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Based on the random finite set (RFS) framework and the probability hypothesis density (PHD) filter, the extended target PHD (ET‐PHD) filter is proposed for multiple extended target tracking. However, the clutter process in the ET‐PHD filter is modelled as Poisson RFS, which is reasonable for only some scenarios in reality. An easily implemented ET‐PHD filter for arbitrary clutter process, has not yet been studied. In this work, another form of the general PHD filter, suitable for arbitrary clutter and target measurement is derived using a probability generating functional (PGFL)‐based RFS framework. The proposed filter is equivalent to the general PHD filter proposed by Clark and Mahler, but it is easily implemented in an arbitrary clutter process because its clutter process is denoted by the probability density function. Then, an arbitrary clutter ET‐PHD (AC‐ET‐PHD) filter is simplified from the general PHD filter derived by us. To reduce the computational complexity of the proposed filter, a variant of the distance partitioning algorithm is put forward. Simulation results show that the AC‐ET‐PHD filter can be applied to the scenarios with the non‐Poisson clutter process, which means that it will be useful for multiple extended targets tracking in a more complicated clutter process. [ABSTRACT FROM AUTHOR]

Abstract This paper considers a class of nonlinear filtering algorithms based on variational Bayesian (VB) theory to settle the unknown measurement noise problem in target tracking system. When the unknown measurement noise is conditionally independent of states, based on the variational idea, estimate of probability density function of state is converted into approximation two probability density functions for both unknown noise and nonlinear states. Then, an iterative algorithm is established to jointly estimate the state and the unknown measurement noise using variational Bayesian inference. Thus, the unknown measurement noise could be estimated as hidden state. The convergence result of the proposed nonlinear probability density function approximation algorithm is also given. The simulation results of typical examples show that the proposed VB based methods have superior performance to these classic algorithms in target tracking problems. [ABSTRACT FROM AUTHOR]

This study deals with the stochastic non-linear dynamic response of functionally graded materials (FGMs) plate with uncertain system properties subjected to time-dependent uniformly distributed transverse load in thermal environments. System properties, such as material properties of each constituent's material, volume fraction index, and transverse load, are taken as uncorrelated random input variables. Material properties are assumed as temperature dependent (TD). The formulation is based on higher-order shear deformation theory (HSDT) with von-Karman non-linear strain kinematics using modified C° continuity. A Newton-Raphson-based non-linear finite element method along with a first-order perturbation technique (FOPT) and Monte Carlo sampling (MCS) is outlined to examine the second-order statistics (mean, standard deviation (SD), and probability density function (PDF)) of the non-linear dynamic response of the FGM plate. The governing dynamic equation is solved by Newmark's time integration scheme. The effects of volume fraction index, load parameters, plate thickness ratios, and temperature changes with random system properties are examined through parametric studies. The present outlined approach is validated with the results available in the literature and by MCS. [ABSTRACT FROM AUTHOR]

In this paper, the influence of a zero and a non-zero boresight pointing errors on the performances of free-space optic transmission over the Málaga atmospheric turbulence channel is considered. Closed form expressions for a zero boresight channel model probability density function, non-zero boresight channel model probability density function, as well as a bit-error rate over a binary phase shift keying modulation transmission are provided. Numerical results for zero and non-zero boresight pointing errors are graphically presented. [ABSTRACT FROM AUTHOR]

Abstract A cantilever with an open surface crack excited by the pedestal movement with the form of the Gaussian white noise was modeled in this paper. The open crack was simplified as an elastic hinge and the modal functions of the cracked cantilever were obtained. The dynamical equation governing the transverse motion was gained with the Kane's method after sufficient consideration of the geometrical nonlinearity and the axial inertia. Both quasi-conservative stochastic averaging method and the improved energy envelop stochastic averaging method which is effective on strong nonlinearity were applied to analyze the responses of the cantilever model theoretically. The stationary probability density function (PDF) of the amplitude, as well as the joint stationary PDF of the displacement and velocity, was investigated. For the purpose of exploring the reliability of the cantilever, the reliability function was established to avoid the potential fatigue or damage. Numerical simulations were carried out to vindicate the theoretical analysis. The advantage of the improved energy envelop stochastic averaging method was explicated by the digital results of both the stationary PDF of the amplitude and the joint PDF of displacement and velocity. Furthermore, the accuracy of the prediction on the reliability given by the theoretical analysis was also vindicated by the numerical simulation (Monte Carlo method) of the reliability. [ABSTRACT FROM AUTHOR]

Abstract Wavelet estimators for a probability density f enjoy many good properties, however they are not 'shape-preserving' in the sense that the final estimate may not be non-negative or integrate to unity. A solution to negativity issues may be to estimate first the square root of f and then square this estimate up. This paper proposes and investigates such an estimation scheme, generalizing to higher dimensions some previous constructions which are valid only in one dimension. The estimation is mainly based on nearest-neighbor-balls. The theoretical properties of the proposed estimator are obtained, and it is shown to reach the optimal rate of convergence uniformly over large classes of densities under mild conditions. Simulations show that the new estimator performs as well in terms of Mean Integrated Squared Error as the classical wavelet estimator and better than it in terms of Mean Squared Hellinger Distance between the estimator and the truth, while automatically producing estimates which are bona fide densities. [ABSTRACT FROM AUTHOR]

We suggest a method of studying the joint probability density (JPD) of an eigenvalue and the associated ‘non-orthogonality overlap factor’ (also known as the ‘eigenvalue condition number’) of the left and right eigenvectors for non-selfadjoint Gaussian random matrices of size N×N. First we derive the general finite N expression for the JPD of a real eigenvalue λ and the associated non-orthogonality factor in the real Ginibre ensemble, and then analyze its ‘bulk’ and ‘edge’ scaling limits. The ensuing distribution is maximally heavy-tailed, so that all integer moments beyond normalization are divergent. A similar calculation for a complex eigenvalue z and the associated non-orthogonality factor in the complex Ginibre ensemble is presented as well and yields a distribution with the finite first moment. Its ‘bulk’ scaling limit yields a distribution whose first moment reproduces the well-known result of Chalker and Mehlig (Phys Rev Lett 81(16):3367-3370, 1998), and we provide the ‘edge’ scaling distribution for this case as well. Our method involves evaluating the ensemble average of products and ratios of integer and half-integer powers of characteristic polynomials for Ginibre matrices, which we perform in the framework of a supersymmetry approach. Our paper complements recent studies by Bourgade and Dubach (The distribution of overlaps between eigenvectors of Ginibre matrices, 2018. arXiv:1801.01219). [ABSTRACT FROM AUTHOR]

Flow regime map is often used in choosing constitutive correlations for the two-phase flow model. The related research mainly concentrates on the vertical upward and horizontal flow, while it is not sufficient in the vertical downward flow. Downward flow is very important as it is frequently encountered in the industrial applications. To enrich the downward flow research, an experiment is performed on a piping system with an inner diameter of 0.1524 m. Four different flow patterns (bubbly flow, cap bubbly flow, churn turbulent flow, and annular flow) are classified with the artificial neural network method. The probability density function (PDF) profile of each flow pattern is discussed. The proposed flow regime map is compared with the other experiments and the effect of the pipe size is discussed. The existing downward flow regime boundary criteria are assessed with the experiment results. It is found that these criteria cannot fit the experiment results well. A set of general boundary criteria are still needed. In this paper, the criteria for the boundary of the bubbly flow, the boundary between the cap bubbly flow and the slug flow, and the boundary of the falling film regime are proposed. They are verified with the experiments on different size pipes. A significant inlet effect on the flow regime boundary is found. The falling film boundary criterion proposed cannot be applied when a sparger is used to inject gas into the downward test section. [ABSTRACT FROM AUTHOR]

In this paper, we study the probability density function, P (c , α , β , n) d c , of the center of mass of the finite n Jacobi unitary ensembles with parameters α > −1 and β > −1; that is the probability that trMn ∈ (c, c + dc), where Mn are n × n matrices drawn from the unitary Jacobi ensembles. We compute the exponential moment generating function of the linear statistics ∑ j = 1 n f ( x j ) ≔ ∑ j = 1 n x j , denoted by M f (λ , α , β , n). The weight function associated with the Jacobi unitary ensembles reads xα(1 − x)β, x ∈ [0, 1]. The moment generating function is the n × n Hankel determinant Dn(λ, α, β) generated by the time-evolved Jacobi weight, namely, w (x; λ, α, β) = xα(1 − x)β e−λx, x ∈ [0, 1], α > −1, β > −1. We think of λ as the time variable in the resulting Toda equations. The non-classical polynomials defined by the monomial expansion, Pn(x, λ) = xn + p(n, λ) xn−1 + ⋯ + Pn(0, λ), orthogonal with respect to w (x, λ, α, β) over [0, 1] play an important role. Taking the time evolution problem studied in Basor et al. [J. Phys. A: Math. Theor. 43, 015204 (2010)], with some change of variables, we obtain a certain auxiliary variable rn(λ), defined by integral over [0, 1] of the product of the unconventional orthogonal polynomials of degree n and n − 1 and w (x; λ, α, β)/x. It is shown that rn(2iez) satisfies a Chazy II equation. There is another auxiliary variable, denoted as Rn(λ), defined by an integral over [0, 1] of the product of two polynomials of degree n multiplied by w (x; λ, α, β)/x. Then Yn(−λ) = 1 − λ/Rn(λ) satisfies a particular Painlevé V: PV(α2/2, − β2/2, 2n + α + β + 1, 1/2). The σn function defined in terms of the λp(n, −λ) plus a translation in λ is the Jimbo–Miwa–Okamoto σ-form of Painlevé V. The continuum approximation, treating the collection of eigenvalues as a charged fluid as in the Dyson Coulomb Fluid, gives an approximation for the moment generating function M f (λ , α , β , n) when n is sufficiently large. Furthermore, we deduce a new expression of M f (λ , α , β , n) when n is finite, in terms the σ function of this is a particular case of Painlevé V. An estimate shows that the moment generating function is a function of exponential type and of order n. From the Paley-Wiener theorem, one deduces that P (c , α , β , n) has compact support [0, n]. This result is easily extended to the β ensembles, as long as the weight w is positive and continuous over [0, 1]. [ABSTRACT FROM AUTHOR]

Numerical schemes for scalar transport and mixing in turbulent flows must be high-order accurate, and observe conservation and boundedness constraints. Discretization accuracy can be evaluated from the truncation error, and assessed by its dispersion and dissipation properties. Dispersion errors can cause violation of physical scalar bounds, whereas numerical dissipation is key to mitigating those violations. Numerical dissipation primarily alters the energy at small scales that are critical to turbulent mixing. Influence of additional dissipation on scalar mixing in large-eddy simulations (LES) of incompressible temporally evolving shear flow is examined in terms of the resolved passive-scalar field, Z ¯ . Scalar fields in flows with different mixing behavior, exhibiting both uniform and non-uniform mixed-fluid composition across a shear layer, are compared for different grid resolutions, subgrid-scale models, and scalar-convection schemes. Scalar mixing is assessed based on resolved passive scalar probability density function (PDF), variance, and spectra. The numerical-dissipation influence on mixing is found to depend on the nature of the flow. Mixing metrics sensitive to numerical dissipation are applied to examine the performance of limiting methods employed to mitigate unphysical scalar excursions. Two approaches, using a linear-scaling limiter for finite-volume schemes and a monotonicity-preserving limiter for finite-difference schemes, are studied. Their performance with respect to accuracy, conservation, and boundedness is discussed. [ABSTRACT FROM AUTHOR]

Over the years, combinations of different methods that use multi-satellites and multi-sensors have been developed for estimating global precipitation. Recently, studies that have evaluated Integrated Multi-satellite Retrievals for GPM (IMERG) Final-Run (FR) version V-03D and other precipitation products have indicated better performance for IMERG-FR compared to other similar products in different climate regimes. This study comprehensively evaluates the two GPM-IMERG products, specifically IMERG-FR and IMERG-Real-Time (RT) late-run, against a dense station network (62 stations) in northeast Austria from mid-March 2015 to the end of January 2016 using different time-scales. Both products are examined against station data in capturing the occurrence and statistical characteristics of precipitation intensity. With regard to probability density functions (PDFs), the satellite precipitation estimate (SPE) products have detected more heavy and extreme precipitation events than the ground measurements. Both precipitation products at all time-scales, except for IMERG-RT 12-hourly and daily precipitation, capture less occurrence of precipitation than the station dataset for light precipitation. This partially explains the under-detection of precipitation events. For all time-scales, both IMERG products' CDFs (Cumulative Distribution Function) are well above that of the stations' precipitation. For lower precipitation levels, IMERG-RT is slightly below the IMERG-FR whereas IMERG-RT is above IMERG-FR at higher precipitation levels. Furthermore, for entire spectrum precipitation rates (P ≥ 0.1 mm), 1, 3, 6-hourly, IMERG-FR did not show a clear improvement of the Bias over IMERG-RT, while for 12-hourly and daily precipitation estimates, the bias in IMERG-FR has improved compared to IMERG-RT. In addition, IMERG-FR shows a considerable improvement in RMSE as compared to IMERG-RT. IMERG-FR, however, systematically underestimates moderate to extreme precipitation and overestimates light precipitation for all time scales against rain-gauges in northeast Austria. When comparing the bias, RMSE, and correlation coefficients, IMERG-FR has outperformed IMERG-RT particularly for 6-hourly, 12-hourly, and daily precipitation. Despite the general low probability of detection (POD) and threat score (TS) and the high false alarm ratio (FAR) within specified precipitation thresholds, the contingency table shows relatively acceptable values of the POD, TS and FAR for precipitation without classification. [ABSTRACT FROM AUTHOR]

Among the combustion products, nitrogen oxides are one of the main contributors to a negative impact on the environment, participating in harmful processes such as tropospheric ozone and acid rains production. The main source of emissions of nitrogen oxides is the human combustion of fossil fuels. Their formation models are investigated and implemented with the goal of obtaining a tool for studying the nitrogen-containing pollutant production. In this work, numerical simulation of solid fuel combustion was carried out on a three-dimensional model of a drop tube furnace by using the commercial software FIRE. It was used for simulating turbulent fluid flow and temperature field, concentrations of the reactants and products, as well as the fluid-particles interaction by numerically solving the integro-differential equations describing these processes. Chemical reactions mechanisms for the formation of nitrogen oxides were implemented by the user functions. To achieve reasonable calculation times for running the simulations, as well as efficient coupling with the turbulent mixing process, the nitrogen scheme is limited to sufficiently few homogeneous reactions and species. Turbulent fluctuations that affect the reaction rates of nitrogen oxides' concentration are modelled by probability density function approach. Results of the implemented model for nitrogen oxides' formation from coal and biomass are compared to the experimental data. Temperature, burnout and nitrogen oxides' concentration profiles are compared, showing satisfactory agreement. The new model allows the simulation of pollutant formation in the real-world applications. [ABSTRACT FROM AUTHOR]

This paper develops a hybrid approach of spectral representation and random function for simulating stationary stochastic vector processes. In the proposed approach, the high-dimensional random variables, included in the original spectral representation (OSR) formula, could be effectively reduced to only two elementary random variables by introducing the random functions that serve as random constraints. Based on this, a satisfactory simulation accuracy can be guaranteed by selecting a small representative point set of the elementary random variables. The probability information of the stochastic excitations can be fully emerged through just several hundred of sample functions generated by the proposed approach. Therefore, combined with the probability density evolution method (PDEM), it could be able to implement dynamic response analysis and reliability assessment of engineering structures. For illustrative purposes, a stochastic turbulence wind velocity field acting on a frame-shear-wall structure is simulated by constructing three types of random functions to demonstrate the accuracy and efficiency of the proposed approach. Careful and in-depth studies concerning the probability density evolution analysis of the wind-induced structure have been conducted so as to better illustrate the application prospects of the proposed approach. Numerical examples also show that the proposed approach possesses a good robustness. [ABSTRACT FROM AUTHOR]

The fitting of θ/2θ and ω peaks in X-ray diffraction curves is shown to be most accurate in the case of using an inverse fourth-degree polynomial or probability density function with Student’s distribution (Pearson type VII function). These functions describe well both the highest-intensity central part of the experimental peak and its low-intensity broadened base caused by X-ray diffuse scattering. The mean microdeformation ε and mean vertical domain size D are determined by the Williamson-Hall method for layers of GaN (ε ≈ 0.00006, D ≈ 200 nm) and Al0.32Ga0.68N (ε = 0.0032 ± 0.0005, D = 24 ± 7 nm). The D value obtained for the Al0.32Ga0.68N layer is most likely to result from the nominal thickness of this layer, which is 11 nm. [ABSTRACT FROM AUTHOR]

This paper proposes a novel and efficient clustering algorithm for probability density functions based on k-medoids. Further, a scheme used for selecting the powerful initial medoids is suggested, which speeds up the computational time significantly. Also, a general proof for convergence of the proposed algorithm is presented. The effectiveness and feasibility of the proposed algorithm are verified and compared with various existing algorithms through both artificial and real datasets in terms of adjusted Rand index, computational time, and iteration number. The numerical results reveal an outstanding performance of the proposed algorithm as well as its potential applications in real life. [ABSTRACT FROM AUTHOR]

Maps on a parameter space for expressing distribution functions are exactly derived from the Perron-Frobenius equations for a generalized Boole transform family. Here the generalized Boole transform family is a one-parameter family of maps, where it is defined on a subset of the real line and its probability distribution function is the Cauchy distribution with some parameters. With this reduction, some relations between the statistical picture and the orbital one are shown. From the viewpoint of information geometry, the parameter space can be identified with a statistical manifold, and then it is shown that the derived maps can be characterized. Also, with an induced symplectic structure from a statistical structure, symplectic and information geometric aspects of the derived maps are discussed. [ABSTRACT FROM AUTHOR]

Current degradation modeling and remaining useful life prediction studies share a common assumption that the degrading systems are not maintained or maintained perfectly (i.e., to an as-good-as new state). This paper concerns the issues of how to model the degradation process and predict the remaining useful life of degrading systems subjected to imperfect maintenance activities, which can restore the health condition of a degrading system to any degradation level between as-good-as new and as-bad-as old. Toward this end, a nonlinear model driven by Wiener process is first proposed to characterize the degradation trajectory of the degrading system subjected to imperfect maintenance, where negative jumps are incorporated to quantify the influence of imperfect maintenance activities on the system’s degradation. Then, the probability density function of the remaining useful life is derived analytically by a space-scale transformation, i.e., transforming the constructed degradation model with negative jumps crossing a constant threshold level to a Wiener process model crossing a random threshold level. To implement the proposed method, unknown parameters in the degradation model are estimated by the maximum likelihood estimation method. Finally, the proposed degradation modeling and remaining useful life prediction method are applied to a practical case of draught fans belonging to a kind of mechanical systems from steel mills. The results reveal that, for a degrading system subjected to imperfect maintenance, our proposed method can obtain more accurate remaining useful life predictions than those of the benchmark model in literature. [ABSTRACT FROM AUTHOR]

We report a non-perturbative study of the effects of shear flows on turbulence reduction in a decaying turbulence in two dimensions. By considering different initial power spectra and shear flows (zonal flows, streamers and zonal flows and streamers combined), we demonstrate how shear flows rapidly generate small scales, leading to a fast damping of turbulence amplitude. In particular, a double exponential decrease in the turbulence amplitude is shown to occur due to an exponential increase in wavenumber. The scaling of the effective dissipation time scale τe, previously taken to be a hybrid time scale τe ∝ τ2/3 Ω τη, is shown to depend on types of shear flow as well as the initial power spectrum. Here, τΩ and τη are shearing and molecular diffusion times, respectively. Furthermore, we present time-dependent Probability Density Functions (PDFs) and discuss the effect of enhanced dissipation on PDFs and a dynamical time scale τ(t), which represents the time scale over which a system passes through statistically different states. [ABSTRACT FROM AUTHOR]

Motivated by high filtration efficiency (mass- and number-based) and low pressure drop requirements for gasoline particulate filters (GPFs), a previously developed heterogeneous multiscale filtration (HMF) model is extended to simulate dynamic filtration characteristics of GPFs. This dynamic HMF model is based on a probability density function (PDF) description of the pore size distribution and classical filtration theory. The microstructure of the porous substrate in a GPF is resolved and included in the model. Fundamental particulate filtration experiments were conducted using an exhaust filtration analysis (EFA) system for model validation. The particulate in the filtration experiments was sampled from a spark-ignition direct-injection (SIDI) gasoline engine. With the dynamic HMF model, evolution of the microscopic characteristics of the substrate (pore size distribution, porosity, permeability, and deposited particulate inside the porous substrate) during filtration can be probed. Also, predicted macroscopic filtration characteristics including particle number concentration and normalized pressure drop show good agreement with the experimental data. The resulting dynamic HMF model can be used to study the dynamic particulate filtration process in GPFs with distinct microstructures, serving as a powerful tool for GPF design and optimization. [ABSTRACT FROM AUTHOR]

Nearest-neighbor queries, which ask for returning the nearest neighbor of a query point in a set of points, are important and widely studied in many fields because of a wide range of applications. In many of these applications, such as sensor databases, location based services, face recognition, and mobile data, the location of data is imprecise. We therefore study nearest-neighbor queries in a probabilistic framework in which the location of each input point and/or query point is specified as a probability density function and the goal is to return the point that minimizes the expected distance, which we refer to as the expected nearest neighbor ( $$\mathop {\mathrm {ENN}}$$ ). We present methods for computing an exact $$\mathop {\mathrm {ENN}}$$ or an $$\varepsilon $$ -approximate $$\mathop {\mathrm {ENN}}$$ , for a given error parameter $$0<\varepsilon < 1$$ , under different distance functions. These methods build a data structure of near-linear size and answer $$\mathop {\mathrm {ENN}}$$ queries in polylogarithmic or sublinear time, depending on the underlying function. As far as we know, these are the first nontrivial methods for answering exact or $$\varepsilon $$ -approximate $$\mathop {\mathrm {ENN}}$$ queries with provable performance guarantees. Moreover, we extend our results to answer exact or $$\varepsilon $$ -approximate k- $$\mathop {\mathrm {ENN}}$$ queries. [ABSTRACT FROM AUTHOR]

This paper investigates the force transmission modes, mainly described by probability density distributions, in non-cohesive dry and cohesive wet granular materials by discrete element modeling. The critical state force transmission patterns are focused on with the contact model effect being analyzed. By shearing relatively dense and loose dry specimens to the critical state in the conventional triaxial loading path, it is observed that there is a unique critical state force transmission mode. There is a universe critical state force distribution pattern for both the normal contact forces and tangential contact forces. Furthermore, it is found that using either the linear Hooke or the non-linear Hertz model does not affect the universe force transmission mode, and it is only related to the grain size distribution. Wet granular materials are also simulated by incorporating a water bridge model. Dense and loose wet granular materials are tested, and the critical state behavior for the wet material is also observed. The critical state strength and void ratio of wet granular materials are higher than those of a non-cohesive material. The critical state inter-particle distribution is altered from that of a non-cohesive material with higher probability in relatively weak forces. Grains in non-cohesive materials are under compressive stresses, and their principal directions are mainly in the axial loading direction. However, for cohesive wet granular materials, some particles are in tension, and the tensile stresses are in the horizontal direction on which the confinement is applied. The additional confinement by the tensile stress explains the macro strength and dilatancy increase in wet samples. [ABSTRACT FROM AUTHOR]

The rise of Performance Based Design methodologies for fire safety engineering has increased the interest of the fire safety community in the concepts of risk and reliability. Practical applications have however been severely hampered by the lack of an efficient unbiased calculation methodology. This is because on the one hand, the distribution types of model output variables in fire safety engineering are not known and traditional distribution types as for example the normal and lognormal distribution may result in unsafe approximations. Therefore unbiased methods must be applied which make no (implicit) assumptions on the PDF type. Traditionally these unbiased methods are based on Monte Carlo simulations. On the other hand, Monte Carlo simulations require a large number of model evaluations and are therefore too computationally expensive when large and nonlinear calculation models are applied, as is common in fire safety engineering. The methodology presented in this paper avoids this deadlock by making an unbiased estimate of the PDF based on only a very limited number of model evaluations. The methodology is known as the Maximum Entropy Multiplicative Dimensional Reduction Method (ME-MDRM) and results in a mathematical formula for the probability density function (PDF) describing the uncertain output variable. The method can be applied with existing models and calculation tools and allows for a parallelization of model evaluations. The example applications given in the paper stem from the field of structural fire safety and illustrate the excellent performance of the method for probabilistic structural fire safety engineering. The ME-MDRM can however be considered applicable to other types of engineering models as well. [ABSTRACT FROM AUTHOR]

Based on the random theory of the probability density evolution method (PDEM), a refined random wheel/rail contact model for train-track-bridge systems is proposed. The trace line method with the law of equal slopes is developed to determine the wheel/rail contact locations, which are described by probability density functions on the wheel/rail surface. The number theoretic method (NTM) and stochastic harmonic functions (SHFs) are employed to generate the representative random track irregularities. Considering the excitation of random track irregularities and the wheel/rail contact geometry, the wheel/rail normal contact force and the tangential creep force are accurately calculated and analyzed using the proposed wheel/rail contact model. A high-speed railway case study is presented to reveal the random dynamic characteristics of a train-track-bridge system and the wheel/rail interaction. The results show that the time-varying probability density evolution functions of the wheel/rail contact points can effectively indicate both the phenomenon of wear on the rail surface and its distribution characteristics. Additionally, some significant conclusions are obtained. [ABSTRACT FROM AUTHOR]

Motivated by that the quartic potential can confined Lévy flights, we investigate the escape rate of an inertial Lévy particle from a truncated quartic potential well via Langevin simulation. The escape rate still depends on the noise intensity in a power-law form in low noise intensity, but the exponent and the inverse coefficient vary significantly for different Lévy indexes compared with previous works. Trimodal structure of the probability density function was found in simulations. The probability density function in a quasi-stable state exhibits transition among unimodal, bimodal, and trimodal structures. A metastable state by stable state approach is developed to calculate the escape rate analytically, which may be applied to extensive escape problems. The theoretical approach is confirmed by Langevin simulation for the Cauchy case of Lévy flight in the applied potential. [ABSTRACT FROM AUTHOR]

The probability density evolution method (PDEM) provides a feasible approach for the dynamic response analysis of nonlinear stochastic structures. The key step in this regard is to solve a generalized density evolution equation (GDEE) in order to establish the probability density function (PDF). Previously, a finite difference method (FDM) has often been resorted to solve the GDEE. However, one may encounter the problem of mesh sensitivity in the application of FDM to the PDEM. To this end, a novel difference-wavelet method that can improve the finite difference result by means of a nonlinear wavelet density estimation method is proposed in the present paper. By exploiting the multi-resolution property of wavelet functions and by choosing the optimal scale at each instant, it is expected that the bothering mesh sensitivity issue in finite difference method can be overcome to some extent and a better probability density result can be obtained. In order to verify the proposed method, a single-degree-of-freedom (SDOF) oscillator and an 8-story frame structure are investigated in detail. The results show the notable superiority of the proposed method to finite difference method. [ABSTRACT FROM AUTHOR]

In an elementary variational treatment of the electronic structure of H2+, Eyring, Walter, and Kimball (EWK) serendipitously discovered charge migration (CM) in 1944. Using an analytic expression for the electronic probability density (EPD), they found that if the electron is initially localized on one of the protons (by taking the initial state to be a superposition of the ground and first excited electronic energy eigenstates), then it oscillates adiabatically between fixed protons with a period T inversely proportional to the energy gap between the eigenstates. At the equilibrium internuclear separation, T = 550.9 as. As shown here, the EWK model also yields an analytic formula for the electronic flux density (EFD). While the EPD indicates where the electron is at any instant, the EFD reveals the pathways the electron follows during its migration. Thus, the EFD complements the EPD, providing valuable new insight into the mechanism of CM. The formula for the EFD is a simple product of a time factor and a spatial factor. This factoring exposes a plethora of spatial-temporal symmetry relations which imply novel and surprising properties. An especially significant finding is that, in contrast to multielectron systems, where electron correlation may play a role in CM, in the EWK model of H2+, CM is due strictly to quantum interference between the ground and first excited electronic states. [ABSTRACT FROM AUTHOR]

We present 21 new radio-continuum detections at catalogued planetary nebula (PN) positions in the Large Magellanic Cloud (LMC) using all presently available data from the Australia Telescope Online Archive at 3, 6, 13 and 20 cm. Additionally, 11 previously detected LMC radio PNe are re-examined with seven detections confirmed and reported here. An additional three PNe from our previous surveys are also studied. The last of the 11 previous detections is now classified as a compact H ii region that makes for a total sample of 31 radio PNe in the LMC. The radio-surface-brightness-to-diameter (&#931--D) relation is parametrized as Σ ∝ D-β. With the available 6 cm Σ–D data, we construct Σ–D samples from 28 LMC PNe and nine Small Magellanic Cloud (SMC) radio-detected PNe. The results of our sampled PNe in the MCs are comparable to previous measurements of the Galactic PNe. We obtain β = 2.9 ± 0.4 for the MC PNe compared to β = 3.1 ± 0.4 for the Galaxy. For a better insight into sample completeness and evolutionary features, we reconstruct the &#931--D data probability density function (PDF). The PDF analysis implies that PNe are not likely to follow linear evolutionary paths. To estimate the significance of sensitivity selection effects, we perform a Monte Carlo sensitivity simulation on the &#931--D data. The results suggest that selection effects are significant for values larger than β ~ 2.6 and that a measured slope of β = 2.9 should correspond to a sensitivity-free value of ~3.4. [ABSTRACT FROM AUTHOR]

Stochastic distribution control (SDC) systems are a group of systems where the outputs considered is the measured probability density function (PDF) of the system output whilst subjected to a normal crisp input. The purpose of the active fault tolerant control of such systems is to use the fault estimation information and other measured information to make the output PDF still track the given distribution when the objective PDF is known. However, if the target PDF is unavailable, the PDF tracking operation will be impossible. Minimum entropy control of the system output can be considered as an alternative strategy. The mean represents the center location of the stochastic variable, and it is reasonable that the minimum entropy fault tolerant controller can be designed subjected to mean constraint. In this paper, using the rational square-root B-spline model for the shape control of the system output probability density function (PDF), a nonlinear adaptive observer based fault diagnosis algorithm is proposed to diagnose the fault. Through the controller reconfiguration, the system entropy subjected to mean restriction can still be minimized when fault occurs. An illustrative example is utilized to demonstrate the use of the minimum entropy fault tolerant control algorithms. [ABSTRACT FROM AUTHOR]

At low densities, the standard ionization history of the intergalactic medium (IGM) predicts a decreasing temperature of the IGM with decreasing density once hydrogen (and helium) reionization is complete. Heating the high-redshift, low-density IGM above the temperature expected from photoheating is difficult, and previous claims of high/rising temperatures in lowdensity regions of the Universe based on the probability density function (PDF) of the opacity in Ly α forest data at 2 < z < 4 have been met with considerable scepticism, particularly since they appear to be in tension with other constraints on the temperature-density relation (TDR). We utilize here an ultrahigh signal-to-noise spectrum of the Quasi-stellar object HE0940-1050 and a novel technique to study the low opacity part of the PDF. We show that there is indeed evidence (at 90 per cent confidence level) that a significant volume fraction of the underdense regions at z ~ 3 has temperatures as high or higher than those at densities comparable to the mean and above. We further demonstrate that this conclusion is nevertheless consistent with measurements of a slope of the TDR in overdense regions that imply a decreasing temperature with decreasing density, as expected if photoheating of ionized hydrogen is the dominant heating process.We briefly discuss implications of our findings for the need to invoke either spatial temperature fluctuations, as expected during helium reionization, or additional processes that heat a significant volume fraction of the low-density IGM. [ABSTRACT FROM AUTHOR]

A parameterization scheme is proposed for the subgrid-scale transport of hydrometeors in an assumed probability density function (PDF) scheme. Joint distributions of vertical velocity and hydrometeor mixing ratios are typically unknown, but marginal (1D) PDFs of these variables are available. The parameterization is developed using high-resolution simulations of continental and tropical deep convection. A 3D cloudresolving model (CRM) providing benchmark solutions has a horizontal grid spacing of 250m and employs the Morrison microphysics scheme, which treats prognostically mass and number mixing ratios for four types of precipitating hydrometeors (rain, graupel, snow, and ice) as well as cloud droplet number mixing ratio. The subgrid-scale hydrometeor transport scheme assumes input given in the form of marginal PDFs of vertical velocity and hydrometeor mixing ratios; in this study, these marginal distributions are provided by the cloudresolving model. Conditional sampling and scaling are then applied to the marginal distributions to account for subplume correlations. The parameterized fluxes tested for four episodes of deep convection show good agreement with benchmark fluxes computed directly from the CRM output. The results demonstrate the potential use of the subgrid-scale hydrometeor transport scheme in an assumed PDF scheme to parameterize the covariances of vertical velocity and hydrometeor mixing ratios. [ABSTRACT FROM AUTHOR]

There are relatively few publications that assess capacity decline in enough commercial cells to quantify cell-to-cell variation, but those that do show a surprisingly wide variability. Capacity curves cross each other often, a challenge for efforts to measure the state of health and predict the remaining useful life (RUL) of individual cells. We analyze capacity fade statistics for 24 commercial pouch cells, providing an estimate for the time to 5% failure. Our data indicate that RUL predictions based on remaining capacity or internal resistance are accurate only once the cells have already sorted themselves into “better” and “worse” ones. Analysis of our failure data, using maximum likelihood techniques, provide uniformly good fits for a variety of definitions of failure with normal and with 2- and 3-parameter Weibull probability density functions, but we argue against using a 3-parameter Weibull function for our data. pdf fitting parameters appear to converge after about 15 failures, although business objectives should ultimately determine whether data from a given number of batteries provides sufficient confidence to end lifecycle testing. Increased efforts to make batteries with more consistent lifetimes should lead to improvements in battery cost and safety. [ABSTRACT FROM AUTHOR]

Data collection process in most observational and experimental studies yield different types of variables, leading to the use of joint models that are capable of handling multiple data types. Evaluation of various statistical techniques that have been developed for mixed data in simulated environments requires concurrent generation of multiple variables. In this article, I present an important augmentation to a unified framework proposed in our previously published work for simultaneously generating binary and nonnormal continuous data given the marginal characteristics and correlation structure, via fifth-order power polynomials that are known to extend the area covered in the skewness-elongation plane and to provide a better approximation to the probability density function of the continuous variables. I evaluate how well the improved methodology performs in comparison to the original one, in a simulated setting with illustrations of algorithmic steps. Although the relative gains for the associational quantities are not substantial, the augmented version appears to better capture the marginal quantities that are pertinent to the higher-order moments, as indicated by very close resemblance between the specified and empirically computed quantities on average. [ABSTRACT FROM AUTHOR]

Results related to the filtered non-premixed turbulent flame statistics are presented. Considering non-premixed turbulent combustion modeling in the framework of large eddy simulations (LES), a turbulent flamelet equation is derived based on the filtered governing equations. Using a direct numerical simulation (DNS) data set, the statistical properties of two important parameters, the scalar dissipation and chemical source term, are analyzed in details. Numerical results show that for the scalar dissipation term, the probability density function (PDF) of the magnitude of the mixture fraction gradient changes is strongly dependent on the filter scale, whereas the PDF of the turbulent diffusivity is almost invariant. Theoretically from the scalar structure function a scaling law can be expected for the conditional mean of the difference between the filtered and nonfiltered scalars with respect to the filter scale, which is important for estimating the chemical source in the subgrid scale models. [ABSTRACT FROM AUTHOR]

A key physical quantity during reionization is the size of H II regions. Previous studies found a characteristic bubble size which increases rapidly during reionization, with apparent agreement between simulations and analytic excursion set theory. Using four different methods, we critically examine this claim. In particular, we introduce the use of the watershed algorithm - widely used for void finding in galaxy surveys - which we show to be an unbiased method with the lowest dispersion and best performance on Monte Carlo realizations of a known bubble size probability density function (PDF).We find that a friends-of-friends algorithm declares most of the ionized volume to be occupied by a network of volume-filling regions connected by narrow tunnels. For methods tuned to detect the volume-filling regions, previous apparent agreement between simulations and theory is spurious, and due to a failure to correctly account for the window function of measurement schemes. The discrepancy is already obvious from visual inspection. Instead, H II regions in simulations are significantly larger (by factors of 10-1000 in volume) than analytic predictions. The size PDF is narrower, and evolves more slowly with time, than predicted. It becomes more sharply peaked as reionization progresses. These effects are likely caused by bubble mergers, which are inadequately modelled by analytic theory. Our results have important consequences for high-redshift 21 cm observations, the mean free path of ionizing photons, and the visibility of Lyα emitters, and point to a fundamental failure in our understanding of the characteristic scales of the reionization process. [ABSTRACT FROM AUTHOR]

The recently published analytic probability density function for the mildly non-linear cosmic density field within spherical cells is used to build a simple but accurate maximum likelihood estimate for the redshift evolution of the variance of the density, which, as expected, is shown to have smaller relative error than the sample variance. This estimator provides a competitive probe for the equation of state of dark energy, reaching a few per cent accuracy on wp and wa for a Euclid-like survey. The corresponding likelihood function can take into account the configuration of the cells via their relative separations. A code to compute one-cell-density probability density functions for arbitrary initial power spectrum, top-hat smoothing and various spherical-collapse dynamics is made available online, so as to provide straightforward means of testing the effect of alternative dark energy models and initial power spectra on the low-redshift matter distribution. [ABSTRACT FROM AUTHOR]

Learning introductory quantum physics is challenging, in part due to the different paradigms in classical mechanics and quantum physics. Classical mechanics is deterministic in that the equations of motion and the initial conditions fully determine a particle's trajectory. Quantum physics is an inherently probabilistic theory in that only probabilities for measurement outcomes can be determined. Prior to studying quantum physics, students will typically have little experience with probabilistic analyses of physical systems, and thus probability may be a conceptual hurdle for introductory quantum physics students. This article describes two interactive simulations developed as part of the QuVis Quantum Mechanics Visualization Project that aim to bridge the gap between classical mechanics and quantum physics using probabilistic analyses of classical systems. The simulations illustrate how a probability density can be obtained for two classical systems well known to students. The key learning goals of the simulations are to introduce students to probability densities and to help students distinguish between a probability and a probability density. The simulations build on previous work by Bao and Redish, who developed an activity that used pseudo-random video frames of a glider in harmonic motion to derive a classical probability density for this system, and a University of Washington quantum mechanics tutorial focusing on probability and probability density for a classical system. Interactive simulations allow students to easily carry out experiments and change variables that would be difficult to do with real equipment, and help students connect multiple representations by showing explicitly how they are linked. The simulations described here only require basic knowledge of algebra and classical mechanics. They run on touchscreen devices as well as desktop computers, and can be run in a standard web browser from the QuVis website or downloaded for offline use. [ABSTRACT FROM AUTHOR]

In this paper we present a framework to model the effect of multiple sources of degradation on deteriorating systems, and to find easy-to-evaluate expressions for important reliability quantities. By considering the system deterioration as a general increasing Lévy process (i.e., a subordinator), which is a process with independent, stationary and non-negative increments, the proposed methodology allows the modeling of shock-based degradation (in the form of compound Poisson processes) with different distributions of shock sizes, progressive degradation as deterministic linear drift plus a Gamma process, and multiple sources of degradation by the superposition of any of these models into the same mathematical formalism. In addition, we obtain expressions for the reliability quantities (reliability function, and probability density and moments of the lifetime L ), and the mean and central moments of the deterioration process X t . Moreover, we present efficient and accurate numerical methods to compute the reliability quantities and to simulate sample paths. Several deterioration models are compared in terms of their reliability quantities, the simulated sample paths and the feasible moments of the deterioration X t . Furthermore, we propose a method to mix different Lévy models that extend the available moments with possible applications to data fitting. The results demonstrate the generality, versatility, efficiency and accuracy of the proposed Lévy degradation framework, which can open a new productive research field in the area of probabilistic degradation models. [ABSTRACT FROM AUTHOR]

The mobilized shear strength ( τ f m ) of a soil is the most important design parameter governing ultimate limit states. The authors have proposed some models and equations to characterize several useful aspects of the mean, variance, and probability density function of τ f m for a two dimensional (2D) spatially variable clay specimen. This paper demonstrates that the models and equations developed for 2D can be extended to three-dimensional (3D) cases. An important class of 3D cases are examined in detail: two horizontal scales of fluctuation (SOF) are equal, and both are larger than the vertical SOF. A key observation for 3D cases is that the averaging effect over the potential slip planes is significantly stronger than that for 2D cases. The upshot is that the phenomenon of the critical SOF (e.g., SOF producing the lowest mean value of τ f m ) is less clear for 3D cases. [ABSTRACT FROM AUTHOR]

The analysis of reliability of complex engineering systems remains a challenge in the field of reliability. It will be even more difficult if correlated random vectors are involved, which is generally the case as practical engineering systems invariably contain parameters that are mutually correlated. A new method for transforming correlated distributions, involving the Nataf transformation, is proposed that avoids the solution of integral equations; the method is based on the Taylor series expansion of the probability density function (PDF) of a bivariate normal distribution resulting in an explicit polynomial equation of the equivalent correlation coefficient. The required numerical results can be obtained efficiently and accurately. The proposed method for transformation of correlated random vectors is useful for developing a method for system reliability including complex systems with correlated random vectors. Based on the complete system failure process (originally defined as the development process of nonlinearity) and the fourth-moment method, the analysis of system reliability for elastic-plastic material avoids the identification of the potential failure modes of the system and their mutual correlations which are required in the traditional methods. Finally, four examples are presented – two examples to illustrate the potential of the new method for transformation of correlated random vectors, and two examples to illustrate the application of the proposed more effective method for system reliability. [ABSTRACT FROM AUTHOR]

With the increasing permeability of photovoltaic (PV) power production, the uncertainties and randomness of PV power have played a critical role in the operation and dispatch of the power grid and amplified the abandon rate of PV power. Consequently, the accuracy of PV power forecast urgently needs to be improved. Based on the amplitude and fluctuation characteristics of the PV power forecast error, a short-term PV output forecast method that considers the error calibration is proposed. Firstly, typical climate categories are defined to classify the historical PV power data. On the one hand, due to the non-negligible diversity of error amplitudes in different categories, the probability density distributions of relative error (RE) are generated for each category. Distribution fitting is performed to simulate probability density function (PDF) curves, and the RE samples are drawn from the fitted curves to obtain the sampling values of the RE. On the other hand, based on the fluctuation characteristic of RE, the recent RE data are utilized to analyze the error fluctuation conditions of the forecast points so as to obtain the compensation values of the RE. The compensation values are adopted to sequence the sampling values by choosing the sampling values closest to the compensation ones to be the fitted values of the RE. On this basis, the fitted values of the RE are employed to correct the forecast values of PV power and improve the forecast accuracy. [ABSTRACT FROM AUTHOR]

Functional reliability is one of the important properties of physical systems provided by reliability of system components, in particular, control logical components. The new approach to fully delay testable circuit design oriented to cut overheads and lengths of circuit paths has been developed. Compact representation of all PDF test pairs is reduced to keeping the corresponding generative vector pairs. The number of generative vector pairs does not exceed the doubled number of internal ROBDD nodes originating from the circuit, while the number of the circuit paths can exponentially depend on the number of these internal nodes. The algorithm of involving the PDF test pair from the proper generative vector pair is suggested. This procedure does not require essential calculations. The algorithm of deriving the generative vector pair has a polynomial complexity. [ABSTRACT FROM AUTHOR]

It usually suffers from long observing time and interference sensitivity when a radar transmits the high-range-resolution stepped-frequency chirp signal. Motivated by this, only partial pulses of the stepped-frequency chirp are utilized. For the obtained incomplete frequency data, a Bayesian model based on transfer learning is proposed to reconstruct the corresponding full-band frequency data. In the training phase, a complex beta process factor analysis (CBPFA) model is utilized to statistically model each aspect-frame from a set of given full-band frequency data, whose probability density function (pdf) can be learned from this CBPFA model. It is important to note that the numbers of factors and dictionaries are automatically learned from the data. The inference of CBPFA can be performed via the variational Bayesian (VB) method. In the reconstruction phase for the incomplete frequency data that “related” to the training samples, its corresponding full-band frequency data can be analytically reconstructed via the compressive sensing (CS) method and Bayesian criterion based on the transfer knowledge of the previous pdfs learned from the training phase. About the “relatedness” between each training frame and the incomplete test frequency data, we utilize the frame condition distribution of incomplete test frequency data to represent. The proposed method is validated on the measured high range resolution (HRR) data. [ABSTRACT FROM PUBLISHER]

A theoretical model is proposed to describe the steady-state average limiting current associated with a single redox moiety (ox or red) trapped in a closed generator-collector system along with excess supporting electrolyte. By "closed" we mean that neither solvent nor solutes can enter or leave the system. The potential difference, EOE - ERE, between the oxidizing electrode (OE) and the reducing electrode (RE) is maintained constant with the values of EOE and ERE chosen so that the operative faradaic electrode processes are very fast, i.e., red = ox + nETe- (kox = ∞) at the OE and ox + nETe- = red (kred = ∞) at the RE. Because there is only a single redox moiety the faradaic process occurs at only one electrode at a time while current at the other electrode is purely capacitive (we refer to this as capacitive coupling). We propose that a two-step process is required to transfer nETqe coulombs (qe is the absolute value of the elemental electronic charge). The first step is associated with diffusion (approximated as a random walk) of a single red moiety to the OE where it is oxidized to ox with a concomitant transfer of qstep1 (= nETqe/(1 + AOECOE/ARECRE)) coulombs; the second step is associated with the diffusion (random walk) of the newly formed single ox moiety to the RE with the concomitant transfer of qstep2 (= nETqe/(1 + ARECRE/AOECOE)) coulombs (ARE,AOE and CRE,COEare the areas (cm2) and differential capacitances (farads cm-2) of the corresponding electrodes). The total charge transferred in the two steps is nETqe(= qstep1 + qstep2). Transport of the redox moiety from one electrode to the other is accomplished by a random walk. The probability density function (pdf) and cumulative density function (CDF) for the duration of a full redox cycle are presented as the analytical solution to a 1-dimensional bounded random-walk problem (confirmed by numerical simulation). These show that tfull, the average time for the full redox cycle (step 1 + step 2), is equal to L2/D where L is the intraelectrode distance and D is the diffusion coefficient. The average steady-state limiting current is shown to be described by the familiar expression for a generator-collector system: ilim = (qstep1 + qstep2)/√tfull = nETqe/√tfull = nETqeD/L2. [ABSTRACT FROM AUTHOR]

A necessary and sufficient form of two-point velocity characteristic function to embody two-point velocity distribution in turbulence is constructed on the mathematical basis of homogeneity and isotropy. This is applied in the first equation (for one-point velocity probability density) of the Monin-Lundgren hierarchy to see its substantial effect on the dynamics of homogeneous isotropic turbulence, the pressure term in which then is proved to vanish, as argued in "One-point velocity statistics in decaying homogeneous isotropic turbulence," Phys. Rev. E 78, 066312 (2008). Furthermore, an approximate form of three-point velocity characteristic function is searched on this basis, so that we obtain a simple closed hierarchy at the second equation stage. Thereby a certain closure method for the hierarchy in homogeneous, isotropic turbulence is illuminated from a new point of view. [ABSTRACT FROM AUTHOR]

We shall study random flights that start in a space of one given dimension and, after performing a definite number of steps, continue to develop in a space of higher dimension. We show that if the difference of the dimension of spaces is even, then the probability density describing the composite flight can be expressed as marginalizations of the probability density associated to a random flight in the space of less dimensions. This dimensional reduction is a consequence of Gegenbauer addition theorem. [ABSTRACT FROM AUTHOR]

In this paper the problem smoothing parameter choice for probability density kernel estimation is considered. The task of optimal smoothing parameter search according to predetermined criteria is formulated. [ABSTRACT FROM AUTHOR]

A flow of physical events (photons, electrons, etc.) is studied. One of the mathematical models of such flows is the Markovian arrival process (MAP) of flow of events. The flow functions are considered under conditions of unextendable dead time period. Explicit expressions for the probability density and joint probability density of interarrival interval of the observable flow are presented considering the effect of the unextendable dead time period. The recurrence relations for the observable flow of events are formulated. [ABSTRACT FROM AUTHOR]

In this communication, we address a numerical method to efficiently calculate a non-zero optimal threshold value for the performance improvement of a natural frequency-based radar target recognition. From the probability of correct classification with respect to a varying threshold value, we define a function, denoted by f(z) in this communication, so that one of the roots of f(z)=0 corresponds to the optimal threshold. The final optimal threshold value is obtained via the Newton iteration. The scheme is validated by comparing the threshold obtained from the Newton iteration with that using the probability density function. [ABSTRACT FROM PUBLISHER]