irreducible unitary representations of su 2 2 we give an explicit classification of the irreducible unitary representations of the simple lie group su 2 2

positive energy unitary irreducible representations of d 6 conformal supersymmetry we give a constructive classification of the positive energy lowest weight unitary irreducible representations of the d 6 superconformal algebras osp 8 2n our results confirm all but one of the conjectures of minwalla for n 1 2 on this classification our main tool is the explicit construction of the norms of the states that have to be checked for positivity we also give the reduction of the four exceptional unitary irreducible representations

on learning multicategory classification with sample queries consider the pattern recognition problem of learning multicategory classification from a labeled sample for instance the problem of learning character recognition where a category corresponds to an alphanumeric letter the classical theory of pattern recognition assumes labeled examples appear according to the unknown underlying pattern class conditional probability distributions where the pattern classes are picked randomly according to their a priori probabilities in this paper we pose the following question can the learning accuracy be improved if labeled examples are independently randomly drawn according to the underlying class conditional probability distributions but the pattern classes are chosen not necessarily according to their a priori probabilities we answer this in the affirmative by showing that there exists a tuning of the sub sample proportions which minimizes a loss criterion the tuning is relative to the intrinsic complexity of the bayes classifier as this complexity depends on the underlying probability distributions which are assumed to be unknown we provide an algorithm which learns the proportions in an on line manner utilizing sample querying which asymptotically minimizes the criterion in practice this algorithm may be used to boost the performance of existing learning classification algorithms by apportioning better sub sample proportions

on twistor transformations and invariant differential operator of simple lie group g2 2 the twistor transformations associated to the simple lie group g2 are described explicitly we consider the double fibration g2 p2 g2 b g2 p1 where p1 and p2 are two parabolic subgroups of g2 and b is a borel subgroup and its local version h2 f h1 where h1 is the heisenberg group of dimension 5 embedded in the coset space g2 p1 f cp1 h1 and h2 contains the nilpotent lie group h2 of step three the baker campbell hausdorff formula is used to parametrize the coset spaces coordinates charts their transition functions and the fibers of the projection as complex curves we write down the relative de rham sequence on f along the fibers and push it down to h1 to get a family of matrix valued differential operators dk then we establish a kind of penrose correspondence for g2 the kernel of dk is isomorphic to the first cohomology of the sheaf o k over h2 we also give the penrose type integral transformation u pf for f o k which gives solutions to equations dku 0 when restricted to the re

kernel methods in machine learning we review machine learning methods employing positive definite kernels these methods formulate learning and estimation problems in a reproducing kernel hilbert space rkhs of functions defined on the data domain expanded in terms of a kernel working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions the latter include nonlinear functions as well as functions defined on nonvectorial data we cover a wide range of methods ranging from binary classifiers to sophisticated methods for estimation with structured data

explicit orbit classification of reducible jordan algebras and freudenthal triple systems we determine explicit orbit representatives of reducible jordan algebras and their corresponding freudenthal triple systems this work has direct application to the classification of extremal black hole solutions of mathcal n 2 4 locally supersymmetric theories of gravity coupled to an arbitrary number of abelian vector multiplets in d 4 5 space time dimensions

on the convergence rate of l p norm multiple kernel learning we derive an upper bound on the local rademacher complexity of lp norm multiple kernel learning which yields a tighter excess risk bound than global approaches previous local approaches analyzed the case p 1 only while our analysis covers all cases 1 p assuming the different feature mappings corresponding to the different kernels to be uncorrelated we also show a lower bound that shows that the bound is tight and derive consequences regarding excess loss namely fast convergence rates of the order o n 1 a where is the minimum eigenvalue decay rate of the individual kernels

the unitary irreducible representations of overline rm so _ 0 4 2 an exhaustive classification of all irreducible harish chandra mathfrak s mathfrak o 4 2 modules integrable to unitarizable projective representations of the conformal group is established by infinitesimal methods the classification is based 1 on the reduction upon the maximal compact subalgebra associated with a lattice of points in 3 and 2 on a set of additional parameters upon which the eigenvalues of central elements of the enveloping algebra depend polynomially

risk bounds of learning processes for levy processes leevy processes refer to a class of stochastic processes for example poisson processes and brownian motions and play an important role in stochastic processes and machine learning therefore it is essential to study risk bounds of the learning process for time dependent samples drawn from a leevy process or briefly called learning process for leevy process it is noteworthy that samples in this learning process are not independently and identically distributed i i d therefore results in traditional statistical learning theory are not applicable or at least cannot be applied directly because they are obtained under the sample i i d assumption in this paper we study risk bounds of the learning process for time dependent samples drawn from a leevy process and then analyze the asymptotical behavior of the learning process in particular we first develop the deviation inequalities and the symmetrization inequality for the learning process by using the resultant inequalities we then obtain the risk bounds based on the covering number finally based on the resulting risk bounds we study the asymptotic convergence and the rate of convergence of the learning process for leevy process meanwhile we also give a comparison to the related results under the sample i i d assumption

