Non-integer Moments Research Articles

Non-uniform Bounds and Edgeworth Expansions in Self-normalized Limit Theorems

We study Edgeworth expansions in limit theorems for self-normalized sums. Non-uniform bounds for expansions in the central limit theorem are established while imposing only minimal moment conditions. Within this result, we address the case of non-integer moments leading to a reduced remainder. Furthermore, we provide non-uniform bounds for expansions in local limit theorems. The enhanced tail accuracy of our non-uniform bounds allows for deriving an Edgeworth-type expansion in the entropic central limit theorem as well as a central limit theorem in total variation distance for self-normalized sums.

Lattice Dynamics, Mechanical Properties, Electronic Structure and Magnetic Properties of Equiatomic Quaternary Heusler Alloys CrTiCoZ (Z = Al, Si) Using First Principles Calculations.

First principles calculations are performed to investigate the thermodynamical stability, dynamical, mechanical, electronic and magnetic properties of CrTiCoZ (Z = Al/Si) novel quaternary Heusler alloys. A Y-type III atomic configuration is found to be the most stable structure for both compounds. The lattice constant values obtained using GGA-PBE approach are 5.9368 Å and 5.7853 Å for CrTiCoAl and CrTiCoSi, respectively. Using the value of elastic moduli for both the compounds, the computed Pugh’s ratio value is 2.5 and 2.7 for CrTiCoAl and CrTiCoSi, respectively, which is higher than 1.75, indicating both the compounds are ductile in nature. The melting temperatures of both compounds are as high as 2142 K and 2420 K for CrTiCoAl and CrTiCoSi, respectively. The electronic structure calculations, using the GGA-PBE approach, show a half metallic behavior of CrTiCoAl. The spin-down channel exhibits a direct band gap of 0.15 eV, whereas the spin-up channel is metallic, making CrTiCoAl a half metallic ferromagnet with 100% spin polarization and an appreciable magnetic moment of −2 μB. However, CrTiCoSi is found to be semi-metallic in the spin-down channel and metallic in the spin-up channel, which leads to a spin polarization of 99.7% with a non-integer magnetic moment of −0.99 μB. The Curie temperature of CrTiCoAl is well above the room temperature (385 K), whereas that of CrTiCoSi is below the room temperature (203 K). Thus, CrTiCoAl is found to be more promising than CrTiCoSi as a spin injector in spintronic devices.

Uncertainty analysis of rock tunnel based on fractional moment and dimensional reduction method

In this study, fractional moment was adopted to estimate the uncertainty in a tunnel excavated in rock. The multiplicative dimensional reduction method (M-DRM) was adopted to approximate the relationship between random variables and response of the tunnel and to predict its behavior. Integer and non-integer (fractional) moments were estimated by Gaussian quadrature based on M-DRM. The maximum entropy principle was also used to estimate the probability distribution of the tunnel response using the optimization technology of Excel Solver based on fractional moment. The example of a circular tunnel in a rock mass subjected to hydrostatic stress is given. This demonstrates the uncertainty analysis procedure applied to the displacements and plastic zone induced by the excavation, for different rock-support pressures and far-field stresses. Statistical moment, standard deviation and probability distribution of the tunnel response all agree well with Monte Carlo simulation (MCS). M-DRM closely approximates the tunnel response, and improves the efficiency of the uncertainty analysis. The proposed method estimates the probability distribution based on frictional moment without any prior knowledge. The results show that the proposed uncertainty analysis method is robust, efficient and precise, and is suitable for a complex practical engineering system as encountered in rock engineering.

Iterative maximum likelihood estimation of the compound inverse Gaussian clutter parameters

This Letter aims at proposing a cost-effective method for estimating the parameters of the compound inverse Gaussian distributed clutter. Under the assumption of the absence of thermal noise, an iterative maximum likelihood estimator (IMLE) is proposed and compared with existing MLE, the [zlog(z)] estimator, the non-integer order moments estimator and the higher order moment estimator. The results obtained show that the IMLE method outperforms all the other methods and has similar estimation performance than the MLE method but requires lower computational time.

Radar CFAR detection in Weibull clutter based on zlog(z) estimator

ABSTRACTIn this paper, the zlog(z) based estimator for constant false alarm rate (CFAR) detection in Weibull clutter is proposed. This estimation method is obtained in terms of the digamma function where the estimates of the shape parameter are determined by the interpolation tool. The non-integer order moments estimator (NIOME) is also given and coincides the zlog(z) estimation results for low values of the moment’s fractional order. Via simulated data, it is shown that the CFAR detection performances based on the zlog(z) estimator have almost similar results as well as the existing maximum likelihood (ML) CFAR detector, but with low time-consuming which is very important in real-time applications.

Mathematical Modelling in Accounting Information Systems

The aim of this paper is to introduce continuous mathematical models for calculating planned cash-flow, based on the calculation of planned accounting data obtained from ERP systems of business entities. Using mathematical models defined in this way, it is possible to predict the values of required statements, or key performance indicators (KPIs), needed to assess the near future of an enterprise. The advantage compared to a discrete model is the possibility of finding the value of a quantity at any (non-integer) moment. This article should be considered as a contribution to the application of mathematical methods in business information systems in a general context.

Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value

In various disordered systems or non-equilibrium dynamical models, the large deviations of some observables have been found to display different scalings for rare values bigger or smaller than the typical value. In the present paper, we revisit the simpler observables based on independent random variables, namely the empirical maximum, the empirical average, the empirical non-integer moments or other additive empirical observables, in order to describe the cases where asymmetric large deviations already occur. The unifying starting point to analyze the large deviations of these various empirical observables is given by the Sanov theorem for the large deviations of the empirical histogram: the rate function corresponds to the relative entropy with respect to the true probability distribution and it can be optimized in the presence of the appropriate constraints. Finally, the physical meaning of large deviations rate functions is discussed from the renormalization perspective.

Mellin polar coordinate moment and its affine invariance

The moment-based method is a fundamental approach to the extraction of affine invariants. However, only integer-order traditional moments can be used to construct affine invariants. No invariants can be constructed by moments with an order lower than 2. Consequently, the obtained invariants are sensitive to noise. In this paper, the moment order is generalized from integer to non-integer. However, the moment order cannot simply be generalized from integer to non-integer to achieve affine invariance. The difficulty of this generalization lies in the fact that the angular factor owing to shearing in the affine transform can hardly be eliminated for non-integer order moments. In order to address this problem, the Mellin polar coordinate moment (MPCM) is proposed, which is directly defined by a repeated integral. The angular factor can easily be eliminated by appropriately selecting a repeated integral. A method is provided for constructing affine invariants by means of MPCMs. The traditional affine moment invariants (AMIs) can be derived in terms of the proposed MPCM. Furthermore, affine invariants constructed with real-order (lower than 2) MPCMs can be derived using the proposed method. These invariants may be more robust to noise than AMIs. Several experiments were conducted to evaluate the proposed method performance.

Parameter estimation of CGIG clutter plus noise using constrained NIOME and MLE approaches

Compound Gaussian inverse Gaussian (CGIG) distribution with thermal noise is well suited for sea clutter. The compact expression of higher order moments estimator (HOME) for CGIG clutter plus noise parameters is derived. When non-integer order moments (NIOM) are considered, numerical integration based on ‘Legendre’ and ‘Laguerre’ polynomials of Gauss quadrature method is employed. This estimator is referred to as constrained NIOM estimator (NIOME)-based method. Using the two first intensity moments, a constrained maximum likelihood estimator (CMLE) is formed by minimising the negative log likelihood function in one variable to obtain the shape parameter. For comparison purposes, computer simulations are presented with known and unknown clutter-to-noise ratio (CNR). In the case of known CNR, the method of moments provides a good fit as well as the NIOME and the CMLE methods. For unknown CNR, regardless the computing time, the HOME method is less accurate than the others. The constrained NIOME method requires numerical optimisation with acceptable calculation time. In most cases, the CMLE approach is accurate, but relatively slow due to the calculation of numerical integration of likelihood functions.

Closed‐form estimators of CGIG distributed parameters

The interpolation method has been employed in non-integer-order moments estimators obtained in a previous work for the parameters of compound-Gaussian clutter model with inverse Gaussian texture (CGIG). Closed-form parameter estimators for the CGIG distributed clutter are derived with no special functions. These are achieved after a simple combination of three basic expressions: non-integer positive moments, non-integer negative moments and Bessel recurrence relation. Simulation results show that the derived estimators are well suited for multilook data and have a comparable accuracy as well as the numerically solved maximum-likelihood approach, but they are much simpler and faster to compute

Parameter estimation for compound‐Gaussian clutter with inverse‐Gaussian texture

This study deals with the estimation problem of the inverse-Gaussian (IG)–compound-Gaussian (CG) distributed clutter parameters. Under the assumption of the absence of thermal noise, non-integer moments, log-moments and maximum-likelihood-based approaches are employed for model parameter estimation in the cases of single pulse and non-coherent integration of N pulses. The forms of the non-integer-order moments estimator and the [z log(z)] estimator are derived in terms of the Bessel and the exponential–integral functions, respectively. These formulas maintain monotonic nature over all values of the ratio between the shape parameter and the mean clutter power. For a single look data, the ML estimate combined with the mean clutter power is constructed in one dimension search for the shape parameter. By accommodating the mean square error metric, the estimation assessments are investigated via simulated and real data. Authors’ results illustrate that the derived non-integer-order moments estimator is asymptotically efficient for the IG–CG distributed clutter parameters particularly in heavy tailed clutter situations.

