an algebraic construction of generalized coherent states for shape invariant potentials generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics we show that this generalized formalism is able to a supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system continuity of labelling resolution of unity temporal stability and action identity b reproduce results already known for shape invariant systems such as harmonic oscillator double anharmonic poschl teller and self similar potentials and c point to a formalism that provides a unified description of the different kind of coherent states for quantum systems

a p q oscillator realization of two parameter quantum algebras it is noted that the study of a quantum algebra sup q 2 with two independent deformation parameters p q leads to a p q oscillator realization for it the analysis is extended to the p q analogues of su 1 1 osp 2 1 and the centreless virasoro algebra the standard single parameter q deformations are obtained in the limit p q

a statistical decision rule with incomplete knowledge about classes abstract a statistical decision rule among m classes is presented when the a priori knowledge about classes is not complete either the number of classes is not the true one or it is not possible to obtain samples from all the possible classes the reject option proposed by chow is extended by defining an ambiguity reject option and a distance reject option these two types of reject can be defined in a parametric as well as in a non parametric way an example is given in r in order to illustrate this rule this method has been developed essentially to solve diagnostic problems

on the variance of the internal path length of generalized digital trees the mellin convolution approach abstract this paper studies a generalization of the internal path length of a random digital search tree to bucket trees which are capable of storing up to b records per node we show that under the assumption of the symmetric bernoulli probabilistic model the expected path length of a tree built from n records is asymptotically n log 2 n a 1 log 2 n n and the variance is c e 1 log 2 n n where a and c depend on b only the continuous functions 1 and e 1 are periodic with mean 0 and period 1 the proofs are analytical and make use of generating functions harmonic sums and the mellin integral transform an important and very general tool for the analysis is mellin s convolution integral these results and techniques are motivated by a paper by flajolet and richmond and by kirschenhofer prodinger and szpankowski

generalized boson algebra and its entangled bipartite coherent states starting with a given generalized boson algebra known as the bosonized version of the quantum super hopf algebra we employ the hopf duality arguments to provide the dually conjugate function algebra with both the hopf algebras being finitely generated we produce a closed form expression of the universal matrix that caps the duality and generalizes the familiar exponential map relating a lie algebra with its corresponding group subsequently using an inverse mellin transform approach the coherent states of single node systems subject to the symmetry are found to be complete with a positive definite integration measure the nonclassical coalgebraic structure of the algebra is found to generate naturally entangled coherent states in bipartite composite systems

new q deformed coherent states with an explicitly known resolution of unity we construct a new family of q deformed coherent states zq where 0 q 1 these states are normalizable on the whole complex plane and continuous in their label z they allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated stieltjes power moment problem and expressed in terms of one of the two jacksons s q exponentials they also permit exact evaluation of matrix elements of physically relevant operators we use this to show that the photon number statistics for the states is sub poissonian and that they exhibit quadrature squeezing as well as an enhanced signal to quantum noise ratio over the conventional coherent state value finally we establish that they are the eigenstates of some deformed boson annihilation operator and study some of their characteristics in deformed quantum optics

profiles of tries tries from retrieval are one of the most popular data structures on words they are pertinent to the internal structure of stored words and several splitting procedures used in diverse contexts the profile of a trie is a parameter that represents the number of nodes either internal or external with the same distance from the root it is a function of the number of strings stored in a trie and the distance from the root several if not all trie parameters such as height size depth shortest path and fill up level can be uniformly analyzed through the external and internal profiles although profiles represent one of the most fundamental parameters of tries they have hardly been studied in the past the analysis of profiles is surprisingly arduous but once it is carried out it reveals unusually intriguing and interesting behavior we present a detailed study of the distribution of the profiles in a trie built over random strings generated by a memoryless source we first derive recurrences satisfied by the expected profiles and solve them asymptotically for all possible ranges of the distance from the root it appears that profiles of tries exhibit several fascinating phenomena when moving from the root to the leaves of a trie the growth of the expected profiles varies near the root the external profiles tend to zero at an exponential rate and then the rate gradually rises to being logarithmic the external profiles then abruptly tend to infinity first logarithmically and then polynomially they then tend polynomially to zero again furthermore the expected profiles of asymmetric tries are oscillating in a range where profiles grow polynomially while symmetric tries are nonoscillating in contrast to most shape parameters of random tries studied previously such a periodic behavior for asymmetric tries implies that the depth satisfies a central limit theorem but not a local limit theorem of the usual form also the widest levels in symmetric tries contain a linear number of nodes differing from the order n sqrt log n for asymmetric tries n being the size of the trees finally it is observed that profiles satisfy central limit theorems when the variance goes unbounded while near the height they are distributed according to poisson laws as a consequence of these results we find typical behaviors of the height shortest path fill up level and depth these results are derived here by methods of analytic algorithmics such as generating functions mellin transform poissonization and de poissonization the saddle point method singularity analysis and uniform asymptotic analysis

