automorphic vector bundles on connected shimura varieties introduction 0 review of terminology concerning shimura varieties 1 the taniyama group the period torsor and conjugates of shimura varieties 2 the compact dual symmetric hermitian space and its conjugates 3 the principal bundle y x statement of the first main theorem 4 automorphic vector bundles 5 conjugates of automorphic vector bundles 6 proof of theorem 3 10 for the symplectic group 7 proof of theorem 3 10 for connected shimura varieties of abelian type 8 first completion of the proof of theorem 3 10 9 second completion of the proof of theorem 3 10 appendix pairs defining connected and nonconnected shimura varieties bibliography 91 93 95 101 105 112 115 118 121 123 124 126 128

higher coherent cohomology and p adic modular forms of singular weights we investigate the p adic properties of higher coherent cohomology of automorphic vector bundles of singular weights on siegel threefolds

fuzzy rule base simplification using multidimensional scaling and constrained optimization this paper proposes a novel approach for the development of highly accurate and interpretable fuzzy models with gaussian fuzzy sets under the framework of multidimensional scaling and non linear constrained optimization upon the assumption that an accurate initial fuzzy model has already been designed we introduce an effective methodology to approximate the similarity measure between fuzzy sets the resulting similarity degrees guide the quantification of the dissimilarity degrees between rule antecedents we then put the multidimensional scaling in place in order to transform the rule antecedents into points in a low dimensional euclidean space the elaboration on the distribution of these points is carried out by means of an objective function based fuzzy clustering using a cluster validity index rules that correspond to points belonging to the same cluster are similar and therefore are unified through a specialized merging process with respect to each dimension the aforementioned merging process acts to create a topology of fuzzy sets that enables us to elicit interpretability constraints which are used to minimize the model s performance index in terms of non linear constrained optimization the established fuzzy model appears to possess a simple and transparent i e interpretable structure while maintaining a highly accurate behavior the overall method is rigorously tested and evaluated through a number of simulation experiments that involve low and high dimensional data sets

improved data visualisation through multiple dissimilarity modelling popular dimension reduction and visualisation algorithms rely on the assumption that input dissimilarities are typically euclidean for instance metric multidimensional scaling t distributed stochastic neighbour embedding and the gaussian process latent variable model it is well known that this assumption does not hold for most datasets and often high dimensional data sits upon a manifold of unknown global geometry we present a method for improving the manifold charting process coupled with elastic mds such that we no longer assume that the manifold is euclidean or of any particular structure we draw on the benefits of different dissimilarity measures allowing for the relative responsibilities under a linear combination to drive the visualisation process

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

improved data visualisation through nonlinear dissimilarity modelling inherent to state of the art dimension reduction algorithms is the assumption that global distances between observations are euclidean despite the potential for altogether non euclidean data manifolds we demonstrate that a non euclidean manifold chart can be approximated by implementing a universal approximator over a dictionary of dissimilarity measures building on recent developments in the field this approach is transferable across domains such that observations can be vectors distributions graphs and time series for instance our novel dissimilarity learning method is illustrated with four standard visualisation datasets showing the benefits over the linear dissimilarity learning approach

extending sammon mapping with bregman divergences the sammon mapping has been one of the most successful nonlinear metric multidimensional scaling methods since its advent in 1969 but effort has been focused on algorithm improvement rather than on the form of the stress function this paper further investigates using left bregman divergences to extend the sammon mapping and by analogy develops right bregman divergences and reveals the mechanism that improves the performance of scaling over the sammon mapping the influence of data space distance preprocessing on optimisation speed is noticed non stress visualisation quality measures are used to compare the configuration quality of the sammon mapping and its extensions using both euclidean distance and graph distance on three data sets

the n cohomology of limits of discrete series abstract let g be a semisimple lie group which has a compact cartan subgroup h let k be a maximal compact subgroup of g containing h and let n be the sum of the negative root spaces of g corresponding to an arbitrary choice of a positive root system of g h we compute the n cohomology of the k finite vectors in a limit of a discrete series representation of g in the special case when is a discrete series representation our result reduces to the result of w schmid in the special case when is a holomorphic limit we interpret its multiplicity in l 2 g cohomologically where is a discrete subgroup of g in the general case we present a conjecture for this multiplicity

on tensor product l functions for quaternion unitary groups and gl 2 over totally real number fields mixed weight cases abstract we prove an algebraicity of special values of tensor product l functions for quaternion unitary groups and gl 2 in the mixed weight cases over any totally real number field as a consequence we can prove a period relation for automorphic forms of quaternion unitary groups of degree 2 and gsp 4 when these automorphic forms correspond under the conjectural jacquet langlands correspondence and a period relation for endoscopic liftings of gsp 4 in very general situations

extending metric multidimensional scaling with bregman divergences sum of weighted square distance errors has been a popular way of defining stress function for metric multidimensional scaling mmds like the sammon mapping in this paper we generalise this popular mmds with bregman divergences as an example we show that the sammon mapping can be thought of as a truncated bregman mmds bmmds and we show that the full bmmds improves upon the sammon mapping on some standard data sets and investigate the reasons underlying this improvement we then extend a well known family of mmds that deploy a strategy of focusing on small distances with bmmds and investigate limitations of the strategy empirically then an opposite strategy is introduced to create another family of bmmds that gives increasing mapping quality a data preprocessing method and a distance matrix preprocessing are introduced

local multidimensional scaling for nonlinear dimension reduction graph drawing and proximity analysis in the past decade there has been a resurgence of interest in nonlinear dimension reduction among new proposals are local linear embedding isomap and kernel principal components analysis which all construct global low dimensional embeddings from local affine or metric information we introduce a competing method called local multidimensional scaling lmds like lle isomap and kpca lmds constructs its global embedding from local information but it uses instead a combination of mds and force directed graph drawing we apply the force paradigm to create localized versions of mds stress functions with a tuning parameter to adjust the strength of nonlocal repulsive forces we solve the problem of tuning parameter selection with a meta criterion that measures how well the sets of k nearest neighbors agree between the data and the embedding tuned lmds seems to be able to outperform mds pca lle isomap and kpca as illustrated with two well known image datasets the meta criterion can also be

comparison between analytic and algebraic constructions of toroidal compactifications of pel type shimura varieties using explicit identications between algebraic and analytic theta functions we compare the algebraic constructions of toroidal compactications by faltings chai and the author with the analytic constructions of toroidal compactications following ash mumford rapport tai as one of the appli cations we obtain the corresponding comparison for fourier jacobi expansions of holomorphic automorphic forms

superconvergent discontinuous galerkin methods for second order elliptic problems we identify discontinuous galerkin methods for second order elliptic problems in several space dimensions having superconvergence properties similar to those of the raviart thomas and the brezzi douglas marini mixed methods these methods use polynomials of degree k 0 for both the potential as well as the flux we show that the approximate flux converges in l2 with the optimal order of k 1 and that the approximate potential and its numerical trace superconverge in l2 like norms to suitably chosen projections of the potential with order k 2 we also apply element by element postprocessing of the approximate solution to obtain new approximations of the flux and the potential the new approximate flux is proven to have normal components continuous across inter element boundaries to converge in l2 with order k 1 and to have a divergence converging in l2 also with order k 1 the new approximate potential is proven to converge with order k 2 in l2 numerical experiments validating these theoretical results are presented