Cosmological General Relativity with Scale Factor and Dark Energy

In this paper the four-dimensional (4-D) space-velocity Cosmological General Relativity of Carmeli is developed by a general solution of the Einstein field equations. The Tolman metric is applied in the form 1 $$ ds^2 = g_{\mu \nu} dx^{\mu} dx^{\nu} = \tau^2 dv^2 -e^{\mu} dr^2 - R^2 \left(d{\theta}^2 + \mbox{sin}^2{\theta} d{\phi}^2 \right), $$ where g _μν is the metric tensor. We use comoving coordinates x ^α = (x ⁰, x ¹, x ², x ³) = (τv, r, θ, ?), where τ is the Hubble-Carmeli time constant, v is the universe expansion velocity and r, θ and ? are the spatial coordinates. We assume that μ and R are each functions of the coordinates τv and r. The vacuum mass density ρ _Λ is defined in terms of a cosmological constant Λ, 2 $$ \rho_{\Lambda} \equiv -\frac{ \Lambda } { \kappa \tau^2 }, $$ where the Carmeli gravitational coupling constant κ = 8πG/c ² τ ², where c is the speed of light in vacuum. This allows the definitions of the effective mass density 3 $$ \rho_{eff} \equiv \rho + \rho_{\Lambda} $$ and effective pressure 4 $$ p_{eff} \equiv p - c \tau \rho_{\Lambda}, $$ where ρ is the mass density and p is the pressure. Then the energy-momentum tensor 5 $$ T_{\mu \nu} = \tau^2 \left[ \left(\rho_{eff} + \frac{p_{eff}} {c \tau} \right) u_{\mu} u_{\nu} - \frac{p_{eff}} {c \tau} g_{\mu \nu} \right], $$ where u _μ = (1,0,0,0) is the 4-velocity. The Einstein field equations are taken in the form 6 $$ R_{\mu \nu} = \kappa \left(T_{\mu \nu} - \frac{1} {2} g_{\mu \nu} T \right), $$ where R _μν is the Ricci tensor, κ = 8πG/c ² τ ² is Carmeli’s gravitation constant, where G is Newton’s constant and the trace T = g ^αβ T _αβ . By solving the field equations (6) a space-velocity cosmology is obtained analogous to the Friedmann-Lemaître-Robertson-Walker space-time cosmology. We choose an equation of state such that 7 $$ p = w_e c \tau \rho, $$ with an evolving state parameter 8 $$ w_e \left(R_v \right) = w_0 + \left(1 - R_v \right) w_a, $$ where R _v = R _v (v) is the scale factor and w ₀ and w _a are constants. Carmeli’s 4-D space-velocity cosmology is derived as a special case.     

An inequality onS wave bound states,with correct coupling constant dependence

We prove that the number ofS wave bound states in a spherically symmetric potentialgV(r) is less than 1 $$g^{1/2} \left[ {\int\limits_0^\infty {r^2 V^ - (r)dr} \int\limits_0^\infty {V^ - (r)dr} } \right]^{1/4}$$ whereV ^? is the attractive part of the potential, in units where ?²/2M=1.     

Boundary regularity for some nonlinear elliptic degenerate equations

We consider the nonlinear elliptic degenerate equation (1) $$ - x^2 \left( {\frac{{\partial ^2 u}}{{\partial x^2 }} + \frac{{\partial ^2 u}}{{\partial y^2 }}} \right) + 2u = f(u)in\Omega _a ,$$ where $$\Omega _a = \left\{ {(x,y) \in \mathbb{R}^2 ,0< x< a,\left| y \right|< a} \right\}$$ for some constanta>0 andf is aC ^∞ functions on ? such thatf(0)=f′(0)=0. Our main result asserts that: ifu∈C \((\bar \Omega _a )\) satisfies (2) $$u(0,y) = 0for\left| y \right|< a,$$ thenx ^?2 u(x,y)∈C ^∞ \(\left( {\bar \Omega _{a/2} } \right)\) and in particularu∈C ^∞ \(\left( {\bar \Omega _{a/2} } \right)\) .     

