periodic solutions for nonlinear systems with mean curvature like operators abstract we give an existence result for a periodic boundary value problem involving mean curvature like operators following a recent work of r manasevich and j mawhin we use an approach based on the leray schauder degree

selecting the individual numerical scale and prioritization method in the analytic hierarchy process a 2 tuple fuzzy linguistic approach the validity of the priority vector used in the analytic hierarchy process ahp relies on two factors the selection of a numerical scale and the selection of a prioritization method the traditional ahp selects only one numerical scale e g the saaty scale and one prioritization method e g the eigenvector method for each particular problem for this traditional selection approach there is disagreement on which numerical scale and prioritization method is better in deriving a priority vector in fact the best numerical scale and the best prioritization method both rely on the content of the pairwise comparison data provided by the ahp decision makers by defining a set of concepts regarding the scale function and the linguistic pairwise comparison matrices lpcms of the priority vector and by using lpcms to unify the format of the input and output of ahp this paper extends the ahp prioritization process under the 2 tuple fuzzy linguistic model based on the extended ahp prioritization process we present two performance measure criteria to evaluate the effect of the numerical scales and prioritization methods we also use the performance measure criteria to develop a 2 tuple fuzzy linguistic multicriteria approach to select the best numerical scales and the best prioritization methods for different lpcms in this paper we call this type of selection the individual selection of the numerical scale and prioritization method we also compare this individual selection with traditional selection by using both random and real data and show better results with individual selection

renyi entropies and fisher informations as measures of nonextensivity in a tsallis setting abstract we study nonextensive statistical scenarios a la tsallis with reference to fisher s information and renyi s entropy a new way of evaluating tsallis generalized expectation values is examined within such a context and is shown to lead to a much better cramer rao bound than the customary procedure a connection between the information measures of fisher s and renyi s is found we show that fisher s measure for translation families remains additive even in a non extensive tsallis setting

assessing priority weights from subsets of pairwise comparisons in multiple criteria optimization problems using a set covering concept we explore the possibility and flexibility of selecting a subset of all possible pairs of comparisons to assess priority weights of the decision criteria different techniques offer different priority weights with different properties they are however interrelated when special subsets such as basic coverings elementary cycles and complete covering are considered this paper provides a framework for unifying existing assessment methods and offers some fundamental theorems which reveal the intricate relationships among the methods over the subsets

power law distributions and fisher s information measure abstract we show that thermodynamic uncertainties tu preserve their form in passing from boltzmann gibbs statistics to tsallis one provided that we express these tu in terms of the appropriate variable conjugate to the temperature in a nonextensive context

deriving partition functions and entropic functionals from thermodynamics abstract given an arbitrary probability distribution and a nondegenerate energy spectrum so that a mean energy u can be computed we derive the partition function z and the entropic functional s that satisfy the basic relation d u t d s the procedure is illustrated by considering examples of typical distributions found currently in nature in particular the power law spectrum is shown to correspond to a critical state associated with tsallis entropy

fuzzy analytic hierarchy process a performance analysis of various algorithms abstract analytical hierarchical process ahp along with fuzzy set theory has been used extensively in the multi criteria decision making mcdm process in which fuzzy numbers are utilized to represent human judgments more realistically over the past couple of decades numerous articles have been published proposing algorithms through which priority vector or weight vector can be calculated from fuzzy comparison matrices the aim of this study is to conduct a comprehensive performance analysis of the most popular algorithms proposed in this domain in terms of accuracy of weights calculated from fuzzy comparison matrices such an analysis is much needed by the researchers and practitioners however none is available an experimental analysis is conducted and the performance of various algorithms are evaluated with varying three parameters i e the size of the comparison matrix the level of fuzziness and the level of inconsistency we found that modified logarithmic least squares method and fuzzy inverse of column sum method ficsm generally outperformed other algorithms while fuzzy extent analysis fea which is the most frequently used algorithm in the literature provides the least accurate results furthermore it was observed that a modified version of fea method significantly improved its performance

an axiomatic characterization of a two parameter extended relative entropy the uniqueness theorem for a two parameter extended relative entropy is proven this result extends our previous one the uniqueness theorem for a one parameter extended relative entropy to a two parameter case in addition the properties of a two parameter extended relative entropy are studied

