improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates the problem of estimating the eigenvalues and eigenvectors of the covariance matrix associated with a multivariate stochastic process is considered the focus is on finite sample size situations whereby the number of observations is limited and comparable in magnitude to the observation dimension using tools from random matrix theory and assuming a certain eigenvalue splitting condition new estimators of the eigenvalues and eigenvectors of the covariance matrix are derived that are shown to be consistent in a more general asymptotic setting than the traditional one indeed these estimators are proven to be consistent not only when the sample size increases without bound for a fixed observation dimension but also when the observation dimension increases to infinity at the same rate as the sample size numerical evaluations indicate that the estimators have an excellent performance in small sample size scenarios where the observation dimension and the sample size are comparable in magnitude

optimal weighting scheme and the role of coupling strength against load failures in degree based weighted interdependent networks abstract the optimal weighting scheme and the role of coupling strength against load failures on symmetrically and asymmetrically coupled interdependent networks were investigated the degree based weighting scheme was extended to interdependent networks with the flow dynamics dominated by global redistribution based on weighted betweenness centrality through contingency analysis of one node removal we demonstrated that there still exists an optimal weighting parameter on interdependent networks but it might shift as compared to the case in isolated networks because of the break of symmetry and it will be easier for the symmetrically and asymmetrically coupled interdependent networks to achieve robustness and better cost configuration against the one node removal induced cascade of load failures when coupling strength was weaker our findings might have great generality for characterizing load failure induced cascading dynamics in real world degree based weighted interdependent networks

compact moduli spaces of del pezzo surfaces and kahler einstein metrics we prove that the gromov hausdorff compactification of the moduli space of kahler einstein del pezzo surfaces in each degree agrees with certain algebro geometric compactification in particular this recovers tian s theorem on the existence of kahler einstein metrics on smooth del pezzo surfaces and classifies all the degenerations of such metrics the proof is based on a combination of both algebraic and differential geometric techniques

rational covariance functions for nonstationary random fields this correspondence propose rational type covariance functions for nonstationary gaussian stochastic processes or random fields which are rational functions of given variograms typically have long range dependence and include some stationary covariance functions as special cases

covariance matrix functions of vector χ 2 random fields in space and time this paper introduces vector multivariate or multiple 2 and rayleigh random functions or random fields on a spatial temporal or spatio temporal index domain and explores their basic properties formulated as a sum of squares of independent gaussian random functions a vector 2 random function has an interesting feature that its finite dimensional laplace transforms are not determined by its own covariance matrix but by that of the underlying gaussian one with the conditionally negative definite matrix as an important building block this paper constructs a class of vector 2 random functions from whose covariance matrices one can easily identify those of the underlying gaussian one so that the resulting vector 2 random function can be easily simulated and analyzed

percolation and epidemics in random clustered networks the social networks that infectious diseases spread along are typically clustered because of the close relation between percolation and epidemic spread the behavior of percolation in such networks gives insight into infectious disease dynamics a number of authors have studied percolation or epidemics in clustered networks but the networks often contain preferential contacts in high degree nodes we introduce a class of random clustered networks and a class of random unclustered networks with the same preferential mixing percolation in the clustered networks reduces the component sizes and increases the epidemic threshold compared to the unclustered networks

a second variation formula for perelman s mathcal w functional along the modified kahler ricci flow we show a quite simple second variation formula for perelman s mathcal w functional along the modified kahler ricci flow over fano manifolds

moduli spaces of critical riemannian metrics with l n 2 norm curvature bounds abstract we consider the moduli space of a class of critical metrics on compact manifolds that includes extremal kahler metrics we show that under the conditions of volume and diameter bounds l n 2 curvature bounds and sobolev constant bounds this moduli space can be compactified by including orbifolds with finitely many orbifold points in dimension 4 these orbifold points are c riemannian orbifold points and in the 4 dimensional extremal kahler case they are also of class c

on a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth on etudie le comportement asymptotique de varietes de riemann on construit des coordonnees a l infini sous certaines conditions intrinseques

