a numerical approach for solving an extended fisher kolomogrov petrovskii piskunov equation in the present paper a numerical method based on finite differences and spline collocation is presented for the numerical solution of a generalized fisher integro differential equation a composite weighted trapezoidal rule is manipulated to handle the numerical integrations which results in a closed form difference scheme a number of test examples are solved to assess the accuracy of the method the numerical solutions obtained indicate that the approach is reliable and yields results compatible with the exact solutions and consistent with other existing numerical methods convergence and stability of the scheme have also been discussed

validity of the third law of thermodynamics for the tsallis entropy bento et al phys rev e 91 022105 2015 recently stated that the tsallis entropy violates the third law of thermodynamics for 0 1 is already inherent in the fundamental incomplete structure of the deformed logarithms and exponentials underlying the tsallis entropy then we provide the complete deformed functions and show that the tsallis entropy conforms to the third law of thermodynamics for both superadditive q 1 regimes finally we remark that the tsallis entropy does not require the use of an escort averaging scheme once it is expressed in terms of the complete deformed functions

when shannon and khinchin meet shore and johnson equivalence of information theory and statistical inference axiomatics we propose a unified framework for both shannon khinchin and shore johnson axiomatic systems we do it by rephrasing shannon khinchine axioms in terms of generalized arithmetics of kolmogorov and nagumo we prove that the two axiomatic schemes yield identical classes of entropic functionals the uffink class of entropies this allows to re establish the entropic parallelism between information theory and statistical inference that has seemed to be broken by the use of non shannonian entropies

renyi entropy yields artificial biases not in the data and incorrect updating due to the finite size data we show that the renyi entropy implies artificial biases not warranted by the data and incorrect updating information due to the finite size of the data despite being additive it is demonstrated that this is so because it does not conform to the system and subset independence axioms of shore and johnson j ieee trans inf theory 26 26 1980 iettaw0018 944810 1109 tit 1980 1056144 we finally show that the escort averaged constraints do not remedy the situation

reciprocity relations and generalized classic entropic quantifiers that lack trace form abstract in this effort we show that the legendre reciprocity relations thermodynamics essential formal feature are respected by general classic entropic functionals even if they are not of trace form nature in contrast with shannon s or tsallis cases further with reference to the maximum entropy maxent variational process we encounter important cases relevant to physical applications currently discussed in the research literature in which the associated reciprocity relations exhibit anomalies we show that these anomalies can be cured by carefully discriminating between apparently equivalent entropic forms

consistent inference of probabilities for reproducible experiments the need for inducing a probability distribution from partial data and the complementary problem of the analysis of an observed distribution in terms of fewer relevant variables occur in many branches of physics for reproducible experiments consistency conditions which must be satisfied by any algorithm for inferring a discrete probability distribution with given averages are formulated the only consistent algorithm is the one leading to the distribution of maximal entropy subject to the given constraints

a spline quasi interpolation based method to obtain the reset voltage in resistive rams in the charge flux domain abstract resistive rams rrams are the most promising devices for near future in terms of non volatile memory applications this new technology needs advances in all the fronts that have to be addressed prior to industrialization one of them is connected with compact modeling i e the development of analytical expressions to account for the most important physical effects that are needed to calculate the current capacitance transient response etc the device models should be accurate and this issue is achieved by implementing the correct physics in a flexible and robust mathematical architecture we will focus on this latter problem here since we will deal with a good numerical approximation of experimental data based on spline quasi interpolation to perform the integrals of the current and voltage as function of time we do so to transform the usual modeling domain consisting of a current voltage representation to a charge flux domain i e the time integral of the current and voltage measured variables in this domain we introduced a new method to obtain the reset voltage of rrams and avoid the effects of the usual measurement noise the main features of the mathematical technique we propose along with practical examples built upon real experimental data will be explained the numerical stability of this new technique is of great interest for the implementation in automatic measurement environments for industrial applications

unidirectional random growth with resetting abstract we review stochastic processes without detailed balance condition and derive their h theorem we obtain stationary distributions and investigate their stability in terms of generalized entropic distances beyond the kullback leibler formula a simple stochastic model with local growth rates and direct resetting to the ground state is investigated and applied to various networks scientific citations and facebook popularity hadronic yields in high energy particle reactions income and wealth distributions biodiversity and settlement size distributions

