On Some Harris-FKG Type Correlation Inequalities for a Non-Attractive Model

We consider the following non-attractive one-dimensional model whose evolution is given by ξ_n+1(x)=ξ_n(x+1)+ξ_n(x?1) (mod 2) with probability p, =0 with probability 1?p. In the present letter, we prove that some Harris-FKG type correlation inequalities hold for this model. Moreover it is shown that a correlation inequality is also correct for the more general non-attractive class.     

关於展开子

本文讨论展开子的一些性质。将展开子A_nrst变换至ξ表示,定义为〈ξ|〉=∑ξ₀^-n-1ξ₁^rξ₂^sξ₃^tA_nrst,立即可以看出〈ξ|〉在洛伦兹变换中的变换,正如标准表示中的变换。由此可以立即证明,在标志洛伦兹群的各种不可约表示的两个量J=-1/2I_klI^kl,I=1/2ε^klmnI_klI_mn中,对於展开子而言,I一定等於零。我们也证明了如果我们要求J的本徵函数〈ξ|〉在各处行为正常,便获得J<0,亦即展开子表示为么正的条件。对於在展开子空间(J,0)及其他空间(I′,J′)中作用的矢量算符,我们作出了计算。选择定则为(i)I′=0,J′=1+J±2(1+J)^1/2;(ii)I′=±(1+J)^1/2i,J′=1+J。我们又证明了ξ^vξ_v?/(?ξ^μ)/(ξ^μ)-(1±(1+J)^1/2)ξ_μ将(J,0)空间变为(1+J±2(1+J)^1/2,0)空间。利用上式中取“-”符号的算符,我们可以构成一个像(-ir_μp^μ+k)ψ=0的波动方程,其中ψ只在两个展开子空间中。     

The exponential interaction in Rn

We prove that the Schwinger functions for the ultraviolet cut-off exponential interaction with euclidean measure exp {;?λ∫Λ:e^αξk(x):dx} dμ₀(ξ/ ∫ exp{?λ∫Λ:e^αξk(x):dx} dμ₀(ξ), λ > 0, converge as the ultraviolet cut-off is removed. The limits are the free Schwinger functions in the case of space-time dimension n ? 3. In the case n = 2 this holds for |α| sufficiently big, whereas for |α| < 2 √π, one has the well-known nontrivial Schwinger functions of the exponential interaction.     

A Level 1 Large-Deviation Principle for the Autocovariances of Uniquely Ergodic Transformations with Additive Noise

A large-deviation principle (LDP) at level 1 for random means of the type $$M_n \equiv \frac{1}{n}\sum\limits_{j = 0}^{n - 1} {Z_j Z_{j + 1} ,{\text{ }}n = 1,2,...}$$ is established. The random process {Z _n} _n≥0 is given by Z _n = Φ(X _n) + ξ _n , n = 0, 1, 2,..., where {X _n} _n≥0 and {ξ _n} _n≥0 are independent random sequences: the former is a stationary process defined by X _n = T ⁿ(X ₀), X ₀ is uniformly distributed on the circle S ¹, T: S ¹ → S ¹ is a continuous, uniquely ergodic transformation preserving the Lebesgue measure on S ¹, and {ξ_n} _n≥0 is a random sequence of independent and identically distributed random variables on S ¹; Φ is a continuous real function. The LDP at level 1 for the means M _n is obtained by using the level 2 LDP for the Markov process {V _n = (X _n, ξ _n , ξ _n+1)} _n≥0 and the contraction principle. For establishing this level 2 LDP, one can consider a more general setting: T: [0, 1) → [0, 1) is a measure-preserving Lebesgue measure, $\Phi :\left[ {0,\left. 1 \right)} \right. \to \mathbb{R}$ is a real measurable function, and ξ _n are independent and identically distributed random variables on $\mathbb{R}$ (for instance, they could have a Gaussian distribution with mean zero and variance σ²). The analogous result for the case of autocovariance of order k is also true.     

