reduced notation inner plethysms and the symmetric group the reduced notation for irreducible representations of the symmetric group s is interpreted in terms of symmetric formal series and vertex operators and is used to prove a number of properties of reduced kronecker products and inner plethysms in an n independent manner conditions for self associativity of kronecker products and inner plethysms are established reduced inner plethysms are developed and applied to the question of non simple phase groups among the symmetric sn and alternating an groups

applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

a generalized kernel approach to dissimilarity based classification usually objects to be classified are represented by features in this paper we discuss an alternative object representation based on dissimilarity values if such distances separate the classes well the nearest neighbor method offers a good solution however dissimilarities used in practice are usually far from ideal and the performance of the nearest neighbor rule suffers from its sensitivity to noisy examples we show that other more global classification techniques are preferable to the nearest neighbor rule in such cases for classification purposes two different ways of using generalized dissimilarity kernels are considered in the first one distances are isometrically embedded in a pseudo euclidean space and the classification task is performed there in the second approach classifiers are built directly on distance kernels both approaches are described theoretically and then compared using experiments with different dissimilarity measures and datasets including degraded data simulating the problem of missing values

extinction in population dynamics we study a generic reaction diffusion model for single species population dynamics that includes reproduction death and competition the population is assumed to be confined in a refuge beyond which conditions are so harsh that they lead to certain extinction standard continuum mean field models in one dimension yield a critical refuge length l c such that a population in a refuge larger than this is assured survival herein we extend the model to take into account the discreteness and finiteness of the population which leads us to a stochastic description we present a particular critical criterion for likely extinction namely that the standard deviation of the population be equal to the mean according to this criterion we find that while survival can no longer be guaranteed for any refuge size for sufficiently weak competition one can make the refuge large enough certainly larger than l c to cause extinction to be unlikely however beyond a certain value of the competition rate parameter it is no longer possible to escape a likelihood of extinction even in an infinite refuge these unavoidable fluctuations therefore have a severe impact on refuge design issues

generalized quantum onsager coefficients from a symmetrized campbell hausdorff expansion the analogue of the classical onsager theory of entropy production is systematically derived for weakly irreversible processes in open quantum systems with finite dimensional hilbert space the dynamics is assumed to be given by a quantum dynamical semigroup with infinitesimal generator of gorini kossakowski sudarshan type the basic spohn formula for entropy production is used to obtain an expansion in terms of powers of the deviation of the initial state relative to the final stationary state of irreversible dynamics to this end an appropriate lie series is constructed from a particular symmetrization procedure applied to the ordinary campbell hausdorff expansion in this way only hermitian contributions by higher order commutators are generated which allow an identification with so called generalized onsager coefficients the explicit derivations concentrate on second third and fourth order coefficients whereas complete detailed expressions are worked out for second and third order in a suitable coherence vector representation of density matrices the results can be given in terms of the dynamical parameters fixing the infinitesimal semigroup generator and in terms of symmetric and antisymmetric structure constants of the lie algebra of su n as an illustration an application to generalized bloch equations for two level systems is studied where the onsager like expansion can be compared with exact results for entropy production we find that convergence is good even for rather large deviations between initial and final state if the calculation includes second and third order coefficients only the formalism presented in this paper generalizes restrictions on admitted final states adopted in much simpler earlier treatments to the most general case of arbitrary unique final states of irreversible processes

topographic mapping of large dissimilarity data sets topographic maps such as the self organizing map som or neural gas ng constitute powerful data mining techniques that allow simultaneously clustering data and inferring their topological structure such that additional features for example browsing become available both methods have been introduced for vectorial data sets they require a classical feature encoding of information often data are available in the form of pairwise distances only such as arise from a kernel matrix a graph or some general dissimilarity measure in such cases ng and som cannot be applied directly in this article we introduce relational topographic maps as an extension of relational clustering algorithms which offer prototype based representations of dissimilarity data to incorporate neighborhood structure these methods are equivalent to the standard vectorial techniques if a euclidean embedding exists while preventing the need to explicitly compute such an embedding extending these techniques for the general case of non euclidean dissimilarities makes possible an interpretation of relational clustering as clustering in pseudo euclidean space we compare the methods to well known clustering methods for proximity data based on deterministic annealing and discuss how far convergence can be guaranteed in the general case relational clustering is quadratic in the number of data points which makes the algorithms infeasible for huge data sets we propose an approximate patch version of relational clustering that runs in linear time the effectiveness of the methods is demonstrated in a number of examples

