on the decay and stability to global solutions of the 3 d inhomogeneous navier stokes equations in this chapter we consider the large time decay and stability to any given global smooth solution of the three dimensional inhomogeneous incompressible navier stokes equations

structural and functional discovery in dynamic networks with non negative matrix factorization time series of graphs are increasingly prevalent in modern data and pose unique challenges to visual exploration and pattern extraction this paper describes the development and application of matrix factorizations for exploration and time varying community detection in time evolving graph sequences the matrix factorization model allows the user to home in on and display interesting underlying structure and its evolution over time the methods are scalable to weighted networks with a large number of time points or nodes and can accommodate sudden changes to graph topology our techniques are demonstrated with several dynamic graph series from both synthetic and real world data including citation and trade networks these examples illustrate how users can steer the techniques and combine them with existing methods to discover and display meaningful patterns in sizable graphs over many time points

global well posedness for the incompressible mhd equations with variable viscosity and conductivity abstract in this paper we consider the initial boundary value problem of the incompressible mhd equations with variable viscosity and conductivity in a smooth bounded domain r 3 we establish the global well posedness of strong solutions in both non vacuum and vacuum cases when the initial velocity field and magnetic field are suitably small in some sense with arbitrarily large initial density in addition we also get some results of the large time behavior in both two cases our results generalize some previous results in some sense

network cross validation for determining the number of communities in network data abstractthe stochastic block model sbm and its variants have been a popular tool for analyzing large network data with community structures in this article we develop an efficient network cross validation ncv approach to determine the number of communities as well as to choose between the regular stochastic block model and the degree corrected block model dcbm the proposed ncv method is based on a block wise node pair splitting technique combined with an integrated step of community recovery using sub blocks of the adjacency matrix we prove that the probability of under selection vanishes as the number of nodes increases under mild conditions satisfied by a wide range of popular community recovery algorithms the solid performance of our method is also demonstrated in extensive simulations and two data examples supplementary materials for this article are available online

lattice boltzmann model for the incompressible navier stokes equation in this paper a lattice boltzmann lb model to simulate incompressible flow is developed the main idea is to explicitly eliminate the terms of o m2 where m is the mach number due to the density fluctuation in the existing lb models in the proposed incompressible lb model the pressure p instead of the mass density is the independent dynamic variable the incompressible navier stokes equations are derived from the incompressible lb model via chapman enskog procedure numerical results of simulations of the plane poiseuille flow driven either by pressure gradient or a fixed velocity profile at entrance as well as of the 2d womersley flow are presented the numerical results are found to be in excellent agreement with theory

incompressible flows with piecewise constant density we investigate the incompressible navier stokes equations with variable density the aim is to prove existence and uniqueness results in the case of discontinuous initial density in dimension n 2 3 assuming only that the initial density is bounded and bounded away from zero and that the initial velocity is smooth enough we get the local in time existence of unique solutions uniqueness holds in any dimension and for a wider class of velocity fields in particular all those results are true for piecewise constant densities with arbitrarily large jumps global results are established in dimension two if the density is close enough to a positive constant and in n dimensions if in addition the initial velocity is small the lagrangian formulation for describing the flow plays a key role in the analysis that is proposed in the present paper

an exponential family of probability distributions for directed graphs abstract directed graph or digraph data arise in many fields especially in contemporary research on structures of social relationships we describe an exponential family of distributions that can be used for analyzing such data a substantive rationale for the general model is presented and several special cases are discussed along with some possible substantive interpretations a computational algorithm based on iterative scaling procedures for use in fitting data is described as are the results of a pilot simulation study an example using previously reported empirical data is worked out in detail an extension to multiple relationship data is discussed briefly

community extraction for social networks analysis of networks and in particular discovering communities within networks has been a focus of recent work in several fields and has diverse applications most community detection methods focus on partitioning the entire network into communities with the expectation of many ties within communities and few ties between however many networks contain nodes that do not fit in with any of the communities and forcing every node into a community can distort results here we propose a new framework that extracts one community at a time allowing for arbitrary structure in the remainder of the network which can include weakly connected nodes the main idea is that the strength of a community should depend on ties between its members and ties to the outside world but not on ties between nonmembers the proposed extraction criterion has a natural probabilistic interpretation in a wide class of models and performs well on simulated and real networks for the case of the block model we establish asymptotic consistency of estimated node labels and propose a hypothesis test for determining the number of communities

