Q-exponential Function Research Articles

Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the q-exponential function

By making use of the concept of basic (or q-) calculus, a subclass S⁎(L,q) of q-starlike functions, which is associated with the q-exponential function, is introduced here in the open unit disk U given byU={z:z∈Cand|z|<1}. The main object of this article is to determine the upper bound of the third-order Hankel determinant H3(1) for functions belonging to the q-starlike function class S⁎(L,q). For validity of our results, relevant connections with those in earlier works are also pointed out.

The personal wealth importance to the intertemporal choice

In behavioral finance, collecting data on amounts and delays has helped research time preference. But, could other data support new conclusions about the discount dynamics? This paper considers only the physical aspects necessary to choose between payments sooner or later. So it is verified that personal wealth is one of the elements that influence the discount shape. Specifically, when the impact of rewards on wealth is low, the dynamics can be simplified due to the low contrast between time averages, allowing different discount functions to be generalized by the q-exponential function. Therefore, the wealth (or income) can be a relevant factor in individuals’ behavior because it influences intertemporal choice’s dynamical aspects.

Non-Extensive Statistical Analysis of Acoustic Emissions Recorded in Marble and Cement Mortar Specimens Under Mechanical Load Until Fracture.

Non-extensive statistical mechanics (NESM), which is a generalization of the traditional Boltzmann-Gibbs statistics, constitutes a theoretical and analytical tool for investigating the irreversible damage evolution processes and fracture mechanisms occurring when materials are subjected to mechanical loading. In this study, NESM is used for the analysis of the acoustic emission (AE) events recorded when marble and cement mortar specimens were subjected to mechanical loading until fracture. In total, AE data originating from four distinct loading protocols are presented. The cumulative distribution of inter-event times (time interval between two consecutive AE events) and the inter-event distances (three-dimensional Euclidian distance between the centers of successive AE events) were examined under the above concept and it was found that NESM is suitable to detect criticality under the terms of mechanical status of a material. This was conducted by evaluating the fitting results of the q-exponential function and the corresponding q-indices of Tsallis entropy and , along with the parameters and . Results support that for AE data recorded from marble and cement mortar specimens of this work, which is in good agreement with the conjecture previously found in seismological data and AE data recorded from Basalt specimens.

Universal properties of primary and secondary cosmic ray energy spectra

Atomic nuclei appearing in cosmic rays (CRs) are typically classified as primary or secondary. However, a better understanding of their origin and propagation properties is still necessary. We analyse the flux of primary (He, C, O) and secondary nuclei (Li, Be, B) detected with rigidity (momentum/charge) between 2 GV and 3 TV by the alpha magnetic spectrometer on the International Space Station. We show that q-exponential distribution functions, as motivated by generalized versions of statistical mechanics with temperature fluctuations, provide excellent fits for the measured flux of all nuclei considered. Primary and secondary fluxes reveal a universal dependence on kinetic energy per nucleon for which the underlying energy distribution functions are solely distinguished by their effective degrees of freedom. All given spectra are characterized by a universal mean temperature parameter ∼200 MeV which agrees with the Hagedorn temperature. Our analysis suggests that QCD scattering processes together with nonequilibrium temperature fluctuations imprint universally onto the measured CR spectra, and produce a similar shape of energy spectra as high energy collider experiments on the Earth.

Explicit Properties of q-Cosine and q-Sine Euler Polynomials Containing Symmetric Structures

In this paper, we introduce q-cosine and q-sine Euler polynomials and determine identities for these polynomials. From these polynomials, we obtain some special properties using a power series of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. We investigate the approximate roots of q-cosine Euler polynomials that help us understand these polynomials. Moreover, we display the approximate roots movements of q-cosine Euler polynomials in a complex plane using the Newton method.

Discount models in intertemporal choice: an empirical analysis

PurposeThe main purpose of this paper is to determine the discount function which better fits the individuals' preferences through the empirical analysis of the different functions used in the field of intertemporal choice.Design/methodology/approachAfter an in-depth revision of the existing literature and unlike most studies which only focus on exponential and hyperbolic discounting, this manuscript compares the adjustment of data to six different discount functions. To do this, the analysis is based on the usual statistical methods, and the non-linear least squares regression, through the algorithm of Gauss-Newton, in order to estimate the models' parameters; finally, the AICc method is used to compare the significance of the six proposed models.FindingsThis paper shows that the so-called q-exponential function deformed by the amount is the model which better explains the individuals' preferences on both delayed gains and losses. To the extent of the authors' knowledge, this is the first time that a function different from the general hyperbola fits better to the individuals' preferences.Originality/valueThis paper contributes to the search of an alternative model able to explain the individual behavior in a more realistic way.

Behaviors of solutions to network diffusion equation with power-nonlinearity

We study a certain class of network diffusion equation with large power-nonlinearity. We first report several interesting behaviors like synchronization and mode-switching, and next analytically clarify their mechanism. Throughout the investigation we demonstrate that the q-exponential function plays an important role and that the nonlinear behaviors are critically concerned with network topology.

Mass distributions of meteorites

ABSTRACTFor at least five decades, the study of the mass distribution of meteorites has been carried out. This study aims to obtain the flux of material that comes to the Earth’s surface. For this, the observational data were modelled statistical distributions of the most varied types, derived from Gibbs entropy. However, it appears that the fragmentation process is probably complex in nature. Given this particularity, we model the mass distribution of meteorites using the q-exponential function, derived from Tsallis non-extensive statistical mechanics. This distribution is capable of modelling the entire observed spectrum of meteorite mass regardless of whether the specimens originate from the fragmentation of a single meteorite, belong to the same mineralogical group or type, or when are separated by collection sites on the Earth’s surface. We suggest that most meteorite samples are incomplete in certain mass ranges due to the action of the so-called gathering bias.

Connection relations for q-Taylor polynomial bases

We derive the connection relations between three q-Taylor polynomial bases. We also derive the connection relations between two of these bases and the continuous q-Hermite polynomials, as well as several new generating functions for the continuous q-Hermite polynomials. Our results also lead to some new integral evaluations. Using one of these connection relations, we give a new proof of a general addition theorem of Suslov for the q-exponential function Eq of Ismail and Zhang.

Research on Payment Preferences for Reverse Mortgage Based on q-Exponential Discount Function

Research on Payment Preferences for Reverse Mortgage Based on q-Exponential Discount Function

Temporal Structure in Haptic Signaling Under a Cooperative Task.

Haptic communication between humans plays an important role in society. Although this form of communication is ubiquitous at all levels of society and of human development, little is known about how synchronized coordination of motion between two persons leads to higher-order cognitive functions used in communication. In this study, we developed a novel experimental paradigm of a coin-collecting task in which participants used their hands to control a rod to jointly collect the coins on the screen. We characterized the haptic interactions between paired participants while they were taking part in a cooperative task. The individual participants first completed this task on their own and then with a randomly assigned partner for the cooperative task. Single participant experiments were used as a baseline to compare results of the paired participants. Forces applied to the rod were translated to four possible haptic states which encode the combination of the haptic interactions. As a next step, pairs of consecutive haptic states were then combined into 16 possible haptic signals which were classified in terms of their temporal patterns using a Tsallis q-exponential function. For paired participants, 80% of the haptic signals could be fit by the Tsallis q-exponential. On the other hand, only 30% of the signals found in the single-participant trials could be fit by the Tsallis q-exponential. This shows a clear difference in the temporal structures of haptic signals when participants are interacting with each other and when they are not. We also found a large difference in the number of haptic signals used by paired participants and singles. Single participants only used 1/4 of the possible haptic signals. Paired participants, on the other hand, used more than half of the possible signals. These results suggest that temporal structures present in haptic communication could be linked to the emergence of language at an evolutionary level.

Q-deformed Gamma function, q-deformed probability distributions and q-deformed statistical physics based on Tsallis’s q-exponential function

In this paper we consider a new q -number from the expansion of the q -exponential function appearing in Tsallis’s non-extensive entropy theory. Using this q -number we define the q -Gamma function and use this to discuss some q -deformed distributions and q -deformed statistical physics.

Dual conformable derivative: Variational approach and nonlinear equations

In this letter, we present a variational approach with a recently proposed form of local deformed derivative, the dual conformable derivative, which leads us to obtain a class of nonlinear equations. The ansatz on the solutions can be mathematically inferred by this dual conformable derivative eigenvalue equation and the q-exponential family functions appear naturally. Also, a clear and natural justification for the appearance of the q → 2 - q-symmetry is given. To show the potential of our variational approach with this type of dual deformed derivative, we obtain the porous medium equation and present insight for the solution in terms of the q-Gaussian. Also, a new dual conformable wave equation, which is nonlinear, is proposed and a solution is built up in terms of q-plane waves. A dual conformable harmonic oscillator equation is also obtained and promptly solved by the natural ansatz. Aspects of the nonlinear Schroedinger equation are also contemplated and one shows that it can be obtained without the need of an additional ϕ-field, from a simple Lagrangian density. The solution to the nonlinear Schroedinger is also expressed in terms of the q-exponential family of functions.

Portfolio theory, information theory and Tsallis statistics

We developed a strategic of optimal portfolio based on information theory and Tsallis statistics. The growth rate of a stock market is defined by using q-deformed functions and we find that the wealth after n days with the optimal portfolio is given by a q-exponential function. In this context, the asymptotic optimality is investigated on causal portfolios, showing advantages of the optimal portfolio over an arbitrary choice of causal portfolios. Finally, we apply the formulation in a small number of stocks in brazilian stock market [B]3 and analyzed the results.

Quantitative Cross-Cultural Similarities and Differences in Social Discounting for Gains and Losses.

