Generating Function Method Research Articles

Measurement of flow harmonics with multi-particle cumulants in Pb+Pb collisions at [Formula: see text]TeV with the ATLAS detector.

ATLAS measurements of the azimuthal anisotropy in lead–lead collisions at sqrt{s_{mathrm {NN}}}=2.76 TeV are shown using a dataset of approximately 7 upmu b^{-1} collected at the LHC in 2010. The measurements are performed for charged particles with transverse momenta 0.5<p_{mathrm {T}}<20 GeV and in the pseudorapidity range |eta |<2.5. The anisotropy is characterized by the Fourier coefficients, mathrm {v}_n, of the charged-particle azimuthal angle distribution for n = 2–4. The Fourier coefficients are evaluated using multi-particle cumulants calculated with the generating function method. Results on the transverse momentum, pseudorapidity and centrality dependence of the mathrm {v}_n coefficients are presented. The elliptic flow, mathrm {v}_2, is obtained from the two-, four-, six- and eight-particle cumulants while higher-order coefficients, mathrm {v}_3 and mathrm {v}_4, are determined with two- and four-particle cumulants. Flow harmonics mathrm {v}_n measured with four-particle cumulants are significantly reduced compared to the measurement involving two-particle cumulants. A comparison to mathrm {v}_n measurements obtained using different analysis methods and previously reported by the LHC experiments is also shown. Results of measurements of flow fluctuations evaluated with multi-particle cumulants are shown as a function of transverse momentum and the collision centrality. Models of the initial spatial geometry and its fluctuations fail to describe the flow fluctuations measurements.

Cattaneo-type subdiffusion-reaction equation.

Subdiffusion in a system in which mobile particles A can chemically react with static particles B according to the rule A+B→B is considered within a persistent random-walk model. This model, which assumes a correlation between successive steps of particles, provides hyperbolic Cattaneo normal diffusion or fractional subdiffusion equations. Starting with the difference equation, which describes a persistent random walk in a system with chemical reactions, using the generating function method and the continuous-time random-walk formalism, we will derive the Cattaneo-type subdiffusion differential equation with fractional time derivatives in which the chemical reactions mentioned above are taken into account. We will also find its solution over a long time limit. Based on the obtained results, we will find the Cattaneo-type subdiffusion-reaction equation in the case in which mobile particles of species A and B can chemically react according to a more complicated rule.

Effect of feed rate of monofunctional monomers on structure of hyperbranched copolymers formed by self-condensing vinyl copolymerization in semibatch reactor

Hyperbranched polymers (HBPs) formed by a self-condensing vinyl copolymerization, SCVCP, of monofunctional monomers M gradually fed into a reactor containing inimers AB*, which had been charged initially, were investigated using a kinetic model. Under the assumption that no cyclization occurred during polymerization, changes in the average degree of polymerization (DP), the degree-of-polymerization dispersity, the degree of branching (DB), and the number of structural units in the hyperbranched copolymers at different feed rates of monomers M were calculated by a generating function method. The distribution of the degree of polymerization could be narrowed with the addition of monomers M, and the degree-of-polymerization dispersity was increased slightly at a slower feed rate parameter of M, ϕ. A higher DB of the HBPs could be obtained at a slower feed rate and a specific stopping ratio of M, θ. For instance, a final DB of approximately 0.62 was attained at ϕ=0.2 and θ=1.2, which was higher than the maximum DB of 0.5 attained in a batch system with monomers M at θ=0.6.

Note on “discrete-like diffraction dynamics in free space”: highlighting the variety of solving procedures

A new family of paraxial optical beams has been recently introduced in the literature [Opt. Express21, 17951 (2013)]10.1364/OE.21.017951OPEXFF1094-4087, having the significant feature of exhibiting discrete-like diffraction patterns reminiscent of those observed in periodic evanescently coupled waveguide lattices. In this connection, we wish to highlight the symmetry properties of the paraxial wave equation, properties that, in our opinion, are still not exploited at their full potentiality, and the effectiveness of differential equation-solving procedures based on the generating-function method.

Mathematical Models of Mb/M/1 Bulk Arrival Queueing System

This paper deals with the study of bulk queueing model with the fixed batch size ‘b’ and customers arrive to the system with Poisson fashion with the rate λ and are severed exponentially with the rate μ. On formulating the mathematical model, we obtain the expressions for mean waiting time in the queue, mean time spent in the system, mean number of customers/work pieces in the queue and in the system by using generating function method. Some numerical illustrations are also obtained by using MATLAB-7 so as to show the applicability of the model under study.DOI: http://dx.doi.org/10.3126/jie.v10i1.10899Journal of the Institute of Engineering, Vol. 10, No. 1, 2014, pp. 184–191

New formulae of products of the Frobenius-Euler polynomials

In this paper, we perform a further investigation for the Frobenius-Euler polynomials. Some new formulae of products of the Frobenius-Euler polynomials are established by applying the generating function methods and some summation transform techniques. It turns out that some corresponding known results are obtained as special cases. MSC:11B68, 05A19.

