*Xiphias gladius Linnaeus*)

**Abstract**

*Xiphias gladius Linnaeus*) length dataset. Full article

A topical collection in *Entropy* (ISSN 1099-4300). This collection belongs to the section "Statistical Mechanics".

Collection Editor
Dr. Antonio M. Scarfone
Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
Interests: nonextensive statistical mechanics; nonlinear fokker-planck equations; geometry information; nonlinear schroedinger equation; quantum groups and quantum algebras; complex systems |

Dear Colleagues,

There is a diffuse belief that statistical properties of physical systems are well described by BoltzmannGibbs statistical mechanics. However, a constantly increasing amount of situations are known to violate the predictions of orthodox statistical mechanics. Systems where these emerging features are observed seem do not fulfill the standard ergodic and mixing properties on which the Boltzmann-Gibbs formalism are founded. In general, these systems are governed by nonlinear dynamics which establishes a deep relation among the parts. As a consequence, they reach a dynamical equilibrium in which the equilibrium probability distribution can differ deeply from the exponential shape typical of the Gibbs distribution.

In the last decades, we assisted to an intense research activity that has modified our understanding of statistical physics, extending and renewing its applicability considerably. Important developments, relating equilibrium and nonequilibrium statistical physics, kinetic theory, information theory and others, have produced a new understanding of the properties of complex systems that requires, in many

cases, the extension of the theory beyond the Boltzmann-Gibbs formalism.

The aim of this collection, is to collect papers in both the foundations and the applications of Statistical Mechanics going outside its traditional application. In particular, foundations regard classical and quantum aspects of statistical physics including generalized entropies, free-scale distributions, information theory, geometry information, nonextensive statistical mechanics, kinetic theory, long-range interactions and small systems. Applications are different and may include biophysics, seismology, econophysics, social systems, physics of networks, physics of risk, traffic flow, complex systems, fractal systems and others.

Specific topics of interest include (but are not limited to): * Generalized entropies * Boltzmann entropy * Renyi entropy * Non linear kinetic * Fokker-Planck equations * Quantum information * Geometry information * Fractal systems * Complex systems * Networks * Econophysics * Sociophysics * Biophysics

Dr. Antonio Maria Scarfone*Collection Editor*

**Manuscript Submission Information**

Manuscripts for the topical collection can be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on this website. The topical collection considers regular research articles, short communications and review articles. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1400 CHF (Swiss Francs).

- non extensive systems
- generalized entropies
- boltzmann entropy
- renyi entropy
- non linear kinetic
- fokker-planck equations
- quantum information
- geometry information
- fractal systems
- complex systems
- networks
- econophysics
- sociophysics
- biophysics

Open AccessArticle
Bounds on Rényi and Shannon Entropies for Finite Mixtures of Multivariate Skew-Normal Distributions: Application to Swordfish (*Xiphias gladius Linnaeus*)
**Abstract **
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Received: 3 August 2016 / Revised: 4 October 2016 / Accepted: 21 October 2016 / Published: 26 October 2016

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Mixture models are in high demand for machine-learning analysis due to their computational tractability, and because they serve as a good approximation for continuous densities. Predominantly, entropy applications have been developed in the context of a mixture of normal densities. In this paper,

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Mixture models are in high demand for machine-learning analysis due to their computational tractability, and because they serve as a good approximation for continuous densities. Predominantly, entropy applications have been developed in the context of a mixture of normal densities. In this paper, we consider a novel class of skew-normal mixture models, whose components capture skewness due to their flexibility. We find upper and lower bounds for Shannon and Rényi entropies for this model. Using such a pair of bounds, a confidence interval for the approximate entropy value can be calculated. In addition, an asymptotic expression for Rényi entropy by Stirling’s approximation is given, and upper and lower bounds are reported using multinomial coefficients and some properties and inequalities of ${L}^{p}$ metric spaces. Simulation studies are then applied to a swordfish (*Xiphias gladius Linnaeus*) length dataset.
Full article

Open AccessArticle
Entropy for the Quantized Field in the Atom-Field Interaction: Initial Thermal Distribution
**Abstract **
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Received: 23 August 2016 / Revised: 19 September 2016 / Accepted: 21 September 2016 / Published: 23 September 2016

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We study the entropy of a quantized field in interaction with a two-level atom (in a pure state) when the field is initially in a mixture of two number states. We then generalise the result for a thermal state; i.e., an (infinite) statistical

