Generating Function Method Research Articles

Developing bulk arrival queuing models with the constant batch policy under uncertainty data using (0 - 1) variables

This paper delves into some significant performance measures (PMs) of a bulk arrival queueing system with constant batch size b, according to arrival rates and service rates being fuzzy parameters. The bulk arrival queuing system deals with observation arrival into the queuing system as a constant group size before allowing individual customers entering to the service. This leads to obtaining a new tool with the aid of generating function methods. The corresponding traditional bulk queueing system model is more convenient under an uncertain environment. The $alpha$-cut approach is applied with the conventional Zadeh's extension principle (ZEP) to transform the triangular membership functions (Mem. Fs) fuzzy queues into a family of conventional bulkqueues. This new model focus on mixed-integer non-linear programming (MINLP) tenders a mathematical computational approach is known as (0 -1) variables. To measures the efficiency of the method, the efficient solution strategy plays a crucial role in the adequate application of these techniques. Furthermore, different stages of the $alpha$-cut intervals were analyzed and the final part of the article gives a numerical solution of the proposed model to achieve practical issues.

Theoretical Design of Blue-Color Phosphorescent Complexes for Organic Light-Emitting Diodes: Emission Intensities and Nonradiative Transition Rate Constants in Ir(ppy)2(acac) Derivatives.

Theoretical calculations of phosphorescent spectra and nonradiative transition (NRT) rate constants for S1 ⇝ T1, T1 ⇝ S0, and S1 ⇝ S0 were carried out to determine the best candidate for a blue-color phosphorescent complex among several derivatives of bis(2-phenylpyridine)(acetylacetonate)iridium(III). The geometries of the ground state (S0), the lowest triplet state (T1), and the lowest excited singlet state (S1) were optimized at the levels of density functional theory, in which B3LYP functionals and SBKJC+p basis sets were used. The NRT rate constants were derived by using a generating function method within the displaced harmonic oscillator model. The results of the calculation for phosphorescence showed that the introduction of F and/or CN substituents at the 4'/6'-th and 5'-th sites in 2-phenylpyridinate (ppy) ligands, respectively, causes a blue shift of the emission spectra. They also suggest that Ir(5-CN,6-F-ppy)2(acac), denoted 3(56) in the text, is a good candidate for a blue-color phosphorescent complex because a blue shift of emission spectra and a moderate intensity are obtained for phosphorescence and, furthermore, this complex is calculated to have a large rate constant for S1 ⇝ T1 and relatively smaller rate constants for T1 ⇝ S0 and S1 ⇝ S0 based on the calculations of spin-orbit coupling and nonadiabatic coupling constants.

Symmetry-resolved entanglement for excited states and two entangling intervals in AdS3/CFT2

We test the proposal of [1] for the holographic computation of the charged moments and the resulting symmetry-resolved entanglement entropy in different excited states, as well as for two entangling intervals. Our holographic computations are performed in U(1) Chern-Simons-Einstein-Hilbert gravity, and are confirmed by independent results in a conformal field theory at large central charge. In particular, we consider two classes of excited states, corresponding to charged and uncharged conical defects in AdS3. In the conformal field theory, these states are generated by the insertion of charged and uncharged heavy operators. We employ the monodromy method to calculate the ensuing four-point function between the heavy operators and the twist fields. For the two-interval case, we derive our results on the AdS and the conformal field theory side, respectively, from the generating function method of [1], as well as the vertex operator algebra. In all cases considered, we find equipartition of entanglement between the different charge sectors. We also clarify an aspect of conformal field theories with a large central charge and hat{mathfrak{u}}{(1)}_k Kac-Moody symmetry used in our calculations, namely the factorization of the Hilbert space into a gravitational Virasoro sector with large central charge, and a hat{mathfrak{u}}{(1)}_k Kac-Moody sector.