digital search trees revisited several algorithms have been proposed which build search trees using digital properties of the search keys a general approach to the study of the average case performance of such algorithms is discussed with particular attention to the analysis of the digital search tree structures of coflman and eve specifically the method leads to the solution of a problem left open by knuth finding the average number of nodes in digital search trees with both sons null the paper may be of interest as a survey and tutorial treatment of the analysis of the three primary digital tree search methods digital search trees radix search tries and patricia tries

an analogue of the unitary displacement operator for the q oscillator using the concept of a non standard hilbert space over the quantum complex plane recently introduced by kowalski et al 1993 we construct a unitary q analogue of the weyl displacement operator we investigate the q displaced vacuum states and show that they exhibit properties analogous to coherent states in the undeformed theory they are however distinct from any q coherent states previously found in the literature

arik coon oscillator with q 1 in the framework of unified q α β γ ν deformation in this paper we study the arik coon oscillator with the main relation aa qa a 1 where q 1 in the framework of the unified q deformed oscillator algebra this oscillator is connected with the askey q 1 hermite polynomials we construct a family of the generalized coherent states associated with these polynomials and give their explicit expression in terms of standard special functions by means of the solution of the appropriate classical moment problem we prove the over completeness of this family of coherent states

asymptotic behavior of the lempel ziv parsing scheme and digital search trees the lempel ziv parsing scheme finds a wide range of applications most notably in data compression and algorithms on words it partitions a sequence of length n into variable phrases such that a new phrase is the shortest substring not seen in the past as a phase the parameter of interest is the number mn of phrases that one can construct from a sequence of length n in this paper for the memoryless source with unequal probabilities of symbols generation we derive the limiting distribution of mn which turns out to be normal this proves a long standing open problem in fact to obtain this result we solved another open problem namely that of establishing the limiting distribution of the internal path length in a digital search tree the latter is a consequence of an asymptotic solution of a multiplicative differential functional equation often arising in the analysis of algorithms on words interestingly enough our findings are proved by a combination of probabilistic techniques such as renewal equation and uniform integrability and analytical techniques such as mellin transform differential functional equations de poissonization and so forth in concluding remarks we indicate a possibility of extending our results to markovian models

possibilistic and probabilistic fuzzy clustering unification within the framework of the non extensive thermostatistics fuzzy clustering algorithms are becoming the major technique in cluster analysis in this paper we consider the fuzzy clustering based on objective functions they can be divided into two categories possibilistic and probabilistic approaches leading to two different function families depending on the conditions required to state that fuzzy clusters are a fuzzy c partition of the input data recently we have presented in menard and eboueya fuzzy sets and systems 27 to be published an axiomatic derivation of the possibilistic and maximum entropy inference mei clustering approaches based upon an unifying principle of physics that of extreme physical information epi defined by frieden physics from fisher information a unification cambridge university press cambridge 1999 here using the same formalism we explicitly give a new criterion in order to provide a theoretical justification of the objective functions constraint terms membership functions and weighting exponent m used in the probabilistic and possibilistic fuzzy clustering moreover we propose an unified framework including the two procedures this approach is inspired by the work of frieden and plastino and plastino and miller physics a 235 577 extending the principle of extremal information in the framework of the non extensive thermostatistics then we show how with the help of epi one can propose extensions of the fcm and possibilistic algorithms

coherent and squeezed states of quantum heisenberg algebras starting from deformed quantum heisenberg lie algebras some realizations are given in terms of the usual creation and annihilation operators of the standard harmonic oscillator then the associated algebra eigenstates are computed and give rise to new classes of deformed coherent and squeezed states they are parametrized by deformed algebra parameters and suitable redefinitions of them as paragrassmann numbers some properties of these deformed states are also analysed

a completeness relation for the q analogue coherent states by q integration q integration is defined for the q oscillator realization of quantum groups this is used to prove a completeness relation for the q analogue of the usual coherent states these states are overcomplete

on the balance property of patricia tries external path length viewpoint abstract in this paper we give exact and asymptotic approximations for the variance of the external path length in a symmetric patricia tree the problem was open up to now we prove that for the binary patricia tree the variance is asymptotically equal to 0 37 n np log 2 n where n is the number of stored records and p x is a periodic function with a very small amplitude this implies that the external path length is asymptotically equal to n log 2 n with probability one i e almost surely these results are next used to show that from the practical average viewpoint the patricia tree does not need to be restructured in order to keep it balanced in general we ask to what extent simpler and more direct algorithms for digital search trees can be expected in practice to match the performance of more complicated worst case asymptotically better ones

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

generation of coherent states of photon added type via pathway of eigenfunctions we obtain and investigate the regular eigenfunctions of simple differential operators xr dr 1 dxr 1 r 1 2 with the eigenvalues equal to 1 with the help of these eigenfunctions we construct a non unitary analogue of a boson displacement operator which will be acting on the vacuum in this way we generate collective quantum states of the fock space which are normalized and equipped with the resolution of unity with the positive weight functions that we obtain explicitly these states are thus coherent states in the sense of klauder they span the truncated fock space without first r lowest lying basis states 0 1 r 1 these states are squeezed sub poissonian in nature and reminiscent of photon added states in agarwal and tara 1991 phys rev a 43 492

new generalized coherent states we construct a new family of boson coherent states using a specially designed function which is a solution of a functional equation de q x dx e q qx with 0 q 1 and e q 0 1 we use this function in place of the usual exponential to generate new coherent states q z from the vacuum which are normalized and continuous in their label z these states allow the resolution of unity and a corresponding weight function is furnished by the exact solution of the associated stieltjes moment problem they also permit exact evaluation of matrix elements of an arbitrary polynomial given as a normally ordered function of boson operators we exemplify this by showing that the photon number statistics for these states is sub poissonian for any q 1 the states q z are squeezed we obtain and discuss their signal to quantum noise ratio the function e q x allows a natural generation of multiboson coherent states of arbitrary multiplicity which is impossible for the usual coherent states for q 1 all the above resul

unified p q α β ν γ deformed oscillator algebra irreducible representations and induced deformed harmonic oscillator a new deformed canonical commutation relation generalizing various known deformations is defined together with its structure function of deformation then the related irreducible representations are characterized and classified finally the discrete spectrum of the corresponding deformed harmonic oscillator hamiltonian is investigated and discussed

new results on the size of tries a precise asymptotic expansion of the variance of the size of a trie built on random binary strings is presented this data structure appears in some hashing schemes and communications protocols the variance is asymptotically linear and numerical results are given the reader is referred to an earlier work for formal proofs

generalized fuzzy c shells clustering and detection of circular and elliptical boundaries abstract the fuzzy c shells fcs algorithm and its adaptive generalization called the adaptive fuzzy c shells afcs algorithm are considered for detection of curved boundaries specifically circular and elliptical the fcs algorithms utilize hyper spherical shells as cluster prototypes thus in two dimensions the prototypes are circles the afcs algorithms consider hyper ellipsoidal shells as prototypes hence the ability to characterize elliptical boundaries the generalization is achieved by allowing the distances to be measured through a norm inducing matrix that is symmetric positive definite each cluster is allowed to have a different matrix which is made a variable of optimization the ability of the algorithms to detect circular and elliptical boundaries in two dimensional data is illustrated through several examples

