invariant algebra and cuspidal representations of finite monoids abstract motivated by the theories of hecke algebras and schur algebras we consider in this paper the algebra c mgofg invariants of a finite monoidmwith unit groupg ifmis a regular balanced monoid we show that c mgis a quasi hereditary algebra in such a case we find the blocks of c mgto be the sections of the blocks of c m we go on to develop a theory of cuspidal representations for balanced monoids we then apply our results to the full transformation semigroup and the multiplicative monoid of triangular matrices over a finite field

graph embedding in vector spaces by node attribute statistics graph based representations are of broad use and applicability in pattern recognition they exhibit however a major drawback with regards to the processing tools that are available in their domain graph embedding into vector spaces is a growing field among the structural pattern recognition community which aims at providing a feature vector representation for every graph and thus enables classical statistical learning machinery to be used on graph based input patterns in this work we propose a novel embedding methodology for graphs with continuous node attributes and unattributed edges the approach presented in this paper is based on statistics of the node labels and the edges between them based on their similarity to a set of representatives we specifically deal with an important issue of this methodology namely the selection of a suitable set of representatives in an experimental evaluation we empirically show the advantages of this novel approach in the context of different classification problems using several databases of graphs

complex representations of finite monoids ii highest weight categories and quivers abstract in this paper we continue our study of complex representations of finite monoids we begin by showing that the complex algebra of a finite regular monoid is a quasi hereditary algebra and we identify the standard and costandard modules we define the concept of a monoid quiver and compute it in terms of the group characters of the standard and costandard modules we use our results to determine the blocks of the complex algebra of the full transformation semigroup we show that there are only two blocks when the degree 3 we also show that when the degree 5 the complex algebra of the full transformation semigroup is not of finite representation type answering negatively a conjecture of ponizovskii

quasi hereditary twisted category algebras we give a sufficient criterion for when a twisted finite category algebra over a field is quasi hereditary in terms of the partially ordered set of l classes in the morphism set of the category we show that this is a common generalisation of a long list of results in the context of ei categories regular monoids brauer algebras temperley lieb algebras partition algebras

the restricted simple lie algebras are of classical or cartan type abstract the classification of the finite dimensional restricted simple lie algebras over an algebraically closed field f of prime characteristic p 7 is announced such an algebra is either of classical type an analogue over f of a finite dimensional simple lie algebra over the complex numbers or of cartan type an analogue over f of one of the infinite lie algebras of cartan over the complex numbers

representation theory of finite semigroups semigroup radicals and formal language theory in this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field generalizing results of rhodes for the field of complex numbers applications are given to obtain many new results as well as easier proofs of several results in the literature involving triangularizability of finite semigroups which semigroups have split basic semigroup algebras two sided semidirect product decompositions of finite monoids unambiguous products of rational languages products of rational languages with counter and cerny s conjecture for an important class of automata

varieties of tori and cartan subalgebras of restricted lie algebras this paper investigates varieties of tori and cartan subalgebras of a finite dimensional restricted lie algebra g p defined over an algebraically closed field k of positive characteristic p we begin by showing that schemes of tori may be used as a tool to retrieve results by a premet on regular cartan subalgebras moreover they give rise to principal fibre bundles whose structure groups coincide with the weyl groups in case g lie g is the lie algebra of a smooth group g for solvable lie algebras varieties of tori are full affine spaces while simple lie algebras of classical or cartan type cannot have varieties of this type in the final sections the quasi projective variety of cartan subalgebras of minimal dimension rk g is shown to be irreducible of dimension dim k g rk g with premet s regular cartan subalgebras belonging to the regular locus

pseudo likelihood methods for community detection in large sparse networks many algorithms have been proposed for fitting network models with communities but most of them do not scale well to large networks and often fail on sparse networks here we propose a new fast pseudo likelihood method for fitting the stochastic block model for networks as well as a variant that allows for an arbitrary degree distribution by conditioning on degrees we show that the algorithms perform well under a range of settings including on very sparse networks and illustrate on the example of a network of political blogs we also propose spectral clustering with perturbations a method of independent interest which works well on sparse networks where regular spectral clustering fails and use it to provide an initial value for pseudo likelihood we prove that pseudo likelihood provides consistent estimates of the communities under a mild condition on the starting value for the case of a block model with two communities