global existence of weak solutions for a viscous two phase model abstract the purpose of this paper is to explore a viscous two phase liquid gas model relevant for well and pipe flow our approach relies on applying suitable modifications of techniques previously used for studying the single phase isothermal navier stokes equations a main issue is the introduction of a novel two phase variant of the potential energy function needed for obtaining fundamental a priori estimates we derive an existence result for weak solutions in a setting where transition to single phase flow is guaranteed not to occur when the initial state is a true mixture of both phases some numerical examples are also included in order to demonstrate characteristic behavior of solutions in particular we illustrate how two phase flow is genuinely different compared to single phase flow concerning the behavior of an initial mass discontinuity

statistical clustering of temporal networks through a dynamic stochastic block model statistical node clustering in discrete time dynamic networks is an emerging field that raises many challenges here we explore statistical properties and frequentist inference in a model that combines a stochastic block model sbm for its static part with independent markov chains for the evolution of the nodes groups through time we model binary data as well as weighted dynamic random graphs with discrete or continuous edges values our approach motivated by the importance of controlling for label switching issues across the different time steps focuses on detecting groups characterized by a stable within group connectivity behavior we study identifiability of the model parameters propose an inference procedure based on a variational expectation maximization algorithm as well as a model selection criterion to select for the number of groups we carefully discuss our initialization strategy which plays an important role in the method and compare our procedure with existing ones on synthetic datasets we also illustrate our approach on dynamic contact networks one of encounters among high school students and two others on animal interactions an implementation of the method is available as a r package called dynsbm

latent space approaches to social network analysis network models are widely used to represent relational information among interacting units in studies of social networks recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation between actors we develop a class of models where the probability of a relation between actors depends on the positions of individuals in an unobserved social space we make inference for the social space within maximum likelihood and bayesian frameworks and propose markov chain monte carlo procedures for making inference on latent positions and the effects of observed covariates we present analyses of three standard datasets from the social networks literature and compare the method to an alternative stochastic blockmodeling approach in addition to improving on model fit for these datasets our method provides a visual and interpretable model based spatial representation of social relationships and improv

global well posedness of the navier stokes equations of an inhomogeneous fluid in the half space with inflow boundary condition abstract in this paper we study the global solvability of the incompressible navier stokes equations of an inhomogeneous fluid in the half space with inflow boundary condition we prove the existence of a solution in l q l p framework when the initial density is close to a positive constant and the initial and boundary data of velocity are in some critical function spaces uniqueness holds true if the given data have slightly more regularity

hypergraph based anomaly detection of high dimensional co occurrences this paper addresses the problem of detecting anomalous multivariate co occurrences using a limited number of unlabeled training observations a novel method based on using a hypergraph representation of the data is proposed to deal with this very high dimensional problem hypergraphs constitute an important extension of graphs which allow edges to connect more than two vertices simultaneously a variational expectation maximization algorithm for detecting anomalies directly on the hypergraph domain without any feature selection or dimensionality reduction is presented the resulting estimate can be used to calculate a measure of anomalousness based on the false discovery rate the algorithm has o np computational complexity where n is the number of training observations and p is the number of potential participants in each co occurrence event this efficiency makes the method ideally suited for very high dimensional settings and requires no tuning bandwidth or regularization parameters the proposed approach is validated on both high dimensional synthetic data and the enron email database where p 75 000 and it is shown that it can outperform other state of the art methods

an improved multiphase lattice boltzmann flux solver for three dimensional flows with large density ratio and high reynolds number an improved multiphase lattice boltzmann flux solver mlbfs is proposed in this work for effective simulation of three dimensional 3d multiphase flows with large density ratio and high reynolds number as a finite volume scheme the mlbfs originally proposed in 27 applies the finite volume method to solve for macroscopic flow variables directly the fluxes are reconstructed locally at each cell interface by using the standard lbm solutions due to the modeling error of the standard lbm the reconstructed fluxes deviate from those in the navier stokes equations and to compensate this error a complex tensor is introduced in the original mlbfs however the computation of the tensor introduces additional complexity and usually needs a relatively thicker interface thickness to maintain numerical stability which makes the solver be complex and inefficient in the 3d case to remove this drawback in this work a theoretical analysis to the formulations obtained from the chapman enskog expansion is conducted it is shown that the modeling error can be effectively removed by modifying the computation of the equilibrium density distribution function with this improvement the proposed 3d mlbfs not only avoids the calculation of the compensation tensor but also is able to maintain numerical stability with very thin interface thickness several benchmark cases including the challenging droplet impacting on a dry surface head on collisions of binary droplets and droplet splashing on a thin film with density ratio 1000 and reynolds number up to 3000 are studied to validate the proposed solver the obtained results agree well with the published data

