Moments Of Arbitrary Order Research Articles

Validation of symmetry-induced high moment velocity and temperature scaling laws in a turbulent channel flow.

The symmetry-based turbulence theory has been used to derive new scaling laws for the streamwise velocity and temperature moments of arbitrary order. For this, it has been applied to an incompressible turbulent channel flow driven by a pressure gradient with a passive scalar equationcoupled in. To derive the scaling laws, symmetries of the classical Navier-Stokes and the thermal energy equationshave been used together with statistical symmetries, i.e., the statistical scaling and translation symmetries of the multipoint moment equations. Specifically, the multipoint moments are built on the instantaneous velocity and temperature fields other than in the classical approach, where moments are based on the fluctuations of these fields. With this instantaneous approach, a linear system of multipoint correlation equationshas been obtained, which greatly simplifies the symmetry analysis. The scaling laws have been derived in the limit of zero viscosity and heat conduction, i.e., Re_{τ}→∞ and Pr>1, and they apply in the center of the channel, i.e., they represent a generalization of the deficit law, thus extending the work of Oberlack etal. [Phys. Rev. Lett. 128, 024502 (2022)0031-900710.1103/PhysRevLett.128.024502]. The scaling laws are all power laws, with the exponent of the high moments all depending exclusively on those of the first and second moments. To validate the new scaling laws, the data from a large number of direct numerical simulations (DNS) for different Reynolds and Prandtl numbers have been used. The results show a very high accuracy of the scaling laws to represent the DNS data. The statistical scaling symmetry of the multipoint moment equations, which characterizes intermittency, has been the key to the new results since it generates a constant in the exponent of the final scaling law. Most important, since this constant is independent of the order of the moments, it clearly indicates anomalous scaling.

Mass moments of functionally graded 2D domains and axisymmetric solids

We present a general methodology to evaluate the mass moments of two-dimensional domains and axisymmetric solids made of functionally graded materials. The approach developed in the paper is based on the sequence of two steps. First, the original domain integrals are converted to integrals extended to the relevant boundary by exploiting Gauss theorem. Second, for domains having a polygonal or circular shape, the boundary integrals are evaluated analytically by providing algebraic expressions that depend upon the parameters defining the density distribution, the position vectors of the vertices of the polygonal domain or the initial and ending points of an arbitrary circular sector, respectively. While the first step refers to moments of arbitrary order, the second step is limited to the most useful quantities for engineering applications, i.e. generalised mass, static moment and inertia tensor. The formulas derived in the paper are validated by means of examples retrieved from the specialised literature for which analytical results are available or have been specifically derived by the authors. Finally, in order to ascertain the computational savings entailed by the use of the proposed analytical formulas with respect to numerical techniques, the mass moments of a longitudinal section of a human femure, made of a functionally graded material and characterised by a linear density distribution, have been computed.

The multipolar elastic fields of ellipsoidal and polytopal plastic inclusions

The article offers a general formulation for the multipolar field expansion of an inclusion. The multipolar moments of arbitrary order for the ellipsoidal inclusion are derived, showing known features of their elastic fields in both the far and near field. Using integer point enumeration theory, the same formulation is extended to all polytopal inclusions, which serve to model faceted inclusions found in, e.g. igneous and metamorphic rocks, particle reinforced composite materials, or as intermetallic precipitates in alloys. A general algorithm for the calculation of the multipolar moments of polytopes of any number of vertices is derived. A number of examples for tetrahedral, cuboidal and dodecahedral inclusions are given. It is shown that the parity of the axial moment function of a polytopal inclusion mirrors the symmetry of the inclusion. This represents the main properties of their elastic fields, including the long- and near-field decay. If the inclusion is symmetric with respect to a given axes, the fields will decay with 1 / r 2 in the far field and 1 / r 4 in the near field. If symmetry along that direction is lost, as is expected in most real inclusions, the decay rate progresses with 1 / r 2 but the near field with 1 / r 3 .

Multi-dimensional backward stochastic differential equations of diagonally quadratic generators: The general result

This paper is devoted to a general solvability of a multi-dimensional backward stochastic differential equation (BSDE) of a diagonally quadratic generator g(t,y,z), by relaxing the assumptions of Hu and Tang [15] on the generator and terminal value. More precisely, the generator g(t,y,z) can have more general growth and continuity in the first unknown variable y in the local solution; while in the global solution, the generator g(t,y,z) can have a skew sub-quadratic but in addition “strictly and diagonally” quadratic growth in the second unknown variable z, or the terminal value can be unbounded but the generator g(t,y,z) is “diagonally dependent” on the variable z (i.e., the i-th component gi of the generator g only depends on the i-th row zi of z for each i=1,⋯,n). Three new results are established on the local and global solutions when the terminal value is bounded and the generator g is subject to some general assumptions. When the terminal value is unbounded but is of exponential moments of arbitrary order, an existence and uniqueness result is given under the assumptions that the generator g(t,y,z) is Lipschitz continuous in y, and varies with z in a “diagonal”, “component-wisely convex or concave”, and “quadratically growing” way, which seems to be the first general solvability of system of quadratic BSDEs with unbounded terminal values. This generalizes and strengthens some existing results via some new ideas.