discontinuous galerkin methods for the helmholtz equation with large wave number this paper develops and analyzes some interior penalty discontinuous galerkin ipdg methods using piecewise linear polynomials for the helmholtz equation with the first order absorbing boundary condition in two and three dimensions it is proved that the proposed discontinuous galerkin methods are stable hence well posed without any mesh constraint for each fixed wave number k optimal order with respect to h error estimate in the broken h 1 norm and suboptimal order estimate in the l 2 norm are derived without any mesh constraint the latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime i e k 2 h gtrsim 1 numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect with respect to k in the error bounds the novelties of the proposed ipdg methods include the following first the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives second the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters since the helmholtz problem is a non hermitian and indefinite linear problem as expected the crucial and the most difficult part of the whole analysis is to establish the stability estimates i e a priori estimates for the numerical solutions to this end the cruxes of our analysis are to establish and to make use of a local version of the rellich identity for the laplacian and to mimic the stability analysis for the pde solutions given in p cummings analysis of finite element based numerical methods for acoustic waves elastic waves and fluid solid interactions in the frequency domain ph d thesis the university of tennessee knoxville tn 2001 p cummings and x feng math models methods appl sci 16 2006 pp 139 160 u hetmaniuk commun math sci 5 2007 pp 665 678

double fold localized multiple matrixized learning machine in this paper we develop an effective multiple matrixized learning machine named double fold localized multiple matrixized learning machine dlmmlm the characteristic of the proposed dlmmlm is that it possesses double folds of local information from data the first fold lies in the whole representation space which consists of different matrix representations it is known that each pattern can be represented by different matrix representations the matrices have their respective representation information and can play different discriminant roles in the final classification therefore from the viewpoint of the whole representation space each matrix has its own local information the second fold is that in each matrix representation learning different patterns represented with the same matrix representation can carry different information therefore in the pattern space with the same matrix size local information of different patterns should be introduced into the classifier design on the whole the advantages of the proposed dlmmlm are i establishing a pattern depended function in the matrixized learning so as to realize different roles of patterns for the first time ii adopting the double fold local information in both the representation space and the pattern space iii proposing a new nonlinear classifier that is different from the state of the art kernelization one and iv getting a tighter empirical generalization risk bound in terms of the rademacher complexity and thus achieving a statistically superior classification performance than those classifiers without the introduction of the double fold local information

structure of indicator function classes with finite vapnik chervonenkis dimensions the vapnik chervonenkis vc dimension is used to measure the complexity of a function class and plays an important role in a variety of fields including artificial neural networks and machine learning one major concern is the relationship between the vc dimension and inherent characteristics of the corresponding function class according to sauer s lemma if the vc dimension of an indicator function class f is equal to d the cardinality of the set fs1n will not be larger than d 0dcnd therefore there naturally arises a question about the vc dimension of an indicator function class what kinds of elements will be contained in the function class f if f has a finite vc dimension in this brief we answer the above question first we investigate the structure of the function class f when the cardinality of the set fs1n reaches the maximum value d 0dcnd based on the derived result we then figure out what kinds of elements will be contained in f if f has a finite vc dimension

rademacher penalties and structural risk minimization we suggest a penalty function to be used in various problems of structural risk minimization this penalty is data dependent and is based on the sup norm of the so called rademacher process indexed by the underlying class of functions sets the standard complexity penalties used in learning problems and based on the vc dimensions of the classes are conservative upper bounds in a probabilistic sense uniformly over the set of all underlying distributions for the penalty we suggest thus for a particular distribution of training examples one can expect better performance of learning algorithms with the data driven rademacher penalties we obtain oracle inequalities for the theoretical risk of estimators obtained by structural minimization of the empirical risk with rademacher penalties the inequalities imply some form of optimality of the empirical risk minimizers we also suggest an iterative approach to structural risk minimization with rademacher penalties in which the hierarchy of classes is not given in advance but is determined in the data driven iterative process of risk minimization we prove probabilistic oracle inequalities for the theoretical risk of the estimators based on this approach as well

managing diversity in regression ensembles ensembles are a widely used and effective technique in machine learning their success is commonly attributed to the degree of disagreement or diversity within the ensemble for ensembles where the individual estimators output crisp class labels this diversity is not well understood and remains an open research issue for ensembles of regression estimators the diversity can be exactly formulated in terms of the covariance between individual estimator outputs and the optimum level is expressed in terms of a bias variance covariance trade off despite this most approaches to learning ensembles use heuristics to encourage the right degree of diversity in this work we show how to explicitly control diversity through the error function the first contribution of this paper is to show that by taking the combination mechanism for the ensemble into account we can derive an error function for each individual that balances ensemble diversity with individual accuracy we show the relationship between this error function and an existing algorithm called negative correlation learning which uses a heuristic penalty term added to the mean squared error function it is demonstrated that these methods control the bias variance covariance trade off systematically and can be utilised with any estimator capable of minimising a quadratic error function for example mlps or rbf networks as a second contribution we derive a strict upper bound on the coefficient of the penalty term which holds for any estimator that can be cast in a generalised linear regression framework with mild assumptions on the basis functions finally we present the results of an empirical study showing significant improvements over simple ensemble learning and finding that this technique is competitive with a variety of methods including boosting bagging mixtures of experts and gaussian processes on a number of tasks

realization of quaternionic discrete series on the unit ball in h d abstract we give a realization for the quaternionic discrete series for the group sp 1 d as the kernel of certain elementary cauchy riemann fueter operator and we compute the corresponding reproducing kernels

quasi conformal actions quaternionic discrete series and twistors su 2 1 and g 2 2 quasi conformal actions were introduced in the physics literature as a generalization of the familiar fractional linear action on the upper half plane to hermitian symmetric tube domains based on arbitrary jordan algebras and further to arbitrary freudenthal triple systems in the mathematics literature quaternionic discrete series unitary representations of real reductive groups in their quaternionic real form were constructed as degree 1 cohomology on the twistor spaces of symmetric quaternionic kahler spaces these two constructions are essentially identical as we show explicitly for the two rank 2 cases su 2 1 and g2 2 we obtain explicit results for certain principal series quaternionic discrete series and minimal representations of these groups including formulas for the lowest k types in various polarizations we expect our results to have applications to topological strings black hole micro state counting and to the theory of automorphic forms

wavenumber explicit convergence analysis for galerkin discretizations of the helmholtz equation we develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems bvps by considering the helmholtz equation at high wavenumber k as our model problem the key element in this theory is a novel k explicit regularity theory for helmholtz bvps that is based on decomposing the solution into two parts the first part has the sobolev regularity properties expected of second order elliptic pdes but features k independent regularity constants the second part is an analytic function for which k explicit bounds for all derivatives are given this decomposition is worked out in detail for several types of bvps namely the helmholtz equation in bounded smooth domains or convex polygonal domains with robin boundary conditions and in exterior domains with dirichlet boundary conditions we present an error analysis for the classical hp version of the finite element method hp fem where the dependence on the mesh width h the approximation order p and the wavenumber k is given explicitly in particular under the assumption that the solution operator for helmholtz problems is polynomially bounded in k it is shown that quasi optimality is obtained under the conditions that kh p is sufficiently small and the polynomial degree p is at least o log k

optimal aggregation of classifiers in statistical learning classification can be considered as nonparametric estimation of sets where the risk is defined by means of a specific distance between sets associated with misclassification error it is shown that the rates of convergence of classifiers depend on two parameters the complexity of the class of candidate sets and the margin parameter the dependence is explicitly given indicating that optimal fast rates approaching o n 1 can be attained where n is the sample size and that the proposed classifiers have the property of robustness to the margin the main result of the paper concerns optimal aggregation of classifiers we suggest a classifier that automatically adapts both to the complexity and to the margin and attains the optimal fast rates up to a logarithmic factor

classification using geometric level sets a variational level set method is developed for the supervised classification problem nonlinear classifier decision boundaries are obtained by minimizing an energy functional that is composed of an empirical risk term with a margin based loss and a geometric regularization term new to machine learning the surface area of the decision boundary this geometric level set classifier is analyzed in terms of consistency and complexity through the calculation of its e entropy for multicategory classification an efficient scheme is developed using a logarithmic number of decision functions in the number of classes rather than the typical linear number of decision functions geometric level set classification yields performance results on benchmark data sets that are competitive with well established methods

simple space time symmetries generalizing conformal field theory we study simple space time symmetry groups g which act on a space time manifold m g h which admits a g invariant global causal structure we classify pairs g m which share the following additional properties of conformal field theory 1 the stability subgroup h of o m is the identity component of a parabolic subgroup of g implying factorization h man where m generalizes lorentz transformations a dilatations and n special conformal transformations 2 special conformal transformations n act trivially on tangent vectors v tom the allowed simple lie groups g are the universal coverings of su m m so 2 d sp l r so 4n and e7 25 and h are particular maximal parabolic subgroups they coincide with the groups of fractional linear transformations of euclidean jordan algebras whose use as generalizations of minkowski space time was advocated by gunaydin mod phys lett a 8 1407 1993 all these groups g admit positive energy representations it will also be shown that the classical confor

pac bayesian analysis of co clustering and beyond we derive pac bayesian generalization bounds for supervised and unsupervised learning models based on clustering such as co clustering matrix tri factorization graphical models graph clustering and pairwise clustering we begin with the analysis of co clustering which is a widely used approach to the analysis of data matrices we distinguish among two tasks in matrix data analysis discriminative prediction of the missing entries in data matrices and estimation of the joint probability distribution of row and column variables in co occurrence matrices we derive pac bayesian generalization bounds for the expected out of sample performance of co clustering based solutions for these two tasks the analysis yields regularization terms that were absent in the previous formulations of co clustering the bounds suggest that the expected performance of co clustering is governed by a trade off between its empirical performance and the mutual information preserved by the cluster variables on row and column ids we derive an iterative projection algorithm for finding a local optimum of this trade off for discriminative prediction tasks this algorithm achieved state of the art performance in the movielens collaborative filtering task our co clustering model can also be seen as matrix tri factorization and the results provide generalization bounds regularization terms and new algorithms for this form of matrix factorization the analysis of co clustering is extended to tree shaped graphical models which can be used to analyze high dimensional tensors according to the bounds the generalization abilities of tree shaped graphical models depend on a trade off between their empirical data fit and the mutual information that is propagated up the tree levels we also formulate weighted graph clustering as a prediction problem given a subset of edge weights we analyze the ability of graph clustering to predict the remaining edge weights the analysis of co clustering easily extends to this problem and suggests that graph clustering should optimize the trade off between empirical data fit and the mutual information that clusters preserve on graph nodes

new concentration inequalities in product spaces we introduce three new ways to measure the distance from a point to a subset of a product space and we prove corresponding concentration inequalities each of them allows to control the fluctuation of a new class of random variables

attribute bagging improving accuracy of classifier ensembles by using random feature subsets we present attribute bagging ab a technique for improving the accuracy and stability of classifier ensembles induced using random subsets of features ab is a wrapper method that can be used with any learning algorithm it establishes an appropriate attribute subset size and then randomly selects subsets of features creating projections of the training set on which the ensemble classifiers are built the induced classifiers are then used for voting this article compares the performance of our ab method with bagging and other algorithms on a hand pose recognition dataset it is shown that ab gives consistently better results than bagging both in accuracy and stability the performance of ensemble voting in bagging and the ab method as a function of the attribute subset size and the number of voters for both weighted and unweighted voting is tested and discussed we also demonstrate that ranking the attribute subsets by their classification accuracy and voting using only the best subsets further improves the resulting performance of the ensemble

harmonic oscillator pre potentials in su 2 lattice gauge theory we write the su 2 lattice gauge theory hamiltonian in d dimensions in terms of pre potentials which are su 2 fundamental doublets of harmonic oscillators the hamiltonian in terms of pre potentials has su 2 u 1 local gauge invariance these pre potentials enable us to solve the su 2 gauss law and characterize the su 2 gauge invariant hilbert space in terms of a set of integers we discuss the consequences of the additional u 1 gauge invariance the extension to su n lattice gauge theory is discussed

convergence analysis for finite element discretizations of the helmholtz equation with dirichlet to neumann boundary conditions a rigorous convergence theory for galerkin methods for a model helmholtz problem in bbb r d d in 1 2 3 is presented general conditions on the approximation properties of the approximation space are stated that ensure quasi optimality of the method as an application of the general theory a full error analysis of the classical hp version of the finite element method is presented for the model problem where the dependence on the mesh width h the approximation order p and the wave number k is given explicitly in particular it is shown that quasi optimality is obtained under the conditions that kh p is sufficiently small and the polynomial degree p is at least o log k

learning by canonical smooth estimation ii learning and choice of model complexity this paper analyzes the properties of a procedure for learning from examples this canonical learner is based on a canonical error estimator developed in part i in learning problems one can observe data that consists of labeled sample points and the goal is to find a model or hypothesis from a set of candidates that will accurately predict the labels of new sample points the expected mismatch between a hypothesis prediction and the actual label of a new sample point is called the hypothesis generalization error we compare the canonical learner with the traditional technique of finding hypotheses that minimize the relative frequency based empirical error estimate it is shown that for a broad class of learning problems the set of cases for which such empirical error minimization works is a proper subset of the cases for which the canonical learner works we derive bounds to show that the number of samples required by these two methods is comparable we also address the issue of how to determine the appropriate complexity for the class of candidate hypotheses