Etude spectrale d'une famille d'opérateurs non-symétriques intervenant dans la théorie des champs de reggeons

In this paper, we study a few spectral properties of a non-symmetrical operator arising in the Gribov theory. The first and second section are devoted to Bargmann's representation and the study of general spectral properties of the operator: $$\begin{gathered} H_{\lambda ',\mu ,\lambda ,\alpha } = \lambda '\sum\limits_{j = 1}^N {A_j^{ * 2} A_j^2 + \mu \sum\limits_{j = 1}^N {A_j^ * A_j + i\lambda \sum\limits_{j = 1}^N {A_j^ * (A_j + A_j^ * )A_j } } } \hfill \\ + \alpha \sum\limits_{j = 1}^{N - 1} {(A_{j + 1}^ * A_j + A_j^ * A_{j + 1} ),} \hfill \\ \end{gathered}$$ whereA* _j andA _j ,j∈[1,N] are the creation and annihilation operators. In the third section, we restrict our study to the case of nul transverse dimension (N=1). Following the study done in [1], we consider the operator: $$H_{\lambda ',\mu ,\lambda } = \lambda 'A^{ * 2} A^2 + \mu A^ * A + i\lambda A^ * (A + A^ * )A,$$ whereA* andA are the creation and annihilation operators. For λ′>0 and λ′²≦μλ′+λ². We prove that the solutions of the equationu′(t)+H _{λ′, μ,λ} u(t)=0 are expandable in series of the eigenvectors ofH _λ′,μ,λ fort>0. In the last section, we show that the smallest eigenvalue σ(α) of the operatorH _{λ′,μ,λ,α} is analytic in α, and thus admits an expansion: σ(α)=σ₀+ασ₁+α²σ₂+..., where σ₀ is the smallest eigenvalue of the operatorH _{λ′,μ,λ,0}.     

A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description

We consider the canonical Gibbs measure associated to aN-vortex system in a bounded domain Λ, at inverse temperature \(\widetilde\beta \) and prove that, in the limitN→∞, \(\widetilde\beta \) /N→β, αN→1, where β∈(?8π, + ∞) (here α denotes the vorticity intensity of each vortex), the one particle distribution function ?^N = ? ^N x,x∈Λ converges to a superposition of solutions ? _α of the following Mean Field Equation: $$\left\{ {\begin{array}{*{20}c} {\varrho _{\beta (x) = } \frac{{e^{ - \beta \psi } }}{{\mathop \smallint \limits_\Lambda e^{ - \beta \psi } }}; - \Delta \psi = \varrho _\beta in\Lambda } \\ {\psi |_{\partial \Lambda } = 0.} \\ \end{array} } \right.$$ Moreover, we study the variational principles associated to Eq. (A.1) and prove thai, when β→?8π⁺, either ?_β → δ _{x 0} (weakly in the sense of measures) wherex ₀ denotes and equilibrium point of a single point vortex in Λ, or ?_β converges to a smooth solution of (A.1) for β=?8π. Examples of both possibilities are given, although we are not able to solve the alternative for a given Λ. Finally, we discuss a possible connection of the present analysis with the 2-D turbulence.     

Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity

We consider the nonlinear Schrödinger equation: (1) $${{i\partial u} \mathord{\left/ {\vphantom {{i\partial u} {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}} = - \Delta u - \left| u \right|^{{4 \mathord{\left/ {\vphantom {4 N}} \right. \kern-\nulldelimiterspace} N}} uandu\left( {0,.} \right) = \varphi \left( . \right),$$ whereu:[0,T)×? ^N →?. For any given pointsx ₁,x ₂,...,x _k in ? ^N , we construct a solution of Eq. (1),u(t), which blows up in a finite timeT at exactlyx ₁,x ₂,...,x _k . In addition, we describe the precise behavior of the solutionu(t) whent→T, at the blow-up points {x ₁,x ₂,...,x _k } and in ? ^N ?{x ₁,x ₂,...,x _k }.     

