harmonic analysis on nilpotent groups and singular integrals iii fractional integration along manifolds in this paper we continue the analysis of integral operators on nilpotent lie groups with kernels supported on submanifolds and or containing an oscillatory factor of polynomial type in our previous articles 13 143 we have essentially considered singular integral operators i e those defined by kernels having the critical degree of homogeneity here we focus our attention on kernels having a higher degree of homogeneity so that locally they have a finite mass but do not have enough decay at infinity more precisely if q is the homogeneous dimension of the group g which in this article we always assume endowed with automorphic dilations we consider kernels k having the same homogeneity as 1x1 q l a where 1x1 denotes a homogeneous gauge and re a 0 these kernels are assumed to be supported on analytic homogeneous submanifolds v we call them fractional integration kernels along v we discuss three interrelated problems concerning fractional integration kernels

moments of the inverse participation ratio for the laplacian on finite regular graphs we investigate the rst and second moments of the inverse participation ratio ipr for all eigenvectors of the laplacian on nite random regular graphs with n vertices and degree z by exactly diagonalizing a large set of z regular graphs we nd that as n becomes large the mean of the inverse participation ratio on each graph when averaged over a large ensemble of graphs approaches the numerical value 3 this universal number is understood as the large n limit of the average of the quartic polynomial corresponding to the ipr over an appropriate n 2 dimensional hypersphere of r n the graph ensemble mean variance of the inverse participation ratio for all graph laplacian eigenvectors deviates from its continuous hypersphere average due to large graph to graph uctuations that arise from the existence of localized modes

harmonic analysis on nilpotent groups and singular integrals ii singular kernels supported on submanifolds on etudie des operateurs de convolution singuliers sur des groupes de lie nilpotents generaux et plus specifiquement des operateurs dont les noyaux ont pour supports des varietes de faible dimension et qui peuvent aussi contenir des facteurs polynomiaux exponentiels oscillatoires

a direct search method for unconstrained quantile based simulation optimization abstract simulation optimization has gained popularity over the decades because of its ability to solve many practical problems that involve profound randomness the methodology development of simulation optimization however is largely concerned with problems whose objective function is mean based performance metric in this paper we propose a direct search method to solve the unconstrained simulation optimization problems with quantile based objective functions because the proposed method does not require gradient estimation in the search process it can be applied to solve many practical problems where the gradient of objective function does not exist or is difficult to estimate we prove that the proposed method possesses desirable convergence guarantee i e the algorithm can converge to the true global optima with probability one an extensive numerical study shows that the performance of the proposed method is promising two illustrative examples are provided in the end to demonstrate the viability of the proposed method in real settings

a smooth nonparametric estimator of a quantile function abstract a smooth alternative to the conventional sample quantile function as a nonparametric estimator of a population quantile function is proposed the proposed estimator is essentially a kernel type of estimator and has the same asymptotic distribution as the conventional sample quantile the mean squared convergence rate of the proposed estimator is also estimated monte carlo studies are conducted to compare the proposed estimator with the sample quantile and the estimator proposed by kaigh and lachenbruch the feasibility of using bootstrap techniques to estimate the optimal window width for the proposed estimator is also considered

quantile estimation with latin hypercube sampling quantiles are often used to measure risk of stochastic systems we examine quantile estimators obtained using simulation with latin hypercube sampling lhs a variance reduction technique that efficiently extends stratified sampling to higher dimensions and produces negatively correlated outputs we consider single sample lhs sslhs which minimizes the variance that can be obtained from lhs and also replicated lhs rlhs we develop a consistent estimator of the asymptotic variance of the sslhs quantile estimator s central limit theorem enabling us to provide the first confidence interval ci for a quantile when applying sslhs for rlhs we construct cis using batching and sectioning on average our rlhs cis are shorter than previous rlhs cis and only slightly wider than the sslhs ci we establish the asymptotic validity of the cis by first proving that the quantile estimators satisfy bahadur representations which show that the quantile estimators can be approximated by linear transformations of e