information measures based on tsallis entropy and geometric considerations for thermodynamic systems abstract an analysis of the thermodynamic behavior of quantum systems can be performed from a geometrical perspective investigating the structure of the state space we have developed such an analysis for nonextensive thermostatistical frameworks making use of the q divergence derived from tsallis entropy generalized expressions for operator variance and covariance are considered in terms of which the fundamental tensor is given

a multicriteria evaluation of methods for obtaining weights from ratio scale matrices abstract ratio scale matrices are commonly used as a vehicle to distinguish among competing alternatives or conflicting criteria the issue of extracting weights out of such matrices has attracted increasing attention in recent years and resulted in the development of several scaling methods this paper presents an attempt to evaluate some of the more known techniques in this area the evaluation procedure includes random generation of a large number of representative matrices which are used to evaluate the various scaling methods according to the specified criteria the performance of the different techniques is analyzed with respect to the proposed criteria and conclusions are drawn

mathematical inequalities for some divergences abstract some natural phenomena are deviating from standard statistical behavior and their study has increased interest in obtaining new definitions of information measures but the steps for deriving the best definition of the entropy of a given dynamical system remain unknown in this paper we introduce some parametric extended divergences combining jeffreys divergence and tsallis entropy defined by generalized logarithmic functions which lead to new inequalities in addition we give lower bounds for one parameter extended fermi dirac and bose einstein divergences finally we establish some inequalities for the tsallis entropy the tsallis relative entropy and some divergences by the use of the young s inequality

exact number of solutions of a prescribed mean curvature equation abstract we study the exact number of positive solutions of the dirichlet problem for the one dimensional prescribed mean curvature equation u 1 u 2 u p u q u 0 u 1 0 where 0 is a parameter and p q satisfy either 1 p q or 0 p q 1

multi attribute decision making a simulation comparison of select methods abstract several methods have been proposed for solving multi attribute decision making problems madm a major criticism of madm is that different techniques may yield different results when applied to the same problem the problem considered in this study consists of a decision matrix input of n criteria weights and ratings of l alternatives on each criterion the comparative performance of some methods has been investigated in a few mostly field studies in this simulation experiment we investigate the performance of eight methods electre topsis multiplicative exponential weighting mew simple additive weighting saw and four versions of ahp original vs geometric scale and right eigenvector vs mean transformation solution simulation parameters are the number of alternatives criteria and their distribution the solutions are analyzed using twelve measures of similarity of performance similarities and differences in the behavior of these methods are investigated dissimilarities in weights produced by these methods become stronger in problems with few alternatives however the corresponding final rankings of the alternatives vary across methods more in problems with many alternatives although less significant the distribution of criterion weights affects the methods differently in general all ahp versions behave similarly and closer to saw than the other methods electre is the least similar to saw except for closer matching the top ranked alternative followed by mew topsis behaves closer to ahp and differently from electre and mew except for problems with few criteria a similar rank reversal experiment produced the following performance order of methods saw and mew best followed by topsis ahps and electre it should be noted that the electre version used was adapted to the common madm problem and therefore it did not take advantage of the method s capabilities in handling problems with ordinal or imprecise information

a unified approach to cramer rao inequalities a unified approach is presented for establishing a broad class of cramer rao inequalities for the location parameter including as special cases the original inequality of cramer and rao as well as an lp version recently established by the authors the new approach allows for generalized moments and fisher information measures to be defined by convex functions that are not necessarily homogeneous in particular it is shown that associated with any log concave random variable whose density satisfies certain boundary conditions is a cramer rao inequality for which the given log concave random variable is the extremal applications to specific instances are also provided

tsallis entropy maximization procedure revisited abstract the proper way of averaging is an important question with regards to tsallis thermostatistics three different procedures have been thus far employed in the pertinent literature the third one i e the tsallis mendes plastino tmp physica a 261 1998 534 normalization procedure exhibits clear advantages with respect to earlier ones in this work we advance a distinct from the tmp one way of handling the lagrange multipliers involved in the extremization process that leads to tsallis statistical operator it is seen that the new approach considerably simplifies the pertinent analysis without losing the beautiful properties of the tsallis mendes plastino formalism

inequalities related to some types of entropies and divergences abstract the aim of this paper is to discuss new results concerning some kinds of parametric extended entropies and divergences as a result of our studies for mathematical properties on entropy and divergence we give new bounds for the tsallis quasilinear entropy and divergence by applying the hermite hadamard inequality we also give bounds for biparametric extended entropies and divergences which have been given in furuichi 2010 in addition we study r q quasilinear entropies and divergences as alternative biparametric extended entropy and divergence and then we give bounds for them finally we obtain inequalities for an extended lin s divergence and some characterizations of fermi dirac entropy and bose einstein entropy