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We study the entropy of a quantized field in interaction with a two-level atom (in a pure state) when the field is initially in a mixture of two number states. We then generalise the result for a thermal state; i.e., an (infinite) statistical mixture of number states. We show that for some specific interaction times, the atom passes its purity to the field and therefore the field entropy decreases from its initial value.
Full article

Open AccessLetter
Combinatorial Intricacies of Labeled Fano Planes
**Abstract **
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Received: 20 July 2016 / Revised: 5 August 2016 / Accepted: 19 August 2016 / Published: 23 August 2016

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Given a seven-element set $X=\{1,2,3,4,5,6,7\}$ , there are 30 ways to define a Fano plane on it. Let us call a line of such a Fano plane—that

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Given a seven-element set $X=\{1,2,3,4,5,6,7\}$ , there are 30 ways to define a Fano plane on it. Let us call a line of such a Fano plane—that is to say an unordered triple from *X*—ordinary or defective, according to whether the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of the order *s*, $0\le s\le 3$ , if there are *s* *defective* lines passing through it. With such structural refinement in mind, the 30 Fano planes are shown to fall into eight distinct types. Out of the total of 35 lines, nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types. It is shown that no labeled Fano plane can have all points of zero-th order, or feature just one point of order two. A connection with prominent configurations in Steiner triple systems is also pointed out.
Full article

Open AccessArticle
Common Probability Patterns Arise from Simple Invariances
**Abstract **

Received: 26 January 2016 / Revised: 29 April 2016 / Accepted: 14 May 2016 / Published: 19 May 2016

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Shift and stretch invariance lead to the exponential-Boltzmann probability distribution. Rotational invariance generates the Gaussian distribution. Particular scaling relations transform the canonical exponential and Gaussian patterns into the variety of commonly observed patterns. The scaling relations themselves arise from the fundamental invariances of

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Shift and stretch invariance lead to the exponential-Boltzmann probability distribution. Rotational invariance generates the Gaussian distribution. Particular scaling relations transform the canonical exponential and Gaussian patterns into the variety of commonly observed patterns. The scaling relations themselves arise from the fundamental invariances of shift, stretch and rotation, plus a few additional invariances. Prior work described the three fundamental invariances as a consequence of the equilibrium canonical ensemble of statistical mechanics or the Jaynesian maximization of information entropy. By contrast, I emphasize the primacy and sufficiency of invariance alone to explain the commonly observed patterns. Primary invariance naturally creates the array of commonly observed scaling relations and associated probability patterns, whereas the classical approaches derived from statistical mechanics or information theory require special assumptions to derive commonly observed scales.
Full article

Open AccessArticle
Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems
**Abstract **

Received: 9 April 2015 / Accepted: 4 May 2015 / Published: 5 May 2015

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It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional (\(S_{BG}\)) is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's {\it Conceptual inadequacy of the Shannon information in quantum measurements}, among many

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It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional (\(S_{BG}\)) is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's {\it Conceptual inadequacy of the Shannon information in quantum measurements}, among many other systems exhibiting various forms of complexity. On the other hand, the Shannon and Khinchin axioms uniquely mandate the BG form \(S_{BG}=-k\sum_i p_i \ln p_i\); the Shore and Johnson axioms follow the same path. Many natural, artificial and social systems have been satisfactorily approached with nonadditive entropies such as the \(S_q=k \frac{1-\sum_i p_i^q}{q-1}\) one (\(q \in {\cal R}; \,S_1=S_{BG}\)), basis of nonextensive statistical mechanics. Consistently, the Shannon 1948 and Khinchine 1953 uniqueness theorems have already been generalized in the literature, by Santos 1997 and Abe 2000 respectively, in order to uniquely mandate \(S_q\). We argue here that the same remains to be done with the Shore and Johnson 1980 axioms. We arrive to this conclusion by analyzing specific classes of strongly correlated complex systems that await such generalization.
Full article

Open AccessArticle
Optimum Accelerated Degradation Tests for the Gamma Degradation Process Case under the Constraint of Total Cost
**Abstract **

Received: 25 February 2015 / Revised: 14 April 2015 / Accepted: 20 April 2015 / Published: 23 April 2015

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An accelerated degradation test (ADT) is regarded as an effective alternative to an accelerated life test in the sense that an ADT can provide more accurate information on product reliability, even when few or no failures may be expected before the end of

