cuntz krieger algebras of infinite graphs and matrices we show that the cuntz krieger algebras of infinite graphs and infinite 0 1 matrices can be approximated by those of finite graphs we then use these approximations to deduce the main uniqueness theorems for cuntz krieger algebras and to compute their k theory since the finite approximating graphs have sinks we have to calculate the k theory of cuntz krieger algebras of graphs with sinks and the direct methods we use to do this should be of independent interest

purely infinite simple c algebras that are principal groupoid c algebras abstract from a suitable groupoid g we show how to construct an amenable principal groupoid whose c algebra is a kirchberg algebra which is kk equivalent to c g using this construction we show by example that many uct kirchberg algebras can be realised as the c algebras of amenable principal groupoids

statistical analysis of multiple sociometric relations abstract loglinear models are adapted for the analysis of multivariate social networks a set of sociometric relations among a group of actors models that focus on the similarities and differences between the relations and models that concentrate on individual actors are discussed this approach allows for the partitioning of the actors into blocks or subgroups some ideas for combining these models are described and the various models and computational methods are applied to the analysis of data for a corporate interlock network of the 25 largest organizations in minneapolis st paul and for a classic network of 18 monks in a cloister the computational techniques all involve variations on the standard iterative proportional fitting algorithm used extensively in the analysis of multidimensional contingency tables

radial symmetry of solution for fractional p laplacian system abstract in this paper we investigate the method of moving planes for fractional p laplacian system we firstly discuss the key ingredients for the method of moving planes such as maximum principle for anti symmetric functions decay at infinity and boundary estimate then we apply the method of moving planes to establish the radial symmetry and the monotonicity of the positive solutions for fractional p laplacian system in a unit ball or in the whole space

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

random networks graphical models and exchangeability we study conditional independence relationships for random networks and their interplay with exchangeability we show that for finitely exchangeable network models the empirical subgraph densities are maximum likelihood estimates of their theoretical counterparts we then characterize all possible markov structures for finitely exchangeable random graphs thereby identifying a new class of markov network models corresponding to bidirected kneser graphs in particular we demonstrate that the fundamental property of dissociatedness corresponds to a markov property for exchangeable networks described by bidirected line graphs finally we study those exchangeable models that are also summarized in the sense that the probability of a network depends only on the degree distribution and we identify a class of models that is dual to the markov graphs of frank and strauss particular emphasis is placed on studying consistency properties of network models under the process of forming subnetworks and we show that the only consistent systems of markov properties correspond to the empty graph the bidirected line graph of the complete graph and the complete graph

symmetry of positive solutions for choquard equations with fractional p laplacian abstract this paper deals with the following choquard equations with fractional p laplacian p s u c n t x 2 t n u q u q 1 in r n u 0 on r n where 0 s t 1 2 p p 1 q and n 2 by constructing a decay at infinity theorem and a narrow region principle we establish the radial symmetry of positive solutions based on the generalized direct method of moving planes

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

a direct method of moving spheres on fractional order equations abstract in this paper we introduce a direct method of moving spheres for the fractional laplacian 2 with 0 2 in which a key ingredient is the narrow region maximum principle as immediate applications we classify non negative solutions for semilinear equations involving the fractional laplacian in r n we prove a non existence result for the prescribing q curvature equation on s n then by combining the direct method of moving planes and moving spheres we establish a liouville type theorem on a half euclidean space we expect to see more applications of this method to many other nonlinear equations involving non local operators

the sliding method for the nonlocal monge ampere operator abstract we develop a sliding method for the nonlocal monge ampere operator we first establish a narrow region principle in bounded domains which plays an important role in the sliding method then we consider the equation with the nonlocal monge ampere operator and derive the monotonicity of solutions in both bounded domains and the whole space we use a new idea estimating the singular integrals defining the nonlocal monge ampere operator along a sequence of approximate maximum points

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

nonexistence results for a class of fractional elliptic boundary value problems abstract in this paper we study a class of fractional elliptic problems of the form s u f x u in u 0 in r n where s 0 1 we prove nonexistence of positive solutions when is star shaped and f is supercritical we also derive a nonexistence result for subcritical f in some unbounded domains the argument relies on the method of moving spheres applied to a reformulated problem using the caffarelli silvestre extension caffarelli and silvestre 2007 11 of a solution of the above problem

integration by parts formula for regional fractional laplacian we obtain the integration by parts formula for the regional fractional laplacian which are generators of symmetric stable processes on a subset of mathbb r n 0 2 in this formula a local operator appears on the boundary connected with the regional fractional laplacian on domain hence this formula can be understood as the green formula for the regional fractional laplacian the similar integration by parts formula is also given for more general nonlocal operators the reflected stable like processes are studied

monotonicity of positive solutions for fractional p systems in unbounded lipschitz domains abstract in this paper without imposing any decay assumptions on the solutions at infinity we first develop an unbounded narrow region principle for the fractional p laplacian systems an essential ingredient to carry on the direct method of moving planes then we combine this direct method with the sliding method to obtain the monotonicity of positive bounded solutions for cooperative systems involving the fractional p laplacian in unbounded lipschitz domains we believe that the new ideas introduced here will become useful tools in studying qualitative properties of solutions for other types of nonlinear nonlocal systems in various unbounded domains

amenability of groupoids arising from partial semigroup actions and topological higher rank graphs we consider the amenability of groupoids g equipped with a group valued cocycle c g q with amenable kernel c 1 e we prove a general result which implies in particular that g is amenable whenever q is amenable and if there is countable set d g such that c g u d q for all u g 0 we show that our result is applicable to groupoids arising from partial semigroup actions we explore these actions in detail and show that these groupoids include those arising from directed graphs higher rank graphs and even topological higher rank graphs we believe our methods yield a nice alternative groupoid approach to these important constructions

maximum principles for the fractional p laplacian and symmetry of solutions abstract in this paper we consider nonlinear equations involving the fractional p laplacian p s u x c n s p p v r n u x u y p 2 u x u y x y n s p d y f x u we prove a maximum principle for anti symmetric functions and obtain other key ingredients for carrying on the method of moving planes such as a variant of the hopf lemma a boundary estimate lemma which plays the role of the narrow region principle then we establish radial symmetry and monotonicity for positive solutions to semilinear equations involving the fractional p laplacian in a unit ball and in the whole space we believe that the methods developed here can be applied to a variety of problems involving nonlinear nonlocal operators

liouville theorems involving the fractional laplacian on a half space abstract let r n be the upper half euclidean space and let be any real number between 0 and 2 consider the following dirichlet problem involving the fractional laplacian 1 2 u u p x r n u 0 x r n instead of using the conventional extension method of caffarelli and silvestre 8 we employ a new and direct approach by studying an equivalent integral equation 2 u x r n g x y u p y d y applying the method of moving planes in integral forms we prove the non existence of positive solutions in the critical and subcritical cases under no restrictions on the growth of the solutions

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

a liouville theorem for a class of fractional systems in r n abstract let 0 2 be any real number in this paper we investigate a class of fractional elliptic systems of the form 2 u x f v x 2 v x g u x x r n u v 0 x r n applying the iteration method and the direct method of moving planes for the fractional laplacian without any decay assumption on the solutions at infinity we prove the liouville theorem of nonnegative solutions under some natural conditions on f and g

the sliding methods for the fractional p laplacian abstract in this paper we develop a sliding method for the fractional p laplacian we first obtain the key ingredient needed in the sliding method in a bounded domain the narrow region principle then using nonlinear equations involving the fractional p laplacian in both bounded domains and in the whole space we illustrate how this new sliding method can be employed to obtain monotonicity of solutions during these processes we introduce a new idea estimating the singular integrals defining the fractional p laplacian along a sequence of approximate maximum points we believe that the new ideas and methods employed here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities

nonexistence of positive solutions for an indefinite fractional p laplacian equation abstract in this paper we consider the following fractional p laplacian equation with indefinite nonlinearities p s u x x 1 u q x x r n where 0 s 1 p q 1 2 we use the direct method of moving planes to prove the solution of above fractional p laplacian equation is monotone increasing along x 1 direction then obtain that the above equation possess no positive bounded solution on the whole space there is no decay condition at infinity on solution for the fractional p laplacian and the traditional approaches of translation and taking limit there no longer work for the nonlocal nonlinear fractional p laplacian here we also cannot find a suitable auxiliary function for this fractional p laplacian to circumvent these difficulties we use the following two estimate to study the property of solution i we estimate the singular integrals defining the fractional p laplacian along a sequence of approximate maxima ii we estimate the lower and upper bound of the fractional p laplacian equation

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

solutions of fully nonlinear nonlocal systems abstract in this paper we consider the system involving fully nonlinear nonlocal operators f u x c n p v r n g u x u y x y n d y f v x f v x c n p v r n g v x v y x y n d y g u x for carrying on the method of moving planes a narrow region principle and a decay at infinity are established then we prove the radial symmetry and monotonicity for positive solutions to the nonlinear system in the whole space furthermore non existence of positive solutions to the nonlinear system on a half space is derived

a direct method of moving planes for the fractional laplacian abstract in this paper we develop a direct method of moving planes for the fractional laplacian instead of using the conventional extension method introduced by caffarelli and silvestre we work directly on the non local operator using the integral defining the fractional laplacian by an elementary approach we first obtain the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space such as strong maximum principles for anti symmetric functions narrow region principles and decay at infinity then using simple examples semi linear equations involving the fractional laplacian we illustrate how this new method of moving planes can be employed to obtain symmetry and non existence of positive solutions we firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities

bayesian models for directed graphs abstract holland and leinhardt 1981 proposed a simple exponential model called the p 1 model for analyzing digraphs that arise in studies of networks a digraph consists of a set of g nodes and a g g adjacency matrix xij where xij 1 if node i relates to node j and xij 0 otherwise the underlying cell probabilities pij pr xij 1 are to be estimated from these dichotomous responses the p 1 model imposes an additive structure on a log odds version of the pij it provides information about the abilities of an individual node to attract and to produce relational ties as well as the tendency of a pair of nodes to reciprocate ties for digraphs of realistic sizes the maximum likelihood estimates mle s of the p 1 exponential parameters are often unsatisfactory particularly when some of the row and column marginal totals of the adjacency matrix are small a bayesian approach using an exchangeable normal prior on the parameters representing the attractiveness and expansiveness characteristic