asymptotic form of the approach to equilibrium in reversible recombination reactions the reversible reactions and are investigated from the exact langevin equations describing our model we set up a systematic approximation scheme to compute the approach of the density of c particles to its equilibrium value we show that for a sufficiently long time t this approach takes the form of a power law for any dimension d the amplitude a is also computed exactly but is expected to be model dependent for uncorrelated initial conditions the c density turns out to be a monotonic time function the cases of correlated initial conditions and unequal diffusion constants are investigated as well in the former correlations may break the monotonicity of the density or in some special cases they may change the long time behaviour for the latter the power law remains valid only the amplitude changes even in the extreme case of immobile c particles we also consider the case of segregated initial condition for which a reaction front is observed and confirm that its width is governed by mean field exponent in any dimension

vertex operators and symmetric functions an algorithm for the calculation of q dependent spin characters of the symmetric group is given with an explicit example of s4 a method for constructing the q analogue of the vertex operators is developed a 1 1 correspondence between the space v of twisted q vertex operators and the ring of q deformed symmetric functions lambda x zq q t is established and a mapping from v to lambda x zq q t is defined a number of relevant theorems are given

feature discovery in non metric pairwise data pairwise proximity data given as similarity or dissimilarity matrix can violate metricity this occurs either due to noise fallible estimates or due to intrinsic non metric features such as they arise from human judgments so far the problem of non metric pairwise data has been tackled by essentially omitting the negative eigenvalues or shifting the spectrum of the associated pseudo covariance matrix for a subsequent embedding however little attention has been paid to the negative part of the spectrum itself in particular no answer was given to whether the directions associated to the negative eigenvalues would at all code variance other than noise related we show by a simple exploratory analysis that the negative eigenvalues can code for relevant structure in the data thus leading to the discovery of new features which were lost by conventional data analysis techniques the information hidden in the negative eigenvalue part of the spectrum is illustrated and discussed for three data sets namely usps handwritten digits text mining and data from cognitive psychology

discrete differential forms for 1 1 dimensional cosmological space times in this article we describe applications of the numerical method of discrete differential forms in computational general relativity gr in particular we consider the initial value problem for vacuum space times that admit plane gravitational waves as described in an earlier paper the discrete differential form approach provides accurate results in spherically symmetric static space times r richter j frauendiener and m vogel classical quantum gravity 24 2007 p 433 and it is manifestly coordinate independent here we extend the method to time dependent systems we use the polarized gowdy solution as a testbed for two numerical schemes one scheme reproduces that solution very well in particular it is stable for a long time and converges quadratically

compact exponential product formulas and operator functional derivative a new scheme for deriving compact expressions of the logarithm of the exponential product is proposed and it is applied to several exponential product formulas a generalization of the dynkin specht wever dsw theorem on free lie elements is given and it is used to study the relation between the traditional method based on the dsw theorem and the present new scheme the concept of the operator functional derivative is also proposed and it is applied to ordered exponentials such as time evolution operators for time dependent hamiltonians

the q deformation of symmetric functions and the symmetric group the q deformation of symmetric functions is introduced leading to q analogues of many well known relationships in the theory of symmetric functions q deformed scalar products are developed and used to define q dependent symmetric functions the symmetric functions commonly associated with the names hall littlewood schur and jack are all special cases of the q deformation of macdonald s 1988 new symmetric functions plambda s t a q analogue of the spin and ordinary characters of sn is given and illustrated by the explicit calculation of examples of q deformed characters the methods used are closely parallel to those of quantum groups

indefinite proximity learning a review efficient learning of a data analysis task strongly depends on the data representation most methods rely on symmetric similarity or dissimilarity representations by means of metric inner products or distances providing easy access to powerful mathematical formalisms like kernel or branch and bound approaches similarities and dissimilarities are however often naturally obtained by nonmetric proximity measures that cannot easily be handled by classical learning algorithms major efforts have been undertaken to provide approaches that can either directly be used for such data or to make standard methods available for these types of data we provide a comprehensive survey for the field of learning with nonmetric proximities first we introduce the formalism used in nonmetric spaces and motivate specific treatments for nonmetric proximity data second we provide a systematization of the various approaches for each category of approaches we provide a comparative discussion of the individual algorithms and address complexity issues and generalization properties in a summarizing section we provide a larger experimental study for the majority of the algorithms on standard data sets we also address the problem of large scale proximity learning which is often overlooked in this context and of major importance to make the method relevant in practice the algorithms we discuss are in general applicable for proximity based clustering one class classification classification regression and embedding approaches in the experimental part we focus on classification tasks