an integrated approach to fuzzy learning vector quantization and fuzzy c means clustering derives an interpretation for a family of competitive learning algorithms and investigates their relationship to fuzzy c means and fuzzy learning vector quantization these algorithms map a set of feature vectors into a set of prototypes associated with a competitive network that performs unsupervised learning derivation of the new algorithms is accomplished by minimizing an average generalized distance between the feature vectors and prototypes using gradient descent a close relationship between the resulting algorithms and fuzzy c means is revealed by investigating the functionals involved it is also shown that the fuzzy c means and fuzzy learning vector quantization algorithms are related to the proposed algorithms if the learning rate at each iteration is selected to satisfy a certain condition

extreme physical information and objective function in fuzzy clustering fuzzy clustering algorithms have been widely studied and applied in a variety of areas they become the major techniques in cluster analysis in this paper we focus on objective function models whose aim is to assign the data to clusters so that a given objective function is optimized we propose a new approach in fuzzy clustering and show how it can be used to obtain a systematic method deriving objective functions this approach is based on a unifying principle of physics that of extreme physical information epi defined by frieden physics from fisher information a unification 1999 the information in question is the trace of the fisher information matrix for the estimation procedure this information is shown to be a physical measure of disorder we use the epi approach for finding the effective and minimal constraint terms in objective functions with the proposed approach we justify the constraint terms defined a priori in the fuzzy c means fcm algorithm and possibilistic and maximum entropy inference approaches indeed these algorithms by contrast offer no such systematic method of finding its constraints moreover in this context the epi approach derives the reason for the extremization of objective functions the resulting formulae have a clearer physical meaning than those obtained by means of classical algorithms the updated equations of our algorithm are identical to those of the possibilistic mei and fcm with regularization approaches

average profile and limiting distribution for a phrase size in the lempel ziv parsing algorithm consider the parsing algorithm developed by lempel and ziv 1978 that partitions a sequence of length n into variable phrases blocks such that a new block is the shortest substring not seen in the past as a phrase in practice the following parameters are of interest number of phrases the size of a phrase the number of phrases of given size and so forth in this paper we focus on the size of a randomly selected phrase and the average number of phrases of a given size the so called average profile of phrase sizes these parameters can be efficiently analyzed through a digital search tree representation for a memoryless source with unequal probabilities of symbols generation the so called asymmetric bernoulli model we prove that the size of a typical phrase is asymptotically normally distributed with mean and variance explicitly computed in terms of digital search trees we prove the normal limiting distribution of the typical depth i e the length of a path from the root to a randomly selected node the latter finding is proved by a technique that belongs to the toolkit of the analytical analysis of algorithms and it seems to be novel in the context of data compression

dobinski type relations and the log normal distribution we consider sequences of generalized bell numbers b n n 1 2 which can be represented by dobinski type summation formulae i e b n 1 c k 0 p k n d k with p k a polynomial d k a function of k and c const they include the standard bell numbers p k k d k k c e their generalizations br r n r 2 3 appearing in the normal ordering of powers of boson monomials p k k r k d k k c e variants of ordered bell numbers bo p n p k k d k p 1 p k c 1 p p 1 2 etc we demonstrate that for t positive integers t 0 b n2 n t is the nth moment of a positive function on 0 which is a weighted infinite sum of log normal distributions

algebraic properties of rogers szego functions i applications in quantum optics by means of a well established algebraic framework rogers szego functions associated with a circular geometry in the complex plane are introduced in the context of q special functions and their properties are discussed in detail the eigenfunctions related to the coherent and phase states emerge from this formalism as infinite expansions of rogers szego functions the coefficients being determined through proper eigenvalue equations in each situation furthermore a complementary study on the robertson schrodinger and symmetrical uncertainty relations for the cosine sine and nondeformed number operators is also conducted corroborating in this way certain features of q deformed coherent states

fuzzy clustering and switching regression models using ambiguity and distance rejects abstract this paper examines how reject options can be used in performing fuzzy clustering and switching regression models we define an objective function in which reject options are introduced to optimization of certain clustering models this approach can be directly applied to any clustering model which can be represented as a functional dependent upon a set of cluster centers the approach can be further generalized for models that require parameters other than the cluster centers two types of reject have been included 1 the ambiguity reject which concerns patterns lying near the cluster boundaries or in the case of switching regression problems the data points which fit several models equally well 2 the distance or error reject dealing with patterns that are far away from all the clusters clustering and fuzzy c regression algorithms such as fcm fuzzy c means and fcrm fuzzy c regression models which use calculus based optimization methods suffer from several drawbacks they are very sensitive to the presence of noise moreover the memberships are relative numbers the membership of a point in a cluster depends on the membership of the point in all other clusters so the cluster centers or estimates for the parameters are poor this can be a serious problem in situations where one wishes to generate membership functions from training data this paper provides answers to these problems to avoid the memberships to be spread across the clusters and to allow the distinction between equally likely and unknown we define partial ambiguity rejects which introduce a discounting process between the classical fcm or fcrm membership functions to improve the performance of our algorithm in the presence of noise we use an amorphous noise cluster defined in demko et al actes des sixiemes rencontres del la societe francophone de classification montpellier france september 1998 to compute these rejects we propose an extension of fcrm algorithm hathaway and bezdek ieee trans fuzzy systems 1 3 1993 195 203 this algorithm is called the fuzzy c 2 regression model fc 2rm preliminary computational experiences on the developed algorithm are encouraging and compare favorably with results from other methods as fcrm and afc algorithms on the same data sets

q graded heisenberg algebras and deformed supersymmetries the notion of q grading on the enveloping algebra generated by products of q deformed heisenberg algebras is introduced for q complex number in the unit disk within this formulation we consider the extension of the notion of supersymmetry in the enveloping algebra we recover the ordinary z2 grading or grassmann parity for associative superalgebra and a modified version of the usual supersymmetry as a specific problem we focus on the interesting limit q 1 for which the arik and coon deformation j math phys 17 524 1976 of the heisenberg algebra allows one to map fermionic modes to bosonic ones in a modified sense different algebraic consequences are discussed

ncm neutrosophic c means clustering algorithm abstract in this paper a new clustering algorithm neutrosophic c means ncm is introduced for uncertain data clustering which is inspired from fuzzy c means and the neutrosophic set framework to derive such a structure a novel suitable objective function is defined and minimized and the clustering problem is formulated as a constrained minimization problem whose solution depends on the objective function in the objective function two new types of rejection have been introduced the ambiguity rejection which concerns the patterns lying near the cluster boundaries and the distance rejection dealing with patterns that are far away from all the clusters these measures are able to manage uncertainty due to imprecise and or incomplete definition of the clusters we conducted several experiments with synthetic and real data sets the results are encouraging and compared favorably with results from other methods as fcm pcm and fpcm algorithms on the same data sets finally the proposed method was applied into image segmentation algorithm the experimental results show that the proposed algorithm can be considered as a promising tool for data clustering and image processing

on the variance of a class of inductive valuations of data structures for digital search let an inductive valuation l on the family of binary tries or patricia tries or digital search trees be defined in the following way l t l tl l tr r t where tl and tr denote the left and right subtrees of t and r depends only on the size the number of records t of t let ln denote l restricted to the trees of size n in theorem 1 we give sufficient conditions on the sequence r t r t for the variance var ln to be of exact order n if the family of tries resp patricia tries resp digital search trees is equipped with the bernoulli model for the symmetric bernoulli model we prove the existence of a continuous periodic function with period 1 such that var ln log2 n n holds

hierarchical dobinski type relations via substitution and the moment problem we consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson a a 1 monomials of the form exp a ra r 1 2 under the composition of their exponential generating functions they turn out to be of sheffer type we demonstrate that two key properties of these sequences remain preserved under substitutional composition a the property of being the solution of the stieltjes moment problem and b the representation of these sequences through infinite series dobinski type relations we present a number of examples of such composition satisfying properties a and b we obtain new dobinski type formulae and solve the associated moment problem for several hierarchically defined combinatorial families of sequences

