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Let f be a holomorphic Hecke eigenform of S L 2 (Z) with weight k. Denote by L(s, f) the automorphic L-function attached to f, and S (t , f) = 1 π arg L (1 2 + i t , f) , where the argument is obtained by continuous variation along the straight line { s ∈ C | ℜ s ≥ 1 2 , ℑ s = t } , starting with the value 0 at infinity. Here, a new zero density estimate of L(s, f) in short intervals is achieved. As an application, the integral moment of S(t, f) is obtained, i.e., for l ∈ Z + and sufficiently large T, ∫ T T + H | S (t , f) | 2 l d t = (2 l) ! l ! (2 π) 2 l H (log log T) l + O (H (log log T) l - 1 2 ) holds for T 15 16 + ε ≤ H ≤ T , which improves the previous result. [ABSTRACT FROM AUTHOR]

Let f be a Gaussian random field on R d and let X be the number of critical points of f contained in a compact subset. A long-standing conjecture is that, under mild regularity and non-degeneracy conditions on f, the random variable X has finite moments. So far, this has been established only for moments of order lower than three. In this paper, we prove the conjecture. Precisely, we show that X has finite moment of order p, as soon as, at any given point, the Taylor polynomial of order p of f is non-degenerate. We present a simple and general approach that is not specific to critical points and we provide various applications. In particular, we show the finiteness of moments of the nodal volumes and the number of critical points of a large class of smooth, or holomorphic, Gaussian fields, including the Bargmann-Fock ensemble. [ABSTRACT FROM AUTHOR]

We construct and study properties of an infinite dimensional analog of Kahane's theory of Gaussian multiplicative chaos (Kahane in Ann Sci Math Quebec 9(2):105-150, 1985). Namely, if H T (ω) is a random field defined w.r.t. space-time white noise B ˙ and integrated w.r.t. Brownian paths in d ≥ 3 , we consider the renormalized exponential μ γ , T , weighted w.r.t. the Wiener measure P 0 (d ω) . We construct the almost sure limit μ γ = lim T → ∞ μ γ , T in the entire weak disorder (subcritical) regime and call it subcritical GMC on the Wiener space. We show that μ γ { ω : lim T → ∞ H T (ω) T (ϕ ⋆ ϕ) (0) ≠ γ } = 0 almost surely, meaning that μ γ is supported almost surely only on γ -thick paths, and consequently, the normalized version is singular w.r.t. the Wiener measure. We then characterize uniquely the limit μ γ w.r.t. the mollification scheme ϕ in the sense of Shamov (J Funct Anal 270:3224–3261, 2016) – we show that the law of B ˙ under the random rooted measure Q μ γ (d B ˙ d ω) = μ γ (d ω , B ˙) P (d B ˙) is the same as the law of the distribution f ↦ B ˙ (f) + γ ∫ 0 ∞ ∫ R d f (s , y) ϕ (ω s - y) d s d y under P ⊗ P 0 . We then determine the fractal properties of the measure around γ -thick paths: - C 2 ≤ lim inf ε ↓ 0 ε 2 log μ ^ γ (‖ ω ‖ < ε) ≤ lim sup ε ↓ 0 sup η ε 2 log μ ^ γ (‖ ω - η ‖ < ε) ≤ - C 1 w.r.t a weighted norm ‖ · ‖ . Here C 1 > 0 and C 2 < ∞ are the uniform upper (resp. pointwise lower) Hölder exponents which are explicit in the entire weak disorder regime. Moreover, they converge to the scaling exponent of the Wiener measure as the disorder approaches zero. Finally, we establish negative and L p ( p > 1 ) moments for the total mass of μ γ in the weak disorder regime. [ABSTRACT FROM AUTHOR]

We propose an algorithm for the closed-form recursive computation of joint moments and cumulants of all orders of k-hop counts in the 1D unit disk random graph model with Poisson distributed vertices. Our approach uses decompositions of k-hop counts into multiple Poisson stochastic integrals. As a consequence, using the Stein and cumulant methods we derive Berry-Esseen bounds for the asymptotic convergence of renormalized k-hop path counts to the normal distribution as the density of Poisson vertices tends to infinity. Computer codes for the recursive symbolic computation of moments and cumulants of any orders are provided as an online resource. [ABSTRACT FROM AUTHOR]

We study the 2k-th moment of central values of the family of Dirichlet L-functions to a fixed prime modulus. We establish sharp lower bounds for all real k ≥ 0 and sharp upper bounds for k in the range 0 ≤ k ≤ 1 . [ABSTRACT FROM AUTHOR]