a renormalization group study of a class of reaction diffusion models with particles input we study a class of reaction diffusion models extrapolating continuously between the pure coagulation diffusion case and the pure annihilation diffusion one with particles input at a rate j for dimension the dynamics strongly depends on the fluctuations while for d 2 the behaviour is mean field like the models are mapped onto a field theory whose properties are studied in a renormalization group approach simple relations are found between the time dependent correlation functions of the different models of the class for the pure coagulation diffusion model the time dependent density is found to be of the form where d is the diffusion constant the critical exponent and are computed to all orders in where d is the dimension of the system while the scaling function is computed to second order in for the one dimensional case an exact analytical solution is provided whose predictions are compared with the results of the renormalization group approach for

higher order operator splitting methods via zassenhaus product formula theory and applications abstract in this paper we contribute higher order operator splitting methods improved by zassenhaus product we apply the contribution to classical and iterative splitting methods the underlying analysis to obtain higher order operator splitting methods is presented while applying the methods to partial differential equations the benefits of balancing time and spatial scales are discussed to accelerate the methods the verification of the improved splitting methods are done with numerical examples an individual handling of each operator with adapted standard higher order time integrators is discussed finally we conclude the higher order operator splitting methods

variational and potential formulation for stochastic partial differential equations recently there has been interest in finding a potential formulation for stochastic partial differential equations spdes the rationale behind this idea lies in obtaining all the dynamical information of the system under study from one single expression in this letter we formally provide a general lagrangian formalism for spdes using the hojman et al method we show that it is possible to write the corresponding effective potential starting from an s equivalent lagrangian and that this potential is able to reproduce all the dynamics of the system once a special differential operator has been applied this procedure can be used to study the complete time evolution and spatial inhomogeneities of the system under consideration and is also suitable for the statistical mechanics description of the problem

stability based validation of clustering solutions data clustering describes a set of frequently employed techniques in exploratory data analysis to extract natural group structure in data such groupings need to be validated to separate the signal in the data from spurious structure in this context finding an appropriate number of clusters is a particularly important model selection question we introduce a measure of cluster stability to assess the validity of a cluster model this stability measure quantifies the reproducibility of clustering solutions on a second sample and it can be interpreted as a classification risk with regard to class labels produced by a clustering algorithm the preferred number of clusters is determined by minimizing this classification risk as a function of the number of clusters convincing results are achieved on simulated as well as gene expression data sets comparisons to other methods demonstrate the competitive performance of our method and its suitability as a general validation tool for clustering solutions in real world problems

classification with nonmetric distances image retrieval and class representation a key problem in appearance based vision is understanding how to use a set of labeled images to classify new images systems that model human performance or that use robust image matching methods often use nonmetric similarity judgments but when the triangle inequality is not obeyed most pattern recognition techniques are not applicable exemplar based nearest neighbor methods can be applied to a wide class of nonmetric similarity functions the key issue however is to find methods for choosing good representatives of a class that accurately characterize it we show that existing condensing techniques are ill suited to deal with nonmetric dataspaces we develop techniques for solving this problem emphasizing two points first we show that the distance between images is not a good measure of how well one image can represent another in nonmetric spaces instead we use the vector correlation between the distances from each image to other previously seen images second we show that in nonmetric spaces boundary points are less significant for capturing the structure of a class than in euclidean spaces we suggest that atypical points may be more important in describing classes we demonstrate the importance of these ideas to learning that generalizes from experience by improving performance we also suggest ways of applying parametric techniques to supervised learning problems that involve a specific nonmetric distance functions showing how to generalize the idea of linear discriminant functions in a way that may be more useful in nonmetric spaces

vertex operators and composite supersymmetric s functions the s function basis for the fock space of vertex operator constructions is considered where states are associated with partitions mod lambda the authors show that the matrix element lambda mod valpha z mod mu is essentially given by the composite s function slambda mu z where the argument z identical to z 1 is replicated mod alpha mod times matrix elements of products of vertex operators lambda mod valpha z v beta w mod mu are given up to functions of z w independent of lambda and mu by the composite supersymmetric s function slambda mu z w where the z w are replicated mod alpha mod mod beta mod times the supersymmetric arguments being those associated with negative roots