exact number of solutions of a one dimensional prescribed mean curvature equation with concave convex nonlinearities abstract in this paper an exact number of solutions is obtained for the one dimensional prescribed mean curvature equation with concave convex nonlinearities in the form of u 1 u 2 u p u q u x 0 0 x 1 u 0 u 1 0 where 0 is a parameter and p q satisfy 0 p 1 q the arguments are based upon a time map method

preference programming and inconsistent interval judgments the problem of derivation of the weights of altematives from pairwise comparison matrices is long standing in this paper lexicographic goal programming lgp has been used to find out weights from pairwise inconsistent interval judgment matrices a number of properties and advantages of lgp as a weight determination technique have been explored an algorithm for identification and modification of inconsistent bounds is also provided the proposed technique has been illustrated by means of numerical examples

on the maximum entropy principle and the minimization of the fisher information in tsallis statistics we give a new proof of the theorems on the maximum entropy principle in tsallis statistics that is we show that the q canonical distribution attains the maximum value of the tsallis entropy subject to the constraint on the q expectation value and the q gaussian distribution attains the maximum value of the tsallis entropy subject to the constraint on the q variance as applications of the non negativity of the tsallis relative entropy without using the lagrange multipliers method in addition we define a q fisher information and then prove a q cramer rao inequality that the q gaussian distribution with special q variances attains the minimum value of the q fisher information

one dimensional prescribed mean curvature equation with exponential nonlinearity abstract we establish the existence and multiplicity results for the one dimensional prescribed mean curvature equation u 1 u 2 e u u 0 0 u l 0 in particular we characterize the precise dependence of the multiplicity of solutions with respect to both the length l of the interval and the value of the parameter

the periodic problem for curvature like equations with asymmetric perturbations we discuss existence and multiplicity of solutions of the periodic problem for the curvature like equation u 0 p a2 u02 0 f t u by means of variational techniques in the space of bounded variation functions as a 0 is allowed both the prescribed curvature equation and the 1 laplace equation are considered we are concerned with the case where the right hand side f of the equation interacts with the beginning of the spectrum of the 1 laplace operator with periodic boundary conditions on 0 t being mainly interested in the situation where ess sup 0 t r f t s may dier from ess inf 0 t r f t s

law of error in tsallis statistics in order to theoretically explain the ubiquitous existence of power law behavior such as chaos and fractals in nature tsallis entropy has been successfully applied to the generalization of the traditional boltzmann gibbs statistics the fundamental information measure of which is shannon entropy tsallis entropy s sub q is a one parameter generalization of shannon entropy s sub 1 in the sense that lim sub q spl rarr 1 s sub q s sub 1 the generalized statistics using tsallis entropy are referred to as tsallis statistics in order to present the law of error in tsallis statistics as a generalization of gauss law of error and prove it mathematically we apply the new multiplication operation determined by q logarithm and q exponential the fundamental functions in tsallis statistics to the definition of the likelihood function in gauss law of error the present maximum likelihood principle mlp leads us to determine the so called q gaussian distribution which coincides with one of the tsallis distributions derived from the maximum entropy principle for tsallis entropy under the second moment constraint

the group consensus based evidential reasoning approach for multiple attributive group decision analysis abstract many multiple attribute decision analysis problems include both quantitative and qualitative attributes with various kinds of uncertainties such as ignorance fuzziness interval data and interval belief degrees an evidential reasoning er approach developed in the 1990s and in recent years can be used to model these problems in this paper the er approach is extended to group consensus gc situations for multiple attributive group decision analysis problems in order to construct and check the gc a compatibility measure between two belief structures is developed first considering two experts utilities the compatibility between their assessments is naturally constructed using the compatibility measure based on the compatibility between two experts assessments the gc at a specific level that may be the attribute level the alternative level or the global level can be constructed and reached after the group analysis and discussion within specified times under the condition of gc we conduct a study on the forming of group assessments for alternatives the achievement of the aggregated utilities of assessment grades and the properties and procedure of the extended er approach an engineering project management software selection problem is solved by the extended er approach to demonstrate its detailed implementation process and its validity and applicability

