the numerical solution of laplace s equation in three dimensions we consider the dirichlet problem for laplace s equation on a simply connected three dimensional region with a smooth boundary this problem is easily converted to the solution of a fredholm integral equation of the second kind based on representing the harmonic solution as a double layer potential function we solve this integral equation formulation by using galerkin s method with spherical harmonics as the basis functions this approach leads to small linear systems and once the galerkin coefficients for the region have been calculated the computation time is small for any particular boundary function the major disadvantage of the method is the calculation of the galerkin coefficients each of which is a four fold integral with a singular integrand theoretical and computational details of the method are presented

a stochastic spectral projection method for solving pdes in domains composed by overlapping discs spheres and half spaces a new mesh free stochastic spectral method based on a spectral inversion of the poisson integral formula is suggested we consider the cases of laplace equation and the system of elasticity equations for domains consisting of overlapping discs half planes spheres and half spaces a randomized projection method we developed in sabelfeld and loshchina 2010 21 is used to solve the system of linear equations approximating the integral equations whose phase space is the intersection surfaces of the domains

complex geometry and the asymptotics of harish chandra modules for real reductive lie groups i let g be a connected semisimple real matrix group it is now apparent that the representation theory of g is intimately connected with the complex geometry of the flag variety 9 by studying appropriate orbit structures on 9 we are naturally led to representation theory in the category of harish chandra modules x or the representation theory of category 9 the jacquet functor j f 9 has proved a useful tool in converting yfw problems into 9 problems which are often more tractable in this paper we advance the philosophy that the complex geometry of 9 associated to yf and 9 interacts in a natural way with the functor j leading to deep new information on the structure of jacquet modules this in turn gives new insight into the structure of certain nilpotent cohomology groups associated to harish chandra modules our techniques are based upon many of the ideas present in the proof of the kazhdan lusztig conjectures and bernstein s proof of the jantzen conjecture

edge distribution method for solving elliptic boundary value problems with boundary singularities elliptic boundary value problems are difficult to treat in the vicinity of singularities i e edges and corners of the boundary the concentration of electrical charge on the edges and corners of a conductor is perhaps the simplest example of such problems here we provide a rapid method for accurate treatment of these problems it utilizes a green s function based implementation of last passage monte carlo diffusion methods this is combined with a diffusion algorithm for the scaling of solutions to the laplace equation near a corner singularity to yield the solution of a benchmark problem the charge distribution near the edge and corner of a conducting cube

localization and standard modules for real semisimple lie groups i the duality theorem in this paper we relate two constructions of representations of semisimple lie groups constructions that appear quite different at first glance homogeneous vector bundles are one source of representations if a real semisimple lie group go acts on a vector bundle e m over a quotient space m go ho then go acts also on the space of sections c m e and on any subspace vcc m e defined by a go invariant system of differential equations ordinary induction so called cohomological induction and the construction of representations by quantizat ion all fit into the framework of homogeneous vector bundles for any complex semisimple lie algebra g there is an equivalence of categories due to beilinson bernstein 1 between modules on the one hand and sheaves of modules over the flag variety x of g on the other in the context of real semisimple lie groups this equivalence of categories associates infinitesimal representations to orbits in the flag variety of the complexified lie algebra orbits not of the group go itself but of the complexification of maximal compact subgroup ko c go we shall show that these beilinson bernstein modules are naturally dual to modules attached to certain homogeneous vector bundles in the special case of a compact group both the beilinson bernstein construction and the construction via homogeneous vector bundles reduce to the borel weil bott theorem our duality theorem is then a particular instance of serre duality

frequency domain system identification using arbitrary signals it is the common conviction that frequency domain system identification suffers from the drawback that it cannot handle arbitrary signals without introducing systematic errors this paper shows that it is possible to deal with nonperiodic signals without any approximation and under the same assumptions as in the time domain by estimating simultaneously some initial conditions and the system model parameters