the calculation of integrals involving b splines by means of recursion relations a procedure is given for constructing the exact integrals involving b splines using recursion relations the recursive integrals form the basic constituents for the exact evaluation of integrals that appear in the calculation of atomic and molecular properties the method can be applied to solve differential equations as well as to produce a complete set of basis functions that may approximate a function arbitrarily well depending on the degree k and the number of b splines that are employed in the approximation the advantage of this method is that the recursions developed over a fixed interval represent exact results for the particular integral involved several examples are also provided to show how these recursion relations can be applied to evaluate integrals involving multiple b splines of same or different degree closed forms of these integrals involve complicated recursion relations

generalized shannon khinchin axioms and uniqueness theorem for pseudo additive entropies abstract we consider the shannon khinchin axiomatic systems for the characterization of generalized entropies such as the sharma mittal and frank daffertshofer entropies we provide the generalization of the shannon khinchin axioms and give the corresponding uniqueness theorem the previous attempts at such axiomatizations are also discussed

nonadditive entropy maximization is inconsistent with bayesian updating the maximum entropy method char22 used to infer probabilistic models from data char22 is a special case of bayes s model inference prescription which in turn is grounded in basic propositional logic by contrast to the maximum entropy method the compatibility of nonadditive entropy maximization with bayes s model inference prescription has never been established here we demonstrate that nonadditive entropy maximization is incompatible with bayesian updating and discuss the immediate implications of this finding we focus our attention on special cases as illustrations

numerical solution of a non classical parabolic problem an integro differential approach a numerical method based on an integro differential formulation and approximation by local interpolating functions is proposed for solving a one dimensional parabolic partial differential equation subject to non classical conditions some specific test problems are solved using the proposed method numerical results obtained indicate that it can give accurate solutions and that it is an interesting and viable alternative to existing numerical methods for solving the class of problems under consideration

thermodynamics from first principles correlations and nonextensivity the standard formulation of thermostatistics being based on the boltzmann gibbs distribution and logarithmic shannon entropy describes idealized uncorrelated systems with extensive energies and short range interactions in this rapid communication we use the fundamental principles of ergodicity via liouville s theorem the self similarity of correlations and the existence of the thermodynamic limit to derive generalized forms of the equilibrium distribution for long range interacting systems significantly our formalism provides a justification for the well studied nonextensive thermostatistics characterized by the tsallis distribution which it includes as a special case we also give the complementary maximum entropy derivation of the same distributions by constrained maximization of the gibbs shannon entropy the consistency between the ergodic and maximum entropy approaches clarifies the use of the latter in the study of correlations and nonextensive thermodynamics

measuring renyi entanglement entropy in quantum monte carlo simulations we develop a quantum monte carlo procedure in the valence bond basis to measure the renyi entanglement entropy of a many body ground state as the expectation value of a unitary swap operator acting on two copies of the system an improved estimator involving the ratio of swap operators for different subregions enables convergence of the entropy in a simulation time polynomial in the system size we demonstrate convergence of the renyi entropy to exact results for a heisenberg chain finally we calculate the scaling of the renyi entropy in the two dimensional heisenberg model and confirm that the neel ground state obeys the expected area law for systems up to linear size l 32

general pseudoadditivity of composable entropy prescribed by the existence of equilibrium the concept of composability states that entropy of the total system composed of independent subsystems is a function of entropies of the subsystems here the most general pseudoadditivity rule for composable entropy is derived based only on the existence of equilibrium

nonadditive entropies yield probability distributions with biases not warranted by the data different quantities that go by the name of entropy are used in variational principles to infer probability distributions from limited data shore and johnson showed that maximizing the boltzmann gibbs form of the entropy ensures that probability distributions inferred satisfy the multiplication rule of probability for independent events in the absence of data coupling such events other types of entropies that violate the shore and johnson axioms including nonadditive entropies such as the tsallis entropy violate this basic consistency requirement here we use the axiomatic framework of shore and johnson to show how such nonadditive entropy functions generate biases in probability distributions that are not warranted by the underlying data

group entropies correlation laws and zeta functions the notion of group entropy is proposed it enables the unification and generaliztion of many different definitions of entropy known in the literature such as those of boltzmann gibbs tsallis abe and kaniadakis other entropic functionals are introduced related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium when the dynamics is weakly chaotic the associated thermostatistics are discussed the mathematical structure underlying our construction is that of formal group theory which provides the general structure of the correlations among particles and dictates the associated entropic functionals as an example of application the role of group entropies in information theory is illustrated and generalizations of the kullback leibler divergence are proposed a new connection between statistical mechanics and zeta functions is established in particular tsallis entropy is related to the classical riemann zeta function