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

analysis of a one dimensional prescribed mean curvature equation with singular nonlinearity abstract in this paper the classical solution set u of the one dimensional prescribed mean curvature equation u 1 u 2 1 u 2 l x l u l u l 0 for 0 and l 0 is analyzed via a time map it is shown that the solution set depends on both parameters and l and undergoes two bifurcations the first is a standard saddle node bifurcation which happens for all l at l the second is a splitting bifurcation namely there exists a value l such that as l transitions from greater than or equal to l to less than l the upper branch of the bifurcation diagram of problem splits into two parts in contrast the solution set of the semilinear version of problem is independent of l and exhibits only a saddle node bifurcation therefore as this analysis suggests the splitting bifurcation is a byproduct of the mean curvature operator coupled with the singular forcing

fundamental properties of tsallis relative entropy fundamental properties for the tsallis relative entropy in both classical and quantum systems are studied as one of our main results we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by hiai and petz the monotonicity of the quantum tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible the generalized tsallis relative entropy is defined and its subadditivity is shown by its joint convexity moreover the generalized peierls bogoliubov inequality is also proven

a group decision making approach based on evidential reasoning for multiple criteria sorting problem with uncertainty a new group decision making approach is developed to address a multiple criteria sorting problem with uncertainty the uncertainty in this paper refers to imprecise evaluations of alternatives with respect to the considered criteria the belief structure and the evidential reasoning approach are employed to represent and aggregate the uncertain evaluations in our approach the preference information elicited from a group of decision makers is composed of the assignment examples of some reference alternatives the disaggregation aggregation paradigm is utilized to infer compatible preference models from these assignment examples to help the group reach an agreement on the assignment of alternatives we propose a consensus reaching process in this process a consensus degree is defined to measure the agreement among the decision makers opinions when the decision makers are not satisfied with the consensus degree possible solutions are explored to help them adjust assignment examples in order to improve the consensus level if the consensus degree arrives at a satisfactory level a linear program is built to determine the collective assignment of alternatives the application of the proposed approach to a customer satisfaction analysis is presented at the end of the paper

singular value decomposition in ahp abstract the analytic hierarchy process ahp has been accepted as a leading multiattribute decision aiding model both by practitioners and academics the foundation of the ahp is the saaty s eigenvector method em and associated inconsistency index that are based on the largest eigenvalue and associated eigenvector of an n n positive reciprocal matrix the elements of the matrix are the decision maker s dm numerical estimates of the preference of n alternatives with respect to a criterion when they are compared pairwise using the 1 9 ahp fundamental comparison scale the components of the normalized eigenvector provide approximations of the unknown weights of the criteria alternatives and the deviation of the largest eigenvector from n yields a measure of how inconsistent the dm is with respect to the pairwise comparisons singular value decomposition svd is an important tool of matrix algebra that has been applied to a number of areas e g principal component analysis canonical correlation in statistics the determination of the moore penrose generalized inverse and low rank approximation of matrices in this paper using the svd and the theory of low rank approximation of a pairwise comparison matrix we offer a new approach for determining the associated weights we prove that the rank one left and right singular vectors that is the vectors associated with the largest singular value yield theoretically justified weights we suggest that an inconsistency measure for these weights is the frobenius norm of the difference between the original pairwise comparison matrix and one formed by the svd determined weights how this measure can be applied in practice as a means of measuring the confidence the dm should place in the svd weights is still an open question we illustrate the svd approach and compare it to the em for some numerical examples

a shannon tsallis transformation we determine a general link between two different solutions of the maxent variational problem namely the ones that correspond to using either shannon s or tsallis entropies in the concomitant variational problem it is shown that the two variations lead to equivalent solutions that take different appearances but contain the same information these solutions are linked by our transformation

a comparative study of the numerical scales and the prioritization methods in ahp abstract in the analytic hierarchy process ahp a decision maker first gives linguistic pairwise comparisons then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them and finally derives a priority vector from the numerical pairwise comparisons in particular the validity of this decision making tool relies on the choice of numerical scale and the design of prioritization method by introducing a set of concepts regarding the linguistic variables and linguistic pairwise comparison matrices lpcms and by defining the deviation measures of lpcms we present two performance measure algorithms to evaluate the numerical scales and the prioritization methods using these performance measure algorithms we compare the most common numerical scales the saaty scale the geometrical scale the ma zheng scale and the salo hamalainen scale and the prioritization methods the eigenvalue method and the logarithmic least squares method in addition we also discuss the parameter of the geometrical scale develop a new prioritization method and construct an optimization model to select the appropriate numerical scales for the ahp decision makers the findings in this paper can help the ahp decision makers select suitable numerical scales and prioritization methods

