Generating Function Method Research Articles

New Classes of Degenerate Unified Polynomials

In this paper, we introduce a class of new classes of degenerate unified polynomials and we show some algebraic and differential properties. This class includes the Appell-type classical polynomials and their most relevant generalizations. Most of the results are proved by using generating function methods and we illustrate our results with some examples.

Binomial-Geometric Mixture and Its Applications

A mixture distribution is a combination of two or more probability distributions; it can be obtained from different distribution families or the same distribution families with different parameters. The underlying distributions may be discrete or continuous, so the resulting mixture probability distribution function should be a mass or density function. In the last few years, there has been great interest in the problem of developing a mixture distribution based on the binomial distribution. This paper uses the probability generating function method to develop a new two-parameter discrete distribution called a binomial-geometric (BG) distribution, a mixture of binomial distribution with the number of trials (parameter <img src=image/13428702_01.gif>) taken after a geometric distribution. The quantile function, moments, moment generating function, Shannon entropy, order statistics, stress-strength reliability and simulating the random sample are some of the statistical highlights of the BG distribution that are explored. The model's parameters are estimated using the maximum likelihood method. To examine the performance of the accuracy of point estimates for BG distribution parameters, the Monte Carlo simulation is performed with different scenarios. Finally, the BG distribution is fitted to two real lifetime count data sets from the medical field. As a result, the proposed BG distribution is an overdispersed right-skewed and can accommodate a constant hazard rate function. The proposed BG distribution is appropriate for modelling the overdispersed right-skewed real-life count data sets and it can be an alternative to the negative binomial and geometric distributions.

Analytical solution of the DGLAP equations using the generating function method

Recently, we generated the evolution equations of the parton distribution functions (PDF) usually used in the hadrons phenomenology by the stochastic modeling of the non-equilibrium statistical mechanics in the momentum space. The evolution equations obtained from the stochastic modeling are the same as the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations, but they are acquired by a mathematical procedure based on the non-equilibrium statistical mechanics and Markov processes. In this paper, we present a new analytical solution of the parton evolution equation (DGLAP equation) for the non-singlet quark distribution function using the generating function method. This method is a general technique for solving the master equation of Markov processes.

On the effect of training simulation point spread function in deep-learning based super-resolution ultrasound imaging

Deep learning (DL)-based super-resolution ultrasound microvascular imaging (SR-UMI) has gained interest in recent years owing to the success of DL in medical imaging applications. Mainstream DL techniques require an abundance of labeled training data. Since high-quality in vivo training data are challenging to obtain for medical imaging, most of the current studies have relied on simulations to generate training data of microbubbles (MBs) flowing in vasculature. This study investigates the effect of different MB point spread function (PSF) generation methods on the performance of DL models for SR-UMI. We used Gaussian shaped PSF (Gaussian-PSF), numerical simulation software (Field II-PSF), and experimentally collected PSFs (Exp-PSF) to generate DL training sets. The training sets were used to train DL models that performed both MB localization- and non-localization-based SR-UMI. The quality of the models trained with different datasets were evaluated based on their in vivo performance (mouse brain, chicken embryo), with optical imaging as ground truth for some applications. We discovered that Exp-PSF achieved the best performance overall, due to its high resemblance to real experimental data. Finally, we present a pipeline for obtaining Exp-PSF and integrating them with ultrasound MB simulation, with a publicly available Exp-PSF dataset provided via Github.

Some new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order $\alpha$ and level $m$ in the variable $x$. Here the degenerate polynomials are a natural extension of the classic polynomials. In more detail, we derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. Most of the results are proved by using generating function methods.

Four Variants of Riemann Zeta Function

By means of the generating function method and Dougall’s formulae for bilateral hypergeometric series, we examine four classes of infinite series, which may be considered as variants of Riemann zeta function. Several summation formulae are established in closed form, which shows remarkably that the values of these series result in multiples of integer powers of π either by rational numbers or by algebraic numbers.

Analysis of batch size-dependent bulk service queue with multiple working vacation

ABSTRACT The present article analyses an infinite buffer batch size-dependent bulk service queue with multiple working vacation (MWV). The customer's arrival pattern follows the Poisson process, and they get the service in batches according to the general bulk service (GBS) rule in regular period (RP) as well as in working vacation period (WVP). After one service in RP, if the queue size is greater or equal to the lower threshold of the GBS rule, then the server performs the service in RP, otherwise, it starts the working vacation (WV) following exponential vacation time distribution. During the WVP, the server serves the customers in batches (if any) at a lower service rate than the usual service rate. The service time in the RP as well as in WVP is generally distributed. At an arbitrary epoch, the joint probabilities of the queue size and batch size with the server in RP as well as in WVP have been obtained using the supplementary variable technique (SVT) and the bivariate generating function method. Finally, some numerical observations are presented to enhance the applicability of the analytical results.

Construction and Dynamics of Efficient High-Order Methods for Nonlinear Systems

In this paper, we derive new multi-parametric families of iterative methods whose orders range from six to eight, for solving nonlinear systems. Based on a generating function method known in the literature, we construct these families in the most general way possible in order to include some well-known methods as special cases. Several applied problems are solved to check the performance of our methods and other existing ones and to verify the theoretical results. It is found that our methods are competitive in performance compared to the other methods. Moreover, the basin of attraction method is introduced for nonlinear systems to confirm our findings and to choose the best performers.

Selective maintenance of multi‐state systems with the repairperson fatigue effect and stochastic break duration

AbstractSystems in the industry are often required to execute a sequence of missions, and the state of the multi‐state systems may deteriorate with the increasing of running time. To improve the system reliability of the next mission, selective maintenance is applied in the limited break. Due to the fatigue effect, the work rate of the repairperson decays with time. However, the existing selective maintenance studies do not consider the impact of fatigue effect on work rate, which leads to underestimating the duration of maintenance actions and overestimating the system reliability. To overcome the above problems, a new selective maintenance model for multi‐state systems is developed to maximize the system reliability of the next mission. First, the decay process of the work rate of a repairperson is described as a piecewise continuous decreasing exponential function of working time. Next, a homogenous discrete‐state continuous‐time Markov process is used to describe the performance capacity degradation of components. Considering the stochastic break duration, based on the state probability distribution of each component obtained by the Markov model, the universal generation function (UGF) method is applied to evaluate the multi‐state system reliability. Then, taking the selected maintenance actions and maintenance priority as decision variables, the resulting selective maintenance optimization problem is solved by a tailored genetic algorithm. Finally, two computational experiments are carried out to demonstrate the effectiveness of the proposed method.

Age Analysis of Status Updating System with Probabilistic Packet Preemption.

The age of information (AoI) metric was proposed to measure the freshness of messages obtained at the terminal node of a status updating system. In this paper, the AoI of a discrete time status updating system with probabilistic packet preemption is investigated by analyzing the steady state of a three-dimensional discrete stochastic process. We assume that the queue used in the system is , which represents that the system size is 2 and the packet in the buffer can be preempted by a fresher packet with probability . Instead of considering the system’s AoI separately, we use a three-dimensional state vector to simultaneously track the real-time changes of the AoI, the age of a packet in the server, and the age of a packet waiting in the buffer. We give the explicit expression of the system’s average AoI and show that the average AoI of the system without packet preemption is obtained by letting . When is set to 1, the mean of the AoI of the system with a queue is obtained as well. Combining the results we have obtained and comparing them with corresponding average continuous AoIs, we propose a possible relationship between the average discrete AoI with the queue and the average continuous AoI with the queue. For each of two extreme cases where and , we also determine the stationary distribution of AoI using the probability generation function (PGF) method. The relations between the average AoI and the packet preemption probability , as well as the AoI’s distribution curves in two extreme cases, are illustrated by numerical simulations. Notice that the probabilistic packet preemption may occur, for example, in an energy harvest (EH) node of a wireless sensor network, where the packet in the buffer can be replaced only when the node collects enough energy. In particular, to exhibit the usefulness of our idea and methods and highlight the merits of considering discrete time systems, in this paper, we provide detailed discussions showing how the results about continuous AoI are derived by analyzing the corresponding discrete time system and how the discrete age analysis is generalized to the system with multiple sources. In terms of packet service process, we also propose an idea to analyze the AoI of a system when the service time distribution is arbitrary.

A comparative study of multiband mamdani fuzzy classification methods for west of Iraq satellite image

In our paper, performance of four fuzzy membership function generation methods was studied. These methods were studied in the context of implementing Mamdani fuzzy classification on a set of satellite images for western Iraqi territory. The first method generate triangulate membership functions using mean, minimum (min) and maximum (max) of histogram attribute values (AV), while peak and standard deviation (STD) of these AV were used in the second. On the other hand, in the third and fourth methods, Gaussian membership functions are generated using same mentioned values in the first and second method respectively. The goal was to generate a Mamdani type fuzzy inference system the membership function (MF) of each fuzzy set and implementing the AV of western Iraqi territory training data sets. A pixel-by-pixel comparison of each method with traditional maximum likelihood method (ML) was made on data sets comprising six bands of satellite imagery of the western Iraqi region taken by the Landsat-5 satellite. Simulation results of these performance comparisons singled out that the method using Gaussian MFs together with peak and STD of the AV as the best achiever with a similarity of 83.16 percent for band (3) of the studied area.

Lagrange-Based Hypergeometric Bernoulli Polynomials

Special polynomials play an important role in several subjects of mathematics, engineering, and theoretical physics. Many problems arising in mathematics, engineering, and mathematical physics are framed in terms of differential equations. In this paper, we introduce the family of the Lagrange-based hypergeometric Bernoulli polynomials via the generating function method. We state some algebraic and differential properties for this family of extensions of the Lagrange-based Bernoulli polynomials, as well as a matrix-inversion formula involving these polynomials. Moreover, a generating relation involving the Stirling numbers of the second kind was derived. In fact, future investigations in this subject could be addressed for the potential applications of these polynomials in the aforementioned disciplines.

Three-parameter interval number theory and its application in power system reliability evaluation

With the increasing integration of renewable energy in the power system, its strong volatility and randomness will greatly affect the planning and operation of the power system. In this situation, compared to the traditional point-value-based reliability assessment results, system operators prefer the results which can give the most likely recommended values and their possible ranges. To this end, in this paper, we present a novel framework to evaluate the reliability of multi-state power systems that incorporates three-parameter interval number theory and improved universal generating function method. In particular, we first improve the three-parameter interval number theory by proposing a similarity-based indicator to describe the distance (or closeness-degree) between two three-parameter interval numbers and redefine the probability that one three-parameter interval number is greater or less than another one. Secondly, we extended the traditional universal generating function to a three-parameter interval number-based universal generating function. Moreover, the three-parameter interval number-based universal generating functions of conventional generators, renewable energy generators and load demand are further established to describe their multi-state characteristics. At last, a three-parameter interval number-based loss of load expectation metric is presented for reliability evaluation. Compared with the existing methods, the proposed framework can well evaluate the reliability of multi-state power systems, and can describe the uncertainty of reliability evaluation results through the three-parameter interval number, thus providing the most likely value of the evaluation result and its fluctuation range at the same time. In the numerical simulation part, a comparison of five different scenarios is performed on an IEEE Reliability Test System. The results show that the proposed method is suitable for the reliability assessment of power systems with high penetration of renewable energy, which can describe the fluctuation range and indicate the most possible value of reliability simultaneously.

Measurement-induced quantum walks.

We investigate a tight-binding quantum walk on a graph. Repeated stroboscopic measurements of the position of the particle yield a measured "trajectory," and a combination of classical and quantum mechanical properties for the walk are observed. We explore the effects of the measurements on the spreading of the packet on a one-dimensional line, showing that except for the Zeno limit, the system converges to Gaussian statistics similarly to a classical random walk. A large deviation analysis and an Edgeworth expansion yield quantum corrections to this normal behavior. We then explore the first passage time to a target state using a generating function method, yielding properties like the quantization of the mean first return time. In particular, we study the effects of certain sampling rates that cause remarkable changes in the behavior in the system, such as divergence of the mean detection time in finite systems and decomposition of the phase space into mutually exclusive regions, an effect that mimics ergodicity breaking, whose origin here is the destructive interference in quantum mechanics. For a quantum walk on a line, we show that in our system the first detection probability decays classically like (time)^{-3/2}. This is dramatically different compared to local measurements, which yield a decay rate of (time)^{-3}, indicating that the exponents of the first passage time depend on the type of measurements used.

Mathematical analysis of a Markovian multi-server feedback queue with a variant of multiple vacations, balking and reneging

In this paper, we analyze a multi-server queue with customers impatience and Bernoulli feedback under a variant of multiple vacations. On arrival, a customer decides whether to join or balk the system, based on the observation of the system size as well as the status of the servers. It is supposed that customer impatience can arise both during busy and vacation period because of the long wait already experienced in the system. The latter can be retained via certain mechanism used by the system. The feedback occurs as returning a part of serviced customers to get a new service. The queue under consideration can be used to model the processes of information transmission in telecommunication networks. We develop the Chapman-Kolmogorov equations for the steady-state probabilities and solve the differential equations by using the probability generating function method. In addition, we obtain explicit expressions of some important system characteristics. Different queueing indices are derived such as the probabilities when the servers are in different states, the mean number of customers served per unit of time, and the average rates of balking and reneging.

Computational and implementational analysis of generating functions for higher order combinatorial numbers and polynomials attached to Dirichlet characters

The main purpose of this paper is to give computational and implementational analysis of generating functions for some extensions of combinatorial numbers and polynomials attached to Dirichlet characters. By using generating function methods and p‐adic q‐integral techniques, it is also aimed to derive some computational formulas for these numbers and polynomials. By implementing both the derived computational formulas and the constructed generating functions in Mathematica, we present some tables and plots for these numbers and polynomials, and their generating functions to illustrate the effects of their parameters for some special cases. Moreover, by making an observation on some results of this paper, we derive some novel computational formulas for the finite sums that contain the Dirichlet characters and falling factorials. We also assess some cases of these special finite sums. In the sequel, we pose some open problems regarding these special finite sums. Apart from these findings, we also give not only Fourier series expansion but also asymptotic estimates for the mentioned combinatorial numbers. In addition, we give some Raabe‐type multiplication formulas for the mentioned combinatorial polynomials. Finally, we give an observation on decompositions of the generating functions attached to any group homomorphism.

Generating Function Method for Calculating the Potentials of Inhomogeneous Polyhedra

We propose a method of constructing analytical, closed-form expressions for electrostatic/Newtonian potentials of non-uniform polyhedral bodies, in which the density distributions are polynomials of coordinates. Possible applications of the proposed method are spread from astronomy to nanotechnology. The method is based on the use of the generating function for the potential. Explicit expressions for the potential are derived in the case of quadratic or cubic coordinate dependence of the density within a polyhedral body.

Enumerating Possible Molecular Formulae in Mass Spectrometry Using a Generating Function Based Method

What molecular formulae correspond to a given mass M? Here, we address this problem - the compomer problem of mass spectrometry. We describe, implement, and test a straightforward and practical aufbau algorithm which leads to the molecular formula tree from which all possible solutions may be obtained. The algorithm was inspired from the generating function method used in combinatorics, to count the number of compomers. The molecular formula tree can be generated in linear time with regards to the number of elements N, and in quadratic time with respect to the mass M; and we demonstrate that the memory requirements do not exceed that of a standard laptop for the formula of a small molecule, say about 10 elements and a mass of 1000 Da. Nevertheless, we still discuss and demonstrate how a range of heuristics can be used to mitigate the memory requirements.

Analysis of a single server queue in a multi-phase random environment with working vacations and customers’ impatience

In this paper, we analyze an M/M/1 queueing system under both single and multiple working vacation policies, multiphase random environment, waiting server, balking and reneging. When the system is in operative phase j = 1,2,...,K, customers are served one by one. Whenever the system becomes empty, the server waits a random amount of time before taking a vacation, causing the system to move to working vacation phase 0 at which new arrivals are served at a lower rate. Using the probability generating function method, we obtain the distribution for the steady-state probabilities of the system. Then, we derive important performance measures of the queueing system. Finally, some numerical examples are illustrated to show the impact of system parameters on performance measures of the queueing system.

Cost Optimization of Queueing System with Working Vacation, Setup, Feedback, Reneging, and Retention of Reneged Customers

In this manuscript, Markovian queueing system with working vacation, Bernoulli schedule interruption, setup time under feedback, reneging of impatient customers, and retention of reneged customers are analyzed. The unsatisfied customers on service completion may either leave the system with probabilities or may rejoin the queue with complementary probabilities during working vacations and regular service, respectively. The waiting customers in the queue may lose patience due to vacations and decide to leave without getting the service with probability q. They may be retained in the system via some convincing mechanisms with probability (1-q). The mean system length, probability of server in various states, mean sojourn time are obtained using the probability generating function method. The MATLAB software is used for representing the observed behavior graphically.