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

basic representations of a2l 2 and dl 1 2 and the polynomial solutions to the reduced bkp hierarchies basic representations of a2l 2 and dl 1 2 are studied the weight vectors are represented in terms of schur s q functions the method to get the polynomial solutions to the reduced bkp hierarchies is shown to be equivalent to a certain rule in the maya game

exponential operators and the algebraic description of quantum confined systems we study the relations and transformations produced by exponential operators the argument of which are dependent on the basic algebraic elements of supersymmetric and shape invariant potential systems and obtain explicit expressions we apply our results to self similar potential systems and to a set of translational shape invariant systems including the morse poschl teller scarf and rosen morse potentials and obtain closed form expressions we show that our results reproduce those obtained for the harmonic oscillator in the appropriate limits

maximal reductions in the baker hausdorff formula a preliminary expression for the baker hausdorff formula is found up to ninth order i e a series expansion of z in terms of multiple commutators where ez exey with x and y noncommuting up to ninth degree in x y by means of complete sets of linear relations between multiple commutators maximal reduction of the number of different multiple commutators in the series are obtained

renormalization group study of thea b diffusion limited reaction thea b diffusion limited reaction with equal initial densitiesa 0 b 0 n0 is studied by means of a field theoretic renormalization group formulation of the problem for dimensiond 2 an effective theory is derived from which the density and correlation functions call be calculated we find the density decays in time as a b sim c sqrt delta dt d 4 ford 4 with n0 c n 0 d 2 wherec is a universal constant andc is nonuniversal the calculation is extended to the case of unequal diffusion constantsd a d b resulting in a new amplitude but the same exponent ford 2 a controlled calculation is not possible but a heuristic argument is presented that the results above give at least the leading term in an e 2 d expansion finally we address reaction zones formed in the steady state by opposing currents ofa andb particles and derive scaling properties

a note on the zassenhaus product formula we provide a simple method for the calculation of the terms cn in the zassenhaus product ea b ea eb n 2 ecn for noncommuting a and b this method has been implemented in a computer program furthermore we formulate a conjecture on how to translate these results into nested commutators this conjecture was checked up to order n 17 using a computer

quantum analysis non commutative differential and integral calculi a new scheme of quantum analysis namely a non commutative calculus of operator derivatives and integrals is introduced this treats differentiation of an operator valued function with respect to the relevant operator in a banach space in this new scheme operator derivatives are expressed in terms of the relevant operator and its inner derivation explicitly derivatives of hyperoperators are also defined some possible applications of the present calculus to quantum statistical physics are briefly discussed

a new type of kinetic critical phenomenon the authors study a new critical phenomenon in an non thermal one dimensional lattice model it is characterised by the transition from stability to instability of kinks between ordered states below the critical point the kinks are stable and move by annihilating random walks above the critical point they are unstable against creation of kink antikink pairs the spontaneous production of pairs is assumed to be absent at the critical point p pcr the density of kinks decreases like n approximately talpha with alpha 0 27 or 0 08 above the critical point the density of kinks in the stationary state is approximately p pcr beta with beta 0 06 or 0 2 possible extensions to two or more dimensions and possible applications are discussed

generative models for similarity based classification a maximum entropy approach to generative similarity based classifiers model is proposed first a descriptive set of similarity statistics is assumed to be sufficient for classification then the class conditional distributions of these descriptive statistics are estimated as the maximum entropy distributions subject to empirical moment constraints the resulting exponential class conditional distributions are used in a maximum a posteriori decision rule forming the similarity discriminant analysis sda classifier simulated and real data experiments compare performance to the k nearest neighbor classifier the nearest centroid classifier and the potential support vector machine psvm

on the convergence and optimization of the baker campbell hausdorff formula abstract in this paper the problem of the convergence of the baker campbell hausdorff series for z log e x e y is revisited we collect some previous results about the convergence domain and present a new estimate which improves all of them we also provide a new expression of the truncated lie presentation of the series up to sixth degree in x and y requiring the minimum number of commutators numerical experiments suggest that a similar accuracy is reached with this approximation at a considerably reduced computational cost

operator splitting methods via the zassenhaus product formula abstract in this paper we contribute an operator splitting method improved by the zassenhaus product zassenhaus products are of fundamental importance for the theory of lie groups and lie algebras while their applications in physics and physical chemistry are important novel applications in cfd computational fluid dynamics arose based on the fact that their sparse matrices can be seen as generators of an underlying lie algebra we apply this to classical splitting and the novel zassenhaus product formula the underlying analysis for obtaining higher order operator splitting methods based on the zassenhaus product is presented the benefits of dealing with sparse matrices given by spatial discretization of the underlying partial differential equations are due to the fact that the higher order commutators are very quickly computable their matrix structures thin out and become nilpotent when applying these methods to convection diffusion reaction equations the benefits of balancing time and spatial scales can be used to accelerate these methods and take into account these sparse matrix structures the verification of the improved splitting methods is done with numerical examples finally we conclude with higher order operator splitting methods