interdependent networks with identical degrees of mutually dependent nodes we study a problem of failure of two interdependent networks in the case of identical degrees of mutually dependent nodes we assume that both networks a and b have the same number of nodes n connected by the bidirectional dependency links establishing a one to one correspondence between the nodes of the two networks in a such a way that the mutually dependent nodes have the same number of connectivity links i e their degrees coincide this implies that both networks have the same degree distribution p k we call such networks correspondently coupled networks ccns we assume that the nodes in each network are randomly connected we define the mutually connected clusters and the mutual giant component as in earlier works on randomly coupled interdependent networks and assume that only the nodes that belong to the mutual giant component remain functional we assume that initially a 1 p fraction of nodes are randomly removed because of an attack or failure and find analytically for an arbitrary p k the fraction of nodes p that belong to the mutual giant component we find that the system undergoes a percolation transition at a certain fraction p p c which is always smaller than p c for randomly coupled networks with the same p k we also find that the system undergoes a first order transition at p c 0 if p k has a finite second moment for the case of scale free networks with 2 0 finally we find that the robustness of ccn increases with the broadness of their degree distribution

organization of modular networks we examine the global organization of heterogeneous equilibrium networks consisting of a number of well distinguished interconnected parts char22 communities or modules we develop an analytical approach allowing us to obtain the statistics of connected components and the intervertex distance distribution in these modular networks and to describe their global organization and structure in particular we study the evolution of the intervertex distance distribution with an increasing number of interlinks connecting two infinitely large uncorrelated networks we demonstrate that even a relatively small number of shortcuts unite the networks into one in more precise terms if the number of interlinks is any finite fraction of the total number of connections then the intervertex distance distribution approaches a ensuremath delta function peaked form and so the network is united

gravitational instantons with faster than quadratic curvature decay iii this is our third paper in a series on the gravitational instantons in this paper we classify alg and alh gravitational instantons in alg case we extend tian yau hein s construction slightly and show that it is the only alg gravitational instanton in alh case we prove a torelli type theorem

heat kernel and curvature bounds in ricci flows with bounded scalar curvature abstract in this paper we analyze ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time dependent constant on such ricci flows we establish a new time derivative bound for solutions to the heat equation based on this bound we solve several open problems 1 distance distortion estimates 2 the existence of a cutoff function 3 gaussian bounds for heat kernels and 4 a backward pseudolocality theorem which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time using the backward pseudolocality theorem we next establish a uniform l 2 curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity we also obtain a stronger e regularity theorem for ricci flows this result is particularly useful in the study of kahler ricci flows on fano manifolds where it can be used to derive certain convergence results

a compactness result for kahler ricci solitons abstract in this paper we prove a compactness result for compact kahler ricci gradient shrinking solitons if m i g i is a sequence of kahler ricci solitons of real dimension n 4 whose curvatures have uniformly bounded l n 2 norms whose ricci curvatures are uniformly bounded from below and g i 1 2 a where is perelman s functional there is a subsequence m i g i converging to a compact orbifold m g with finitely many isolated singularities where g is a kahler ricci soliton metric in an orbifold sense satisfies a soliton equation away from singular points and smoothly extends in some gauge to a metric satisfying kahler ricci soliton equation in a lifting around singular points

vector time frequency ar models for nonstationary multivariate random processes we introduce the vector time frequency autoregressive vtfar model for a parsimonious parametric description of nonstationary vector random processes the vtfar model generalizes the recently proposed scalar tfar model to the multivariate case it is physically meaningful because nonstationarity and spectral correlation are represented in terms of frequency shifts and it is parsimonious for the practically relevant class of underspread vector processes i e nonstationary vector processes with rapidly decaying correlation in time and frequency for vector processes with decaying correlation across the signals we introduce a variant of the vtfar model with banded parameter matrices furthermore we present a vtfar parameter estimator that is based on a system of linear equations with two level block toeplitz structure and we develop an efficient order recursive algorithm for solving these equations we also present information criteria for estimating the vtfar model order and the matrix bandwidth of the banded vtfar model the performance of the proposed vtfar parameter and order estimators is assessed through numerical simulations finally an application to nonstationary multivariate spectral analysis is presented

universal robustness characteristic of weighted networks against cascading failure we investigate the cascading failure on weighted complex networks by adopting a local weighted flow redistribution rule where the weight of an edge is k i k j theta with k i and k j being the degrees of the nodes connected by the edge assume that a failed edge leads only to a redistribution of the flow passing through it to its neighboring edges we found that the weighted complex network reaches the strongest robustness level when the weight parameter theta 1 where the robustness is quantified by a transition from normal state to collapse we determined that this is a universal phenomenon for all typical network models such as small world and scale free networks we then confirm by theoretical predictions this universal robustness characteristic observed in simulations we furthermore explore the statistical characteristics of the avalanche size of a network thus obtaining a power law avalanche size distribution together with a tunable exponent by varying theta our findings have great generality for characterizing cascading failure induced disasters in nature