The generalized hyperbolic distributions are fast becoming the most popular models for financial returns. In recent years, many variants of generalized hyperbolic distributions have been proposed in the literature. This paper provides a review of generalized hyperbolic and related distributions, including software available for them. A simulation study and real data applications are presented to compare some of the reviewed distributions. [ABSTRACT FROM AUTHOR]

We introduce a new unified extension of the integral form of Euler's beta function with a MacDonald function in the integrand and establish functional upper bounds for it. We use this definition to extend as well the Gaussian and Kummer's confluent hypergeometric functions, for which we provide bounding inequalities. Moreover, we use our extension of the beta function to define a new probability distribution, for which we establish raw moments and moment inequalities and, as by-products, Turán inequalities for the initially defined extended beta function. [ABSTRACT FROM AUTHOR]

In this paper, we derive the exact expressions, as well as recurrence relations, for the single and product moments of the order statistics (OSs) and record values of the Topp-Leone Lomax (TLLo) distribution proposed by Oguntunde et al. (Wirel Person Commun 109:349–360, 2019). In addition, we study the corresponding L-moments. We estimate the distribution parameters through these L-moments and compare our results to those obtained in other models. Subsequently, motivated by the study of Okorie and Nadarajah (Wirel Person Commun 115:589–596, 2020), we fit the flood level data to the TLLo distribution. Finally, we reveal a relationship between the entropy and the upper and lower record values, as well as OSs. [ABSTRACT FROM AUTHOR]

The geometric condition that n complex numbers are the vertices of a regular polygon is translated into algebraic equations that symmetric polynomials in these numbers must satisfy. [ABSTRACT FROM AUTHOR]

The derivation of a gas-dynamic model of the radiative conductivity of a weakly ionized gas based on an analysis of the electron kinetics is presented. The gas is formed by the impact ionization of rarefied air by fast primary electrons. The distribution function of slow secondary electrons is studied by the local numerical solution of the kinetic equation. The revealed properties of the distribution function are used to derive equations for the concentration, drift velocity, and specific energy of slow electrons. [ABSTRACT FROM AUTHOR]

Macroeconomic researchers use a variety of estimators to parameterise their models empirically. One such is FIML; another is indirect inference (II). One form of indirect inference is 'informal' whereby data features are 'targeted' by the model — i.e. parameters are chosen so that model-simulated features replicate the data features closely. Monte Carlo experiments show that in the small samples prevalent in macro data, both FIML informal II produce high bias, while formal II, in which the joint probability of the data- generated auxiliary model is maximised under the model simulated distribution, produces low bias. They also show that FII gets this low bias from its high power in rejecting misspecified models, which comes in turn from the fact that this distribution is restricted by the model-specified parameters, so sharply distinguishing it from rival misspecified models. [ABSTRACT FROM AUTHOR]

The Lupaş q-analogue, R n , q , is historically the first known q-version of the Bernstein operator. It has been studied extensively in different aspects by a number of authors during the last decades. In this work, the following issues related to the image of the Lupaş q-analogue are discussed: A new explicit formula for the moments has been derived, independence of the image R n,q from the parameter q has been examined, the diagonalizability of operator R n,q has been proved, and the fact that R n,q does not preserve modulus of continuity has been established. [ABSTRACT FROM AUTHOR]

We study the different properties of the Rayleigh distribution. These include the descriptive properties, reliability properties and stochastic orders. Next, we consider seven different estimation methods for estimating the parameter, namely: maximum likelihood estimation, matching moments estimation, maximum product of spacing estimation, ordinary least squares estimation, Cramér-von Mises estimation, Anderson-Darling estimation and right-tail Anderson-Darling estimation. A simulation study is done to assess the performance of these methods of estimation and the results shows that all the estimators are mostly efficient and consistent. Finally, using the method of maximum likelihood estimation, we demonstrate the applicability of the Rayleigh distribution by modelling the Netherlands’s COVID-19 mortality rate data as an example. [ABSTRACT FROM AUTHOR]