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

probabilistic error bounds for simulation quantile estimators quantile estimation has become increasingly important particularly in the financial industry where value at risk var has emerged as a standard measurement tool for controlling portfolio risk in this paper we analyze the probability that a simulation based quantile estimator fails to lie in a prespecified neighborhood of the true quantile first we show that this error probability converges to zero exponentially fast with sample size for negatively dependent sampling then we consider stratified quantile estimators and show that the error probability for these estimators can be guaranteed to be 0 with sufficiently large but finite sample size these estimators however require sample sizes that grow exponentially in the problem dimension numerical experiments on a simple var example illustrate the potential for variance reduction

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

the morphology of nodal lines random waves versus percolation in this paper we investigate the properties of nodal structures in random wave fields and in particular we scrutinize their recently proposed connection with short range percolation models we propose a measure which shows the difference between monochromatic random waves which are characterized by long range correlations and gaussian fields with short range correlations which are naturally assumed to be better modelled by percolation theory we also study the relevance of the quantities which we compute to the probability that nodal lines are in the vicinity of a given reference line

the absolute continuity of convolutions of orbital measures in symmetric spaces abstract we characterize the absolute continuity of convolution products of orbital measures on the classical irreducible riemannian symmetric spaces g k of cartan type iii where g is a non compact connected lie group and k is a compact connected subgroup by the orbital measures we mean the uniform measures supported on the double cosets kzk in g the characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements z a consequence of our work is to show that the convolution product of any rank g k continuous k bi invariant measures is absolutely continuous in any of these symmetric spaces other than those whose restricted root system is type a n or d 3 when rank g k 1 is needed

quantum chaos on discrete graphs adapting a method developed for the study of quantum chaos on quantum metric graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76 spectral functions and trace formulae for discrete laplacians on graphs are derived this is achieved by expressing the spectral secular equation in terms of the periodic orbits of the graph and obtaining functions which belong to the class of functions proposed originally by ihara 1966 j mat soc japan 18 219 and expanded by subsequent authors stark and terras 1996 adv math 121 124 kotani and sunada 2000 j math sci univ tokyo 7 7 finally a model of classical dynamics on the discrete graph is proposed it is analogous to the corresponding classical dynamics derived for quantum graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76

the smoothness of convolutions of zonal measures on compact symmetric spaces abstract we prove that for every irreducible compact symmetric space g c k of rank r the convolution of any 2 r 1 continuous k bi invariant measures is absolutely continuous with respect to the haar measure on g c we also prove that the convolution of r 1 continuous k invariant measures on the 1 eigenspace in the cartan decomposition of the lie algebra of g c is absolutely continuous with respect to lebesgue measure these results are nearly sharp

percolation on finite graphs and isoperimetric inequalities consider a uniform expanders family gn with a uniform bound on the degrees it is shown that for any p and c 0 a random subgraph of gn obtained by retaining each edge randomly and independently with probability p will have at most one cluster of size at least c gn with probability going to one uniformly in p the method from ajtai komlos and szemeredi combinatorica 2 1982 1 7 is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth as well as a simple proof of a result of kesten about the critical probability for bond percolation in high dimensions several problems and conjectures regarding percolation on finite transitive graphs are presented

eigenvectors of the discrete laplacian on regular graphs a statistical approach in an attempt to characterize the structure of eigenvectors of random regular graphs we investigate the correlations between the components of the eigenvectors associated with different vertices in addition we provide numerical observations suggesting that the eigenvectors follow a gaussian distribution following this assumption we reconstruct some properties of the nodal structure which were observed in numerical simulations but were not explained so far dakel et al 2007 approx random pp 436 48

geometric characterization of nodal domains the area to perimeter ratio in an attempt to characterize the distribution of forms and shapes of nodal domains in wavefunctions we define a geometric parameter the ratio between the area of a domain and its perimeter measured in units of the wavelength we show that the distribution function can distinguish between domains in which the classical dynamics is regular or chaotic for separable surfaces we compute the limiting distribution and show that it is supported on a compact interval which is independent of the properties of the surface in systems which are chaotic or in random waves the area to perimeter distribution has substantially different features which we study numerically we compare the features of the distribution for chaotic wavefunctions with the predictions of the percolation model to find agreement but only for nodal domains which are big with respect to the wavelength scale this work is also closely related to and provides a new point of view on isoperimetric inequalities