optimal cluster preserving embedding of nonmetric proximity data for several major applications of data analysis objects are often not represented as feature vectors in a vector space but rather by a matrix gathering pairwise proximities such pairwise data often violates metricity and therefore cannot be naturally embedded in a vector space concerning the problem of unsupervised structure detection or clustering in this paper a new embedding method for pairwise data into euclidean vector spaces is introduced we show that all clustering methods which are invariant under additive shifts of the pairwise proximities can be reformulated as grouping problems in euclidian spaces the most prominent property of this constant shift embedding framework is the complete preservation of the cluster structure in the embedding space restating pairwise clustering problems in vector spaces has several important consequences such as the statistical description of the clusters by way of cluster prototypes the generic extension of the grouping procedure to a discriminative prediction rule and the applicability of standard preprocessing methods like denoising or dimensionality reduction

the baker campbell hausdorff formula and nested commutator identities the coefficients of the baker campbell hausdorff expansion are calculated by using various methods comparison of the results yields several remarkable identities satisfied by multiple commutators which in turn allow us to greatly simplify the form of the expansion up to order eight

on the relevance of linear discriminative features linear discriminant analysis lda has been widely used to extract linear features for classification in real applications the usefulness of the extracted features usually needs to be confirmed using an error rate of classification embedded in a classifier little attention has been paid to whether and how discriminative features themselves can be interpreted as indicators of usefulness we refer to this as relevance i e the capability of discriminative features to characterize the contribution of the original variables to classification we approach the relevance by considering how it could be lost while extracting optimal discriminative features then the discrepancy between the relevance and optimality of discriminative features is shown to originate from the angle between the space spanned by eigenvectors of the within class scattering matrix and the primary space in which the original variables reside in particular for a given dataset the larger the angle the less evident is the relevance discovered from optimal discriminative features furthermore the relevance and optimality are regarded as two constraint conditions or a tradeoff in order to extract relevant discriminative features at last a simulated experiment is used to show how the relevance is lost when the angle is changed experimental results based on both usps handwritten digitals and pie face databases show that a maximum margin criterion is a reasonable compromise between the relevance and optimality since it approximates the averaged class margin using euclidean distance measured in the primary space

effective action for stochastic partial differential equations stochastic partial differential equations spdes are the basic tool for modeling systems where noise is important spdes are used for models of turbulence pattern formation and the structural development of the universe itself it is reasonably well known that certain spdes can be manipulated to be equivalent to nonquantum field theories that nevertheless exhibit deep and important relationships with quantum field theory in this paper we systematically extend these ideas we set up a functional integral formalism and demonstrate how to extract all the one loop physics for an arbitrary spde subject to arbitrary gaussian noise it is extremely important to realize that gaussian noise does not imply that the field variables undergo gaussian fluctuations and that these nonquantum field theories are fully interacting the limitation to one loop is not as serious as might be supposed experience with quantum field theories qfts has taught us that one loop physics is often quite adequate to give a good description of the salient issues the limitation to one loop does however offer marked technical advantages because at one loop almost any field theory can be rendered finite using zeta function technology we can sidestep the complications inherent in the martin siggia rose formalism the spde analog of the becchi rouet stora tyutin formalism used in qft and instead focus attention on a minimalist approach that uses only the physical fields this direct approach is the spde analog of canonical quantization using physical fields after setting up the general formalism for the characteristic functional partition function we show how to define the effective action to all loops and then focus on the one loop effective action and its specialization to constant fields the effective potential the physical interpretation of the effective action and effective potential for spdes is addressed and we show that key features carry over from qft to the case of spdes an important result is that the amplitude of the two point function governing the noise acts as the loop counting parameter and is the analog of planck s constant in this spde context we derive a general expression for the one loop effective potential of an arbitrary spde subject to translation invariant gaussian noise and compare this with the one loop potential for qft

on the information and representation of non euclidean pairwise data two common data representations are mostly used in intelligent data analysis namely the vectorial and the pairwise representation pairwise data which satisfy the restrictive conditions of euclidean spaces can be faithfully translated into a euclidean vectorial representation by embedding non metric pairwise data with violations of symmetry reflexivity or triangle inequality pose a substantial conceptual problem for pattern recognition since the amount of predictive structural information beyond what can be measured by embeddings is unclear we show by systematic modeling of non euclidean pairwise data that there exists metric violations which can carry valuable problem specific information furthermore euclidean and non metric data can be unified on the level of structural information contained in the data stable component analysis selects linear subspaces which are particularly insensitive to data fluctuations experimental results from different domains support our pattern recognition strategy

from time ordered products to magnus expansion recent work by lam on the decomposition of time ordered products appearing in the time evolution operator in terms of sums of products of nested commutators is endowed with a recursive algorithm a sufficient condition for absolute convergence is subsequently obtained connection with magnus expansion is established

