monte carlo studies of three dimensional two step restricted self avoiding walks two step restricted walk tsrw models are a class of restricted self avoiding walk saw where in addition to the self avoidance constraint certain restrictions are placed upon each pair of successive steps in this paper we explore the relationship between the restrictions and the scaling of the average size of walks in three dimensional models we use the pruned enriched rosenbluth method algorithm to perform monte carlo studies in five representative tsrw models in three dimensions the results present strong numerical evidence that all non trivial tsrw models in three dimensions have the same size scaling behaviour as unrestricted saws this is in contrast to two dimensions where several universality classes are accepted to exist in particular we find no rule analogous to the spiral walk of two dimensions

frequency operator for anharmonic oscillators by perturbation theory we obtain approximate analytical expressions for the frequency operator for anharmonic oscillators by means of perturbation theory and a unitary transformation

fractal weyl laws in discrete models of chaotic scattering we analyse simple models of quantum chaotic scattering namely quantized open baker s maps we numerically compute the density of quantum resonances in the semiclassical regime this density satisfies a fractal weyl law where the exponent is governed by the fractal dimension of the set of trapped trajectories this type of behaviour is also expected in the physically more relevant case of hamiltonian chaotic scattering within a simplified model we are able to rigorously prove this weyl law and compute quantities related to the coherent transport through the system namely the conductance and shot noise the latter is close to the prediction of random matrix theory

lie symmetries of multidimensional difference equations a method is presented for calculating the lie point symmetries of a scalar difference equation on a two dimensional lattice the symmetry transformations act on the equations and on the lattice they take solutions into solutions and can be used to perform symmetry reduction the method generalizes the one presented in a recent publication for the case of ordinary difference equations in turn it can easily be generalized to difference systems involving an arbitrary number of dependent and independent variables

a recurrent probabilistic neural network with dimensionality reduction based on time series discriminant component analysis this paper proposes a probabilistic neural network nn developed on the basis of time series discriminant component analysis tsdca that can be used to classify high dimensional time series patterns tsdca involves the compression of high dimensional time series into a lower dimensional space using a set of orthogonal transformations and the calculation of posterior probabilities based on a continuous density hidden markov model with a gaussian mixture model expressed in the reduced dimensional space the analysis can be incorporated into an nn which is named a time series discriminant component network tsdcn so that parameters of dimensionality reduction and classification can be obtained simultaneously as network coefficients according to a backpropagation through time based learning algorithm with the lagrange multiplier method the tsdcn is considered to enable high accuracy classification of high dimensional time series patterns and to reduce the computation time taken for network training the validity of the tsdcn is demonstrated for high dimensional artificial data and electroencephalogram signals in the experiments conducted during the study

categorical entropy for fourier mukai transforms on generic abelian surfaces abstract in this note we shall compute the categorical entropy of an autoequivalence on a generic abelian surface

real space renormalization group approach to driven diffusive systems we introduce a real space renormalization group procedure for driven diffusive systems which predicts both steady state and dynamic properties we apply the method to the boundary driven asymmetric simple exclusion process and recover exact results for the steady state phase diagram as well as the crossovers in the relaxation dynamics for each phase

evaluation of the coulomb force via the fredholm integral equation the electrostatic force between two conducting spheres is solved exactly from coulomb s law the force is evaluated in two steps i the charge distribution on each of the spheres is calculated as a function of separation distance by way of a solution to the fredholm integral equation ii the effect of the known charge distribution is integrated to obtain the electrostatic force evaluation of the electrostatic force is fast because the series expression for the charge distribution is super converging a characteristic trait for some fredholm type equations

random matrix ensembles with column row constraints ii we analyze statistical properties of a complex system subjected to conditions which manifests through specific constraints on the column row sum of the matrix elements of its hermitian operators the presence of additional constraints besides real symmetric nature leads to new correlations among their eigenfunctions hinders a complete delocalization of dynamics and affects the eigenvalues too the statistical analysis of the latter indicates the presence of a new universality class analogous to that of a special type of brownian ensemble appearing between poisson and gaussian orthogonal ensemble

distributions for the eigenvalues of large random matrices generated from four manifolds in this paper we will first study the spectrum of certain large euclidean random matrices the entries of these matrices are functions of n random points lp norm uniformly sampled in n dimensional lp ball or lp sphere under the setting n n 0 as n and n tend to it is shown that the empirical distributions of eigenvalues of normalized euclidean random matrices generated from the above two manifolds both converge weakly to semicircle law we also investigate certain sample covariance type matrices whose entries are determined by n random positions from an n dimensional lp ellipsoid or its surface when n and n with n n 0 we prove that their empirical spectral distribution converge to the same fixed distribution

toward bayes optimal linear dimension reduction dimension reduction is the process of transforming multidimensional vectors into a low dimensional space in pattern recognition it is often desired that this task be performed without significant loss of classification information the bayes error is an ideal criterion for this purpose however it is known to be notoriously difficult for mathematical treatment consequently suboptimal criteria have been used in practice we propose an alternative criterion based on the estimate of the bayes error that is hopefully closer to the optimal criterion than the criteria currently in use an algorithm for linear dimension reduction based on this criterion is conceived and implemented experiments demonstrate its superior performance in comparison with conventional algorithms

curvilinear component analysis a self organizing neural network for nonlinear mapping of data sets we present a new strategy called curvilinear component analysis cca for dimensionality reduction and representation of multidimensional data sets the principle of cca is a self organized neural network performing two tasks vector quantization vq of the submanifold in the data set input space and nonlinear projection p of these quantizing vectors toward an output space providing a revealing unfolding of the submanifold after learning the network has the ability to continuously map any new point from one space into another forward mapping of new points in the input space or backward mapping of an arbitrary position in the output space

motions in a bose condensate x new results on the stability of axisymmetric solitary waves of the gross pitaevskii equation the stability of the axisymmetric solitary waves of the gross pitaevskii gp equation is investigated the implicitly restarted arnoldi method for banded matrices with shift invert is used to solve the linearized spectral stability problem the rarefaction solitary waves on the upper branch of the jones roberts dispersion curve are shown to be unstable to axisymmetric infinitesimal perturbations whereas the solitary waves on the lower branch and all two dimensional solitary waves are linearly stable the growth rates of the instabilities on the upper branch are so small that an arbitrarily specified initial perturbation of a rarefaction wave at first usually evolves towards the upper branch as it acoustically radiates away its excess energy this is demonstrated through numerical integrations of the gp equation starting from an initial state consisting of an unstable rarefaction wave and random non axisymmetric noise the resulting solution evolves towards and remains for a significant time in the vicinity of an unperturbed unstable rarefaction wave it is shown however that ultimately or for an initial state extremely close to the upper branch the solution evolves onto the lower branch or is completely dissipated as sound it is shown how density depletions in uniform and trapped condensates can generate rarefaction waves and a simple method is suggested by which such waves can be created in the laboratory

gauge transformation of the third kind for u 1 invariant coupled schrodinger equations we consider a large class of canonical coupled nonlinear schrodinger equations invariant over the action of the u 1 group the most general nonlinearity is taken into account through a matrix that without loss of generality can be separated into the sum of a hermitian matrix and an anti hermitian matrix the u 1 symmetry implies the existence of a set of continuity equations for the conserved densities where the corresponding currents have in general a nonlinear structure for this class of coupled schrodinger equations we introduce a nonlinear gauge transformation which changes the nonlinear matrix into another one purely hermitian consequently the currents are transformed in the standard bilinear form generalization to noncanonical systems is also discussed some examples are presented to illustrate the applicability of the method

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

semiclassical expansions in the toda hierarchy and the hermitian matrix model an iterative algorithm for determining a class of solutions of the dispersionful 2 toda hierarchy characterized by string equations is developed this class includes the solution which underlies the large n limit of the hermitian matrix model in the one cut case it is also shown how the double scaling limit can be naturally formulated in this scheme

two matrix model with semiclassical potentials and extended whitham hierarchy we consider the two matrix model with potentials whose derivatives are arbitrary rational functions of fixed pole structure and the support of the spectra of the matrices are union of intervals hard edges we derive an explicit formula for the planar limit of the free energy and we derive a calculus which allows us to compute derivatives of arbitrarily high order by extending classical rauch s variational formulae the four point correlation functions are explicitly worked out the formalism extends naturally to the computation of residue formulae for the tau function of the so called universal whitham hierarchy studied mainly by i krichever our setting extends the moduli space in that there are certain extra data

bayes optimality motivated linear and multilayered perceptron based dimensionality reduction dimensionality reduction is the process of mapping high dimension patterns to a lower dimension subspace when done prior to classification estimates obtained in the lower dimension subspace are more reliable for some classifiers there is also an improvement in performance due to the removal of the diluting effect of redundant information a majority of the present approaches to dimensionality reduction are based on scatter matrices or other statistics of the data which do not directly correlate to classification accuracy the optimality criteria of choice for the purposes of classification is the bayes error usually however bayes error is difficult to express analytically we propose an optimality criteria based on an approximation of the bayes error and use it to formulate a linear and a nonlinear method of dimensionality reduction the nonlinear method we propose relies on using a multilayered perceptron which produces as output the lower dimensional representation it thus differs from autoassociative like multilayered perceptrons which have been proposed and used for dimensionality reduction our results show that the nonlinear method is as anticipated superior to the linear method in that it can perform unfolding of a nonlinear manifold in addition the nonlinear method we propose provides substantially better lower dimension representation for classification purposes than fisher s linear discriminant fld and two other nonlinear methods of dimensionality reduction that are often used

the lattice schwarzian kdv equation and its symmetries in this paper we present a set of results on the symmetries of the lattice schwarzian korteweg de vries 1skdv equation we construct the lie point symmetries and using its associated spectral problem an infinite sequence of generalized symmetries and master symmetries we finally show that we can use master symmetries of the 1skdv equation to construct non autonomous non integrable generalized symmetries

the quantum probability equation iii perturbing in the energy time independent perturbation theory in the scattering domain developed on the basis of the quantum probability equation milward g c and wilkin c 2001 j phys a math gen 34 5101 is extended to study the energy dependence of the s wave scattering phase shifts and probability density to second order in the energy change at low energies the bethe formula for the effective range is reproduced and an analogous formula for the shape dependent term is also derived the energy dependence of the probability density at short distances is important in final state interaction theory and the method allows the investigation of some of the prescriptions used there results from the present approach are illustrated with soluble models

partition function for multi cut matrix models we consider the partition function z n of a random matrix model with polynomial potential v t 1 t 2 2 t 2d 2d it is known that the second logarithmic derivative of z n with respect to the times t k can be expressed in terms of the recurrence coefficients of the related orthogonal polynomials an explicit formula for the recurrence coefficients of the orthogonal polynomials in the limit n oo for multi cut regular v has been derived in 10 through the riemann hilbert approach the expression for z n in the limit n has been derived in 7 through a mean field approach we show that the above asymptotic formulae satisfy the same relations that hold for finite n

uncertainty relations in deformation quantization robertson and hadamard robertson theorems on non negative definite hermitian forms are generalized to an arbitrary ordered field these results are then applied to the case of formal power series fields and the heisenberg robertson robertson schrodinger and trace uncertainty relations in deformation quantization are found some conditions under which the uncertainty relations are minimized are also given

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

random matrices close to hermitian or unitary overview of methods and results the paper discusses recent progress in understanding statistical properties of eigenvalues of weakly non hermitian and non unitary random matrices the first type of ensembles is of the form i with being a large random n n hermitian matrix with independent entries deformed by a certain anti hermitian n n matrix i satisfying in the limit of large dimension n the condition tr 2 n tr 2 here can be either a random or just a fixed given hermitian matrix ensembles of such a type with 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics related models are used to mimic complex spectra of the dirac operator with chemical potential in the context of quantum chromodynamics ensembles of the second type arising naturally in scattering theory of discrete time systems are formed by n n matrices with complex entries such that i for 0 this coincides with the circular unitary ensemble and 0 i describes deviation from unitarity our result amounts to answering statistically the following old question given the singular values of a matrix describe the locus of its eigenvalues we systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non trivial crossover from wigner dyson statistics of real unimodular eigenvalues typical of hermitian unitary matrices to ginibre statistics in the complex plane typical of ensembles with strong non hermiticity tr 2 tr 2 when n finally we discuss scarce results available on eigenvector statistics for weakly non hermitian random matrices

a unified fluctuation formula for one cut β ensembles of random matrices using a coulomb gas approach we compute the generating function of the covariances of power traces for one cut ensembles of random matrices in the limit of large matrix size this formula depends only on the support of the spectral density and is therefore universal for a large class of models this allows us to derive a closed form expression for the limiting covariances of an arbitrary one cut ensemble as particular cases of the main result we consider the classical gaussian wishart and jacobi ensembles for which we derive previously available results as well as new ones within a unified simple framework we also discuss the connections between the problem of trace fluctuations for the gaussian unitary ensemble and the enumeration of planar maps

reply to comment on integrable kondo impurity in one dimensional q deformed t j models this is a reply to the comment by p schlottmann and a a zvyagin pacs numbers 71 10 fd 71 27 75 10 jm in their comment 1 schlottmann and zvyagin raise several issues regarding the nature of integrable impurities in one dimensional quantum lattice models and claim to expose false statements in our recent paper 2 in order to address these issues in a pedagogical manner we feel that it is appropriate to discuss these questions in terms of models based on the gl 2 1 invariant solution of the yang baxter equation however it is important from the outset to make it clear that the arguments we present below are general and apply to other classes of models the first point that we would like to make is that it is claimed in 1 that there are two approaches to the algebraic bethe ansatz however it appears to us that approach i described in point 2 of 1 is the coordinate bethe ansatz and we do not understand why this should be referred to as an algebraic bethe ansatz in the coordinate bethe ansatz approach one starts with a prescribed hamiltonian and then solves the schrodinger equation to obtain the two particle scattering matrices and the particle impurity scattering matrix together these scattering matrices form the monodromy matrix it is impossible to infer the hamiltonian from such a monodromy matrix in this context we do not feel that point 1 of 1 answers our query about the existence of the impurity monodromy matrix in the algebraic approach ii of point 2 in 1 hereafter we focus our attention on this case the solution of the yang baxter equation r12 u v r13 u r23 v r23 v r13 u r12 u v 1 associated with the lie superalgebra gl 2 1 is an operator r u end v v where v is a three dimensional z2 graded space with one bosonic and two fermionic degrees of freedom explicitly this operator takes the form r u u i i p 2 where p is the z2 graded permutation operator for the purposes of constructing integrable one dimensional quantum systems on a closed lattice it is usual to introduce the yang baxter 0305 4470 02 296197 05 30 00 2002 iop publishing ltd printed in the uk 6197

an integrating factor matrix method to find first integrals in this paper we develop an integrating factor matrix method to derive conditions for the existence of first integrals we use this novel method to obtain first integrals along with the conditions for their existence for two and three dimensional lotka volterra systems with constant terms the results are compared to previous results obtained by other methods

resurgence in quasi classical scattering in quasi classical studies of closed systems e g a billiard resurgence means that the contribution of long periodic orbits to the spectral determinant can be expressed in terms of composites of short orbits and the resulting expression for the determinant is manifestly real the question has thus long been posed whether something like resurgence applies to a scattering system with its resonances we find here a resurgent expression for wigner s r matrix which gives the s matrix by a cayley transform in which long scattering pseudo orbits are expressed in terms of composites of short pseudo orbits both scattering and periodic and the result is manifestly hermitian giving a unitary expression for the s matrix this is particularly useful in the case that the resonance width is comparable with the resonance spacing the pseudo orbits are defined in terms of a fictitious and to some extent arbitrary closed reference system we relate the results to other formulations we give a simple but non trivial approximation for a particular example which illustrates the phenomenon of resonance trapping

replica equivalence in the edwards anderson model after introducing and discussing the link overlap between spin configurations we show that the edwards anderson model has a replica equivalent quenched equilibrium state a property introduced by parisi in the description of the mean field spin glass phase which generalizes ultrametricity our method is based on the control of fluctuations through the property of stochastic stability and works for all the finite dimensional spin glass models

eigenvalues distribution for products of independent spherical ensembles we consider the product of independent spherical ensembles by the special structure of eigenvalues as a rotation invariant determinant point process we show that the empirical spectral distribution of the product converges with probability one to a non random distribution and the limiting eigenvalue distribution is a power of spherical law we also present an interesting correspondence between the eigenvalues of three classes of random matrix ensembles and zeros of random polynomials

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

the diamagnetic coulomb problem an eigenvalue problem with two singularities by using the angular oblate spheroidal functions as basis functions a bounded wavefunction is constructed in the singularities of the schrodinger equation of the diamagnetic coulomb problem with infinite nuclear mass the expansion in terms of these functions is a model to resolve singularities in an eigenvalue problem of non separable partial differential equations of non relativistic quantum mechanics a comprehensive asymptotic analysis reveals the complete set of asymptotic solutions makes possible a uniform numerical treatment of the bound autoionizing continuum and continuum levels and indicates how to find hitherto unknown low lying stationary levels an example the splitting of the ground level has been found numerically by an iterative shooting method

fractional step dimensionality reduction linear projections for dimensionality reduction computed using linear discriminant analysis lda are commonly based on optimization of certain separability criteria in the output space the resulting optimization problem is linear but these separability criteria are not directly related to the classification accuracy in the output space consequently a trial and error procedure has to be invoked experimenting with different separability criteria that differ in the weighting function used and selecting the one that performed best on the training set often even the best weighting function among the trial choices results in poor classification of data in the subspace in this short paper we introduce the concept of fractional dimensionality and develop an incremental procedure called the fractional step lda f lda to reduce the dimensionality in fractional steps the f lda algorithm is more robust to the selection of weighting function and for any given weighting function it finds a subspace in which the classification accuracy is higher than that obtained using lda

on the integrability of a new discrete nonlinear schrodinger equation we consider the nonlinear schrodinger equation on the lattice introduced by leon and manna two years ago to describe the slowly varying envelope approximation of some nonlinear differential difference equations onlinear klein gordon chain etc we show that this equation does not admit local generalized symmetries of order greater than three in such a way we prove that the leon and manna discrete nonlinear schrodinger equation does not have the same integrability properties as the toda lattice equation from which it has been derived at the end we provide some reasoning to justify the result obtained

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

continuous symmetries of the lattice potential kdv equation in this paper we present a set of results on the integration and on the symmetries of the lattice potential korteweg de vries lpkdv equation using its associated spectral problem we construct the soliton solutions and the lax technique enables us to provide infinite sequences of generalized symmetries finally using a discrete symmetry of the lpkdv equation we construct a large class of non autonomous generalized symmetries

asymptotics of finite system lyapunov exponents for some random matrix ensembles for products pn of n random matrices of size there is a natural notion of finite n lyapunov exponents in the case of standard gaussian random matrices with real complex or real quaternion elements and extended to the general variance case for methods known for the computation of are used to compute the large n form of the variances of the exponents analogous calculations are performed in the case that the matrices making up pn are products of sub blocks of random unitary matrices with haar measure furthermore we make some remarks relating to the coincidence of the lyapunov exponents and the stability exponents relating to the eigenvalues of pn

complex curve of the two matrix model and its tau function we study the hermitian and normal two matrix models in planar approximation for an arbitrary number of eigenvalue supports its planar graph interpretation is given the study reveals a general structure of the underlying analytic complex curve different from the hyperelliptic curve of the one matrix model the matrix model quantities are expressed through the periods of meromorphic generating differential on this curve and the partition function of the multiple support solution as a function of filling numbers and coefficients of the matrix potential is shown to be a quasiclassical tau function the relation to 1 supersymmetric yang mills theories is discussed a general class of solvable multi matrix models with tree like interactions is considered

matrix models with hard walls geometry and solutions we discuss various aspects of most general multisupport solutions to matrix models in the presence of hard walls i e in the case where the eigenvalue support is confined to subdomains of the real axis the structure of the solution at the leading order is described by semiclassical or generalized whitham krichever hierarchies as in the unrestricted case derivatives of tau functions for these solutions are associated with families of riemann surfaces with possible double points and satisfy the witten dijkgraaf verlinde verlinde equations we then develop the diagrammatic technique for finding free energy of this model in all orders of the t hooft expansion in the reciprocal matrix size generalizing the feynman diagrammatic technique for the hermitian one matrix model due to eynard