the simulation tabulation method for classical diffusion monte carlo abstract many important classes of problems in materials science and biotechnology require the solution of the laplace or poisson equation in disordered two phase domains in which the phase interface is extensive and convoluted green s function first passage gffp methods solve such problems efficiently by generalizing the walk on spheres wos method to allow first passage fp domains to be not just spheres but a wide variety of geometrical shapes in particular this solves the difficulty of slow convergence with wos by allowing fp domains that contain patches of the phase interface previous studies accomplished this by using geometries for which the green s function was available in quasi analytic form here we extend these studies by using the simulation tabulation st method we simulate and then tabulate surface green s functions that cannot be obtained analytically the st method is applied to the solc stockmayer model with zero potential to the mean trapping rate of a diffusing particle in a domain of nonoverlapping spherical traps and to the effective conductivity for perfectly insulating nonoverlapping spherical inclusions in a matrix of finite conductivity in all cases this class of algorithms provides the most efficient methods known to solve these problems to high accuracy

hochschild and cyclic homology of quantum groups for an arbitrary complex linear semisimple lie groupg we consider hopf algebras of the deformations of the formal and algebraic functions ong the hochschild and cyclic homology of these hopf algebras are computed when the value of the deformation parameter is generic

improved extrapolation technique in the boundary element method to find the capacitances of the unit square and cube extrapolation of the numbernof subdivisions to infinity in the boundary element method involves fitting computed results to a polynomial in 1 n a technique of choosing the sizes of the subdivisions in such a way that the terms of lowest order in 1 nare eliminated giving a more accurate extrapolation is described the technique is used to establish the capacitances of the unit square and cube with relative errors of 3 10 7and 3 10 6respectively

projective modules in the category _ self duality given a parabolic subalgebra ps of a complex semisimple lie algebra there is a naturally defined category s of g modules which includes all the g modules induced from finite dimensional ps modules this paper treats the question of whether certain projective modules in s are isomorphic to their dual modules the projectives in question are the projective covers of those simple modules occurring in the socles of generalized verma modules their self duality is proved in a number of cases and additional information is obtained on the generalized verma modules

extrapolated surface charge method for capacity calculation of polygons and polyhedra effectiveness of extrapolation in calculating the electric capacities of polygons and polyhedra by scm surface charge method is represented in the case of a square it is divided into n2 small squares as treated by maxwell n 6 empirically extrapolation function of the form a1n a2n2 1 log n n 2 logn n2 is found to give the best result with the accuracy of more than six decimal figures at n 28 in conventional methods without extrapolation forbiddingly large n 105 should be needed to obtain the same accuracy extrapolation without logarithmic terms 1 2 0 does not work well thus extrapolation using a logarithmic series and successive refinement leads to both accurate solutions and a saving in computational time the origin of logarithmic terms is studied the result of a numerical experiment suggests that logarithmic terms are needed when there are sharp edges in the configuration

representation type of the blocks of category os abstract in this paper the authors investigate the representation type of the blocks of the relative parabolic category o s for complex semisimple lie algebras a complete classification of the blocks corresponding to regular weights is given the main results of the paper provide a classification of the blocks in the mixed case when the simple roots corresponding to the singular set and s do not meet

graded characters of induced representations ii classification of principal series modules for complex groups let g be a connected complex reductive algebraic group which we view here as a real lie group by taking its complex points and then ignoring its complex structure the corresponding real lie group obtained in this way will be denoted g the purpose of this paper is the classification of principal series modules for g with integral regular infinitesimal character up to isomorphism of harish chandra modules using translation functors this parametrizes all lie algebra modules that occur as the space of k finite sections of an integral regular line bundle on the flag manifold of g see 2 14 here the term integral regular means that we consider only line bundles that are gr equivariant and whose space of c sections have an integral regular infinitesimal character the same as that of a finite dimensional g module our main result is the proof in this complex case of a conjecture that was stated in 4 see os we also give some estimates on ext groups 2 16 at the end of section 2