fast k most similar neighbor classifier for mixed data tree k msn the k nearest neighbor k nn classifier has been a widely used nonparametric technique in pattern recognition because of its simplicity and good performance in order to decide the class of a new prototype the k nn classifier performs an exhaustive comparison between the prototype to classify and the prototypes in the training set t however when t is large the exhaustive comparison is expensive for this reason many fast k nn classifiers have been developed some of them are based on a tree structure which is created during a preprocessing phase using the prototypes in t then in a search phase the tree is traversed to find the nearest neighbor the speed up is obtained while the exploration of some parts of the tree is avoided using pruning rules which are usually based on the triangle inequality however in soft sciences as medicine geology sociology etc the prototypes are usually described by numerical and categorical attributes mixed data and sometimes the comparison function for computing the similarity between prototypes does not satisfy metric properties therefore in this work an approximate fast k most similar neighbor classifier for mixed data and similarity functions that do not satisfy metric properties based on a tree structure tree k msn is proposed some experiments with synthetic and real data are presented

general theory of fractal path integrals with applications to many body theories and statistical physics a general scheme of fractal decomposition of exponential operators is presented in any order m namely exp x a b sm x o xm 1 for any positive integer m where sm x et1a et2b et3a et4b etma with finite m depending on m a general recursive scheme of construction of tj is given explicitly it is proven that some of tj should be negative for m 3 and for any finite m nonexistence theorem of positive decomposition general systematic decomposition criterions based on a new type of time ordering are also formulated the decomposition exp x a b sm x n n o xm 1 nm yields a new efficient approach to quantum monte carlo simulations

fluctuation effects and multiscaling of the reaction diffusion front for a b to oe we consider the properties of the diffusion controlled reaction a b to oe in the steady state where fixed currents of a and b particles are maintained at opposite edges of the system using renormalization group methods we explicitly calculate the asymptotic forms of the reaction front and particle densities as expansions in jd 1 mod x mod d 1 1 where j are the equal applied currents and d the equal diffusion constants for the asymptotic densities of the minority species we find in addition to the expected exponential decay fluctuation induced power law tails which for d 2 we find similar power laws with mu d 3 but with non universal coefficients logarithmic corrections occur in d 2 these results imply that in the time dependent case with segregated initial conditions the moments integral mod x mod qr x t dx fail to satisfy simple scaling for q mu 1 finally it is shown that the fluctuation induced wandering of the position of the reaction front centre may be neglected for large enough systems

hall littlewood symmetric functions and the bkp equation the connection between solutions of the bkp equation and hall littlewood symmetric functions is utilised in a unified approach to soliton and polynomial solutions this is analogous to the wronskian formulation of the solution of the kp equation as a by product two novel expressions for certain hall littlewood functions in terms of pfaffians are derived

effect of static disorder and reactant segregation on the a b 0 reaction we derive the long time behavior of the a b to emptyset reaction in two dimensions finding a universal exponent and prefactor in the absence of disorder sufficiently singular disorder leads to a sub diffusion limited reaction and a continuously variable decay exponent pattern matching between the reactant segregation and the disorder is not strong enough to affect the long time decay

characters of hecke algebras hn q of type an 1 the connection between the ocneanu trace on hn q and schur functions leads to a simple method for calculating the irreducible characters of the hecke algebras hn q the characters appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to schur functions

convergence of unitary and complex decompositions of exponential operators the convergence of the fractal unitary and complex decompositions of exponential operators is proved for hermitian operators in a banach space

an efficient algorithm for computing the baker campbell hausdorff series and some of its applications we provide a new algorithm for generating the baker campbell hausdorff bch series z log exey in an arbitrary generalized hall basis of the free lie algebra l x y generated by x and y it is based on the close relationship of l x y with a lie algebraic structure of labeled rooted trees with this algorithm the computation of the bch series up to degree of 20 111 013 independent elements in l x y takes less than 15min on a personal computer and requires 1 5gbytes of memory we also address the issue of the convergence of the series providing an optimal convergence domain when x and y are real or complex matrices

goldberg s theorem and the baker campbell hausdorff formula abstract goldberg s theorem duke math j 23 1956 13 is applied to expressing the logarithm of exponential product in terms of free lie elements using hyperoperators namely inner derivations a symmetric property of the expansion coefficients in the formal power series is also discussed