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An accelerated degradation test (ADT) is regarded as an effective alternative to an accelerated life test in the sense that an ADT can provide more accurate information on product reliability, even when few or no failures may be expected before the end of a practical test period. In this paper, statistical methods for optimal designing ADT plans are developed assuming that the degradation characteristic follows a gamma process (GP). The GP-based approach has an advantage that it can deal with more frequently encountered situations in which the degradation should always be nonnegative and strictly increasing over time. The optimal ADT plan is developed under the total experimental cost constraint by determining the optimal settings of variables such as the number of measurements, the measurement times, the test stress levels and the number of units allocated to each stress level such that the asymptotic variance of the maximum likelihood estimator of the *q*-th quantile of the lifetime distribution at the use condition is minimized. In addition, compromise plans are developed to provide means to check the adequacy of the assumed acceleration model. Finally, sensitivity analysis procedures for assessing the effects of the uncertainties in the pre-estimates of unknown parameters are illustrated with an example.
Full article

Open AccessArticle
Extreme Value Laws for Superstatistics
**Abstract **

Received: 11 September 2014 / Revised: 15 October 2014 / Accepted: 15 October 2014 / Published: 20 October 2014

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We study the extreme value distribution of stochastic processes modeled by superstatistics. Classical extreme value theory asserts that (under mild asymptotic independence assumptions) only three possible limit distributions are possible, namely: Gumbel, Fréchet and Weibull distribution. On the other hand, superstatistics contains three

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We study the extreme value distribution of stochastic processes modeled by superstatistics. Classical extreme value theory asserts that (under mild asymptotic independence assumptions) only three possible limit distributions are possible, namely: Gumbel, Fréchet and Weibull distribution. On the other hand, superstatistics contains three important universality classes, namely *χ*^{2}-superstatistics, inverse* χ*^{2 }-superstatistics, and lognormal superstatistics, all maximizing different effective entropy measures. We investigate how the three classes of extreme value theory are related to the three classes of superstatistics. We show that for any superstatistical process whose local equilibrium distribution does not live on a finite support, the Weibull distribution cannot occur. Under the above mild asymptotic independence assumptions, we also show that *χ*^{2}-superstatistics generally leads an extreme value statistics described by a Fréchet distribution, whereas inverse *χ*^{2 }-superstatistics, as well as lognormal superstatistics, lead to an extreme value statistics associated with the Gumbel distribution.
Full article

Open AccessArticle
Nonlinearities in Elliptic Curve Authentication
**Abstract **

Received: 2 September 2014 / Accepted: 19 September 2014 / Published: 25 September 2014

Cited by 5 | PDF Full-text (233 KB) | HTML Full-text | XML Full-text
In order to construct the border solutions for nonsupersingular elliptic curve equations, some common used models need to be adapted from linear treated cases for use in particular nonlinear cases. There are some approaches that conclude with these solutions. Optimization in this area

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In order to construct the border solutions for nonsupersingular elliptic curve equations, some common used models need to be adapted from linear treated cases for use in particular nonlinear cases. There are some approaches that conclude with these solutions. Optimization in this area means finding the majority of points on the elliptic curve and minimizing the time to compute the solution in contrast with the necessary time to compute the inverse solution. We can compute the positive solution of PDE (partial differential equation) like oscillations of f(s)/s around the principal eigenvalue λ_{1} of -Δ in ${H}_{0}^{1}\left(\Omega \right)$.Translating mathematics into cryptographic applications will be relevant in everyday life, where in there are situations in which two parts that communicate need a third part to confirm this process. For example, if two persons want to agree on something they need an impartial person to confirm this agreement, like a notary. This third part does not influence in anyway the communication process. It is just a witness to the agreement. We present a system where the communicating parties do not authenticate one another. Each party authenticates itself to a third part who also sends the keys for the encryption/decryption process. Another advantage of such a system is that if someone (sender) wants to transmit messages to more than one person (receivers), he needs only one authentication, unlike the classic systems where he would need to authenticate himself to each receiver. We propose an authentication method based on zero-knowledge and elliptic curves.
Full article

Open AccessArticle
Measures of Causality in Complex Datasets with Application to Financial Data
**Abstract **

Received: 8 January 2014 / Revised: 10 March 2014 / Accepted: 8 April 2014 / Published: 24 April 2014

