Factorial Moments Research Articles

Scaling and correlations in foreign exchange market

We observe that the distribution of the relative return, describing the variation of a certain currency, of 74 global currencies obeys a power-law. By using the random matrix theory we find that the distribution of eigenvalues of correlation matrix of relative return also follows a power-law. Using a scaled factorial moment we investigate the distribution of correlation coefficients of the relative return and observe intermittence phenomenon. Furthermore, we define the influence strength for a certain currency, which reflects the influence of its price change to the community interested. By doing that, we find that the distribution of influence strength is again a power-law. Beyond that, we compare the influence strength of Chinese Yuan (RMB) to those of other seven important currencies, which may have some interesting indications.

Moment analysis and Zipf law

The moment analysis method and nuclear Zipf's law of fragment size distributions are reviewed to study nuclear disassembly. In this report, we present a compilation of both theoretical and experimental studies on moment analysis and Zipf law performed so far. The relationship of both methods to a possible critical behavior or phase transition of nuclear disassembly is discussed. In addition, scaled factorial moments and intermittency are reviewed.

On the degree distribution of the nodes in increasing trees

Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i.e., labellings of the nodes by distinct integers of the set { 1 , … , n } in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular interest: recursive trees, plane-oriented recursive trees and binary increasing trees. We study the quantity degree of node j in a random tree of size n and give closed formulae for the probability distribution and all factorial moments for those subclass of tree families, which can be constructed via a tree evolution process. Furthermore limiting distribution results of this parameter are given, which completely characterize the phase change behavior depending on the growth of j compared to n.

A comparative study of multifractal moments in relativistic heavy-ion collisions

A study of the fractal structure of relativistic shower particles produced in the interactions of 28Si and 12C nuclei at 4.5 A GeV/c with nuclear emulsion using the methods of Takagi moments, Tq, factorial moments, Fq, and modified multifractal moments, Gq, has been performed. The dependences of these moments on the number of bins M are found to follow power law behaviour for the experimental data in pseudorapidity phase space. The calculated values of the generalized fractal dimensions, Dq, are found to decrease with the increasing order of moments, q. This observation indicates the existence of multifractality and a self-similar cascade mechanism in multiparticle production. The multifractal Bernoulli representation Dq = D∞ + c ln q/(q − 1) is found to support the linear dependence of Dq on ln q/(q − 1). Finally, from the analysis of the data it can be concluded that no universality in the values of the multifractal specific heat, c, calculated from Tq, Fq and Gq moments is found. Some consistency seems to be observed in the values of the specific heat obtained from the Takagi method.

Excitation energy dependence of pion fluctuation: an in-depth study

This paper investigates the so-called excitation energy dependence (i.e. classification of fluctuation pattern of pions on the basis of number of grey tracks) at different projectile energies. The different data sets thus obtained correspond to different degrees of target excitation. The fluctuation pattern of pions is studied in two dimensions considering anisotropy of phase space. The parameter characterizing the anisotropy of phase is the Hurst exponent. The analysis is performed using a very rigorous method by fitting one-dimensional second-order factorial moment saturation curves. The study reveals that the effective fluctuation strength depends on target excitation. It is also observed that the target excitation dependence of fluctuation of pions is different at different projectile energies. Recent investigations of the RHIC data have revealed the fact that the behaviour of multiplicity fluctuations with collision centrality is non-monotonic. Our investigation also reveals the same fact.

Critical Opalescence in Baryonic QCD Matter

We show that critical opalescence, a clear signature of second-order phase transition in conventional matter, manifests itself as critical intermittency in QCD matter produced in experiments with nuclei. This behavior is revealed in transverse momentum spectra as a pattern of power laws in factorial moments, to all orders, associated with baryon production. This phenomenon together with a similar effect in the isoscalar sector of pions (sigma mode) provide us with a set of observables associated with the search for the QCD critical point in experiments with nuclei at high energies.

