co-doped oxyfluoride glass

The ultraviolet upconversion luminescence of Tm³⁺ ions sensitized by Yb³⁺ ions in oxyfluoride glass when excited by a 975 nm diode laser was studied in this paper. One typical ultraviolet upconversion luminescence lines positioned at 362.3 nm was found. It can be attributed to the five-photon upconversion luminescence transition of ¹D₂ → ³H₆. Several visible upconversion luminescence lines at 451.1 nm, (477.9 nm, 462.5 nm), 648.7 nm, (680.5 nm, 699.5 nm) and (777.5 nm, 800.7 nm) were found also, which results from the fluorescence transitions of five-photon ¹D₂ → ³F₄, three-photon ¹G₄ → ³H₆, three-photon ¹G₄ → ³F₄, two-photon ³F₃ → ³H₆ and two-photon ³H₄ → ³H₆ of Tm³⁺ ion, respectively. The theoretical analysis suggests that the upconversion mechanism of the 362.3 nm ¹D₂ → ³H₆ upconversion luminescence is the cross energy transfer of {³H₄(Tm³⁺) → ³F₄(Tm³⁺), ¹G₄(Tm³⁺) → ¹D₂(Tm³⁺)} and {¹G₄(Tm³⁺) → ³F₄(Tm³⁺), ³H₄(Tm³⁺) → ¹D₂(Tm³⁺)} between Tm³⁺ ions. In addition, the upconversion luminescence of ¹G₄ and ³H₄ state results from the sequential energy transfer {²F_5/2(Yb³⁺) → ²F_7/2(Yb³⁺), ³H₄(Tm³⁺) → ¹G₄(Tm³⁺)} and {²F_5/2(Yb³⁺) → ²F_7/2(Yb³⁺), ³F₄(Tm³⁺) → ³F₂(Tm³⁺)} from Yb³⁺ ions to Tm³⁺ions, respectively.     

Acoustic scattering by a penetrable spheroid

The scattering of a plane acoustic wave from a penetrable prolate or oblate spheroid is considered. Two different methods are used for the evaluation. In the first, the pressure field is expressed in terms of spheroidal wave functions. In the second, a shape perturbation method, the field is expressed in terms of spherical wave functions only, while the equation of the spheroidal boundary is given in spherical coordinates. Analytical expressions are obtained for the scattered pressure field and the various scattering cross-sections when the solution is specialized to small values of the eccentricity h = d/(2a), (h ≪ 1), with d being the interfocal distance of the spheroid and 2a the length of its rotation axis. In this case, exact, closed-form expressions are obtained for the expansion coefficients g ⁽²⁾ and g ⁽⁴⁾ in the relation S(h) = S(0)[1 + g ⁽²⁾ h ² + g ⁽⁴⁾ h ⁴ + O(h ⁶)] expressing the scattered field and the scattering cross-sections. Numerical results are given for various values of the parameters. Published in Russian in Akusticheskiĭ Zhurnal, 2008, Vol. 54, No. 2, pp. 189–204. The text was submitted by the authors in English.     

Integration on then th power of a hyperbolic space in terms of invariants under diagonal action of isometries (Lorentz transformations)

The integral of a function over then'th power of hyperbolicd-dimensional spaceH is decomposed into integration along each orbit under diagonal action onH ⁿ of the isometry groupG onH, followed by integration over the orbit space, parametrized in terms of a complete set of invariants. The Jacobian entering in this last integral is expressed explicitly in terms of certain determinants. When viewingH as a half-hyperboloid in _d+1 ,G is induced by the homogeneous Lorentz groupO (1,d) acting on _d+1 .     

Effect of electron-phonon interaction on the knight shift

The spin reorientations in HoFeO₃ were studied. It was found that above 34 K the Fe³⁺ magnetic moments leave the (ac) plane of the orthorhombic lattice by a continuous rotation toward a finite angle, probably in the (bc) plane:G _z F _x −G _zy F _x . Then above 45 K they start returning to the (ac) plane in a jumpwise manner to go on with the well known continuous reorientationG _zx F _xz −G _x F _z towards thea-axis.