asymptotic behavior of densities for two particle annihilating random walks consider the system of particles on d where particles are of two types a andb and execute simple random walks in continuous time particles do not interact with their own type but when ana particle meets ab particle both disappear i e are annihilated this system serves as a model for the chemical reactiona b inert we analyze the limiting behavior of the densities a t and b t when the initial state is given by homogeneous poisson random fields we prove that for equal initial densities a 0 b 0 there is a change in behavior fromd 4 where a t b t c t d 4 tod 4 where a t b t c tast for unequal initial densities a 0 b 0 a t e c l ind 1 a t e ct logt ind 2 and a t e ct ind 3 the termc depends on the initial densities and changes withd techniques are from interacting particle systems the behavior for this two particle annihilation process has similarities to those for coalescing random walks a a a and annihilating random walks a a inert the analysis of the present process is made considerably more difficult by the lack of comparison with an attractive particle system

renormalisation of fluctuation effects in the a a to a reaction the field theoretical description of the a a to a reaction is exactly renormalised to all orders in perturbation theory it is confirmed that it belong to the same universality class as the a a to oe reaction and that the concentration decay like t d 2 for space dimensionality d smaller than two and like ln t t for d 2 are exact results

a unified approach to pattern recognition abstract the paper is an outline of a new approach to pattern recognition developed by the author a fuller introduction to the approach will appear soon 1 within the proposed framework the two principal approaches to pattern recognition vector and syntactic are unified

wasserstein distances in the analysis of time series and dynamical systems abstract a new approach based on wasserstein distances which are numerical costs of an optimal transportation problem allows us to analyze nonlinear phenomena in a robust manner the long term behavior is reconstructed from time series resulting in a probability distribution over phase space each pair of probability distributions is then assigned a numerical distance that quantifies the differences in their dynamical properties from the totality of all these distances a low dimensional representation in a euclidean space is derived in which the time series can be classified and statistically analyzed this representation shows the functional relationships between the dynamical systems under study it allows us to assess synchronization properties and also offers a new way of numerical bifurcation analysis the statistical techniques for this distance based analysis of dynamical systems are presented filling a gap in the literature and their application is discussed in a few examples of datasets arising in physiology and neuroscience and in the well known henon system

dynkin s method of computing the terms of the baker campbell hausdorff series the infinite series for log exp x exp y for noncommuting x and y is expressible in terms of iterated commutators of x and y except for the linear term x y dynkin derived an explicit expression for the terms as a sum of iterated commutators over a certain set of sequences this paper presents a practical algorithm for applying dynkin s formula and gives several illustrative examples

ricci flow embedding for rectifying non euclidean dissimilarity data pairwise dissimilarity representations are frequently used as an alternative to feature vectors in pattern recognition one of the problems encountered in the analysis of such data is that the dissimilarities are rarely euclidean while statistical learning algorithms often rely on euclidean dissimilarities such non euclidean dissimilarities are often corrected or a consistent euclidean geometry is imposed on them via embedding this paper commences by reviewing the available algorithms for analysing non euclidean dissimilarity data the novel contribution is to show how the ricci flow can be used to embed and rectify non euclidean dissimilarity data according to our representation the data is distributed over a manifold consisting of patches each patch has a locally uniform curvature and this curvature is iteratively modified by the ricci flow the raw dissimilarities are the geodesic distances on the manifold rectified euclidean dissimilarities are obtained using the ricci flow to flatten the curved manifold by modifying the individual patch curvatures we use two algorithmic components to implement this idea firstly we apply the ricci flow independently to a set of surface patches that cover the manifold second we use curvature regularisation to impose consistency on the curvatures of the arrangement of different surface patches we perform experiments on three real world datasets and use these to determine the importance of the different algorithmic components i e ricci flow and curvature regularisation we conclude that curvature regularisation is an essential step needed to control the stability of the piecewise arrangement of patches under the ricci flow

on the numerical integration of ordinary differential equations by symmetric composition methods differential equations of the form dot x x a b are considered where the vector fields a and b can be integrated exactly enabling numerical integration of x by composition of the flows of a and b various symmetric compositions are investigated for order complexity and reversibility free lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order a new more accurate way of applying the methods thus obtained to compositions of an arbitrary first order integrator is described and tested the determining equations are explored and new methods up to 100 times more accurate at constant work than those previously known are given

a theory of proximity based clustering structure detection by optimization abstract in this paper a systematic optimization approach for clustering proximity or similarity data is developed starting from fundamental invariance and robustness properties a set of axioms is proposed and discussed to distinguish different cluster compactness and separation criteria the approach covers the case of sparse proximity matrices and is extended to nested partitionings for hierarchical data clustering to solve the associated optimization problems a rigorous mathematical framework for deterministic annealing and mean field approximation is presented efficient optimization heuristics are derived in a canonical way which also clarifies the relation to stochastic optimization by gibbs sampling similarity based clustering techniques have a broad range of possible applications in computer vision pattern recognition and data analysis as a major practical application we present a novel approach to the problem of unsupervised texture segmentation which relies on statistical tests as a measure of homogeneity the quality of the algorithms is empirically evaluated on a large collection of brodatz like micro texture mondrians and on a set of real word images to demonstrate the broad usefulness of the theory of proximity based clustering the performances of different criteria and algorithms are compared on an information retrieval task for a document database the superiority of optimization algorithms for clustering is supported by extensive experiments

linear time computation of similarity measures for sequential data efficient and expressive comparison of sequences is an essential procedure for learning with sequential data in this article we propose a generic framework for computation of similarity measures for sequences covering various kernel distance and non metric similarity functions the basis for comparison is embedding of sequences using a formal language such as a set of natural words k grams or all contiguous subsequences as realizations of the framework we provide linear time algorithms of different complexity and capabilities using sorted arrays tries and suffix trees as underlying data structures experiments on data sets from bioinformatics text processing and computer security illustrate the efficiency of the proposed algorithms enabling peak performances of up to 106 pairwise comparisons per second the utility of distances and non metric similarity measures for sequences as alternatives to string kernels is demonstrated in applications of text categorization network intrusion detection and transcription site recognition in dna

a simple expression for the terms in the baker campbell hausdorff series a simple expression is derived for the terms in the baker campbell hausdorff series one formulation of the result involves a finite number of operations with matrices of rational numbers generalizations are discussed

the baker campbell hausdorff formula and the convergence of the magnus expansion the authors show that for a wide class of dynamical systems described by hamiltonians of the form usually considered in time dependent perturbation theory the divergence of the magnus expansion in the schrodinger picture for large time intervals is due to pole singularities inherent to the baker campbell hausdorff formula

consensus based formation control laws for systems on lie groups this paper designs the consensus based formation control laws for multi agent systems defined on lie groups the specific results are illustrated on se 2 and se 3 the special euclidean groups of rigid body motions and each of which shares the geometric structure of a lie group the paper develops the control laws for the problem concerned under the assumption that the communication topologies of the multi agent systems are spanning tree shaped ring shaped and complete respectively at last several examples are included in the numerical simulations to verify the theoretical results

heat kernel regularization of the effective action for stochastic reaction diffusion equations the presence of fluctuations and nonlinear interactions can lead to scale dependence in the parameters appearing in stochastic differential equations stochastic dynamics can be formulated in terms of functional integrals in this paper we apply the heat kernel method to study the short distance renormalizability of a stochastic polynomial reaction diffusion equation with real additive noise we calculate the one loop effective action and its ultraviolet scale dependent divergences we show that for white noise a polynomial reaction diffusion equation is one loop finite in d 0 and d 1 and is one loop renormalizable in d 2 and d 3 space dimensions we obtain the one loop renormalization group equations and find they run with scale only in d 2

quantum monte carlo methods and general decomposition theory of exponential operators and symplectic integrators abstract the present paper reviews briefly the basic ideas of the quantum monte carlo methods the general decomposition theory of ordered exponentials and symplectic integrators

graph transduction as a noncooperative game graph transduction is a popular class of semisupervised learning techniques that aims to estimate a classification function defined over a graph of labeled and unlabeled data points the general idea is to propagate the provided label information to unlabeled nodes in a consistent way in contrast to the traditional view in which the process of label propagation is defined as a graph laplacian regularization this article proposes a radically different perspective based on game theoretic notions within the proposed framework the transduction problem is formulated in terms of a noncooperative multiplayer game whereby equilibria correspond to consistent labelings of the data an attractive feature of this formulation is that it is inherently a multiclass approach and imposes no constraint whatsoever on the structure of the pairwise similarity matrix being able to naturally deal with asymmetric and negative similarities alike experiments on a number of real world problems demonstrate that the proposed approach performs well compared with state of the art algorithms and it can deal effectively with various types of similarity relations

variations on a theme of jost and pais abstract we explore the extent to which a variant of a celebrated formula due to jost and pais which reduces the fredholm perturbation determinant associated with the schrodinger operator on a half line to a simple wronski determinant of appropriate distributional solutions of the underlying schrodinger equation generalizes to higher dimensions in this multi dimensional extension the half line is replaced by an open set r n n n n 2 where has a compact nonempty boundary satisfying certain regularity conditions our variant involves ratios of perturbation determinants corresponding to dirichlet and neumann boundary conditions on and invokes the corresponding dirichlet to neumann map as a result we succeed in reducing a certain ratio of modified fredholm perturbation determinants associated with operators in l 2 d n x n n to modified fredholm determinants associated with operators in l 2 d n 1 n 2 applications involving the birman schwinger principle and eigenvalue counting functions are discussed