identification of dynamic errors in variables models this paper deals with the identifiability of scalar dynamic errors in variables models characterized by rational spectra the hypothesis of causality for the underlying dynamic system is taken into account by making use of stochastic realization theory and of the structural properties of state space representations it is shown that under mild assumptions the model is uniquely identified

the homomorphisms between scalar generalized verma modules associated to maximal parabolic subalgebras let g be a finite dimensional simple lie algebra over c we classify the homomorphisms between g modules induced from one dimensional modules of maximal parabolic subalgebras 1

hydrodynamic friction and the capacitance of arbitrarily shaped objects the translational friction coefficient and the capacitance of a variety of objects are calculated with a probabilistic method involving hitting the probed objects with random walks launched from an enclosing spherical surface this method is applied to exactly solvable examples to test the program accuracy and to physically important and analytically intractable examples cube chain of spheres at the vertices of self avoiding and random walks etc large fluctuations in the friction of polymer chains with a random coil structure are found to give large deviations from the mean field kirkwood riseman theory and hydrodynamic fluctuation effects are found to diminish with the chain swelling accompanying excluded volume interaction capacity applications are reviewed and our probabilistic estimates of polymer friction are compared with previous calculations using alternative methods transients to the capacity and related properties are expressed in terms of fluctuations in the wiener sausage volume volume swept out by a brownian particle where a repeated visit to a spatial region does not contribute to the volume increase in time

bgg algebras and the bgg reciprocity principle the bgg reciprocity principle is a famous result in the representation theary of complex semisimple lie algebras proved by bernstein gelfand and gelfand in bgg 21 it may be viewed as an analogue for the category 6 of the brauer reciprocity principle in the modular representation theory of finite groups and of humphreys reciprocity principle in the representation theory of restricted lie algebras indeed the result of humphreys served as motivation for the bgg result similar results have since been proved in a number of other settings by various approaches for all these results one works in a suitable category of modules in which each simple module l has a projective cover p in addition there is a family of intermediate modules mi such that each p has a filtration with m s as quotients then the appropriate reciprocity principle states that m occurs as a filtration quotient in p as often as l occurs as a composition factor in mi in the case of the category 0 the simple modules are the simple highest weight modules the category is introduced as a setting for projective covers to exist and the intermediate modules are the verma modules the proof of bgg reciprocity in bgg 21 is fairly straightforward once the basic facts about the category 0 are developed the crucial point is the construction of enough projective modules in 0 once these are constructed the existence of projective covers the existence of suitable filtrations on these and the reciprocity principle are ail proved with reference to the construction however another proof of reciprocity exists which is more natural in the sense that it makes no further reference to the construction of projective modules this proof can be used in some of the other settings as well although in any given setting no drastic simplifications result i believe it is still worthwhile to draw attention to this more conceptual approach the properties of the category 0 which are needed for the proof may be

parabolic kazhdan lusztig r polynomials for quasi minuscule quotients abstract we give explicit combinatorial formulas for the parabolic kazhdan lusztig r polynomials of the quasi minuscule quotients of the classical weyl groups as an application of our results we obtain explicit combinatorial formulas for certain sums and alternating sums of ordinary kazhdan lusztig r polynomials

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

n cohomology of simple highest weight modules on walls and purity letg p b be respectively a complex connected linear algebraic semisimple group a parabolic subgroup and a borel subgroup the first main result is the following theorem let be a pure complex ong b smooth with respect to bruhat cells then its restriction to anyp orbit is pure as well of the same weight as a consequence we are able to compute then cohomology of simple highest weight modules on walls

on functors associated to a simple root abstract associated to a simple root of a finite dimensional complex semisimple lie algebra there are several endofunctors defined by arkhipov enright frenkel irving jantzen joseph mathieu vogan and zuckerman on the bgg category o we study their relations compute cohomologies of their derived functors and describe the monoid generated by arkhipov s and joseph s functors and the monoid generated by irving s functors it turns out that the endomorphism rings of all elements in these monoids are isomorphic we prove that the functors give rise to an action of the singular braid monoid on the bounded derived category of o 0 we also use arkhipov s joseph s and irving s functors to produce new generalized tilting modules