Theoretical study of the effect of hydrogen adsorption on the stability and electronic properties of hybrid monolayers

We have applied first-principle calculations, based on the density functional theory, to analyze the stability, electronic and magnetic properties of monolayers of graphene with a nanodomain of boron nitride (h-BN) with different geometries. To this end, we have investigated the effects of the adsorption of one and two hydrogen atoms by different atoms at the edge of the h-BN nanodomain, where we used the GGA approximation for the functional exchange and correlation. Specifically, we found that the structures with triangular-shaped nanodomains are the most stable and have a non-integer magnetic moment upon adsorption of the hydrogen atoms. We have also shown that both the electronic and magnetic properties can be influenced by the type of atom which adsorbs the hydrogen at the edge of the nanodomain. Additionally, we have demonstrated that some structures can quench the spin that would be generated by the adsorption of the hydrogen atoms.

Discontinuous payoff option pricing by Mellin transform: A probabilistic approach

The Mellin transform technique is applied for solving the Black-Scholes equation with time-dependent parameters and discontinuous payoff. We show that the option pricing is equivalent to recovering a probability density function on the positive real axis based on its moments, which are integer or fractional Mellin transform values. Then the Mellin transform can be effectively inverted from a collection of appropriately chosen fractional (i.e. non-integer) moments by means of the Maximum Entropy (MaxEnt) method. An accurate option pricing is guaranteed by previous theoretical results about MaxEnt distributions constrained by fractional moments. We prove that typical drawbacks of other numerical techniques, such as Finite Difference schemes, are bypassed exploiting the Mellin transform properties. An example involving discretely monitored barrier options is illustrated and the accuracy, efficiency and time consuming are discussed.

Estimating the Pareto plus noise distribution parameters using non‐integer order moments and [zlog(z)] approaches

Pareto plus noise clutter distribution has been introduced recently as a good candidate model for X-band high resolution maritime clutter returns. In this study, the authors derive a non-integer order moments estimator (NIOME) and [zlog(z)] based estimator to find the parameters of this distribution in the case of non-coherent integration of N-pulses. For this, the authors first develop an asymptotic formula of moments with non-integer order which is expressed in terms of the gamma and the generalised hypergeometric functions. Then, the authors derive [zlog(z)] based approach as a function of the log based moments and the generalised hypergeometric function. By accommodating a non-integer moments and moments of orders one and two, the proposed estimators are given so that non-linear estimates of the shape parameter are achieved using numerical computations. Through synthetic and real data, the authors show numerous examples demonstrating the applicability of the estimation procedures as a function of clutter-plus-noise parameters. The obtained results illustrate that the proposed NIOME method is asymptotically efficient especially for multiple looks case and high sample size.

Finite size corrections in the random energy model and the replica approach

We present a systematic and exact way of computing finite size corrections for the random energy model, in its low temperature phase. We obtain explicit (though complicated) expressions for the finite size corrections of the overlap functions. In its low temperature phase, the random energy model is known to exhibit Parisi's broken symmetry of replicas. The finite size corrections given by our exact calculation can be reproduced using replicas if we make specific assumptions about the fluctuations (with negative variances!) of the number and sizes of the blocks when replica symmetry is broken. As an alternative we show that the exact expression for the non-integer moments of the partition function can be written in terms of coupled contour integrals over what can be thought of as ‘complex replica numbers’. Parisi's one step replica symmetry breaking arises naturally from the saddle point of these integrals without making any ansatz or using the replica method. The fluctuations of the ‘complex replica numbers’ near the saddle point in the imaginary direction correspond to the negative variances we observed in the replica calculation. Finally our approach allows one to see why some apparently diverging series or integrals are harmless.

First principles study of structural, electronic and magnetic properties of ferromagnetic Bi2Fe4O9

The structural, electronic, and magnetic properties of ferromagnetic bismuth ferrate (Bi2Fe4O9) are investigated using density functional theory (DFT). Different exchange–correlation (xc) functionals (LSDA, PBE-GGA, PBEsol-GGA and WC-GGA) are tested to calculate various properties of Bi2Fe4O9. All the exchange correlation functionals accurately describe the structural properties of Bi2Fe4O9 but fail to calculate its correct band structure. The calculated band structure improves when DFT+U method is employed. The PBE-GGA+U calculations predict a bandgap of 2.1eV for the ferromagnetic Bi2Fe4O9 in close agreement with the experiment. The calculated density of states shows appreciable hybridization between Fe-3d states and O-2p states along with minor overlap between Bi-6p and O-2p states. The magnetic properties of Bi2Fe4O9 are primarily due to Fe3+ ions, each of which has a non-integer magnetic moment of approximately 3.9μB. The values of the induced magnetic moments at the other atomic sites depend on their local environments.

Approximation an estimate of the ss-distributions stability factor

The problem of approximating a stability factor estimate of alpha-stable distributions, obtained by the method of fractional moments,has been considered. Such distributions are widely used in models of stochastic processes, describing a wide range of processes and phenomena. The analysis of existing methods for estimating parameters of stable distributions has been carried out. One of the new and promising methods for solving the problem under consideration is the method of non-integer (fractional) moments. It has been noted that the formula for calculating the stability factor estimate, developed in accordance with this method, contains a non-elementary and rarely used function (inverse to a gamma function), that significantly complicates using such estimate in applied problems. The problem of approximating the stability factor estimate has been set and successfully solved in the paper. The original dependence has been approximated with a simple fractional-linear function with quite a sufficient practical accuracy. The suggested approximation has given an opportunity of adjusting an asymptotic variance estimate of the parameter under estimation regarding its true value. As a result, the discrepancy between a theoretical estimate and the data of numerical experiments has been eliminated. The conducted numerical modeling has fully justified obtained results.

Closure of non-integer moments arising in multiphase flow phenomena

In this paper we consider closure problem for non-integer moments of form 〈xp〉,p∈R which are encountered in the moment evolution equations of many multiphase flow systems involving polydispersed particles, heat and mass transfer, chemical reactions, agglomeration or break-up. Two different categories of methods are considered: the first method is based on the reconstruction of the underlying PDF using Laguerre polynomials and the other is based on the direct calculation of non-integer moments using the fractional derivatives of moment generating function (MGF). By applying the results of fractional calculus an explicit equation is derived to express non-integer moments as a function of any arbitrary number of integer moments. The proposed methods are tested on several highly non-Gaussian analytical PDFs in addition to experimental agglomeration data and direct numerical simulation of fluid–particle turbulent multiphase flows.

Magnetic Properties of Ni Alloys for Superconducting Wire Substrates: A First-Principles Study

Ni-based alloys, especially Ni–W alloys, are widely used as substrates for high-temperature superconducting (HTSC) wires. Ni–W alloys have many useful properties, including, strength, flexibility, durability, and the ability to texture thin films grown on them. Ni metal has an fcc structure, and is well known to exhibit band magnetism on account of the splitting of its Ni 3d bands, which produce a non-integer magnetic moment per Ni atom (magnetic moment of Pure Ni: 0.6 µB/atom). However, the magnetic moment in a Ni-based alloy substrate interferes with the superconducting behavior of the film. Consequently, suitable alloys with reduced ferromagnetism are required. In this paper, we report first-principles calculations of the magnetic moments of a series of Ni-based alloys by the Korringa–Kohn–Rostoker coherent-potential approximation (KKR-CPA) method within local-density approximation (LDA). Using the KKR-CPA method, it is possible to treat any given Ni alloy composition. Our calculation results reproduce the experimental data well, showing the concentration dependences of the magnetic moments of various Ni alloys. The series of calculations of Ni 3d metal, Ni 4d metal and Ni 5d metal alloys reveal early 4d and early 5d metals (i.e., Nb, Mo, Ta, and W) successfully lower the magnetic moments. For these elements, the magnetic moments of the alloys become zero at around 10 at. % owing to the strong hybridization between the Ni 3d band and the Nb/Mo 4d or Ta/W 5d band around the Fermi level.

Disordered pinning models and copolymers: Beyond annealed bounds

We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by Garel et al. [Europhys. Lett. 8 (1989) 9–13], pinning and wetting models in various dimensions, and the Poland–Scheraga model of DNA denaturation. We prove a new variational upper bound for the free energy via an estimation of noninteger moments of the partition function. As an application, we show that for strong disorder the quenched critical point differs from the annealed one, for example, if the disorder distribution is Gaussian. In particular, for pinning models with loop exponent 0<α<1/2 this implies the existence of a transition from weak to strong disorder. For the copolymer model, under a (restrictive) condition on the law of the underlying renewal, we show that the critical point coincides with the one predicted via renormalization group arguments in the theoretical physics literature. A stronger result holds for a “reduced wetting model” introduced by Bodineau and Giacomin [J. Statist. Phys. 117 (2004) 801–818]: without restrictions on the law of the underlying renewal, the critical point coincides with the corresponding renormalization group prediction.