spectral statistics for unitary transfer matrices of binary graphs quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits it is proposed here to generalize this approach by considering arbitrary directed graphs with unitary transfer matrices an exponentially increasing contribution to the form factor is identified when performing a diagonal summation over periodic orbit degeneracy classes a special class of graphs so called binary graphs is studied in more detail for these the conditions for periodic orbit pairs to be correlated including correlations due to the unitarity of the transfer matrix can be given explicitly using combinatorial techniques it is possible to perform the summation over correlated periodic orbit pair contributions to the form factor for some low dimensional cases gradual convergence towards random matrix results is observed when increasing the number of vertices of the binary graphs

pattern characteristics of an evolution between two classes pattern recognition involves two phases learning and recognition using fuzzy theory the first phase consists mainly in learning membership functions of classes the recognition phase or diagnosis consists in computing membership degrees of the data to classes and deciding in which class the data fits more appropriately in real time processes the system may evolve from one class to another the main idea of this paper is to propose a method that is able to learn and recognize this evolution the learning phase is based on fuzzy interval regressions the second part presents a method that gives a diagnosis depending on the percentage of evolution a decision method is suggested at the end

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

quantum mechanical potentials related to the prime numbers and riemann zeros prime numbers are the building blocks of our arithmetic however their distribution still poses fundamental questions riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the nontrivial zeros of the riemann ensuremath zeta s function according to the hilbert p olya conjecture there exists a hermitian operator of which the eigenvalues coincide with the real parts of the nontrivial zeros of ensuremath zeta s this idea has encouraged physicists to examine the properties of such possible operators and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems we apply the mar ifmmode check c else v c fi henko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the ensuremath zeta s function we demonstrate the multifractal nature of these potentials by measuring the r enyi dimension of their graphs our results offer hope for further analytical progress

classifier design with incomplete knowledge abstract pattern classification is a well studied problem in which the identity of an unknown pattern is determined to be one of m classes spanning the pattern space a new criterion function the inck incomplete knowledge criterion function is proposed which approximates the error probability when the m classes do not span the entire pattern space this criterion is based on the probabilistic measures obtained from a modified version of dubuisson and masson s statistical decision rule with reject the error probability or conversely the probability of correct classification can be determined without having complete knowledge about the class distributions the attractiveness of this criterion is that it is highly correlated with the statistical decision rule with reject which can be used by the classifier thus the criterion provides a good indication of the classifier performance that can be expected with the reduced feature space a classifier design is presented which incorporates the new criterion function as well it is described how the rejection classes of the statistical decision rule with reject can be used in a multiresolution classifier design to determine the class identity of patterns which cannot be determined at a given resolution experimental evidence is provided which demonstrates the advantages of the inck criterion function used in conjunction with a multiresolution classifier design

possibilistic instance based learning a method of instance based learning is introduced which makes use of possibility theory and fuzzy sets particularly a possibilistic version of the similarity guided extrapolation principle underlying the instance based learning paradigm is proposed this version is compared to the commonly used probabilistic approach from a methodological point of view moreover aspects of knowledge representation such as the modeling of uncertainty are discussed taking the possibilistic extrapolation principle as a point of departure an instance based learning procedure is outlined which includes the handling of incomplete information methods for reducing storage requirements and the adaptation of the influence of stored cases according to their typicality first theoretical and experimental results showing the efficiency of possibilistic instance based learning are presented as well

fractal potentials from energy levels we analyze the reconstruction by wu and sprung phys rev e 48 2595 1993 of a fractal one dimensional potential the quantum spectrum of which reproduces the first 500 nontrivial zeros of the riemann ensuremath zeta function our construction is based on a spectrum with gaussian unitary ensemble statistics as far as the nearest neighbor spacing distribution is concerned our results show that a reliable estimate of the fractal dimension of the potential necessitates a very large number of levels

semiclassical form factor for chaotic systems with spin we study the properties of the two point spectral form factor for classically chaotic systems with spin in the semiclassical limit with a suitable semiclassical trace formula as our principal tool to this end we introduce a regularized form factor and discuss the limit in which the so called diagonal approximation can be recovered the incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics provided this skew product is mixing we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory

quantum scars of classical closed orbits in phase space the way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x q p through the spectral wigner function w x e this is a sum over wigner functions of eigenstates within a range of energy e in the classical limit w is concentrated on the energy surface and smoothly distributed over it closed orbits provide oscillatory corrections scars for which explicit semiclassical formulae are calculated each scar is a fringe pattern decorating the orbit as x moves off the energy surface the fringes form an airy pattern with spacing of order h as x moves off the closed orbit the fringes form a complex gaussian with spacing h

a unified view of class selection with probabilistic classifiers the possibility of selecting a subset of classes instead of one unique class for assignation is of great interest in many decision making systems selecting a subset of classes instead of singleton allows to reduce the error rate and to propose a reduced set to another classifier or an expert this second step provides additional information and therefore increases the quality of the result in this paper a unified view of the problem of class selection with probabilistic classifiers is presented the proposed framework based on the evaluation of the probabilistic equivalence allows to retrieve class selective frameworks that have been proposed in the literature we also describe an approach in which the decision rules are compared by the help of a normalized area under the error selection curve it allows to get a relative independence of the performance of a classifier without reject option and thus a reliable class selection decision rule evaluation the power of this generic proposition is demonstrated by evaluating and comparing it to several state of the art methods on nine real world datasets and four different probabilistic classifiers a unified framework of class selection by the use of probabilistic equivalence is presented a learning procedure for adapting the probabilistic equivalence is proposed a normalized performance measure for class selection is described experiments on real world datasets using different classifiers show the statistical significance of the proposition

the bakerian lecture 1987 quantum chaology bounded or driven classical systems often exhibit chaos exponential instability that persists but their quantum counterparts do not nevertheless there are new regimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic and this motivates the following definition definition quantum chaology is the study of semiclassical but nonclassical behaviour characteristic of systems whose classical motion exhibits chaos this is illustrated by the statistics of energy levels on scales comparable with the mean level spacing of order h n for n freedoms these fall into universality classes for classically chaotic systems the statistics are those of random matrices real symmetric or complex hermitian depending on the presence or absence of time reversal symmetry for classically regular ones the statistics are poisson on larger scales of order h i e classically small but semiclassically large universality breaks down these phenomena are being explained by representing spectra in terms of classical closed orbits universal spectral behaviour has its origin in very long orbits non universal behaviour depends only on short ones

a characterization of the prime graphs of solvable groups abstract let g denote the set of prime divisors of the order of a finite group g the prime graph of g denoted g is the graph with vertex set g with edges p q e g if and only if there exists an element of order pq in g in this paper we prove that a graph is isomorphic to the prime graph of a solvable group if and only if its complement is 3 colorable and triangle free we then introduce the idea of a minimal prime graph we prove that there exists an infinite class of solvable groups whose prime graphs are minimal we prove the 3k conjecture on prime divisors in element orders for solvable groups with minimal prime graphs and we show that solvable groups whose prime graphs are minimal have fitting length 3 or 4

probabilistic modeling and recognition of 3 d objects this paper introduces a uniform statistical framework for both 3 d and 2 d object recognition using intensity images as input data the theoretical part provides a mathematical tool for stochastic modeling the algorithmic part introduces methods for automatic model generation localization and recognition of objects 2 d images are used for learning the statistical appearance of 3 d objectss both the depth information and the matching between image and model features are missing for model generation the implied incomplete data estimation problem is solved by the expectation maximization algorithm this leads to a novel class of algorithms for automatic model generation from projections the estimation of pose parameters corresponds to a non linear maximum likelihood estimation problem which is solved by a global optimization procedure classification is done by the bayesian decision rule this work includes the experimental evaluation of the various facets of the presented approach an empirical evaluation of learning algorithms and the comparison of different pose estimation algorithms show the feasibility of the proposed probabilistic framework

resummation and the semiclassical theory of spectral statistics we address the question as to why in the semiclassical limit classically chaotic systems generically exhibit universal quantum spectral statistics coincident with those of random matrix theory to do so we use a semiclassical resummation formalism that explicitly preserves the unitarity of the quantum time evolution by incorporating duality relations between short and long classical orbits this allows us to obtain both the non oscillatory and the oscillatory contributions to spectral correlation functions within a unified framework thus overcoming a significant problem in previous approaches in addition our results extend beyond the universal regime to describe the system specific approach to the semiclassical limit

what are the differences between bayesian classifiers and mutual information classifiers in this paper both bayesian and mutual information classifiers are examined for binary classifications with or without a reject option the general decision rules are derived for bayesian classifiers with distinctions on error types and reject types a formal analysis is conducted to reveal the parameter redundancy of cost terms when abstaining classifications are enforced the redundancy implies an intrinsic problem of nonconsistency for interpreting cost terms if no data are given to the cost terms we demonstrate the weakness of bayesian classifiers in class imbalanced classifications on the contrary mutual information classifiers are able to provide an objective solution from the given data which shows a reasonable balance among error types and reject types numerical examples of using two types of classifiers are given for confirming the differences including the extremely class imbalanced cases finally we briefly summarize the bayesian and mutual information classifiers in terms of their application advantages and disadvantages respectively

learning valued preference structures for solving classification problems this paper introduces a new approach to classification which combines pairwise decomposition techniques with ideas and tools from fuzzy preference modeling more specifically our approach first decomposes a polychotomous classification problem involving m classes into an ensemble of binary problems one for each ordered pair of classes the corresponding classifiers are trained on the relevant subsets of the transformed original training data in the classification phase a new query is submitted to every binary learner the output of each classifier is interpreted as a fuzzy degree of preference for the first in comparison with the second class by combining the outputs of all classifiers one thus obtains a fuzzy preference relation which is taken as a point of departure for the final classification decision this way the problem of classification is effectively reduced to a problem of decision making based on a fuzzy preference relation corresponding techniques which have been investigated quite intensively in the field of fuzzy set theory hence become amenable to the task of classification in particular by decomposing a preference relation into a strict preference an indifference and an incomparability relation this approach allows one to quantify different types of uncertainty in classification and thereby supports sophisticated classification and postprocessing strategies

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

a k order fuzzy or operator for pattern classification with k order ambiguity rejection in pattern recognition the membership of an object to classes is often measured by labels this article mainly deals with the mathematical foundations of labels combination operators built on t norms that extend previous ambiguity measures of objects by dealing not only with two classes ambiguities but also with k classes k lying between 1 and the number of classes c mathematical properties of this family of combination operators are established and a weighted extension is proposed allowing to give more or less importance to a given class a classifier with reject options built on the proposed measure is presented and applied on synthetic data a critical analysis of the results led to derivate some new operators by aggregating previous measures a modified classifier is proposed and applied to synthetic data as well as to standard real data

action correlations and random matrix theory the correlations in the spectra of quantum systems are intimately related to correlations which are of genuine classical origin and which appear in the spectra of actions of the classical periodic orbits of the corresponding classical systems we review this duality and the semiclassical theory which brings it about the conjecture that the quantum spectral statistics are described in terms of random matrix theory leads to the proposition that the classical two point correlation function is also given in terms of a universal function we study in detail the spectrum of actions of the baker map and use it to illustrate the steps needed to reveal the classical correlations their origin and their relation to symbolic dynamics

decision rule for pattern classification by integrating interval feature values pattern classification based on bayesian statistical decision theory needs a complete knowledge of the probability laws to perform the classification in the actual pattern classification however it is generally impossible to get the complete knowledge as constant feature values are influenced by noise therefore it is necessary to construct more flexible and robust theory for pattern classification in this paper a pattern classification theory using feature values defined on closed interval is formalized in the framework of dempster shafer measure then in order to make up the lack of information an integration algorithm is proposed which integrates the information observed by several information sources with considering source values

riemann zeros prime numbers and fractal potentials using two distinct inversion techniques the local one dimensional potentials for the riemann zeros and prime number sequence are reconstructed we establish that both inversion techniques when applied to the same set of levels lead to the same fractal potential this provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one dimension we also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be d 1 5 for the riemann zeros and d 1 8 for the prime numbers this result is somewhat surprising since the nearest neighbor spacings of the riemann zeros are known to be chaotically distributed whereas the primes obey almost poissonlike statistics our findings show that the fractal dimension is dependent on both level statistics and spectral rigidity delta 3 of the energy levels

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

the analog of huppert s conjecture on character codegrees abstract let g be a finite group and be a character of g we denote the codegree of to be codeg g ker 1 we define codeg g max codeg irr g where codeg is the set of prime divisors of codeg and g to be the set of prime divisors of g in this note we show that there exists a fixed constant k such that g k codeg g this answers an open question in 12

spectral statistics in chaotic systems with a point interaction we consider quantum systems with a chaotic classical limit that are perturbed by a point like scatterer the spectral form factor k for these systems is evaluated semiclassically in terms of periodic and diffractive orbits it is shown for order 2 and 3 that off diagonal contributions to the form factor which involve diffractive orbits cancel exactly the diagonal contributions from diffractive orbits implying that the perturbation by the scatterer does not change the spectral statistic we further show that parametric spectral statistics for these systems are universal for small changes of the strength of the scatterer

four fuzzy supervised classification methods for discriminating classes of non convex shape our work deals with modelling and optimising industrial processes such as metal cutting with high speed machining in this field we have chosen to use fuzzy supervised classification methods in order to design a diagnosis system or a process monitoring module the problem we currently meet concerns the shape of the classes we generally obtain these shapes are often non convex and non separable by a hyperplane for these reasons we focus on fuzzy supervised classification methods in order to discriminate these classes the choice of a method is not obvious and we perform a comparative study the two classical methods tested were the fuzzy k nearest neighbours method and a method based on distributed fuzzy rules furthermore we propose two adaptations of the fuzzy pattern matching algorithm called fuzzy pattern matching with exponential function and fuzzy pattern matching multidensity after some refresher on supervised classification the four tested methods are detailed and compared according to the following criteria quality of the discrimination computation time and ability to decide the response of each classifier is illustrated by membership level curves and the quality of diagnosis is studied by the introduction of membership and ambiguity rejects

optimal decision rule with class selective rejection and performance constraints the problem of defining a decision rule which takes into account performance constraints and class selective rejection is formalized in a general framework in the proposed formulation the problem is defined using three kinds of criteria the first is the cost to be minimized which defines the objective function the second are the decision options determined by the admissible assignment classes or subsets of classes and the third are the performance constraints the optimal decision rule within the statistical decision theory framework is obtained by solving the stated optimization problem two examples are provided to illustrate the formulation and the decision rule is obtained

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

quantum chaos in one dimension in this work we investigate the inverse of the celebrated bohigas giannoni schmit conjecture using two inversion methods we compute a one dimensional potential whose lowest n eigenvalues obey random matrix statistics our numerical results indicate that in the asymptotic limit n yields infinity the solution is nowhere differentiable and most probably nowhere continuous thus such a counterexample does not exist

riemann zeros and a fractal potential the nontrivial riemann zeroes are reproduced using a one dimensional local potential model a close look at the potential suggests that it has a fractal structure of dimension d 1 5

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

moments of the wigner delay times the wigner time delay is a measure of the time spent by a particle inside the scattering region of an open system for chaotic systems the statistics of the individual delay times whose average is the wigner time delay are thought to be well described by random matrix theory here we present a semiclassical derivation showing the validity of random matrix results in order to simplify the semiclassical treatment we express the moments of the delay times in terms of correlation functions of scattering matrices at different energies in the semiclassical approximation the elements of the scattering matrix are given in terms of the classical scattering trajectories requiring one to study correlations between sets of such trajectories we describe the structure of correlated sets of trajectories and formulate the rules for their evaluation to the leading order in inverse channel number this allows us to derive a polynomial equation satisfied by the generating function of the moments along with showing the agreement of our semiclassical results with the moments predicted by random matrix theory we infer that the scattering matrix is unitary to all orders in the semiclassical approximation

a family of measures for best top n class selective decision rules when classes strongly overlap in the feature space or when some classes are not known in advance the performance of a classifier heavily decreases to overcome this problem the reject option has been introduced it simply consists in withdrawing the decision and let another classifier or an expert take the decision whenever exclusively classifying is not reliable enough the classification problem is then a matter of class selection from none to all classes in this paper we propose a family of measures suitable to define such decision rules it is based on a new family of operators that are able to detect blocks of similar values within a set of numbers in the unit interval the soft labels of an incoming pattern to be classified using a single threshold experiments on synthetic and real datasets available in the public domain show the efficiency of our approach

periodic orbits and a correlation function for the semiclassical density of states through the rule that there is one quantum state per planck cell in phase space classical mechanics supplies an average semiclassical density of states for a system with bound motion can it supply more refined averages such as the correlation function for the density of states i shall argue that for the two extreme cases of integrable classical motion and ergodic classical motion information can indeed be obtained on not quite this but a closely related correlation function the results for the two types of motion are markedly different

semiclassical form factor for spectral and matrix element fluctuations of multidimensional chaotic systems we present a semiclassical calculation of the generalized form factor kab tau which characterizes the fluctuations of matrix elements of the operators a and b in the eigenbasis of the hamiltonian of a chaotic system our approach is based on some recently developed techniques for the spectral form factor of systems with hyperbolic and ergodic underlying classical dynamics and f 2 degrees of freedom that allow us to go beyond the diagonal approximation first we extend these techniques to systems with f 2 then we use these results to calculate kab tau we show that the dependence on the rescaled time tau time in units of the heisenberg time is universal for both the spectral and the generalized form factor furthermore we derive a relation between kab tau and the classical time correlation function of the weyl symbols of a and b