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This article investigates the causality structure of financial time series. We concentrate on three main approaches to measuring causality: linear Granger causality, kernel generalisations of Granger causality (based on ridge regression and the Hilbert–Schmidt norm of the cross-covariance operator) and transfer entropy, examining

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This article investigates the causality structure of financial time series. We concentrate on three main approaches to measuring causality: linear Granger causality, kernel generalisations of Granger causality (based on ridge regression and the Hilbert–Schmidt norm of the cross-covariance operator) and transfer entropy, examining each method and comparing their theoretical properties, with special attention given to the ability to capture nonlinear causality. We also present the theoretical benefits of applying non-symmetrical measures rather than symmetrical measures of dependence. We apply the measures to a range of simulated and real data. The simulated data sets were generated with linear and several types of nonlinear dependence, using bivariate, as well as multivariate settings. An application to real-world financial data highlights the practical difficulties, as well as the potential of the methods. We use two real data sets: (1) U.S. inflation and one-month Libor; (2) S&P data and exchange rates for the following currencies: AUDJPY, CADJPY, NZDJPY, AUDCHF, CADCHF, NZDCHF. Overall, we reach the conclusion that no single method can be recognised as the best in all circumstances, and each of the methods has its domain of best applicability. We also highlight areas for improvement and future research.
Full article

Open AccessArticle
From Random Motion of Hamiltonian Systems to Boltzmann’s *H* Theorem and Second Law of Thermodynamics: a Pathway by Path Probability
**Abstract **

Received: 25 November 2013 / Revised: 18 December 2013 / Accepted: 23 January 2014 / Published: 13 February 2014

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A numerical experiment of ideal stochastic motion of a particle subject to conservative forces and Gaussian noise reveals that the path probability depends exponentially on action. This distribution implies a fundamental principle generalizing the least action principle of the Hamiltonian/Lagrangian mechanics and yields

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A numerical experiment of ideal stochastic motion of a particle subject to conservative forces and Gaussian noise reveals that the path probability depends exponentially on action. This distribution implies a fundamental principle generalizing the least action principle of the Hamiltonian/Lagrangian mechanics and yields an extended formalism of mechanics for random dynamics. Within this theory, Liouville’s theorem of conservation of phase density distribution must be modified to allow time evolution of phase density and consequently the Boltzmann *H* theorem. We argue that the gap between the regular Newtonian dynamics and the random dynamics was not considered in the criticisms of the *H* theorem.
Full article

Open AccessReview
Statistical Mechanics Ideas and Techniques Applied to Selected Problems in Ecology
**Abstract **

Received: 22 August 2013 / Revised: 15 November 2013 / Accepted: 20 November 2013 / Published: 27 November 2013

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Ecosystem dynamics provides an interesting arena for the application of a plethora concepts and techniques from statistical mechanics. Here I review three examples corresponding each one to an important problem in ecology. First, I start with an analytical derivation of clumpy patterns for

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Ecosystem dynamics provides an interesting arena for the application of a plethora concepts and techniques from statistical mechanics. Here I review three examples corresponding each one to an important problem in ecology. First, I start with an analytical derivation of clumpy patterns for species relative abundances (SRA) empirically observed in several ecological communities involving a high number *n* of species, a phenomenon which have puzzled ecologists for decades. An interesting point is that this derivation uses results obtained from a statistical mechanics model for ferromagnets. Second, going beyond the mean field approximation, I study the spatial version of a popular ecological model involving just one species representing vegetation. The goal is to address the phenomena of catastrophic shifts—gradual cumulative variations in some control parameter that suddenly lead to an abrupt change in the system—illustrating it by means of the process of desertification of arid lands. The focus is on the aggregation processes and the effects of diffusion that combined lead to the formation of non trivial spatial vegetation patterns. It is shown that different quantities—like the variance, the two-point correlation function and the patchiness—may serve as early warnings for the desertification of arid lands. Remarkably, in the onset of a desertification transition the distribution of vegetation patches exhibits scale invariance typical of many physical systems in the vicinity a phase transition. I comment on similarities of and differences between these catastrophic shifts and paradigmatic thermodynamic phase transitions like the liquid-vapor change of state for a fluid. Third, I analyze the case of many species interacting in space. I choose tropical forests, which are mega-diverse ecosystems that exhibit remarkable dynamics. Therefore these ecosystems represent a research paradigm both for studies of complex systems dynamics as well as to unveil the mechanisms responsible for the assembly of species-rich communities. The more classical equilibrium approaches are compared versus non-equilibrium ones and in particular I discuss a recently introduced cellular automaton model in which species compete both locally in physical space and along a niche axis.
Full article

Open AccessArticle
The κ-Generalizations of Stirling Approximation and Multinominal Coefficients
**Abstract **
Stirling approximation of the factorials and multinominal coefficients are generalized based on the κ-generalized functions introduced by Kaniadakis. We have related the κ-generalized multinominal coefficients to the κ-entropy by introducing a new κ-product operation, which exists only when κ ≠ 0.
Full article

Received: 20 August 2013 / Revised: 20 November 2013 / Accepted: 21 November 2013 / Published: 26 November 2013

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Open AccessArticle
Dynamics of Instantaneous Condensation in the ZRP Conditioned on an Atypical Current
**Abstract **

Received: 2 September 2013 / Revised: 30 October 2013 / Accepted: 5 November 2013 / Published: 19 November 2013

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Using a generalized Doob’s h-transform we consider the zero-range process (ZRP) conditioned to carry an atypical current, with focus on the regime where the Gallavotti-Cohen symmetry loses its validity. For a single site we compute explicitly the boundary injection and absorption rates of

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Using a generalized Doob’s h-transform we consider the zero-range process (ZRP) conditioned to carry an atypical current, with focus on the regime where the Gallavotti-Cohen symmetry loses its validity. For a single site we compute explicitly the boundary injection and absorption rates of an effective process which maps to a biased random walk. Our approach provides a direct probabilistic confirmation of the theory of “instantaneous condensation” which was proposed some while ago to explain the dynamical origin of the the failure of the Gallavotti-Cohen symmetry for high currents in the ZRP. However, it turns out that for stochastic dynamics with infinite state space care needs to be taken in the application of the Doob’s transform—we discuss in detail the sense in which the effective dynamics can be interpreted as “typical” for different regimes of the current phase diagram.
Full article

Open AccessReview
Statistical Mechanics and Information-Theoretic Perspectives on Complexity in the Earth System
**Abstract **

Received: 16 September 2013 / Revised: 22 October 2013 / Accepted: 22 October 2013 / Published: 7 November 2013

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This review provides a summary of methods originated in (non-equilibrium) statistical mechanics and information theory, which have recently found successful applications to quantitatively studying complexity in various components of the complex system Earth. Specifically, we discuss two classes of methods: (i) entropies of

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This review provides a summary of methods originated in (non-equilibrium) statistical mechanics and information theory, which have recently found successful applications to quantitatively studying complexity in various components of the complex system Earth. Specifically, we discuss two classes of methods: (i) entropies of different kinds (e.g., on the one hand classical Shannon and R´enyi entropies, as well as non-extensive Tsallis entropy based on symbolic dynamics techniques and, on the other hand, approximate entropy, sample entropy and fuzzy entropy); and (ii) measures of statistical interdependence and causality (e.g., mutual information and generalizations thereof, transfer entropy, momentary information transfer). We review a number of applications and case studies utilizing the above-mentioned methodological approaches for studying contemporary problems in some exemplary fields of the Earth sciences, highlighting the potentials of different techniques.
Full article

Open AccessArticle
Group Invariance of Information Geometry on *q*-Gaussian Distributions Induced by Beta-Divergence
**Abstract **

Received: 6 October 2013 / Revised: 26 October 2013 / Accepted: 29 October 2013 / Published: 4 November 2013

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We demonstrate that the *q*-exponential family particularly admits natural geometrical structures among deformed exponential families. The property is the invariance of structures with respect to a general linear group, which transitively acts on the space of positive definite matrices. We prove this

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We demonstrate that the *q*-exponential family particularly admits natural geometrical structures among deformed exponential families. The property is the invariance of structures with respect to a general linear group, which transitively acts on the space of positive definite matrices. We prove this property via the correspondence between information geometry induced by a deformed potential on the space and the one induced by what we call *β-divergence* defined on the q-exponential family with *q *=* β* + 1. The results are fundamental in robust multivariate analysis using the q-Gaussian family.
Full article

Open AccessArticle
Law of Multiplicative Error and Its Generalization to the Correlated Observations Represented by the q-Product
**Abstract **

Received: 8 September 2013 / Revised: 6 October 2013 / Accepted: 21 October 2013 / Published: 28 October 2013

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The law of multiplicative error is presented for independent observations and correlated observations represented by the q-product, respectively. We obtain the standard log-normal distribution in the former case and the log-q-normal distribution in the latter case. Queirós’ q-log normal distribution is also reconsidered

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The law of multiplicative error is presented for independent observations and correlated observations represented by the q-product, respectively. We obtain the standard log-normal distribution in the former case and the log-q-normal distribution in the latter case. Queirós’ q-log normal distribution is also reconsidered in the framework of the law of error. These results are presented with mathematical conditions to give rise to these distributions.
Full article

Open AccessArticle
The Phase Space Elementary Cell in Classical and Generalized Statistics
**Abstract **

Received: 5 September 2013 / Revised: 25 September 2013 / Accepted: 27 September 2013 / Published: 15 October 2013

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In the past, the phase-space elementary cell of a non-quantized system was set equal to the third power of the Planck constant; in fact, it is not a necessary assumption. We discuss how the phase space volume, the number of states and the

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In the past, the phase-space elementary cell of a non-quantized system was set equal to the third power of the Planck constant; in fact, it is not a necessary assumption. We discuss how the phase space volume, the number of states and the elementary-cell volume of a system of non-interacting N particles, changes when an interaction is switched on and the system becomes or evolves to a system of correlated non-Boltzmann particles and derives the appropriate expressions. Even if we assume that nowadays the volume of the elementary cell is equal to the cube of the Planck constant, h3, at least for quantum systems, we show that there is a correspondence between different values of h in the past, with important and, in principle, measurable cosmological and astrophysical consequences, and systems with an effective smaller (or even larger) phase-space volume described by non-extensive generalized statistics.
Full article

Open AccessReview
Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions
**Abstract **

Received: 29 July 2013 / Revised: 17 September 2013 / Accepted: 17 September 2013 / Published: 25 September 2013

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We present the main features of the mathematical theory generated by the* ** * *κ*-deformed exponential function ${\mathrm{exp}}_{k}\left(x\right)\text{}=\text{}{(\sqrt{1\text{}+\text{}{k}^{2}{x}^{2}}\text{}+\text{}kx)}^{\frac{1}{k}}$, with 0 *≤ **κ < *1, developed

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We present the main features of the mathematical theory generated by the* ** * *κ*-deformed exponential function ${\mathrm{exp}}_{k}\left(x\right)\text{}=\text{}{(\sqrt{1\text{}+\text{}{k}^{2}{x}^{2}}\text{}+\text{}kx)}^{\frac{1}{k}}$, with 0 *≤ **κ < *1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The *κ*-mathematics has its roots in special relativity and furnishes the theoretical foundations of the *κ*-statistical mechanics predicting power law tailed statistical distributions, which have been observed experimentally in many physical, natural and artificial systems. After introducing the *κ*-algebra, we present the associated *κ*-differential and *κ*-integral calculus. Then, we obtain the corresponding *κ*-exponential and *κ*-logarithm functions and give the *κ*-version of the main functions of the ordinary mathematics.
Full article

Open AccessArticle
Examples of the Application of Nonparametric Information Geometry to Statistical Physics
**Abstract **

Received: 15 August 2013 / Revised: 13 September 2013 / Accepted: 16 September 2013 / Published: 25 September 2013

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We review a nonparametric version of Amari’s information geometry in which the set of positive probability densities on a given sample space is endowed with an atlas of charts to form a differentiable manifold modeled on Orlicz Banach spaces. This nonparametric setting is

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We review a nonparametric version of Amari’s information geometry in which the set of positive probability densities on a given sample space is endowed with an atlas of charts to form a differentiable manifold modeled on Orlicz Banach spaces. This nonparametric setting is used to discuss the setting of typical problems in machine learning and statistical physics, such as black-box optimization, Kullback-Leibler divergence, Boltzmann-Gibbs entropy and the Boltzmann equation.
Full article

Open AccessArticle
Solutions of Some Nonlinear Diffusion Equations and Generalized Entropy Framework
**Abstract **

Received: 1 August 2013 / Revised: 26 August 2013 / Accepted: 11 September 2013 / Published: 18 September 2013

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We investigate solutions of a generalized diffusion equation that contains nonlinear terms in the presence of external forces and reaction terms. The solutions found here can have a compact or long tail behavior and can be expressed in terms of the *q*-exponential