system identification using nonstationary signals the conventional method for identifying the transfer function of an unknown linear system consists of a least squares fit of its input to its output it is equivalent to identifying the frequency response of the system by calculating the empirical cross spectrum between the system s input and output divided by the empirical auto spectrum of the input process however if the additive noise at the system s output is correlated with the input process e g in case of environmental noise that affects both system s input and output the method may suffer from a severe bias effect we present a modification of the cross spectral method that exploits nonstationary features in the data in order to circumvent bias effects caused by correlated stationary noise the proposed method is particularly attractive to problems of multichannel signal enhancement and noise cancellation when the desired signal is nonstationary in nature e g speech or image

analysis of the maximum likelihood total least squares and principal component approaches for frequency response function estimation this paper considers the problem of estimation frequency response functions frfs for a single input single output siso system in the presence of additive noise on both input and output measurements it demonstrates that principle component analysis pca can be employed to solve such problems and demonstrates that this is equivalent to the methods based on total least squares tls frf estimation is also cast as a problem in statistical inference and the use of the principle of maximum likelihood ml leads to a novel development of a generalised tls scheme this analysis also provides a framework within which one can compute asymptotic expressions for the variance of such estimators

transfer function and transient estimation by gaussian process regression in the frequency domain inspired by the recent promising developments of bayesian learning techniques in the context of system identification this paper proposes a transfer function estimator based on gaussian process regression contrary to existing kernel based impulse response estimators a frequency domain approach is adopted this leads to a formulation and implementation which is seamlessly valid for both continuous and discrete time systems and which conveniently enables the selection of the frequency band of interest a pragmatic approach is proposed in an output error framework from sampled input and output data the transient is dealt with by estimating it simultaneously with the transfer function modelling the transfer function and the transient as gaussian processes allows for the incorporation of relevant prior knowledge on the system in the form of suitably designed kernels the ss stable spline and dc diagonal correlated kernels from the literature are translated to the frequency domain and are proven to impose the stability of the estimated transfer function specifically the estimates are shown to be stable rational functions in the frequency variable the hyperparameters of the kernel are tuned via marginal likelihood maximisation the comparison between the proposed method and three existing methods from the literature the regularised finite impulse response rfir estimator the local polynomial method lpm and the local rational method for frequency response function estimation is illustrated on simulations on multiple case studies

estimation of fekete points we aim here at presenting a new procedure to numerically estimate the fekete points of a wide variety of compact sets in r 3 we understand the fekete point problem in terms of the identification of near equilibrium configurations for a potential energy that depends on the relative position of n particles the compact sets for which our procedure works are basically the finite union of piecewise regular surfaces and curves in order to determine a good initial configuration to start the search of the fekete points of these objects we construct a sequence of approximating regular surfaces our algorithm is based on the concept of disequilibrium degree which is defined from a physical interpretation of the behavior of a system of particles when they search for a minimum energy configuration moreover the algorithm is efficient and robust independently of the considered compact set as well as of the kernel used to define the energy the numerical experimentation carried out suggests that a local minimum can be localized with a computational cost of order less than n 3

maximum likelihood estimation of the non parametric frf for pulse like excitations this technical note introduces the closed form maximum likelihood estimator for estimating the coefficients of the non parametric frequency response function from system identification experiments it is assumed that the experiments consist of repeated pulse excitations and that both the excitation and system response are measured which leads to an error in variables setting monte carlo simulations indicate that the estimator achieves efficiency at low signal to noise ratios with only few measurements comparison with the least squares estimator shows that better unbiased results are obtained

equivariant sheaves on flag varieties we show that the borel equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves this proves a conjecture of soergel and lunts in the case of flag varieties