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

latent space approaches to social network analysis network models are widely used to represent relational information among interacting units in studies of social networks recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation between actors we develop a class of models where the probability of a relation between actors depends on the positions of individuals in an unobserved social space we make inference for the social space within maximum likelihood and bayesian frameworks and propose markov chain monte carlo procedures for making inference on latent positions and the effects of observed covariates we present analyses of three standard datasets from the social networks literature and compare the method to an alternative stochastic blockmodeling approach in addition to improving on model fit for these datasets our method provides a visual and interpretable model based spatial representation of social relationships and improv

hybrid adaptive wavelet neuro fuzzy system for chaotic time series identification in the paper a five layers architecture of hybrid wavelet neuro fuzzy system which is using the adaptive w neurons as the nodes is proposed w neuron is a neuron which structure is similar to a radial basis functions network but instead of conventional radial basis functions we used multidimensional adaptive wavelet activation membership functions the distinctive feature of the proposed system is usage of the wavelets as membership functions in the antecedent layer and the adaptive multidimensional wavelets as activation functions in the consequent layer that can tune not only the dilation and translation parameters but also its own form during the learning process the learning algorithms for all antecedent and consequent functions parameters that have both following and filtering properties are proposed the experimental results have shown that this wavelet neuro fuzzy system has improved approximation properties and has a higher learning rate in comparison with usual wavelet neuro fuzzy networks the proposed hybrid wavelet neuro fuzzy system can be used to solve tasks of diagnosis forecasting emulation and identification of nonlinear chaotic and stochastic non stationary processes

the representation theory of the monoid of all partial functions on a set and related monoids as ei category algebras the ordinary quiver of an algebra a is a graph that contains information about the algebra s representations we give a description of the quiver of cptn the algebra of the monoid of all partial functions on n elements our description uses an isomorphism between cptn and the algebra of the epimorphism category en whose objects are the subsets of 1 n and morphism are all total epimorphisms this is an extension of a well known isomorphism of the algebra of isn the monoid of all partial injective maps on n elements and the algebra of the groupoid of all bijections between subsets of an n element set the quiver of the category algebra is described using results of margolis steinberg and li on the quiver of ei categories we use the same technique to compute the quiver of other natural transformation monoids we also show that the algebra cptn has three blocks for n 1 and we give a natural description of the descending loewy series of cptn in the category form

technical communique convergence analysis of recursive identification algorithms with forgetting factor a convergence analysis of a modified version of the least squares recursive identification algorithm with forgetting factor is given it is shown that the parametric distance converges to a zero mean random variable it is also shown that under persistent excitation condition on both system input and output the condition number of the adaptation gain matrix is bounded the variance of the parametric distance is bounded by the product of the noise variance times the upper bound of the condition number of the gain matrix this is done by normalizing the measurement vector entering in the identification algorithm and by using a forgetting factor verifying l t 0

sandwich matrices solomon algebras and kazhdan lusztig polynomials sandwich matrices have proved to be of importance in semigroup theory for the last 50 years the work of the author on algebraic monoids leads to sandwich matrices in group theory in this paper we find some connections between sandwich matrices and the hecke algebras for monoids introduced recently by louis solomon at the local level we then obtain an explicit isomorphism between solomon s hecke algebra and the complex monoid algebra of the renner monoid in the simplest case of monoids associated with a borel subgroup we find that the entries of the inverse of the sandwich matrix as well as those of the related structure matrix of solomon s hecke algebra are almost the polynomials r x y associated with the kazhdan lusztig polynomials

character theory of monoids over an arbitrary field abstract the basic character theory of finite monoids over the complex numbers was developed in the sixties and seventies based on work of munn ponizovski mcalister rhodes and zalcstein in particular mcalister determined the space of functions spanned by the irreducible characters of a finite monoid over c and the ring of virtual characters in this paper we present the corresponding results over an arbitrary field as a consequence we obtain a quick proof of the theorem of berstel and reutenauer that the characteristic function of a regular cyclic language is a virtual character of the free monoid this is a crucial ingredient in their proof of the rationality of the zeta function of a sofic shift in symbolic dynamics