The generalized Kaiser–Bessel window function is defined via the modified Bessel function of the first kind and arises frequently in tomographic image reconstruction. In this paper, we study in details the properties of the Kaiser–Bessel distribution, which we define via the symmetric form of the generalized Kaiser–Bessel window function. The Kaiser–Bessel distribution resembles to the Bessel distribution of McKay of the first type, it is a platykurtic or sub-Gaussian distribution, it is not infinitely divisible in the classical sense and it is an extension of the Wigner's semicircle, parabolic and n-sphere distributions, as well as of the ultra-spherical (or hyper-spherical) and power semicircle distributions. We deduce the moments and absolute moments of this distribution and we find its characteristic and moment generating function in two different ways. In addition, we find its cumulative distribution function in three different ways and we deduce a recurrence relation for the moments and absolute moments. Moreover, by using a formula of Ismail and May on quotient of modified Bessel functions of the first kind, we deduce a closed-form expression for the differential entropy. We also prove that the Kaiser–Bessel distribution belongs to the family of log-concave and geometrically concave distributions, and we study in details the monotonicity and convexity properties of the probability density function with respect to the argument and each of the parameters. In the study of the monotonicity with respect to one of the parameters we complement a known result of Gronwall concerning the logarithmic derivative of modified Bessel functions of the first kind. Finally, we also present a modified method of moments to estimate the parameters of the Kaiser–Bessel distribution, and by using the classical rejection method we present two algorithms for sampling independent continuous random variables of Kaiser–Bessel distribution. The paper is closed with conclusions and proposals for future works. [ABSTRACT FROM AUTHOR]

The goal of this research is to create a new general family of Topp-Leone distributions called the Topp-Leone Cauchy Family (TLC), which is exceedingly versatile and results from a careful merging of the Topp-Leone and Cauchy distribution families. Some of the new family's theoretical properties are investigated using specific results on stochastic functions, quantile functions and associated measures, generic moments, probability weighted moments, and Shannon entropy. A parametric statistical model is built from a specific member of the family. The maximum likelihood technique is used to estimate the model's unknown parameters. Furthermore, to emphasize the new family's practical potential, we applied our model to two real-world data sets and compared it to existing rival models. [ABSTRACT FROM AUTHOR]

Nadarajah and Haghighi [1] introduced the alternative to the gamma, Weibull and exponentiated exponential distribution with zero mode and allows for increasing, decreasing and constant hazard rates as an extension of exponential distribution. Later Lemonte [2] described the three parameters as exponentiated of the distribution of Nadarajah and Haghighi [1] and given the name as exponentiated Nadarajah–Haghighi (ENH) distribution. Lemonte [2] obtained the rational properties and application of the ENH distribution as moments, entropies, maximum likelihood estimation and stress–strength estimation etc. In this paper, we have obtained the exact expression for single and product moments of order statistics arising from ENH distribution. By using these expressions, we have computed the means of order statistics for and for arbitrarily chosen parameter values. Furthermore, the practical application of our findings was well-suited by the linear combination of these order moments as L-moments. Finally, we interpreted the findings and their future work. [ABSTRACT FROM AUTHOR]

In this article, by using the notion of quantum calculus, we define a new type Szász–Mirakjan operators based on the q-integers. We derive a recurrence formula and calculate the moments Φ n , q (t m ; x) for m = 0 , 1 , 2 and the central moments Φ n , q ((t − x) m ; x) for m = 1 , 2 . We give estimation for the first and second-order central moments. We present a Korovkin type approximation theorem and give a local approximation theorem by using modulus of continuity. We obtain a local direct estimate for the new Szász–Mirakjan operators in terms of Lipschitz-type maximal function of order α. Finally, we prove a Korovkin type weighted approximation theorem. [ABSTRACT FROM AUTHOR]

We correct the statements of the non-critical convergence theorems in Gonzalez et al. (Probab Theory Rel Fields, 2022), principally correcting the recursive constants that appear in the limits. [ABSTRACT FROM AUTHOR]

Blur is an image degradation that makes object recognition challenging. Restoration approaches solve this problem via image deblurring, deep learning methods rely on the augmentation of training sets. Invariants with respect to blur offer an alternative way of describing and recognising blurred images without any deblurring and data augmentation. In this paper, we present an original theory of blur invariants. Unlike all previous attempts, the new theory requires no prior knowledge of the blur type. The invariants are constructed in the Fourier domain by means of orthogonal projection operators and moment expansion is used for efficient and stable computation. Applying a general substitution rule, combined invariants to blur and spatial transformations are easy to construct and use. Experimental comparison to Convolutional Neural Networks shows the advantages of the proposed theory. [ABSTRACT FROM AUTHOR]

We focus on an R d -valued discrete time branching random walk with immigration in a random environment, where the environment ξ = (ξ n) is a stationary and ergodic sequence indexed by time n ∈ N . For the process, the partition function is the moment generating function of the counting measure describing the dispersion of individuals at time n. Let Z n (t) be the partition function of the process with immigration, and Z ¯ n (t) be that of the process without immigration. For t ∈ R d fixed, we are interested in the relationships and differences between the moments of Z n (t) and those of E [ Z ¯ n (t) | ξ ] . We show asymptotic properties of annealed moments and quenched moments of all orders of Z n (t) and discover the differences from the corresponding moments of E [ Z ¯ n (t) | ξ ] . Then, with the help of the moments of Z n (t) , we establish large and moderate deviation results associated with the free energy 1 n log Z n (t) . [ABSTRACT FROM AUTHOR]

In this article, the discrete distribution named Poisson distribution was compounded with a continuous distribution, i.e., NXLindley distribution to generate the Poisson–NXLindley distribution. Its basic mathematical properties were explored and a general expression for its ordinary factorial moments, moment generating function, dispersion index, coefficient of variation, skewness, and kurtosis. The proposed distribution is over-dispersed since its mean is less than the variance. This feature opens a new opportunity to model over-dispersed count datasets. The maximum likelihood approach is used for the estimation of the unknown parameter of the proposed model. Three real-life datasets are used to show the applicability of the new distribution. Additionally, the count regression model based on the Poisson–NXLindley distribution is introduced. The new model efficiently analyzed considered datasets than the competitive one-parameter discrete distribution. [ABSTRACT FROM AUTHOR]

This study introduces the exponentiated additive Teissier-Exponential distribution, a three-parameter continuous model. The current model can be used in reliability engineering and survival analysis because it exhibits an increasing and bathtub-shaped failure rate. Several distributional characteristics of the model are derived and thoroughly examined, including moments, quantile function, stress-strength reliability, moments of residual life function, entropy, and order statistics. The maximum likelihood approach to estimating the model parameters is derived and simulations with various parameter settings are carried out. By fitting a dataset representing the lifetimes of 20 electronic components with bathtub failure rates based on reliability and comparing the results with several distributions with different parameters ranging from one to six, it is possible to demonstrate the applicability of the proposed distribution. The application's outcome demonstrates that the model, as supported by several goodness-of-fit tests, provides a good fit to the dataset when compared to some standard distributions and some distributions derived through mixture, exponentiation, and additive techniques. [ABSTRACT FROM AUTHOR]

Order statistic plays an important role in statistical theoretical and practical problems. In this paper, we derive algebraic expressions for both single and product moments of order statistics from generalized Topp–Leone (GTL) distribution. These expressions will be useful for computational purposes. Further, based on order statistics, we have obtained maximum likelihood estimators of the model parameters and uniformly minimum variance unbiased estimator for the model parameters of the GTL distribution. Finally, based on order statistics, Monte Carlo simulation study has been carried out to compare the performances of the proposed methods in terms of bias and mean square error, and a real life data set has been analyzed for illustrative purposes. [ABSTRACT FROM AUTHOR]

Objective: The objective of this study was to measure the forces and moments exerted by direct printed aligners (DPAs) with varying facial and lingual aligner surface thicknesses, in all three planes of space, during lingual movement of a maxillary central incisor. Materials and methods: An in vitro experimental setup was used to quantify forces and moments experienced by a programmed tooth to be moved and by adjacent anchor teeth, during lingual movement of a maxillary central incisor. DPAs were directly 3D-printed with Tera Harz TC-85 (Graphy Inc., Seoul, South Korea) clear photocurable resin in 100-µm layers. Three multi-axis sensors were used to measure the moments and forces generated by 0.50 mm thick DPAs modified with labial and lingual surface thicknesses of 1.00 mm in selective locations. The sensors were connected to three maxillary incisors (the upper left central, the upper right central, and the upper left lateral incisors) during 0.50 mm of programmed lingual bodily movement of the upper left central incisor. Moment-to-force ratios were calculated for all three incisors. Aligners were benchtop tested in a temperature-controlled chamber at intra-oral temperature to simulate intra-oral conditions. Results: The results showed that increased facial thickness of DPAs slightly reduced force levels on the upper left central incisor compared to DPAs of uniform thickness of 0.50 mm. Additionally, increasing the lingual thickness of adjacent teeth reduced force and moment side effects on the adjacent teeth. DPAs can produce moment-to-force ratios indicative of controlled tipping. Conclusions: Targeted increases in thickness of direct 3D-printed aligners change the magnitude of forces and moments generated, albeit in complex patterns that are difficult to predict. The ability to vary labiolingual thicknesses of DPAs is promising to optimize the prescribed orthodontic movements while minimizing unwanted tooth movements, thereby increasing the predictability of tooth movements. [ABSTRACT FROM AUTHOR]

