corrigendum to approximation by cp smooth lipschitz functions on banach spaces j math anal appl 315 2006 599 605 in this erratum we recover the results from an earlier paper of the author s which contained a gap specifically we prove that if x is a banach space with an unconditional basis and admits a c p smooth lipschitz bump function and y is a convex subset of x then any uniformly continuous function f y r can be uniformly approximated by lipschitz c p smooth functions k x r also if z is any banach space and f x z is l lipschitz then the approximates k x z can be chosen cl lipschitz and c p smooth for some constant c depending only on x

cavity approach to the spectral density of sparse symmetric random matrices the spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method we consider two cases matrices whose associated graphs are locally treelike and sparse covariance matrices we derive a closed set of equations from which the density of eigenvalues can be efficiently calculated within this approach the wigner semicircle law for gaussian matrices and the marcenko pastur law for covariance matrices are recovered easily our results are compared with numerical diagonalization showing excellent agreement

spectra of sparse random matrices we compute the spectral density for ensembles of sparse symmetric random matrices using replica our formulation of the replica symmetric ansatz shares the symmetries of that suggested in a seminal paper by rodgers and bray symmetry with respect to permutation of replica and rotation symmetry in the space of replica but uses a different representation in terms of superpositions of gaussians it gives rise to a pair of integral equations which can be solved by a stochastic population dynamics algorithm remarkably our representation allows us to identify pure point contributions to the spectral density related to the existence of normalizable eigenstates our approach is not restricted to matrices defined on graphs with poissonian degree distribution matrices defined on regular random graphs or on scale free graphs are easily handled we also look at matrices with row constraints such as discrete graph laplacians our approach naturally allows us to unfold the total density of states into contributions coming from vertices of different local coordinations and an example of such an unfolding is presented our results are well corroborated by numerical diagonalization studies of large finite random matrices

random walks on graphs ideas techniques and results random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology in recent years relevant mathematical results have been obtained in this field and new ideas have been introduced which can be fruitfully extended to different areas and disciplines here we aim at giving a brief but comprehensive perspective of these progresses with a particular emphasis on physical aspects

a reference free cuscore chart for dynamic mean change detection and a unified framework for charting performance comparison to detect and estimate nonconstant time varying mean shifts statistical process control spc tools such as the cumulative score cuscore and generalized likelihood ratio test glrt charts have recently been proposed however their efficiency is based on previous and exact knowledge of a reference pattern in this article a reference free cuscore rfcuscore chart is proposed that can trace and detect dynamic mean changes quickly without knowing the reference pattern in addition a unified framework that contains most of the control charts is presented and applied for a theoretical comparison of the rfcuscore cuscore glrt and cusum charts in detecting dynamic mean changes moreover numerical simulations and a real example are used to illustrate and verify the results both theoretical analysis and numerical results show that the rfcuscore chart performs not only robustly but also quickly in detecting both small and large dynamic mean changes

whitney extension theorems for convex functions of the classes c1 and c1 ω let c be a subset of n not necessarily convex f c r be a function and g c n be a uniformly continuous function with modulus of continuity we provide a necessary and sufficient condition on f g for the existence of a convex function f cc1 n such that f f on c and f g on c with a good control of the modulus of continuity of f in terms of that of g on the other hand assuming that c is compact we also solve a similar problem for the class of c1 convex functions on n with a good control of the lipschitz constants of the extensions namely lip f g finally we give a geometrical application concerning interpolation of compact subsets k of n by boundaries of c1 or c1 1 convex bodies with prescribed outer normals on k

likelihood based ewma charts for monitoring poisson count data with time varying sample sizes many applications involve monitoring incidence rates of the poisson distribution when the sample size varies over time recently a couple of cumulative sum and exponentially weighted moving average ewma control charts have been proposed to tackle this problem by taking the varying sample size into consideration however we argue that some of these charts which perform quite well in terms of average run length arl may not be appealing in practice because they have rather unsatisfactory run length distributions with some charts the specified in control ic arl is attained with elevated probabilities of very short and very long runs as compared with a geometric distribution this is reflected in a larger run length standard deviation than that of a geometric distribution and an elevated probability of false alarms with short runs which in turn hurt an operator s confidence in valid alarms furthermore with many charts the ic arl exhibits considerable variations with different patterns of sam

