Moments Of Arbitrary Order Research Articles

First passages for a search by a swarm of independent random searchers

In this paper we study some aspects of search for an immobile target by a swarm ofN non-communicating, randomly moving searchers (numbered by the indexk,k = 1, 2,..., N), which all start their random motion simultaneously at the same point in space. For eachrealization of the search process, we record the unordered set of time moments{τk}, whereτk is the time of thefirst passage of the kth searcher to the location of the target. Clearly,τks are independent, identically distributed random variables with the same distribution functionΨ(τ). We evaluate thenthe distribution P(ω) of the random variable , where is the ensemble-averaged realization-dependent first passage time. We show thatP(ω) exhibits quite a non-trivial and sometimes a counterintuitive behavior. Wedemonstrate that in some well-studied cases (e.g. Brownian motion in finited-dimensionaldomains) the mean first passage time is not a robust measure of the search efficiency, despite the factthat Ψ(τ) has moments of arbitrary order. This implies, in particular, that even in this simplest case(not to mention complex systems and/or anomalous diffusion) first passage data extractedfrom a single-particle tracking should be regarded with appropriate caution because of thesignificant sample-to-sample fluctuations.

Position-momentum uncertainty relations based on moments of arbitrary order

The position-momentum uncertainty-like inequality based on moments of arbitrary order for $d$-dimensional quantum systems, which is a generalization of the celebrated Heisenberg formulation of the uncertainty principle, is improved here by use of the R\'enyi-entropy-based uncertainty relation. The accuracy of the resulting lower bound is physico-computationally analyzed for the two main prototypes in $d$-dimensional physics: the hydrogenic and oscillator-like systems.

Higher order statistics in the annulus square billiard: transport and polyspectra

Classical transport in a doubly connected polygonal billiard, i.e. the annulus square billiard, is considered. Dynamical properties of the billiard flow with a fixed initial direction are analyzed by means of the moments of arbitrary order of the number of revolutions around the inner square, accumulated by the particles during the evolution. An ‘anomalous’ diffusion is found: the moment of order q exhibits an algebraic growth in time with an exponent different from q/2, like in the normal case. Transport features are related to spectral properties of the system, which are reconstructed by Fourier transforming time correlation functions. An analytic estimate for the growth exponent of integer-order moments is derived as a function of the scaling index at zero frequency of the spectral measure, associated with the angle spanned by the particles. The nth-order moment is expressed in terms of a multiple-time correlation function, depending on n − 1 time intervals, which is shown to be linked to higher order density spectra (polyspectra), by a generalization of the Wiener–Khinchin theorem. Analytic results are confirmed by numerical simulations.

Analytical derivation of moment equations in stochastic chemical kinetics

The master probability equation captures the dynamic behavior of a variety of stochastic phenomena that can be modeled as Markov processes. Analytical solutions to the master equation are hard to come by though because they require the enumeration of all possible states and the determination of the transition probabilities between any two states. These two tasks quickly become intractable for all but the simplest of systems. Instead of determining how the probability distribution changes in time, we can express the master probability distribution as a function of its moments, and, we can then write transient equations for the probability distribution moments. In 1949, Moyal defined the derivative, or jump, moments of the master probability distribution. These are measures of the rate of change in the probability distribution moment values, i.e. what the impact is of any given transition between states on the moment values. In this paper we present a general scheme for deriving analytical moment equations for any N-dimensional Markov process as a function of the jump moments. Importantly, we propose a scheme to derive analytical expressions for the jump moments for any N-dimensional Markov process. To better illustrate the concepts, we focus on stochastic chemical kinetics models for which we derive analytical relations for jump moments of arbitrary order. Chemical kinetics models are widely used to capture the dynamic behavior of biological systems. The elements in the jump moment expressions are a function of the stoichiometric matrix and the reaction propensities, i.e. the probabilistic reaction rates. We use two toy examples, a linear and a non-linear set of reactions, to demonstrate the applicability and limitations of the scheme. Finally, we provide an estimate on the minimum number of moments necessary to obtain statistical significant data that would uniquely determine the dynamics of the underlying stochastic chemical kinetic system. The first two moments only provide limited information, especially when complex, non-linear dynamics are involved.

Analysis of the busy period in threshold control system

Consideration was given to the controllable Markov queuing system with nonhomogeneous servers and threshold policy of server activation depending on the queue length. By the busy period is meant the time interval between the instant of customer arrival to the empty system and the instant of service completion when the system again becomes free. The system busy period and the number of serviced customers were analyzed. Recurrent relations for the distribution density in terms of the Laplace transform and generating function, as well as formulas for calculation of the corresponding arbitrary-order moments, were obtained.