numerical solution of the heat equation with nonlocal boundary conditions abstract the method for the heat equation with nonlocal boundary conditions is discussed in this paper the unconditional stability is proved for 1 2 subject to a condition that is much weaker than the one assumed in a paper by ekolin due to the nonlocal boundary conditions the systems of linear equations generated by the method have a coefficient matrix that is tridiagonal except its first and last rows three efficient algorithms for solving this kind of linear systems are presented a simple numerical example is given to compare their efficiency

on renyi information for ergodic diffusion processes in this paper we derive explicit formulas of the renyi information shannon entropy and song measure for the invariant density of one dimensional ergodic diffusion processes in particular the diffusion models considered include the hyperbolic the generalized inverse gaussian the pearson the exponential family and a new class of skew t diffusions

strong correlations between the exponent α and the particle number for a renyi monoatomic gas in gibbs statistical mechanics appealing to the 1902 gibbs formalism for classical statistical mechanics sm the first sm axiomatic theory ever that successfully explained equilibrium thermodynamics we show that already at the classical level there is a strong correlation between renyi s exponent and the number of particles for very simple systems no reference to heat baths is needed for such a purpose

maximum entropy principle in statistical inference case for non shannonian entropies in this letter we show that the shore johnson axioms for the maximum entropy principle in statistical estimation theory account for a considerably wider class of entropic functional than previously thought apart from a formal side of the proof where a one parameter class of admissible entropies is identified we substantiate our point by analyzing the effect of weak correlations and by discussing two pertinent examples two qubit quantum system and transverse momentum behavior of hadrons in high energy proton proton collisions

toward a relative q entropy abstract we address the question and related controversy of the formulation of the q entropy and its relative entropy counterpart for models described by continuous non discrete sets of variables we notice that an l p normalized functional proposed by lutwak yang zhang lyz which is essentially a variation of a properly normalized relative renyi entropy up to a logarithm has extremal properties that make it an attractive candidate which can be used to construct such a relative q entropy we comment on the extremizing probability distributions of this lyz functional its relation to the escort distributions a generalized fisher information and the corresponding cramer rao inequality we point out potential physical implications of the lyz entropic functional and of its extremal distributions

numerical solution of a parabolic equation with non local boundary specifications the parabolic partial differential equations with non local boundary specifications model various physical problems numerical schemes are developed for obtaining approximate solutions to the initial boundary value problem for one dimensional second order linear parabolic partial differential equation with non local boundary specifications replacing boundary conditions the method of lines semi discretization approach will be used to transform the model partial differential equation into a system of first order linear ordinary differential equations odes the spatial derivative in the pde is approximated by a finite difference approximation the solution of the resulting system of first order odes satisfies a recurrence relation which involves a matrix exponential function numerical techniques are developed by approximating the exponential matrix function in this recurrence relation the new algorithms are tested on two problems from the literature the central processor unit times needed are also considered

on some entropy functionals derived from renyi information divergence we consider the maximum entropy problems associated with renyi q entropy subject to two kinds of constraints on expected values the constraints considered are a constraint on the standard expectation and a constraint on the generalized expectation as encountered in nonextensive statistics the optimum maximum entropy probability distributions which can exhibit a power law behaviour are derived and characterized the renyi entropy of the optimum distributions can be viewed as a function of the constraint this defines two families of entropy functionals in the space of possible expected values general properties of these functionals including nonnegativity minimum convexity are documented their relationships as well as numerical aspects are also discussed finally we work out some specific cases for the reference measure q x and recover in a limit case some well known entropies

a spline collocation approach for a generalized parabolic problem subject to non classical conditions abstract a numerical method based on finite differences and spline collocation is proposed for solving a generalized parabolic problem subject to non classical conditions it is shown numerically that this scheme has second order convergence the stability conditions relating the time step size to mesh sizes were examined through numerical examples it turns out that the stability of the scheme in certain cases is subjected to a stability condition and depends on the nature of the problem a number of specific test problems are solved to assess the efficiency and performance of the method the numerical results obtained indicate that the approach is viable and yields accurate solutions when compared with the existing closed form solutions and generates results consistent with recent numerical approaches

generalized nonadditive entropies and quantum entanglement we examine the inference of quantum density operators from incomplete information by means of the maximization of general nonadditive entropic forms extended thermodynamic relations are given when applied to a bipartite spin 1 2 system the formalism allows one to avoid fake entanglement for data based on the bell clauser horne shimony holt observable and in general on any set of bell constraints particular results obtained with the tsallis entropy and with an introduced exponential entropic form are also discussed

