Search Results (63)

A previous study determined criteria ensuring that a probability distribution supported in positive integers is the limiting conditional law of a subcritical Markov branching process. It is known that there is an close connection between the limiting conditional law and the stationary measure of the transition semigroup. This paper revisits that theme of by seeking tractable criteria ensuring that a sequence on positive integers is the stationary measure of a subcritical or critical Markov branching process. These criteria are illustrated with several examples. The subcritical case motivates consideration of the Sibuya distribution, leading to the demonstration that members of a certain family of complete Bernstein functions, in fact, are Thorin–Bernstein. The critical case involves deriving a notion of the limiting law of population size given that extinction occurs at a precise future time. Examples are given, and some show an interesting relation between stationary measures and Hausdorff moment sequences. Full article

This study investigates the computation of fractional and higher integer-order moments for a stochastic process governed by a one-dimensional, non-homogeneous linear stochastic differential equation (SDE) driven by fractional Brownian motion (fBm). Unlike conventional approaches relying on moment-generating functions or Fokker–Planck equations, which often yield intractable expressions, we derive explicit closed-form formulas for these moments. Our methodology leverages the Wick–Itô calculus (fractional Itô formula) and the properties of Hermite polynomials to express moments efficiently. Additionally, we establish a recurrence relation for moment computation and propose an alternative approach based on generalized binomial expansions. To validate our findings, Monte Carlo simulations are performed, demonstrating a high degree of accuracy between theoretical and empirical results. The proposed framework provides novel insights into stochastic processes with long-memory properties, with potential applications in statistical inference, mathematical finance, and physical modeling of anomalous diffusion. Full article

In this paper, we propose an approach for constructing quasiparticle-like asymptotic solutions within the weak diffusion approximation for the generalized population Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation, which incorporates nonlocal quadratic competitive losses and a fractal time derivative of non-integer order ($\alpha$, where $0<\alpha \le 1$). This approach is based on the semiclassical approximation and the principles of the Maslov method. The fractal time derivative is introduced in the framework of $F^{\alpha }$ calculus. The Fisher–KPP equation is decomposed into a system of nonlinear equations that describe the dynamics of interacting quasiparticles within classes of trajectory-concentrated functions. A key element in constructing approximate quasiparticle solutions is the interplay between the dynamical system of quasiparticle moments and an auxiliary linear system of equations, which is coupled with the nonlinear system. General constructions are illustrated through examples that examine the effect of the fractal parameter ($\alpha$) on quasiparticle behavior. Full article

Front distribution centers are extensively employed in E-commerce distribution networks to shorten the delivery time, thereby stimulating customers’ purchase intentions and enhancing customer loyalty. When a customer places an order, the designated front distribution center quickly processes it to ensure prompt delivery. If the front distribution center is out of stock, the order will be fulfilled by its corresponding regional distribution center, which will result in a longer delivery time. Once the regional distribution center is also out of stock, a lost sale occurs. This paper improves a distributionally robust allocation model aimed at enhancing the fulfillment rates of front distribution centers while also preserving the overall fulfillment rate within the region. We reformulate this distributionally robust allocation model into an equivalent mixed-integer linear programming model and develop a corresponding approximation algorithm. Through numerical experiments, we comprehensively reveal the impact of moment information in demand forecasting on the distributionally robust fulfillment rate improvement algorithm by discovering how demand forecasting influences the allocation rule and how forecasted variance influences the fulfillment rates at fixed or changing inventory levels. Full article

In real-life inter-related time series, the counting responses of different entities are commonly influenced by some time-dependent covariates, while the individual counting series may exhibit different levels of mutual over- or under-dispersion or mixed levels of over- and under-dispersion. In the current literature, there is still no flexible bivariate time series process that can model series of data of such types. This paper introduces a bivariate integer-valued autoregressive of order 1 (BINAR(1)) model with COM-Poisson innovations under time-dependent moments that can accommodate different levels of over- and under-dispersion. Another particularity of the proposed model is that the cross-correlation between the series is induced locally by relating the current observation of one series with the previous-lagged observation of the other series. The estimation of the model parameters is conducted via a Generalized Quasi-Likelihood (GQL) approach. The proposed model is applied to different real-life series problems in Mauritius, including transport, finance, and socio-economic sectors. Full article