efficient computations for high density ratio rising bubble flows using a diffused interface coupled lattice boltzmann level set scheme a mass conserving diffused interface coupled lattice boltzmann level set scheme is proposed and numerically studied for the simulation of high density and viscosity ratio multiphase flows the approach is based on the pressure evolution formulation of the lattice boltzmann equation which is then coupled with the level set equation to capture a diffused level set function multiple relaxation time collision and isotropic force discretization techniques are employed to further reinforce the stability and accuracy of the lbm solver and a monolithic approach is employed to convect the level set function with minimal computations to preserve a smooth density profile we present extensive investigations for numerical accuracy through performing benchmark simulations for rising bubble problems it is observed that the proposed scheme is successful in producing accurate results as compared to those from high precision 2d finite element solutions in hysing et al 2009 offering a remarkable improvement in mass conservation and characteristic benchmark quantities upon our previous sharp interface coupled model described in safi and turek 2014 moreover parallel scalability of the coupled lb scheme is shown to be preserved through performing efficient cpu and gpgpu based computations

existence results for viscous polytropic fluids with vacuum abstract we consider the full navier stokes equations for viscous polytropic fluids with nonnegative thermal conductivity we prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition the initial density need not be positive and may vanish in an open set moreover our results hold for both bounded and unbounded domains

attribute fusion in a latent process model for time series of graphs hypothesis testing on time series of attributed graphs has applications in diverse areas e g social network analysis wherein vertices represent individual actors or organizations connectome inference wherein vertices are neurons or brain regions and text processing wherein vertices represent authors or documents we consider the problem of anomaly change point detection given the latent process model for time series of graphs with categorical attributes on the edges presented in n h lee and c e priebe a latent process model for time series of attributed random graphs statist inference stoch process vol 14 pp 231 253 2011 various attributed graph invariants are considered and their power for detection as a function of a linear fusion parameter is presented our main result is that inferential performance in mathematically tractable first order and second order approximation models does provide guidance for methodological choices applicable to the exact realistic but intractable model furthermore to the extent that the exact model is realistic we may tentatively conclude that approximation model investigations have some bearing on real data applications

stochastic blockmodels for directed graphs abstract holland and leinhardt 1981 proposed the p 1 model for the analysis of binary directed graph data in network studies such a model provides information about the attractiveness and expansiveness of the individual nodes in the network as well as the tendency of a pair of nodes to reciprocate relational ties when the nodes are a priori partitioned into subgroups based on attributes such as race and sex the density of ties from one subgroup to another can differ considerably from that relating another pair of subgroups thus creating a situation called blocking in social networks the p 1 model completely ignores this extra piece of information and is therefore unable to explain the block structure blockmodels that are simple extensions of the p 1 model are proposed specifically for such data an iterative scaling algorithm is presented for fitting the model parameters by maximum likelihood the methodology is illustrated in detail on two empirical examples

a conservative level set method for two phase flow ii in this paper we continue to develop and study the conservative level set method for incompressible two phase flow with surface tension introduced in j comput phys 210 2005 225 246 we formulate a modification of the reinitialization and present a theoretical study of what kind of conservation we can expect of the method a finite element discretization is presented as well as an adaptive mesh control procedure numerical experiments relevant for problems in petroleum engineering and material science are presented for these problems the surface tension is strong and conservation of mass is important problems in both two and three dimensions with uniform as well as non uniform grids are studied from these calculations convergence and conservation is studied good conservation and convergence are observed

on the global well posedness of 2 d inhomogeneous incompressible navier stokes system with variable viscous coefficient abstract given solenoidal vector u 0 h 2 h 1 r 2 0 1 l 2 r 2 and 0 l w 1 r r 2 with a positive lower bound for 0 1 2 and 2 r 2 1 2 we prove that 2 d incompressible inhomogeneous navier stokes system 1 1 has a unique global solution provided that the viscous coefficient 0 is close enough to 1 in the l norm compared to the size of and the norms of the initial data with smoother initial data we can prove the propagation of regularities for such solutions furthermore for 1 p 4 if 0 1 u 0 belongs to the critical besov spaces b p 1 2 p r 2 b p 1 1 2 p l 2 r 2 and the b p 1 2 p r 2 norm of 0 1 is sufficiently small compared to the exponential of u 0 l 2 2 u 0 b p 1 1 2 p we prove the global well posedness of 1 1 in the scaling invariant spaces finally for initial data in the almost critical besov spaces we prove the global well posedness of 1 1 under the assumption that the l norm of 0 1 is sufficiently small