Option Pricing Where the Underlying Assets Follow a Gram/Charlier Density of Arbitrary Order

If a probability distribution is sufficiently close to a normal distribution, its density can be approximated by a Gram/Charlier Series A expansion. In option pricing, this has been used fit risk-neutral asset price distributions to the implied volatility smile, ensuring an arbitrage-free interpolation of implied volatilities across exercise prices. However, the existing literature is restricted to truncating the series expansion after the fourth moment. This paper presents an option pricing formula in terms of the full (untruncated) series and discusses a fitting algorithm, which ensures that a series truncated at a moment of arbitrary order represents a valid probability density. While it is well known that valid densities resulting from truncated Gram/Charlier Series A expansions do not always have sufficient flexibility to fit all market-observed option prices perfectly, this paper demonstrates that option pricing in a model based on these densities is as tractable as the (far less flexible) original model of Black and Scholes (1973), allowing non-trivial higher moments such as skewness, excess kurtosis and so on to be incorporated into the pricing of exotic options: Generalising the Gram/Charlier Series A approach to the multiperiod, multivariate case, a model calibrated to standard option prices is developed, in which a large class of exotic payoffs can be priced in closed form. Furthermore, this approach, when applied to a foreign exchange option market involving several currencies, can be used to ensure that the volatility smiles for options on the cross exchange rate are constructed in a consistent, arbitrage-free manner.

MOMENTS OF THE DURATION OF BUSY PERIODS OF MX/G/1/n SYSTEMS

We derive a simple recursion to compute moments of arbitrary order of the duration of busy periods of MX/G/1/n systems starting with an arbitrary number of customers in the system.

Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff

The spatially homogeneous Boltzmann equation without angular cutoff is discussed on the regularity of solutions for the modified hard potential and Debye-Yukawa potential. When the angular singularity of the cross section is moderate, any weak solution having the finite mass, energy and entropy lies in the Sobolev space of infinite order for any positive time, while for the general potentials, it lies in the Schwartz space if it has moments of arbitrary order. The main ingredients of the proof are the suitable choice of the mollifiers composed of pseudo-differential operators and the sharp estimates of the commutators of the Boltzmann collision operator and pseudo-differential operators. The method developed here also provides some new estimates on the collision operator.

Application of Chebyshev polynomials to the theory of thin bodies

The application of shifted Chebyshev polynomials of the second kind to the construction of the thin-body theory is considered. Some basic and additional recurrence relations for Chebyshev polynomials are given. Arbitrary-order moments are obtained for the first and second derivatives of a tensor field and for some expressions. Several equations of motion expressed in terms of the moments of displacement and rotation vectors are derived for the moment theory; a number of constitutive relations of the zeroth approximation are obtained.

Energy evolution in the time-dependent harmonic oscillator with arbitrary external forcing

The classical Hamiltonian system of time-dependent harmonic oscillator driven by an arbi- trary external time-dependent force is considered. Exact analytical solution of the corresponding equations of motion is constructed in the framework of the technique based on WKB approach. Energy evolution for the ensemble of uniformly distributed w.r.t, the canonical angle initial conditions on the initial invariant torus is studied. Exact expressions for the energy moments of arbitrary order taken at arbitrary time moment are analytically derived. Corresponding character- istic function is analytically constructed in the form of infinite series and numerically evaluated for certain values of the system parameters. Energy distribution function is numerically obtained in some particular cases. In the limit of small initial ensemble's energy the relevant formula for the energy distribution function is analytically derived. Keywords: classical dynamics, Hamiltonian systems, linear oscillator, energy distribution func- tion, WKB approach.

Higher Order Moments and Conditional Asymptotics of the Batch Markovian Arrival Process

We consider the number of arrivals in a Batch Markovian Arrival Process (BMAP) and derive matrix analytic expressions for its moments of arbitrary order. These expressions consist of decomposition formulas connected to the semigroup structure of the moments, forward and backward differential equations, and recursive as well as direct integral formulas. This extends earlier work by Narayana and Neuts on the first two factorial moment matrices. We next turn to the terminating BMAP, i.e., a BMAP with an absorbing state in which no arrivals occur. We consider the asymptotic behavior of the moments conditional on the process not yet having terminated. We show that the conditional mean and variance possess affine asymptotics and derive the coefficients explicitly. Finally, we discuss how parts of our work also apply to the more general class of Rational Arrival Processes (RAPs).

Analytic distribution functions for an ion plasma crossing an MHD shock

Context. Very recently, we derived the Boltzmann equation for a collisionless plasma crossing an MHD shock, such as that found in astrophysical situations. Aims. We aim to solve the Boltzmann equation analytically, to obtain exact downstream velocity distribution functions for a given upstream distribution function and an MHD compression ratio. Methods. We apply the general Boltzmann equation to the case of a purely perpendicular shock and solve it using a separation Ansatz. Results. We obtain a simple,'analytical relation between the upstream and downstream velocity distribution functions, which is valid for all physical plasmas. Having obtained the desired relation, we also derive an analytical relation for MHD velocity moments of arbitrary order.

Wigner Distribution Moments Measured as Intensity Moments in Separable First-Order Optical Systems

It is shown how all global Wigner distribution moments of arbitrary order can be measured as intensity moments in the output plane of an appropriate number of separable first-order optical systems (generally anamorphic ones). The minimum number of such systems that are needed for the determination of these moments is derived.

Stochastic approaches to uncertainty quantification in CFD simulations

This paper discusses two stochastic approaches to computing the propagation of uncertainty in numerical simulations: polynomial chaos and stochastic collocation. Chebyshev polynomials are used in both cases for the conventional, deterministic portion of the discretization in physical space. For the stochastic parameters, polynomial chaos utilizes a Galerkin approximation based upon expansions in Hermite polynomials, whereas stochastic collocation rests upon a novel transformation between the stochastic space and an artificial space. In our present implementation of stochastic collocation, Legendre interpolating polynomials are employed. These methods are discussed in the specific context of a quasi-one-dimensional nozzle flow with uncertainty in inlet conditions and nozzle shape. It is shown that both stochastic approaches efficiently handle uncertainty propagation. Furthermore, these approaches enable computation of statistical moments of arbitrary order in a much more effective way than other usual techniques such as the Monte Carlo simulation or perturbation methods. The numerical results indicate that the stochastic collocation method is substantially more efficient than the full Galerkin, polynomial chaos method. Moreover, the stochastic collocation method extends readily to highly nonlinear equations. An important application is to the stochastic Riemann problem, which is of particular interest for spectral discontinuous Galerkin methods.

Resonant transfer of excitation between two molecules using Maxwell fields

The matrix element for the resonant transfer of excitation between two molecules possessing electric and magnetic multipole moments of arbitrary order is calculated using quantum electrodynamical response theory. A prerequisite of the method is the functional form for the lth order linear electric and magnetic multipole dependent electric displacement and magnetic field operators in the neighborhood of a molecule, whose derivation is also given. The initially unexcited species is viewed as a test body accepting energy resonantly via coupling to the Maxwell fields of the excited multipole source molecule. The generalized electric-electric multipole contribution to the matrix element is shown to agree with an earlier calculation using time-dependent perturbation theory. As an application involving both electric and magnetic terms, the rate of excitation transfer between two chiral molecules is computed and found to depend on the handedness of each species.

A general formula for the rate of resonant transfer of energy between two electric multipole moments of arbitrary order using molecular quantum electrodynamics

A general expression is derived for the matrix element for the resonant transfer of energy between an initially excited donor species and an acceptor moiety in the ground state, with each entity possessing an electric multipole moment of arbitrary order. In the quantum electrodynamical framework employed, the coupling between the pair is mediated by the exchange of a single virtual photon. The probability amplitude found from second-order perturbation theory is a product of the electric moments located at each center and the resonant multipole-multipole interaction tensor. Using the Fermi golden rule, a general formula for the rate of energy transfer is obtained. As an illustration of the efficacy of the theory developed, rates of excitation energy exchange are calculated for systems interacting through dipole-quadrupole, dipole-octupole, quadrupole-quadrupole, and the familiar dipole-dipole coupling. For each of the cases examined, the near- and far-zone limits of the migration rate are calculated from the result valid for all donor-acceptor separations beyond wave function overlap. Expression of the octupole contribution to the transfer rate in terms of its irreducible components of weights 1 and 3 leads to new features. The octupole weight-1 term is found to contribute only when the interaction is retarded, while the dipole-octupole weight-1 contribution appears as a higher-order correction term to the dipole-dipole rate. Order of magnitude estimates are given for the contributions of dipole-quadrupole and dipole-octupole terms relative to the leading dipole-dipole rate for near-, intermediate-, and far-zone separations to further understand the role played by higher multipole moments in the transfer of excitation and the mechanism dominating the process.

Generalized expressions for resonant excitation transfer and retarded dispersion energy shifts obtained using multipolar quantum electrodynamics

AbstractGeneralized expressions are presented within the framework of molecular quantum electrodynamics for intermolecular interactions arising from the exchange of one or two virtual photons. When one species is excited while the other is in the ground state, the former process corresponds to resonant transfer of excitation. The rate of energy exchange for molecules possessing electric multipole moments of arbitrary order is given in terms of the resonant multipole–multipole coupling tensor. This latter quantity is then used in the calculation of the generalized dispersion energy shift between a pair of ground‐state electric multipole polarizable molecules of arbitrary order via the interaction of induced multipole moments at each center. The results obtained offer an efficient alternative means of calculating higher multipole contributions to intermolecular processes. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

DELAY ANALYSIS OF A MULTIMEDIA ATM MULTIPLEXER WITH HOMOGENEOUS ARRIVALS

In this paper we present a delay analysis of an ATM multiplexer handling multimedia traffic of two classes: real time class, e.g., live audio and video and nonreal time class, e.g., file transfers. Packets arrive into the multiplexer in batches of general size and are output one at a time. The arrivals are homogeneous in the sense that either the arriving batch is all class-1 packets or all class-2 packets. The multiplexer is modeled as a priority discrete time queueing system with batch arrivals and geometric service. The homogeneity of the arrivals is reflected by the addition of an indicator random variable into the model. The analysis is comprehensive in that it covers the three main delay measures: the unfinished work, the busy period and the waiting time. For these measures, probability generating functions are derived, from which moments of arbitrary orders can be obtained. However, only the first moments are obtained in the article to produce expected values. Presented at the end is a special case when the arriving batches are of Poissonian size. Some numerical values are obtained and plotted for this case.

An Identity Relating Moments of Functionals of Convex Hulls

Denote by $K_n$ the convex hull of $n$ independent random points distributed uniformly in a convex body $K$ in $\R^d$, by $V_n$ the volume of $K_n$, by $D_n$ the volume of $K\backslash K_n$, and by $N_n$ the number of vertices of $K_n$. A well-known identity due to Efron relates the expected volume ${\it ED}_n$---and thus ${\it EV}_n$---to the expected number ${\it EN}_{n+1}$. This identity is extended from expected values to higher moments. The planar case of the arising identity for the variances provides in a simple way the corrected version of a central limit theorem for $D_n$ by Cabo and Groeneboom ($K$ being a convex polygon) and an improvement of a central limit theorem for $D_n$ by Hsing ($K$ being a circular disk). Estimates of $\var D_n$ ($K$ being a two-dimensional smooth convex body) and $\var N_n$ ($K$ being a $d$-dimensional smooth convex body, $d\geq 4$) are obtained. The identity for moments of arbitrary order shows that the distribution of $N_n$ determines ${\it EV}_{n-1}, {\it EV}_{n-2}^2,\dots, {\it EV}_{d+1}^{n-d-1}$. Reversely it is proved that these $n-d-1$ moments determine the distribution of $N_n$ entirely. The resulting formula for the probability that $N_n=k\ (k=d+1,\dots , n)$ appears to be new for $k\geq d+2$ and yields an answer to a question raised by Baryshnikov. For $k=d+1$ the formula reduces to an identity which has been repeatedly pointed out.

The force–moments on a deforming body introduced impulsively into an inviscid incompressible fluid with uniform vorticity

This paper is concerned with the derivation of dynamical equations for freely deforming bodies with more than six degrees of freedom which are immersed in an inviscid incompressible fluid. Following Proudman's pioneering work for a sphere our method is applied to a fluid with uniform vorticity but otherwise arbitrary non-uniform strain-rate at the instant after the body has been impulsively introduced into the fluid. The rotational disturbance field is consequently zero thus enabling the generalised force–moments of arbitrary order to be determined from a Laplace problem through the use of Green's theorem and generalised Kirchhoff potentials. An infinite system of equations is obtained each which contains an inertial term, given by the rate of change of the generalised Kelvin Impulse, a generalised lift, a deformation-induced surface momentum flux and a surface kinetic energy. The assumption of an impulsive start places no constraint on the use of our force–moment formulae in irrotational flow but they can only be applied at the starting instant in rotational flow or, when the strain-rate is weak, for early times in the body's motion. Nonetheless, the start conditions for the rotational case can be created experimentally and be applied to initially free tumbling bodies when they start to deform. This newly identified equation system provides the foundation for new analytical and numerical approaches to the macroscopic modelling of freely deforming bodies and bubbly two-phase flow. In particular, the equations show that the added masses are not sufficient to characterise the body's geometry and that independent geometric constants are also required, here referred to as the added Kirchhoff energies. Finally, the zero- and first-order force–moment equations are used to derive the force and torque that apply to bodies with six degrees of freedom and their analytic forms are shown to agree with independent results for arbitrarily shaped deforming bodies in both rotational and irrotational flows.