nodal domains statistics a criterion for quantum chaos we consider the distribution of the properly normalized numbers of nodal domains of wave functions in 2d quantum billiards we show that these distributions distinguish clearly between systems with integrable separable or chaotic underlying classical dynamics and for each case the limiting distribution is universal system independent thus a new criterion for quantum chaos is provided by the statistics of the wave functions which complements the well established criterion based on spectral statistics

quantile estimation with adaptive importance sampling we introduce new quantile estimators with adaptive importance sampling the adaptive estimators are based on weighted samples that are neither independent nor identically distributed using a new law of iterated logarithm for martingales we prove the convergence of the adaptive quantile estimators for general distributions with nonunique quantiles thereby extending the work of feldman and tucker we illustrate the algorithm with an example from credit portfolio risk analysis

l2 singular dichotomy for orbital measures of classical compact lie groups abstract we prove that for any classical compact simple connected lie group g the g invariant orbital measures supported on non trivial conjugacy classes satisfy a surprising l 2 singular dichotomy either h k l 2 g or h k is singular to the haar measure on g the minimum exponent k for which h k l 2 is specified it depends on lie properties of the element h g as a corollary we complete the solution to a classical problem to determine the minimum exponent k such that k l 1 g for all central continuous measures on g our approach to the singularity problem is geometric and involves studying the size of tangent spaces to the products of the conjugacy classes

computing densities for markov chains via simulation we introduce a new class of density estimators termed look ahead density estimators for performance measures associated with a markov chain look ahead density estimators are given for both transient and steady state quantities look ahead density estimators converge faster especially in multidimensional problems and empirically give visually superior results relative to more standard estimators such as kernel density estimators several numerical examples that demonstrate the potential applicability of look ahead density estimation are given

control variates for probability and quantile estimation in stochastic systems quantiles indicate the level of system performance that can be delivered with a specified probability while probabilities indicate the likelihood that a specified level of system performance can be achieved we present new estimators for use in simulation experiments designed to estimate such quantiles or probabilities of system performance all of the estimators exploit control variates to increase their precision which is especially important when extreme quantiles in the tails of the distribution of system performance or extreme probabilities near zero or one are of interest control variates are auxiliary random variables with known properties in this case known quantiles and a strong stochastic association with the performance measure of interest since transforming a control variate can increase its effectiveness we propose both continuous and discrete approximations to the optimal variance minimizing transformation for estimating probabilities and then invert the probability estimators to obtain corresponding quantile estimators we also propose a direct control variate quantile estimator that is not based on inverting a probability estimator an empirical study using queueing inventory and project planning examples shows that substantial reductions in mean squared error can be obtained when estimating the 0 9 0 95 and 0 99 quantiles

norms of characters and fourier series on compact lie groups abstract we prove new estimates for the p norms of irreducible characters of compact lie groups these estimates are applied to give negative results on p mean convergence of fourier series on compact lie groups

reclaiming quasimonte carlo efficiency in portfolio value at risk simulation through fourier transform quasimonte carlo methods overcome the problem of sample clustering in regular monte carlo simulation and have been shown to improve simulation efficiency in the derivatives pricing literature when the price is expressed as a multidimensional integration and the integrand is suitably smooth for portfolio value at risk var problems the distribution of portfolio value change is based on the expectation of an indicator function hence the integrand is discontinuous the purpose of this paper is to smooth the expectation estimation of an indicator function via fourier transform so that the faster convergence rate of quasimonte carlo methods can be reclaimed theoretically under fairly mild assumptions the simulation of portfolio value at risk is fast and accurate numerical examples elucidate the advantage of the proposed approach over regular monte carlo and quasimonte carlo methods

l2 singular dichotomy for orbital measures of classical simple lie algebras we prove that the g invariant orbital measures supported on adjoint orbits in the lie algebra of a classical compact connected simple lie group satisfy a smoothness dichotomy either k is singular to lebesgue measure or k l 2 the minimum k for which k l 2 is specified and is also the minimum k such that the k fold sum of the orbit has positive measure

quantum ergodicity on regular graphs we give three different proofs of the main result of anantharaman and le masson duke math j 164 4 723 765 2015 establishing quantum ergodicity a form of delocalization for eigenfunctions of the laplacian on large regular graphs of fixed degree these three proofs are much shorter than the original one quite different from one another and we feel that each of the four proofs sheds a different light on the problem the goal of this exploration is to find a proof that could be adapted for other models of interest in mathematical physics such as the anderson model on large regular graphs regular graphs with weighted edges or possibly certain models of non regular graphs a source of optimism in this direction is that we are able to extend the last proof to the case of anisotropic random walks on large regular graphs