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ( q = 1 ) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont–Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally, we show that one of our methods can be applied to deal with the moments of Askey–Wilson polynomials. [ABSTRACT FROM AUTHOR]

A combinatorial theory for type R I orthogonal polynomials is given. The ingredients include weighted generalized Motzkin paths, moments, continued fractions, determinants, and histories. Several explicit examples in the Askey scheme are given. [ABSTRACT FROM AUTHOR]

Given a , b ∈ R , m = min { a , b } and M = max { a , b } , we consider the orthogonal polynomials associated with nontrivial positive measures ϕ for which s u p p (ϕ) ⊂ (- ∞ , m ] ∪ [ M , ∞) and (x - a) d ϕ (x) = - (x - b) d ϕ (- x + a + b). For this class of measures, formulas in order to compute the moments, as well as formulas for the weights and nodes in the associated Gaussian quadrature rules are provided. We also show that the QD-algorithm can be applied in order to generate new orthogonal polynomials in a simple way. Several examples are given to illustrate the results obtained. [ABSTRACT FROM AUTHOR]

Inverted modified Lindley distribution was developed without considering any special function or additional parameters in its formulation. This model provides better fit than exponential and Lindley distributions and it was suitable for modeling increasing, reverse bathtub (unimodal) and constant shaped hazard rate function. This article emphasize the estimation of a one-parameter inverted modified Lindley distribution by using order statistics. First, the explicit expressions for single and product moments of order statistics from this distribution are derived. We have also tabulated the first four inverse moments of order statistics for various values of the parameter. In the part of estimation, we first utilize the maximum likelihood (ML) estimator and approximate confidence interval to obtain the model parameter based on order statistics. Based on order statistics, a simulation study is carried out to check the efficiency of the estimator. Two real life data sets, have been analyzed for order statistics to demonstrate how the proposed methods may work in practice. [ABSTRACT FROM AUTHOR]

Extensive experiments illustrate that moments and their derivations can act as effective fingerprint features for specific emitter identification. Nevertheless, the lack of mechanistic explanation restricts the moment-based fingerprint features to a trial-based and data-driven technique. To make up for theoretical weakness and enhance generalization ability, we analytically investigate how intentional modulation and unintentional modulation affect moments. A framework for extracting moment-based fingerprint features is proposed through fine-segmenting slices. Fingerprint features are extracted, followed by segmenting signals into a combination of sinewaves and calculating their moments. The proposed framework shows advantages in mechanism interpretability and generalizing ability. Simulations and experiments verified the correctness and effectiveness of the proposed framework. [ABSTRACT FROM AUTHOR]

Suppose that X = (X t , t ≥ 0) is either a superprocess or a branching Markov process on a general space E, with non-local branching mechanism and probabilities P δ x , when issued from a unit mass at x ∈ E . For a general setting in which the first moment semigroup of X displays a Perron–Frobenius type behaviour, we show that, for k ≥ 2 and any positive bounded measurable function f on E, lim t → ∞ g k (t) E δ x [ ⟨ f , X t ⟩ k ] = C k (x , f) , where the constant C k (x , f) can be identified in terms of the principal right eigenfunction and left eigenmeasure and g k (t) is an appropriate deterministic normalisation, which can be identified explicitly as either polynomial in t or exponential in t, depending on whether X is a critical, supercritical or subcritical process. The method we employ is extremely robust and we are able to extract similarly precise results that additionally give us the moment growth with time of ∫ 0 t ⟨ f , X t ⟩ d s , for bounded measurable f on E. [ABSTRACT FROM AUTHOR]

We derive formulas for the moments of the ruin time in a Lévy risk model and use these to determine the asymptotic behavior of the moments of the ruin time as the initial capital tends to infinity. In the special case of the perturbed Cramér-Lundberg model with phase-type or even exponentially distributed claims, we explicitly compute the first two moments of the ruin time. All our considerations distinguish between the profitable and the unprofitable setting. [ABSTRACT FROM AUTHOR]