performance of cusum control schemes for serially correlated observations this article discusses situations in which one is interested in evaluating the run length characteristics of a cumulative sum control scheme when the underlying data show the presence of serial correlation in practical applications situations of this type are common in problems associated with monitoring such characteristics of data as forecasting errors measures of model adequacy and variance components the discussed problem is also relevant in situations in which data transformations are used to reduce the magnitude of serial correlation the basic idea of analysis involves replacing the sequence of serially correlated observations by a sequence of independent and identically distributed observations for which the runlength characteristics of interest are roughly the same applications of the proposed method for several classes of processes arising in the area of statistical process control are discussed in detail and it is shown that it leads to approximations that can be considered acceptable in

the state of statistical process control as we proceed into the 21st century 2000 the state of statistical process control as we proceed into the 21st century journal of the american statistical association vol 95 no 451 pp 992 998

target problem on small world networks in this work we focus on reactions on small world networks swn s disordered graphs of much recent interest we study the target problem since it allows an exact solution on regular lattices on swn s we find that the decay of the targets for which we extend the formalism to disordered lattices is again related to s n the mean number of distinct sites visited in n steps although the s n vs n dependence changes here drastically in going from regular linear chains to their swn

trapping of random walks on small world networks we investigate the trapping of random walkers on small world networks swn s irregular graphs we derive bounds for the survival probability phi swn n and display its analysis through cumulant expansions computer simulations are performed for large swns we show that in the limit of infinite sizes trapping on swns is equivalent to trapping on a certain class of random trees which are grown during the random walk

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

on the efficiency and robustness of discrete proportional integral control schemes feedback control schemes have been widely used in process industries for many years they are also increasingly being used in the discrete parts manufacturing industry in recent years proportionalintegral pi schemes are especially popular primarily because of their simple structure and ease of implementation this article studies the efficiency and robustness properties of discrete pi schemes under some commonly encountered situations for process disturbance we consider the stationary arma 1 1 model and the nonstationary arima 1 1 1 model process dynamics is studied under a first order dynamic model including the special case of pure gain the efficiency of pi schemes is compared with that of minimum mean squared error mmse schemes under these models the pi schemes are seen to be quite efficient over a broad range of the parameter space furthermore the pi schemes are much more robust than mmse schemes to model misspecihcations especially the presence of first order nonstationarity th

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

decision theoretic optimality of the cusum procedure suppose x 1 x 2 are independent random variables such that for some unknown each of x 1 x 1 is distributed according to f 0 while x x 1 are all distributed according to f 1 we prove a result of moustakides that claims that the cusum procedures are optimal in the sense of lorden we do that by proving that the procedures are bayes for some stochastic mechanism of generating

multivariate statistical process control using lasso this article develops a new multivariate statistical process control spc methodology based on adapting the lasso variable selection method to the spc problem the lasso method has the sparsity property of being able to select exactly the set of nonzero regression coefficients in multivariate regression modeling which is especially useful in cases where the number of nonzero coefficients is small in multivariate spc applications process mean vectors often shift in a small number of components our primary goals are to detect such a shift as soon as it occurs and to identify the shifted mean components using this connection between the two problems we propose a lasso based multivariate test statistic and then integrate this statistic into the multivariate ewma charting scheme for phase ii multivariate process monitoring we show that this approach balances protection against various shift levels and shift directions and thus provides an effective tool for multivariate spc applications this article

number of common sites visited by n random walkers we compute analytically the mean number of common sites w n t visited by n independent random walkers each of length t and all starting at the origin at t 0 in d dimensions we show that in the n d plane there are three distinct regimes for the asymptotic large t growth of w n t these three regimes are separated by two critical lines d 2 and d d c n 2n n 1 in the n d plane for d d c n w n t const as t exactly at the critical dimensions there are logarithmic corrections for d 2 we get w n t t ln t n while for d d c n w n t ln t for large t our analytical predictions are verified in numerical simulations

optimal stopping times for detecting changes in distributions it is shown that page s stopping time is optimal for the detection of changes in distributions in a well defined sense this work is a generalization of an existing result where it was shown that page s stopping time is optimal asymptotically

a robust likelihood cumulative sum chart in practice the cumulative sum cusum control chart is often used to detect small shifts in the mean of a normally distributed process but it performs poorly for thick tailed processes and for large shifts this article provides a robust likelihood cumulative sum rlcusum chart that discounts outliers and yet has the ability to detect large shifts quickly the new chart is motivated by the likelihood underpinnings of the cusum it is based on the likelihood of a variate constructed to ensure robust performance of the resulting chart the new chart is compared with the conventional cusum in terms of average run length with the finding that the rlcusum is much better than the conventional cusum especially for large shifts for small shifts the two charts are essentially equivalent we study the properties of the conditional limiting distribution a final application of the rlcusum in assessing livestock disease shows that the method is applicable in process control

a new non parametric stationarity test of time series in the time domain summary we propose a new double order selection test for checking second order stationarity of a time series to develop the test a sequence of systematic samples is defined via walsh functions then the deviations of the autocovariances based on these systematic samples from the corresponding autocovariances of the whole time series are calculated and the uniform asymptotic joint normality of these deviations over different systematic samples is obtained with a double order selection scheme our test statistic is constructed by combining the deviations at different lags in the systematic samples the null asymptotic distribution of the statistic proposed is derived and the consistency of the test is shown under fixed and local alternatives simulation studies demonstrate well behaved finite sample properties of the method proposed comparisons with some existing tests in terms of power are given both analytically and empirically in addition the method proposed is applied to check the stationarity assumption of a chemical process viscosity readings data set

invariance of closed convex cones for stochastic partial differential equations abstract the goal of this paper is to clarify when a closed convex cone is invariant for a stochastic partial differential equation spde driven by a wiener process and a poisson random measure and to provide conditions on the parameters of the spde which are necessary and sufficient

quantitative geometry rather than describing a traditional field of mathematics quantitative geometry refers to a modern way of thinking about geometry and its applications that occur in a wide range of mathematical disciplines the overarching theme here is that certain geometric problems can only be formulated after the introduction of auxiliary quantitative parameters and asking for their asymptotic or approximate behavior this leads to meaningful questions that do not necessarily have counterparts in classical qualitative geometric investigations a viewpoint that allows one to uncover deep and useful structural information that appears only in the quantitative regime quantitative reasoning allows one to tame complicated objects and phenomena that are unwieldy if one insists on exact measurements it is often the case that a rich yet structured picture emerges only if one asks the correct questions by allowing for controlled errors

replica theory for learning curves for gaussian processes on random graphs we use a statistical physics approach to derive accurate predictions for the challenging problem of predicting the performance of gaussian process regression performance is quantified by the learning curve defined as the average error versus number of training examples we assume the gaussian process prior is defined by a random walk kernel inputs are vertices on a random graph and the outputs are noisy function values we show that replica techniques can be used to obtain exact performance predictions in the limit of large graphs after first rewriting the average error in terms of a graphical model conventionally the gaussian process kernel is only globally normalized so that the prior variance can differ between vertices as a more principled alternative we also consider local normalization where the prior variance is uniform the normalization constants for the prior then have to be defined as thermal averages in an unnormalized model and this requires the introduction of a second auxiliary set of replicas our results for both types of kernel normalization apply generically to all random graph ensembles constrained by a fixed but arbitrary degree distribution we compare with numerically simulated learning curves and find excellent agreement a significant improvement over existing approximations

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

run length distributions of special cause control charts for correlated processes we derive run length distributions of the special cause control chart proposed by alwan and roberts for correlated observations given that the assignable cause to be detected is a shift in the process mean both recursive and closed form solutions are derived for the run length distribution average run length arl and standard deviation of the run length srl for any ar p process and approximate solutions are derived for the more general arma p q processes the expressions derived do not depend on the type of shift in the process mean numerical results are illustrated for the arl and srl of the arma l l model given that the shift in the mean is a step shift these results show that the arl and srl of the specialcause control chart are relatively smaller when the process is negatively rather than positively autocorrelated regardless of the sign of the autocorrelation the shape of the probability mass function of the run length reveals that the probability of detecting shifts very early is subs

revealing the microstructure of the giant component in random graph ensembles the microstructure of the giant component of the erd s renyi network and other configuration model networks is analyzed using generating function methods while configuration model networks are uncorrelated the giant component exhibits a degree distribution which is different from the overall degree distribution of the network and includes degree degree correlations of all orders we present exact analytical results for the degree distributions as well as higher order degree degree correlations on the giant components of configuration model networks we show that the degree degree correlations are essential for the integrity of the giant component in the sense that the degree distribution alone cannot guarantee that it will consist of a single connected component to demonstrate the importance and broad applicability of these results we apply them to the study of the distribution of shortest path lengths on the giant component percolation on the giant component and spectra of sparse matrices defined on the giant component we show that by using the degree distribution on the giant component one obtains high quality results for these properties which can be further improved by taking the degree degree correlations into account this suggests that many existing methods currently used for the analysis of the whole network can be adapted in a straightforward fashion to yield results conditioned on the giant component