control variates for quantile estimation new point and interval estimators for quantiles that employ a control variate are introduced the properties of these estimators do not depend on the usual assumption of joint normality between the random variable of interest and the control illustrative examples for queueing and stochastic activity network models are given in those examples the new estimators are superior to the standard estimator in terms of the mean squared error of the point estimator and the length of the confidence interval

oscillatory singular integrals and harmonic analysis on nilpotent groups abstract several related classes of operators on nilpotent lie groups are considered these operators involve the following features i oscillatory factors that are exponentials of imaginary polynomials ii convolutions with singular kernels supported on lower dimensional submanifolds iii validity in the general context not requiring the existence of dilations that are automorphisms

integrated variance reduction strategies for simulation we develop strategies for integrated use of certain well known variance reduction techniques to estimate a mean response in a finite horizon simulation experiment the building blocks for these integrated variance reduction strategies are the techniques of conditional expectation correlation induction including antithetic variates and latin hypercube sampling and control variates all pairings of these techniques are examined for each integrated strategy we establish sufficient conditions under which that strategy will yield a smaller response variance than its constituent variance reduction techniques will yield individually we also provide asymptotic variance comparisons between many of the methods discussed with emphasis on integrated strategies that incorporate latin hypercube sampling an experimental performance evaluation reveals that in the simulation of stochastic activity networks substantial variance reductions can be achieved with these integrated strategies both the theoretical and experimental results indicate that superior performance is obtained via joint application of the techniques of conditional expectation and latin hypercube sampling

top eigenpair statistics for weighted sparse graphs we develop a formalism to compute the statistics of the top eigenpair of weighted sparse graphs with finite mean connectivity and bounded maximal degree framing the problem in terms of optimisation of a quadratic form on the sphere and introducing a fictitious temperature we employ the cavity and replica methods to find the solution in terms of self consistent equations for auxiliary probability density functions which can be solved by population dynamics this derivation allows us to identify and unpack the individual contributions to the top eigenvector s components coming from nodes of degree k the analytical results are in perfect agreement with numerical diagonalisation of large weighted adjacency matrices and are further cross checked on the cases of random regular graphs and sparse markov transition matrices for unbiased random walks

the statistics of the points where nodal lines intersect a reference curve we study the intersection points of a fixed planar curve with the nodal set of a translationally invariant and isotropic gaussian random field r and the zeros of its normal derivative across the curve the intersection points form a discrete random process which is the object of this study the field probability distribution function is completely specified by the correlation g r r r r given an arbitrary g r r we compute the two point correlation function of the point process on the line and derive other statistical measures repulsion rigidity which characterize the short and long range correlations of the intersection points we use these statistical measures to quantitatively characterize the complex patterns displayed by various kinds of nodal networks we apply these statistics in particular to nodal patterns of random waves and of eigenfunctions of chaotic billiards of special interest is the observation that for monochromatic random waves the number variance of the intersections with long straight segments grows like lln l as opposed to the linear growth predicted by the percolation model which was successfully used to predict other long range nodal properties of that field

lp lq estimates for orbital measures and radon transforms on compact lie groups and lie algebras abstract let be an invariant measure on a regular orbit in a compact lie group or in a lie algebra we prove sharp l p l q estimates for the convolution operators defined through we also obtain similar results for the related radon transform on the lie algebra

correlation induction techniques for estimating quantiles in simulation experiments a simulation based quantile estimator measures the level of system performance that can be delivered with a prespecified probability to estimate selected quantiles of the response of a finite horizon simulation we develop procedures based on correlation induction techniques for variance reduction with emphasis on antithetic variates and latin hypercube sampling these procedures achieve improved precision by controlling the simulation s random number inputs as an integral part of the experimental design the proposed multiple sample quantile estimator is the average of negatively correlated quantile estimators computed from disjoint samples of the simulation response where negative correlation is induced between corresponding responses in different samples while mutual independence of responses is maintained within each sample the proposed single sample quantile estimator is computed from negatively correlated simulation responses within one all inclusive sample the single sample estimator based on latin hypercube sampling is shown to be asymptotically normal and unbiased with smaller variance than the comparable directsimulation estimator based on independent replications similar asymptotic comparisons of the multiple sample and directsimulation estimators focus on bias and mean square error monte carlo results suggest that the proposed procedures can yield significant reductions in bias variance and mean square error when estimating quantiles of the completion time of a stochastic activity network