The growing penetration of renewable energy sources (RES) has exacerbated operational flexibility deficiencies in modern power systems under time-varying conditions. To address the limitations of existing flexibility management approaches, which often exhibit excessive conservatism or risk exposure in managing supply–demand uncertainties, this study introduces a data-driven distributionally robust optimization (DRO) framework for power system scheduling. The methodology comprises three key phases: First, a meteorologically aware uncertainty characterization model is developed using Copula theory, explicitly capturing spatiotemporal correlations in wind and PV power outputs. System flexibility requirements are quantified through integrated scenario-interval analysis, augmented by flexibility adjustment factors (FAFs) that mathematically describe heterogeneous resource participation in multi-scale flexibility provision. These innovations facilitate the formulation of physics-informed flexibility equilibrium constraints. Second, a two-stage DRO model is established, incorporating demand-side resources such as electric vehicle fleets as flexibility providers. The optimization objective aims to minimize total operational costs, encompassing resource activation expenses and flexibility deficit penalties. To strike a balance between robustness and reduced conservatism, polyhedral ambiguity sets bounded by generalized moment constraints are employed, leveraging Wasserstein metric-based probability density regularization to diminish the probabilities of extreme scenarios. Third, the bilevel optimization structure is transformed into a solvable mixed-integer programming problem using a zero-sum game equivalence. This problem is subsequently solved using an enhanced column-and-constraint generation (C&CG) algorithm with adaptive cut generation. Finally, simulation results demonstrate that the proposed model positively impacts the flexibility margin and economy of the power system, compared to traditional uncertainty models. Full article

Spherical tanks are widely utilized in process industries due to their substantial storage capacity. These industries’ inherent challenges necessitate using highly efficient controllers to manage various process parameters, especially given their nonlinear behavior. This paper proposes the Approximate Generalized Time Moments (AGTM) optimization technique for designing the parameters of multi-model fractional-order controllers for regulating the output (liquid level) of a real-time nonlinear spherical tank. System identification for different regions of the nonlinear process is here innovatively conducted using a black-box model, which is determined to be nonlinear and approximated as a First Order Plus Dead Time (FOPDT) system over each region. Both model identification and controller design are performed in simulation and real-time using a National Instruments NI DAQmx 6211 Data Acquisition (DAQ) card (NI SYSTEMS INDIA PVT. LTD., Bangalore Karnataka, India) and MATLAB/SIMULINK software (MATLAB R2021a). The performance of the overall algorithm is evaluated through simulation and experimental testing, with several setpoints and load changes, and is compared to the performance of other algorithms tuned within the same framework. While traditional approaches, such as integer-order controllers or linear approximations, often struggle to provide consistent performance across the operating range of spherical tanks, it is originally shown how the combination of multi-model fractional-order controller design—AGTM optimization method—GA for expansion point selection and index minimization has benefits in specifically controlling a (difficult to be controlled) nonlinear process. Full article

The lognormal moment sequence is considered. Using the fractional moments technique, it is first proved that the lognormal has the largest differential entropy among the infinite positively supported probability densities with the same lognormal-moments. Then, relying on previous theoretical results on entropy convergence obtained by the authors concerning the indeterminate Stieltjes moment problem, the lognormal distribution is accurately reconstructed by the maximum entropy technique using only its integer moment sequence, although it is not uniquely determined by moments. Full article

In recent decades, the study of discrete distributions has received increasing attention in the field of statistics, mainly because discrete distributions can model a wide range of count data. One common distribution used for modeling count data, for instance, is the negative binomial distribution (NBD), which performs well with over-dispersed data. In this paper, a new count distribution is introduced, called the conflation of negative binomial and logarithmic distributions, which is formed by conflating the negative binomial and logarithmic distributions, resulting in a distribution that possesses some of the properties of negative binomial and logarithmic distributions. The distribution has two parameters and is verified by a positive integer. Two modifications are proposed to the distribution, which includes zero as a support point. The new distribution is valuable from a theoretical perspective since it is a member of the weighted negative binomial distribution family. In addition, the distribution differs from the NBD in the sense that the probability of lower counts is inflated. This study discusses the characteristics of the proposed distribution and its modified versions, such as moments, probability generating functions, likelihood stochastic ordering, log-concavity, and unimodality properties. Real-world data are used to evaluate the performance of the proposed models against other models. All computations shown in this paper were produced using the R programming language. Full article

In many real-life instances, time series of counts are often exposed to the dispersion phenomenon while at the same time being influenced by some explanatory variables. This paper takes into account these two issues by assuming that the series of counts follow an observation-driven first-order integer-valued moving-average structure (INMA(1)) where the innovation terms are COM-Poisson (CMP) distributed under a link function characterized by time-independent covariates. The second part of the paper constitutes estimating the regression effects, dispersion and the serial parameters using popular estimation methods, mainly the Conditional Least Squares (CLS), Generalized Method of Moments and Generalized Quasi-Likelihood (GQL) approaches. The performance of these estimation methods is compared via simulation experiments under different levels of dispersion. Additionally, the suggested model is used to examine time series accident data from actual incidents in several Mauritius locations. Full article

The first part of this work provides explicit solutions for two integral equations; both are solved by means of Fourier transform. In the second part of this paper, sufficient conditions for the existence and uniqueness of the solutions satisfying sandwich constraints for two types of full moment problems are provided. The only given data are the moments of all positive integer orders of the solution and two other linear, not necessarily positive, constraints on it. Under natural assumptions, all the linear solutions are continuous. With their value in the subspace of polynomials being given by the moment conditions, the uniqueness follows. When the involved linear solutions and constraints are positive, the sufficient conditions mentioned above are also necessary. This is achieved in the third part of the paper. All these conditions are written in terms of quadratic expressions. Full article

To gain a sustained competitive advantage, organizations such as UPS, Fedex, Amazon, etc., began to seek for industry 5.0 innovative autonomous delivery options for the last mile. Autonomous unmanned aerial vehicles are a promising alternative for the logistics industry. The fact that drones are propelled by green renewable energy source fits the companies’ need to become sustainable, replacing their fuel truck fleets, especially for traveling to remote rural locations to deliver small packages, but a major obstacle is the necessity for charging stations which is well documented in the literature. Therefore, the current research embarks on devising a novel yet practical piece of technology adopting the simplicity approach of direct flights to destinations. The analysis showcases the application for a network of warehouses and hospitals in Israel while controlling costs. Given the products in the case study are medical, direct flight has the potential to save lives when every moment counts. Hydrogen cell technology allows long-range flying without refueling, and it is both vibration-free which is essential for sensitive medical equipment and environmentally friendly in terms of air pollution and silence in urban areas. Importantly, hydrogen cells are lighter, with higher energy density than batteries, which makes them ideal for drone usage to reduce weight, maintain a longer life, and enable faster charging, all of which minimize downtime. Also, hydrogen sourcing is low-cost and unlimited compared to lithium-ion material which needs to be mined. The case study investigates an Israeli entrepreneurial company, Gadfin, which builds a vertical takeoff-and-landing-type of drone with folded wings that enable higher speed for the delivery of refrigerated medical cargo, blood, organs for transplant, and more to hospitals in partnership with the Israeli medical logistic conglomerate, SAREL. An analysis of shipping optimization (concerning the number and type of drone) is conducted using a mixed-integer linear programming technique based on various types of constraints such as traveling distance, parcel weight, the amount of flight controllers and daily number of flights allowed in order to not overcrowd the airspace. Importantly, the discussion assesses the ecosystem’s variety of risks and commensurate safety mechanisms for advancing a newly shaped landscape of drones in an Israeli tight airspace to establish a network of national routes for drone traffic. The conclusion of this research cautions limitations to overcome as the utilization of drones expand and offers future research avenues. Full article

The aim of this paper is to consider the indeterminate Stieltjes moment problem together with all its probability density functions that have the positive real or the entire real axis as support. As a consequence of the concavity of the entropy function in both cases, there is one such density that has the largest entropy: we call it $f_{\mathrm{hmax}}$, the largest entropy density. We will prove that the Jaynes maximum entropy density (MaxEnt), constrained by an increasing number of integer moments, converges in entropy to the largest entropy density $f_{\mathrm{hmax}}$. Note that this kind of convergence implies convergence almost everywhere, with remarkable consequences in real applications in terms of the reliability of the results obtained by the MaxEnt approximation of the underlying unknown distribution, both for the determinate and the indeterminate case. Full article

In the case of certain chemical compounds, especially organic ones, electrons can be delocalized between different atoms within the molecule. These resulting bonds, known as resonance bonds, pose a challenge not only in theoretical descriptions of the studied system but also present difficulties in simulating such systems using molecular dynamics methods. In computer simulations of such systems, it is often common practice to use fractional bonds as an averaged value across equivalent structures, known as a resonance hybrid. This paper presents the results of the analysis of five forms of C₆₀ fullerene polymorphs: one with all bonds being resonance, three with all bonds being integer (singles and doubles in different configurations), one with the majority of bonds being integer (singles and doubles), and ten bonds (within two opposite pentagons) valued at one and a half. The analysis involved the Shannon entropy value for bond length distributions and the eigenfrequency of intrinsic vibrations (first vibrational mode), reflecting the stiffness of the entire structure. The maps of the electrostatic potential distribution around the investigated structures are presented and the dipole moment was estimated. Introducing asymmetry in bond redistribution by incorporating mixed bonds (integer and partial), in contrast to variants with equivalent bonds, resulted in a significant change in the examined observables. Full article

We deal with absolutely continuous probability distributions with finite all-positive integer-order moments. It is well known that any such distribution is either uniquely determined by its moments (M-determinate), or it is non-unique (M-indeterminate). In this paper, we follow the maximum entropy approach and establish a new criterion for the M-indeterminacy of distributions on the positive half-line (Stieltjes case). Useful corollaries are derived for M-indeterminate distributions on the whole real line (Hamburger case). We show how the maximum entropy is related to the symmetry property and the M-indeterminacy. Full article