Phase-type distributions model the time until absorption in continuous or discrete-time Markov chains on a finite state space. The multivariate phase-type distributions have diverse and important applications by modeling rewards accumulated at visited states. However, even moderately sized state spaces make the traditional matrix-based equations computationally infeasible. State spaces of phase-type distributions are often large but sparse, with only a few transitions from a state. This sparseness makes a graph-based representation of the phase-type distribution more natural and efficient than the traditional matrix-based representation. In this paper, we develop graph-based algorithms for analyzing phase-type distributions. In addition to algorithms for state space construction, reward transformation, and moments calculation, we give algorithms for the marginal distribution functions of multivariate phase-type distributions and for the state probability vector of the underlying Markov chains of both time-homogeneous and time-inhomogeneous phase-type distributions. The algorithms are available as a numerically stable and memory-efficient open source software package written in C named ptdalgorithms. This library exposes all methods in the programming languages C and R. We compare the running time of ptdalgorithms to the fastest tools using a traditional matrix-based formulation. This comparison includes the computation of the probability distribution, which is usually computed by exponentiation of the sub-intensity or sub-transition matrix. We also compare time spent calculating the moments of (multivariate) phase-type distributions usually defined by inversion of the same matrices. The numerical results of our graph-based and traditional matrix-based methods are identical, and our graph-based algorithms are often orders of magnitudes faster. Finally, we demonstrate with a classic problem from population genetics how ptdalgorithms serves as a much faster, simpler, and completely general modeling alternative. [ABSTRACT FROM AUTHOR]

Mohsin, Abbas and Khan [Stochastic Environmental Research and Risk Assessment, 2024, doi: 10.1007/s00477–024-02787-z] used a generalized Pareto-exponential distribution to model extreme events involving exponentially decaying variables. In this letter, we show that the maximum likelihood procedure used in simulation studies as well as the data application is not correct. We derive the corrected version. We also derive simpler expressions for mathematical properties of the generalized Pareto-exponential distribution. [ABSTRACT FROM AUTHOR]

In this paper, we study a problem posed by Lasserre (Discrete Comput Geom 50(3):673–678, 2013). Namely, if G = { x ∈ R n : g (x) ≤ 1 } is the compact sublevel set of a non-negative k-homogeneous polynomial g of n variables, we seek to find the minimum number of moments required to determine or to recover g. For polynomials of degree two, we can recover g computing eigenvalues and eigenvectors associated to a matrix M ∈ R n × n . For polynomials of degree k, we only prove that g is determined by all its n + k - 1 k moments of degree k [ABSTRACT FROM AUTHOR]

In the present work, a compact, fast, and yet accurate algorithm is developed to calculate the numerical values of the Ambartsumian-Chandrasekhar's H -function for isotropic scattering and its moments on the basis of the double exponential (DE) formula of Takahashi and Mori (RIMS, Kyoto Univ., 9:721, 1974). The main improvement made in the new method is an elimination of the iterative procedure for automatic adjustment of the step-size of integrations carried out with the DE-formula. Instead, a set of optimal values for the upper limit of integration T max and the number of division points N T to specify the step-size of the quadrature is predetermined for calculations of the H -function with a 15-digit accuracy, and also another for the evaluations of the moments with an accuracy of 14-digits or better. FORTRAN90 subroutines HFISCA for the H -function and HFMOMENT for the moments of arbitrary degrees are subsequently constructed (their source codes and a driver together with a sample set of output are shown in Appendix of this paper). Tables of sample calculations of the H -function and its moments of degree −1 through 6 carried out by these programs are also presented. The routines HFISCA and HFMOMENT should prove useful not only in astrophysical applications but also in other disciplines of science such as the electron transports in condensed matter and remote-sensing data analyses. A request for a copy of the Fortran 90 source code of the program can be made by writing to kawabata@rs.kagu.tus.ac.jp. [ABSTRACT FROM AUTHOR]

The paper deals with the semi-exponential type Gauss–Weierstrass operators. The central moments of these operators are constant functions. We estimate some direct results. We also provide a modification of such operators so as the preserve two exponential functions, we estimate weighted approximation, quantitative asymptotic formula for the modified form of operators. Later, a comparison is stated, that shows that in a certain sense and for certain family of illustrative functions the modified form of the semi-exponential Gauss–Weierstrass operators approximates better than the classical ones. Finally, we give some numerical results, which help us better understand the theoretical results obtained in the previous sections. [ABSTRACT FROM AUTHOR]