The Inviscid Burgers Equation with Initial Value of Poissonian Type

We study the discontinuities (shocks) of the solution to the Burgers equation in the limit of vanishing viscosity (the inviscid limit) when the initial value is the opposite of the standard Poisson process p. We show that this solution is only defined for t ε (0, 1). Let T ₀ = 0 and T _n , n≧1, be the successive jumps of p. We prove that for all M > 0 the inviscid limit is characterized on the region x ε (-∞, M], t ε (0, 1) by the increasing process $N(t) = \sup \{ n \in \mathbb{N} {\text{| }}M + nt > T_n \} $ and the random set I(x) = {n ε {0,..., N(t)}‖T _n -nt≦x<T _n+1 - nt}. The positions of shocks are given in a precise manner. We give the distribution of N(t) and also the distribution of its first jump. We also prove similar results when the initial value is u _μ(y, 0) = -μp(y/μ²) + μ^-1 max(y, 0), μ ε (0, 1).     

Improved calculation of the structure functions of the nucleon in the O(4, 2)-infinite multiplet model

A new investigation of the inelastic electon scattering from proton in the O(4, 2) models is presented. The resultant explicit structure functions in the limit satisfy scaling, F₁(ξ) ≠ 0 (σ_T ≠ 0), the Drell-Yan relation F₂(ξ) ～ (1?ξ)³ and, approximately, the Callan-Gross relation F₂(ξ) ≈ 2ξF₁(ξ).     

变分法与无规及高阶无规位相近似法(Ⅰ)

本文目的是指明,无规与高阶无规位相近似法(以下简称RPA及HRPA)也可由一变分法推得,由此可使我们能更清楚地了解RPA及HRPA的久期方程不具有厄密性的原因,此外变分法还自然地指出了一个使相应的久期方程会具有厄密性的途径以及一个确定粒子填充数的方法。     

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.     

Asymptotic behavior of Mayer cluster sums for the one-dimensional Ising model

The properties of the high-field polynomialsL _n (u) for the one-dimensional spin 1/2 Ising model are investigated. [The polynomialsL _n (u) are essentially lattice gas analogues of the Mayer cluster integralsb _n (T) for a continuum gas.] It is shown thatu ^?1 L _n (u) can be expressed in terms of a shifted Jacobi polynomial of degreen?1. From this result it follows thatu ^?1 L _n (u); n=1, 2,... is a set of orthogonal polynomials in the interval (0, 1) with a weight functionω(u)=u, andu ^?1 L _n (u) hasn?1 simple zerosu _n (v); v=1, 2,...,n?1 which all lie in the interval 0<u<1. Next the detailed behavior ofL _n (u) asn→∞ is studied. In particular, various asymptotic expansions forL _n (u) are derived which areuniformly valid in the intervalsu<0, 0<u<1, andu>1. These expansions are then used to analyze the asymptotic properties of the zeros {u _n (v); v=1, 2,...,n?1}. It is found that $$\begin{array}{*{20}c} {u_n (v) \sim \tfrac{1}{4}({{j_{1,v} } \mathord{\left/ {\vphantom {{j_{1,v} } n}} \right. \kern-\nulldelimiterspace} n})^2 [1 - ({{j_{1,v}^2 } \mathord{\left/ {\vphantom {{j_{1,v}^2 } {12}}} \right. \kern-\nulldelimiterspace} {12}})n^{ - 1} + ({{j_{1,v}^2 } \mathord{\left/ {\vphantom {{j_{1,v}^2 } {700)( - 3 + 2j_{1,v}^2 )n^{ - 4} }}} \right. \kern-\nulldelimiterspace} {700)( - 3 + 2j_{1,v}^2 )n^{ - 4} }}} \\ { + ({{j_{1,v}^2 } \mathord{\left/ {\vphantom {{j_{1,v}^2 } {20160)(40 + 4j_{1,v}^2 - j_{1,v}^4 }}} \right. \kern-\nulldelimiterspace} {20160)(40 + 4j_{1,v}^2 - j_{1,v}^4 }})n^{ - 6} + \cdot \cdot \cdot ]} \\ {u_n (n - v) \sim 1 - ({{j_{0,v}^2 } \mathord{\left/ {\vphantom {{j_{0,v}^2 } 4}} \right. \kern-\nulldelimiterspace} 4})n^{ - 2} + ({{j_{0,v}^2 } \mathord{\left/ {\vphantom {{j_{0,v}^2 } {48)( - 2 + j_{0,v}^2 )n^{ - 4} }}} \right. \kern-\nulldelimiterspace} {48)( - 2 + j_{0,v}^2 )n^{ - 4} }}} \\ { + ({{j_{0,v}^2 } \mathord{\left/ {\vphantom {{j_{0,v}^2 } {2880)(2 + 9j_{0,v}^2 - 2j_{0,v}^4 )n^{ - 6} + \cdot \cdot \cdot }}} \right. \kern-\nulldelimiterspace} {2880)(2 + 9j_{0,v}^2 - 2j_{0,v}^4 )n^{ - 6} + \cdot \cdot \cdot }}} \\ \end{array} $$ asn→∞v fixed, wherej _k,v denotes thevth zero of the Bessel functionJ _k(^z)     

Theory of the froissaron exchange

The results of the multi-pomeron exchange theory considered earlier, are shown to depend crucially on the threshold behaviour of the pomeron contribution at t = 4μ_π², under the condition α(0) ? 1 = Δ > α′(0) 4μ_π². The t-channel partial wave f(t) of the multi-pomeron exchange is calculated. In the limit ξ · Δ > 1, where ξ = ln s/4μ², it corresponds to the scattering on a black disk of expanding radius b₀ ≈ a · ξ where a = s/2μ_π ? α′2μ_π. Due to the threshold singularity influence, it does not violate the t-channel unitarity condition. At a sufficiently small value of the froissaron coupling constant g₀₀ ～ Δ³/a², the theory is shown to be simultaneously s-unitary.     

On the free energy of the hopfield model

The general theory of inhomogeneous mean-field systems of Raggio and Werner provides a variational expression for the (almost sure) limiting free energy density of the Hopfield model $$H_{N,p}^{\{ \xi \} } (S) = - \frac{1}{{2N}}\sum\limits_{i,j = 1}^N {\sum\limits_{\mu = 1}^N {\xi _i^\mu \xi _j^\mu S_i S_j } } $$ for Ising spinsS _i andp random patterns ξ^μ=(ξ ₁ ^μ ,ξ ₂ ^μ ,...,ξ _N ^μ ) under the assumption that $$\mathop {\lim }\limits_{N \to \gamma } N^{ - 1} \sum\limits_{i = 1}^N {\delta _{\xi _i } = \lambda ,} \xi _i = (\xi _i^1 ,\xi _i^2 ,...,\xi _i^p )$$ exists (almost surely) in the space of probability measures overp copies of {?1, 1}. Including an “external field” term ?ξ _μ ^p h^μμξ _i=1 ^N ξ _i ^μ S_i, we give a number of general properties of the free-energy density and compute it for (a)p=2 in general and (b)p arbitrary when λ is uniform and at most the two componentsh ^μ1 andh ^μ2 are nonzero, obtaining the (almost sure) formula $$f(\beta ,h) = \tfrac{1}{2}f^{ew} (\beta ,h^{\mu _1 } + h^{\mu _2 } ) + \tfrac{1}{2}f^{ew} (\beta ,h^{\mu _1 } - h^{\mu _2 } )$$ for the free energy, wheref ^cw denotes the limiting free energy density of the Curie-Weiss model with unit interaction constant. In both cases, we obtain explicit formulas for the limiting (almost sure) values of the so-called overlap parameters $$m_N^\mu (\beta ,h) = N^{ - 1} \sum\limits_{i = 1}^N {\xi _i^\mu \left\langle {S_i } \right\rangle } $$ in terms of the Curie-Weiss magnetizations. For the general i.i.d. case with Prob {ξ _i ^μ =±1}=(1/2)±?, we obtain the lower bound 1+4?²(p?1) for the temperatureT _c separating the trivial free regime where the overlap vector is zero from the nontrivial regime where it is nonzero. This lower bound is exact forp=2, or ε=0, or ε=±1/2. Forp=2 we identify an intermediate temperature region between T_*=1?4?² and T_c=1+4?² where the overlap vector is homogeneous (i.e., all its components are equal) and nonzero.T _* marks the transition to the nonhomogeneous regime where the components of the overlap vector are distinct. We conjecture that the homogeneous nonzero regime exists forp≥3 and that T_*=max{1?4?²(p?1),0}.     