Multipolar quantum electrodynamics of localized charge-current distributions: Spectral theory and renormalization

We formulate a nonrelativistic quantum field theory to model interactions between quantized electromagnetic fields and localized charge-current distributions. The electronic degrees of freedom are encoded in microscopic polarization and magnetization field operators whose moments are identified with the multipole moments of the charge-current distribution. The multipolar Hamiltonian is obtained from the minimal coupling Hamiltonian through a unitary transformation, often referred to as the Power-Zienau-Woolley transformation; we renormalize this Hamiltonian using perturbation theory, the result of which is used to compute the leading-order radiative corrections to the electronic energy levels due to interactions between the electrons and quantum vacuum fluctuations in the electromagnetic field. Our renormalized energy shift constitutes a generalization of the Lamb shift in atomic hydrogen, valid for general localized assemblies of atoms and molecules, possibly with net charge but absent a free current. By expanding the fields in a series of multipole moments, our results can be used to study contributions to this energy shift coming from specific multipole moments of arbitrary order.

Slip and jump coefficients for general gas–surface interactions according to the moment method

We develop a moment method based on the Hermite series of the arbitrary order to calculate viscous-slip, thermal-slip, and temperature-jump coefficients for general gas-surface scattering kernels. Under some usual assumptions of scattering kernels, the solvability is obtained by showing the positive definiteness of the symmetric coefficient matrix in the boundary conditions. For gas flows with the Cercignani–Lampis gas–surface interaction and inverse-power-law intermolecular potentials, the model can capture the slip and jump coefficients accurately with elegant analytic expressions. On the one hand, the proposed method can apply to the cases of arbitrary order moments with increasing accuracy. On the other hand, the explicit formulas for low-order situations are simpler and more accurate than some existing results in references. Therefore, one may apply these formulas in slip and jump conditions to improve the accuracy of macroscopic fluid dynamic models for gas flows.

Self-similar Profiles for Homoenergetic Solutions of the Boltzmann Equation for Non-cutoff Maxwell Molecules

We consider a modified Boltzmann equation which contains, together with the collision operator, an additional drift term which is characterized by a matrix A. Furthermore, we consider a Maxwell gas, where the collision kernel has an angular singularity. Such an equation is used in the study of homoenergetic solutions to the Boltzmann equation. Under smallness assumptions on the drift term, we prove that the longtime asymptotics is given by self-similar solutions. We work in the framework of measure-valued solutions with finite moments of order p>2 and show existence, uniqueness and stability of these self-similar solutions for sufficiently small A. Furthermore, we prove that they have finite moments of arbitrary order if A is small enough. In addition, the singular collision operator allows to prove smoothness of these self-similar solutions. Finally, we study the asymptotics of particular homoenergetic solutions. This extends previous results from the cutoff case to non-cutoff Maxwell gases.

Quadratic [formula omitted]-BSDEs with convex generators and unbounded terminal conditions

In this paper, we first study one-dimensional quadratic backward stochastic differential equations driven by G-Brownian motions (G-BSDEs) with unbounded terminal values. With the help of a θ-method of Briand and Hu (2008) and nonlinear stochastic analysis techniques, we propose an approximation procedure to prove existence and uniqueness result when the generator is convex (or concave) and terminal value is of exponential moments of arbitrary order. Finally, we also establish the well-posedness of multi-dimensional G-BSDEs with diagonally quadratic generators.

Moment-based representation of the diffusion inside the brain from reduced DMRI acquisitions: Generalized AMURA.

AMURA (Apparent Measures Using Reduced Acquisitions) was originally proposed as a method to infer micro-structural information from single-shell acquisitions in diffusion MRI. It reduces the number of samples needed and the computational complexity of the estimation of diffusion properties of tissues by assuming the diffusion anisotropy is roughly independent on the b-value. This simplification allows the computation of simplified expressions and makes it compatible with standard acquisition protocols commonly used even in clinical practice. The present work proposes an extension of AMURA that allows the calculation of general moments of the diffusion signals that can be applied to describe the diffusion process with higher accuracy. We provide simplified expressions to analytically compute a set of scalar indices as moments of arbitrary orders over either the whole 3-D space, particular directions, or particular planes. The existing metrics previously proposed for AMURA (RTOP, RTPP and RTAP) are now special cases of this generalization. An extensive set of experiments is performed on public data and a clinical clase acquired with a standard type acquisition. The new metrics provide additional information about the diffusion processes inside the brain.