estimation and prediction for stochastic blockstructures a statistical approach to a posteriori blockmodeling for digraphs and valued digraphs is proposed the probability model assumes that the vertices of the digraph are partitioned into several unobserved latent classes and that the probability distribution of the relation between two vertices depends only on the classes to which they belong a bayesian estimator based on gibbs sampling is proposed the basic model is not identified because class labels are arbitrary the resulting identifiability problems are solved by restricting inference to the posterior distributions of invariant functions of the parameters and the vertex class membership in addition models are considered where class labels are identified by prior distributions for the class membership of some of the vertices the model is illustrated by an example from the social networks literature kapferer s tailor shop

convergence results for continuous time adaptive stochastic filtering algorithms abstract the adaptive stochastic filtering problem for gaussian processes is considered the self tuning synthesis procedure is used to derive two algorithms for this problem almost sure convergence for the parameter estimate and the filtering error will be established the convergence analysis is based on an almost supermartingale convergence lemma that allows a stochastic lyapunov like approach

simple lie algebras of small characteristic v the non melikian case abstract let l be a finite dimensional simple lie algebra over an algebraically closed field f of characteristic p 3 we prove in this paper that if for every torus t of maximal dimension in the p envelope of adl in derl the centralizer of t in adl acts triangulably on l then l is either classical or of cartan type as a consequence we obtain that any finite dimensional simple lie algebra over an algebraically closed field of characteristic p 5 is either classical or of cartan type this settles the last remaining case of the generalized kostrikin shafarevich conjecture the case where p 7

covariate assisted spectral clustering biological and social systems consist of myriad interacting units the interactions can be represented in the form of a graph or network measurements of these graphs can reveal the underlying structure of these interactions which provides insight into the systems that generated the graphs moreover in applications such as connectomics social networks and genomics graph data are accompanied by contextualizing measures on each node we utilize these node covariates to help uncover latent communities in a graph using a modification of spectral clustering statistical guarantees are provided under a joint mixture model that we call the node contextualized stochastic blockmodel including a bound on the misclustering rate the bound is used to derive conditions for achieving perfect clustering for most simulated cases covariate assisted spectral clustering yields results superior both to regularized spectral clustering without node covariates and to an adaptation of canonical correlation analysis we apply our clustering method to large brain graphs derived from diffusion mri data using the node locations or neurological region membership as covariates in both cases covariate assisted spectral clustering yields clusters that are easier to interpret neurologically

the canonical compactification of a finite group of lie type let g be a finite group of lie type we construct a finite monoid m having g as the group of units m has properties analogous to the canonical compactification of a reductive group the complex representation theory of m yields harish chandra s philosophy of cuspidal representations of g the main purpose of this paper is to determine the irreducible modular representations of m we then show that all the irreducible modular representations of g come via the 1942 work of clifford from the one dimensional representations of the maximal subgroups of m this yields a semigroup approach to the modular representation theory of g via the full rank factorizations of the sandwich matrices of m we then determine the irreducible modular representations of any finite monoid of lie type

non parametric sparse additive auto regressive network models consider a multi variate time series x t t 0 t where x t in mathbb r d which may represent spike train responses for multiple neurons in a brain crime event data across multiple regions and many others an important challenge associated with these time series models is to estimate an influence network between the d variables especially when d is large meaning we are in the high dimensional setting prior work has focused on parametric vector auto regressive models however parametric approaches are somewhat restrictive in practice in this paper we use the non parametric sparse additive model spam framework to address this challenge using a combination of beta and phi mixing properties of markov chains and empirical process techniques for reproducing kernel hilbert spaces rkhss we provide upper bounds on mean squared error in terms of the sparsity s the logarithm of the dimension log d the number of time points t and the smoothness of the rkhss our rates are sharp up to logarithm factors in many cases we also provide numerical experiments that support our theoretical results and display the potential advantages of using our non parametric spam framework for a chicago crime data set

optimal self tuning filtering prediction and smoothing for discrete multivariable processes an algorithm is proposed for self tuning optimal fixed lag smoothing or filtering for linear discrete time multivariable processes a z transfer function solution to the discrete multivariable estimation problem is first presented this solution involves spectral factorization of polynomial matrices and assumes knowledge of the process parameters and the noise statistics the assumption is then made that the signal generating process and noise statistics are unknown the problem is reformulated so that the model is in an innovations signal form and implicit self tuning estimation algorithms are proposed the parameters of the innovation model of the process can be estimated using an extended kalman filter or alternatively extended recursive least squares these estimated parameters are used directly in the calculation of the predicted smoothed or filtered estimates the approach is an attempt to generalize the work of hagander and wittenmark