schubert cells and representation theory let g be a linear real reductive group and let p be a parabolic subgroup the bruhat decomposition of g gives a cellular decomposition of the generalized flag manifold x g p originating in the work of schubert on grassmann manifolds this cellular decomposition was used by ehresmann eh to give a proof of the basis theorem for the integral cohomology of grassmannians since ehresmann it has been clear that for a generalized flag manifold x a more detailed understanding of this decomposition by generalized schubert cells would provide a determination of the integral co homology of x the aim of this article is to give a representation theoretic determination of the differentials in the schubert cell decomposition of x and thereby obtain a combinatorial description of the integral co homology of x our primary tool will be the infinite dimensional representation theory of g if g and p are complex groups elementary considerations show that all schubert cells define non zero integral homology classes and that none are torsion moreover the cohomology has the well known connection to finitedimensional representation theory ct ko1 ko2 bt for real groups g and p even in low dimensions torsion can be present in the homology of real flag manifolds thus the major new development in this paper is that the infinite dimensional representation theory of g detects which real schubert cells define integral in particular torsion classes in the co homology the innovation on the representation theoretic side that makes possible the consideration of co homology with integer coefficients is the geometric formulation of representation theory introduced by beilinson bernstein to compute homology we use a topological technique that applies to spaces with filtration let x p be the union of schubert cells whose dimension

the random walk on the boundary method for calculating capacitance in this paper we present the random walk on the boundary method for the rapid solution of integral equations that arise in electrostatics and related areas this method is a monte carlo method based on the construction of a markov chain that is readily interpreted as a random walk along the boundary over which integration in the integral equation is taken to illustrate the usefulness of this technique we apply it to the computation of the capacitance of the unit cube obtaining the capacitance of the cube usually requires computing the charge density and this problem has been used as a benchmark by many in the field for algorithms of this kind here the random walk on the boundary method does not require charge density computation and obtains the capacitance of the cube within a statistical error of 2 7 10 7 the most accurate estimate to date

identification of multivariable dynamic errors in variables system with arbitrary inputs the present work deals with the identification of multivariable linear dynamic system from noisy input output observations where the input signal is arbitrary and the input output noises are mutually correlated a frequency domain identification framework is developed in which the consistent estimator of the multivariable plant model parameters and of the input output noise covariance matrix is defined as the solution of a set of normal equations and sufficient conditions for the uniqueness of the parameter estimate are established based on the rank property of the matrix of the normal equation the uncertainty bound of the parameter estimates is constructed and compared with the cramer rao lower bound the proposed methodology is validated on a simulated multivariable dynamic system

errors in variables identification of dynamic systems excited by arbitrary non white input abstract this work deals with the identification of dynamic systems from noisy input output observations where the noise free input is not parameterized the basic assumptions made are 1 the dynamic system can be modeled by a discrete or continuous time rational transfer function model 2 the temporal input output disturbances are mutually independent identically distributed noises and 3 the input power spectrum is non white not necessarily rational and is modeled nonparametrically the system identifiability is guaranteed by exploiting the non white spectrum property of the noise free input a frequency domain identification strategy is developed to estimate consistently the plant model parameters and the input output noise variances the uncertainty bound of the estimates is calculated and compared to the cramer rao lower bound the efficiency of the proposed algorithm is illustrated on numerical examples

jacquet modules and dirac cohomology abstract we show that dirac cohomology of the jacquet module of a harish chandra module is a harish chandra module for the corresponding levi subgroup we obtain an explicit formula of dirac cohomology of the jacquet module for most of the principal series based on our determination of dirac cohomology of irreducible generalized verma modules with regular infinitesimal characters

nonparametric identification of linear dynamic errors in variables systems the present work handles the nonparametric identification of linear dynamic systems within an errors in variables framework where the input is arbitrary and both the input and output disturbing noises are white with unknown variances using the property that the frequency response function and the system leakage term can be locally approximated very well by a low order degree polynomial a frequency domain estimator is developed which gives consistent estimates for the frequency response function and the input output noise variances the consistency and uniqueness of the estimator are theoretically analyzed under mild conditions and uncertainty bounds are also provided the proposed method is finally validated on a simulated linear dynamic system