one parameter class of uncertainty relations based on entropy power we use the concept of entropy power to derive a one parameter class of information theoretic uncertainty relations for pairs of conjugate observables in an infinite dimensional hilbert space this class constitutes an infinite tower of higher order statistics uncertainty relations which allows one in principle to determine the shape of the underlying information distribution function by measuring the relevant entropy powers we illustrate the capability of this class by discussing two examples superpositions of vacuum and squeezed states and the cauchy type heavy tailed wave function

on q non extensive statistics with non tsallisian entropy we combine an axiomatics of renyi with the qq deformed version of khinchin axioms to obtain a measure of information i e entropy which accounts both for systems with embedded self similarity and non extensivity we show that the entropy thus obtained is uniquely solved in terms of a one parameter family of information measures the ensuing maximal entropy distribution is phrased in terms of a special function known as the lambert w function we analyze the corresponding high and low temperature asymptotics and reveal a non trivial structure of the parameter space salient issues such as concavity and schur concavity of the new entropy are also discussed

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

dimensional regularization of renyi s statistical mechanics abstract we show that typical renyi s statistical mechanics quantifiers exhibit poles we are referring to the partition function z and the mean energy u renyi s entropy is characterized by a real parameter the poles emerge in a numerable set of rational numbers belonging to the line physical effects of these poles are studied by appeal to dimensional regularization as usual interesting effects are found as for instance gravitational ones in particular negative specific heats

relative entropies in conformal field theory relative entropy is a measure of distinguishability for quantum states and it plays a central role in quantum information theory the family of renyi entropies generalizes to renyi relative entropies that include as special cases most entropy measures used in quantum information theory we construct a euclidean path integral approach to renyi relative entropies in conformal field theory then compute the fidelity and the relative entropy of states in one spatial dimension at zero and finite temperature using a replica trick in contrast to the entanglement entropy the relative entropy is free of ultraviolet divergences and is obtained as a limit of certain correlation functions the relative entropy of two states provides an upper bound on their trace distance

some properties of renyi entropy and renyi entropy rate in this paper we define the conditional renyi entropy and show that the so called chain rule holds for the renyi entropy then we introduce a relation for the rate of renyi entropy and use it to derive the rate of the renyi entropy for an irreducible aperiodic markov chain we also show that the bound for the renyi entropy rate is simply the shannon entropy rate

generalized statistics yet another generalization we provide a unifying axiomatics for renyi s entropy and non extensive entropy of tsallis it is shown that the resulting entropy coincides with csiszar s measure of directed divergence known from communication theory

maxent second variation and generalized statistics abstract there are two kinds of tsallis probability distributions heavy tail ones and compact support distributions we show here by appeal to functional analysis tools that for lower bound hamiltonians the second variation s analysis of the entropic functional guarantees that the heavy tail q distribution constitutes a maximum of tsallis entropy on the other hand in the compact support instance a case by case analysis is necessary in order to tackle the issue

statistical mechanics based on renyi entropy abstract in this work we show that it is possible to obtain a generalized statistical mechanics thermostatistics based on renyi entropy to be maximized with adequate constraints the equilibrium probability distribution thus obtained has a very interesting property indeed it reminds us the statistical distribution proposed by tsallis known to conveniently describe a variety of phenomena in nonextensive systems moreover some examples are worked out in order to illustrate the main features of the herein introduced formalism

towards information theory for q nonextensive statistics without q deformed distributions in this paper we extend our recent results p jizba t arimitsu physica a 340 2004 110 on q nonextensive statistics with non tsallis entropies in particular we combine an axiomatics of renyi with the q deformed version of khinchin axioms to obtain the entropy which accounts both for systems with embedded self similarity and q nonextensivity we find that this entropy can be uniquely solved in terms of a one parameter family of information measures the corresponding entropy maximizer is expressible via a special function known under the name of the lambert w function we analyze the corresponding high and low temperature asymptotics and make some remarks on the possible applications r 2006 elsevier b v all rights reserved

asymptotically scale invariant occupancy of phase space makes the entropy sq extensive phase space can be constructed for n equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated if they are locally correlated we expect the boltzmann gibbs entropy s bg k i p i ln p i to be extensive i e s bg n n for n in particular if they are independent s bg is strictly additive i e s bg n ns bg 1 n however if the subsystems are globally correlated we expect for a vast class of systems the entropy s q k 1 i p q i q 1 with s 1 s bg for some special value of q 1 to be the one which is extensive i e s q n n for n another concept which is relevant is strict or asymptotic scale freedom or scale invariance defined as the situation for which all marginal probabilities of the n system coincide or asymptotically approach for n the joint probabilities of the n 1 system if each subsystem is a binary one scale freedom is guaranteed by what we hereafter refer to as the leibnitz rule i e the sum of two successive joint probabilities of the n system coincides or asymptotically approaches the corresponding joint probability of the n 1 system the kinds of interplay of these various concepts are illustrated in several examples one of them justifies the title of this paper we conjecture that these mechanisms are deeply related to the very frequent emergence in natural and artificial complex systems of scale free structures and to their connections with nonextensive statistical mechanics summarizing we have shown that for asymptotically scale invariant systems it is s q with q 1 and not s bg the entropy which matches standard clausius like prescriptions of classical thermodynamics

a spline collocation approach for the numerical solution of a generalized nonlinear klein gordon equation in this paper a finite element collocation approach using cubic b splines is employed for the numerical solution of a generalized form of the nonlinear klein gordon equation the efficiency of the method is tested on a number of examples that represent special cases of the extended equation including the sine gordon equation the numerical results are compared with existing numerical and analytic solutions and the outcomes confirm that the scheme yields accurate and reliable results even when few nodes are used at the time levels

penalized spline models for functional principal component analysis summary we propose an iterative estimation procedure for performing functional principal component analysis the procedure aims at functional or longitudinal data where the repeated measurements from the same subject are correlated an increasingly popular smoothing approach penalized spline regression is used to represent the mean function this allows straightforward incorporation of covariates and simple implementation of approximate inference procedures for coefficients for the handling of the within subject correlation we develop an iterative procedure which reduces the dependence between the repeated measurements that are made for the same subject the resulting data after iteration are theoretically shown to be asymptotically equivalent in probability to a set of independent data this suggests that the general theory of penalized spline regression that has been developed for independent data can also be applied to functional data the effectiveness of the proposed procedure is demonstrated via a simulation study and an application to yeast cell cycle gene expression data

tsallis thermostatics as a statistical physics of random chains in this paper we point out that the generalized statistics of tsallis havrda charvat can be conveniently used as a conceptual framework for statistical treatment of random chains in particular we use the path integral approach to show that the ensuing partition function can be identified with the partition function of a fluctuating oriented random loop of arbitrary length and shape in a background scalar potential to put some meat on the bare bones we illustrate this with two statistical systems schultz zimm polymer and relativistic particle further salient issues such as the projective special linear group psl 2 r transformation properties of tsallis inverse temperature parameter and a grand canonical ensemble of fluctuating random loops related to the tsallis havrda charvat statistics are also briefly discussed

principal components analysis of sampled functions this paper describes a technique for principal components analysis of data consisting ofn functions each observed atp argument values this problem arises particularly in the analysis of longitudinal data in which some behavior of a number of subjects is measured at a number of points in time in such cases information about the behavior of one or more derivatives of the function being sampled can often be very useful as for example in the analysis of growth or learning curves it is shown that the use of derivative information is equivalent to a change of metric for the row space in classical principal components analysis the reproducing kernel for the hilbert space of functions plays a central role and defines the best interpolating functions which are generalized spline functions an example is offered of how sensitivity to derivative information can reveal interesting aspects of the data

family of additive entropy functions out of thermodynamic limit we derive a one parametric family of entropy functions that respect the additivity condition and which describe effects of finiteness of statistical systems in particular distribution functions with long tails this one parametric family is different from the tsallis entropies and is a convex combination of the boltzmann gibbs shannon entropy and the entropy function proposed by burg an example of how longer tails are described within the present approach is worked out for the canonical ensemble we also discuss a possible origin of a hidden statistical dependence and give explicit recipes on how to construct corresponding generalizations of the master equation

troublesome aspects of the renyi maxent treatment we study in great detail the possible existence of a renyi associated thermodynamics with negative results in particular we uncover a hidden relation in renyi s variational problem maxent this relation connects the two associated lagrange multipliers canonical ensemble with the mean energy u and the renyi parameter as a consequence of such relation we obtain anomalous renyi maxent thermodynamic results

generalized molecular chaos hypothesis and the h theorem problem of constraints and amendment of nonextensive statistical mechanics quite unexpectedly kinetic theory is found to specify the correct definition of average value to be employed in nonextensive statistical mechanics it is shown that the normal average is consistent with the generalized stosszahlansatz i e molecular chaos hypothesis and the associated h theorem whereas the q average widely used in the relevant literature is not in the course of the analysis the distributions with finite cutoff factors are rigorously treated accordingly the formulation of nonextensive statistical mechanics is amended based on the normal average in addition the shore johnson theorem which supports the use of the q average is carefully re examined and it is found that one of the axioms may not be appropriate for systems to be treated within the framework of nonextensive statistical mechanics

fractional generalized cauchy process this paper presents a fractional generalized cauchy process fgcp with an additive and a multiplicative gaussian white noise for describing subordinated anomalous fluctuations the fgcp displays intermittent dynamics on random time durations whose analytical representation is given by the ito over stochastic integral the associated probability density function is given by a generalized cauchy distribution at the stationary state a fractional feynman kac formula is utilized to show that weak ergodicity breaking of the fgcp depends on the existence of the subordinator and or the divergence of variance

a note on the definition of deformed exponential and logarithm functions the recent generalizations of the boltzmann gibbs statistics mathematically rely on the deformed logarithmic and exponential functions defined through some deformation parameters in the present work we investigate whether a deformed logarithmic exponential map is a bijection from r r set of positive real numbers all real numbers to r r as their undeformed counterparts we show that their inverse map exists only in some subsets of the aforementioned co domains furthermore we present conditions which a generalized deformed function has to satisfy so that the most important properties of the ordinary functions are preserved the fulfillment of these conditions permits us to determine the validity interval of the deformation parameters we finally apply our analysis to tsallis q deformed functions and discuss the interval of concavity of the renyi entropy

zeroth law compatibility of nonadditive thermodynamics nonextensive thermodynamics is criticized by the statement that the zeroth law cannot be satisfied with nonadditive composition rules corrected in this paper we determine the general functional form of those nonadditive composition rules that are compatible with the zeroth law of thermodynamics we find that this general form is additive for the formal logarithms of the original quantities and the familiar relations of thermodynamics apply to these our result offers a possible solution to the long standing questions about equilibrium between extensive and nonextensive systems or systems with different nonextensivity parameters

route from discreteness to the continuum for the tsallis q entropy the existence and exact form of the continuum expression of the discrete nonlogarithmic q entropy is an important open problem in generalized thermostatistics since its possible lack implies that nonlogarithmic q entropy is irrelevant for the continuous classical systems in this work we show how the discrete nonlogarithmic q entropy in fact converges in the continuous limit and the negative of the q entropy with continuous variables is demonstrated to lead to the csiszar type q relative entropy just as the relation between the continuous boltzmann gibbs expression and the kullback leibler relative entropy as a result we conclude that there is no obstacle for the applicability of the q entropy to the continuous classical physical systems

numerical solution of a parabolic equation subject to specification of energy in this paper special attention is devoted to one particular problem the diffusion equation subject to the specification of mass in a portion of the domain which has been studied quite extensively both analytically and numerically in recent years parabolic partial differential equations with a non local constraint in place of one of the standard boundary specifications feature in the mathematical modeling of many phenomena in this paper the application of the method of lines mol to such problems is considered the mol semi discretization approach will be used to transform the model partial differential equation into a system of first order linear ordinary differential equations odes the mol is a method of solving pdes by discretizing the equation with respect to all but one variable usually time the spatial partial derivative is approximated by a finite difference method the solution of the resulting system of first order odes satisfies a recurrence relation which involves a matrix exponential function numerical techniques are developed by approximating the exponential matrix function in this recurrence relation the matrix exponential function is approximated by a rational approximation consisting of four parameters new parallel algorithms are developed using the resulting approximation numerical experiments on two challenging examples are presented to illustrate the performance of the algorithms

properties of cross entropy minimization the principle of minimum cross entropy minimum directed divergence minimum discrimination information is a general method of inference about an unknown probability density when there exists a prior estimate of the density and new information in the form of constraints on expected values various fundamental properties of cross entropy minimization are proven and collected in one place cross entropy s well known properties as an information measure are extended and strengthened when one of the densities involved is the result of cross entropy minimization the interplay between properties of cross entropy minimization as an inference procedure and properties of cross entropy as an information measure is pointed out examples are included and general analytic and computational methods of finding minimum cross entropy probability densities are discussed

third law of thermodynamics as a key test of generalized entropies the laws of thermodynamics constrain the formulation of statistical mechanics at the microscopic level the third law of thermodynamics states that the entropy must vanish at absolute zero temperature for systems with nondegenerate ground states in equilibrium conversely the entropy can vanish only at absolute zero temperature here we ask whether or not generalized entropies satisfy this fundamental property we propose a direct analytical procedure to test if a generalized entropy satisfies the third law assuming only very general assumptions for the entropy s and energy u of an arbitrary n level classical system mathematically the method relies on exact calculation of ds du in terms of the microstate probabilities pi to illustrate this approach we present exact results for the two best known generalizations of statistical mechanics specifically we study the kaniadakis entropy s which is additive and the tsallis entropy sq which is nonadditive we show that the kaniadakis entropy correctly satisfies the third law only for 1 1 thereby shedding light on why is conventionally restricted to this interval surprisingly however the tsallis entropy violates the third law for q 1 finally we give a concrete example of the power of our proposed method by applying it to a paradigmatic system the one dimensional ferromagnetic ising model with nearest neighbor interactions

tsallis power laws and finite baths with negative heat capacity it is often stated that heat baths with finite degrees of freedom i e finite baths are sources of tsallis distributions for classical hamiltonian systems by using well known fundamental statistical mechanics expressions we rigorously show that tsallis distributions with fat tails are possible only for finite baths with constant negative heat capacity while constant positive heat capacity finite baths yield decays with sharp cutoff with no fat tails however the correspondence between tsallis distributions and finite baths holds at the expense of violating the equipartition theorem for finite classical systems at equilibrium we comment on the implications of the finite bath for the recent attempts towards a q generalized central limit theorem

optimal smoothing and interpolating splines with constraints abstract this paper considers the problem for designing optimal smoothing and interpolating splines with equality and or inequality constraints the splines are constituted by employing normalized uniform b splines as the basis functions namely as weighted sum of shifted b splines of degree k then a central issue is to determine an optimal vector of the so called control points by employing such an approach it is shown that various types of constraints are formulated as linear function of the control points and the problems reduce to quadratic programming problems we demonstrate the effectiveness and usefulness by numerical examples including approximation of probability density functions approximation of discontinuous functions and trajectory planning

the validity of the third law of thermodynamics for the renyi and homogeneous entropies abstract bento et al 2015 recently proposed to use the third law of thermodynamics as a test for the generalized entropies since the third law should be respected by any statistical mechanical entropy regardless of the explicit form of the hamiltonian we first consider the renyi entropy which is the sole additive generalized entropy and show that it violates the third law for q 1 interestingly this is exactly the interval where the renyi entropy is neither concave nor convex we then consider the homogeneous entropy and show that it also violates the third law for 0 q 1 this implies that the homogeneous entropy cannot be used as a generalized statistical mechanical entropy since the aforementioned interval is the only interval for which the homogeneous entropy yields normalizable distributions

affinity based extension of non extensive entropy and statistical mechanics abstract tsallis non extensive entropy is extended to incorporate the dependence on affinities between the microstates of a system at the core of our construction of the extended entropy h is the concept of the effective number of dissimilar states termed the effective diversity it is a unique integrated measure derived from the probability distribution among states and the affinities between states the effective diversity is related to the extended entropy through the boltzmann s equation like relation h ln q in terms of the tsallis q logarithm a new principle called the nesting principle is established stating that the effective diversity remains invariant under arbitrary grouping of the constituent states it is shown that this invariance property holds only for q 2 however the invariance is recovered for general q in the zero affinity limit i e the tsallis and boltzmann gibbs case using the affinity based extended tsallis entropy the microcanonical and the canonical ensembles are constructed in the presence of general between state affinities it is shown that the classic postulate of equal a priori probabilities no longer holds but is modified by affinity dependent terms as an illustration a two level system is investigated by the extended canonical method which manifests that the thermal behaviour of the thermodynamic quantities at equilibrium are affected by the between state affinity furthermore some applications and implications of the affinity based extended diversity entropy for information theory and biodiversity theory are addressed in appendices

comment on renyi entropy yields artificial biases not in the data and incorrect updating due to the finite size data in their recent paper phys rev e 99 032134 2019 2470 004510 1103 physreve 99 032134 oikonomou and bagci have argued that renyi entropy is ill suited for inference purposes because it is not consistent with the shore johnson axioms of statistical estimation theory in this comment we seek to clarify the latter statement by showing that there are several issues in oikonomou s and bagci s reasonings which lead to erroneous conclusions when all these issues are properly accounted for no violation of shore johnson axioms is found

penalized pca approaches for b spline expansions of smooth functional data functional principal component analysis fpca is a dimension reduction technique that explains the dependence structure of a functional data set in terms of uncorrelated variables in many applications the data are a set of smooth functions observed with error in these cases the principal components are difficult to interpret because the estimated weight functions have a lot of variability and lack of smoothness the most common way to solve this problem is based on penalizing the roughness of a function by its integrated squared d order derivative two alternative forms of penalized fpca based on b spline basis expansions of sample curves and a simpler discrete penalty that measures the roughness of a function by summing squared d order differences between adjacent b spline coefficients p spline penalty are proposed in this paper the main difference between both smoothed fpca approaches is that the first uses the p spline penalty in the least squares approximation of the sample curves in terms of a b spline basis meanwhile the second introduces the p spline penalty in the orthonormality constraint of the algorithm that computes the principal components leave one out cross validation is adapted to select the smoothing parameter for these two smoothed fpca approaches a simulation study and an application with chemometric functional data are developed to test the performance of the proposed smoothed approaches and to compare the results with non penalized fpca and regularized fpca