Social discounting is when resource allocation decreases as social distance increases. Studies fitting different quantitative models to social discounting data have shown that a q-exponential function based on Tsallis' statistics best fits loss data, whereas a hyperbolic power function best fits gain data. However, a social discounting sign effect, where losses are discounted less than equivalent gains, has not been consistently found. This study fit four different quantitative social discounting models to gain and loss data for 40 United States (US) participants. We compared quantitative model fits to previous studies collected with Japanese and German participants to determine if (1) different quantitative social discounting models best fit loss and gain data, (2) US participants discounted less gains than Japanese participants, but not losses, and (3) US participants showed the sign effect. Results showed that the q-exponential function and the hyperbolic power function best fit median loss and gain data, respectively. There were no significant absolute differences between cultures for gains or losses, and US participants showed a robust sign effect. While most results for US participants were consistent with previous data, future cross-cultural social discounting studies are needed that manipulate sign as well as reward magnitude to determine best quantitative model fits. Social discounting results are also discussed in relation to their application to important health behaviors such as smoking and obesity.

Energy Balance in Avalanche Models for Solar Flares

Abstract The distributions of solar flare energies and waiting times have not been described simultaneously by a single physical model, yet. In this research, we investigate whether recent avalanche models can describe the distributions for both the released energies and waiting times of flares in an active region. Flaring events are simulated using the modified Lu and Hamilton model and also the optimized model. Applying a probability balance equation approach, we study the statistics of the simulated flaring events and investigate the origin of the observed power law in the flare frequency-size distribution. The results indicate that the power law originates in the distribution of transition rates (the distribution of the probabilities of transitions between different energies) rather than the distribution of the energy of the active region. It is also observed that the waiting-time distribution of simulated flaring events follows a q-exponential function, which approximates a simple Poisson distribution.

Non-Lévy stable random walk propagators for a non-Markovian walk with both superdiffusive and subdiffusive regimes

One of the most important quantities of interest in the theory of diffusion and transport is the random walk propagator. For Markovian processes, such as the standard Brownian random walk and Lévy flights, the functional form of the random walk propagator is well understood. Similarly, for certain kinds of simple non-Markovian processes, such as Lévy walks, the problem can be mapped to a solvable Markovian model. However, more complicated non-Markovian walks pose a challenge. Here we study a non-Markovian model that is rich enough to exhibit superdiffusion, normal diffusion and subdiffusion regimes (Kumar, Harbola, and Lindenberg (2010)). We numerically estimate propagators for this model and obtain good fits with a family of non-Lévy propagators based on the Tsallis q-exponential function. We conclude that stops and restrictions play similar roles in the long time limit of the propagator.

Non-extensive statistical analysis of acoustic emissions series recorded during the uniaxial compression of brittle rocks

Non-extensive statistical physics (NESP) provides the theoretical tool to investigate complex systems that appear in different length-scales and violate the classical Boltzmann–Gibbs statistics. In the present study, we apply the concept of NESP to analyze the Acoustic Emissions (AE) series recorded during uniaxial compression of brittle rocks, such as marble and sandstone specimens. In both types of rock samples, the cumulative distribution of inter-event times (interval times between successive AE hits) are well described by the q-exponential function. The variations of the q-index of Tsallis entropy and of the βq parameter with the applied stress follow distinct behavior in each type of rock, indicating in both cases the entrance of the system to the critical stage of fracture. The analysis of the experimental data suggests that NESP is suitable for describing AE activity in rock samples under compression, in a similar way to Earth’s seismic activity.

A Deformed Exponential Statistical Manifold.

Consider a probability measure and the set of -equivalent strictly positive probability densities. To endow with a structure of a -Banach manifold we use the -connection by an open arc, where is a deformed exponential function which assumes zero until a certain point and from then on is strictly increasing. This deformed exponential function has as particular cases the q-deformed exponential and -exponential functions. Moreover, we find the tangent space of at a point p, and as a consequence the tangent bundle of . We define a divergence using the q-exponential function and we prove that this divergence is related to the q-divergence already known from the literature. We also show that q-exponential and -exponential functions can be used to generalize of Rényi divergence.

Multi-scale entropy analysis and Hurst exponent

Several methods exist for measuring the complexity in a system through analysis of its associated time series. Multi-scale entropy appears as a successful method on this matter. It has been applied in many disciplines with great achievements. For example by analysis of the bio-signals, we are able to diagnose various diseases. However, in most versions for the multi-scale entropy the examined time series is analyzed qualitatively. In this study, we try to present a quantitative picture for the multi-scale entropy analysis. Particularly, we focus on finding relation between the result of the multi-scale analysis and the Hurst exponent which quantifies the persistence in time series. For this purpose, the fractional Gaussian noise time series with different Hurst exponents are analyzed by the multi-scale entropy method and the results are fitted to a decreasing q-exponential function. We observe remarkable relation between the function parameters and Hurst exponent. This function can simulate the result of analysis for the white noise to the 1∕f noise.