regularity of einstein manifolds and the codimension 4 conjecture in this paper we are concerned with the regularity of noncollapsed riemannian manifolds m n g with bounded ricci curvature as well as their gromov hausdor limit spaces m n dj dgh x d where dj denotes the riemannian distance our main result is a solution to the codimension 4 conjecture namely thatx is smooth away from a closed subset of codimension 4 we combine this result with the ideas of quantitative stratication to prove a priori l q estimates on the full curvaturejrmj for all q v 0 and diam m d contains at most a nite number of dieomorphism classes a local version is used to show that noncollapsed 4 manifolds with bounded ricci curvature have a priori l 2 riemannian curvature estimates

communication and correlation among communities given a network and a partition in communities we consider the issues how communities influence each other and when two given communities do communicate specifically we address these questions in the context of small world networks where an arbitrary quenched graph is given and long range connections are randomly added we prove that among the communities a superposition principle applies and gives rise to a natural generalization of the effective field theory already presented by m ostilli and j f f mendes phys rev e 78 031102 2008 n 1 which here n 1 consists in a sort of effective tap thouless anderson and palmer equations in which each community plays the role of a microscopic spin the relative susceptibilities derived from these equations calculated at finite or zero temperature where the method provides an effective percolation theory give us the answers to the above issues unlike the case n 1 asymmetries among the communities may lead via the tap like structure of the equations to many metastable states whose number in the case of negative shortcuts among the communities may grow exponentially fast with n as examples we consider the n viana bray communities model and the n one dimensional small world communities model despite being the simplest ones the relevance of these models in network theory as e g in social networks is crucial and no analytic solution were known until now connections between percolation and the fractal dimension of a network are also discussed finally as an inverse problem we show how from the relative susceptibilities a natural and efficient method to detect the community structure of a generic network arises

on the perelman s reduced entropy and ricci flat manifolds with maximal volume growth in this paper we study the ricci flat manifolds with maximal volume growth using perelman s reduced volume of ricci flow we show that if m n g is an noncompact complete ricci flat manifold with maximal volume growth satisfying rm x rightarrow 0 as d x d g x p rightarrow infty then m n has the quadratic curvature decay some applications to this result are also presented

finiteness theorems for nonnegatively curved vector bundles we prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds more generally we obtain finiteness results for open riemannian manifolds whose topology is concentrated on compact domains of bounded geometry

critical metrics on connected sums of einstein four manifolds abstract we develop a gluing procedure designed to obtain canonical metrics on connected sums of einstein four manifolds the main application is an existence result using two well known einstein manifolds as building blocks the fubini study metric on cp 2 and the product metric on s 2 s 2 using these metrics in various gluing configurations toric invariant critical metrics are found on connected sums for a specific riemannian functional which depends on the global geometry of the factors furthermore using certain quotients of s 2 s 2 as one of the gluing factors critical metrics on several non simply connected manifolds are also obtained

ricci curvature bounds and einstein metrics on compact manifolds where f mo m1 is a homeomorphism and dil f is the dilatation of f given by dil f supx x2 dist f x f x2 dist x1 x2 if mo and m1 are not homeomorphic define dl mo mi oo gromov 20 proves the remarkable result that the space of compact riemannian manifolds f a 3 d of sectional curvature iki 3 0 and diameter dm v and diameter dm c iki dm vm1 in particular gromov s compactness theorem may be strengthened to the statement that f a v d is c1 l compact in the lipschitz topology in this paper we study the question of lipschitz convergence of compact riemannian manifolds with bounds imposed on the ricci curvature ric in

covariance matrices for second order vector random fields in space and time this paper deals with vector or multivariate random fields in space and or time with second order moments for which a framework is needed for specifying not only the properties of each component but also the possible cross relationships among the components we derive basic properties of the covariance matrix function of the vector random field and propose three approaches to construct covariance matrix functions for gaussian or non gaussian random fields the first approach is to take derivatives of a univariate covariance function the second one is to work on the univariate random field whose index domain is in a higher dimension and the third one is based on the scale mixture of separable spatio temporal covariance matrix functions to illustrate these methods many parametric or semiparametric examples are formulated

percolation in networks composed of connectivity and dependency links networks composed from both connectivity and dependency links were found to be more vulnerable compared to classical networks with only connectivity links their percolation transition is usually of a first order compared to the second order transition found in classical networks we analytically analyze the effect of different distributions of dependencies links on the robustness of networks for a random erdos renyi er network with average degree k that is divided into dependency clusters of size s the fraction of nodes that belong to the giant component p is given by p p s 1 1 exp kpp s where 1 p is the initial fraction of removed nodes our general result coincides with the known erdos renyi equation for random networks for s 1 for networks with poissonian distribution of dependency links we find that p is given by p f k p p e s 1 pf k p p 1 where f k p p 1 exp kpp and s is the mean value of the size of dependency clusters for networks with gaussian distribution of dependency links we show how the average and width of the distribution affect the robustness of the networks

chi 2 random fields in space and time this paper introduces some 2 random functions which are not homogeneous stationary in general but contain the stationary case as a special case and whose correlation structures are nonparametric and flexible so that many parametric forms can be specified the mean square differentiability of such 2 random function is discussed we also make the comparison of two 2 random functions through their finite dimensional laplace transforms

removable singularities in yang mills fields we show that a field satisfying the yang mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite we obtain the result that every yang mills field overr4 with bounded functional l2 norm may be obtained from a field ons4 r4 hodge or coulomb gauges are constructed for general small fields in arbitrary dimensions including 4

gap phenomena and curvature estimates for conformally compact einstein manifolds author s li gang qing jie shi yuguang abstract in this paper we first use the result in 12 to remove the assumption of the l 2 boundedness of weyl curvature in the gap theorem in 9 and then obtain a gap theorem for a class of conformally compact einstein manifolds with very large renormalized volume we also uses the blow up method to derive curvature estimates for conformally compact einstein manifolds with large renormalized volume the second part of this paper is on conformally compact einstein manifolds with conformal infinities of large yamabe constants based on the idea in 15 we manage to give the complete proof of the relative volume inequality 1 9 on conformally compact einstein manifolds therefore we obtain the complete proof of the rigidity theorem for conformally compact einstein manifolds in general dimensions with no spin structure assumption cf 29 15 as well as the new curvature pinch estimates for conformally compact einstein manifolds with conformal infinities of very large yamabe constant we also derive the curvature estimates for conformally compact einstein manifolds with conformal infinities of large yamabe constant

heterogeneous bond percolation on multitype networks with an application to epidemic dynamics considerable attention has been paid in recent years to the use of networks in modeling complex real world systems among the many dynamical processes involving networks propagation processes in which the final state can be obtained by studying the underlying network percolation properties have raised formidable interest in this paper we present a bond percolation model of multitype networks with an arbitrary joint degree distribution that allows heterogeneity in the edge occupation probability as previously demonstrated the multitype approach allows many nontrivial mixing patterns such as assortativity and clustering between nodes we derive a number of useful statistical properties of multitype networks as well as a general phase transition criterion we also demonstrate that a number of previous models based on probability generating functions are special cases of the proposed formalism we further show that the multitype approach by naturally allowing heterogeneity in the bond occupation probability overcomes some of the correlation issues encountered by previous models we illustrate this point in the context of contact network epidemiology

space of ricci flows i based on the compactness of the moduli of non collapsed calabi yau spaces with mild singularities we set up a structure theory for polarized kahler ricci flows with proper geo metric bounds our theory is a generalization of the structure theory of non collapsed kahler einstein manifolds as applications we prove the hamilton tian conjecture and the partial c 0 conjecture of tian

first and second passage times of rayleigh processes corresp the first and second passage times of a stationary rayleigh process r t a are discussed r t a represents the envelope of a stationary random process consisting of a sinusoidal signal of amplitude and frequency f 0 plus stationary gaussian noise of unit variance having a narrow band power spectral density which is symmetrical about f 0 approximate integral equations are developed whose solutions yield approximate probability densities concerning the first and second passage times of r t a the resulting probability functions are presented in graphs for the case when the power spectral density of the noise is gaussian related results concerning the approximate distribution function of the absolute minimum or absolute maximum of r t a in the closed interval 0 tau are also presented the exact probability densities are expressed in the form of an infinite series of multiple integrals

cascade of failures in coupled network systems with multiple support dependence relations we study both analytically and numerically the cascade of failures in two coupled network systems a and b where multiple support dependence relations are randomly built between nodes of networks a and b in our model we assume that each node in one network can function only if it has at least a single support link connecting it to a functional node in the other network we assume that networks a and b have i sizes n a and n b ii degree distributions of connectivity links p a k and p b k iii degree distributions of support links p a k and p b k and iv random attack removes 1 r a n a and 1 r b n b nodes form the networks a and b respectively we find the fractions of nodes a and b which remain functional giant component at the end of the cascade process in networks a and b in terms of the generating functions of the degree distributions of their connectivity and support links in a special case of erd s renyi networks with average degrees a and b in networks a and b respectively and poisson distributions of support links with average degrees a and b in networks a and b respectively a r a 1 exp a b 1 exp a a and b r b 1 exp b a 1 exp b b in the limit of a and b both networks become independent and our model becomes equivalent to a random attack on a single erd s renyi network we also test our theory on two coupled scale free networks and find good agreement with the simulations

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

gluing eguchi hanson metrics and a question of page in 1978 gibbons pope and page proposed a physical picture for the ricci flat kahler metrics on the k3 surface based on a gluing construction in this construction one starts from a flat torus with 16 orbifold points and resolves the orbifold singularities by gluing in 16 small eguchi hanson manifolds that all have the same orientation this construction was carried out rigorously by topiwala lebrun singer and donaldson in 1981 page asked whether the above construction can be modified by reversing the orientations of some of the eguchi hanson manifolds this is a subtle question if successful this construction would produce einstein metrics that are neither kahler nor self dual in this paper we focus on a configuration of maximal symmetry involving eight small eguchi hanson manifolds of each orientation that are arranged according to a chessboard pattern by analyzing the interactions between eguchi hanson manifolds with opposite orientation we identify a nonvanishing obstruction to the gluing problem thereby destroying any hope of producing a metric of zero ricci curvature in this way using this obstruction we are able to understand the dynamics of such metrics under ricci flow as long as the eguchi hanson manifolds remain small in particular for the configuration described above we obtain an ancient solution to the ricci flow with the property that the maximum of the riemann curvature tensor blows up at a rate of t 1 2 while the maximum of the ricci curvature converges to 0 2016 wiley periodicals inc

kahler einstein metrics on complex surfaces with c 1 0 various estimates of the lower bound of the holomorphic invariant m defined in t are given here by using branched coverings potential estimates and lelong numbers of positive d closed 1 1 currents of certain type etc these estimates are then applied to produce kahler einstein metrics on complex surfaces withc1 0 in particular we prove that there are kahler einstein structures withc1 0 on any manifold of differential type cp 2 overline ncp 2 3 leqq n leqq 8

removing point singularities of riemannian manifolds we study the behavior of geodesics passing through a point singularity of a riemannian manifold in particular we show that if the curvature does not blow up too rapidly near the singularity then the singularity is at worst an orbifold singularity the idea is to construct the exponential map centered at a singularity since there is no tangent space at the singularity a surrogate is needed we show that the vector space of radially parallel vector fields is well defined and that there is a correspondence between unit radially parallel vector fields and geodesics emanating from the singular point

a gradient estimate for all positive solutions of the conjugate heat equation under ricci flow abstract we establish a point wise gradient estimate for all positive solutions of the conjugate heat equation this contrasts to perelman s point wise gradient estimate which works mainly for the fundamental solution rather than all solutions like perelman s estimate the most general form of our gradient estimate does not require any curvature assumption moreover assuming only lower bound on the ricci curvature we also prove a localized gradient estimate similar to the li yau estimate for the linear schrodinger heat equation the main difference with the linear case is that no assumptions on the derivatives of the potential scalar curvature are needed a classical harnack inequality follows

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

intrinsic limits of dimensionality and richness in random multipath fields we study the dimensions or degrees of freedom of farfield multipath that is observed in a limited source free region of space the multipath fields are studied as solutions to the wave equation in an infinite dimensional vector space we prove two universal upper bounds on the truncation error of fixed and random multipath fields a direct consequence of the derived bounds is that both fixed and random multipath fields have an effective finite dimension for circular and spherical spatial regions we show that this finite dimension is proportional to the radius and area of the region respectively we use the karhunen loegraveve kl expansion of random multipath fields to quantify the notion of multipath richness the multipath richness is defined as the number of significant eigenvalues in the kl expansion that achieve 99 of the total multipath energy we establish a lower bound on the largest eigenvalue this lower bound quantifies to some extent the well known reduction of multipath richness with reducing the angular power spread of multipath angular power spectrum

properties of the solutions of the conjugate heat equations in this paper we consider the class cal a of those solutions u x t to the conjugate heat equation partial over partial t u delta u ru on compact k ahler manifolds m with c 1 0 where g t changes by the unnormalized k ahler ricci flow blowing up at t infty which satisfy perelman s differential harnack inequality 6 on 0 t we show cal a is nonempty if mathop rm ric nolimits g t le c over t t which is always true if we have a type i singularity we prove the solution u x t satisfies the elliptic type harnack inequality with the constants that are uniform in time if the flow g t has a type i singularity at t then cal a has exactly one element