ipcdgm and multiscale ipdpgm for the helmholtz problem with large wave number abstract in this paper based on the two nested spaces of piecewise linear polynomials on fine and coarse meshes with sizes h and h respectively which are continuous in macro elements but discontinuous across interior edges faces of them an interior penalty continuous discontinuous galerkin method ipcdgm and a multiscale interior penalty discontinuous petrov galerkin method msipdpgm are proposed to solve the helmholtz problem with large wave number k and homogeneous robin boundary condition this paper devotes to analyzing the preasymptotic error of the two methods separately in order to reduce the pollution errors efficiently the two methods not only include the penalty terms on jumps of function on coarse mesh edges faces but also add the penalty terms on jumps of first order normal derivatives on fine mesh edges faces the error of the ipcdg solution in the broken h 1 norm is bounded by o k h k 3 h 2 by splitting into coarse and fine scales and basing on the ipcdg variational formulation the msipdpgm s trial and test spaces are constructed with the macro corrected bases due to the exponential decay of the correctors the corrected bases are obtained by solving the local problems on localized subdomains of size l h l being the oversampling parameter the preasymptotic error analysis of msipdpgm shows that if k h and log k l fall below some constants and if the fine mesh size h is sufficient small the errors of numerical solutions in the broken h 1 norm can be dominated by o h 3 without pollution effect numerical tests are provided to verify the theoretical findings and advantages of the two methods

harmonic analysis on spannors the spannor representation of the universal covering group of the conformal group of minkowski space time is studied herein it is an example of an indecomposable representation various parallelizations for spannors are described normalized k finite basis fields for the spannors are introduced and a computation of the actions of the scale generator and other generators of the conformal group on these basis fields is given the irreducibility of all irreducible composition factors is established and it is decided which ones are infinitesimally unitary two sets of generators for maximal abelian subalgebras of the enveloping algebra of the conformal group are introduced the action of one of the sets on k finite basis fields in the spannor representation is studied this is important for understanding of the problem of degeneracy in the spannor representation since k types occur with multiplicity 2 many calculations make use of the lowest highest weight module structures of the unitarizable and positive negative energy irreducible composition factors and thus many of our results and techniques can be used to initiate a study of the quantum deformations of the associated lie algebra representations results on the poincare content of the unitarizable irreducible composition factors are stated and unitary equivalences between certain of these factors are established

minimal representations spherical vectors and exceptional theta series theta series for exceptional groups have been suggested as a possible description of the eleven dimensional quantum supermembrane we present explicit formulae for these automorphic forms whenever the underlying lie group g is split or complex and simply laced specifically we review and construct explicitly the minimal representation of g generalizing the schrodinger representation of symplectic groups we compute the spherical vector in this representation i e the wave function invariant under the maximal compact subgroup which plays the role of the summand in the automorphic theta series we also determine the spherical vector over the complex field we outline how the spherical vector over the p adic number fields provides the summation measure in the theta series postponing its determination to a sequel of this work the simplicity of our result is suggestive of a new born infeld like description of the membrane where u duality is realized non linearly our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

minimal unitary representation of su 2 2 and its deformations as massless conformal fields and their supersymmetric extensions we study the minimal unitary representation minrep of so 4 2 over a hilbert space of functions of three variables obtained by quantizing its quasiconformal action on a five dimensional space the minrep of so 4 2 which coincides with the minrep of su 2 2 similarly constructed corresponds to a massless conformal scalar in four space time dimensions there exists a one parameter family of deformations of the minrep of su 2 2 for positive negative integer values of the deformation parameter one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in 0 2 2 0 representation of the sl 2 c subgroup we construct the supersymmetric extensions of the minrep of su 2 2 and its deformations to those of su 2 2 n the minimal unitary supermultiplet of su 2 2 4 in the undeformed case simply corresponds to the massless n 4 yang mills supermultiplet in four dimensions for each given nonzero integer value of one obtains a uniq

a novel multi view learning developed from single view patterns the existing multi view learning mvl learns how to process patterns with multiple information sources in generalization this mvl is proven to have a significant advantage over the usual single view learning svl however in most real world cases we only have single source patterns to which the existing mvl is unable to be directly applied this paper aims to develop a new mvl technique for single source patterns to this end we first reshape the original vector representation of single source patterns into multiple matrix representations in doing so we can change the original architecture of a given base classifier into different sub ones each newly generated sub classifier can classify the patterns represented with the matrix here each sub classifier is taken as one view of the original base classifier as a result a set of sub classifiers with different views are come into being then one joint rather than separated learning process for the multi view sub classifiers is developed in practice the original base classifier employs the vector pattern oriented ho kashyap classifier with regularization learning called mhks as a paradigm which is not limited to mhks thus the proposed joint multi view learning is named as multiv mhks finally the feasibility and effectiveness of the proposed multiv mhks is demonstrated by the experimental results on benchmark data sets more importantly we have demonstrated that the proposed multi view approach generally has a tighter generalization risk bound than its single view one in terms of the rademacher complexity analysis

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

conformal and quasiconformal realizations of exceptional lie groups we present a nonlinear realization of e8 8 on a space of 57 dimensions which is quasiconformal in the sense that it leaves invariant a suitably defined light cone in 57 this realization which is related to the freudenthal triple system associated with the unique exceptional jordan algebra over the split octonions contains previous conformal realizations of the lower rank exceptional lie groups on generalized space times and in particular a conformal realization of e7 7 on 27 which we exhibit explicitly possible applications of our results to supergravity and m theory are briefly mentioned

preasymptotic error analysis of cip fem and fem for helmholtz equation with high wave number part ii hp version in this paper which is the second in a series of two the preasymptotic error analysis of the continuous interior penalty finite element method cip fem and the fem for the helmholtz equation in two and three dimensions is continued while part i contained results on the linear cip fem and fem the present part deals with approximation spaces of order p ge 1 by using a modified duality argument preasymptotic error estimates are derived for both methods under the condition of frac kh p le c 0 frac p k frac 1 p 1 where k is the wave number h is the mesh size and c 0 is a constant independent of k h p and the penalty parameters it is shown that the pollution errors of both methods in h 1 norm are o k 2p 1 h 2p if p o 1 and are o frac k p 2 frac kh sigma p 2p if the exact solution u in h 2 omega which coincide with existent dispersion analyses for the fem on cartesian grids here sigma is a constant independent of k h p and the penalty par

elementary representations and intertwining operators for su 2 2 i the structure of the group su 2 2 and of its lie algebra is studied in detail the results will be applied in subsequent parts devoted to the explicit construction of elementary representations of su 2 2 induced from different parabolic subgroups and of the intertwining operators between these representations a summary of some results of parts ii and iii is given

convergence of a discontinuous galerkin multiscale method we present a discontinuous galerkin multiscale method for second order elliptic problems and prove convergence we consider a heterogeneous and highly varying diffusion coefficient in l infty omega mathbb r d times d sym with uniform spectral bounds without any assumption on scale separation or periodicity the multiscale method uses a corrected basis that is computed on patches subdomains the error due to truncation of the corrected basis decreases exponentially with the size of the patches hence to achieve an algebraic convergence rate of the multiscale solution on a uniform mesh with mesh size h to a reference solution it is sufficient to choose the patch sizes mathcal o h log h we also discuss a way to further localize the corrected basis to elementwise support improved convergence rate can be achieved depending on the piecewise regularity of the forcing function linear convergence in energy norm and quadratic convergence in the l 2 norm is obtained independently of the

unitary representations of non compact supergroups we give a general theory for the construction of oscillator like unitary irreducible representations uirs of non compact supergroups in a super fock space this construction applies to all non compact supergroupsg whose coset spaceg k with respect to their maximal compact subsupergroupk is hermitean supersymmetric we illustrate our method with the example of su m p n q by giving its oscillator like uirs in a particle state basis as well as supercoherent state basis the same class of uirs can also be realized over the super hilbert spaces of holomorphic functions of az variable labelling the coherent states

multiple rank multi linear svm for matrix data classification matrices or more generally multi way arrays tensors are common forms of data that are encountered in a wide range of real applications how to classify this kind of data is an important research topic for both pattern recognition and machine learning in this paper by analyzing the relationship between two famous traditional classification approaches i e svm and stm a novel tensor based method i e multiple rank multi linear svm mrmlsvm is proposed different from traditional vector based and tensor based methods multiple rank left and right projecting vectors are employed to construct decision boundary and establish margin function we reveal that the rank of transformation can be regarded as a tradeoff parameter to balance the capacity of learning and generalization in essence we also proposed an effective approach to solve the proposed non convex optimization problem the convergence behavior initialization computational complexity and parameter determination problems are analyzed compared with vector based classification methods mrmlsvm achieves higher accuracy and has lower computational complexity compared with traditional supervised tensor based methods mrmlsvm performs better for matrix data classification promising experimental results on various kinds of data sets are provided to show the effectiveness of our method highlightswe have proposed a novel method i e mrmlsvm for matrix data classification we have revealed the essence of mrmlsvm from the view of learning theory we have provided an effective way to solve the proposed non convex problem our proposed approach is a common model and mrmlsvm is just an instance

improved sample complexity estimates for statistical learning control of uncertain systems probabilistic methods and statistical learning theory have been shown to provide approximate solutions to difficult control problems unfortunately the number of samples required in order to guarantee stringent performance levels may be prohibitively large this paper introduces bootstrap learning methods and the concept of stopping times to drastically reduce the bound on the number of samples required to achieve a performance level we then apply these results to obtain more efficient algorithms which probabilistically guarantee stability and robustness levels when designing controllers for uncertain systems

wavenumber explicit analysis of a dpg method for the multidimensional helmholtz equation we study the properties of a novel discontinuous petrov galerkin dpg method for acoustic wave propagation the method yields hermitian positive definite matrices and has good pre asymptotic stability properties numerically we find that the method exhibits negligible phase errors otherwise known as pollution errors even in the lowest order case theoretically we are able to prove error estimates that explicitly show the dependencies with respect to the wavenumber the mesh size h and the polynomial degree p but the current state of the theory does not fully explain the remarkably good numerical phase errors theoretically comparisons are made with several other recent works that gave wave number explicit estimates numerically comparisons are made with the standard finite element method and its recent modification for wave propagation with clever quadratures the new dpg method is designed following the previously established principles of optimal test functions in addition to the nonstandard test functions in this work we also use a nonstandard wave number dependent norm on both the test and trial space to obtain our error estimates

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

group representations arising from lorentz conformal geometry abstract it is shown that there exist conformally covariant differential operators d 2 l k of all even orders 2 l on differential forms of all orders k in the double cover n of the n dimensional compactified minkowski space n these act as intertwining differential operators for natural representations of o 2 n the conformal group of n for even n the resulting decompositions of differential form representations of o 2 n the orthochronous conformal group produce infinite families of unitary representations the most interesting of which are carried by positive mass squared positive frequency quotients for 2l n 2k physically these generalize unitary representations of the conformal group associated with the modified wave operator d 2 0 n 2 2 2 and the maxwell operator on vector potentials d 2 n 2 2 d all the representation spaces produced unitary and nonunitary may be viewed as infinite systems of harmonic oscillators as a by product of the spectral resolution of the d 2 l k one gets some striking wave propagative properties for all of the equations d 2 l k 0 including huygens principle in the curved spacetime n the operators d 2 l k have not been seen before except in the special cases k 0 or n and k n 2 2 l 1 the maxwell operator thus much new information is obtained even in the physical case n 4

martingale inequalities and np complete problems we use martingale inequalities in the probabilistic analysis of stochastic models of np complete problems we give examples of applications to the bin packing and traveling salesman problems in both cases we obtain sharp bounds on pr un eun t where un is the objective function value of the problem with n random data

hyperkahler and quaternionic kahler geometry a quaternion hermitian manifold of dimension at least 12 with closed fundamental 4 form is shown to be quaternionic kahler a similar result is proved for 8 manifolds hyperkahler metrics are constructed on the fundamental quaternionic line bundle with the zero section removed of a quaternionic kahler manifold indefinite if the scalar curvature is negative this construction is compatible with the quaternionic kahler and hyperkahier quotient constructions and allows quaternionic kahler geometry to be subsumed into the theory of hyperkahler manifolds it is shown that the hyperkahler metrics that arise admit a certain type of su 2 action possess functions which are kahler potentials for each of the complex structures simultaneously and determine quaternionic kahler structures via a variant of the moment map construction quaternionic kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperkahler metrics with circle actions on complex line bundles over kahler einstein complex contact manifolds nilpotent orbits in a complex semi simple lie algebra with the hyperkahler metrics defined by kronheimer are shown to give rise to quaternionic kahler metrics and various examples of these metrics are identified it is shown that any quaternionic kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds the twistor space structure of the projectivised nilpotent orbits is studied

su n irreducible schwinger bosons we construct su n irreducible schwinger bosons satisfying certain u n 1 constraints which implement the symmetries of su n young tableaues as a result all su n irreducible representations are simple monomials of n 1 types of su n irreducible schwinger bosons further we show that these representations are free of multiplicity problems thus all su n representations are made as simple as su 2

a new lda based face recognition system which can solve the small sample size problem a new lda based face recognition system is presented in this paper linear discriminant analysis lda is one of the most popular linear projection techniques for feature extraction the major drawback of applying lda is that it may encounter the small sample size problem in this paper we propose a new lda based technique which can solve the small sample size problem we also prove that the most expressive vectors derived in the null space of the within class scatter matrix using principal component analysis pca are equal to the optimal discriminant vectors derived in the original space using lda the experimental results show that the new lda process improves the performance of a face recognition system signi cantly 2000 pattern recognition society published by elsevier science ltd all rights reserved

unitary highest weight representations of noncompact supergroups the oscillator method for the construction of unitary highest or lowest weight representations of noncompact groups and supergroups is generalized within this generalization the method yields unitary highest weight representations of all simple supergroups whose even subgroups are in the form of a direct product of a compact group with a simple noncompact group the method is illustrated by studying in detail the unitary highest weight representations of the supergroup osp 2n 1 2m r the generalized supercoherent states associated with these unitary representations are also defined

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

quantum field theory on quantized bergman domain we present an oscillator realization of discrete series representations of group su 2 2 we give formulas for the coherent state star product quantization of a bergman domain d a formulation of a regularized noncommutative scalar field on a quantized d is given

localization of elliptic multiscale problems this paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients the basis functions are solutions of local problems on vertex patches the error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches hence on a uniform mesh of size h patches of diameter h log 1 h are sufficient to preserve a linear rate of convergence in h without pre asymptotic or resonance effects the analysis does not rely on regularity of the solution or scale separation in the coefficient this result motivates new and justifies old classes of variational multiscale methods see more at http www ams org journals mcom 2014 83 290 s0025 5718 2014 02868 8 sthash z2ccfxig dpuf

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

the unitary irreducible representations of su 2 1 this paper analyzes the irreducible unitary representations of su 2 1 in a basis labeled as p q jmy where p q correspond to quantum numbers associated with the quadratic and cubic casimir operators j m label states of the su 2 subgroup and y labels the quantum number with respect to the u 1 subgroup all the irreducible representations are found and the allowed range of these quantum numbers for each representation are given the results are expressed in the form of diagrams that show the allowed values in a j y plot for fixed values of p q a p q plot is also provided that indicates the allowed values of these quantum numbers

quaternionic analysis representation theory and physics we develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group we first review the cauchy fueter and poisson formulas and explain their representation theoretic meaning the requirement of unitarity of representations leads us to the extensions of these formulas in the minkowski space which can be viewed as another real form of quaternions representation theory also suggests a quaternionic version of the cauchy formula for the second order pole remarkably the derivative appearing in the complex case is replaced by the maxwell equations in the quaternionic counterpart we also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory such as the spectrum of the hydrogen atom polarization of vacuum one loop feynman integrals we also make some further conjectures the main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis representation theory and four dimensional physics

oscillator like unitary representations of non compact groups with a jordan structure and the non compact groups of supergravity we give a general bosonic construction of oscillator like unitary irreducible representations uir of non compact groups whose coset spaces with respect to their maximal compact subgroups are hermitian symmetric with the exception of e7 7 they include all the non compact invariance groups of extended supergravity theories in four dimensions these representations have the remarkable property that each uir is uniquely determined by an irreducible representation of the maximal compact subgroup we study the connection between our construction the hermitian symmetric spaces and the tits koecher construction of the lie algebras of corresponding groups we then give the bosonic construction of the lie algebra ofe7 7 in su 8 so 8 and u 7 bases and study its properties application of our method toe7 7 leads to reducible unitary representations

algorithm dependent generalization bounds for multi task learning often tasks are collected for multi task learning mtl because they share similar feature structures based on this observation in this paper we present novel algorithm dependent generalization bounds for mtl by exploiting the notion of algorithmic stability we focus on the performance of one particular task and the average performance over multiple tasks by analyzing the generalization ability of a common parameter that is shared in mtl when focusing on one particular task with the help of a mild assumption on the feature structures we interpret the function of the other tasks as a regularizer that produces a specific inductive bias the algorithm for learning the common parameter as well as the predictor is thereby uniformly stable with respect to the domain of the particular task and has a generalization bound with a fast convergence rate of order mathcal o 1 n where n is the sample size of the particular task when focusing on the average performance over multiple tasks we prove that a similar inductive bias exists under certain conditions on the feature structures thus the corresponding algorithm for learning the common parameter is also uniformly stable with respect to the domains of the multiple tasks and its generalization bound is of the order mathcal o 1 t where t is the number of tasks these theoretical analyses naturally show that the similarity of feature structures in mtl will lead to specific regularizations for predicting which enables the learning algorithms to generalize fast and correctly from a few examples

symmetry analysis of wave equation on static spherically symmetric spacetimes with higher symmetries symmetry analysis of wave equation on all static spherically symmetric spacetimes admitting maximal isometry groups g10 or g7 or g6 is carried out symmetry algebras of the wave equation are found and their structural information in the sense of iwasawa decomposition is obtained joint invariants of appropriate subalgebras are utilized to obtain many exact solutions of the wave equation on static spherically symmetric spacetimes

preasymptotic error analysis of higher order fem and cip fem for helmholtz equation with high wave number a preasymptotic error analysis of the finite element method fem and some continuous interior penalty finite element method cip fem for the helmholtz equation in two and three dimensions is proposed h 1 and l 2 error estimates with explicit dependence on the wave number k are derived in particular it is shown that if k 2p 1 h 2p is sufficiently small then the pollution errors of both methods in h 1 norm are bounded by o k 2p 1 h 2p which coincides with the phase error of the fem obtained by existent dispersion analyses on cartesian grids where h is the mesh size and p is the order of the approximation space and is fixed the cip fem extends the classical one by adding more penalty terms on jumps of higher up to p th order normal derivatives in order to reduce efficiently the pollution errors of higher order methods numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the cip fem in reducing the pollution effect

generalization error of linear discriminant analysis in spatially correlated sensor networks generalization error the probability of error of a detection rule learned from training samples on new unseen samples is a fundamental quantity to be characterized however characterizations of generalization error in the statistical learning theory literature are often loose and practically unusable for optimizing detection systems in this work focusing on learning linear discriminant analysis detection rules from spatially correlated sensor measurements a tight generalization error approximation is developed that can be used to optimize the parameters of a sensor network detection system as such the approximation is used to optimize network settings the approximation is also used to derive a detection error exponent and select an optimal subset of deployed sensor nodes a gauss markov random field is used to model correlation and weak laws of large numbers in geometric probability are employed in the analysis