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We investigate solutions of a generalized diffusion equation that contains nonlinear terms in the presence of external forces and reaction terms. The solutions found here can have a compact or long tail behavior and can be expressed in terms of the *q*-exponential functions present in the Tsallis framework. In the case of the long-tailed behavior, in the asymptotic limit, these solutions can also be connected with the L´evy distributions. In addition, from the results presented here, a rich class of diffusive processes, including normal and anomalous ones, can be obtained.
Full article

Open AccessArticle
Deformed Exponentials and Applications to Finance
**Abstract **

Received: 29 July 2013 / Revised: 22 August 2013 / Accepted: 26 August 2013 / Published: 2 September 2013

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We illustrate some financial applications of the Tsallis and Kaniadakis deformed exponential. The minimization of the corresponding deformed divergence is discussed as a criterion to select a pricing measure in the valuation problems of incomplete markets. Moreover, heavy-tailed models for price processes are

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We illustrate some financial applications of the Tsallis and Kaniadakis deformed exponential. The minimization of the corresponding deformed divergence is discussed as a criterion to select a pricing measure in the valuation problems of incomplete markets. Moreover, heavy-tailed models for price processes are proposed, which generalized the well-known Black and Scholes model.
Full article

Open AccessArticle
Kinetic Theory Microstructure Modeling in Concentrated Suspensions
**Abstract **

Received: 3 June 2013 / Revised: 6 July 2013 / Accepted: 11 July 2013 / Published: 19 July 2013

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When suspensions involving rigid rods become too concentrated, standard dilute theories fail to describe their behavior. Rich microstructures involving complex clusters are observed, and no model allows describing its kinematics and rheological effects. In previous works the authors propose a first attempt to

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When suspensions involving rigid rods become too concentrated, standard dilute theories fail to describe their behavior. Rich microstructures involving complex clusters are observed, and no model allows describing its kinematics and rheological effects. In previous works the authors propose a first attempt to describe such clusters from a micromechanical model, but neither its validity nor the rheological effects were addressed. Later, authors applied this model for fitting the rheological measurements in concentrated suspensions of carbon nanotubes (CNTs) by assuming a rheo-thinning behavior at the constitutive law level. However, three major issues were never addressed until now: (i) the validation of the micromechanical model by direct numerical simulation; (ii) the establishment of a general enough multi-scale kinetic theory description, taking into account interaction, diffusion and elastic effects; and (iii) proposing a numerical technique able to solve the kinetic theory description. This paper focuses on these three major issues, proving the validity of the micromechanical model, establishing a multi-scale kinetic theory description and, then, solving it by using an advanced and efficient separated representation of the cluster distribution function. These three aspects, never until now addressed in the past, constitute the main originality and the major contribution of the present paper.
Full article

Open AccessArticle
Statistical Properties of the Foreign Exchange Network at Different Time Scales: Evidence from Detrended Cross-Correlation Coefficient and Minimum Spanning Tree
**Abstract **

Received: 15 April 2013 / Revised: 24 April 2013 / Accepted: 26 April 2013 / Published: 6 May 2013

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We investigate the statistical properties of the foreign exchange (FX) network at different time scales by two approaches, namely the methods of detrended cross-correlation coefficient (DCCA coefficient) and minimum spanning tree (MST). The daily FX rates of 44 major currencies in the period

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We investigate the statistical properties of the foreign exchange (FX) network at different time scales by two approaches, namely the methods of detrended cross-correlation coefficient (DCCA coefficient) and minimum spanning tree (MST). The daily FX rates of 44 major currencies in the period of 2007–2012 are chosen as the empirical data. Based on the analysis of statistical properties of cross-correlation coefficients, we find that the cross-correlation coefficients of the FX market are fat-tailed. By examining three MSTs at three special time scales (i.e., the minimum, medium, and maximum scales), we come to some conclusions: USD and EUR are confirmed as the predominant world currencies; the Middle East cluster is very stable while the Asian cluster and the Latin America cluster are not stable in the MSTs; the Commonwealth cluster is also found in the MSTs. By studying four evaluation criteria, we find that the MSTs of the FX market present diverse topological and statistical properties at different time scales. The scale-free behavior is observed in the FX network at most of time scales. We also find that most of links in the FX network survive from one time scale to the next.
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