The static multipole moments of Δ1236(3/2,3/2) from superconvergence inNγ → Δπ scattering

Saturating superconvergence sum rules inNγ→Δπ scattering byN andΔ, we are able to relate the (isoscalar) dipole magnetic moment \(\tilde \mu _\Delta\) and the quadrupole electric moment \(\tilde Q_\Delta\) of the isobarΔ to the electric charge \(\tilde Z_\Delta\) and the dipole magnetic momentμ _N of the nucleonN. The numerical results are: \(\tilde \mu _\Delta \equiv \mu _{\Delta ^ + } + \mu _{\Delta ^0 } = 3.26\) (in unitse/2M)=2.48 (in unitse/2m), and \(\tilde Q_\Delta \equiv Q_{\Delta ^ + } + Q_{\Delta ^0 } = 0.050\) (in unitse/M ²)=0.029 (in unitse/m ²), whereM(m) is the mass ofΔ(N). Neglecting the pion mass and takingM=m,μ _n /μ _p =?2/3, we get theSU ₆ result μ_Δ+=μ _p .     

Uniform boundedness of conditional gauge and Schrödinger equations

We prove that for a bounded domainD ?R ⁿ withC ² boundary and \(q \in K_n^{loc} (n \geqq 3) if E^x \exp \int\limits_0^{\tau _D } {q(x_t )dt} \mathop \ddag \limits_--- \infty \) inD, then $$\mathop {\sup }\limits_{\mathop {x \in D}\limits_{z \in \partial D} } E_z^x \exp \int\limits_0^{\tau _D } {q(x_t )dt}< + \infty $$ ({x _t : Brownian motion}) The important corollary of this result is that if the Schrödinger equation Δ/2u+qu=0 has a strictly positive solution onD, then for anyD ₀ ? ?D, there exists a constantC=C(n,q,D,D ₀) such that for anyf εL ¹(?D, σ), (σ: area measure on ?D) we have $$\mathop {\sup |}\limits_{x \in D_0 } u_f (x)| \mathop< \limits_ = C\int\limits_{\partial D} {|f(y)|\sigma (dy)} $$ whereu _f is the solution of the Schrödinger equation corresponding to the boundary valuef. To prove the main result we set up the following estimate inequalities on the Poisson kernelK(x,z) corresponding to the Laplace operator: $$C_1 \frac{{d(x,\partial D)}}{{|x - z|^n }}\mathop< \limits_ = K(x,z)\mathop< \limits_ = C_2 \frac{{d(x,\partial D)}}{{|x - z|^n }},x \in D,z \in \partial D$$ whereC ₁ andC ₂ are constants depending onn andD.     

Light quark masses in quantum chromodynamics and chiral symmetry breaking

We derive model independent lower bounds for the sums of effective quark masses \(\bar m_u + \bar m_d \) and \(\bar m_u + \bar m_s \) . The bounds follow from the combination of the spectral representation properties of the hadronic axial currents two-point functions and their behavior in the deep euclidean region (known from a perturbative QCD calculation to two loops and the leading non-perturbative contribution). The bounds incorporate PCAC in the Nambu-Goldstone version. If we define the invariant masses \(\hat m\) by $$\bar m_i = \hat m_i \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^{{{\gamma _1 } \mathord{\left/ {\vphantom {{\gamma _1 } {\beta _1 }}} \right. \kern-\nulldelimiterspace} {\beta _1 }}} $$ and <F ²> is the vacuum expectation value of $$F^2 = \Sigma _a F_{(a)}^{\mu v} F_{\mu v(a)} $$ , we find, e.g., $$\hat m_u + \hat m_d \geqq \sqrt {\frac{{2\pi }}{3} \cdot \frac{{8f_\pi m_\pi ^2 }}{{3\left\langle {\alpha _s F^2 } \right\rangle ^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} $$ ; with the value <α _u F ²?0.04GeV⁴, recently suggested by various analysis, this gives $$\hat m_u + \hat m_d \geqq 35MeV$$ . The corresponding bounds on \(\bar m_u + \bar m_s \) are obtained replacingm _π ² f _π bym _K ² f _K . The PCAC relation can be inverted, and we get upper bounds on the spontaneous masses, \(\hat \mu \) : $$\hat \mu \leqq 170MeV$$ where \(\hat \mu \) is defined by $$\left\langle {\bar \psi \psi } \right\rangle \left( {Q^2 } \right) = \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^d \hat \mu ^3 ,d = {{12} \mathord{\left/ {\vphantom {{12} {\left( {33 - 2n_f } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {33 - 2n_f } \right)}}$$ .     