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

estimating quantile sensitivities quantiles of a random performance serve as important alternatives to the usual expected value they are used in the financial industry as measures of risk and in the service industry as measures of service quality to manage the quantile of a performance we need to know how changes in the input parameters affect the output quantiles which are called quantile sensitivities in this paper we show that the quantile sensitivities can be written in the form of conditional expectations based on the conditional expectation form we first propose an infinitesimal perturbation analysis ipa estimator the ipa estimator is asymptotically unbiased but it is not consistent we then obtain a consistent estimator by dividing data into batches and averaging the ipa estimates of all batches the estimator satisfies a central limit theorem for the i i d data and the rate of convergence is strictly slower than n 1 3 the numerical results show that the estimator works well for practical problems

sums of adjoint orbits and l2 singular dichotomy for su m abstract let g be a real compact connected simple lie group and g its lie algebra we study the problem of determining from root data when a sum of adjoint orbits in g or a product of conjugacy classes in g contains an open set our general methods allow us to determine exactly which sums of adjoint orbits in su m and products of conjugacy classes in su m contain an open set in terms of the highest multiplicities of eigenvalues for su m and su m we show l 2 singular dichotomy the convolution of invariant measures on adjoint orbits or conjugacy classes is either singular to haar measure or in l 2

variance reduction via lattice rules this is a review article on lattice methods for multiple integration over the unit hypercube with a variance reduction viewpoint it also contains some new results and ideas the aim is to examine the basic principles supporting these methods and how they can be used effectively for the simulation models that are typically encountered in the area of management science these models can usually be reformulated as integration problems over the unit hypercube with a large sometimes infinite number of dimensions we examine selection criteria for the lattice rules and suggest criteria which take into account the quality of the projections of the lattices over selected low dimensional subspaces the criteria are strongly related to those used for selecting linear congruential and multiple recursive random number generators numerical examples illustrate the effectiveness of the approach

percolating level sets of the adjacency eigenvectors of d regular graphs one of the most surprising discoveries in quantum chaos was that nodal domains of eigenfunctions of quantum chaotic billiards and maps in the semi classical limit display critical percolation here we extend these studies to the level sets of the adjacency eigenvectors of d regular graphs numerical computations show that the statistics of the largest level sets the maximal connected components of the graph for which the eigenvector exceeds a prescribed value depend critically on the level the critical level is a function of the eigenvalue and the degree d to explain the observed behaviour we study a random gaussian waves ensemble over the d regular tree for this model we prove the existence of a critical threshold using the local tree property of d regular graphs and assuming the local applicability of the random waves model we can compute the critical percolation level and reproduce numerical simulations these results support the random waves model for random regular graphs suggested in 1 and provide an extension to bogomolny s percolation model 2 for two dimensional chaotic billiards

quantile and tolerance interval estimation in simulation this paper discusses two sequential procedures to construct proportional half width confidence intervals for a simulation estimator of the steady state quantile and tolerance intervals for a stationary stochastic process having the reasonable property that the autocorrelation of the underlying process approaches zero with increasing lag at each quantile to be estimated the marginal cumulative distribution function must be absolutely continuous in some neighborhood of that quantile with a positive continuous probability density function these algorithms sequentially increase the simulation run length so that the quantile and tolerance interval estimates satisfy pre specified precision requirements an experimental performance evaluation demonstrates the validity of these procedures

random wavefunctions and percolation recently it was conjectured that nodal domains of random wavefunctions are adequately described by critical percolation theory in this paper we strengthen this conjecture in two respects first we show that though wavefunction correlations decay slowly a careful use of harris criterion confirms that these correlations are unessential and nodal domains of random wavefunctions belong to the same universality class as non correlated critical percolation second we argue that level domains of random wavefunctions are described by the non critical percolation model

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized