unipotent elements and generalized exponential maps abstract let g be a simple and simply connected algebraic group over an algebraically closed field k of characteristic p 0 assume that p is good for the root system of g and that the covering map g s c g is separable in previous work we proved the existence of a not necessarily unique springer isomorphism for g that behaved like the exponential map on the restricted nullcone of g in the present paper we give a formal definition of these maps which we call generalized exponential maps we provide an explicit and uniform construction of such maps for all root systems demonstrate their existence over z p and give a complete parameterization of all such maps one application is that this gives a uniform approach to dealing with the saturation problem for a unipotent element u in g providing a new proof of the known result that u lies inside a subgroup of c g u that is isomorphic to a truncated witt group we also develop a number of other explicit and new computations for g and for g this paper grew out of an attempt to answer a series of questions posed to us by p deligne who also contributed several of the new ideas that appear here

outlier detection with the kernelized spatial depth function statistical depth functions provide from the deepest point a center outward ordering of multidimensional data in this sense depth functions can measure the extremeness or outlyingness of a data point with respect to a given data set hence they can detect outliers observations that appear extreme relative to the rest of the observations of the various statistical depths the spatial depth is especially appealing because of its computational efficiency and mathematical tractability in this article we propose a novel statistical depth the kernelized spatial depth ksd which generalizes the spatial depth via positive definite kernels by choosing a proper kernel the ksd can capture the local structure of a data set while the spatial depth fails we demonstrate this by the half moon data and the ring shaped data based on the ksd we propose a novel outlier detection algorithm by which an observation with a depth value less than a threshold is declared as an outlier the proposed algorithm is simple in structure the threshold is the only one parameter for a given kernel it applies to a one class learning setting in which normal observations are given as the training data as well as to a missing label scenario where the training set consists of a mixture of normal observations and outliers with unknown labels we give upper bounds on the false alarm probability of a depth based detector these upper bounds can be used to determine the threshold we perform extensive experiments on synthetic data and data sets from real applications the proposed outlier detector is compared with existing methods the ksd outlier detector demonstrates a competitive performance

component groups of unipotent centralizers in good characteristic abstract let g be a connected reductive group over an algebraically closed field of good characteristic for u g unipotent we describe the conjugacy classes in the component group a u of the centralizer of u our results extend work of the second author done for simple adjoint g over the complex numbers when g is simple and adjoint the previous work of the second author makes our description combinatorial and explicit moreover it turns out that knowledge of the conjugacy classes suffices to determine the group structure of a u thus we obtain the result previously known through case checking that the structure of the component group a u is independent of good characteristic

online estimation of the geometric median in hilbert spaces nonasymptotic confidence balls estimation procedures based on recursive algorithms are interesting and powerful techniques that are able to deal rapidly with very large samples of high dimensional data the collected data may be contaminated by noise so that robust location indicators such as the geometric median may be preferred to the mean in this context an estimator of the geometric median based on a fast and efficient averaged nonlinear stochastic gradient algorithm has been developed by bernoulli19 2013 18 43 this work aims at studying more precisely the nonasymptotic behavior of this nonlinear algorithm by giving nonasymptotic confidence balls in general separable hilbert spaces this new result is based on the derivation of improved l 2 rates of convergence as well as an exponential inequality for the nearly martingale terms of the recursive nonlinear robbins monro algorithm

double centralizers of unipotent elements in simple algebraic groups of type g2 f4 and e6 let u be a unipotent element of a simple algebraic group g over a field k of characteristic p we develop a method for computing the connected component of z c g u in the cases where p is positive and both it and the rank of g are small enough the method is then carried out for g of type g 2 f 4 and e 6 in bad characteristic c 2013 elsevier inc all rights reserved

statistically self similar fractals letx be a complete separable bounded metric space and a borel probability measure on the space con x n of alln tuples of contractions ofx with the topology of pointwise convergence then there exists a unique self similar probability measurep on the spacek x of all non empty compact subsets ofx here a measurep onk x is called self similar if for every borel setb k x p b int p n left k o k n 1 left bigcup limits i 0 n 1 s i k i in b right right d mu s o s n 1 if for a e s 0 sn 1 eachs i has an inverse which satisfies a lipschitz condition then there is an 0 such that forp a e k k x the hausdorff dimensionh dim k is equal to ifx d is compact and has non empty interior and if a e s 0 sn 1 consists of similarities which satisfy a certain disjointness condition w r t x then is determined by the equation int sum limits i 0 n 1 lip s i alpha d mu s o s n 1 1 where lip s i denotes the smallest lipschitz constant fors i under fairly general assumptions the dimensional hausdorff measure ofp a e k k x equals o if andx are chosen in a rather special way thenp a e k k x is the graph of a homeomorphism of 0 1 or a curve or the graph of a continuous function

the multivariate l1 median and associated data depth this paper gives three related results i a new simple fast monotonically converging algorithm for deriving the l1 median of a data cloud in d a problem that can be traced to fermat and has fascinated applied mathematicians for over three centuries ii a new general definition for depth functions as functions of multivariate medians so that different definitions of medians will correspondingly give rise to different dept functions and iii a simple closed form formula of the l1 depth function for a given data cloud in d

monge kantorovich depth quantiles ranks and signs we propose new concepts of statistical depth multivariate quantiles ranks and signs based on canonical transportation maps between a distributionof interest on ird and a reference distribution on the d dimensionalunit ball the new depth concept called monge kantorovich depth specializesto halfspace depth in the case of elliptical distributions but for more generaldistributions differs from the latter in the ability for its contours to account fornon convex features of the distribution of interest we propose empirical counterpartsto the population versions of those monge kantorovich depth contours quantiles ranks and signs and show their consistency by establishing a uniformconvergence property for transport maps which is of independent interest

general notions of statistical depth function statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data here we introduce several general structures for depth functions classify many existing examples as special cases and establish results on the possession or lack thereof of four key properties desirable for depth functions in general roughly speaking these properties may be described as affine invariance maximality at center monotonicity relative to deepest point and vanishing at infinity this provides a more systematic basis for selection of a depth function in particular from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors

on local intrinsic dimension estimation and its applications in this paper we present multiple novel applications for local intrinsic dimension estimation there has been much work done on estimating the global dimension of a data set typically for the purposes of dimensionality reduction we show that by estimating dimension locally we are able to extend the uses of dimension estimation to many applications which are not possible with global dimension estimation additionally we show that local dimension estimation can be used to obtain a better global dimension estimate alleviating the negative bias that is common to all known dimension estimation algorithms we illustrate local dimension estimation s uses towards additional applications such as learning on statistical manifolds network anomaly detection clustering and image segmentation

inferring a tree from lowest common ancestors with an application to the optimization of relational expressions we present an algorithm for constructing a tree to satisfy a set of lineage constraints on common ancestors we then apply this algorithm to synthesize a relational algebra expression from a simple tableau a problem arising in the theory of relational databases

unipotent centralizers in algebraic groups let g be a simple algebraic group over an algebraically closed field k of characteristic p 0 where p is a good prime for g and let u g be a unipotent element we study the embedding of u in abelian unipotent subgroups and certain reductive subgroups of g and use this information to obtain information on cg u of particular interest are z cg u and the reductive part of cg u the process of embedding a unipotent element u g in a connected abelian unipotent group is sometimes called saturation in previous work saturation results were achieved when u p where it was shown that there is a unique 1dimensional unipotent group the saturation of u which contains u and which is contained in a restricted sometimes called good a1 subgroup of g here an a1 subgroup of g is restricted if all weights on the adjoint module of g are at most 2p 2 it was shown in that these a1 subgroups provide the basis for a variant of the steinberg tensor product theorem hence the name it was also shown in that cg u can be factored as a product of the unipotent radical and the centralizer of a restricted a1 subgroup containing u in this paper we extend these results to cover unipotent elements of arbitrary order we describe a certain 1 dimensional torus which normalizes cg u and then decompose z cg u into indecomposable summands each invariant under the torus and such that u is contained in one of the summands say w if u p p we show that the saturation of up described above coincides with the subgroup of elements of order at most p in w we next establish the existence of a pair of reductive groups j r such that each is the centralizer of the other u is a semiregular unipotent element of j and r is the reductive part of cg u these subgroups provide insight into a number of issues surrounding unipotent elements for instance the reductive part of cg u appears as the centralizer of a reductive group and this is helpful

rayleigh processes real trees and root growth with re grafting the real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree aldous s brownian continuum random tree the random tree like object naturally associated with a standard brownian excursion may be thought of as a random compact real tree the continuum random tree is a scaling limit as n of both a critical galton watson tree conditioned to have total population size n as well as a uniform random rooted combinatorial tree with n vertices the aldous broder algorithm is a markov chain on the space of rooted combinatorial trees with n vertices that has the uniform tree as its stationary distribution we construct and study a markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as n of the aldous broder chain a key technical ingredient in this work is the use of a pointed gromov hausdorff distance to metrize the space of rooted compact real trees berkeley statistics technical report no 654 february 2004 revised october 2004 to appear in probability theory and related fields

unipotent elements tilting modules and saturation the results in this paper were motivated by work of serre 19 where semisimplicity results for representations of arbitrary groups in positive characteristic were established a key ingredient in the arguments was the notion of saturation where one embeds a unipotent element of prime order of a simple algebraic group in a 1 dimensional unipotent subgroup one of our goals here is to show that saturation can be achieved rather generally and to establish a uniqueness result for the resulting unipotent subgroups as a by product we show for a simple algebraic group in good characteristic a unipotent element of prime order is contained in a particularly nice subgroup of type a1 and that the lie algebra of the ambient algebraic group usually affords a tilting module for this subgroup the tilting decompositions are applied to show that the centralizers of the unipotent element a corresponding 1 dimensional unipotent group and its lie algebra all coincide and are closely related to the centralizer of the a1 subgroup we also establish a convenient factorization of the centralizer of the unipotent element results for finite groups of lie type are also obtained

faithful representations of sl2 over truncated witt vectors abstract let 2 be the six dimensional linear algebraic k group sl 2 w 2 where w 2 is the ring of witt vectors of length two over the algebraically closed field k of characteristic p 2 then the minimal dimension of a faithful rational k representation of 2 is p 3

extremal depth for functional data and applications abstractwe propose a new notion called extremal depth ed for functional data discuss its properties and compare its performance with existing concepts the proposed notion is based on a measure of extreme outlyingness ed has several desirable properties that are not shared by other notions and is especially well suited for obtaining central regions of functional data and function spaces in particular a the central region achieves the nominal desired simultaneous coverage probability b there is a correspondence between ed based simultaneous central regions and appropriate pointwise central regions and c the method is resistant to certain classes of functional outliers the article examines the performance of ed and compares it with other depth notions its usefulness is demonstrated through applications to constructing central regions functional boxplots outlier detection and simultaneous confidence bands in regression problems supplementary materials for this article are availab

central limit theorems for the spectra of classes of random fractals we discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal the continuum random tree in the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit we will show examples from a class of random fractals generated from dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a central limit theorem the brownian continuum random tree can also be viewed as a random fractal generated by a dirichlet distribution the first order term in the spectral asymptotics is known almost surely and here we show that there is a central limit theorem describing the fluctuations about this though the positivity of the variance arising in the central limit theorem is left open in both cases these fractals can be described through a general crump mode jagers branching process and we exploit this connection to establish our central limit theorems for the higher order terms in the spectral asymptotics our main tool is a central limit theorem for such general branching processes which we prove under conditions which are weaker than those previously known

speeding up mcmc by efficient data subsampling we propose subsampling markov chain monte carlo mcmc an mcmc framework where the likelihood function for n observations is estimated from a random subset of m observations we introduce a highly

nonparametric imputation by data depth abstractwe present single imputation method for missing values which borrows the idea of data depth a measure of centrality defined for an arbitrary point of a space with respect to a probability d

permutation tests for equality of distributions in high dimensional settings motivated by applications in high dimensional settings we suggest a test of the hypothesis h 0 that two sampled distributions are identical it is assumed that two independent datasets are drawn from the respective populations which may be very general in particular the distributions may be multivariate or infinite dimensional in the latter case representing for example the distributions of random functions from one euclidean space to another our test uses a measure of distance between data this measure should be symmetric but need not satisfy the triangle inequality so it is not essential that it be a metric the test is based on ranking the pooled dataset with respect to the distance and relative to any fixed data value and repeating this operation for each fixed datum a permutation argument enables a critical point to be chosen such that the test has concisely known significance level conditional on the set of all pairwise distances

a nonparametric regression model with tree structured response developments in science and technology over the last two decades has motivated the study of complex data objects in this article we consider the topological properties of a population of tree structured objects our interest centers on modeling the relationship between a tree structured response and other covariates for tree structured objects this poses serious challenges since most regression methods rely on linear operations in euclidean space we generalize the notion of nonparametric regression to the case of a tree structured response variable in addition we develop a fast algorithm and give its theoretical justification we implement the proposed method to analyze a dataset of human brain artery trees an important lesson is that smoothing in the full tree space can reveal much deeper scientific insights than the simple smoothing of summary statistics this article has supplementary materials online

a geometric approach to visualization of variability in functional data abstractwe propose a new method for the construction and visualization of boxplot type displays for functional data we use a recent functional data analysis framework based on a representation of functions called square root slope functions to decompose observed variation in functional data into three main components amplitude phase and vertical translation we then construct separate displays for each component using the geometry and metric of each representation space based on a novel definition of the median the two quartiles and extreme observations the outlyingness of functional data is a very complex concept thus we propose to identify outliers based on any of the three main components after decomposition we provide a variety of visualization tools for the proposed boxplot type displays including surface plots we evaluate the proposed method using extensive simulations and then focus our attention on three real data applications including exploratory data analysis of sea surface tempe

on a functional contraction method methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space our approach is an extension of the so called contraction method to the space c 0 1 of continuous functions endowed with uniform topology and the space d 0 1 of cadlag functions with the skorokhod topology the contraction method originated from the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences it is based on stochastic fixed point equations where probability metrics can be used to obtain contraction properties and allow the application of banach s fixed point theorem we develop the use of the zolotarev metrics on the spaces c 0 1 and d 0 1 in this context applications are given in particular a short proof of donsker s functional limit theorem is derived and recurrences arising in the probabilistic analysis of algorithms are discussed

on the concept of depth for functional data the statistical analysis of functional data is a growing need in many research areas in particular a robust methodology is important to study curves which are the output of many experiments in applied statistics as a starting point for this robust analysis we propose analyze and apply a new definition of depth for functional observations based on the graphic representation of the curves given a collection of functions it establishes the centrality of an observation and provides a natural center outward ordering of the sample curves robust statistics such as the median function or a trimmed mean function can be defined from this depth definition its finite dimensional version provides a new depth for multivariate data that is computationally feasible and useful for studying high dimensional observations thus this new depth is also suitable for complex observations such as microarray data images and those arising in some recent marketing and financial studies natural properties of these

characterization stability and convergence of hierarchical clustering methods we study hierarchical clustering schemes under an axiomatic view we show that within this framework one can prove a theorem analogous to one of kleinberg 2002 in which one obtains an existence and uniqueness theorem instead of a non existence result we explore further properties of this unique scheme stability and convergence are established we represent dendrograms as ultrametric spaces and use tools from metric geometry namely the gromov hausdorff distance to quantify the degree to which perturbations in the input metric space affect the result of hierarchical methods

spatial medians depth functions and multivariate jensen s inequality for any given partial order in a d dimensional euclidean space under mild regularity assumptions we show that the intersection of closed generalized intervals containing more than 1 2 of the probability mass is a non empty compact interval this property is shared with common intervals on real line where the intersection is the median set of the underlying probability distribution so obtained multivariate medians with respect to a partial order can be observed as special cases of centers of distribution in the sense of type d depth functions introduced by y zuo and r serfling em ann stat bf 28 2000 461 482 we show that the halfspace depth function can be realized via compact convex sets or for example closed balls in place of halfspaces and discuss structural properties of halfspace and related depth functions and their centers among other things we prove that in general the maximal guaranteed depth is frac 1 d 1 as an application of these results we provide a jensen s type inequality for functions of several variables with medians in place of expectations which is an extension of the previous work by m merkle em stat prob letters bf 71 2005 277 281

simulating size constrained galton watson trees we discuss various methods for generating random galton watson trees conditional on their sizes being equal to n a linear expected time algorithm is proposed

asymptotic data analysis on manifolds given an m dimensional compact submanifold m of euclidean space r s the concept of mean location of a distribution related to mean or expected vector is generalized to more general r s valued functionals including median location which is derived from the spatial median the asymptotic statistical inference for general functionals of distributions on such submanifolds is elaborated convergence properties are studied in relation to the behavior of the underlying distributions with respect to the cutlocus an application is given in the context of independent but not identically distributed samples in particular to a multisample setup

robust functional estimation using the median and spherical principal components summary we present robust estimators for the mean and the principal components of a stochastic process in l2 r robustness and asymptotic properties of the estimators are studied theoretically by simulation and by example it is shown that the proposed estimators are generally more robust to outliers than the commonly used sample mean and principal components although their properties depend on the spacings of the eigenvalues of the covariance function

sub principal homomorphisms in positive characteristic let g be a reductive group over an algebraically closed field of characteristic p and let u g be a unipotent element of order p suppose that p is a good prime for g we show in this paper that there is a homomorphism sl 2 k g whose image contains u this result was first obtained by d testerman j algebra 1995 using case considerations for each type of simple group and using in some cases computer calculations with explicit representatives for the unipotent orbits the proof we give is free of case considerations except in its dependence on the bala carter theorem our construction of generalizes the construction of a principal homomorphism made by j p serre in invent math 1996 in particular is obtained by reduction modulo from a homomorphism of group schemes over a valuation ring in a number field this permits us to show moreover that the weight spaces of a maximal torus of sl 2 k on lie g are the same as in characteristic 0 the existence of a with this property was previously obtained again using case considerations by lawther and testerman memoirs ams 1999 and has been applied in some recent work of g seitz invent math 2000

subgroups of typea1containing semiregular unipotent elements keywords simple algebraic groups connected simple subgroups unipotent elements exceptional groups of lie type reference ctg article 1997 001doi 10 1006 jabr 1997 7099 record created on 2008 12 16 modified on 2017 05 12

witt groups and unipotent elements in algebraic groups let g be a semisimple algebraic group defined over an algebraically closed field k of good characteristic p 0 let u be a unipotent element of g of order p t for some t n in this paper it is shown that u lies in a closed subgroup of g isomorphic to the it witt group w t k which is a t dimensional connected abelian unipotent algebraic group 2000 mathematics subject classification 20g15

the spatial distribution in infinite dimensional spaces and related quantiles and depths the spatial distribution has been widely used to develop various nonparametric procedures for finite dimensional multivariate data in this paper we investigate the concept of spatial distribution for data in infinite dimensional banach spaces many technical difficulties are encountered in such spaces that are primarily due to the non compactness of the closed unit ball in this work we prove some glivenko cantelli and donsker type results for the empirical spatial distribution process in infinite dimensional spaces the spatial quantiles in such spaces can be obtained by inverting the spatial distribution function a bahadur type asymptotic linear representation and the associated weak convergence results for the sample spatial quantiles in infinite dimensional spaces are derived a study of the asymptotic efficiency of the sample spatial median relative to the sample mean is carried out for some standard probability distributions in function spaces the spatial distribution can be used to define the spatial depth in infinite dimensional banach spaces and we study the asymptotic properties of the empirical spatial depth in such spaces we also demonstrate the spatial quantiles and the spatial depth using some real and simulated functional data

variations on a theme of steinberg abstract the steinberg tensor product theorem is a fundamental tool for studying irreducible representations of simple algebraic groups over fields of positive characteristic this paper is concerned with extending the result replacing the target group sl v by an arbitrary simple algebraic group

inhomogeneous continuum random trees and the entrance boundary of the additive coalescent regard an element of the set of ranked discrete distributions x1 x2 x1 x2 0 i x i 1 as a fragmentation of unit mass into clusters of masses x i the additive coalescent is the valued markov process in which pairs of clusters of masses x i x j merge into a cluster of mass x i x j at rate x i x j aldous and pitman 1998 showed that a version of this process starting from time with infinitesimally small clusters can be constructed from the brownian continuum random tree of aldous 1991 1993 by poisson splitting along the skeleton of the tree in this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees

recursive self similarity for random trees random triangulations and brownian excursion recursive self similarity for a random object is the property of being decomposable into independent rescaled copies of the original object certain random combinatorial objects trees and triangulations possess approximate versions of recursive self similarity and then their continuous limits possess exact recursive self similarity in particular since the limit continuum random tree can be identified with brownian excursion we get a nonobvious recursive self similarity property for brownian excursion

a1 type overgroups of elements of order p in semisimple algebraic groups and the associated finite groups keywords semisimple algebraic groups closed connected subgroups finite groups fixed point subgroups reference ctg article 1995 001doi 10 1006 jabr 1995 1285 record created on 2008 12 16 modified on 2017 05 12

penalty based aggregation of multidimensional data abstract research in aggregation theory is nowadays still mostly focused on algorithms to summarize tuples consisting of observations in some real interval or of diverse general ordered structures of course in practice of information processing many other data types between these two extreme cases are worth inspecting this contribution deals with the aggregation of lists of data points in r d for arbitrary d 1 even though particular functions aiming to summarize multidimensional data have been discussed by researchers in data analysis computational statistics and geometry there is clearly a need to provide a comprehensive and unified model in which their properties like equivariances to geometric transformations internality and monotonicity may be studied at an appropriate level of generality the proposed penalty based approach serves as a common framework for all idempotent information aggregation methods including componentwise functions pairwise distance minimizers and data depth based medians it also allows for deriving many new practically useful tools

optimally weighted l2 distance for functional data many techniques of functional data analysis require choosing a measure of distance between functions with the most common choice being l2 distance in this article we show that using a weighted l2 distance with a judiciously chosen weight function can improve the performance of various statistical methods for functional data including k medoids clustering nonparametric classification and permutation testing assuming a quadratically penalized e g spline basis representation for the functional data we consider three nontrivial weight functions design density weights inverse variance weights and a new weight function that minimizes the coefficient of variation of the resulting squared distance by means of an efficient iterative procedure the benefits of weighting in particular with the proposed weight function are demonstrated both in simulation studies and in applications to the berkeley growth data and a functional magnetic resonance imaging data set

random stable laminations of the disk we study large random dissections of polygons we consider random dissections of a regular polygon with n sides which are chosen according to boltzmann weights in the domain of attraction of a stable law of index 2 1 2 as n goes to infinity we prove that these random dissections converge in distribution towards a random compact set called the random stable lamination if 2 we recover aldous brownian triangulation however if 2 1 2 large faces remain in the limit and a dierent random compact set appears we show that the random stable lamination can be coded by the continuous time height function associated to the normalized excursion of a strictly stable spectrally positive levy process of index using this coding we establish that the hausdor dimension of the stable random lamination is almost surely 2 1

infinitesimal 1 parameter subgroups and cohomology this is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p 0 whereas sfb is concerned with detection of cohomology classes the present paper introduces the graded algebra k vr g of functions on the scheme of infinitesimal 1 parameter subgroups of height r on an affine group scheme g and demonstrates that this algebra is essentially a retract of h g k provided that g is an infinitesimal group scheme of height kr this work is a continuation of f s in which the existence of certain universal extension classes was established thereby enabling the proof of finite generation of h g k for any finite group scheme g over k the role of the scheme of infinitesimal 1 parameter subgroups of g was foreshadowed in f p where h g 1 k was shown to be isomorphic to the coordinate algebra of the scheme of p nilpotent elements of g lie g for g a smooth reductive group g 1 the first frobenius kernel of g and p char k sufficiently large indeed p nilpotent elements of g correspond precisely to infinitesimal 1 parameter subgroups of g 1 much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1 parameter subgroups in 1 we construct the affine scheme vr g of homomorphisms from ga r to g which we call the scheme of infinitesimal 1 parameter subgroups of height r in g in the special case r 1 this is the scheme of p nilpotent elements of the p restricted lie algebra lie g for various classical groups g vr g is the scheme of r tuples of p nilpotent pairwise commuting elements of lie g more generally an embedding g c gln determines a closed embedding of vr g into the scheme of r tuples of p nilpotent pairwise commuting elements of gl1 lie gln the relationship between k vr g the coordinate algebra of vr g and h g k is introduced in theorem 1 14 a natural homomorphism of graded k algebras

statistical tests for large tree structured data we develop a general statistical framework for the analysis and inference of large tree structured data with a focus on developing asymptotic goodness of fit tests we first propose a consistent statistical model for binary trees from which we develop a class of invariant tests using the model for binary trees we then construct tests for general trees by using the distributional properties of the continuum random tree which arises as the invariant limit for a broad class of models for tree structured data based on conditioned galton watson processes the test statistics for the goodness of fit tests are simple to compute and are asymptotically distributed as 2 and f random variables we illustrate our methods on an important application of detecting tumour heterogeneity in brain cancer we use a novel approach with tree based representations of magnetic resonance images and employ the developed tests to ascertain tumor heterogeneity between two groups of patients

trend representation based log density regularization system for portfolio optimization abstract portfolio optimization po has been catching more and more attention in the artificial intelligence and the machine learning communities in this paper we propose a novel trend representation based log density regularization trlr system for portfolio optimization its novelty falls into two aspects first it introduces a log density regularization to the increasing factor of portfolio which is seldom addressed by previous po systems it reflects a relationship between the portfolio and the price relative at an equilibrium point second trlr exploits a novel trend representation by taking the time variable as regressor in a weighted ridge regression hence trlr captures price trend patterns effectively extensive experiments conducted on 5 benchmark datasets from real world financial markets demonstrate that trlr achieves significantly better performance than other state of the art strategies and runs fast which shows its effectiveness and efficiency for large scale applications

nilpotent orbits in good characteristic and the kempf rousseau theory let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 image and suppose that p is a good prime for the root system of g in this paper we give a fairly short conceptual proof of pommerening s theorem pommerening j algebra 49 1977 525 536 j algebra 65 1980 373 398 which states that any nilpotent element in image is richardson in a distinguished parabolic subalgebra of the lie algebra of a levi subgroup of g as a by product we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent g orbits in image for earlier proofs of this theorem see kawanaka invent math 84 1986 575 616 premet trans amer math soc 347 1995 2961 2988 spaltenstein j fac sci univ tokyo sect ia math 31 1984 283 286 we extend recent results of sommers internal math res notices 11 1998 539 562 to reductive lie algebras of good characteristic thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in image and unipotent elements in g earlier computations of these groups in positive characteristics relied mostly on work of mizuno j fac sci univ tokyo sect ia math 24 1977 525 563 tokyo j math 3 1980 391 459 our approach is based on the theory of optimal parabolic subgroups for g unstable vectors also known as the kempf rousseau theory which provides a good substitute for the image theory prominent in the characteristic zero case

support varieties for modules over chevalley groups and classical lie algebras let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 g 1 be the first frobenius kernel and g f p be the corresponding finite chevalley group let m be a rational g module in this paper we relate the support variety of m over the first frobenius kernel with the support variety of m over the group algebra kg f p this provides an answer to a question of parshall applications of our new techniques are presented which allow us to extend results of alperin mason and janiszczak jantzen and to calculate the dimensions of support varieties for finite chevalley groups

convergence analysis of generalized iteratively reweighted least squares algorithms on convex function spaces the computation of robust regression estimates often relies on minimization of a convex functional on a convex set in this paper we discuss a general technique for a large class of convex functionals to compute the minimizers iteratively which is closely related to majorization minimization algorithms our approach is based on a quadratic approximation of the functional to be minimized and includes the iteratively reweighted least squares algorithm as a special case we prove convergence on convex function spaces for general coercive and convex functionals f and derive geometric convergence in certain unconstrained settings the algorithm is applied to total variation tv penalized quantile regression and is compared with a step size corrected newton raphson algorithm it is found that typically in the first steps the iteratively reweighted least squares algorithm performs significantly better whereas the newton type method outpaces the former only after many iterations finally in the setting of bivariate regression with unimodality constraints we illustrate how this algorithm allows one to utilize highly efficient algorithms for special quadratic programs in more complex settings

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion