A phase cell cluster expansion for Euclidean field theories

We adapt the cluster expansion first used to treat infrared problems for lattice models (a mass zero cluster expansion) to the usual field theory situation. The field is expanded in terms of special block spin functions and the cluster expansion given in terms of the expansion coefficients (phase cell variables); the cluster expansion expresses correlation functions in terms of contributions from finite coupled subsets of these variables. Most of the present work is carried through in d space time dimensions (for φ₂⁴ the details of the cluster expansion are pursued and convergence is proven). Thus most of the results in the present work will apply to a treatment of φ₃⁴ to which we hope to return in a succeeding paper. Of particular interest in this paper is a substitute for the stability of the vacuum bound appropriate to this cluster expansion (for d = 2 and d = 3), and a new method for performing estimates with tree graphs. The phase cell cluster expansions have the renormalization group incorporated intimately into their structure. We hope they will be useful ultimately in treating four dimensional field theories.     

FixedJ spectral distribution in large shell model spaces

A method is develloped to exactly calculate the fixedJ configuration centroid energies and widths. The resulting approximate level densities for²⁰⁴Pb and⁰²⁰Pb show large departures with respect to the gaussian approximate level densities. The goodness of the polynomial expansion of fixedJ configuration moments is studied and a simple improvement to this approximation is proposed which gives very good results.     

Some properties of the two-point functions in a dilatationally covariant field theory

The object of our concern are some properties of the two-point functions in a model of dilatationally covariant field theory. We examine the one- and two-dimensional irreducible representations of the dilatation group. For the one-dimensional case we obtain either a massless free field theory or a theory of an interacting field which does not contribute on the mass shell μ=p²=0 and is characterized by a spectral function μ^r₊?1. In the two-dimensional case both fields differ from the free field, their spectral functions ρ_ij(μ) do not vanish identically and are products of two factors, a polynomial of order up to two in ln μ and μ^r₊?-1. The differences between the case of internal symmetry and the case of dilatations are emphasized. The formula for the form factor in the Araki-Haag limit is given.     

A dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos – capturing all scales of random modes on independent grids

In stochastic computations, or uncertainty quantification methods, the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion. The size of this system increases drastically when the number of independent random variables and/or order of polynomial chaos expansions increases. This is invariably the case for large scale simulations and/or problems involving steep gradients and other multiscale features; such features are variously reflected on each solution component or random/uncertainty mode requiring the development of adaptive methods for their accurate resolution. In this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space-refinement on physical space that allows all scales of each solution component to be refined independently of the rest. We exemplify this using the convection–diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method. Thus we establish a new, elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions.     

Perturbation calculations for the eight-vertex model around the Ising point for T → Tc+. II

The dispersion expansion for the two-point electric correlation function in the eight-vertex model is calculated to first order in the four-spin coupling in the scaling limit in the high-temperature regime. It is found that there is a close relation between the T_c⁻ and T_c⁺ spin-spin-energy density correlation function in the Ising model. This is used to simplify tremendously the calculation in the T_c⁺ case; the result can be obtained from the T_c⁻ result by deleting two variables in the expansion.     

On the determination of the proton RMS-radius from electron scattering data

It is shown that the proton rms radius should be determined from fitting a polynomial of second order to the low-q ² form factors. The commonly used polynomial of first order yields radius values which are too small. The proton rms radius has been redetermined from an analysis of the electron scattering data measured at three laboratories. The best fit value is 〈r _E ² 〉^1/2=0.87±0.02 fm.     

Modified variational calculation of the zero field mobility of polar semiconductors

This paper describes a method of calculating the low field mobility if there exists no universal relaxation time. Here, the drift term in the distribution function is eliminated from the equations connecting the first two terms of the Legendre expansion of the distribution function, and the momentum balance is applied to determine the parameter of a one-parametric trial function used for the isotropic part of the distribution function. The mobility is then calculated via the energy balance relation. Forn-type GaAs the zero field lattice mobilities ofμ _[000]=12,100 cm²/Vsec andμ _[000]=160 cm²/ Vsec at 300 °K are obtained in this way.     

A one-dimensional plume plasma expansion: Self-similar approach

The expansion of a plasma induced by laser ablation is investigated using a single-fluid model combined with Saha?s equation. The space coordinates x and time t are combined to a one self-similar variable ξ=x/(ct). To obtain ordinary differential equations, two different transformations for the density are used. The density profiles during the expansion are found to be completely different, one corresponds to the common results i.e., the density decreases monotonically with ξ, while with the second transformation, the profile shows a density increasing for certain interval of the self-similar variable. This effect is enhanced with higher initial ionization fraction. The role of the initial velocity which corresponds to the start of the expansion from an unperturbed plasma or from an expansion already going on is pointed out.     

A convergent expansion about mean field theory: I. The expansion

We give a convergent expansion for nearly Gaussian quantum field theory in the multiphase region. The expansion combines (1) an expansion in phase boundaries, (2) a cluster expansion, and (3) a perturbation expansion to isolate dominant behavior. We study in detail the ground state of the $\text{P}$(φ)₂ = (λφ⁴ ? φ² ? μφ)₂ model, with ∥ μ ∥ ? λ² ? 1. The ground state is close to the classical free field, obtained by replacing $\text{P}$(φ) by the quadratic mean field polynomial $\text{P}$_c(φ), tangent to $\text{P}$ at a global minimum. Selecting one minimum gives a pure phase (ergodic ground state) satisfying the Wightman-Osterwalder-Schrader axioms with a positive mass. We also establish analyticity in λ for μ = 0 in the sector ∥ Im λ ∥ < ? Re λ ? 1, for ? ? 1.     

Exact and approximate solutions to Schrödinger’s equation with decatic potentials

The one-dimensional Schrödinger’s equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential’s parameters, we show that the decatic polynomial potential V (x) = ax ¹⁰ + bx ⁸ + cx ⁶ + dx ⁴ + ex ², a > 0 is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for corresponding energy-dependent polynomial solutions are given in detail. It is also shown that these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy eigenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schrödinger equation. Further analysis for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.     

Resonance production in the γγ→ππ process and low energy parameters

The amplitudes of the γγ→ππ reaction are obtained describing the process up to the energy of 1.5 GeV. It is shown that in thef-meson region there exists large background with the isospinI=2. This background and the backgrounds of the known mechanisms deal with the threshold expansion of the amplitude. This allows one to give polarizability estimate of the pion (α +β) ^π+ =0.45·10^?+fm, (α+β) ^π0. We predict a very small cross-section for the production of a broad ε (700) by using the low energy normalization.     

Calibration of CR39 detectors with new system for Fe26+ ion beam and measurement of total charge changing cross-section in Al target

In the present study, CR39 track etch detector was calibrated with a new system and the total charge changing cross-section of 300 A MeV Fe²⁶⁺ ion beam in aluminum target was measured. The CR39 nuclear track detectors were used to identify the incident charged particles and their fragments using an optical microscope DM6000 M and automated image analyzer system installed with Leica QWin Plus software. The CR39 detectors before and after the target were calibrated and found to have the same charge response; the charge resolution in both of the detectors were 0.19e and 0.20e, respectively. The calibration points were fitted with a polynomial of degree one and all the points are within the limits of the experimental errors. The response functions were also obtained and fitted with a polynomial of degree three which are quite good throughout Z/β = 4.6 to 41.4. The value of the total charge changing cross-section is σ_tot = (1663 ± 236) mb. The total charge changing cross-section was compared with the experimental results of others and also fitted by the Bradt-Peters geometrical cross-section.     

The infrared spectrum of the CO dimer

The infrared spectrum of (¹²C¹⁸O)₂ has been studied for the first time using a tunable diode laser spectrometer in the 2095 cm⁻¹ region to probe a pulsed supersonic jet expansion. Very dilute gas mixtures of CO in He were used, resulting in small consumption of ¹²C¹⁸O sample gas, as well as cold and simple spectra. The results were analyzed using a term value scheme to obtain model-independent energies for 7 rotational levels belonging to 2 stacks in the lower state, v_CO=0, and 22 levels belonging to 7 stacks in the upper state, v_CO=1. The two ground state isomers of the CO dimer were found to be separated by only 0.639 cm⁻¹ for (¹²C¹⁸O)₂. These results provide a foundation for future studies of the millimeter wave spectrum.     

Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree − 3

In this paper, we study the polynomial integrability of natural Hamiltonian systems with two degrees of freedom having a homogeneous potential of degree k given either by a polynomial, or by an inverse of a polynomial. For k=−2,−1,…,3,4, their polynomial integrability has been characterized. Here, we have two main results. First, we characterize the polynomial integrability of those Hamiltonian systems with homogeneous potential of degree −3. Second, we extend a relation between the nontrivial eigenvalues of the Hessian of the potential calculated at a Darboux point to a family of Hamiltonian systems with potentials given by an inverse of a homogeneous polynomial. This relation was known for such Hamiltonian systems with homogeneous polynomial potentials. Finally, we present three open problems related with the polynomial integrability of Hamiltonian systems with a rational potential.     

Fat Fisher zeroes

We show that it is possible to determine the locus of Fisher zeroes in the thermodynamic limit for the Ising model on planar (“fat”) φ⁴ random graphs and their dual quadrangulations by matching up the real part of the high and low temperature branches of the expression for the free energy. The form of this expression for the free energy also means that series expansion results for the zeroes may be obtained with rather less effort than might appear necessary at first sight by simply reverting the series expansion of a function g(z) which appears in the solution and taking a logarithm.Unlike regular 2D lattices where numerous unphysical critical points exist with non-standard exponents, the Ising model on planar φ⁴ graphs displays only the physical transition at c=exp(−2β)=1/4 and a mirror transition at c=−1/4 both with KPZ/DDK exponents (α=−1, β=1/2, γ=2). The relation between the φ⁴ locus and that of the dual quadrangulations is akin to that between the (regular) triangular and honeycomb lattices since there is no self-duality.     

Studies of the zero-field splitting and g-factor for the defect model of Cr in α-LiIO3 crystals at low temperature

The EPR zero-field splitting parameters D and g-factors for Cr³⁺ in α-LiIO₃ single crystal, taking into account both the effect of lattice distortion and two Li⁺ vacancies, have been investigated using a complete diagonalization method (CDM) for 3d³ ions in a trigonal symmetry crystal field. The theoretical results (D=−0.60876 cm⁻¹, g_∥=1.9641, g_⊥=1.9682) are in excellent agreement with experimental results (D=−0.6099(3) cm⁻¹g_∥=1.965±0.001, g_⊥=1.971±0.002). In addition, Macfarlane's perturbation expressions lead to results almost identical with the CDM for Cr³⁺ in an α-LiIO₃ single crystal.     

Structure and high pressure behaviour of organic crystals based on aromatically substituted 1,3,4-oxadiazoles

The crystalline structure of a new compound containing the 1,3,4-oxadiazole moiety, 4-(5-methyl-1,3,4-oxadiazole-2yl-)-N,N′-dimethyl-phenylamine (MODPA) was determined. It shows a monoclinic structure with space group P2₁/c and lattice parameters: a=1.02997(6), b=0.64840(4), c=1.58117(10) nm and β=99.4820(10)°. To study the intermolecular interactions in oxadiazole containing organic crystals, X-ray studies on MODPA and 2,5-diphenyl-1,3,4-oxadiazole (DPO) were performed up to 5 GPa at room temperature. The Murnaghan equation of state is used to describe the compression behaviour of both substances. From these results, the bulk modulus and its pressure derivative were determined. The values obtained are: K₀=6.3 GPa and K′₀=6.8 for MODPA and K₀=7.3 GPa and K′₀=6.7 for DPO. Additionally, measurements under increasing temperature at ambient pressure were carried out to evaluate the thermal expansion coefficient: α=1.8×10⁻⁴ K⁻¹ for MODPA and α=1.9×10⁻⁴ K⁻¹ for DPO.     

Nilpotent normal form for divergence-free vector fields and volume-preserving maps

We study the normal forms for incompressible flows and maps in the neighborhood of an equilibrium or fixed point with a triple eigenvalue. We prove that when a divergence-free vector field in R³ has nilpotent linearization with maximal Jordan block then, to arbitrary degree, coordinates can be chosen so that the nonlinear terms occur as a single function of two variables in the third component. The analogue for volume-preserving diffeomorphisms gives an optimal normal form in which the truncation of the normal form at any degree gives an exactly volume-preserving map whose inverse is also polynomial with the same degree.     

Separable expansion of the t-matrix in the 3S1-3D1 channel

A recently proposed separable expansion for the t-matrix for local potentials is extended to the coupled channel of nucleon-nucleon interactions. The simple method yields a rather convergent separable expansion, with simple form factors, and the requirements of two particle unitarity and time reversal symmetry are maintained for approximations of any rank. The method is illustrated for the simplified Reid soft-core potential in the ³S₁-³D₁ channel, and the results compared with a recent calculation of Pieper.