The number distribution of neutrons and gamma photons generated in a multiplying sample

The subject of this paper is an analytical derivation of the full probability distribution of the number of neutrons and photons generated in a sample with internal multiplication by one source emission event (spontaneous fission), and its comparison with Monte Carlo calculations. We derive recursive analytic expressions for the probability distributions P ( n ) up to values of n = N for which the cumulative probability ∑ n = 0 N P ( n ) is equal to or larger than 0.99. The derivation is achieved using the symbolic computation language Mathematica. With the introduction of a modified factorial moment of the number of neutrons and gamma photons generated in fission, the resulting expressions could be brought to a formally equivalent form with those for the factorial moments of the total number of neutrons and photons generated in the sample. The results were compared to Monte Carlo calculations, and excellent agreement was found between the analytical results and the simulations. The results show that the probability distributions change with increasing sample mass in such a way that the “bulk” of the distribution changes only slightly, but a tail develops for higher n values. This tail is the main reason for the increase of the factorial moments with increasing sample mass, an effect that was observed in earlier studies.

Neutron fluctuations in a multiplying medium randomly varying in time

The master equation approach, which has traditionally been used for the calculation of neutron fluctuations in multiplying systems with constant parameters, is extended to a case when the parameters of the system change randomly in time. A forward type master equation is considered for the case of a multiplying system whose properties jump randomly between two discrete states, both with and without a stationary external source. The first two factorial moments are calculated, including the covariance. This model can be considered as the unification of stochastic methods that were used either in a constant multiplying medium via the master equation technique, or in a fluctuating medium via the Langevin technique. The results obtained show a much richer characteristic of the zero power noise than that in constant systems. The results are relevant in medium power subcritical nuclear systems where the zero power noise is still significant, but they also have a bearing on all types of branching processes, such as evolution of biological systems, spreading of epidemics etc, which are set in a time-varying environment.

Study of Non-Statistical Fluctuations in Relativistic Nuclear Collisions

An attempt is made to study the existence of non-statistical fluctuations of relativistic particles using the method of scaled factorial moments (SFMs) in the interactions of 28 Si and 12 C nuclei ...

Statistical properties of multiphoton time-dependent three-boson coupled oscillators

We investigate the quantum statistics of three time-dependent coupled oscillators in the presence of multiphoton processes. The system is connected with the two-atom multiphoton Tavis-Cummings model. The solution of the Heisenberg equations of the motion is obtained in a compact form. We assume that the modes are initially prepared in coherent states, and we discuss nonclassical phenomena (squeezing and sub-Poissonian behavior). Further, we examine the joint quasi-distribution functions as well as photon-number distribution and its factorial moments. The system has shown that the nonclassical effect is apparent in compound modes (1,3) and (2,3). Moreover, the superstructure phenomenon is observed when the photon transition is increased.

Pronounced Fluctuations of Target Fragments in ForwardHemisphere Only in Ultra Relativistic Nuclear Collision

Dynamical fluctuation of target evaporated black particles isinvestigated in both forward and backward hemispheres within theframework of multi-dimensional factorial moment methodology using thebrilliant concept of the Hurst exponent. We analyse the black particlesemitted in 32S–AgBr interactions at 200 AGeV and it isevident that the dynamical fluctuation in the backward hemisphere isself-affine. In the forward hemisphere, dynamical fluctuation isself-similar but not self-affine. However, study indicates that thefluctuation in the forward hemisphere is more pronounced than that inbackward hemisphere.

On a unified mixture distribution

In this article, a new mixture distribution associated with Fox–Wright generalized hypergeometric function has been studied, which generalizes many mixture distributions investigated earlier by many authors. Some basic functions associated with the probability density function of the mixture distribution, such as kth moments, characteristic function and factorial moments are derived. Some special cases are also pointed out.

QCD coherence and correlations of particles with restricted momenta in hadronic Z decays

QCD coherence effects are studied based on measurements of correlations of particles with either restricted transverse momenta, pT<pTcut, where pT is defined with respect to the thrust axis, or restricted absolute momenta, p equiv |p| < pcut, using about four million hadronic Z decays recorded at LEP with the OPAL detector. The correlations are analyzed in terms of normalized factorial and cumulant moments. The analysis is inspired by analytical QCD calculations which, in conjunction with Local Parton-Hadron Duality (LPHD), predict that, due to colour coherence, the multiplicity distribution of particles with restricted transverse momenta should become Poissonian as pTcut decreases. The expected correlation pattern is indeed observed down to pTcut approx 1GeV but not at lower transverse momenta. Furthermore, for pcut to 0 GeV a strong rise is observed in the data, in disagreement with theoretical expectation. The Monte Carlo models reproduce well the measurements at large pTcut and pcut but underestimate their magnitudes at the lowest momenta. The e+e- data are also compared to the measurements in deep-inelastic e+p collisions. Our study indicates difficulties with the LPHD hypothesis when applied to many-particle inclusive observables of soft hadrons.

Evidence of Strong Pion Fluctuation in Jet-Like Events in32S–AgBr Interactions

We presents the analysis of ring-like and jet-like events in terms ofScaled Factorial Moments (SFMs) in one-dimensional space up to thesixth order of moments for 32S–AgBr interactions at 200 AGeVenergy. The study reveals a power law growth of SFMs withdecreasing bin width of the azimuthal distribution in the32S–AgBr data at 200 AGeV in both the cases. It is seen thatstrong intermittent pattern is revealed only in jet-like events.

Space and energy dependent theoretical formula for the third order neutron correlation technique

We have studied measurement techniques of subcriticality by using the neutron correlation techniques. Among various techniques, we have paid attention to the third order neutron correlation technique, which was proposed by Furuhashi originally and utilizes the third order fluctuations of neutron counts. By using this technique, we can obtain the absolute value of subcriticality with the aid of factorial moments of fission neutron ν and external source neutron q, without prior knowledge of the prompt neutron decay constant at a reference state. However, in order to apply the third order neutron correlation technique to actual experiments, we must consider the spatial and neutron energy effects in this technique. For this purpose, we derived more generalized theoretical formulas of the third order neutron correlation technique that took account of the spatial and neutron energy effects by virtue of the α-eigenfunction expansion. Making use of the newly derived formulas, we propose the practical formulas and the experimental procedures based on the fundamental mode approximation.

Descendants in Increasing Trees

Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size $n$ all its increasing labellings, i.$\,$e. labellings of the nodes by distinct integers of the set $\{1, \dots, n\}$ in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular interest: recursive trees, plane-oriented recursive trees and binary increasing trees. We study the quantity number of descendants of node $j$ in a random tree of size $n$ and give closed formulæ for the probability distribution and all factorial moments for those subclass of tree families, which can be constructed via an insertion process. Furthermore limiting distribution results of this parameter are given.

Label-based parameters in increasing trees

Grown simple families of increasing trees are a subclass of increasing trees, which can be constructed by an insertion process. Three such tree families contained in the grown simple families of increasing trees are of particular interest: $\textit{recursive trees}$, $\textit{plane-oriented recursive trees}$ and $\textit{binary increasing trees}$. Here we present a general approach for the analysis of a number of label-based parameters in a random grown simple increasing tree of size $n$ as, e.g., $\textit{the degree of the node labeled j}$, $\textit{the subtree-size of the node labeled j}$, etc. Further we apply the approach to the random variable $X_{n,j,a}$, which counts the number of size-$a$ branches attached to the node labeled $j$ (= subtrees of size $a$ rooted at the children of the node labeled $j$) in a random grown simple increasing tree of size $n$. We can give closed formulæ for the probability distribution and the factorial moments. Furthermore limiting distribution results for $X_{n,j,a}$ are given dependent on the growth behavior of $j=j(n)$ compared to $n$.

Level crossings and other level functionals of stationary Gaussian processes

This paper presents a synthesis on the mathematical work done on level crossings of stationary Gaussian processes, with some extensions. The main results [(factorial) moments, representation into the Wiener Chaos, asymptotic results, rate of convergence, local time and number of crossings] are described, as well as the different approaches [normal comparison method, Rice method, Stein-Chen method, a general m-dependent method] used to obtain them; these methods are also very useful in the general context of Gaussian fields. Finally some extensions [time occupation functionals, number of maxima in an interval, process indexed by a bidimensional set] are proposed, illustrating the generality of the methods. A large inventory of papers and books on the subject ends the survey.

EVIDENCE OF FRACTAL BEHAVIOR OF PIONS AND PROTONS IN HIGH ENERGY INTERACTIONS — AN EXPERIMENTAL INVESTIGATION

Different fractal methods (Factorial moment, Takagi moment) have been used for the analysis of fractal behavior of multiparticle production in π-- AgBr interaction at 350 GeV/c of compound multiplicity distribution in emission angle (cos θ) space and azimuthal angle (ϕ) space. The generalized fractal dimensions Dq are determined according to two methods. Results indicate multifractal behavior in compound multiplicity distribution. From the knowledge of Dq, the multifractal specific heat is calculated. Results are compared with the corresponding results of shower multiplicity.

Simple derivations of properties of counting processes associated with Markov renewal processes

A simple, widely applicable method is described for determining factorial moments of N̂ t , the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂ t , and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.