Several characterizations of Bessel functions and their applications

Abstract The present work deals with the mathematical investigation of some generalizations of Bessel functions. The main motive of this paper is to show that the generating function can be employed efficiently to obtain certain results for special functions. The complex form of Bessel functions is introduced by means of the generating function. Certain enthralling properties for complex Bessel functions are investigated using the generating function method. By considering separately the real and the imaginary part of complex Bessel functions, we get respectively cosine-Bessel functions and sine-Bessel functions for which several novel identities and Jacobi–Anger expansions are established. Also, the generating function of degenerate Bessel functions is investigated and certain novel identities are obtained for them. A hybrid form of degenerate Bessel functions, namely, of degenerate Fubini–Bessel functions, is constructed using the replacement technique. Finally, the explicit forms of the real and the imaginary part of complex Bessel functions are established by a hypergeometric approach.

Statistical mechanics and thermodynamics for typical hyperbranched polymerization systems

<p indent="0mm">We summarize statistical thermodynamics of two typical polymerization systems to relate thermodynamic quantities to the properties of hyperbranched polymers (HBPs) such as average degree of polymerization (DP), radius of gyration, and branching density. An attempt is made to obtain the regulation mechanism of structure and properties of HBPs. Starting from the physical viewpoint, the isothermal compressibility has been associated with the weight average DP of polymers that is interpreted as the spatial correlation of monomers due to polymerization. The subchain is employed to derive the mean square radius of gyration and hydrodynamics radius, and a calculation is implemented in terms of the adjacent matrix of an HBP. Furthermore, the scaling behavior of the system is studied to give the generalized scaling law. The average properties of HBPs under the semi-batch mode are also presented by means of the generation function method. In addition, a Monte Carlo simulation method based on the growth generation is proposed, by which the spatial structure of an HBP can be easily analyzed in detail. As an application, the effect of cyclization, nonequal reactivity, and solvent quality on the polymerization is also carried out.

Reliability evaluation of a k-out-of-n(G)-subsystem based multi-state phased mission system with common bus performance sharing subjected to common cause failures

Multi-state phased mission systems (MS-PMSs) with common bus performance sharing widely exist in practice. Their subsystems are often designed as redundant k-out-of-n(G) systems to ensure the reliable operation of the whole system, such as power generating systems. However, common cause failures (CCFs) of components can decrease the reliability of such redundant subsystems. Considering the above two factors, we propose a new reliability model for the MS-PMS with k-out-of-n(G) subsystems and performance sharing subject to CCFs. Each subsystem contains multiple multi-state components, whose behaviors are analyzed using the Markov chain. Components in each subsystem may belong to a CCF group, and each CCF group suffers multiple independent CCF modes. We propose a Binary Indicator Universal Generating Function (BIUGF) method to evaluate the system reliability by adding an indicator variable and a pointer vector in the traditional universal generating function method, and the impacts of CCFs are taken into account using an implicit analysis method. The method is illustrated and validated through applications to power supply systems, which justify the effectiveness of the proposed method.

Coalescense with arbitrary-parameter kernels and monodisperse initial conditions: A study within combinatorial framework

For this work, we studied a finite system of discreet-size aggregating particles for two types of kernels with arbitrary parameters, a condensation (or branched-chain polymerization) kernel, $K(i,j)=(A+i)(A+j)$, and a linear combination of the constant and additive kernels, $K(i,j)=A+i+j$. They were solved under monodisperse initial conditions in the combinatorial approach where discreet time is counted as subsequent states of the system. A generating function method and Lagrange inversion were used for derivations. Expressions for an average number of clusters of a given size and its corresponding standard deviation were obtained and tested against numerical simulation. High precision of the theoretical predictions can be observed for a wide range of $A$ and coagulation stages, excepting post-gel phase in the case of the condensation kernel (a giant cluster presence is preserved). For appropriate $A$, these two kernels reproduced known results of the constant, additive and product kernels. Beside a previously solved linear-chain kernel, they extend the number of arbitrary-parameter kernels solved in the combinatorial approach.

Response Time Distribution in a Tandem Pair of Queues with Batch Processing

Response time density is obtained in a tandem pair of Markovian queues with both batch arrivals and batch departures. The method uses conditional forward and reversed node sojourn times and derives the Laplace transform of the response time probability density function in the case that batch sizes are finite. The result is derived by a generating function method that takes into account that the path is not overtake-free in the sense that the tagged task being tracked is affected by later arrivals at the second queue. A novel aspect of the method is that a vector of generating functions is solved for, rather than a single scalar-valued function, which requires investigation of the singularities of a certain matrix. A recurrence formula is derived to obtain arbitrary moments of response time by differentiation of the Laplace transform at the origin, and these can be computed rapidly by iteration. Numerical results for the first four moments of response time are displayed for some sample networks that have product-form solutions for their equilibrium queue length probabilities, along with the densities themselves by numerical inversion of the Laplace transform. Corresponding approximations are also obtained for (non-product-form) pairs of “raw” batch-queues—with no special arrivals—and validated against regenerative simulation, which indicates good accuracy. The methods are appropriate for modeling bursty internet and cloud traffic and a possible role in energy-saving is considered.

A Unified Theoretical Treatment on Statistical Properties of the Semi-batch Self-condensing Vinyl Polymerization System

We present a novel generating function (GF) method for the self-condensing vinyl polymerization (SCVP) system with any initial distribution of preexisted polymers. Such a method was proven to be especially useful to investigate the semi-batch SCVP system allowing a sequence of feeding operations during the polymerization. Consequently, the number-, weight-, and z-average molecular weights as well as dispersity index of hyperbranched polymers can be explicitly given, which are determined by predetermined feeding details and conversions in each polymerization step. These analytical results are further confirmed by the corresponding Monte Carlo simulations. Therefore, the present GF method has provided a unified treatment to the semi-batch SCVP system. Accordingly, hyperbranched polymers with desired properties can be prepared by designing feeding details and presetting conversions at each step based on the present GF method.

Age-dependent branching processes and applications to the Luria-Delbruck experiment

Microbial populations adapt to their environment by acquiring advantageous mutations, but in the early twentieth century, questions about how these organisms acquire mutations arose. The experiment of Salvador Luria and Max Delbr&amp;uuml;ck that won them a Nobel Prize in 1969 confirmed that mutations don't occur out of necessity, but instead can occur many generations before there is a selective advantage, and thus organisms follow Darwinian evolution instead of Lamarckian. Since then, new areas of research involving microbial evolution have spawned as a result of their experiment. Determining the mutation rate of a cell is one such area. Probability distributions that determine the number of mutants in a large population have been derived by Lea, Coulson, and Haldane. However, not much work has been done when time of cell division is dependent on the cell age, and even less so when cell division is asymmetric, which is the case in most microbial populations. Using probability generating function methods, we rigorously construct a probability distribution for the cell population size given a life-span distribution for both mother and daughter cells, and then determine its asymptotic growth rate. We use this to construct a probability distribution for the number of mutants in a large cell population, which can be used with likelihood methods to estimate the cell mutation rate.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/56/abstr.html

The friendship paradox in real and model networks

AbstractThe friendship paradox is the observation that the degrees of the neighbours of a node in any network will, on average, be greater than the degree of the node itself. In common parlance, your friends have more friends than you do. In this article, we develop the mathematical theory of the friendship paradox, both in general as well as for specific model networks, focusing not only on average behaviour but also on variation about the average and using generating function methods to calculate full distributions of quantities of interest. We compare the predictions of our theory with measurements on a large number of real-world network datasets and find remarkably good agreement. We also develop equivalent theory for the generalized friendship paradox, which compares characteristics of nodes other than degree to those of their neighbours.

Generation of hospital emergency department layouts based on generative adversarial networks

Based on generative adversarial networks (GANs), a type of deep learning model, this paper takes the functional layout of the emergency departments (EDs) of general hospitals as the research object, combines the hierarchical design concepts and proposes an intelligent functional layout generation method for EDs. It aims to explore the application of intelligent algorithms in architectural design and build an intelligent design method to solve the generation problem of ED layouts. The specific process of this method is as follows: First, 120 sets of ED drawings with excellent layouts are collected and labelled. Second, three of the most representative GAN frameworks including deep convolutional generative adversarial network (DCGAN), image-to-image translation with conditional adversarial network (pix2pix) and cycle-consistent adversarial network (CycleGAN) are chosen to establish training models for the ED layout. Finally, the rationality of the generated results is analysed from an architectural perspective, while the loss functions and trend for the generator and discriminator are compared from the algorithmic perspective. The analysis of the three GANs’ results shows that these models can autonomously generate new ED function layouts, of which the DCGAN results are the most flexible but the image quality is not ideal. The pix2pix outputs have the highest image quality, but the dataset has strict constraints. The CycleGAN has loose requirements on the dataset and yields ideal results with strong applicability. Some scholars have used pix2pix to explore the generation of apartment floor plans in recent years. However, this study established three different GAN models for hospitals for the first time, and compared their generation results to explore the applicability of different GANs. • Propose an intelligent functional layout generation method for emergency departments of general hospitals. • Three of the most representative generative adversarial networks including deep convolutional generative adversarial network, image-to-image translation with conditional adversarial network and cycle-consistent adversarial network are chosen to establish training models for the emergency departments layouts. • An intelligent hierarchical training approach to better generate emergency departments layouts through two intermediate layouts.

Two problems of binomial sums involving harmonic numbers

Two open problems recently proposed by Xi and Luo (Adv. Differ. Equ. 2021:38, 2021) are resolved by evaluating explicitly three binomial sums involving harmonic numbers, that are realized mainly by utilizing the generating function method and symmetric functions.

Automatic translation of spoken English based on improved machine learning algorithms

The inherent shortcomings of the automatic translation system for spoken English are its lower accuracy and more errors. In order to improve the efficiency of automatic translation of spoken English, this paper builds an automatic translation model of spoken English based on improved machine learning algorithms based on improved machine learning algorithms. Moreover, on the basis of summarizing the signal transformation methods, feature extraction, and detection function generation methods of the existing note starting point detection algorithms, this paper proposes a note starting point detection framework inspired by the speech knowledge base, and proposes the partial note fluctuation characteristics under this framework. In addition, this paper extracts the characteristics of partial tone fluctuations to detect the starting point of the note, which can improve the performance of the detection algorithm. In addition, after the model is constructed, this paper designs experiments to analyze the performance of the model constructed in this paper and counts the research results. The research results show that the translation accuracy and translation speed of the model constructed in this paper can meet actual needs.

Vibrational signatures for the identification of single-photon emitters in hexagonal boron nitride

Color centers in h-BN are among the brightest emission centers known yet the origins of these emission centers are not well understood. Here, using first-principles calculations in combination with the generating function method, we systematically elucidate the coupling of specific defects to the vibrational degrees of freedom. We show that the lineshape of many defects exhibits strong coupling to high frequency phonon modes and that C$_{\text{N}}$, C$_{\text{B}}$, C$_{\text{B}}$-C$_{\text{N}}$ dimer and V$_{\text{B}}$ can be associated with experimental lineshapes. Our detailed theoretical study serves as a guide to identify optically active defects in h-BN that can suit specific applications in photonic-based quantum technologies, such as single photon emitters, hybrid spin-photon interfaces, or spin-mechanics interfaces.

A novel power system operation simulation method considering multi-state models of coal-fired power unit: a case study of coal-fired power transformation in a certain region of China

Coal-fired power has played an important role in the power systems, but its operational characteristics, especially the flexibility characteristics is modelled relatively simple in the current operation simulations. Given the background of a rapid development of renewable energy and a sustainable transformation of power systems in China, it is essential to build a refined model of coal-fired power units that can be incorporated into an operation simulation method. First, this study provides an analysis on the flexibility characteristics of coal-fired power units, including the minimum stable output, ramp limit and minimum up/down time. Subsequently, a multi-state probabilistic model of coal-fired power units is proposed. This study also constructs a sequential operation simulation method based on the universal generation function (UGF) method that incorporates multiple states of coal-fired power units. Finally, this study provides a development and transformation pathway for coal-fired power plants in the North Hebei (NH) region and suggestions for promoting a sustainable development of the power system based on the proposed method and multi-state model.

Novel Binary-Addition Tree Algorithm for Reliability Evaluation of Acyclic Multistate Information Networks

Acyclic multistate information network (AMIN), a variant of multistate information network (MIN) that is not dependent on the conservation laws of flow, plays an important role in the current-day network communications because many modern network structures are based on AMIN, e.g., social networks, local area networks (LANs), and 4G/5G networks. An effective evaluation of the network reliability of an AMIN serves as a major and primary metric for determining the performance and quality of the overall network. The network reliability, which was shown to be NP-hard, has been successfully resolved by the universal generation function method (UGFM). However, the UGFM can only address small-scale problems owing to the limitations of the current-day computer memories. To overcome this problem, an improved and enhanced binary-addition tree algorithm (BAT) is proposed in this study to effectively evaluate and analyze the reliability of the AMIN. The performance of the proposed BAT is validated via a few examples.

First principles theoretical spectroscopy of methylene blue: Between limitations of time-dependent density functional theory approximations and its realistic description in the solvent.

Methylene blue [3,7-Bis(di-methylamino) phenothiazin-5-ium chloride] is a phenothiazine dye with applications as a sensitizer for photodynamic therapy, photoantimicrobials, and dye-sensitized solar cells. Time-dependent density functional theory (TDDFT), based on (semi)local and global hybrid exchange-correlation functionals, fails to correctly describe its spectral features due to known limitations for describing optical excitations of π-conjugated systems. Here, we use TDDFT with a non-empirical optimally tuned range-separated hybrid functional to explore the optical excitations of gas phase and solvated methylene blue. We compute solvated configurations using molecular dynamics and an iterative procedure to account for explicit solute polarization. We rationalize and validate that by extrapolating the optimized range separation parameter to an infinite amount of solvating molecules, the optical gap of methylene blue is well described. Moreover, this method allows us to resolve contributions from solvent-solute intermolecular interactions and dielectric screening. We validate our results by comparing them to first-principles calculations based on the GW+Bethe-Salpeter equation approach and experiment. Vibronic calculations using TDDFT and the generating function method account for the spectra's subbands and bring the computed transition energies to within 0.15 eV of the experimental data. This methodology is expected to perform equivalently well for describing solvated spectra of π-conjugated systems.

Analysis of infinite buffer general bulk service queue with state dependent balking

This paper investigates the effect of impatient phenomena of the arriving customers in a bulk service queue, where inputs are flowing into the system according to the Poisson process and are served in groups according to the 'general bulk service' (GBS) rule. The service time of a group of customers follows exponential distribution. On arrival, a customer decides whether to join or balk the system, based on the observation of the system size and status of the server, i.e., whether server is busy or idle. The steady state joint probability distribution of the number of customers in the queue as well as with the server is obtained by using the probability generating function method, which is based on the roots of the characteristic equations formed using probability generating function for steady state joint probabilities. Finally, various performance measures, such as, average queue length, average waiting time, probability that the server is busy, average queue length when server is busy, etc., have been obtained. The paper ends with several numerical discussions to demonstrate the effect of certain model parameters on the key performance measures.

Analysis of infinite buffer general bulk service queue with state dependent balking

This paper investigates the effect of impatient phenomena of the arriving customers in a bulk service queue, where inputs are flowing into the system according to the Poisson process and are served in groups according to the 'general bulk service' (GBS) rule. The service time of a group of customers follows exponential distribution. On arrival, a customer decides whether to join or balk the system, based on the observation of the system size and status of the server, i.e., whether server is busy or idle. The steady state joint probability distribution of the number of customers in the queue as well as with the server is obtained by using the probability generating function method, which is based on the roots of the characteristic equations formed using probability generating function for steady state joint probabilities. Finally, various performance measures, such as, average queue length, average waiting time, probability that the server is busy, average queue length when server is busy, etc., have been obtained. The paper ends with several numerical discussions to demonstrate the effect of certain model parameters on the key performance measures.