gtm the generative topographic mapping latent variable models represent the probability density of data in a space of several dimensions in terms of a smaller number of latent or hidden variables a familiar example is factor analysis which is based on a linear transformations between the latent space and the data space in this paper we introduce a form of non linear latent variable model called the generative topographic mapping for which the parameters of the model can be determined using the em algorithm gtm provides a principled alternative to the widely used self organizing map som of kohonen 1982 and overcomes most of the significant limitations of the som we demonstrate the performance of the gtm algorithm on a toy problem and on simulated data from flow diagnostics for a multi phase oil pipeline

multiplicity one representations of divergence free lie algebras abstract divergence free lie algebras also known as the special lie algebras of cartan type are lie algebras of volume preserving transformation groups they are simple in generic case dokovic and zhao found a certain graded generalization of them in this paper we classify all the irreducible and indecomposable multiplicity one modules of the simple generalized divergence free lie algebras

dimensionality reduction using principal component analysis applied to the gradient the cost of dimensionality reduction in aerodynamic design applications involving high dimensional design spaces and computational fluid dynamics is often prohibitive in an attempt to overcome this challenge a new method for dimensionality reduction is presented that scales as p log p where p is the number of design variables it works by taking advantage of adjoint design methods to compute the covariance matrix of the gradient this information is then used with principal component analysis to develop a linear transformation that allows an aerodynamic optimization problem to be reformulated in an equivalent coordinate system of lower dimensionality to demonstrate its feasibility the method is tested on a two dimensional staggered airfoil problem intentionally chosen as an abstraction of a more realistic overwing nacelle integration problem and shown to exhibit similarities with the latter results show that the method rivals typical screening methods used in aerospace engineering in terms of eff

doubly z graded lie algebras containing a virasoro algebra abstract in this paper we study z z graded lie algebras a i j zai j over a characteristic 0 field f with dim ai j 1 for each i and j where z denotes the integers we shall assume that a satisfies the following three properties i vir the centerless virasoro algebra ii 0 and ad and ad act faithfully on the subspace iii dim dim 1 and is generated by and the main result of the paper is the classification of these algebras more precisely a is isomorphic to a simple block algebra l a which has basis ei j i j z z 1 a 0 2 a 0 and subject to the bracket formula for some suitable constant a f 0 1 2 or is isomorphic to a one or two dimensional central extension of l a with a f 0 1 2 or a is isomorphic a one dimensional derivation extension of l 1 and l 2 or a two dimensional extension of l 1 by a center element and a derivation

the virasoro algebra three theroems are proved with suitable hypotheses in each case characterizations are found for the virasoro algebra for some of its representations and for the ramond neveu schwarz superalgebra built around the virasoro algebra

probabilistic non linear principal component analysis with gaussian process latent variable models summarising a high dimensional data set with a low dimensional embedding is a standard approach for exploring its structure in this paper we provide an overview of some existing techniques for discovering such embeddings we then introduce a novel probabilistic interpretation of principal component analysis pca that we term dual probabilistic pca dppca the dppca model has the additional advantage that the linear mappings from the embedded space can easily be non linearised through gaussian processes we refer to this model as a gaussian process latent variable model gp lvm through analysis of the gp lvm objective function we relate the model to popular spectral techniques such as kernel pca and multidimensional scaling we then review a practical algorithm for gp lvms in the context of large data sets and develop it to also handle discrete valued data and missing attributes we demonstrate the model on a range of real world and artificially generated data sets

two dimensional generalized toda lattice the zero curvature representation is obtained for the two dimensional generalized toda lattices connected with semisimple lie algebras the reduction group and conservation laws are found and the mass spectrum is calculated