estimation and prediction for stochastic blockstructures a statistical approach to a posteriori blockmodeling for digraphs and valued digraphs is proposed the probability model assumes that the vertices of the digraph are partitioned into several unobserved latent classes and that the probability distribution of the relation between two vertices depends only on the classes to which they belong a bayesian estimator based on gibbs sampling is proposed the basic model is not identified because class labels are arbitrary the resulting identifiability problems are solved by restricting inference to the posterior distributions of invariant functions of the parameters and the vertex class membership in addition models are considered where class labels are identified by prior distributions for the class membership of some of the vertices the model is illustrated by an example from the social networks literature kapferer s tailor shop

phase transitions for detecting latent geometry in random graphs random graphs with latent geometric structure are popular models of social and biological networks with applications ranging from network user profiling to circuit design these graphs are also of purely theoretical interest within computer science probability and statistics a fundamental initial question regarding these models is when are these random graphs affected by their latent geometry and when are they indistinguishable from simpler models without latent structure such as the erd s renyi graph mathcal g n p we address this question for two of the most well studied models of random graphs with latent geometry the random intersection and random geometric graph our results are as follows a the random intersection graph is defined by sampling n random sets s 1 s 2 ldots s n by including each element of a set of size d in each s i independently with probability delta and including the edge i j if s i cap s j ne emptyset we prove that the random intersection graph converges in total variation to an erd s renyi graph if d tilde omega n 3 and does not if d o n 3 for both dense and sparse edge densities p this resolves an open problem in fill et al random struct algorithms 16 2 156 176 2000 rybarczyk random struct algorithms 38 1 2 205 234 2011 and kim et al random struct algorithms 52 4 662 679 2018 the same result was obtained simultaneously and independently by bubeck et al when random intersection graphs lose geometry manuscript 2019 b we strengthen the preceding argument to show that the matrix of random intersection sizes s i cap s j converges in total variation to a symmetric matrix with independent poisson entries this yields the first total variation convergence result for tau random intersection graphs where the edge i j is included if s i cap s j ge tau more precisely our results imply that if p is bounded away from 1 then the tau random intersection graph with edge density p converges to mathcal g n p if d omega tau 3 n 3 c the random geometric graph on mathbb s d 1 is defined by sampling x 1 x 2 dots x n uniformly at random from mathbb s d 1 and including the edge i j if vert x i x j vert 2 le tau a result of bubeck et al random struct algorithms 49 503 532 2016 showed that this model converges to mathcal g n p in total variation where p is chosen so that the models have matching edge densities as long as d omega n 3 it was conjectured in bubeck et al 2016 that this threshold decreases drastically for p small we make the first progress towards this conjecture by showing convergence if d tilde omega left min pn 3 p 2 n 7 2 right our proofs are a hybrid of combinatorial arguments direct couplings and applications of information inequalities previous upper bounds on the total variation distance between random graphs with latent geometry and mathcal g n p have typically not been both combinatorial and information theoretic while this interplay is essential to the sharpness of our bounds

unique solvability for the density dependent navier stokes equations abstract in this paper we consider the incompressible navier stokes equations with a density dependent viscosity in a bounded domain of r n n 2 3 we prove the local existence of unique strong solutions for all initial data satisfying a natural compatibility condition this condition is also necessary for a very general initial data moreover we provide a blow up criterion for the regularity of the strong solution for these results the initial density need not be strictly positive it may vanish in an open subset of

an efficient lattice boltzmann multiphase model for 3d flows with large density ratios at high reynolds numbers we report on the development implementation and validation of a new lattice boltzmann method lbm for the numerical simulation of three dimensional multiphase flows here with only two components with both high density ratio and high reynolds number this method is based in part on but aims at achieving a higher computational efficiency than inamuro et al s model inamuro et al 2004 here we use a lbm to solve both a pressureless navier stokes equation in which the implementation of viscous terms is improved and a pressure poisson equation using different distribution functions and a d3q19 lattice scheme additionally we propose a new diffusive interface capturing method based on the cahn hilliard equation which is also solved with a lbm to achieve maximum efficiency the entire model is implemented and solved on a heavily parallel gpgpu co processor the proposed algorithm is applied to several test cases such as a splashing droplet a rising bubble and a braking ocean wave in all cases numerical results are found to agree very well with reference data and or to converge with the discretization