the optimum class selective rejection rule class selective rejection is an extension of simple rejection that is when an input pattern cannot be reliably assigned to one of the n classes in an n class problem it is assigned to a subset of classes that are most likely to issue the pattern instead of simply being rejected by selecting more classes the risk of making an error can be reduced at the price of subsequently having a larger remaining number of classes the optimality of class selective rejection is therefore defined as the best trade off between error rate and average number of selected classes formally the trade off study is embedded in the framework of decision theory the average expected loss is expressed as a linear combination of error rate and average number of classes the minimization of the average expected loss therefore provides the best trade off the complexity of the resulting optimum rule is reduced via a discrete convex minimization to be linear in the number of classes upper bounds on error rate and average number of classes are derived an example is provided to illustrate various aspects of the optimum decision rule finally the implications of the new decision rule are discussed

ecmdd evidential c medoids clustering with multiple prototypes in this work a new prototype based clustering method named evidential c medoids ecmdd which belongs to the family of medoid based clustering for proximity data is proposed as an extension of fuzzy c medoids fcmdd on the theoretical framework of belief functions in the application of fcmdd and original ecmdd a single medoid prototype which is supposed to belong to the object set is utilized to represent one class for the sake of clarity this kind of ecmdd using a single medoid is denoted by secmdd in real clustering applications using only one pattern to capture or interpret a class may not adequately model different types of group structure and hence limits the clustering performance in order to address this problem a variation of ecmdd using multiple weighted medoids denoted by wecmdd is presented unlike secmdd in wecmdd objects in each cluster carry various weights describing their degree of representativeness for that class this mechanism enables each class to be represented by more than one object experimental results in synthetic and real data sets clearly demonstrate the superiority of secmdd and wecmdd moreover the clustering results by wecmdd can provide richer information for the inner structure of the detected classes with the help of prototype weights

eigenkets of the q deformed creation operator by using the contour integral representation of function and the technique of integration within an ordered product of operators we point out that the q deformed creation operator possesses the eigenkets a set of new completeness and orthogonality relations composed of the kets and bras which are not mutually hermitian conjugates are derived application of the completeness relation in constructing the generalized p representation of density operator is demonstrated

on completeness of barut girardello coherent states of suq 1 1 algebra the completeness of the barut girardello coherent states of the quantized suq 1 1 algebra is studied using the inverse mellin transform the integration measure for the resolution of the unit operator is obtained the measure is found to be positive definite

mittag leffler coherent states we create a family of boson coherent states using the functions of mittag leffler ml e z 0 and their generalizations e z 0 instead of exponentials these states are shown to satisfy the usual requirements of normalizability continuity in the label and the resolution of unity with a positive weight function this last quantity is found for arbitrary 0 as a solution of an associated stieltjes moment problem in addition for m 1 2 3 and 1 corresponding to em z we propose and analyse special q deformations 0 q1 of the functions em z which serve as a tool to define q deformed coherent states of ml type we provide the expressions for expectation values of physical quantities for all the above states we discuss physical properties of these states noting that they are squeezed the ml coherent states are sub poissonian in nature whereas the q deformed ml states can be sub and super poissonian depending on q all these states are shown to be eigenstates of deformed boson operators whose commutation relations are given

new p q μ ν f deformed states we construct a new family of deformed states these states are normalizable on the whole complex plane and continuous in their label z under fixed values of parameters they allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated stieltjes power moment problem in addition to the mathematical characteristics the quantum statistical properties of these states are analytically and numerically discussed in detail in the context of conventional as well as deformed quantum optics we find that for non deformed photons the states exhibit quadrature squeezing and their photon number statistics is sub poissonian on the other hand for deformed photons the states are super poissonian and no quadrature squeezing occurs

an analytic approach to the asymptotic variance of trie statistics and related structures we develop analytic tools for the asymptotics of general trie statistics which are particularly advantageous for clarifying the asymptotic variance many concrete examples are discussed for which new fourier expansions are given the tools are also useful for other splitting processes with an underlying binomial distribution we specially highlight philippe flajolet s contribution in the analysis of these random structures