Turbulence Statistics of Arbitrary Moments of Wall-Bounded Shear Flows: A Symmetry Approach.

The calculation of turbulence statistics is considered the key unsolved problem of fluid mechanics, i.e., precisely the computation of arbitrary statistical velocity moments from first principles alone. Using symmetry theory, we derive turbulent scaling laws for moments of arbitrary order in two regions of a turbulent channel flow. Besides the classical scaling symmetries of space and time, the key symmetries for the present work reflect the two well-known characteristics of turbulent flows: non-Gaussianity and intermittency. To validate the new scaling laws we made a new simulation at an unprecedented friction Reynolds number of 10 000, large enough to test the new scaling laws. Two key results appear as an application of symmetry theory, which allowed us to generate symmetry invariant solutions for arbitrary orders of moments for the underlying infinite set of moment equations. First, we show that in the sense of the generalization of the deficit law all moments of the streamwise velocity in the channel center follow a power-law scaling, with exponents depending on the first and second moments alone. Second, we show that the logarithmic law of the mean streamwise velocity in wall-bounded flows is indeed a valid solution of the moment equations, and further, all higher moments in this region follow a power law, where the scaling exponent of the second moment determines all higher moments. With this we give a first complete mathematical framework for all moments in the log region, which was first discovered about 100years ago.

Analytical study on the evolutionary behavior of transfer matrix element moments in strongly coupled multimode systems.

In this paper, the evolutionary behavior of the transfer matrix element moments in strongly coupled multimode systems is studied analytically. The randomly coupled multimode system is modeled by a set of coupled stochastic differential equations (SDEs), which can be used to find the coupled ordinary differential equations (ODEs) for the arbitrary order moments of the matrix elements. Since the ODEs are with the constant coefficients, it is possible to obtain the analytical solutions. The asymptotic behaviors of the solutions are investigated by comparing with the existing results derived from the property of the Haar matrix, and a perfect agreement is observed. The evolutionary behaviors of the transfer matrix element moments computed by the analytical formulas have excellent match with the Monte Carlo simulation results. The analytical method can be highly beneficiary for the multimode system design and analysis.

Moment method for the Boltzmann equation of reactive quaternary gaseous mixture

We are interested in solving the Boltzmann equation of chemically reacting rarefied gas flows using the Grad’s-14 moment method. We first propose a novel mathematical model that describes the collision dynamics of chemically reacting hard spheres. Using the collision model, we present an algorithm to compute the moments of the Boltzmann collision operator. Our algorithm is general in the sense that it can be used to compute arbitrary order moments of the collision operator and not just the moments included in the Grad’s-14 moment system. For a first-order chemical kinetics, we derive reaction rates for a chemical reaction outside of equilibrium thereby, extending the Arrhenius law that is valid only in equilibrium. We show that the derived reaction rates (i) are consistent in the sense that at equilibrium, we recover the Arrhenius law and (ii) have an explicit dependence on the scalar fourteenth moment, highlighting the importance of considering a fourteen moment system rather than a thirteen one. Through numerical experiments we study the relaxation of the Grad’s-14 moment system to the equilibrium state.

Unified and non-recursive formulas for moments of the normal distribution with stripe truncation

Stripe truncation in the normal distribution is introduced such that the variable is truncated when it is located on several intervals like stripes. This truncation includes single, double and elliptical truncation as special cases. Then, the moments and absolute moments of arbitrary orders for the deviation of the truncated variable from an arbitrary reference point are derived using closed-form formulas based on the incomplete gamma function. The corresponding absolute moments of non-integer valued orders are also derived employing the parabolic cylinder distribution. The formulas do not suffer from difficulties associated with the conventional recursive methods for higher-order moments.

Efficient and accurate EAP imaging from multi-shell dMRI with micro-structure adaptive convolution kernels and dual Fourier Integral Transforms (MiSFIT)

A number of computational techniques have been lately devised to image the Ensemble Average Propagator (EAP) within the white matter of the brain, propelled by the deployment of multi-shell acquisition protocols and databases: approaches like Mean Apparent Propagator Imaging (MAP-MRI) and its Laplacian-regularized version (MAPL) aim at describing the low frequency spectrum of the EAP (limited by the maximum b-value acquired) and afterwards computing scalar indices that embed useful descriptions of the white matter, e. g. the Return-to-Origin, Plane, or Axis Probabilities (RTOP, RTPP, RTAP). These methods resort to a non-parametric, bandwidth limited representation of the EAP that implies fitting a set of 3-D basis functions in a large-scale optimization problem. We propose a semi-parametric approach inspired by signal theory: the EAP is approximated as the spherical convolution of a Micro-Structure adaptive Gaussian kernel with a non-parametric orientation histogram, which aims at representing the low-frequency response of an ensemble of coherent sets of fiber bundles at the white matter. This way, the optimization involves just the 2 to 3 parameters that describe the kernel, making our approach far more efficient than the related state of the art. We devise dual Fourier domains Integral Transforms to analytically compute RTxP-like scalar indices as moments of arbitrary orders over either the whole 3-D space, particular directions, or particular planes. The so-called MiSFIT is both time efficient (a typical multi-shell data set can be processed in roughly one minute) and accurate: it provides estimates of widely validated indices like RTOP, RTPP, and RTAP comparable to MAPL for a wide variety of white matter configurations.