smooth extension of functions on a certain class of non separable banach spaces let us consider a banach space x with the property that every real valued lipschitz function f can be uniformly approximated by a lipschitz c1 smooth function g with lip g clip f with c depending only on the space x this is the case for a banach space x bi lipschitz homeomorphic to a subset of c0 for some set such that the coordinate functions of the homeomorphism are c1 smooth hajek and johanis 2010 then we prove that for every closed subspace y x and every c1 smooth lipschitz function f y r there is a c1 smooth lipschitz respectively extension of f to x we also study c1 smooth extensions of real valued functions defined on closed subsets of x these results extend those given in azagra et al 2010 4 to the class of non separable banach spaces satisfying the above property

nonparametric dynamic curve monitoring rapid sequential comparison between the longitudinal pattern of a given subject and a target pattern has become increasingly important in modern scientific research for detecting abnormal activitie

surveillance strategies for detecting changepoint in incidence rate based on exponentially weighted moving average methods surveillance is a major issue in the today s world the need to develop adequate surveillance strategies is genuine in many spheres of human activity in this article we focus on a specific problem of surveillance and discuss some related statistical issues this specific problem deals with a possible change in the incidence rate of an event to a higher value when a system is studied at discrete time points we apply the exponentially weighted moving average methods for detecting an increased incidence rate per exposure unit of an event different measures of evaluation suitable in different types of applications such as the expected delay out of control average run length and probability of successful detection are studied analytical bounds are provided for those measures of evaluation results from intensive simulations indicate that the analytical bounds perform fairly well when the weight parameter is large

random walks on the bethe lattice we obtain random walk statistics for a nearest neighbor polya walk on a bethe lattice infinite cayley tree of coordination numberz and show how a random walk problem for a particular inhomogeneous bethe lattice may be solved exactly we question the common assertion that the bethe lattice is an infinite dimensional system

a generalized ewma control chart and its comparison with the optimal ewma cusum and glr schemes it is known that both the optimal exponentially weighted moving average ewma and cumulative sum cusum control charts are based on a given reference value delta which for the cusum chart is the magnitude of a shift in the mean to be detected quickly in this paper a generalized ewma control chart gewma which does not depend on delta is proposed for detecting the mean shift we compare theoretically the gewma control chart with the optimal ewma cusum and the generalized likelihood ratio glr control charts the results of the comparison in which the in control average run length approaches infinity show that the gewma control chart is better than the optimal ewma control chart in detecting a mean shift of any size and is also better than the cusum control chart in detecting the mean shift which is not in the interval 0 7842 delta 1 3798 delta moreover the glr control chart has the best performance in detecting mean shift among the four control charts except when detecting a particular mean shift delta when the in control average run length approaches infinity

volume growth of some uniformly contractible riemannian manifolds it is shown that if a uniformly contractible riemannian n manifold m g is k quasi isometric to an n dimensional normed space v n cdot k 1 then liminf r rightarrow infty frac vol g ball r r n omega n geq frac 1 k 2n where n is the volume of the unit euclidean ball in particular if m is uniformly contractible and d gh m d g v n cdot infty then m has at least euclidean volume growth this corollary covers an earlier result by burago and ivanov our results are motivated by a volume growth theorem contained in gromov s book gromov in progress in mathematics vol 152 birkhauser boston 1999 p 256 we give a detailed proof of this theorem using the same argument we also derive a generalization of the theorem which is pointed out by gromov

a least angle regression control chart for multidimensional data in multidimensional applications it is very rare that all variables shift at the same time a statistical process control procedure would have superior efficiency when limited to the subset of variables likely responsible for the out of control conditions the key idea of this article consists of combining a variable selection method with a multivariate control chart to detect changes in both the mean and variability of a multidimensional process with gaussian errors in particular we develop a control chart for phase ii monitoring which integrates the least angle regression algorithm with a multivariate exponentially weighted moving average comparisons with related multivariate control schemes demonstrate the efficiency of the proposed control chart in a wide range of practical applications including profile and multistage process monitoring further the proposed scheme may also provide valuable diagnostic information for fault isolation supplemental materials including an r package are available

a note on closed isometric embeddings abstract a famous theorem due to nash assures that every riemannian manifold can be embedded isometrically into some euclidean space e n an interesting question is whether for a complete manifold m we can find a closed isometric embedding this note gives the affirmative answer to this question asked to the author by paolo piccione

aperiodic tilings positive scalar curvature and amenability of spaces the object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers we shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties the most interesting result characterizes when h uf x vanishes as an analogue and strengthening of f0lner s amenability criterion for groups in terms of isoperimetric inequalities see 4 one can view this result as producing a successful infinite ponzi scheme on any nonamenable space each point with only finite resources gives to some of its neighbors some of these resources yet receives more from the remaining neighbors as one can imagine this is useful for eliminating obstructions on noncompact spaces this has a number of applications we present two of them the first produces tilings that are unbalanced on any nonamenable polyhedron unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically unfortunately imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of euclidean space penrose tilings are necessarily not accessible to our method on the other hand most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion the second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature we prove a converse to a theorem of roe we show that for any nonamenable group f one can find a spin manifold with fundamental group f with nonzero a genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi isometry class

the average number of distinct sites visited by a random walker on random graphs we study the linear large n behaviour of the average number of distinct sites s n visited by a random walker after n steps on a large random graph an expression for the graph topology dependent prefactor b in s n bn is proposed we use generating function techniques to relate this prefactor to the graph adjacency matrix and then devise message passing equations to calculate its value numerical simulations are performed to evaluate the agreement between the message passing predictions and random walk simulations on random graphs scaling with system size and average graph connectivity are also analysed

random walk kernels and learning curves for gaussian process regression on random graphs we consider learning on graphs guided by kernels that encode similarity between vertices our focus is on random walk kernels the analogues of squared exponential kernels in euclidean spaces we show that on large locally treelike graphs these have some counter intuitive properties specifically in the limit of large kernel lengthscales we consider using these kernels as covariance functions of gaussian processes in this situation one typically scales the prior globally to normalise the average of the prior variance across vertices we demonstrate that in contrast to the euclidean case this generically leads to significant variation in the prior variance across vertices which is undesirable from a probabilistic modelling point of view we suggest the random walk kernel should be normalised locally so that each vertex has the same prior variance and analyse the consequences of this by studying learning curves for gaussian process regression numerical calculations as well as novel theoretical predictions for the learning curves using belief propagation show that one obtains distinctly different probabilistic models depending on the choice of normalisation our method for predicting the learning curves using belief propagation is significantly more accurate than previous approximations and should become exact in the limit of large random graphs

a smooth variational principle with applications to hamilton jacobi equations in infinite dimensions abstract we prove that if x is a banach space which admits a smooth lipschitzian bump function then for every lower semicontinuous bounded below function there exists a lipschitzian smooth function g on x such that f g attains its strong minimum on x thus extending a result of borwein and preiss we then show how the above result can be used to obtain existence and uniqueness results of viscosity solutions of hamilton jacobi equations in infinite dimensional banach spaces a without assuming the radon nikodym property

approximation by lipschitz c p smooth functions on weakly compactly generated banach spaces abstract it is shown that on weakly compactly generated banach spaces which admit a lipschitz c p smooth bump function one can uniformly approximate uniformly continuous bounded real valued functions by lipschitz c p smooth functions this provides a lipschitz version of the classical approximation results of godefroy troyanski whitfield and zizler

distribution of shortest path lengths in subcritical erdős renyi networks networks that are fragmented into small disconnected components are prevalent in a large variety of systems these include the secure communication networks of commercial enterprises government agencies and illicit organizations as well as networks that suffered multiple failures attacks or epidemics the structural and statistical properties of such networks resemble those of subcritical random networks which consist of finite components whose sizes are nonextensive surprisingly such networks do not exhibit the small world property that is typical in supercritical random networks where the mean distance between pairs of nodes scales logarithmically with the network size unlike supercritical networks whose structure has been studied extensively subcritical networks have attracted relatively little attention a special feature of these networks is that the statistical and geometric properties vary between different components and depend on their sizes and topologies the overall statistics of the network can be obtained by a summation over all the components with suitable weights we use a topological expansion to perform a systematic analysis of the degree distribution and the distribution of shortest path lengths dspl on components of given sizes and topologies in subcritical erd ifmmode mbox h o else h o fi s r enyi er networks from this expansion we obtain an exact analytical expression for the dspl of the entire subcritical network in the asymptotic limit the dspl which accounts for all the pairs of nodes that reside on the same finite component fc is found to follow a geometric distribution of the form p mathrm fc l ensuremath ell ll ensuremath infty 1 ensuremath c c ensuremath ell ensuremath 1 where cl1 is the mean degree using computer simulations we calculate the dspl in subcritical er networks of increasing sizes and confirm the convergence to this asymptotic result we also obtain exact asymptotic results for the mean distance ensuremath langle l ensuremath rangle mathrm fc and for the standard deviation of the dspl ensuremath sigma l mathrm fc and show that the simulation results converge to these asymptotic results using the duality relations between subcritical and supercritical er networks we obtain the dspl on the nongiant components of er networks above the percolation transition

approximating functions on stratified sets we investigate smooth approximations of functions with prescribed gradient behavior on a distinguished stratified subset of the domain as an application we outline how our results yield important consequences for a recently introduced class of stochastic processes called the matrix valued bessel processes

analytic hadamard states calderon projectors and wick rotation near analytic cauchy surfaces we consider the klein gordon equation on analytic spacetimes with an analytic cauchy surface in this setting we prove the existence of pure analytic hadamard states the proof is based on considering an elliptic operator obtained by wick rotating the klein gordon operator in a neighborhood of a cauchy hypersurface the cauchy data of hadamard two point functions are constructed as calderon projectors suitably generalized if the hypersurface is non compact for the elliptic operator

smooth approximation of lipschitz functions on riemannian manifolds abstract we show that for every lipschitz function f defined on a separable riemannian manifold m possibly of infinite dimension for every continuous e m 0 and for every positive number r 0 there exists a c smooth lipschitz function g m r such that f p g p e p for every p m and lip g lip f r consequently every separable riemannian manifold is uniformly bumpable we also present some applications of this result such as a general version for separable riemannian manifolds of deville godefroy zizler s smooth variational principle

on some problems on smooth approximation and smooth extension of lipschitz functions on banach finsler manifolds abstract let us consider a riemannian manifold m either separable or non separable we prove that for every e 0 every lipschitz function f m r can be uniformly approximated by a lipschitz c 1 smooth function g with lip g lip f e as a consequence every riemannian manifold is uniformly bumpable these results extend to the non separable setting those given in 1 for separable riemannian manifolds the results are presented in the context of c l finsler manifolds modeled on banach spaces sufficient conditions are given on the finsler manifold m and the banach space x where m is modeled so that every lipschitz function f m r can be uniformly approximated by a lipschitz c k smooth function g with lip g c lip f for some c depending only on x some applications of these results are also given as well as a characterization on the separable case of the class of c l finsler manifolds satisfying the above property of approximation finally we give sufficient conditions on the c 1 finsler manifold m and x to ensure the existence of lipschitz and c 1 smooth extensions of every real valued function f defined on a submanifold n of m provided f is c 1 smooth on n and lipschitz with the metric induced by m

using the generalized likelihood ratio statistic for sequential detection of a change point we study sequential detection of a change point using the generalized likelihood ratio statistic for the special case of detecting a change in a normal mean with known variance we give approximations to the average run lengths and compare our procedure to standard cusum tests and combined cusum shewhart tests several examples indicating extensions to problems involving multiple parameters are discussed

quantitative null cobordism author s chambers gregory r dotterrer dominic manin fedor weinberger shmuel abstract for a given null cobordant riemannian n manifold how does the minimal geometric complexity of a null cobordism depend on the geometric complexity of the manifold in gro99 gromov conjectured that this dependence should be linear we show that it is at most a polynomial whose degree depends on n this construction relies on another of independent interest take x and y to be sufficiently nice compact metric spaces such as riemannian manifolds or simplicial complexes suppose y is simply connected and rationally homotopy equivalent to a product of eilenberg maclane spaces for example any simply connected lie group then two homotopic l lipschitz maps f g x rightarrow y are homotopic via a cl lipschitz homotopy we present a counterexample to show that this is not true for larger classes of spaces y

quantitative algebraic topology and lipschitz homotopy we consider when it is possible to bound the lipschitz constant a priori in a homotopy between lipschitz maps if one wants uniform bounds this is essentially a finiteness condition on homotopy this contrasts strongly with the question of whether one can homotop the maps through lipschitz maps we also give an application to cobordism and discuss analogous isotopy questions