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

constrained design optimization using generative topographic mapping high dimensional design optimization problems involving complex and time consuming solvers present computational challenges and are expensive to execute even though surrogate models can replace these expensive problems with simpler models the initial design of experiment for constructing these models effectively is still exponential to the dimension of the problem traditional screening methods in optimization reduce the dimension of the problem by discarding variables which is undesirable in this paper a latent variable model called generative topographic mapping is proposed to reduce the dimension of the problem so as to facilitate an optimization search in a low dimensional space without removing any variables from the design problem the method works by transforming high dimensional data to be embedded on a low dimensional manifold it is demonstrated on a two dimensional branin function subjected to nonlinear constraints and then applied to real engineering constrained optimization problems of a

semigroup graded algebras and codimension growth of graded polynomial identities abstract we show that if t is any of four semigroups of two elements that are not groups there exists a finite dimensional associative t graded algebra over a field of characteristic 0 such that the codimensions of its graded polynomial identities have a non integer exponent of growth in particular we provide an example of a finite dimensional graded simple semigroup graded algebra over an algebraically closed field of characteristic 0 with a non integer graded pi exponent which is strictly less than the dimension of the algebra however if t is a left or right zero band and the t graded algebra is unital or t is a cancellative semigroup then the t graded algebra satisfies the graded analog of amitsur s conjecture i e there exists an integer graded pi exponent moreover in the first case it turns out that the ordinary and the graded pi exponents coincide in addition we consider related problems on the structure of semigroup graded algebras

classification of irreducible harish chandra modules over gap p virasoro algebras abstract we prove that any irreducible harish chandra module for a class of lie algebras which we call gap p virasoro algebras must be a highest weight module a lowest weight module or a module of intermediate series these algebras are closely related to the heisenberg virasoro algebra and the algebra of derivations over a quantum torus they also contain subalgebras which are isomorphic to the virasoro algebra vir but graded by p z unlike vir by z

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

graded polynomial identities and codimensions computing the exponential growth abstract let g be a finite abelian group and a a g graded algebra over a field of characteristic zero this paper is devoted to a quantitative study of the graded polynomial identities satisfied by a we study the asymptotic behavior of c n g a n 1 2 the sequence of graded codimensions of a and we prove that if a satisfies an ordinary polynomial identity lim n c n g a n exists and is an integer we give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple g z 2 graded algebra related to a

factorial sampling plans for preliminary computational experiments a computational model is a representation of some physical or other system of interest first expressed mathematically and then implemented in the form of a computer program it may be viewed as a function of inputs that when evaluated produces outputs motivation for this article comes from computational models that are deterministic complicated enough to make classical mathematical analysis impractical and that have a moderate to large number of inputs the problem of designing computational experiments to determine which inputs have important effects on an output is considered the proposed experimental plans are composed of individually randomized one factor at a time designs and data analysis is based on the resulting random sample of observed elementary effects those changes in an output due solely to changes in a particular input advantages of this approach include a lack of reliance on assumptions of relative sparsity of important inputs monotonicity of outputs with respect to inputs or ad

a global version of grozman s theorem let x be a manifold the classification of all equivariant bilinear maps between tensor density modules over x has been investigated by grozman funct anal appl 14 2 58 59 1980 who has provided a full classification for those which are differential operators here we investigate the same question without the hypothesis that the maps are differential operators in our paper the geometric context is algebraic geometry and the manifold x is the circle text spec mathbb c z z 1 our main motivation comes from the fact that such a classification is required to complete the proof of the main result of iohara and mathieu proc lond math soc 2012 in press indeed it requires to also include the case of deformations of tensor density modules

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

asymptotics of h identities for associative algebras with an h invariant radical abstract we prove the existence of the hopf pi exponent for finite dimensional associative algebras a with a generalized hopf action of an associative algebra h with 1 over an algebraically closed field of characteristic 0 assuming only the invariance of the jacobson radical j a under the h action and the existence of the decomposition of a j a into the sum of h simple algebras as a consequence we show that the analog of amitsur s conjecture holds for g codimensions of finite dimensional associative algebras over a field of characteristic 0 with an action of an arbitrary group g by automorphisms and anti automorphisms and for differential codimensions of finite dimensional associative algebras with an action of an arbitrary lie algebra by derivations

dimension reduction for aerodynamic design optimization the search for an optimal design in a high dimensional design space of a multivariate problem requires a sample size proportional or even exponential to the number of variables of the problem this curse of dimensionality places a computational burden on the cost of optimization especially when the problem uses expensive high fidelity simulations and may force one to try to reduce the dimensions of a problem traditional variable screening techniques reduce the dimensionality of the problem by removing variables that seem irrelevant to the design problem this practice fails when all the variables are equally relevant in the problem or when some variables are relevant only in some parts of the design space the present work describes a dimension reduction method called generative topographic mapping based on non linear latent models which transform a high dimensional data set into a low dimensional latent space without removing any variables it is first illustrated on a two dimensional branin function and then applied to a thirty dimensional airfoil problem the method is then compared with a global optimizer a genetic algorithm other dimension reduction methods principle component analysis and gaussian process latent variable models and with kriging surrogate models the method improves when the initial sample used for dimension reduction is filtered to contain only good designs

the generalized additional symmetries of the two toda lattice hierarchy the generalized additional symmetries of the two toda lattice hierarchy are investigated in this paper the algebraic structure of this generalized additional symmetry is showed as w w and the actions of the generalized additional symmetries on function are also discussed by restricting the two toda lattice hierarchy to the semi infinite case

graded polynomial identities group actions and exponential growth of lie algebras abstract consider a finite dimensional lie algebra l with an action of a finite group g over a field of characteristic 0 we prove the analog of amitsur s conjecture on asymptotic behavior for codimensions of polynomial g identities of l as a consequence we prove the analog of amitsur s conjecture for graded codimensions of any finite dimensional lie algebra graded by a finite abelian group

toda lattice hierarchy and generalized string equations string equations of thepth generalized kontsevich model and the compactifiedc 1 string theory are re examined in the language of the toda lattice hierarchy as opposed to a hypothesis postulated in the literature the generalized kontsevich model atp 1 does not coincide with thec 1 string theory at selfdual radius a broader family of solutions of the toda lattice hierarchy including these models is constructed and shown to satisfy generalized string equations the status of a variety ofc 1 string models is discussed in this new framework

identities of graded algebras and codimension growth let a g g a g be a g graded associative algebra over a field of characteristic zero in this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component a e to that of the whole of a in the case where the support of the grading is finite we prove the conjecture in several natural cases one of them being the case where a is finite dimensional and a e has polynomial growth

gauge theory of w symmetry the w algebra is a higher spin extension of the virasoro algebra in this paper we construct the gauge theory of w symmetry in terms of its representations

geometric filtration using proper orthogonal decomposition for aerodynamic design optimization when carrying out design searches traditional variable screening techniques can find it extremely difficult to distinguish between important and unimportant variables this is particularly true when only a small number of simulations is combined with a parameterization which results in a large number of variables of seemingly equal importance here the authors present a variable reduction technique which employs proper orthogonal decomposition to filter out undesirable or badly performing geometries from an optimization process unlike traditional screening techniques the presented method operates at the geometric level instead of the variable level the filtering process uses the designs which result from a geometry parameterization instead of the variables which control the parameterization the method is shown to perform well in the optimization of a two dimensional airfoil for the minimization of drag to lift ratio producing designs better than those resulting from traditional kriging based surrogate model optimization and with a significant reduction in surrogate tuning cost

screening by conference designs summary screening experiments aim to identify the relevant variables within some process potentially depending on a large number of variables in this paper we introduce a new class of experimental designs called edge designs these designs allow a model independent estimate of the set of relevant variables thus providing more robustness than traditional designs we give a bound on the determinant of the information matrix of certain edge designs and show that a large class of edge designs meeting this bound can be constructed from conference matrices we also show that the resulting conference designs have an optimal space exploration property which guards against unexpected nonlinearities we survey the existence of and constructions for conference matrices and for n 50 variables give explicit such matrices when n is a prime and references to explicit constructions otherwise

optimized nonuniform rational b spline geometrical representation for aerodynamic design of wings the geometric representation and parameterization used in an aerodynamic wing design process determines the number of design variables and influences the smoothness of the wing representation in an attempt to reduce the number of design variables while preserving good smoothness properties the present research investigates the performance of an optimized nonuniform rational b spline nurbs geometrical representation for the aerodynamic design of wings as a first step an approach is described whereby optimal spatial positions and weights of a fixed number of nurbs control points is determined using a quasi newton optimization algorithm to approximate a general airfoil section the resulting optimized nurbs representation significantly reduces the number of design variables needed to define accurately a wing section while ensuring good smoothness properties in a second step the nurbs control point positions and weights are used as design variables in an aerodynamic optimization problem this methodology results in a rapid and robust design process as illustrated by examples of aerodynamic optimization for two and three dimensional cases

choosing the sample size of a computer experiment a practical guide we provide reasons and evidence supporting the informal rule that the number of runs for an effective initial computer experiment should be about 10 times the input dimension our arguments quantify two key characteristics of computer codes that affect the sample size required for a desired level of accuracy when approximating the code via a gaussian process gp the first characteristic is the total sensitivity of a code output variable to all input variables the second corresponds to the way this total sensitivity is distributed across the input variables specifically the possible presence of a few prominent input factors and many impotent ones i e effect sparsity both measures relate directly to the correlation structure in the gp approximation of the code in this way the article moves toward a more formal treatment of sample size for a computer experiment the evidence supporting these arguments stems primarily from a simulation study and via specific codes modeling climate and ligand activa

irreducible representations of virasoro toroidal lie algebras toroidal lie algebras and their vertex operator representations were introduced in mey and a class of indecomposable modules were investigated in this work we extend the toroidal algebra by the virasoro algebra thus constructing a semi direct product algebra containing the toroidal algebra as an ideal and the virasoro algebra as a subalgebra with the use of vertex operators and certain oscillator representations of the virasoro algebra it is proved that the corresponding fock space gives rise to a class of irreducible modules for the virasoro toroidal algebra

classification of simple lie algebras on a lattice let z for some n 1 the aim of the paper is to classify all simple graded lie algebras l l such that diml 1 for all the classification involves two affine lie algebras namely a 1 1 and a 2 2 and a familly w parametrized by a dense open set of the space of all embeddings c the family wl of generalized witt algebras indexed by all embeddings l c appears as a sub family in general the algebras w are described as lie algebras of symbols of twisted pseudo differential operators

on additional symmetries of the kp hierarchy and sato s backlund transformation a short proof is given to the fact that the additional symmetries of the kp hierarchy defined by their action on pseudodifferential operators according to fuchssteiner chen lee lin orlov shulman coincide with those defined by their action on functions as sato s backlund transformations a new simple formula for the generator of additional symmetries is also presented

searching for important factors in simulation models with many factors sequential bifurcation abstract this paper deals with the problem of screening that is how to find the important factors in simulation models that have many for example 300 factors also called simulation parameters or input variables screening assumes that only a few factors are really important parsimony principle this paper solves the screening problem by a novel technique called sequential bifurcation this technique is both effective and efficient that is it does find all important factors yet it requires relatively few simulation runs the technique is demonstrated through a realistic case study concerning a complicated simulation model called image this simulation models the greenhouse phenomenon the worldwide increase of temperatures this case study gives surprising results the technique identifies some factors as being important that the ecological experts initially thought to be unimportant sequential bifurcation assumes that the input output behavior of the simulation model may be approximated by a first order polynomial main effects possibly augmented with interactions between factors the technique is sequential that is it specifies and analyzes simulation runs one after the other

principal manifolds and nonlinear dimensionality reduction via tangent space alignment we present a new algorithm for manifold learning and nonlinear dimensionality reduction based on a set of unorganized data points sampled with noise from a parameterized manifold the local geometry of the manifold is learned by constructing an approximation for the tangent space at each data point and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold we also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases we illustrate our algorithm using curves and surfaces both in two dimensional three dimensional 2d 3d euclidean spaces and in higher dimensional euclidean spaces we also address several theoretical and algorithmic issues for further research and improvements

weight modules with a finite dimensional weight space over truncated virasoro algebras truncated virasoro algebras can be viewed as quotient algebras of the loop virasoro algebra in this paper we prove that any weight module m with a finite dimensional weight space over any truncated virasoro algebra l f vir c t 1 fc t 1 has a harish chandra submodule where vir is the virasoro algebra and f is a nonzero and not invertible laurent polynomial in c t 1 more precisely if m has a nontrivial upper bounded vir submodule then it has a highest weight l f submodule if m has a nontrivial lower bounded vir submodule then it has a lowest weight l f submodule if m is nontrivial over l f and does not contain any nontrivial upper bounded nor any nontrivial lower bounded vir submodules then it has a nontrivial harish chandra l f submodule s such that dims 1 for all supp s 0 as a corollary any irreducible weight module with a finite dimensional weight space over l f is a harish chandra module