Dynamics of Moments of Arbitrary Order for Stochastic Poisson Compressions

The explicit dynamics of the moments for the GKSL equation is obtained. In our case the GKSL equation corresponds to Poisson stochastic processes which lead to unitary jumps. We consider squeeze operators as the unitary jumps.

On upper bounds on expectations of gOSs based on DFR and DFRA distributions

We focus on the problem of establishing the optimal upper bounds on generalized order statistics which are based on the underlying cdf belonging to the family of distributions with decreasing failure rate and decreasing failure rate on the average. This issue has been previously considered by Bieniek [Projection bounds on expectations of generalized order statistics from DFR and DFRA families, Statistics, 2006; 40: 339–351], who established upper nonnegative mean-variance bounds with use of the projections of the compositions of density functions of the uniform generalized order statistic and the exponential distribution function onto the properly chosen convex cones. In this paper we obtain possibly negative upper bounds, by improving the zero bounds obtained by Bieniek for some particular cases of gOSs. We express the bounds in the scale units generated by the central absolute moments of arbitrary orders. We also describe the attainability conditions.

Exact Multipolar Decompositions with Applications in Nanophotonics

AbstractThe multipolar decomposition of electromagnetic sources is an important tool for the study of light–matter interactions in general, and optical materials in particular. Here, a report is given on recent progress in the multipolar decomposition of electromagnetic sources. First, the exact and simple expressions for the multipolar moments of electric current density distributions are reviewed, and then, the results are extended to multipolar moments of magnetization current density distributions due to intrinsic spin. The consideration of both electric and magnetic sources allows to establish the conditions for sources of pure handedness. Scripts are provided that facilitate the computation of multipolar moments of arbitrary order. The work and the included examples of use are placed in the context of nanophotonics and metamaterials, and an outlook for applications in these and other fields is provided.

A Space-Based Method for the Generation of a Schwartz Function with Infinitely Many Vanishing Moments of Higher Order with Applications in Image Processing

In this article we construct a function with infinitely many vanishing (generalized) moments. This is motivated by an application to the Taylorlet transform which is based on the continuous shearlet transform. It can detect curvature and other higher order geometric information of singularities in addition to their position and the direction. For a robust detection of these features a function with higher order vanishing moments, $$\int _\mathbb {R}g(x^k)x^mdx=0$$ , is needed. We show that the presented construction produces an explicit formula of a function with $$\infty $$ many vanishing moments of arbitrary order and thus allows for a robust detection of certain geometric features. The construction has an inherent connection to q-calculus, the Euler function and the partition function.

Evolution of arbitrary moments of radiant intensity distribution for partially coherent general beams in atmospheric turbulence

The evolution law of arbitrary order moments of the Wigner distribution function, which can be applied to the different spatial power spectra, is obtained for partially coherent general beams propagating in atmospheric turbulence using the extended Huygens–Fresnel principle. A coupling coefficient of radiant intensity distribution (RID) in turbulence is introduced. Analytical expressions of the evolution of the first five-order moments, kurtosis parameter, coupling coefficient of RID for general beams in turbulence are derived, and the formulas are applied to Airy beams. Results show that there exist two types for general beams in turbulence. A larger value of kurtosis parameter for Airy beams also reveals that coupling effect due to turbulence is stronger. Both theoretical analysis and numerical results show that the maximum value of kurtosis parameter for an Airy beam in turbulence is independent of turbulence strength parameter and is only determined by inner scale of turbulence. Relative angular spread, kurtosis and coupling coefficient are less influenced by turbulence for Airy beams with a smaller decay factor and a smaller initial width of the first lobe.

Fundamental solutions of the time fractional diffusion-wave and parabolic Dirac operators

In this paper we study the multidimensional time fractional diffusion-wave equation where the time fractional derivative is in the Caputo sense with order β∈]0,2]. Applying operational techniques via Fourier and Mellin transforms we obtain an integral representation of the fundamental solution (FS) of the time fractional diffusion-wave operator. Series representations of the FS are explicitly obtained for any dimension. From these we derive the FS for the time fractional parabolic Dirac operator in the form of integral and series representation. Fractional moments of arbitrary order γ>0 are also computed. To illustrate our results we present and discuss some plots of the FS for some particular values of the dimension and of the fractional parameter.