q ν deformation of generalized basic hypergeometric states we provide with a q deformation of the generalized hypergeometric coherent states defined such that the normalization function is given by generalized basic hypergeometric functions these states are eigenstates of suitably defined deformed lowering operators we study the domain of convergence of the corresponding normalization function on the basis of these states we investigate generalized basic hypergeometric husimi distributions and corresponding phase distributions as well as new analytic basis representations of arbitrary quantum states in bargmann and hardy spaces the quantum statistical properties of the states such as photon counting statistics and quadrature squeezing are analytically and numerically discussed in the framework of conventional quantum optics

the fuzzy c 2 means solving the ambiguity rejection in clustering abstract in this paper we deal with the clustering problem whose goal consists of computing a partition of a family of patterns into disjoint classes the method that we propose is formulated as a constrained minimization problem whose solution depends on a fuzzy objective function in which reject options are introduced two types of rejection have been included the ambiguity rejection which concerns patterns lying near the class boundaries and the distance rejection dealing with patterns that are far away from all the classes to compute these rejections we propose an extension of the fuzzy c means fcm algorithm of bezdek pattern recognition with fuzzy objective function algorithms plenum press new york 1981 this algorithm is called the fuzzy c 2 means fc 2m these measures allow to manage uncertainty due both to imprecise and incomplete definition of the classes the advantages of our method are 1 the degree of membership to the reject classes for a pattern xk are learned in the iterative clustering problem 2 it is not necessary to compute other characteristics to determine the reject and ambiguity degrees 3 the partial ambiguity rejections introduce a discounting process between the classical fcm membership functions in order to avoid the memberships to be spread across the classes 4 the membership functions are more immune to noise and correspond more closely to the notion of compatibility preliminary computational experiences on the developed algorithm are encouraging and compared favorably with results from other methods as fcm fpcm and f c 1 m fuzzy c 1 means clustering with solely distance rejection algorithms on the same data sets the differences in the performance can be attributed to the fact that ambiguous patterns are less accounted in for the computing of the centers

a q analogue of bargmann space and its scalar product a q analogue of bargmann space is defined using the properties of coherent states associated with a pair of q deformed bosons the space consists of a class of entire functions of a complex variable z and has a reproducing kernel on this space the q boson creation and annihilation operators are represented as multiplication by z and q differentiation with respect to z respectively a q integral analogue of bargmann s scalar product is defined involving the q exponential as a weight function associated with this is a completeness relation for the q coherent states

ecm an evidential version of the fuzzy c means algorithm a new clustering method for object data called ecm evidential c means is introduced in the theoretical framework of belief functions it is based on the concept of credal partition extending those of hard fuzzy and possibilistic ones to derive such a structure a suitable objective function is minimized using an fcm like algorithm a validity index allowing the determination of the proper number of clusters is also proposed experiments with synthetic and real data sets show that the proposed algorithm can be considered as a promising tool in the field of exploratory statistics

on 2 protected nodes in random digital trees in this paper we consider the number of 2 protected nodes in random digital trees results for the mean and variance of this number for tries have been obtained by gaither et al 2012 11 and gaither and ward 2013 10 and for the mean in digital search trees by du and prodinger 2012 5 in this short note we show that these previous results and extensions such as the variance in digital search trees and limit laws in both cases can be derived in a systematic way by recent approaches of fuchs et al 2012 2014 8 15 and fuchs and lee 2014 9 interestingly the results for the moments we obtain by our approach are quite different from the previous ones and contain divergent series which have values by appealing to the theory of abel summability we also show that our tools apply to patricia tries for which the number of 2 protected nodes has not been investigated so far

partial match retrieval of multidimensional data a precise analysis of partial match retrieval of multidimensional data is presented the structures considered here are multidimensional search trees k d trees and digital tries k d tries as well as structures designed for efficient retrieval of information stored on external devices the methods used include a detailed study of a differential system around a regular singular point in conjunction with suitable contour integration techniques for the analysis of k d trees and properties of the mellin integral transform for k d tries and extendible cell algorithms