In 1905, Pearson proposed the following: "A man starts from a point 0 and walks l step in a straight line, then he turns any angle whatever and walks l step in a straight line. He repeats this process n times. I require the probability that after n steps he is at a distance r and r + dr from the starting point o." In this paper, we will present some basic properties and characterizations of the distribution of the distance for 2 and 3 steps. [ABSTRACT FROM AUTHOR]

Hawkes processes have been widely studied, but their many probability properties are still difficult to obtain, including their moments. In the paper, we shall give the moments for two classes of linear Hawkes processes with Gamma decay kernel and compound Gamma decay kernel functions by employing the method proposed by Cui et al. (2020), and the relationship between our results and those obtained by employing Dynkin's formula is studied. Finally, the computation complexity of numbers of first-order linear differential equations is considered. [ABSTRACT FROM AUTHOR]

Probability densities that are not uniquely determined by their moments are said to be "moment-indeterminate," or "M-indeterminate." Determining whether or not a density is M-indeterminate, or how to generate an M-indeterminate density, is a challenging problem with a long history. Quantum mechanics is inherently probabilistic, yet the way in which probability densities are obtained is dramatically different in comparison with standard probability theory, involving complex wave functions and operators, among other aspects. Nevertheless, the end results are standard probabilistic quantities, such as expectation values, moments and probability density functions. We show that the quantum mechanics procedure to obtain densities leads to a simple method to generate an infinite number of M-indeterminate densities. Different self-adjoint operators can lead to new classes of M-indeterminate densities. Depending on the operator, the method can produce densities that are of the Stieltjes class or new formulations that are not of the Stieltjes class. As such, the method complements and extends existing approaches and opens up new avenues for further development. The method applies to continuous and discrete probability densities. A number of examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a two parameter extension of exponentiated gamma distribution. We explicitly derive the closed form expressions of the moments, mode and quantiles of the proposed distribution. L-moments and coefficients of skewness and kurtosis are obtained using the quantile function. Other important properties including identifiability, entropy, stochastic orderings, stress-strength reliability and differential equations associated with the distribution are also discussed. We briefly describe different estimation procedures namely, the method of maximum likelihood estimation, moment estimation, maximum product of spacings estimation, ordinary and weighted least squares estimation, and Cramér–von-Mises estimation along with an extensive simulation study for comparing their performance. An application of modeling trees growth data is presented to show the adequacy of the proposed distribution over the distributions existing in the literature. A parametric regression model based on the proposed distribution is introduced and used to establish a regression model for the volume, diameter and height of the trees. [ABSTRACT FROM AUTHOR]

Let p be a large prime number and 1 ≤ N < p. For an integer λ denote J (λ) = J (λ ; N , p) the number of representations n ! ≡ λ (mod p) , 1 ≤ n ≤ N. In the present paper we obtain a nontrivial upper bound for the moment ∑ λ J c (λ) , where c ≥ 1 is a fixed real number. We apply this bound to some congruences that involve factorials and intervals. [ABSTRACT FROM AUTHOR]

The continuous lognormal distribution has been used to model discrete count data, which is clearly not appropriate. In this paper, we introduce two discrete versions of the continuous lognormal distribution. We study their mathematical properties and estimation issues. Two real data applications show superior performance of the discrete versions over the continuous counter parts. [ABSTRACT FROM AUTHOR]

A new class of distributions motivated by systems having both series and parallel structures is introduced. Some mathematical properties of the new class (including the moment generating function, moments and order statistics) are derived. Estimation is addressed by the maximum likelihood method and the performance of the estimators assessed by a simulation study. An illustration using three failure data sets shows the usefulness of the new class. [ABSTRACT FROM AUTHOR]

In this paper, relations between some kinds of cumulative entropies and moments of order statistics are established. By using some characterizations and the symmetry of a non-negative and absolutely continuous random variable X, lower and upper bounds for entropies are obtained and illustrative examples are given. By the relations with the moments of order statistics, a method is shown to compute an estimate of cumulative entropies and an application to testing whether data are exponentially distributed is outlined. [ABSTRACT FROM AUTHOR]

The current study performed the laboratory flume to examine the velocity distribution and the velocity moments of turbulent flow over a sand-gravel mixture bed. The 3D instantaneous velocity data of water is collected by using acoustic doppler velocimeter at the test section which will provide an important data related to the flow turbulence. The parameters of turbulence measured that the lower value of longitudinal velocity (velocity along flow direction) is observed close to the bed surface and increases with the flow depth. The lower magnitude of von Karman constant is achieved than the universal value. The damping of Reynolds shear stress observed close to the boundary wall indicating lower exchange of flow energy towards the boundary and vice versa. The sand-gravel bed influenced the turbulence intensities with higher magnitude in the streamwise than those in vertical direction. The velocity moments which show the structure function of fluctuating components of velocity, has been analysed. The present study also analysed the flow anisotropy, and the flatness factor in the flow. [ABSTRACT FROM AUTHOR]

Andrews, Lewis, and Lovejoy introduced the partition function PD(n) as the number of partitions of n with designated summands. They proved that P D (3 n + 2) is divisible by 3. Later, Chen, Ji, Jin, and the author introduced the pd-rank for partitions with designated summands and gave a combinatorial interpretation of the congruence. In this paper, we define a crank named the d-crank of partitions with designated summands to provide a new combinatorial interpretation for the congruence of Andrews, Lewis, and Lovejoy for P D (3 n + 2) . In addition, we derive the generating functions for the d-crank differences of partitions with designated summands modulo 2, 3, 4, and 6. Based on these generating functions, we obtain an equality and some inequalities for d-cranks of partitions with designated summands. We also introduce the d-crank moments weighted by the parity of d-cranks μ 2 k , d (- 1 , n) and show the positivity of (- 1) n μ 2 k , d (- 1 , n) . [ABSTRACT FROM AUTHOR]

We provide a new method to approximate a (possibly discontinuous) function using Christoffel–Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the graph of the function. Such an input is available when approximating weak (or measure-valued) solution of optimal control problems, entropy solutions to nonlinear hyperbolic PDEs, or using numerical integration from finitely many evaluations of the function. While most of the existing methods construct a piecewise polynomial approximation, we construct a semi-algebraic approximation whose estimation and evaluation can be performed efficiently. An appealing feature of this method is that it deals with nonsmoothness implicitly so that a single scheme can be used to treat smooth or nonsmooth functions without any prior knowledge. On the theoretical side, we prove pointwise convergence almost everywhere as well as convergence in the Lebesgue one norm under broad assumptions. Using more restrictive assumptions, we obtain explicit convergence rates. We illustrate our approach on various examples from control and approximation. In particular, we observe empirically that our method does not suffer from the Gibbs phenomenon when approximating discontinuous functions. [ABSTRACT FROM AUTHOR]

In this paper we introduce the exponential sampling Durrmeyer series. We discuss pointwise and uniform convergence properties and an asymptotic formula of Voronovskaja type. Quantitative results are given, using the usual modulus of continuity for uniformly continuous functions. Some examples are also described. [ABSTRACT FROM AUTHOR]

This paper considers the representation of odd moments of the distribution of a four-step uniform random walk in even dimensions, which are based on both linear combinations of two constants representable as contiguous very well-poised generalized hypergeometric series and as even moments of the square of the complete elliptic integral of the first kind. Neither constants are currently available in closed form. New symmetries are found in the critical values of the L-series of two underlying cusp forms, providing a sense in which one of the constants has a formal counterpart. The significant roles this constant and its counterpart play in multidisciplinary contexts is described. The results unblock the problem of representing them in terms of lower-order generalized hypergeometric series, offering progress towards identifying their closed forms. The same approach facilitates a canonical characterization of the hypergeometry of the parbelos, adding to the characterizations outlined by Campbell, D'Aurozio and Sondow (2020, The American Mathematical Monthly127(1), 23-32). The paper also connects the econometric problem of characterizing the bias in the canonical autoregressive model under the unit root hypothesis to very well-poised generalized hypergeometric series. The confluence of ideas presented reflects a multidisciplinarity that accords with the approach and philosophy of Prasanta Chandra Mahalanobis. [ABSTRACT FROM AUTHOR]

We study the problem of P. L. Chebyshev (proposed in 1883) concerning the extreme values of moments of nonnegative polynomials with weight on the interval at a fixed zero moment, as well as this problem in a more general form. In the case of the first moment , the problem was solved by P. L. Chebyshev (1883) in the case of unit weight and by G. Szegö (1927) for an arbitrary weight. We have previously obtained a solution to Chebyshev's problem for moments of odd order, which is largely based on the monotonicity of the function , . The function is not monotone on the interval , and the problem for moments of even order becomes more difficult. The paper provides a solution to Chebyshev's problem on the largest values of moments of even order for polynomials of even degree. The problem of the smallest value of the second moment for polynomials of even degree is solved under an additional condition for the weight. [ABSTRACT FROM AUTHOR]