a stable discretization of the lattice boltzmann equation for simulation of incompressible two phase flows at high density ratio a stable discretization of the lattice boltzmann equation lbe for non ideal gases is presented for simulation of incompressible two phase flows having high density and viscosity ratios the stiffness of the discretized forcing terms in lbe for non ideal gases is known to trigger severe numerical instability and restrict practical application of the lbe method use of a proper pressure updating scheme is also crucial to the stability of the lbe method because of non negligible pressure variation across the phase interface to deal with these issues we propose a stable discretization scheme which assumes the low mach number approximation and utilizes the stress and potential forms of the surface tension force the incompressible transformation and the consistent discretization of the intermolecular forcing terms the proposed stable discretization scheme is applied to simulate 1 d advection equation with a source term a stationary droplet droplet oscillation and droplet splashing and deposition on a thin film at a density ratio of 1000 with varying reynolds numbers the numerical solutions of stationary and oscillatory droplets agree well with analytic solutions including the laplace s law the time history of the spread factor of the liquid sheet emitted after the droplet impact also follows the known spreading power law

on the well posedness of 3 d inhomogeneous incompressible navier stokes equations with variable viscosity abstract in this paper we mainly study the well posedness for the 3 d inhomogeneous incompressible navier stokes equations with variable viscosity with some smallness assumption on the bmo norm of the initial density we first get the local well posedness of 1 1 in the critical besov spaces moreover if the viscosity coefficient is a constant we can extend this local solution to be a global one our theorem implies that we have successfully extended the integrability index p of the initial velocity which has been obtained by abidi gui and zhang in 3 burtea in 8 and zhai and yin in 32 to approach the ideal one i e 1 p 6 the main novelty of this work is to apply the crw theorem obtained by coifman rochberg weiss in 11 to get a new a priori estimate for an elliptic equation with variable coefficients the uniqueness of the solution also relies on a lagrangian approach as in 16 17 18

logit models and logistic regressions for social networks i an introduction to markov graphs and p spanning nearly sixty years of research statistical network analysis has passed through at least two generations of researchers and models beginning in the late 1930 s the first generation of research dealt with the distribution of various network statistics under a variety of null models the second generation beginning in the 1970 s and continuing into the 1980 s concerned models usually for probabilities of relational ties among very small subsets of actors in which various simple substantive tendencies were parameterized much of this research most of which utilized log linear models first appeared in applied statistics publications

global solution to the 3 d density dependent incompressible flow of liquid crystals abstract in this paper we consider the global wellposedness of the 3 d density dependent incompressible flow of liquid crystals in the critical besov spaces we first get the local solution for this system without assumptions of small density variation and then extend this local solution to be global under the smallness assumptions on initial density initial orientation field and the horizontal components of the initial velocity field this result implies the global well posedness of this system with large initial vertical velocity component

optimal graphon estimation in cut distance consider the twin problems of estimating the connection probability matrix of an inhomogeneous random graph and the graphon of a w random graph we establish the minimax estimation rates with respect to the cut metric for classes of block constant matrices and step function graphons surprisingly our results imply that from the minimax point of view the raw data that is the adjacency matrix of the observed graph is already optimal and more involved procedures cannot improve the convergence rates for this metric this phenomenon contrasts with optimal rates of convergence with respect to other classical distances for graphons such as the l 1 or l 2 metrics

benchmark computations for 3d two phase flows following our previous work on the application of the diffuse interface coupled lattice boltzmann level set lb ls approach to benchmark computations for 2d rising bubble simulations this paper investigates the performance of the coupled scheme in 3d two phase flows in particular the use of different lattice stencils e g d3q15 d3q19 and d3q27 is studied and the results for 3d rising bubble simulations are compared with regard to isotropy and accuracy against those obtained by finite element and finite difference solutions of the navierstokes equations it is shown that the method can eventually recover the benchmark solutions provided that the interface region is aptly refined by the underlying lattice following the benchmark simulations the application of the method in solving other numerically subtle problems e g binary droplet collision and droplet splashing on wet surface under high re and we numbers is presented moreover implementations on general purpose gpus are pursued where the computations are adaptively refined around the critical parts of the flow

covariate assisted spectral clustering biological and social systems consist of myriad interacting units the interactions can be represented in the form of a graph or network measurements of these graphs can reveal the underlying structure of these interactions which provides insight into the systems that generated the graphs moreover in applications such as connectomics social networks and genomics graph data are accompanied by contextualizing measures on each node we utilize these node covariates to help uncover latent communities in a graph using a modification of spectral clustering statistical guarantees are provided under a joint mixture model that we call the node contextualized stochastic blockmodel including a bound on the misclustering rate the bound is used to derive conditions for achieving perfect clustering for most simulated cases covariate assisted spectral clustering yields results superior both to regularized spectral clustering without node covariates and to an adaptation of canonical correlation analysis we apply our clustering method to large brain graphs derived from diffusion mri data using the node locations or neurological region membership as covariates in both cases covariate assisted spectral clustering yields clusters that are easier to interpret neurologically

optimal control for multi phase fluid stokes problems optimal control for a system consistent of the viscosity dependent stokes equations coupled with a transport equation for the viscosity is studied motivated by a lack of sufficient regularity of the adjoint equations artificial diffusion is introduced to the transport equation the asymptotic behavior of the regularized system is investigated optimality conditions for the regularized optimal control problems are obtained and again the asymptotic behavior is analyzed the lack of uniqueness of solutions to the underlying system is another source of difficulties for the problem under investigation

on the global well posedness of 2 d boussinesq system with variable viscosity abstract in this paper we investigate the global well posedness of 2 d boussinesq system which has variable kinematic viscosity and with thermal conductivity of d with general initial data provided that the viscosity coefficient is sufficiently close to some positive constant in l norm

fujita kato theorem for the 3 d inhomogeneous navier stokes equations abstract in this paper we prove the global existence and uniqueness of the solution with discontinuous density for the 3 d inhomogeneous navier stokes equations if the initial data 0 u 0 l r 3 h s r 3 with s 1 2 satisfies 0 c 0 0 c 0 u 0 h 1 2 e for some small e 0 depending only on c 0 c 0 furthermore we introduce the dual method to show that if u 0 l p r 3 for p 6 5 2 the velocity satisfies the decay estimate k u t l 2 c 1 t k 2 p for t 1 k 0 1 with p 3 2 1 p 1 2

regularity of weak solutions of compressible isentropic self gravitating fluid abstract in this paper the regularity of weak solutions to the three dimensional compressible isentropic self gravitating viscous fluids is considered in periodic domain inspired by the idea of desjardins 1997 3 4 we get the higher integrability of density and we remove the additional assumption of l 0 t l q 0 in the work of choe and jin 2003 finally we prove the supremum of the density is finite and the infimum of the density is positive in the domain 0 t t 3 for a positive time t moreover moser type iteration scheme is used to obtain the l norm estimate of the velocity

a bayesian multidimensional scaling procedure for the spatial analysis of revealed choice data we present a new bayesian formulation of a vector multidimensional scaling procedure for the spatial analysis of binary choice data the gibbs sampler is gainfully employed to estimate the posterior distribution of the specified scalar products bilinear model parameters the computational procedure allows for the explicit estimation of a covariance matrix which can accommodate violations of iia due to context effects in addition posterior standard errors can be estimated which reflect differential degrees of consumer choice uncertainty and or brand position instability a marketing application concerning the analysis of consumers consideration sets for luxury automobiles is provided to illustrate the use of the proposed methodology

global regularity for the initial boundary value problem of the 2 d boussinesq system with variable viscosity and thermal diffusivity abstract we prove the global existence of strong solution to the initial boundary value problem of the 2 d boussinesq system and 3 d infinite prandtl number model with viscosity and thermal conductivity depending on the temperature

on the wellposedness of three dimensional inhomogeneous navier stokes equations in the critical spaces we prove the local wellposedness of three dimensional incompressible inhomogeneous navier stokes equations with initial data in the critical besov spaces without assumptions of small density variation furthermore if the initial velocity field is small enough in the critical besov space dot b 1 2 2 1 mathbb r 3 this system has a unique global solution

bayesian multidimensional scaling and choice of dimension multidimensional scaling is widely used to handle data that consist of similarity or dissimilarity measures between pairs of objects we deal with two major problems in metric multidimensional scaling configuration of objects and determination of the dimension of object configuration within a bayesian framework a markov chain monte carlo algorithm is proposed for object configuration along with a simple bayesian criterion called mdsic for choosing their dimension simulation results are presented as are real data our method provides better results than does classical multidimensional scaling and alscal for object configuration and mdsic seems to work well for dimension choice in the examples considered

network cross validation by edge sampling summary while many statistical models and methods are now available for network analysis resampling of network data remains a challenging problem cross validation is a useful general tool for model selection and parameter tuning but it is not directly applicable to networks since splitting network nodes into groups requires deleting edges and destroys some of the network structure in this paper we propose a new network resampling strategy based on splitting node pairs rather than nodes that is applicable to cross validation for a wide range of network model selection tasks we provide theoretical justification for our method in a general setting and examples of how the method can be used in specific network model selection and parameter tuning tasks numerical results on simulated networks and on a statisticians citation network show that the proposed cross validation approach works well for model selection

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

matrix estimation by universal singular value thresholding consider the problem of estimating the entries of a large matrix when the observed entries are noisy versions of a small random fraction of the original entries this problem has received widespread attention in recent times especially after the pioneering works of emmanuel candes and collaborators this paper introduces a simple estimation procedure called universal singular value thresholding usvt that works for any matrix that has a little bit of structure surprisingly this simple estimator achieves the minimax error rate up to a constant factor the method is applied to solve problems related to low rank matrix estimation blockmodels distance matrix completion latent space models positive definite matrix completion graphon estimation and generalized bradley terry models for pairwise comparison

pseudolikelihood estimation for social networks abstract interest in log linear modeling for social network data has grown steadily since holland and leinhardt 1981 proposed their p 1 model that model was designed for a single binary relationship directed graph representing interactions between individuals it assumed that interactions between pairs of individuals are mutually independent subsequent work has extended the model in various ways including block modeling and the case of dependence between pairs of individuals in empirical work it would often be desirable to fit a wide variety of these models as the differences in predictions or goodness of fit are likely to provide insights into the data this has not been common practice however because estimation for some of the models has been difficult and the maximum likelihood schemes developed for others involve different computer programs not always available in standard packages the focus of this article is on a general estimation technique that maximizes the pseudolikelihood the pro

global strong solutions to the 3d incompressible mhd equations with density dependent viscosity abstract this paper is concerned with the 3d incompressible mhd equations with density dependent viscosity in smooth bounded domains through time weighted a priori estimates the global existence of strong solutions is established under the assumption that the initial energy is suitably small this generalizes previous results for the 3d navier stokes equations in huang and wang 2015 and zhang 2015 which need u 0 l 2 to be small the initial vacuum is allowed

latent surface models for networks using aggregated relational data despite increased interest across a range of scientific applications in modeling and understanding social network structure collecting complete network data remains logistically and financially challenging especially in the social sciences this article introduces a latent surface representation of social network structure for partially observed network data we derive a multivariate measure of expected latent distance between an observed actor and unobserved actors with given features we also draw novel parallels between our work and dependent data in spatial and ecological statistics we demonstrate the contribution of our model using a random digit dial telephone survey and a multiyear prospective study of the relationship between network structure and the spread of infectious disease the model proposed here is related to previous network models which represents high dimensional structure through a projection to a low dimensional latent geometric surface encoding dependence as distance in the spa

an existence theorem for the multifluid navier stokes problem abstract existence of a weak solution of the navier stokes problem describing a multifluid flow is proved the velocity fields associated to each fluid solves the navier stokes equations in a time dependent domain classical immiscibility conditions on the varying fluids interfaces are taken into account by a new formulation of the problem this formulation was introduced in 5 and used in numerical computations this paper follows a previous one where an existence theorem for stokes multifluid problems was derived 7

spectral clustering and the high dimensional stochastic blockmodel networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors social ne tworks representing people who communicate with each other are one example communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks spectral clustering is a popular and computationally feasi ble method to discover these communities the stochastic block model holland et al 1983 is a social network model with well defined communities each node is a member of one community for a network generated from the stochastic block model we bound the number of nodes misclus tered by spectral clustering the asymptotic results in th is paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes hence the name high dimensional in order to study spectral clustering under the stochastic block model we first show that under the more general latent space model the eigenvectors of the normalized graph laplacian asymptotically converge to the eigenvectors of a population normal ized graph laplacian aside from the implication for spectral clustering this provides insight into a graph visualization technique our method of studying the eigenvectors of random matrices is original ams 2000 subject classifications primary 62h30 62h25 secondary 60b20

convergence of numerical approximations of the incompressible navier stokes equations with variable density and viscosity we consider numerical approximations of incompressible newtonian fluids having variable possibly discontinuous density and viscosity since solutions of the equations with variable density and viscosity may not be unique numerical schemes may not converge if the solution is unique then approximate solutions computed using the discontinuous galerkin method to approximate the convection of the density and stable finite element approximations of the momentum equation converge to the solution if the solution is not unique a subsequence of these approximate solutions will converge to a solution

large global solutions to 3 d inhomogeneous navier stokes equations slowly varying in one variable abstract motivated by chemin and gallagher 2010 8 we consider the global wellposedness to the 3 d incompressible inhomogeneous navier stokes equations with large initial velocity slowly varying in one space variable in particular we proved that when the initial density is close enough to a positive constant then given divergence free initial velocity field of the type v 0 h w 0 h w 0 3 x h x 3 as that in chemin and gallagher 2010 8 for the classical navier stokes system we shall prove the global wellposedness of ins for sufficiently small the main difficulty here lies in the fact that we will have to obtain the l 1 r lip r 3 estimate for convection velocity in the transport equation of ins toward this and due to the strong anisotropic properties of the approximate solutions we will have to work in the framework of anisotropic type besov spaces here

on the well posedness of 2 d incompressible navier stokes equations with variable viscosity in critical spaces abstract in this paper we first prove the local well posedness of the 2 d incompressible navier stokes equations with variable viscosity in critical besov spaces with negative regularity indices without smallness assumption on the variation of the density the key is to prove for p 1 4 and a b p 1 2 p r 2 that the solution mapping h a f to the 2 d elliptic equation div 1 a div f is bounded on b p 1 2 p 1 r 2 more precisely we prove that b p 1 2 p 1 c 1 a b p 1 2 p 2 f b p 1 2 p 1 the proof of the uniqueness of solution to 1 2 relies on a lagrangian approach 15 16 17 when the viscosity coefficient is a positive constant we prove that 1 2 is globally well posed

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

a lattice boltzmann model for multiphase flows with large density ratio a lattice boltzmann model for simulating multiphase flows with large density ratios is described in this paper the method is easily implemented it does not require solving the poisson equation and does not involve the complex treatments of derivative terms the interface capturing equation is recovered without any additional terms as compared to other methods m r swift w r osborn j m yeomans lattice boltzmann simulation of liquid gas and binary fluid systems phys rev e 54 1996 5041 5052 t inamuro t ogata s tajima n konishi a lattice boltzmann method for incompressible two phase flows with large density differences j comput phys 198 2004 628 644 t lee c l lin a stable discretization of the lattice boltzmann equation for simulation of incompressible two phase flows at high density ratio j comput phys 206 2005 16 47 besides it requires less discrete velocities as a result its efficiency could be greatly improved especially in 3d applications it is validated by several cases a bubble in a stationary flow and the capillary wave the numerical surface tension obtained from the laplace law and the interface profile agrees very well with the respective analytical solution the method is further verified by its application to capillary wave and the bubble rising under buoyancy with comparison to other methods all the numerical experiments show that the present approach can be used to model multiphase flows with large density ratios

sparse maximum entropy random graphs with a given power law degree distribution even though power law or close to power law degree distributions are ubiquitously observed in a great variety of large real networks the mathematically satisfactory treatment of random power law graphs satisfying basic statistical requirements of realism is still lacking these requirements are sparsity exchangeability projectivity and unbiasedness the last requirement states that entropy of the graph ensemble must be maximized under the degree distribution constraints here we prove that the hypersoft configuration model belonging to the class of random graphs with latent hyperparameters also known as inhomogeneous random graphs or w random graphs is an ensemble of random power law graphs that are sparse unbiased and either exchangeable or projective the proof of their unbiasedness relies on generalized graphons and on mapping the problem of maximization of the normalized gibbs entropy of a random graph ensemble to the graphon entropy maximization problem showing that the two entropies converge to each other in the large graph limit

global regularity of 3d nonhomogeneous incompressible magneto micropolar system with the density dependent viscosity abstract this paper is concerned with the 3d nonhomogeneous incompressible magneto micropolar equations with density dependent viscosity and vacuum in smooth bounded domains we prove that the system has a unique global smooth solution under the assumption that the initial energy is suitably small

universally consistent vertex classification for latent positions graphs in this work we show that using the eigen decomposition of the adjacency matrix we can consistently estimate feature maps for latent position graphs with positive definite link function provided that the latent positions are i i d from some distribution f we then consider the exploitation task of vertex classification where the link

regularity results for two dimensional flows of multiphase viscous fluids global regularity results for weak solutions of the navier stokes equations for two dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive lower bounds on the initial density as an application we deduce regularity properties for the integral curves of the corresponding velocity field finally we prove regularity results in the small for strong solutions

testing for nodal dependence in relational data matrices relational data are often represented as a square matrix the entries of which record the relationships between pairs of objects many statistical methods for the analysis of such data assume some degree of similarity or dependence between objects in terms of the way they relate to each other however formal tests for such dependence have not been developed we provide a test for such dependence using the framework of the matrix normal model a type of multivariate normal distribution parameterized in terms of row and column specific covariance matrices we develop a likelihood ratio test lrt for row and column dependence based on the observation of a single relational data matrix we obtain a reference distribution for the lrt statistic thereby providing an exact test for the presence of row or column correlations in a square relational data matrix additionally we provide extensions of the test to accommodate common features of such data such as undefined diagonal entries a nonzero mean multipl

global well posedness for the incompressible navier stokes equations with density dependent viscosity coefficient abstract this paper is concerned with an initial boundary value problem of the incompressible navier stokes equations with density dependent viscosity in a smooth bounded domain r 3 the global well posedness of strong solutions with large oscillations is established in both non vacuum and vacuum cases provided the initial velocity is suitably small in certain sense it is worth pointing out that there isn t any smallness condition on the density and its gradient