Magnetic properties and peculiarity of magnetic states in dilute antiferromagnets Mn1−x Zn x F2

The dependence of magnetic moment and susceptibility on temperature, magnetic field and frequency of some single crystals Mn_1?x Zn _x F₂ (x≈x _e=0.75—percolation limit) were experimentally investigated. Our experiments show that (Bazhan and Petrov 1984; Cowleyet al 1984; Villain 1984) in these crystals the nonequilibrium magnetic state of spinglass type with finite correlation length appears as temperature decreasesT<T in weak magnetic fields. This state is determined by fluctuation magnetic moments √nμ (wheren is the number of magnetic ions, corresponding to finite correlation length andμ the magnetic moment Mn⁺¹). In the experiments in low magnetic fields and frequencies there are no peculiarities in the magnetic susceptibility temperature dependence atT≠T _f. At temperaturesT>T _f andT<T _f magnetic susceptibility is determined by 1 $$\chi \left( {T > T_f } \right) = \frac{{N\left\langle \mu \right\rangle ^2 }}{{3k\left( {T + \theta } \right)}} = \frac{N}{n}\frac{{\left\langle {\sqrt n \mu } \right\rangle ^2 }}{{3k\left( {T + \theta } \right)}} = \chi \left( {T< T_f } \right)$$ . In strong magnetic fields and large frequencies there are peculiarities in thex(T) dependence atT=T _f. AtT<T _f and strong magnetic fieldsX(T)=x ₀ andT<T _f and at large frequenciesx(T)=x ₀+α/T. The dependences of magnetic susceptibility on the frequency are determined by the magnetic system relaxation. Calculations and comparison with experiments show that the relaxation of the investigated magnetic systems atT<T _f follows the relaxation lawM(t)=M(0) exp[?(t/τ) ^r ], suggested in Palmeret al (1984) for spin-glasses relaxation taking into account the time relaxation distributionτ ₀....τ _max in the system and its ‘hierarchically’ dynamics.     

Spin waves in paramagnetic Boltzmann gases

As the temperature is lowered we get an interesting temperature region? _d?T?? ²/mr ₀ ² (where? _d is the quantum degeneracy temperature,m the mass of a gas molecule,r ₀ the radius of interparticle interaction) in which the thermal de Broglie wavelength Λ of a particle is considerably greater than its sizer ₀ though Λ turns out to be less than the mean interparticle distanceN ^?1/3?Λ?r ₀. Although the gas molecules obey the classical Boltzmann-Maxwell statistics the system as a whole begins to exhibit a larger number of essentially quantum macroscopic collective features. One of the most interesting and dramatic features is the possibility of propagation of weakly damped spin oscillations in spin-polarized gases (external magnetic field, optical pumping). Such oscillations can propagate both in the low-frequencyθτ?1 regime and the high frequencyθτ?1. The last case is highly non-trivial for a Boltzmann gas with a short range interaction between particles. The weakness of relaxation damping of spin modes implies that the degree of polarization is high enough 1>/|α|?|a|/Λ, whereα=(N ₊?N _?)N,a is the two-particles-wave scattering length. Under these conditions the spectrum of magnons has the form (Bashkin 1981, 1984; Lhuillier and Laloe 1982) 1 $$\omega = \Omega _H + \left( {{{K^2 \nu _{\rm T}^2 } \mathord{\left/ {\vphantom {{K^2 \nu _{\rm T}^2 } {\Omega _{int} }}} \right. \kern-\nulldelimiterspace} {\Omega _{int} }}} \right)\left( {{{1 - i} \mathord{\left/ {\vphantom {{1 - i} {\Omega _{int} }}} \right. \kern-\nulldelimiterspace} {\Omega _{int} }}\tau } \right), \Omega _{int} = {{ - 4\pi ahN\alpha } \mathord{\left/ {\vphantom {{ - 4\pi ahN\alpha } m}} \right. \kern-\nulldelimiterspace} m}, \nu _{\rm T}^2 = {T \mathord{\left/ {\vphantom {T m}} \right. \kern-\nulldelimiterspace} m}$$ where Ω _H is the Larmor precession frequency for spins in the magnetic fieldH. Collisionless Landau damping restricts the region of applicability of (1) with not too large wave vectorsKv _T?|Ω_int|. The existence of collective spin waves has been experimentally confirmed in NMR-experiments with gaseous atomic hydrogen H↑ (Johnsonet al 1984). The presence of undamped spin oscillations means automatically the existence of long range correlations for transverse magnetization. Such correlations decrease with the distance according to the power law 2 $$\delta _{ik} \left( r \right) = 2\left| a \right|\frac{{\left( {\beta N\alpha } \right)^2 }}{\gamma }\delta _{ik} $$ . Hereβ is the molecule magnetic moment. Spin waves being even damped can nevertheless reveal themselves atT?? ²/mr ₀ ² or when |α|?r ₀/Λ. The first experimental discovery or damped spin waves in gaseous³He↑ has been done in Nacheret al 1984. Oscillations of magnetization can also propagate in some condensed media such as liquid³He-⁴He solutions, semimagnetic semiconductors etc.     

Universal estimate of the gap for the Kac operator in the convex case

The aim of this paper is to prove that ifV is a strictly convex potential with quadratic behavior at ∞, then the quotient μ₂/μ₁ between the largest eigenvalue and the second eigenvalue of the Kac operator defined on L²(? ^m ) by exp ?V(x)/2 · exp Δ_x · exp ?V(x)/2 where Δ_x is the Laplacian on ? ^m satisfies the condition: $${{\mu _2 } \mathord{\left/ {\vphantom {{\mu _2 } {\mu _1 {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} \mathord{\left/ {\vphantom {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} {2,}}} \right. \kern-\nulldelimiterspace} {2,}}}}} \right. \kern-\nulldelimiterspace} {\mu _1 {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} \mathord{\left/ {\vphantom {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} {2,}}} \right. \kern-\nulldelimiterspace} {2,}}}}$$ where σ is such that HessV(x)≥σ>0.     

Construction of Euclidean (QED)2 via lattice gauge theory

Let ν=det_ren(1+K _g ) be the renormalized Matthews-Salam determinant of (QED)₂, where \(K_g = ieA_{g,} S = \left( {\sum {\gamma _\mu \partial } _\mu + m} \right)^{ - 1} \) is euclidean fermion propagator of one of the following boundary conditions: (1) free, (2) periodic at ?Λ, Λ=[?L/2;L/2]², (3) anti-periodic at ?Λ, and \(A_g (x) = (\sum \gamma _\mu A_\mu (x))g(x)\) . Hereg(x)=1 ifxεΛ₀=[?r/2,r/2]² с Λ and 0 otherwise. Then we show

1. νεL ^p (dμ(A)), p>0. Further we prove a new determinant inequality which holds for the QED, QCD-type models containing fermions. This enables us to prove:
2. Z(Λ₀)=∫νdμ(A)≦exp[c|Λ₀|]. Similar volume dependence is shown for the Schwinger functions.

     

The ground state energy of a Bose gas with Coulomb interaction

LetH _N be the 2N particle Hamiltonian $$\begin{array}{*{20}c} {H_N = \sum\limits_{i = 1}^{2N} {( - \Delta _\iota ) + \sum\limits_{i< j = 1}^N {\left| {x_i - x_j } \right|^{ - 1} + } \sum\limits_{i< j = 1}^N {\left| {x_{i + N} - x_{j + N} } \right|^{ - 1} } } } \\ { - \sum\limits_{i,j< j = 1}^N {\left| {x_i - x_{j + N} } \right|^{ - 1} ,} } \\ \end{array} $$ whereΔ _i is the Laplacian in the variablex _i ∈?³, 1≦i≦2N. The operatorH _N is assumed to act on wave functionsΨ(x ₁, ...,x _N ;x _N+1, ...,x _2N ) which are symmetric in the variables (x ₁, ...,x _N ) and (x _N+1, ...,x _2N ). SupposeΨ is supported in a setΛ ^2N , whereΛ is a cube in ?³. It is shown that if a normalized wave functionΨ can be written as a product of two wave functions $$\psi (x_1 ,...,x_N ;x_{N + 1} ,...,x_{2N} ) = \psi _1 (x_2 ,...,x_N )\psi _2 (x_{N + 1} ,...,x_{2N} ),$$ and the density of particles inΛ is constant, then 〈Ψ|H _N |Ψ〉≧?CN ^7/5 for some universal constantC.     

Radiative Σ± decays and search for neutral currents

The decay modesΣ ^±→nπ ^± γ, Σ ⁺→pγ,Σ ⁺ →pe ⁺ e ^}- were studied in the 81 cm Saclay hydrogen bubble chamber. In the radiative decayΣ ^±→nπ ^± γ only low momentum pions which stop in the chamber were accepted. We obtain the following branching ratios: (1) $$\frac{{\Gamma {\text{(}}\sum ^{\text{ + }} \to n\pi ^ + \gamma , p_{\pi + }^*< 110{\text{ MeV/c)}}}}{{\Gamma {\text{(}}\sum ^{\text{ + }} \to n\pi ^ + )}} = (2.7 \pm 0.5) \times 10^{ - 4} ,$$ (2) $$\frac{{\Gamma {\text{(}}\sum ^ - \to n\pi ^ - \gamma , p_{\pi - }^*< 110{\text{ MeV/c)}}}}{{\Gamma {\text{(}}\sum ^ - \to n\pi ^ - )}} = (1.0 \pm 0.2) \times 10^{ - 4} ,$$ (3) $$\frac{{\Gamma {\text{(}}\sum ^ + \to p\gamma {\text{)}}}}{{\Gamma {\text{(}}\sum ^ + \to p\pi ^0 )}} = (2.1 \pm 0.3) \times 10^{ - 3} ,$$ (4) $$\frac{{\Gamma {\text{(}}\sum ^ + \to pe^ + e^ - {\text{)}}}}{{\Gamma {\text{(}}\sum ^ + \to p\pi ^0 )}} = (1.5 \pm 0.9) \times 10^{ - 5} .$$ The radiative branching ratios (1) and (2) agree well with theoretical calculations and confirm very strongly the assignmentS wave toΣ ^? →nπ ^? andP wave toΣ ⁺→nπ ⁺ decay. The branching ratio (4) is based on 3 events with very low invariant masses of the electron-positron pair, being most probably radiative decays with internal conversion of theγ-ray. Combining (3) and (4) we obtain for the conversion coefficientρ: in agreement with predictions from electrodynamics.     

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}.     

Relative x-ray transition probabilities to theK-shell forZ=81, 92, 94, and 96

Accurate intensity measurements of the majorK x-ray groups have been performed with high resolution Ge(Li) detectors in singles and coincidence arrangements and with a high-purity Ge detector of the intrinsic type. Previously reportedK x-ray intensities forZ=96 are in error due to the presence of a 121.5 keV γ-ray in the decay of²⁴⁹Cf. The present results are as follows: forZ=81,K _α2/K _α1=0.589±0.008, \(K_{\beta _1^\prime } /K_{\alpha ^1 } = 0.344 \pm 0.008, K_{\beta _2^\prime } /K_{\alpha _1 } = 0.102 \pm 0.004\) , andK _β/K _α=0.281±0.006; forZ=92 \(K_{\alpha _2 } /K_{\alpha _1 } = 0.611 \pm 0.008,K_{\beta _1^\prime } /K_{\alpha _1 } = 0.365 \pm 0.008, K_{\beta _2^\prime } /K_{\alpha _1 } = 0.125 \pm 0.004\) , andK _β/K _α=0.300±0.006; forZ=94, \(K_{\alpha _2 } /K_{\alpha _1 } = 0.610 \pm 0.008, K_{\beta _1^\prime } /K_{\alpha _1 } = 0.369 \pm 0.010, K_{\beta _2^\prime } /K_{\alpha _1 } = 0.127 \pm 0.004\) , andK _β/K _α=0.308±0.008; and forZ=96, \(K_{\alpha _2 } /K_{\alpha _1 } = 0.627 \pm 0.008, K_{\beta _1^\prime } /K_{\alpha _1 } = 0.372 \pm 0.009, K_{\beta _2^\prime } /K_{\alpha _1 } = 0.133 \pm 0.005\) , andK _β/K _α=0.310±0.008. The error limits are the 2σ statistical errors to which a systematic error in the detector efficiencies has been added linearly. The present results are compared with recent theoretical calculations.     

An analytical expression for the specific capacity of lithium ion cells

The concentration of lithium ions in the cathode of lithium ion cells has been obtained by solving the materials balance equation $$\frac{{\partial c}}{{\partial t}} = \varepsilon ^{1/2} D\frac{{\partial ^2 c}}{{\partial x^2 }} + \frac{{aj_n (1--t_ + )}}{\varepsilon }$$ by Laplace transform. On the assumption that the cell is fully discharged when there are zero lithium ions at the current collector of the cathode, the discharge timet _d is obtained as $$\tau = \frac{{r^2 }}{{\pi ^2 \varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{{r^2 }}\left( {\frac{{\varepsilon ^{1/2} }}{J} + \frac{{r^2 }}{6}} \right)} \right]$$ which, when substituted into the equationC=It _d /M, whereI is the discharge current andM is the mass of the separator and positive electrode, an analytical expression for the specific capacity of the lithium cell is given as $$C = \frac{{IL_c ^2 }}{{\pi {\rm M}D\varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{2}\left( {\frac{{FDc_0 \varepsilon ^{3/2} }}{{I(1 - t_ + )L_c }} + \frac{1}{6}} \right)} \right]$$      

Eine scheinbare Abschwächung der Lokalitätsbedingung. II

If for a relativistic field theory the expectation values of the commutator (Ω|[A (x),A(y)]|Ω) vanish in space-like direction like exp {? const|(x-y ²|^α/2#x007D; with α>1 for sufficiently many vectors Ω, it follows thatA(x) is a local field. Or more precisely: For a hermitean, scalar, tempered fieldA(x) the locality axiom can be replaced by the following conditions 1. For any natural numbern there exist a) a configurationX(n): $$X_1 ,...,X_{n - 1} X_1^i = \cdot \cdot \cdot = X_{n - 1}^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^1 - X_{i + 1}^1 )} \right]^2 + \left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^2 - X_{i + 1}^2 )} \right]^2 > 0\) for all λ _i ≧0i=1,...,n?2, \(\sum\limits_{i = 1}^{n - 2} {\lambda _i > 0} \) , b) neighbourhoods of theX _i 's:U _i (X _i )?R ⁴ i=1,...,n?1 (in the euclidean topology ofR ⁴) and c) a real number α>1 such that for all points (x):x ₁, ...,x _n?1:x _i ∈U _i (X _r ) there are positive constantsC ⁽ⁿ⁾{(x)},h ⁽ⁿ⁾{(x)} with: $$\left| {\left\langle {\left[ {A(x_1 )...A(x_{n - 1} ),A(x_n )} \right]} \right\rangle } \right|< C^{(n)} \left\{ {(x)} \right\}\exp \left\{ { - h^{(n)} \left\{ {(x)} \right\}r^\alpha } \right\}forx_n = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ r \\ \end{array} } \right),r > 1.$$ 2. For any natural numbern there exist a) a configurationY(n): $$Y_2 ,Y_3 ,...,Y_n Y_3^i = \cdot \cdot \cdot = Y_n^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^1 - Y_{i{\text{ + 1}}}^{\text{1}} } )} \right]^2 + \left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^2 - Y_{i{\text{ + 1}}}^{\text{2}} } )} \right]^2 > 0\) for all μ _i ≧0,i=3, ...,n?1, \(\sum\limits_{i = 3}^{n - 1} {\mu _i > 0} \) , b) neighbourhoods of theY _i 's:V _i(Y _i )?R ⁴ i=2, ...,n (in the euclidean topology ofR ⁴) and c) a real number β>1 such that for all points (y):y ₂, ...,y _n y _i ∈V _i (Y _i there are positive constantsC _(n){(y)},h _(n){(y)} and a real number γ_(n){(y)∈a closed subset ofR?{0}?{1} with: γ_(n){(y)}\y ₂,y ₃, ...,y _n totally space-like in the order 2, 3, ...,n and $$\left| {\left\langle {\left[ {A(x_1 ),A(x_2 )} \right]A(y_3 )...A(y_n )} \right\rangle } \right|< C_{(n)} \left\{ {(y)} \right\}\exp \left\{ { - h_{(n)} \left\{ {(y)} \right\}r^\beta } \right\}$$ for \(x_1 = \gamma _{(n)} \left\{ {(y)} \right\}r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right),x_2 = y_2 - [1 - \gamma _{(n)} \{ (y)\} ]r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right)\) and for sufficiently large values ofr.