Fe deep level optical spectroscopy in InP

The optical cross-section σ_n⁰(hv) and σ_p⁰(hv) associated with the (Fe³⁺ ? Fe²⁺) deep level have been measured by Deep Level Optical Spectroscopy in n-type Fe doped samples of InP. Optical transitions are interpreted as transitions from the Fe²⁺ ground state to the Γ and L point minima of the conduction band for σ_n⁰(hv) and from the valence band to the ground and excited state for Fe²⁺ for σ_p(hv). A theoretical model which accounts for the main features of the experimental data is proposed.     

An experimental study of the K+ → πoμ+ν decay dalitz plot

A sample of K⁺_μ3 events, detected in the CERN 1.1 m³ heavy-liquid bubble chamber, has been used to investigate the q² dependence of the form factors, giving ξ(0) = ?1.1 ± 1.0 and ξ(6.6m²_π) = ?0.34 ± 0.20, for λ₊ = 0.027. A three parameter fit gives a value for λ₊ = 0.025 ± 0.017 in good agreement with the preceding K_e3 analysis.     

Tm and Tm-Tb-doped germanate glasses for S-band amplifiers

The mechanism involved in the Tm³⁺(³F₄)→Tb³⁺(⁷F_0,1,2) energy transfer as a function of the Tb concentration was investigated in Tm:Tb-doped germanate (GLKZ) glass. The experimental transfer rate was determined from the best fit of the ³F₄ luminescence decay due to the Tm→Tb energy transfer using the Burshtein model. The result showed that the 1700 nm emission from ³F₄ can be completely quenched by 0.8 mol% of Tb³⁺. As a consequence, the ⁷F₃ state of Tb³⁺ interacts with the ³H₄ upper excited state of Tm³⁺ slighting decreasing its population. The effective amplification coefficient β(cm⁻¹) that depends on the population density difference Δn=n(³H₄)-n(³F₄) involved in the optical transition of Tm³⁺ (S-band) was calculated by solving the rate equations of the system for continuous pumping with laser at 792 nm, using the Runge-Kutta numerical method including terms of fourth order. The population density inversion Δn as a function of Tb³⁺ concentration was calculated by computational simulation for three pumping intensities, 0.2, 2.2 and 4.4 kWcm⁻². These calculations were performed using the experimental Tm→Tb transfer rates and the optical constants of the Tm (0.1 mol%) system. It was demonstrated that 0.2 mol% of Tb³⁺ propitiates best population density inversion of Tm³⁺ maximizing the amplification coefficient of Tm-doped (0.1 mol%) GLKZ glass when operating as laser intensity amplification at 1.47 μm.     

New bounds for dilepton production from heavy neutral leptons

Bounds on 〈E_?ⁿ〉/〈E₊ⁿ〈, 〉E₊E_?〈/〉E₂²〈 and 〈E₊E_?〉/〈E₊〉〈E_?〉 are direved for the processes ν_μN → μ^?μ⁺(e⁺) + X and μN → μ^?μ⁺ + X if dileptons are mediated by a $\text{spin}\text{-}\text{1}\text{2}$ heavy neutral lepton L₀. The bounds are shown to be independent of the production mechanism and mass of L₀. Useful conditional bounds are obtained relating the bounded quantities, which give information about the structure of the weak current responsible for L₀ decay.     

Some aspects of lambda production in the exclusive reactions K−p→Λ0 + n pions at 8.25 GeV/c

Cross sections are given for the various exclusive reactions K^?p→Λ⁰ + n pions, as well as for quasi two-body final states involving ?⁰, ω⁰ and Y₁¹(1385) resonance production. The general features of Λ⁰ production are presented as a function of the pion multiplicity n. Production of Y₁¹⁺(1385) is clearly observed at all multiplicities while the Y₁^1?(1385) signals grow with the multiplicity, as expected in a non-exotic exchange picture. The polarisation of the Λ⁰ is consistent with zero everywhere, except when it is a decay product of Y₁¹(1385), when non-zero values are found for odd values of n. The reactions Λ⁰ + 2π and Λ⁰ + 3π are analysed in terms of the Plahte-Roberts model and good overall agreement is obtained for the various effective mass distributions and the $\text{p}{}_{\mathrm{<mtext>L</mtext>}}{}^1$, p_T and cos θ distributions for the individual particles.     