Sums of products of Apostol-Bernoulli and Apostol-Euler polynomials

In this paper, a further investigation for the Apostol-Bernoulli and Apostol-Euler polynomials and numbers is performed. Some closed formulae of sums of products of any number of Apostol-Bernoulli and Apostol-Euler polynomials and numbers are established by applying the generating function methods and some summation transform techniques. It turns out that some well-known results are derived as special cases. MSC:11B68, 05A19.

Minimizing Bypass Transportation Expenses in Linear Multistate Consecutively-Connected Systems

Many continuous transportation systems can be represented as multi-state linear consecutively connected systems consisting of N+1 linearly ordered nodes. Some of these nodes contain statistically independent multistate elements with different characteristics. Each element j can provide a connection between the node to which it belongs and X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> next nodes, where X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> is a discrete random variable with known probability mass function. If the system contains nodes not connected with any previous node, then gaps exist that require bypass transportation solutions associated with considerable expenses. An algorithm based on the universal generating function method is suggested for evaluating the expected value of these expenses. A problem of finding the multi-state element allocation that minimizes the expected bypass transportation expenses is formulated and solved. Illustrative examples are presented.

The critical load of scale-free fault-tolerant topology in wireless sensor networks for cascading failures

Considering the cascading failures of scale-free fault-tolerant topology in wireless sensor networks (WSNs), based on the power function of load and the fixed capacity, a new load redistribution model is proposed under a single random node failure. Adopting the probability generating function method, the relationship between the load and the largest connected component after cascading failures on scale-free fault-tolerant topology is studied. Then the critical load triggering large-scale cascading failure is obtained. This discovery reveals the cascading failure principium of scale-free fault-tolerant topology in WSNs. It may have practical implications for controlling cascading failure disasters of scale-free fault-tolerant topology in the single random node failure.

Generating Function Methods for Coefficient-Varying Generalized Hamiltonian Systems

AbstractThe generating function methods have been applied successfully to generalized Hamiltonian systems with constant or invertible Poisson-structure matrices. In this paper, we extend these results and present the generating function methods preserving the Poisson structures for generalized Hamiltonian systems with general variable Poisson-structure matrices. In particular, some obtained Poisson schemes are applied efficiently to some dynamical systems which can be written into generalized Hamiltonian systems (such as generalized Lotka-Volterra systems, Robbins equations and so on).

Comparison of Mamdani and Sugeno Fuzzy Inference Systems for Prediction (With Application to Prices of Fund in Egypt)

This paper outlines the basic difference between the Mamdani/Sugeno Fuzzy inference systems (FIS) and the actual values. The main motivation behind this research is to assess which approach provides the best performance for predicting prices of Fund. Due to the importance of performance in Economy, the Mamdani and Sugeno models are compared using four types of membership function (MF) generation methods: the Triangular, Trapezoidal, Gaussian and Gbell. Fuzzy inference systems (Mamdani and Sugeno fuzzy mo dels) can be used to predict the weekly prices of Fund for the Egyptian Market. The application results indicate that Sugeno model is better than that of Mamdani. The results of the two fuzzy inference systems (FIS) are compared.

Spectral probabilities of top-down tandem mass spectra

BackgroundIn mass spectrometry-based proteomics, the statistical significance of a peptide-spectrum or protein-spectrum match is an important indicator of the correctness of the peptide or protein identification. In bottom-up mass spectrometry, probabilistic models, such as the generating function method, have been successfully applied to compute the statistical significance of peptide-spectrum matches for short peptides containing no post-translational modifications. As top-down mass spectrometry, which often identifies intact proteins with post-translational modifications, becomes available in many laboratories, the estimation of statistical significance of top-down protein identification results has come into great demand.ResultsIn this paper, we study an extended generating function method for accurately computing the statistical significance of protein-spectrum matches with post-translational modifications. Experiments show that the extended generating function method achieves high accuracy in computing spectral probabilities and false discovery rates.ConclusionsThe extended generating function method is a non-trivial extension of the generating function method for bottom-up mass spectrometry. It can be used to choose the correct protein-spectrum match from several candidate protein-spectrum matches for a spectrum, as well as separate correct protein-spectrum matches from incorrect ones identified from a large number of tandem mass spectra.

Analysis of an M/M/cQueueing System with Impatient Customers and Synchronous Vacations

We consider an M/M/cqueueing system with impatient customers and a synchronous vacation policy, where customer impatience is due to the servers’ vacation. Whenever a system becomes empty, all the servers take a vacation. If the system is still empty, when the vacation ends, all the servers take another vacation; otherwise, they return to serve the queue. We develop the balance equations for the steady-state probabilities and solve the equations by using the probability generating function method. We obtain explicit expressions of some important performance measures by means of the two indexes. Based on these, we obtain some results about limiting behavior for some performance measures. We derive closed-form expressions of some important performance measures for two special cases. Finally, some numerical results are also presented.

High-Order Perturbation Theory for Incident Flux Response Expansion Methods

In this work, a high-order perturbation method is developed to generate/compute response functions for the coarse mesh transport (COMET) method, which provides whole-core neutron transport solutions to various reactor core types. In this approach, the response functions are first generated at a reference eigenvalue point, and the response functions at an arbitrary eigenvalue are computed as a high-order perturbation from the reference point. The method has been tested at both the lattice level (response function generation) and the core level (whole-core transport calculations). At the lattice level, it is found that the response functions predicted by the perturbation method agree very well with those directly computed by the Monte Carlo method. The average relative difference in the surface-to-surface response functions is 0.29% to 0.46% when the eigenvalue k ranges from 0.6667 to 1.5. In whole-core transport calculations, the COMET calculations using the high-order perturbation method are almost identical to those using the interpolation method. The eigenvalue, assembly, and pin fission densities predicted by COMET agree very well with the MCNP reference solution. This indicates that the high-order perturbation response function generation method can achieve the same accuracy as the interpolation method while significantly improving the computational efficiency of the precomputation phase.

Photon counting statistics of a single molecule under pump–probe fields

We consider the photon statistical properties emitted from a two-level quantum system under pump–probe driving. The line shapes and Mandel's Q parameters are studied for a single quantum system based on the generating function method developed recently. The control of photon emission from a single quantum system driven simultaneously by a driven field and a tunable probe field is investigated. Our theoretical results and the experimental results are in good agreement.

Hierarchical scale-free network is fragile against random failure

We investigate site percolation in a hierarchical scale-free network known as the Dorogovtsev-Goltsev-Mendes network. We use the generating function method to show that the percolation threshold is 1, i.e., the system is not in the percolating phase when the occupation probability is less than 1. The present result is contrasted to bond percolation in the same network of which the percolation threshold is zero. We also show that the percolation threshold of intentional attacks is 1. Our results suggest that this hierarchical scale-free network is very fragile against both random failure and intentional attacks. Such a structural defect is common in many hierarchical network models.

Statistical theory for self-condensing vinyl polymerization system with any initial distribution

A generation function method is used to obtain some average polymer physical quantities in the self-condensing vinyl polymerization (SCVP) system with any initial distribution. For the system under study, the generation function is explicitly derived in terms of the corresponding differential kinetic equations. As an application, two SCVP systems with uniform and dispersed prepolymers were discussed, respectively. As a result, the number-, weight- and Z-average degrees of polymerization and polydispersity index were presented. It is shown that, on the one hand, the initial distribution plays an important role in these physical quantities, and on the other hand, the generation function method is found to be very convenient in the calculations of average polymer quantities. It is expected that the present method can be applied in more complex SCVP systems with such as semi-batch process, multi-components and unequal reactivity.

Entropy Production along Dominant Pathway of Nonequilibrium Phase Transition in Mesoscopic Chemical System

We consider a bistable mesoscopic chemical reaction system and calculate entropy production along the dominant pathway during nonequilibrium phase transition. Using probability generating function method and eikonal approximation, we first convert the chemical master equation into the classical Hamilton-Jacobi equation, and then find the dominant pathways between two steady states in the phase space by calculating zero-energy trajectories. We find that entropy productions are related to the actions of the forward and backward dominant pathways. At the coexistence point where the stabilities of the two steady states are equivalent, both the system entropy change and the medium entropy change are zero; whereas at non-coexistence point both of them are nonzero.

A Novel Cut-Based Universal Generating Function Method

A multistate information network (MIN) is a generalization of the tree-structured multistate-state system that does not satisfy the flow conservation law. The current known universal generating function methods (UGFMs, here called pUGFMs) used to evaluate acyclic MIN (AMIN) reliability are based on the connectedness between node 1 and the targets, i.e. find all possible paths between node 1 and the targets. A novel cut-based UGFM (cUGFM) is proposed for the AMIN reliability problem. The proposed cUGFM is based on the disconnectedness between node 1 and the targets, i.e. find all possible cuts between node 1 and the targets. It provides a flexible, novel method to calculate reliability and unreliability. The computational complexity of the proposed algorithm is also analyzed and compared with the best-known existing methods. Finally, three benchmark examples are given to illustrate how the exact one-to-all-target-subset AMIN reliability is calculated using the proposed cUGFM.

New recurrence formulae for the Apostol-Bernoulli and Apostol-Euler polynomials

The main purpose of this paper is by using the generating function methods and some combinatorial techniques to establish some new recurrence formulae for the Apostol-Bernoulli and Apostol-Euler polynomials. It turns out that some known results in (He and Wang in Adv. Differ. Equ. 2012:209, 2012) are obtained as special cases. MSC:11B68, 05A19.