Moments Of Arbitrary Order Research Articles

Periodic orbit theory of strongly anomalous transport

We establish a deterministic technique to investigate transport moments of arbitrary order. The theory is applied to the analysis of different kinds of intermittent one-dimensional maps and the Lorentz gas with infinite horizon: the typical appearance of phase transitions in the spectrum of transport exponents is explained.

Correlation-based phase space beam characterization.

A generalized correlation-based definition for moments of arbitrary order is introduced that can also accommodate mixed spatial and angular moment. Moreover, a transformation law forthese moments for propagation through linear optical systems is derived. This law has the same form as the corresponding propagation law of the moments defined in terms of the Wigner distribution function. The correlation-based moments can be used to fully characterize beams of arbitrary states of coherence.

Wigner distribution moments in fractional Fourier transform systems.

It is shown how all global Wigner distribution moments of arbitrary order in the output plane of a (generally anamorphic) two-dimensional fractional Fourier transform system can be expressed in terms of the moments in the input plane. Since Wigner distribution moments are identical to derivatives of the ambiguity function at the origin, a similar relation holds for these derivatives. The general input-output relationship is then broken down into a number of rotation-type input-output relationships between certain combinations of moments. It is shown how the Wigner distribution moments (or ambiguity function derivatives) can be measured as intensity moments in the output planes of a set of appropriate fractional Fourier transform systems and thus be derived from the corresponding fractional power spectra. The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived. As an important by-product we get a number of moment combinations that are invariant under (anamorphic) fractional Fourier transformation.

Statistical moments of the sound field propagating in a random, refractive medium near an impedance boundary.

Propagation of a monochromatic sound field in a refractive and turbulent medium near an impedance boundary is considered. Starting from the parabolic equation for a moving medium and using the Markov approximation, a closed equation for the statistical moments of arbitrary order of the sound-pressure field is derived. Numerical methods for directly solving the first- and second-moment versions of this equation are formulated. The first-moment formulation is very similar to parabolic equations (PEs) that are now widely used to calculate sound fields for particular realizations of a random medium. The second-moment formulation involves a large, fringed tridiagonal matrix, which is solved using iterative refinement and Cholesky factorization. The solution is computationally intensive and currently restricted to low frequencies. As an example, the first and second moments are directly calculated for upwind and downwind propagation of a sound wave through a turbulent atmosphere. For these cases, predictions from the second-moment PE were statistically indistinguishable from the result of 40 random trials calculated with a standard Crank-Nicholson PE, although the second-moment PE yielded smoother results due to its ensemble-average nature.

Maximum-entropy technique with logarithmic constraints: Estimation of atomic radial densities

Maximum-entropy (ME) approximations to density functions involving logarithmic constraints are studied. It is proved the existence and uniqueness of the ME approximation constrained by the normalization, the geometric mean and (i) a moment of arbitrary order, or (ii) the logarithmic uncertainty. A numerical analysis of the accuracy of these ME approximations is carried out for the radial electron densities of neutral atoms in both position and momentum spaces.

Evolution equations for arbitrary moments of the orientation distribution of rigid-rod molecules

The kinetic theory for rigid-rod molecules has been successful in describing a variety of experimental observations, from extremely dilute concentrations to the liquid crystalline limit. The theory is formulated in terms of an orientation distribution function, whose tensorial moments can be related to measurable properties. We introduce here a general relation from which evolution equations for moments of arbitrary order can easily be derived. The relation incorporates the effects of flow, magnetic and electric fields (permanent and induced dipoles), as well as “nematic” interactions necessary to describe liquid crystalline transitions. Relevant features of the equation(s) with regard to possible closure strategies are briefly discussed.

Perturbative QCD description of multiparticle correlations in quark and gluon jets

The QCD evolution equations in Modified Leading Log Approximation for the factorial moments of the multiplicity distribution in quark and gluon jets are numerically solved with initial conditions at threshold by fully taking into account the energy conservation law. After applying Local Parton Hadron Duality as hadronization prescription, a consistent quantitative description of available experimental data for factorial cumulants and factorial moments of arbitrary order and for their ratio both in quark and gluon jets and in e + e − annihilation is achieved.

SPECTRAL APPROXIMATION FOR WIND INDUCED STRUCTURAL VIBRATION STUDIES

Abstract. In this paper an efficient procedure for determining spectral moments of arbitrary order of the response of multi-degree-of-freedom (MDOF) linear systems under linearized pressure actions due to wind turbulence is presented. The procedure is initiated by decomposing the n variate incoherent wind field velocity into a sum of n independent totally coherent wind field velocities. Then, the use of a standard discretization procedure for approximating the frequency dependent eigenvalues and eigenvectors of the power spectral density (PSD) matrix leads to a polynomial representation of each component of the PSD matrix. Based on this representation, the spectral moments can be evaluated in a closed form without resorting to numerical quadrature. An illustrative example is included.

Near and far field optical beam characterization using the fractional Fourier transform

A general diffraction formula, valid in the near and far field regions, is derived based on the fractional Fourier transform. The beam moments of arbitrary order are derived and some analytical examples are given for a Gaussian beam.

On Moments of the Inverted Wishart Distribution

Moments of arbitrary order for the inverted Wishart distribution are obtained with the help of a factorization theorem, moments for normally distributed variables and inverse moments for chi-squared variables. Expressions are given in a recursive as well as a non-recursive manner.

Distribution and Density Approximations of a Maximum Likelihood Estimator in the Growth Curve Model

Moments of arbitrary order are obtained and expansions of the distribution and density functions of the maximum likelihood estimator of the mean structure in the Growth Curve model are considered. Results are given for expansions of arbitrary order. The results are obtained with the help of normally distributed variables and inverted Wishart variables.

Tomographic reconstruction of the density operator from its normally ordered moments.

The reconstruction of the density operator from the tomographic data (rotated quadrature components) via the normally ordered moments of the density operator is investigated. It is shown how arbitrary normally ordered moments of arbitrary order n can be obtained from the quadrature components for n+1 discrete angles that can be chosen arbitrarily. An integration over the angles of the rotated quadrature components multiplied by discrete phase factors is not necessary and uses more than the minimally necessary information about the rotated quadrature components. \textcopyright{} 1996 The American Physical Society.

Destruction of higher-order squeezing by thermal noise

We examine the influence of thermal noise on several non-classical properties of squeezed number states. In order to evaluate arbitrary-order moments of both amplitude and quadrature operators of the superposition, we use a normal-ordering technique based on McCoy's theorem. Our analytical results are compact formulae involving Gauss hypergeometric functions. We find that the thermal mean occupancy sufficient to destroy squeezing to any order is less than . Other non-classical features of a squeezed number state, such as pairwise oscillations in the photon number distribution, sub-Poissonian statistics and amplitude-squared squeezing disappear completely above this threshold.

Green's function approach to the theory of hydrodynamic interactions in suspensions

A new approach to the theory of many-sphere hydrodynamic interactions in suspensions is developed starting from the linearized Navier-Stokes equations reformulated in terms of the induced force formalism. This approach is based on the idea of the direct series expansion of the Green's function involved which leads in a very straightforward and controlled manner to the presentation of the velocity moments by means of the irreducible multipoles of the induced forces. To utilize this idea the calculation of the Green's function surface and volume moments of arbitrary order is performed using the method of generating functions. In principle, this offers the possibility to evaluate the translational and rotational mobility tensors in a power series expansions in two natural (for the system under consideration) parameters, namely, a/R and αa where a is a typical radius of the spheres, R a typical distance between spheres and α is expressed through the fluid density π and the viscosity of the fluid η as α=−iωρ/η. The correspondence between our and earlier theories of hydrodynamic interactions is discussed up to the seventh order in parameter (a/R) in the quasistatic limit and up to the third order in combination of both parameters, a/R and αa, in the case of finite frequencies.

The Wigner distribution function of self-Fourier functions

Particular forms of the Wigner distribution function and its moments of arbitrary order are derived for a generalized definition of the self-Fourier functions. The general form of the optical system matrix for which an input self-Fourier function transforms to an output self-Fourier function is found. The bright soliton solution of the electromagnetic wave is treated as an example of a generalized self-Fourier function.

Time invariant scaling in discrete fragmentation processes

Linear rate equations are used to describe the cascading decay of an initial heavy cluster into fragments. We consider moments of arbitrary orders of the mass multiplicity spectrum and derive scaling properties pertaining to their time evolution. We suggest that the mass weighted multiplicity is a suitable observable for the discovery of scaling. Numerical tests validate such properties, even for moderate values of the initial mass (nuclei, percolation clusters, etc.). Finite size effects can be simply parametrized.

Closure of multivariate t and related distributions

Let T : R n→ R k be a Studentizing transformation of the type T( Y ) = ν 1 2 A′Y /‖ B′Y ‖, and let GM( n) consist of scale mixtures of Gaussian laws on ≎ R k; 1 ⩽ k ⩽ n≎. It is shown that GM( n) is closed under T(·). Concentration properties of these distributions are studied via peakedness on varying their parameters, showing that peakedness may be diminished, preserved, or increased under T(·). Random ratios having moments of arbitrary order are exhibited when neither the numerator nor denominator has moments of any order.

Experimental observation of the rotation properties of atomic multipoles

The resonant interaction of optical radiation with atomic vapors can excite the atoms into anisotropic states with long lifetimes. The conventional description of these states uses an expansion in a basis of multipole moments, which are represented by irreducible tensor operators. We demonstrate an experimental method that permits the excitation of these atomic multipole moments in specific, experimentally controllable spatial orientations. Complementary optical methods permit the observation of multipole moments of arbitrary order and a direct measurement of their spatial orientation. We demonstrate these procedures in the ground state of atomic sodium, using optical pumping to polarize the angular-momentum substates. We show how the experimental results can be used to separate signal contributions that originate from different multipole moments.

Ergodicity of hedging control policies in single-part multiple-state manufacturing systems

The Markov renewal viewpoint of single-part/multiple-state manufacturing systems under hedging control policies, subjected to a constant rate of demand for parts, is used to derive sufficient criteria to guarantee the ergodicity of the resulting parts surplus process. The criteria are simple and directly verifiable from an analysis of the Markov chain characterizing the dynamics of the discrete manufacturing system production state. A key intermediate result in reaching these criteria is a potentially very useful system of linear differential equations with boundary conditions which can be generalized to permit computation of moments of arbitrary order for sojourn times in the regions between successive hedging points in the parts surplus space.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Central moments of ion implantation distributions derived by the backward Boltzmann transport equation compared with Monte Carlo simulations

The authors study solutions to the backward Boltzmann transport equation (BBTE) specialized to equations governing moments of the distribution of ions implanted into amorphous targets. A central moment integral equation set has been derived starting from the classical plane source BBTE for non-central moments. A full generator equation is provided to allow construction of equation sets of an arbitrary size, thus allowing computation of moments of arbitrary order. A BBTE solver program has been written that uses the residual correction technique proposed by Winterbon (1970). A simple means is presented to allow direct incorporation of Biersack's two-parameter 'magic formula' into a BBTE solver program. Results for non-central and central moment integral equation sets are compared with Monte Carlo simulations, using three different formulae for the mean free flight path between collisions. Comparisons are performed for the ions B and As, implanted into the target a-Si, over the energy range 1 keV-1 MeV. The central moment integral equation set is found to have superior convergence properties to the non-central moment equation set. For As ions implanted into a-Si, at energies below approximately 30 keV, significant differences are observed, for third- and fourth-order moments, when using alternative versions for the mean free flight path. Third- and fourth-order moments derived using one- and two-parameter scattering mechanisms also show significant differences over the same energy range.