Representations and inequalities for Ising model Ursell functions

We describe and investigate representations for the Ursell functionu _n of a family ofn random variables {σ _i}. The representations involve independent but identically distributed copies of the family. We apply one of these representations in the case that the random variables are spins of a finite ferromagnetic Ising model with quadratic Hamiltonian to show that (?1) ^n/2+1 u _n(σ ₁, ...,σ _n) ≧ 0 forn=2, 4, and 6 by proving the stronger statement \(( - 1 )^{\frac{n}{2} + 1} \frac{{\partial ^m }}{{\partial J_{i1j1} \cdots \partial J_{imjm} }}Z^{\frac{n}{2}} u_n \left| {_{J = 0} } \right. \geqq {}^\backprime 0\) forn=2, 4, and 6, theJ _ij being coupling constants in the Hamiltonian andZ the partition function. For generaln we combine this result with various reductions to show that sufficiently simple derivatives of (?1) ^n/2+1 Z ^n/2u_n, evaluated at zero coupling, are nonnegative. In particular, we conclude that (?1) ^n/2+1 u _n ≧ 0 if all couplings are nonzero and the inverse temperature β is sufficiently small or sufficiently large, though this result is not uniform in the ordern or the system size. In an appendix we give a simple proof of recent inequalities which boundn-spin expectations by sums of products of simpler expectations.     

Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators

We consider the class of matrix h-pseudodifferential operators Op _h (a) with symbols a = (a _ij ) _i,j=1 ^N , where the coefficients a _ij ∈ C ^∞(? _x ⁿ × ? _ξ ⁿ ? C(0, 1] satisfy the estimates |? _x ^β g6 _ξ ^α α _ij (x, ξ, h)| ? C _αβ 〈ξ〉 ^m and 〈ξ〉 = (1 + |ξ|²)^1/2 for every multi-indices α, β. We also assume that a _ij (x, ξ) is analytically continued with respect to ξ to a tube domain ? ⁿ + i $ \mathcal{B} We consider the class of matrix h-pseudodifferential operators Op _h (a) with symbols a = (a _ij ) _i,j=1^N, where the coefficients a _ij ∈ C ^∞(ℝ _x ⁿ × ℝ _ξ ⁿ ⊗ C(0, 1] satisfy the estimates |ϖ _x ^β g6 _ξ ^α α _ij (x, ξ, h)| ⩽ C _αβ 〈ξ〉 ^m and 〈ξ〉 = (1 + |ξ|²)^1/2 for every multi-indices α, β. We also assume that a _ij (x, ξ) is analytically continued with respect to ξ to a tube domain ℝ ⁿ + i , where is a bounded domain in ℝ ⁿ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let H _h ^s (ℝ ⁿ , ℂ ^N ) be the space of distributions with values in ℂ ^N which is equipped with the norm , let Ω ⊂ ℝ ⁿ be a bounded open set, let v ∈ C ^∞(ℝ ⁿ ), let ▿v(x) ∈ for any x ∈ Ω, and let . Let u _h (∈ H _h ^s (ℝ ⁿ ,‒ ^N )) be a solution of the equation Op _h (α)u = 0. In this case, for every ϕ ∈ C ₀^∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h ₀ > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h ₀]:

((1))

We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schr?dinger, Dirac, and square-root Klein-Gordon operators. To the memory of Professor V. A. Borovikov     

Three-dimensional ordering of impurity-doped TMMC in a magnetic field

The three-dimensional ordering temperatures of the quasi-one-dimensional antiferromagnet (CH₃)₄NMnCl₃ containing 0, 0.13, 0.25 and 0.48 at% Cu²⁺ ions were studied as a function of an external magnetic field up to 60 kOe by means of proton NMR. A strong increase of T_N(H) with H was found even for the impure system. The T_N versus H curve for the pure system cam be explained by introducing a simple form for the effect of the field on the correlation length, ξ(T, H) = ξ(T) × [1+(λJ/kT)H²], while that for the impure system is unlike the one for the pure system.