tsallis nonextensive thermostatistics and fisher s information measure abstract the connection between fisher s ideas concerning information measures and nonextensive thermostatistics net is investigated the cramer rao bound is generalized to a net environment a relationship between fisher s information and tsallis entropy is established

geometrical aspects of a generalized statistical mechanics abstract we discuss here the use of generalized forms of entropy taken as information measures to characterize phase transitions and critical behavior in thermodynamic systems our study is based on geometric considerations pertaining to the space of parameters that describe statistical mechanics models the thermodynamics of the system is the focus of attention in this geometric context

scaling binary comparison matrices a comment on narasimhan s proposal and other methods abstract recently narasimhan has proposed an adjustment to saaty s solution for scaling square judgment matrices consisting of pairwise ratio comparisons he recommends that prior to the application of saaty s solution the judgment matrix be adjusted to satisfy the supertransitivity requirement the present note shows that the proposed adjustment is equivalent to an alternative solution to this problem based upon calculation of the geometric means of the matrix rows thus the proposed adjustment should be regarded as an alternative to saaty s solution rather than a preliminary stage finally four solutions to this scaling problem are compared and the need for further research in this area is reiterated

geometry of escort distributions given an original distribution its statistical and probabilistic attributes may be scanned using the associated escort distribution introduced by beck and schlogl and employed in the formulation of nonextensive statistical mechanics here the geometric structure of the one parameter family of the escort distributions is studied based on the kullback leibler divergence and the relevant fisher metric it is shown that the fisher metric is given in terms of the generalized bit variance which measures fluctuations of the crowding index of a multifractal the cramer rao inequality leads to a fundamental limit for the precision of the statistical estimate of the order of the escort distribution we also show quantitatively that it is inappropriate to use the original distribution instead of the escort distribution for calculating the expectation values of physical quantities in nonextensive statistical mechanics

multicriteria compensability analysis abstract passive and active compensability multicriteria analysis pacman is a multicriteria methodology based on a decision maker dm oriented concept of compensation called compensability the preliminary stage of pacman is compensability analysis this is the procedure that translates information about intercriteria compensability provided by the dm into analytical form in this paper a theoretical modelization of compensability analysis is proposed codification of information is carried out by building the so called compensatory functions for each ordered pair of criteria distinguishing the compensating or active criterion from the compensated or passive one some technical features of these functions are analyzed and a methodology for their construction is presented on the basis of compensability analysis the compensatory strength of each criterion can be evaluated by computing its active and passive compensatory powers thus obtaining a partial preorder on the set of criteria moreover this type of approach to intercriteria relations allows one to estimate the dm s global aptitude to compensate among criteria

estimating criterion weights using eigenvectors a comparative study abstract in addition to saaty s eigenvector there are infinitely many eigen weight vectors which can be constructed for any given data of estimated weight ratios as the judgment of the ratios are dependent on personal experience learning situations and state of mind inconsistencies and degree of easiness or judgment on these individual ratios can be different we study the properties of the different eigen weight vectors including that of saaty and that recently proposed by cogger and yu a general framework for the construction of eigen weight vectors incorporating that confidence in obtaining the individual ratios can be different will be proposed and discussed

information theoretical properties of tsallis entropies a chain rule and a subadditivity for the entropy of type which is one of the nonadditive entropies were derived by daroczy in this paper we study the further relations among tsallis type entropies which are typical nonadditive entropies the chain rule is generalized by showing it for tsallis relative entropy and the nonadditive entropy we show some inequalities related to tsallis entropies especially the strong subadditivity for tsallis type entropies and the subadditivity for the nonadditive entropies the subadditivity and the strong subadditivity naturally lead to define tsallis mutual entropy and tsallis conditional mutual entropy respectively and then we show again chain rules for tsallis mutual entropies we give properties of entropic distances in terms of tsallis entropies finally we show parametrically extended results based on information theory

nonlinear systems with singular vector φ laplacian under the hartman type condition based on a continuation principle we give some existence results of solutions for the nonlinear problem u f t u u 0 u t u 0 u t involving singular vector laplacian with b a rn 0 a or rn b a 0 a a homeomorphism under the hartman type condition or the modified hartman type condition

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