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According to the Ray-Knight description, the Brownian local time in some conditional probability space is a diffusion process with respect to spatial variable. This diffusion has a local time. Thus the following definition of local time of the original Brownian local time seems to be natural: such a process is called the second-order Brownian local time. The paper studies the Laplace transform of the distribution of the Brownian local time of the second order. [ABSTRACT FROM AUTHOR]

A family of parsimonious contaminated shifted asymmetric Laplace mixtures is developed for unsupervised classification of asymmetric clusters in the presence of outliers and noise. A series of constraints are applied to a modified factor analyzer structure of the component scale matrices, yielding a family of twelve models. Application of the modified factor analyzer structure and these parsimonious constraints makes these models effective for the analysis of high-dimensional data by reducing the number of free parameters that need to be estimated. A variant of the expectation-maximization algorithm is developed for parameter estimation with convergence issues being discussed and addressed. Popular model selection criteria like the Bayesian information criterion and the integrated complete likelihood (ICL) are utilized, and a novel modification to the ICL is also considered. Through a series of simulation studies and real data analyses, that includes comparisons to well-established methods, we demonstrate the improvements in classification performance found using the proposed family of models. [ABSTRACT FROM AUTHOR]

We present a novel depth from focus technique. Following prior work, our pipeline starts with a focal stack and an estimation of the amount of defocus as given by, for instance, the ring difference filter. To improve robustness to outliers while avoiding to rely on costly nonlinear optimizations, we propose an original scheme that linearly scans the profile over a fixed size window, searching for the best peak within each window using a linearized least-squares Laplace regression. As a post-process, depth estimates with low confidence are reconstructed though an adaptive moving least squares filter. We show how to objectively evaluate the performance of our approach by generating synthetic focal stacks from which the reconstructed depth maps can be compared to ground truth. Our results show that our method achieves higher accuracy than previous nonlinear Laplace regression technique, while being orders of magnitude faster. [ABSTRACT FROM AUTHOR]

Motivated by queueing applications, we study various reflected autoregressive processes with dependencies. Among others, we study cases where the interarrival and service times are proportionally dependent on additive and/or subtracting delay, as well as cases where interarrival times depend on whether the service duration of the previous arrival exceeds or not a random threshold. Moreover, we study cases where the autoregressive parameter is constant as well as a discrete or a continuous random variable. More general dependence structures are also discussed. Our primary aim is to investigate a broad class of recursions of autoregressive type for which several independence assumptions are lifted and for which a detailed exact analysis is provided. We provide expressions for the Laplace transform of the waiting time distribution of a customer in the system in terms of an infinite sum of products of known Laplace transforms. An integer-valued reflected autoregressive process that can be used to model a novel retrial queueing system with impatient customers and a general dependence structure is also considered. For such a model, we provide expressions for the probability generating function of the stationary orbit queue length distribution in terms of an infinite sum of products of known generating functions. A first attempt towards a multidimensional setting is also considered. [ABSTRACT FROM AUTHOR]

We propose novel goodness-of-fit tests for the Weibull distribution with unknown parameters. These tests are based on an alternative characterizing representation of the Laplace transform related to the density approach in the context of Stein's method. Asymptotic theory of the tests is derived, including the limit null distribution, the behaviour under contiguous alternatives, the validity of the parametric bootstrap procedure, and consistency of the tests against a large class of alternatives. A Monte Carlo simulation study shows the competitiveness of the new procedure. Finally, the procedure is applied to real data examples taken from the materials science. [ABSTRACT FROM AUTHOR]

Various load balancing policies are known to achieve vanishing waiting times in the large-scale limit, that is, when the number of servers tends to infinity. These policies either require a communication overhead of one message per job or require job size information. Load balancing policies with an overhead below one message per job are called hyperscalable policies. While these policies often have bounded queue length in the large-scale limit and work well when the overhead is somewhat below one, they show poor performance when the communication overhead becomes small, that is, the mean response time tends to infinity when the overhead tends to zero even at low loads. In this paper, we introduce a hyperscalable load balancing policy, called Join-Up-To(m), that remains effective even when the communication overhead tends to zero. To study its performance under general job size distributions, we make use of the "queue at the cavity" approach. We provide explicit results for the first two moments of the response time, the generating function of the queue length distribution and the Laplace transform of the response time. These results show that the mean response time only depends on the first two moments of the job size distribution. [ABSTRACT FROM AUTHOR]

In this work, a procedure to obtain two-port admittance models of buried cables illuminated by incident electromagnetic fields for the analysis of transients in the frequency domain is presented. The procedure is based on cascading two-port admittance models of various cable segments, avoiding the numerical problems that chain matrix representations may have. To obtain results in the time domain, the Numerical Laplace Transform method is used. [ABSTRACT FROM AUTHOR]

The objective of this research is to propose a fuzzy solution for a specific type of uncertain fractional descriptor linear systems referred to as granular fractional descriptor linear systems (gr-FFDLSs). The coefficients, boundary conditions, and parameters in gr-FFDLSs are represented as fuzzy numbers. This type of system also encompasses the concept of fractional order granular Caputo (gr-Caputo) fuzzy derivative. To achieve this goal, the gr-Caputo fuzzy derivative of order n - 1 < β < n , where n is a natural number, is defined. Furthermore, the granular fuzzy Laplace transforms of the fractional order gr-Caputo fuzzy derivative and fractional order granular fuzzy integrals are introduced. Using the definition of fractional order gr-Caputo fuzzy derivative, fuzzy Drazin inverse matrix, and Laplace transformation, the solution to the state equation of granular descriptor fuzzy fractional linear systems is obtained. Two techniques, the fuzzy shuffle algorithm and the fuzzy Drazin inverse method, are presented to find the fuzzy solution of gr-FFDLSs. To demonstrate the effectiveness of these techniques, two examples are provided to show that they can produce accurate solutions for gr-FFDLSs. [ABSTRACT FROM AUTHOR]

In this paper, we derive asymptotic crack tip fields during damage and tearing of a neo-Hookean sheet. The material damage near the crack tip is characterized by an exponential damage mode inspired by the damage distribution observed in a phase field model in Part I. The asymptotic governing equations consist of a quasi-linear eigenproblem for the deformed coordinates which reduce to the eigenproblem for a Laplace equation in the case of a purely elastic neo-Hookean sheet. The asymptotic fields are complemented by finite element results for a long-notched strip under uniaxial stretch. The asymptotic analysis indicates some interesting scaling relations and distribution features of the crack tip fields that are consistent with those observed in the phase field modelling. In particular, the stress component perpendicular to crack faces, σ 22 , vanishes at the crack tip instead of approaching infinity as observed in a purely elastic neo-Hookean sheet. Its magnitude is proportional to the critical energy release rate G f and inversely proportional to the internal length scale in a scaled coordinate (scaled by the internal length scale ℓ ), i.e., σ 22 ∼ G f ℓ . The near-tip crack faces become more blunted in comparison to a purely elastic neo-Hookean sheet. [ABSTRACT FROM AUTHOR]

The paper presents a three-dimensional hydrodynamic mathematical model that takes into account the processes of salt and heat transport in the Sea of Azov and makes it possible to obtain three-dimensional fields of water flow velocities, pressure, sea water density, salinity, and temperature. The model includes the Navier–Stokes equations of motion, the continuity equation in the case of variable density, and the equations of heat and salt transport. Boundary and initial conditions are specified. The diffusion–convection–reaction equation is approximated in time using weighted schemes. Approximation in spatial variables in the problem of determining the velocity field of the aquatic environment was performed by the balance method, taking into account the filling factors of the control regions. The initial distributions of salinity and temperature, which have a sufficient degree of smoothness at the points of setting their values, was performed using the Laplace equation. Maps of salinity and temperature of the Sea of Azov were obtained by interpolation and superimposing the boundaries of the region. Stationary regimes of heat and salt transfer have been studied. Based on the results of monitoring the water area, three-dimensional mathematical models of the movement of the aquatic environment were built, designed to predict possible scenarios for the development of the Azov Sea ecosystem, making it possible to study how anaerobic contamination areas emerge and take timely measures to localize them. To refine the input data of the mathematical model, an algorithm based on the Kalman filter was developed and implemented, which made it possible to obtain the minimum variance of the unbiased estimate of the state of the dynamic system. The elements of the software package are listed, making it possible to simulate hydrodynamic processes in shallow water bodies with complex spatial structures of currents with the transport of salts and heat. [ABSTRACT FROM AUTHOR]

In this paper, Chaotic implanted opposition-based-quantum equipoise state (CIOQES) and Ascidiacea (CIOQAA) algorithms are applied to solve the loss diminishing problem. In the projected process, the swapping of capricious data is done with the illogical statistics stimulated by Laplace distribution to augment the aid of the possibility of creation level inside the exploration area. Opposition-based learning is one of the persuasive methods to enhance the convergence swiftness of methods. The prospering application of the opposition-based learning involves appraisal of opposite populace and prevailing populace in the parallel generation to determine the greater entrant elucidation. Quantum mechanics has been combined with the algorithms to emulate the analogous performance with the certain stage as they route in a credible powdered of median. Chaotic sequences are integrated into algorithms to augment the exploration and exploitation. Tinkerbell chaotic map engendering standards are implemented. Chaotic implanted opposition-based-quantum equipoise state (CIOQES) algorithm is based on the idea of quantum mechanics about the quantity, inbound, exodus, and fabrication in the governor capacity. Equipoise state algorithm possesses elements as entrant elucidation, and element's location is designated as deliberation. Every element's aptness function is calculated to define the equipoise state of populaces for picking the preeminent solution. Chaotic implanted opposition-based-quantum Ascidiacea (CIOQAA) algorithm is designed to solve the problem. Probing attitudes of Ascidiacea for nutrition spring is imitated to formulate Ascidiacea algorithm. Single end of the Ascidiacea body is continuously resolutely attached to surface like rock or coral. Ascidiacea bottom is corrugated, and core-like postponements will clutch the shallow. Proposed chaotic implanted opposition based-quantum equipoise state (CIOQES) and Ascidiacea (CIOQAA) algorithms are corroborated in G01–G24 benchmark functions, IEEE 30, 57, 118, 300 and 354 bus test systems. Diminished power loss, curtailed power discrepancy, and increased power constancy have been achieved. [ABSTRACT FROM AUTHOR]

The paper focuses on deriving the macroscale viscoelastic constitutive laws using asymptotic expansion method. Both the differential and integral form of the linear viscoelastic constitutive relation of the phases is used in deriving the effective incremental potential and effective constitutive relation, respectively. The integral form is handled by considering the correspondence principle and the Laplace–Carson (LC) transform. A closed-form expression for the effective viscoelastic properties in LC domain is obtained by means of the asymptotic homogenization method (AHM). In addition, AHM coupled with finite element simulation of a representative volume element with periodic boundary conditions is used (AHM + FE). The last step in both approaches is the numerical inversion to the time domain. Solution in time domain is obtained with numerical Laplace inversion algorithms. In case of the differential form, using variational approach, the effective incremental potential in time domain is directly obtained using mean-field method. Different homogenization approaches are exemplified for evaluation of the effective relaxation behavior of composite (viscoelastic matrix reinforced by unidirectional elastic fibers), and they are compared. In the approaches based on LC transform, effective modulus and Poisson's ratio agree well with each other for any property contrast and fiber volume fraction. However, in case of relatively low property contrast, mean field overpredicts as compared to LC approaches in the fiber direction, whereas at relatively higher property contrast, it is vice versa. The difference increases at higher volume fractions due to synergistic effect of the error due to geometrical assumptions involved in the localization tensor and interaction effects of the fiber inclusions. A good agreement in all directions is observed among the three schemes at intermediate volume fractions and property contrast. This study serves as benchmark for further theoretical improvements and experimental investigations. [ABSTRACT FROM AUTHOR]

The B-meson light-cone distribution amplitude is an important non- perturbative quantity arising in the factorization of the amplitudes for many exclusive decays of B mesons, such as B− → γℓ− ν ¯ . We reconsider the renormalization-group (RG) equation satisfied by this function and present its solution at next-to-leading order (NLO) in RG-improved perturbation theory in Laplace space and, for the first time, in momentum space and the so-called diagonal (or dual) space. Since the information needed to describe the B decay processes at leading order in ΛQCD/mb is most directly contained in the distribution amplitude in Laplace space evaluated near the origin, we propose a convenient parameterization of this object in terms of a small set of uncorrelated hadronic parameters. Using recent results on the three-loop anomalous dimension for heavy-light current operators, we derive an expression for the convolution integral appearing in the B− → γℓ− ν ¯ factorization formula that is explicitly scale independent, and we evaluate this formula at (approximate) NNLO. [ABSTRACT FROM AUTHOR]

Time series forecasting is ubiquitous in various scientific and industrial domains. Powered by recurrent and convolutional and self-attention mechanism, deep learning exhibits high efficacy in time series forecasting. However, the existing forecasting methods are suffering some limitations. For example, recurrent neural networks are limited by the gradient vanishing problem, convolutional neural networks cost more parameters, and self-attention has a defect in capturing local dependencies. What's more, they all rely on time invariant or stationary since they leverage parameter sharing by repeating a set of fixed architectures with fixed parameters over time or space. To address the above issues, in this paper we propose a novel time-variant framework named Self-Attention-based Time-Variant Neural Networks (SATVNN), generally capable of capturing dynamic changes of time series on different scales more accurately with its time-variant structure and consisting of self-attention blocks that seek to better capture the dynamic changes of recent data, with the help of Gaussian distribution, Laplace distribution and a novel Cauchy distribution, respectively. SATVNN obviously outperforms the classical time series prediction methods and the state-of-the-art deep learning models on lots of widely used real-world datasets. [ABSTRACT FROM AUTHOR]

Growth mixture modeling is a common tool for longitudinal data analysis. One of the key assumptions of traditional growth mixture modeling is that repeated measures within each class are normally distributed. When this normality assumption is violated, traditional growth mixture modeling may provide misleading model estimation results and suffer from nonconvergence. In this article, we propose a robust approach to growth mixture modeling based on conditional medians and use Bayesian methods for model estimation and inferences. A simulation study is conducted to evaluate the performance of this approach. It is found that the new approach has a higher convergence rate and less biased parameter estimation than the traditional growth mixture modeling approach when data are skewed or have outliers. An empirical data analysis is also provided to illustrate how the proposed method can be applied in practice. [ABSTRACT FROM AUTHOR]

The cubature Kalman filter (CKF) and its square-root version, namely the square-root cubature Kalman filter (SCKF), are commonly used nonlinear estimators under Gaussian noises. Particularly, the latter has the added advantage of low computational complexity and guaranteed positive semi-definiteness. However, the estimation accuracy often degrades substantially when the measurements are contaminated by outliers. This paper proposes a robust Laplace ℓ 1 square-root cubature Kalman filter (LSCKF). The proposed filter employs the heavy-tailed Laplace distribution to model the measurement noises and solves the maximum posterior estimation using the majorization minimization (MM) approach and Gauss–Newton method. Besides, the filter is derived in square root and uses the orthogonal transformations to realize reliable computation of state estimates. Therefore, the new filter not only has good robustness against measurement outliers, but also retains the advantages of high numerical stability. Two numerical simulations are performed to test the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The application of the potential method to finding the stationary distribution of the number of customers in the Mn/Gn/1/r and Mn/Gn/1/∞ queueing systems with threshold operation strateges is proposed. The dependences of the input flow intensity on the number of customers are considered, which are characteristic both for closed systems, which are models of the reliability theory, and for queueing systems with random rarefaction of the input flow. Service intensity control strategies are constructed on the assumption that the intensity can vary at the time when servicing starts. Formulas to determine Laplace transforms of the distribution of the number of customers in the system during the busy period and for calculating the average duration of the busy period are obtained. [ABSTRACT FROM AUTHOR]

As a part of the multi-source cooperative navigation scheme, data fusion has a significant impact on the quality of state estimation. Particle filtering has gradually become the focus of many fusion methods due to its unique theoretical advantages in nonlinear non-Gaussian systems. However, the particle degradation and the resulting sample impoverishment restrict its application in complex engineering scenarios. In this paper, a robust cubature fission particle filter (RCFPF) is proposed to deal with these problems. First, in the framework of cubature rule, Huber function is used to combine the L2 norm and L1 norm to improve the importance density function(IDF), suppress the observation noise. Meanwhile, the proposed distribution(PD) is further optimized by combining the Gaussian distribution with Laplace distribution to alleviate particle degradation. Second, the particle swarm is fissioned before resampling, and the particle weight is reconstructed by fission of high weight particles and covering low weight particles to inhibit sample impoverishment. The vehicle experiments of multi-source cooperative navigation show that the proposed algorithm achieves better test results in accuracy and robustness than extended Kalman filter (EKF), strong tracking particle filter (STPF), and cubature particle filter (CPF). [ABSTRACT FROM AUTHOR]

Trapping and transmission of waves through an acoustic waveguide with a set of perforated cross-walls are studied. Eigenvalues of the corresponding spectral Neumann problem for the Laplace operator are found under geometric symmetry conditions. Almost complete transmission of the piston wave through a system of small holes (an inverted Weinstein anomaly) is achieved by fine tuning of the distance between the cross-walls with a diverse configuration of the connecting holes. A criterion for the possibility of this anomaly is obtained. Related topics—in particular, primitive wave filters and the pinhole-camera effect—are discussed. [ABSTRACT FROM AUTHOR]

Estimation of low or high conditional quantiles is called for in many applications, but commonly encountered data sparsity at the tails of distributions makes this a challenging task. We develop a Bayesian joint-quantile regression method to borrow information across tail quantiles through a linear approximation of quantile coefficients. Motivated by a working likelihood linked to the asymmetric Laplace distributions, we propose a new Bayesian estimator for high quantiles by using a delayed rejection and adaptive Metropolis and Gibbs algorithm. We demonstrate through numerical studies that the proposed estimator is generally more stable and efficient than conventional methods for estimating tail quantiles, especially at small and modest sample sizes. [ABSTRACT FROM AUTHOR]

Full-waveform inversion (FWI) is a powerful methodology employed in estimating subsurface physical parameters. FWI is classically formulated as a data-fitting problem based on minimizing the squared l 2 -norm of the difference between the observed data and modeled data (i.e., the data residuals or errors). The FWI based on l 2 -norm (hereafter classical FWI) stems from the assumption that the errors follow a Gaussian probability distribution. However, since, in most cases the error is non-Gaussian for non-linear problems and therefore, the classical FWI tends to fail. In this work, we consider the Tsallis q-deformation of the exponential function, which is widely used in the field of statistical physics, to derive an alternative misfit function to mitigate the FWI sensitivity to spurious measurements (outliers). In this regard, we formulated the FWI from a q-generalization of the Laplace probability distribution (or q-Laplace distribution). Application on a typical Brazilian pre-salt velocity model with an acoustic synthetic noisy-data is presented, in which we compare the performance of our proposal against the classical FWI. In addition, we compare also our methodology with the robust misfit function based on l 1 -norm, which is based on the Laplace distribution. The results show that our proposal outperforms classical FWI, and also the robust FWI based on l 1 -norm. The results also reveal that our proposal accelerates the data inversion process when the data is polluted by outliers. [ABSTRACT FROM AUTHOR]

We propose a method for studying the reliability of series systems with redundancy and recovery strategies, which provide for a change in the repair intensity depending on the number of failed units. The case of the exponential distribution of the time to failure of units, the non-exponential distribution of the recovery time, and one repair channel are considered. Formulas for determining Laplace transforms of the distribution of the number of failed units during the busy period of the repair channel and of the distribution functions of the busy period and to calculate the stationary indices are obtained. [ABSTRACT FROM AUTHOR]

The solution to the time-dependent 3D hydroelastic problem of vibrations of a thin elastic plate floating on the surface of a heavy fluid of finite depth under the action of unsteady external pressure is obtained within the framework of the linear formulation. The effect of the longitudinal, transverse, and shear compression of the plate on the development of flexural-gravity waves is studied. As examples, the periodic and pulse impact of a load is considered. [ABSTRACT FROM AUTHOR]

In this paper, the shape mixtures of the skew Laplace normal (SMSLN) distribution is introduced as a flexible extension of the skew Laplace normal distribution which is also a heavy-tailed distribution. The SMSLN distribution includes an extra shape parameter, which controls skewness and kurtosis. Some distributional properties of this distribution are derived. Besides, we propose finite mixtures of SMSLN distributions to model both skewness and heavy-tailedness in heterogeneous data sets. The maximum likelihood estimators for parameters of interests are obtained via the expectation–maximization algorithm. We also give a simulation study and examine a real data example for the numerical illustration of proposed estimators. [ABSTRACT FROM AUTHOR]

Quantile regression is a powerful tool for modeling non-Gaussian data, and also for modeling different quantiles of the probability distributions of the responses. We propose a Bayesian approach of estimating the quantiles of multivariate longitudinal data where the responses contain excess zeros. We consider a Tobit regression approach, where the latent responses are estimated using a linear mixed model. The longitudinal dependence and the correlations among different (latent) responses are modeled by the subject-specific vector of random effects. We consider a mixture representation of the Asymmetric Laplace Distribution (ALD), and develop an efficient MCMC algorithm for estimating the model parameters. The proposed approach is used for analyzing data from the health and retirement study (HRS) conducted by the University of Michigan, USA; where we jointly model (i) out-of-pocket medical expenditures, (ii) total financial assets, and (iii) total financial debt for the aged subjects, and estimate the effects of different covariates on these responses across different quantiles. Simulation studies are performed for assessing the operating characteristics of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper proposes a method for solving the stress field of a deep circular tunnel affected by an unsteady temperature field. The tunnel is idealized as an infinite domain problem, and the solution of the unsteady temperature field can be regarded as solving the temperature field in the infinite domain under the first type of boundary conditions. The temperature field distribution inside the tunnel surrounding rock is obtained by the Laplace transform method. On this basis, through the stress condition at the boundary of the tunnel and the zero displacement boundary condition at infinity, the solution equations expressed by the analytic functions are listed, and then the power series method of the complex variable function method is used to obtain the desired analytic functions. From this, the temperature stress field caused by the temperature change can be calculated (the displacement field caused by the temperature change is also derived). By superimposing this stress field upon the surrounding rock stress caused by in situ stress, the total stress field inside the surrounding rock can be obtained. The example shows that when the time t is small and the range of the numerical model is suitably large, the results of the analytical method and the numerical method agree well. This paper also discusses two cases in which the tunnel boundary temperature is greater than and less than the initial temperature of the surrounding rock. [ABSTRACT FROM AUTHOR]

In this paper, a new model-based Bayesian compressive sensing technique for non-stationary images is proposed. Our algorithm is based on the recently addressed triplet Markov fields (TMF) model. The TMF model is well appropriate for non-stationary image processing, owing to the introduction of a third random field which reflects different non-stationarity of images. Furthermore, TMF can extract the interactions among the neighboring sites of an image in a more complete way than the classic hidden Markov models do. In this paper, the inter-scale dependencies between the wavelet coefficients is exploited explicitly in the proposed TMF model, which results in the wavelet domain TMF model. Our proposed model considers the intra- and inter-scale dependencies and the non-stationarity of images simultaneously. Also, we have developed our proposed algorithm for both Gaussian and non-Gaussian measurement noises, and we have modeled the non-Gaussianity of the noise via Laplace distribution. To approximate the posterior distributions of the hidden variables, we resort to a variational Bayesian expectation–maximization algorithm. Simulation results in both the optical and synthetic aperture radar images show that this model leads to an improvement over state-of-the-art algorithms in terms of the reconstruction error and the structural similarity. [ABSTRACT FROM AUTHOR]

We study a modified Markovian bulk-arrival and bulk-service queue incorporating general state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated, and the relationship between this stopped queue and the full queueing model is examined and exploited. Using this relationship, the equilibrium behaviour for the full queueing process is studied and the probability generating function of the equilibrium distribution is obtained. Queue length behaviour is also examined, and the Laplace transform of the queue length distribution is presented. The important questions regarding hitting times and busy period distributions are answered in detail, and the Laplace transforms of these distributions are presented. Further properties regarding the busy period distributions including expectation and conditional expectation of busy periods are also explored. [ABSTRACT FROM AUTHOR]

The performance of the minimax variance (in the Huber sense) robust M-estimates of a location parameter designed in the class of symmetric distributions with bounded interquantile ranges is studied both on large and small samples. Although the general structure of these minimax M-estimates has been described by Huber (1981) long ago, they were not studied as main attention was focused on the estimates designed for the various neighborhoods of a Gaussian. In this work, the proposed estimates are compared to the sample mean, sample median, and the classical robust Huber and Hampel M-estimates under the Gaussian, contaminated Gaussian, Laplace, and Cauchy distributions. The performance of an estimate is measured by its efficiency, bias, and mean squared error. The following conclusions are made: (i) under the Gaussian, Laplace, Cauchy, and moderately contaminated Gaussian distributions, the proposed minimax M-estimates outperform the robust Huber and Hampel M-estimates with respect to asymptotic efficiency; (ii) under heavily contaminated Gaussian distributions, the Huber and Hampel M-estimates are slightly better; (iii) on small samples, these classical robust estimates also slightly outperform the proposed minimax estimates in mean squared error. [ABSTRACT FROM AUTHOR]

Introduction: Time-of-flight (TOF) positron emission tomography (PET) scanners can provide significant benefits by improving the noise properties of reconstructed images. In order to achieve this, the timing response of the scanner needs to be modelled as part of the reconstruction process. This is currently achieved using Gaussian TOF kernels. However, the timing measurements do not necessarily follow a Gaussian distribution. In ultra-fast timing resolutions, the depth of interaction of the γ-photon and the photon travel spread (PTS) in the crystal volume become increasingly significant factors for the timing performance. The PTS of a single photon can be approximated better by a truncated exponential distribution. Therefore, we computed the corresponding TOF kernel as a modified Laplace distribution for long crystals. The obtained (CTR) kernels could be more appropriate to model the joint probability of the two in-coincidenceγ-photons. In this paper, we investigate the impact of using a CTR kernel vs. Gaussian kernels in TOF reconstruction using Monte Carlo generated data. Materials and methods: The geometry and physics of a PET scanner with two timing configurations, (a) idealised timing resolution, in which only the PTS contributed in the CTR, and (b) with a range of ultra-fast timings, were simulated. In order to assess the role of the crystal thickness, different crystal lengths were considered. The evaluation took place in terms of Kullback–Leibler (K-L) distance between the proposed model and the simulated timing response, contrast recovery (CRC) and spatial resolution. The reconstructions were performed using STIR image reconstruction toolbox. Results: Results for the idealised scanner showed that the CTR kernel was in excellent agreement with the simulated time differences. In terms of K-L distance outperformed the a fitted normal distribution for all tested crystal sizes. In the case of the ultra-fast configurations, a convolution kernel between the CTR and a Gaussian showed the best agreement with the simulated data below 40 ps timing resolution. In terms of CRC, the CTR kernel demonstrated improvements, with values that ranged up to 3.8% better CRC for the thickest crystal. In terms of spatial resolution, evaluated at the 60th iteration, the use of CTR kernel showed a modest improvement of the peek-to-valley ratios up to 1% for the 10-mm crystal, while for larger crystals, a clear trend was not observed. In addition, we showed that edge artefacts can appear in the reconstructed images when the timing kernel used for the reconstruction is not carefully optimised. Further iterations, can help improve the edge artefacts. [ABSTRACT FROM AUTHOR]

In VTI media, the conventional inversion methods based on the existing approximation formulas are difficult to accurately estimate the anisotropic parameters of reservoirs, even more so for unconventional reservoirs with strong seismic anisotropy. Theoretically, the above problems can be solved by utilizing the exact reflection coefficients equations. However, their complicated expression increases the difficulty in calculating the Jacobian matrix when applying them to the Bayesian deterministic inversion. Therefore, the new reduced approximation equations starting from the exact equations are derived here by linearizing the slowness expressions. The relatively simple form and satisfactory calculation accuracy make the reduced equations easy to apply for inversion while ensuring the accuracy of the inversion results. In addition, the blockiness constraint, which follows the differentiable Laplace distribution, is added to the prior model to improve contrasts between layers. Then, the concept of GLI and an iterative reweighted least-squares algorithm is combined to solve the objective function. Lastly, we obtain the iterative solution expression of the elastic parameters and anisotropy parameters and achieve nonlinear AVA inversion based on the reduced equations. The test results of synthetic data and field data show that the proposed method can accurately obtain the VTI parameters from prestack AVA seismic data. [ABSTRACT FROM AUTHOR]

Newton certainly regarded his second law of motion in the Principia as a fundamental axiom of mechanics. Yet the works that came after the Principia, the major treatises on the foundations of mechanics in the eighteenth century—by Varignon, Hermann, Euler, Maclaurin, d'Alembert, Euler (again), Lagrange, and Laplace—do not record, cite, discuss, or even mention the Principia's statement of the second law. Nevertheless, the present study shows that all of these scientists do in fact assume the principle that the Principia's second law asserts as a fundamental axiom in their mechanics. (For what that second law asserts, we rely on Newton's own testimony.) Some, like Varignon and Hermann, assume the axiom implicitly, apparently unaware that any assumption is being made, while others, like Maclaurin and Euler, assume the axiom explicitly, apparently unaware that the assertion assumed is the second law as Newton himself understood it. But in every case these scientists employ the principle asserted by the Principia's second law fundamentally, unaware that they should be citing Neutonus, Prin., Phil. Nat. Math., Lex II. [ABSTRACT FROM AUTHOR]

Recently sparse coding based on regression analysis has been widely used in face recognition research. Most existing regression methods add an extra constraint factor to the coding residual to make the fidelity term in the l 2 loss approach the Gaussian or Laplace distribution. But the essence of these methods is that only the fidelity term of l 1 loss or l 2 loss is used. In this paper, weighted Huber constrained sparse coding (WHCSC) is used to study the robustness of face recognition in occluded environments, and alternating direction method of multipliers is used to solve the problem of model minimization. In WHCSC, we propose a sparse coding with weight learning and use Huber loss to determine whether the fidelity is a l 2 loss or l 1 loss. For the WHCSC model, the two kinds of classification modes and the two kinds of weight coefficients are further studied for the intra-class difference and the inter-class difference in the face image classification. Through a large number of experiments on a public face database, WHCSC shows strong robustness in face occlusion, corrosion and illumination changes comparing to the state-of-the-art methods. [ABSTRACT FROM AUTHOR]

We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, and generalized Mittag–Leffler laws. In particular, we prove that the generalized Linnik distribution is a normal scale mixture with the generalized Mittag–Leffler mixing distribution. Based on these representations, we prove some limit theorems for a wide class of statistics constructed from samples with random sized including, e.g., random sums of independent random variables with finite variances, in which the generalized Linnik distribution plays the role of the limit law. Thus we demonstrate that the scheme of geometric (or, in general, negative binomial) summation is by far not the only asymptotic setting (even for sums of independent random variables) in which the generalized Linnik law appears as the limit distribution. [ABSTRACT FROM AUTHOR]

The probabilistic neural network (PNN) is an efficient approach that can compute nonlinear decision boundaries, widely used for classification. In this paper, the often used Gaussian distribution function is replaced by a new probability density function which provides a new variant of the PNN method. Most of the higher-dimensional data are statistically found to be not from the normal distribution, and hence, we have replaced it by the symmetric Laplace distribution. Further, the estimation of the smoothing parameter in the proposed PNN model is carried out with three different evolutionary algorithms, namely bat algorithm (BA), grey wolf optimizer (GWO), and whale optimization algorithm (WOA) with a novel fitness function. These different proposed PNN models with variable smoothing parameter estimation methods are tested on five different benchmark data sets. The performance of proposed three Laplace distribution-based variants of PNN incorporated with BA, GWO, and WOA are reported and compared with Gaussian-based variants of PNN and also other commonly used classifiers: the conventional PNN, extreme learning machine, and K-nearest neighbor in terms of measurement accuracy. The results demonstrate that the proposed approaches using evolutionary algorithms can provide as much as a ten percent increase in accuracy over the conventional PNN method. [ABSTRACT FROM AUTHOR]

Probabilistic neural network (PNN) is a single-pass feed-forward neural network with the capability of providing nonlinear decision boundaries. In this work, we propose the modifications to the existing PNN approach. Contributions are threefold: First, symmetric Laplace distribution has been used instead of Gaussian distribution in the pattern layer of the PNN approach. Second, a new weight coefficients' estimation method has been introduced between the pattern and summation layers of PNN. Third, a novel convex fitness function has been designed for the bat algorithm to obtain an optimal smoothing parameter vector. The designed fitness function maintains a balance between the sensitivity and specificity values. The performance of the proposed algorithm "bat algorithm-based weighted Laplacian probabilistic neural network" is compared with that of conventional PNN, weighted PNN, extreme learning method, optimal pruned extreme learning machine and K-nearest neighbor. Experimental evaluation is carried out on eleven benchmark data sets using different performance measures such as accuracy, sensitivity, specificity and Youden's index. The comparative performance evaluation of the proposed model has been made with the PNN, and other standard algorithms outperform the compared approaches in terms of classification accuracy. Friedman test is used for statistical evaluation of the proposed approach. [ABSTRACT FROM AUTHOR]

In practice, we often encounter situations where a sample size is not defined in advance and can be a random value. In the present paper, we derive second-order Chebyshev–Edgeworth and Cornish–Fisher expansions based of Student's t- and Laplace distributions and their quantiles for samples with random size of a special kind. This derivation uses a general transfer theorem, which allows us to construct asymptotic expansions for distributions of randomly normalized statistics from the distributions of the considered nonrandomly normalized statistics and of the random size of the underlying sample. Recently, interest in Cornish–Fisher expansions has increased because of study in risk management. Widespread risk measure Value at Risk (VaR) substantially depends on quantiles of the loss function, which is connected with description of investment portfolio of financial instruments. [ABSTRACT FROM AUTHOR]

In French mechanical treatises of the nineteenth century, Newton's second law of motion was frequently derived from a relativity principle. The origin of this trend is found in ingenious arguments by Huygens and Laplace, with intermediate contributions by Euler and d'Alembert. The derivations initially relied on Galilean relativity and impulsive forces. After Bélanger's Cours de mécanique of 1847, they employed continuous forces and a stronger relativity with respect to any commonly impressed motion. The name "principle of relative motions" and the very idea of using this principle as a constructive tool were born in this context. The consequences of Poincaré's and Einstein's awareness of this approach are analyzed. Lastly, the legitimacy and significance of a relativity-based derivation of Newton's second law are briefly discussed in a more philosophical vein. [ABSTRACT FROM AUTHOR]

Differentially private spatial decompositions split the whole domain into sub-domains recursively to generate a hierarchical private tree and add Laplace noise to each node's points count. However the Laplace distribution is symmetric about the origin, the mean of a large number of queries may cancel the Laplace noise and reveal privacy. Existing methods take the solution by limiting the number of queries. But in private tree the points count of intermediate node may be real since the summation of all its descendants may cancel the Laplace noise. To address these problems of differentially private spatial decompositions, we propose a more secure algorithm to make the Laplace noise not be canceled. That splits the domains depending on its real points count not noisy, and only adds indefeasible Laplace noise to leaves. That the i th randomly selected leaf of one intermediate node is added noise by β − i + 1 + 1 + β (β − i + 1) + β L a p (λ) . We also propose the definition of Lapmin(λ) whose absolute value is not greater than Sensitivity(f). It is proved that adding Lapmin(λ) noise to query answer guarantees both differential privacy and minimal relative error comparing with unlimited Laplace noise. The experiment results show that our algorithm performs better both on synthetic and real datasets with higher security and data utility, and the noises costs is highly decreased. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of the maximum likelihood estimates (MLEs) of the parameters of Laplace Birnbaum–Saunders distribution based on Type-I, Type-II and hybrid censored samples. We first derive the relationship between the MLEs of the two parameters and then discuss the monotonicity property of the profile likelihood function. Numerical iterative procedure is then discussed for determining the MLEs of the parameters. Finally, for illustrative purpose, we analyze one real data from the literature and present some graphical illustrations of the approach. [ABSTRACT FROM AUTHOR]

In this paper, we continue our analysis of spatial versions of agent-based models for the dynamics of money that have been introduced in the statistical physics literature, focusing on two models with debts. Both models consist of systems of economical agents located on a finite connected graph representing a social network. Each agent is characterized by the number of coins she has, which can be negative in case she is in debt, and each monetary transaction consists in one coin moving from one agent to one of her neighbors. In the first model, that we name the model with individual debt limit, the agents are allowed to individually borrow up to a fixed number of coins. In the second model, that we name the model with collective debt limit, agents can borrow coins from a central bank as long as the bank is not empty, with reimbursements occurring each time an agent in debt receives a coin. Based on numerical simulations of the models on complete graphs, it was conjectured that, in the large population/temperature limits, the distribution of money converges to a shifted exponential distribution for the model with individual debt limit, and to an asymmetric Laplace distribution for the model with collective debt limit. In this paper, we prove exact formulas for the distribution of money that are valid for all possible social networks. Taking the large population/temperature limits in the formula found for the model with individual debt limit, we prove convergence to the shifted exponential distribution, thus establishing the first conjecture. Simplifying the formula found for the model with collective debt limit is more complicated, but using a computer to plot this formula shows an almost perfect fit with the Laplace distribution, which strongly supports the second conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we study a wide and flexible family of discrete distributions, the so-called generalized negative binomial (GNB) distributions, which are mixed Poisson distributions with the mixing laws belonging to the class of generalized gamma (GG) distributions. This family was introduced by E.W. Stacy as a particular family of lifetime distributions containing gamma, exponential power, and Weibull distributions. These distributions seem to be very promising in the statistical description of many real phenomena and very convenient and almost universal models for the description of statistical regularities in discrete data. We study analytic properties of GNB distributions. We prove that a GG distribution is a mixed exponential distribution if and only if the shape and exponent power parameters are no greater than one. We write the mixing distribution explicitly as a scale mixture of strictly stable laws concentrated on the nonnegative half-line. As a corollary, we obtain a representation for the GNB distribution as a mixed geometric distribution. We consider the corresponding scheme of Bernoulli trials with random probability of success. Within this scheme, we prove a random analog of the Poisson theorem establishing the convergence of mixed binomial distributions to mixed Poisson laws. We prove limit theorems for random sums of independent random variables in which the number of summands has the GNB distribution and the summands have both light- and heavy-tailed distributions. We obtain that the class of limit laws is wide enough and includes the so-called generalized variance gamma distributions. We obtain various representations for the limit laws in terms of mixtures of Mittag-Leffler, Linnik, or Laplace distributions. We prove limit theorems establishing the convergence of the distributions of statistics constructed from samples with random sizes obeying the GNB distribution to generalized variance gamma distributions. We also discuss some applications of GNB distributions in meteorology. [ABSTRACT FROM AUTHOR]

The nontransitivity problem of the stochastic precedence relation for three independent random variables with distributions from a given class of continuous distributions is studied. Originally, this issue was formulated in one problem of strength theory. In recent time, nontransitivity has become a popular topic of research for the so-called nontransitive dice. Some criteria are first developed and then applied for proving that nontransitivity may not hold for many classical continuous distributions (uniform, exponential, Gaussian, logistic, Laplace, Cauchy, Simpson, one-parameter Weibull and others). The case of all distributions with a polynomial density on the unit interval is considered separately. Some promising directions of further investigations on the subject are outlined. [ABSTRACT FROM AUTHOR]

Key message: Current genome-enabled prediction models assumed errors normally distributed, which are sensitive to outliers. We propose a model with errors assumed to follow a Laplace distribution to deal better with outliers. Current genome-enabled prediction models use regressions that fit the expected value (mean) of a response variable with errors assumed normally distributed, which are often sensitive to outliers, either genetic or environmental. For this reason, we propose a robust Bayesian genome median regression (BGMR) model that fits regressions to the medians of a distribution, with errors assumed to follow a Laplace distribution to deal better with outliers. The BGMR model was evaluated under a Bayesian framework with Markov Chain Monte Carlo sampling using a location–scale mixture representation of the Laplace distribution. The BGMR was implemented with two simulated and two real genomic data sets, and we compared its prediction performance with that of a conventional genomic best linear unbiased prediction (GBLUP) model and the Laplace maximum a posteriori (LMAP) method. The prediction accuracies of BGMR were higher than those of the GBLUP and LMAP methods when there were outliers. The BGMR model could be useful to breeders who need to predict and select genotypes based on data with unknown outliers. [ABSTRACT FROM AUTHOR]

A pipe is located in a semi-infinite two-dimensional porous medium partially covered by a tarp. A constant pressure gradient may be imposed at infinity on the medium while air is pumped into the pipe. The governing Laplace equation is solved by extending the famous Keldysh-Sedov formulae to doubly connected domains by reduction of the mixed problem to a Riemann-Hilbert problem with discontinuous coefficients; the solution is obtained by combining factorization and functional equations. Then, the influence of the various geometrical and flow parameters is presented and discussed. [ABSTRACT FROM AUTHOR]

We consider the transmission problem for the Laplace equation on an infinite three-dimensional wedge, determining the complex parameters for which the problem is well-posed, and characterizing the infinite multiplicity nature of the spectrum. This is carried out in two formulations leading to rather different spectral pictures. One formulation is in terms of square integrable boundary data, the other is in terms of finite energy solutions. We use the layer potential method, which requires the harmonic analysis of a non-commutative non-unimodular group associated with the wedge. [ABSTRACT FROM AUTHOR]

The most crucial feature of excitation transfer in low-temperature plasma forming the physical basis for operation of contemporary lighting devices is participation of resonant photons in the process. The widely used approximation of complete frequency redistribution assumes independence of the emitted photon frequency from frequency of the absorbed one. Averaged over frequency distribution in a resonance line, the natural transfer equation with the exponential kernel becomes the equation with an inverse power kernel, which in case of an infinite homogeneous medium is asymptotically equivalent to the Laplace equation of non-integer (fractional) order, considered in this article. [ABSTRACT FROM AUTHOR]

Abstract: We construct a Rayleigh approximation for multilayer particles the layer boundaries of which are nonconfocal spheroids. The geometry of the problem is taken into account to the maximum extent by representing the field potentials inside nonconfocal shells as expansions in terms of spheroidal harmonics in different coordinate systems in which the surfaces of the layers are coordinate. To sew two expansions inside each layer, we use relations between spheroidal harmonics of the Laplace equation in systems with different focal lengths that we obtained. The extended boundary conditions method (ЕВСМ) and the separation of variables method (SVM) prove to be equivalent, because they yield the same results. The polarizability of the particle and, therefore, the characteristics of the scattered radiation are written in terms of infinite-dimensional matrices, the elements of which are determined either explicitly or in the form of finite sums. In particular cases of confocal spheroids, this solution is completely consistent with the known results. [ABSTRACT FROM AUTHOR]

We introduce a multiclass single-server queueing system in which the arrival rates depend on the current job in service. The system is characterized by a matrix of arrival rates in lieu of a vector of arrival rates. Our proposed model departs from existing state-dependent queueing models in which the parameters depend primarily on the number of jobs in the system rather than on the job in service. We formulate the queueing model and its corresponding fluid model and proceed to obtain necessary and sufficient conditions for stability via fluid models. Utilizing the natural connection with the multitype Galton-Watson processes, the Laplace-Stieltjes transform of busy periods in the system is given. We conclude with tail asymptotics for the busy period for heavy-tailed service time distributions for the regularly varying case. [ABSTRACT FROM AUTHOR]