generalization of shannon s theorem for tsallis entropy by using the assumptions that the entropy must i be a continuous function of the probabilities pi pi 0 1 i only ii be a monotonic increasing function of the number of states w in the case of equiprobability iii satisfy the pseudoadditivity relation sq a b k sq a k sq b k 1 q sq a sq b k2 a and b being two independent systems q r and k a positive constant and iv satisfy the relation sq pi sq pl pm pl qsq pi pl pm qsq pi pm where pl pm 1 pl i 1wlpi and pm i wl 1wpi we prove along shannon s lines that the unique function that satisfies all these properties is the generalized tsallis entropy sq k 1 i 1wpiq q 1

exact time dependent solutions of the renyi fokker planck equation and the fokker planck equations related to the entropies proposed by sharma and mittal abstract exact time dependent solutions representing generalized one dimensional ornstein uhlenbeck processes are derived from nonlinear fokker planck equations and the corresponding diffusion processes are studied the corresponding systems are related to the renyi entropy and entropies proposed by sharma and mittal in the former case normal diffusion is found with a diffusion rate depending on the order of the renyi entropy in the latter case we find anomalous diffusion processes

smoothed functional principal components analysis by choice of norm the principal components analysis of functional data is often enhanced by the use of smoothing it is shown that an attractive method of incorporating smoothing is to replace the usual l 2 orthonormality constraint on the principal components by orthonormality with respect to an inner product that takes account of the roughness of the functions the method is easily implemented in practice by making use of appropriate function transforms fourier transforms for periodic data and standard principal components analysis programs several alternative possible interpretations of the smoothed principal components as obtained by the method are presented some theoretical properties of the method are discussed the estimates are shown to be consistent under appropriate conditions and asymptotic expansion techniques are used to investigate their bias and variance properties these indicate that the form of smoothing proposed is advantageous under mild conditions indeed milder than those for existing methods of smoothed functional principal components analysis the choice of smoothing parameter by cross validation is discussed the methodology of the paper is illustrated by an application to a biomechanical data set obtained in the study of the behaviour of the human thumb forefinger system

generalized statistical mechanics connection with thermodynamics the authors show the manner through which the recent generalization by tsallis 1988 of the boltzmann gibbs statistics becomes consistent with a generalized thermodynamics preserving the legendre transformation framework of standard thermodynamics in addition to that they generalize the shannon additivity

reply to comment on renyi entropy yields artificial biases not in the data and incorrect updating due to the finite size data we reply to the preceding comment by jizba and korbel jizba and korbel phys rev e 100 026101 2019 10 1103 physreve 100 026101 by first pointing out that the schur concavity proposed by them falls short of identifying the correct intervals of normalization for the optimum probability distribution even though normalization is a necessary ingredient in the entropy maximization procedure second their treatment of the subset independence axiom requires a modification of the lagrange multipliers one begins with thereby rendering the optimization less trustworthy we also explicitly demonstrate that the renyi entropy violates the subset independence axiom and compare it with the shannon entropy third the composition rule offered by jizba and korbel is shown to yield probability distributions even without a need for the entropy maximization procedure at the expense of creating artificial bias in the data

using multivariate adaptive regression splines and multilayer perceptron networks to evaluate paper manufactured using eucalyptus globulus abstract using advanced machine learning techniques as an alternative to conventional double entry volume equations known as classical allometric models regression models of the inside bark volume dependent variable for standing eucalyptus globulus trunks or main stems have been built as a function of the following three independent variables age height and outside bark diameter at breast height d the allometric models of volume biomass or carbon support the estimation of carbon storage in forests and agroforestry systems on one hand this paper presents the construction of allometric models of the inside bark volume for e globulus trees on the other hand the experimental observed data age height d and inside bark volume for 142 trees e globulus were measured and a nonlinear model was built using a data mining methodology based on multivariate adaptive regression splines mars technique and multilayer perceptron networks mlp for regression problems coefficients of determination and furnival s indices indicate the superiority of the mars technique over the allometric regression models and the mlp network the agreement of the mars model with observed data confirmed the good performance of the same one finally conclusions of this innovative research are exposed

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators