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This study provides the first evidence of market efficiency of drug indices, especially cannabis and tobacco, which are known in finance as sin markets. The multifractal detrended fluctuation analysis (MFDFA) is employed on the daily data of six cannabis and one tobacco indices in order to measure efficiency by quantifying the intensity of self-similarity. The findings confirm multifractality in all sample series. Interestingly, Dow Jones Tobacco (DJUSTB) Index shows the highest multifractality, demonstrating the lowest efficiency, whereas S&P/TSX Cannabis (SPTXCAN) Index is the most efficient of all the time series under analysis, with the lowest multifractality levels. Only the North American Marijuana (NAMMAR), Cannabis World Index Gross Total Return (CANWLDGR) and DJUSTB show persistent behavior. These findings could be of interest to policymakers and regulators to establish new reforms to improve the efficiency of these markets, as well as for actual and potential investors.

We assume the market price to diffuse in a hierarchical comb of barriers, the heights of which represent the importance of new information entering the market. We find fat tails with the desired exponent for the price change distribution, and effective multifractality for intermediate times.

It has become popular recently to apply the multifractal formalism of statistical physics (scaling analysis of structure functions and f(α) singularity spectrum analysis) to financial data. The outcome of such studies is a nonlinear shape of the structure function and a nontrivial behavior of the spectrum. Eventually, this literature has moved from basic data analysis to estimation of particular variants of multifractal models for asset returns via fitting of the empirical τ(q) and f(α) functions. Here, we reinvestigate earlier claims of multifractality using four long time series of important financial markets. Taking the recently proposed multifractal models of asset returns as our starting point, we show that the typical "scaling estimators" used in the physics literature are unable to distinguish between spurious and "true" multiscaling of financial data. Designing explicit tests for multiscaling, we can in no case reject the null hypothesis that the apparent curvature of both the scaling function and the Hölder spectrum are spuriously generated by the particular fat-tailed distribution of financial data. Given the well-known overwhelming evidence in favor of different degrees of long-term dependence in the powers of returns, we interpret this inability to reject the null hypothesis of multiscaling as a lack of discriminatory power of the standard approach rather than as a true rejection of multiscaling. However, the complete "failure" of the multifractal apparatus in this setting also raises the question whether results in other areas (like geophysics) suffer from similar shortcomings of the traditional methodology.

We introduce kernel smoothing method to extract the global trend of a time series and remove short time scales variations and fluctuations from it. A multifractal detrended fluctuation analysis (MF-DFA) shows that the multifractality nature of TEPIX returns time series is due to both fatness of the probability density function of returns and long range correlations between them. MF-DFA results help us to understand how genetic algorithm and kernel smoothing methods act. Then we utilize a recently developed genetic algorithm for carrying out successful forecasts of the trend in financial time series and deriving a functional form of Tehran price index (TEPIX) that best approximates the time variability of it. The final model is mainly dominated by a linear relationship with the most recent past value, while contributions from nonlinear terms to the total forecasting performance are rather small.

The scaling properties of the avalanches' interevent time series in a modified Olami–Feder–Christensen (OFC) earthquake model on assortative scale-free networks have been investigated. We analyze the interevent time of avalanches with size larger than or equal to S_th by using rescaled range (R/S) analysis and detrended fluctuation analysis (DFA) methods. Our results demonstrate positive correlation and persistent feature of the interevent time series. The multifractal detrended fluctuation analysis (MF-DFA) has evidenced differences among the avalanches' interevents interval with different S_th. We determine generalized Hurst exponent and singularity spectrum and find that the interevent time series have multifractal nature.

We investigate multifractal properties for an Abelian directed model of self-organized criticality that describes the growth of droplets inside a cloud and the subsequent rainfall. The probability distribution of events of the model satisfies finite-size scaling. We obtain the singularity spectra f(α) associated with temporal records for avalanche size and for the potential energy, defined by the total sum of the product between mass and height of each site. The measure defined by avalanche size has a clear cut multifractal character, while the obtained f(α) for potential energy may include a spurious branch.

A Weierstrass–Mandelbrot function (WMF) model with Morlet wavelets is investigated. Its control relationships are derived quantitatively after proving the convergence of the controlled WMF model. Based on these relationships, it is shown that the scope of the WMF series increases with three parameters of the Morlet wavelets. But other parameters have opposite effect on the scope of the series. The results of simulated examples demonstrate the effectiveness of the control method. Moreover, two statistical characteristics of the series are obtained as the parameters change: One is multifractality of the series of the controlled WMF model, and the other is the Hurst exponent whose value stands for the long-time memory effect on the series.

Control system is an inevitable element of any industrial installation. Its quality affects overall process performance significantly. The assessment, whether control system needs any improvement or not, requires relevant and constructive measures. There are various methods, like time domain based, Minimum Variance, Gaussian and non-Gaussian statistical factors, fractal and entropy indexes. Majority of approaches use time series of control variables. They are able to cover many phenomena. But process complexities and human interventions cause effects that are hardly visible for standard measures. It is shown that the signals originating from industrial installations have multifractal properties and such an analysis may extend standard approach to further observations. The work is based on industrial and simulation data. The analysis delivers additional insight into the properties of control system and the process. It helps to discover internal dependencies and human factors, which are hardly detectable.

This paper presents evidence of intermittency in target fragmentation in ²⁸Si-AgBr interactions at 14.5 AGeV in emission angle space and azimuthal angle space separately and also in two-dimensional (cos θ,ϕ) space. From the intermittency exponent, the anomalous fractal dimension d_q is calculated and the variation of d_q with the order q is investigated. The authors also compare the results with those obtained from ¹⁶O-AgBr interactions at 60 AGeV and at 2.1 AGeV and ³²S-AgBr interactions at 200 AGeV to study the energy dependence of intermittent type of fluctuations in emission angle and azimuthal angle space separately. Comparing the intermittency exponents in one- and two-dimensions it is further seen that intermittency exponent has a higher value in two-dimensional space.

Modified G_q-moments have been employed to study multifractal behavior of multiplicity fluctuations in ²⁸Si-AgBr collisions at 14.6 AGeV. The connection between the G_q-moments and the normalized factorial moments F_q has also been investigated. Power law dependences of the moments on the number of bins M have been observed in the data. The generalized dimensions D_q and the multifractal specific heat for our data have also been determined.

We study the intermittent and fractal behavior of the emission spectrum of target-associated fast particles from ⁸⁴Kr-AgBr interactions at 0.95 GeV/A. We used χ(cos θ) as the phase space variable. Normalized or scaled factorial moments studies for various orders of moment q = 2–5, confirmed the intermittent behavior. G_q moments have a fractal interpretation in their fluctuation in phase space. Our studies reveal the multi-fractal behavior of the emission spectrum.

This paper presents an analysis of medium energy target-associated protons emitted in ²⁸Si-AgBr collisions at 14.6 AGeV. Evidence of non-statistical fluctuations and multifractality in the data has been found. Values of the multifractal specific heat for our data have also been determined.

In this paper the study of existance of non-statistical fluctuations using the method of scaled factorial, F_q, moments (SFMs) has been utilized to probe the distribution of relativistic shower particles produced in ²⁸Si-emulsion collisions at 14.6 A GeV. The analysis indicates that the anomalous fractal dimension, d_q, does not favour the origin of any exotic phenomenon. Finally, our data do not show a clear evidence for the existence of non-thermal phase transitions.

Analysis of the experimental data on 14.5A GeV/c ²⁸Si-nucleus interactions yields a linear relationship between ln〈n^q〉 and ln〈n〉, for q = 2, 3, 4 and 5 and also between and ln〈n〉, where n is the multiplicity in a given psedorapiodity bin. Generalized dimensions, D_q are also determined from the slopes of the linear variations of ln〈n^q〉 with ln〈n〉. These results are supported reasonably well by the predictions of the Lund Model of nuclear collisions, FRITIOF.

Nonstatistical fluctuations in the density distribution of shower tracks coming out of ³²S–Ag/Br interaction at an incident momentum of 200A GeV/c, have been characterized by using the intermittency and multifractality techniques. A sample of ³²S–Ag/Br nuclear emulsion events has been partitioned into three subsamples with respect to the shower track multiplicity in such a way that one can roughly estimate the average centrality for each subsample. Thereby, the centrality dependence of a set of relevant intermittency and multifractal issues has been investigated.

An attempt is made to study multifractal specific heat from F_q, modified G_q and Takagi moments, which can provide some interesting information about event-by-event fluctuations in heavy-ion collisions at high energies. It is an established fact that constancy of the specific heat (CSH) of solids and gases helps explain thermodynamical behavior of intermittency and multifractality in A–A collisions at different projectile energies. It is worthmentioning that the present paper deals with multifractal specific heat, c, extracted from F_q, modified G_q and Takagi moments for the experimental as well as FRITIOF generated data on 14.5A GeV/c²⁸Si-nucleus interactions. The results obtained using the three approaches are compared for both the data sets. The observed power law behavior reflects self-similar property of the multiparticle production process. The common feature of the multifractal specific heat obtained using the three approaches is that there is no systematic variation in its value for different orders of the moments. Furthermore, the value of c is found to be different for both the data sets and no strong target mass dependence is observed in any of the cases. A noteworthy observation is that a non-zero finite value of the multifractal specific heat may be considered a good signal for the presence of multifractality in the relativistic charged particle multiplicity distribution.

Entropy, dimensions and other multifractal characteristics of multiplicity distributions of relativistic charged hadrons produced in ion–ion collisions at SPS energies are investigated. The analysis of the experimental data is carried out in terms of phase space bin-size dependence of multiplicity distributions following the Takagi’s approach. Yet another method is also followed to study the multifractality which, is not related to the bin-width and (or) the detector resolution, rather involves multiplicity distribution of charged particles in full phase space in terms of information entropy and its generalization, Rényi’s order-$q$ information entropy. The findings reveal the presence of multifractal structure — a remarkable property of the fluctuations. Nearly constant values of multifractal specific heat “$c$” estimated by the two different methods of analysis followed indicate that the parameter “$c$” may be used as a universal characteristic of the particle production in high energy collisions. The results obtained from the analysis of the experimental data agree well with the predictions of Monte Carlo model AMPT.

Visibility graphs (VGs), horizontal visibility graphs (HVGs) and the sandbox algorithm (SB) are applied for multifractal characterization of complex network systems that are converted from time series measurements, are used to characterize the fluctuations in pseudorapidity densities of singly charged particles produced in ${}^{16}$O–AgBr interactions at 60AGeV. The work presents the analysis of ring-like and jet-like events in terms of multifractality characterization of ${}^{16}$O–AgBr interactions at 60AGeV. We systematically compared the experimental events of both ring- and jet-like events with Monte Carlo (MC) simulated events. The investigation reveals that the multifractal spectrum for the jet-like events and ring-like events is different and can be distinguished. The ring-like events have its parameters slightly higher than that of jet-like events. Further analysis shows that the strength of the nonstatistical fluctuations is larger for ring-like events than those of jet-like events. The SB method presented here appears to be more useful than the conventional methods used for multifractal characterization.

We have introduced the complex network analysis techniques such as Visibility Graph (VG), Horizontal Visibility Graph (HVG) and Sandbox (SB) algorithm as a mapping between complex network and time series measurements. This study explores the application of complex network-based analysis techniques to proton–proton $(pp)$ collisions data at $\sqrt{s}=13$, 7, 5.02 and 2.76TeV to unravel the underlying dynamics of the particles and their interactions. To ensure the robustness and comprehensive analysis, we have incorporated UrQMD-, PYTHIA- and AMPT-simulated data sets of $pp$ interactions at LHC energies that enable a thorough investigation of the underlying phenomena of the interactions. We have analyzed the fluctuations in pseudorapidity densities of singly charged particles created in $pp$ interactions at LHC energies. We have also investigated some properties like symmetry scaling, correlation, clustering and scale-freeness among the produced particles. The findings from this study contribute to our understanding of the complex dynamics within $pp$ collisions and have implications for the field of high-energy physics and particle phenomenology.

We measure the fractal dimension of an African plant that is widely cultivated as an ornamental – the Asparagus plumosus. This plant presents self-similarity, remarkable in at least two different scalings. In the following, we present the results obtained by analyzing this plant via the box counting method for three different scalings. We show in a quantitative way that this species is a fractal.