mobius functions and semigroup representation theory ii character formulas and multiplicities abstract we generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using rota s theory of mobius inversion the technique works for a large class of semigroups including inverse semigroups semigroups with commuting idempotents idempotent semigroups and semigroups with basic algebras using these tools we are able to give a complete description of the spectra of random walks on finite semigroups admitting a faithful representation by upper triangular matrices over the complex numbers these include the random walks on chambers of hyperplane arrangements studied by bidigare hanlon rockmore brown and diaconis applications are also given to decomposing tensor powers and exterior products of rook matrix representations of inverse semigroups generalizing and simplifying earlier results of solomon for the rook monoid

multimodal manifold analysis by simultaneous diagonalization of laplacians we construct an extension of spectral and diffusion geometry to multiple modalities through simultaneous diagonalization of laplacian matrices this naturally extends classical data analysis tools based on spectral geometry such as diffusion maps and spectral clustering we provide several synthetic and real examples of manifold learning object classification and clustering showing that the joint spectral geometry better captures the inherent structure of multi modal data we also show the relation of many previous approaches for multimodal manifold analysis to our framework

classification of the restricted simple lie algebras soit l une algebre de lie simple restreinte de dimension finie sur un corps algebriquement clos f de caracteristique p 7 alors l est de type classique ou de cartan

spectral clustering and the high dimensional stochastic blockmodel networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors social ne tworks representing people who communicate with each other are one example communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks spectral clustering is a popular and computationally feasi ble method to discover these communities the stochastic block model holland et al 1983 is a social network model with well defined communities each node is a member of one community for a network generated from the stochastic block model we bound the number of nodes misclus tered by spectral clustering the asymptotic results in th is paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes hence the name high dimensional in order to study spectral clustering under the stochastic block model we first show that under the more general latent space model the eigenvectors of the normalized graph laplacian asymptotically converge to the eigenvectors of a population normal ized graph laplacian aside from the implication for spectral clustering this provides insight into a graph visualization technique our method of studying the eigenvectors of random matrices is original ams 2000 subject classifications primary 62h30 62h25 secondary 60b20

toral rank one lie algebras let l be a finite dimensional lie algebra over an algebraically closed field f of characteristic p 0 assume that h is a cartan subalgebra of l we say that l has toral rank m relative to h if dim pa m where a is the set of roots with respect to h p is the prime field and pa is the p vector space spanned by a in this work we study lie algebras of toral rank one the motivation to investigate this class of algebras comes from the problem of classifying simple lie algebras if a is a finite dimensional simple lie algebra with cartan subalgebra h and if ci is a root of h then the subalgebra a ciep a has toral rank one relative to h in 12 133 wilson has shown that if a simple algebra a has characteristic p 7 then the restricted closure w of h in der a has the form r t z where i is the nil radical of b and t is the maximal torus which is central in j it follows that ad r on each root space a can be simultaneously upper triangularized indeed i is a lie algebra of nilpotent transformations on a and thus by 8 theorem 3 31 it annihilates a nonzero vector the space of such vectors is t invariant and so t i has a common eigenvector u on a one can proceed inductively on ajfu to argue that each root space a has a basis relative to which ad r is upper triangular with that in mind we say in what follows that a cartan subalgebra his triangularizable if there is a basis for each root space relative to which ad h is upper triangular we show in section 1 that if l is a lie algebra having a toral rank one cartan subalgebra h and if h is triangularizable then l modulo its maximal solvable ideal either is 0 or consists of a simple toral rank one lie algebra s together with derivations of s in h in the first case we say that the core of l is 0 and in the second case that the core of l is s wilson 141 has classified the simple toral rank one lie algebras using

a globally convergent adaptive predictor this paper establishes global convergence for an adaptive prediction algorithm key features of the algorithm are i a priori residuals are used in the regression vector ii the input signal need not be persistently exciting iii only an upper bound is required on the system order iv no monitoring or projection procedure is required to guarantee stability and v arbitrary feedback is allowed between output and input