on a parabolic problem with nonlinear term in a half space and global behavior of solutions abstract we take up the existence and global behavior of positive continuous solutions of the following nonlinear parabolic equation u u u u t 0 in r n 0 n 2 with boundary conditions u 0 on r n 0 and u x 0 u 0 x the nonlinear term is required to satisfy some conditions related to a functional class p r n which we introduce in this paper and will be called parabolic kato class in the half space our approach is based on potential theory

existence of positive solutions for some nonlinear parabolic systems in exteriors domains we prove some existence results of positive continuous solutions to the semilinear parabolic system u u t p x t g v v v t q x t f u in an unbounded domain d with compact boundary subject to some dirichlet conditions where and are nonnegative parameters the functions f g are nonnegative continuous monotone on 0 and the potentials p q are nonnegative and satisfy some hypotheses related to the parabolic kato class j d

global solutions for a nonlinear absorption parabolic problem in exterior domains abstract we prove the existence of positive continuous solutions with a precise global behavior for the following nonlinear absorption parabolic equation u u u u t 0 in d 0 with boundary conditions u 0 on d 0 and u x 0 u 0 x where d is an exterior domain of r n n 3 the nonlinear term is required to satisfy some conditions related to the parabolic kato class introduced in l m atoug and l riahi 2006 5 and the initial data u 0 may be unbounded our approach is based on some potential theory arguments

conditional gauge and potential theory for the schrodinger operator on etend le theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers on obtient des resultats de theorie du potentiel pour l equation de schrodinger

benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities many complex networks display a mesoscopic structure with groups of nodes sharing many links with the other nodes in their group and comparatively few with nodes of different groups this feature is known as community structure and encodes precious information about the organization and the function of the nodes many algorithms have been proposed but it is not yet clear how they should be tested recently we have proposed a general class of undirected and unweighted benchmark graphs with heterogeneous distributions of node degree and community size an increasing attention has been recently devoted to develop algorithms able to consider the direction and the weight of the links which require suitable benchmark graphs for testing in this paper we extend the basic ideas behind our previous benchmark to generate directed and weighted networks with built in community structure we also consider the possibility that nodes belong to more communities a feature occurring in real systems such as social networks as a practical application we show how modularity optimization performs on our benchmark

on the expected discounted penalty function and optimal dividend strategy for a risk model with random incomes and interclaim dependent claim sizes in this paper we consider a risk model with dependence between claim sizes and interclaim arrivals in contrast with the classical risk model where the premium process is a linear function of time we consider a dependent risk model where the aggregate premium process is a compound poisson process moreover there is a constant barrier strategy in this model the integral equations for the expected discounted penalty function and the expected discounted dividend payments until ruin are obtained in particular when the individual stochastic premium amount is exponentially distributed it is proved that both the expected discounted penalty function and the expected discounted dividend payments until ruin satisfy the volterra integral equations furthermore the representations of the solutions are derived respectively in addition when the individual stochastic premium amount and claim amount are exponentially distributed we can get the explicit expressions for the laplace transform of the ruin time and the expected discounted dividend payments until ruin finally the optimal barrier is presented under the condition of maximizing the expectation of the difference between discounted dividends until ruin and the deficit at ruin

performance of modularity maximization in practical contexts although widely used in practice the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts here we present a broad characterization of its performance in such situations first we revisit and clarify the resolution limit phenomenon for modularity maximization second we show that the modularity function q exhibits extreme degeneracies it typically admits an exponential number of distinct high scoring solutions and typically lacks a clear global maximum third we derive the limiting behavior of the maximum modularity q max for one model of infinitely modular networks showing that it depends strongly both on the size of the network and on the number of modules it contains finally using three real world metabolic networks as examples we show that the degenerate solutions can fundamentally disagree on many but not all partition properties such as the composition of the largest modules and the distribution of module sizes these results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts they also explain why many heuristics are often successful at finding high scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network we conclude by discussing avenues for mitigating some of these behaviors such as combining information from many degenerate solutions or using generative models

gaussian bounds for propagators perturbed by potentials abstract we develop the perturbation theory for propagators with the objective to prove gaussian bounds let u be a strongly continuous propagator i e a family of operators describing the solutions of a non autonomous evolution equation on an l p space and assume that u is positive and satisfies gaussian upper and lower bounds let v be a time dependent potential satisfying certain miyadera conditions with respect to u we show that then the perturbed propagator enjoys gaussian upper and lower bounds as well to prepare the necessary tools we extend the perturbation theory of strongly continuous propagators and the theory of absorption propagators

asymptotic behavior of positive solutions for heat equation with nonlinear term abstract we study the nonlinear parabolic equation u u u u t 0 in r n 0 with boundary condition u x 0 u 0 x not necessarily bounded function the nonlinearity x t u is required to satisfy some conditions related to the parabolic kato class p r n while allowing existence of positive solutions of the equation and continuity of such solutions our approach is based on potential theory tools

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

on a risk model with stochastic premiums income and dependence between income and loss labbe and sendova 2009 9 consider a compound poisson risk model with stochastic premiums income in this paper we extend their model by assuming that there exists a specific dependence structure among the claim sizes interclaim times and premium sizes assume that the distributions of the premium sizes and interclaim times are controlled by the claim sizes when the individual premium sizes are exponentially distributed the laplace transforms and defective renewal equations for the gerber shiu discounted penalty functions are obtained when the individual premium sizes have rational laplace transforms we show that the laplace transforms for the discounted penalty functions can also be obtained

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

asymptotics for a delay claim risk model with diffusion dependence structures and constant force of interest abstract in the paper we consider an insurance risk model perturbed by diffusion with constant force of interest where each main claim may induce a delayed claim called a by claim if the main claims and by claims form a sequence of pairwise quasi asymptotically independent random variables with long tails and dominatedly varying tails and the main claim arrival process is an arbitrary counting process we obtain a uniformly asymptotic formula for finite time ruin probability for times in a finite interval particularly with a certain dependence structure among the inter arrival times of main claims the formula holds uniformly for all times when the claim sizes are consistently varying tailed where the result obtained also covers an asymptotic formula for the infinite time ruin probability

stability based validation of clustering solutions data clustering describes a set of frequently employed techniques in exploratory data analysis to extract natural group structure in data such groupings need to be validated to separate the signal in the data from spurious structure in this context finding an appropriate number of clusters is a particularly important model selection question we introduce a measure of cluster stability to assess the validity of a cluster model this stability measure quantifies the reproducibility of clustering solutions on a second sample and it can be interpreted as a classification risk with regard to class labels produced by a clustering algorithm the preferred number of clusters is determined by minimizing this classification risk as a function of the number of clusters convincing results are achieved on simulated as well as gene expression data sets comparisons to other methods demonstrate the competitive performance of our method and its suitability as a general validation tool for clustering solutions in real world problems

a hybrid clustering method for identifying high density clusters abstract a hybrid method for clustering multivariate observations is proposed which combines elements of the k means and the single linkage clustering techniques one purpose of the proposed method is to discover the high density clusters given a random sample of size n from some underlying population a high density cluster at level c in a population with density f is defined as a maximal connected set of points x with f x c this clustering procedure is practicable for very large numbers of observations and is shown to be consistent under certain regularity conditions in one dimension

the expected discounted penalty function under a renewal risk model with stochastic income abstract in this paper we consider a renewal risk model with stochastic premiums income we assume that the premium number process and the claim number process are a poisson process and a generalized erlang n processes respectively when the individual stochastic premium sizes are exponentially distributed the laplace transform and a defective renewal equation for the gerber shiu discounted penalty function are obtained furthermore the discounted joint distribution of the surplus just before ruin and the deficit at ruin is given when the claim size distributions belong to the rational family the explicit expression of the gerber shiu discounted penalty function is derived finally a specific example is provided

the expected discounted penalty at ruin in the risk process with random income we extend the classical risk model to the case where the premium income process based on a poisson process is no longer a linear function we examine the expected discounted value of a penalty at ruin which is considered as a function of the initial surplus we derive a defective renewal equation satisfied by the discounted penalty function the solution of this renewal equation is then given the associated compound geometric distribution is also studied

on a perturbed by diffusion compound poisson risk model with delayed claims and multi layer dividend strategy in this paper we consider an extension to the compound poisson risk process perturbed by diffusion in which two types of dependent claims main claims and by claims are incorporated every by claim is induced by the main claim and may be delayed for one time period with a certain probability an integro differential equation system for the gerber shiu expected discounted penalty functions is derived and solved by proving that the gerber shiu function satisfies some defective renewal equation an exact representation for the solution of this equation is derived through an associated compound geometric distribution and an analytic expression for this quantity is given when both the claim and the by claim amounts belong to the rational family of distributions further the same risk model is considered in the presence of a multi layer dividend strategy a system of integro differential equations for the expected discounted penalty functions depending on the current surplus level with certain initial and boundary conditions is obtained to solve this we derive a general solution to a certain second order integro differential equation system this solution is obtained by transforming this system to a volterra type system of integral equations of second kind which is solved by using laplace transforms provided an explicit expression for the gerber shiu functions depending on the current surplus level finally numerical results for the ruin probability are given to illustrate the applicability of our results

the expected discounted penalty function under a risk model with stochastic income quantities of interest in ruin theory are investigated under the general framework of the expected discounted penalty function assuming a risk model where both premiums and claims follow compound poisson processes both a defective renewal equation and an integral equation satisfied by the expected discounted penalty function are established some implications that these equations have on particular quantities such as the discounted deficit and the probability of ultimate ruin are illustrated finally the case when premiums have erlang n b distribution and the distribution of the claims is arbitrary is investigated in more depth throughout the paper specific examples where claims and premiums have particular distributions are provided

ganc greedy agglomerative normalized cut for graph clustering this paper describes a graph clustering algorithm that aims to minimize the normalized cut criterion and has a model order selection procedure the performance of the proposed algorithm is comparable to spectral approaches in terms of minimizing normalized cut however unlike spectral approaches the proposed algorithm scales to graphs with millions of nodes and edges the algorithm consists of three components that are processed sequentially a greedy agglomerative hierarchical clustering procedure model order selection and a local refinement for a graph of n nodes and o n edges the computational complexity of the algorithm is o nlog 2n a major improvement over the o n 3 complexity of spectral methods experiments are performed on real and synthetic networks to demonstrate the scalability of the proposed approach the effectiveness of the model order selection procedure and the performance of the proposed algorithm in terms of minimizing the normalized cut metric

on the ruin probability for nonhomogeneous claims and arbitrary inter claim revenues recently raducan et al 2015 obtained recursive formulas for the ruin probability of a surplus process at or before claim instants under the assumptions that the claim sizes are independent nonhomogeneous erlang distributed and independent of the inter claim times i e the times between two successive claims which are assumed to be independent identically distributed i i d following an erlang or a mixture of exponentials distribution in this paper we extend these formulas to the more general case when the inter claim times are i i d nonnegative random variables following an arbitrary distribution we also present numerical results based on the new recursions discuss some computational aspects and state a conjecture that connects the ordering of the claims arrival with the magnitude of the corresponding ruin probabilities

simultaneous classification and multidimensional scaling with external information for the exploratory analysis of a matrix of proximities or dis similarities between objects one often uses cluster analysis ca or multidimensional scaling mds solutions resulting from such analyses are sometimes interpreted using external information on the objects usually the procedures of ca mds and using external information are carried out independently and sequentially although combinations of two of the three procedures ca and mds or multidimensional scaling and using external information have been proposed in the literature the present paper offers a procedure that combines all three procedures in one analysis using a model that describes a partition of objects with cluster centroids represented in a low dimensional space which in turn is related to the information in the external variables a simulation study is carried out to demonstrate that the method works satisfactorily for data with a known underlying structure also to illustrate the method it is applied to two empirical data sets

on the expected discounted penalty function for the compound poisson risk model with delayed claims in this paper we consider an extension to the compound poisson risk model for which the occurrence of the claim may be delayed two kinds of dependent claims main claims and by claims are defined where every by claim is induced by the main claim and may be delayed with a certain probability both the expected discounted penalty functions with zero initial surplus and the laplace transforms of the expected discounted penalty functions are obtained from an integro differential equations system we prove that the expected discounted penalty function satisfies a defective renewal equation an exact representation for the solution of this equation is derived through an associated compound geometric distribution and an analytic expression for this quantity is given for when the claim amounts from both classes are exponentially distributed moreover the closed form expressions for the ruin probability and the distribution function of the surplus before ruin are obtained we prove that the ruin probability for this risk model decreases as the probability of the delay of by claims increases finally numerical results are also provided to illustrate the applicability of our main result and the impact of the delay of by claims on the expected discounted penalty functions

conditional transformation of drift formula and potential theory for 1 2 b using conditional brownian motion and the transformation of drift formula of cameron martin girsarov maruyama we give integral conditions on a vector fieldb which imply the harmonic measures and green functions for 1 2 and 1 2 b on a bounded lipschitz domaind are equivalent by equivalent we mean there exist two sided inequalities with constants depending only onb andd this enables one to conclude the potential theory for 1 2 b ond and 1 2 ond are the same

finding community structure in very large networks the discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community but most methods proposed so far are unsuitable for very large networks because of their computational cost here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms its running time on a network with n vertices and m edges is o md phantom rule 0 2em 0ex mathrm log phantom rule 0 2em 0ex n where d is the depth of the dendrogram describing the community structure many real world networks are sparse and hierarchical with m ensuremath sim n and d ensuremath sim mathrm log phantom rule 0 2em 0ex n in which case our algorithm runs in essentially linear time o n phantom rule 0 2em 0ex mathrm log 2 phantom rule 0 2em 0ex n as an example of the application of this algorithm we use it to analyze a network of items for sale on the web site of a large on line retailer items in the network being linked if they are frequently purchased by the same buyer the network has more than 400 000 vertices and 2 ifmmode times else texttimes fi 10 6 edges we show that our algorithm can extract meaningful communities from this network revealing large scale patterns present in the purchasing habits of customers

keller osserman a priori estimates and the harnack inequality for the evolution p laplace equation with singular absorption term abstract in this paper we study quasilinear equations of type u t d i v u p 2 u v x f u 0 p 2 u 0 despite the lack of comparison principle we prove a priori estimates of keller osserman type using these estimates we give a proof of the intrinsic harnack inequality

the perturbed sparre andersen model with a threshold dividend strategy in this paper we consider a sparre andersen model perturbed by diffusion with generalized erlang n distributed inter claim times and a threshold dividend strategy integro differential equations with certain boundary conditions for the moment generation function and the mth moment of the present value of all dividends until ruin are derived we also derive integro differential equations with boundary conditions for the gerber shiu functions the special case where the inter claim times are erlang 2 distributed and the claim size distribution is exponential is considered in some details

the quantizing effect of potentials on the critical number of reaction diffusion equations we give an almost complete characterization of the relation between the critical fujita number of the equation u vu up ut 0 and the potentials v behaving like a 1 d x b here a 0 and b is any real number the only type of v not covered is when v x a 1 d x b with b 2 and a is not small it is interesting to note that when v a 1 d x 2 we are at a border line case where the critical exponent depending on a can vary from 1 to for all other v except the ones the theorem does not cover the critical fujita number takes only three discrete values 1 and 1 2n in the euclidean case we also obtain global estimates of schrodinger heat kernels and a liouville theorem for a semilinear elliptic equation

on a perturbed sparre andersen risk model with multi layer dividend strategy in this paper we consider a perturbed sparre andersen risk model in which the inter claim times are generalized erlang n distributed under the multi layer dividend strategy piece wise integro differential equations for the discounted penalty functions are derived and a recursive approach is applied to express the solutions a numerical example to calculate the ruin probabilities is given to illustrate the solution procedure

a monte carlo study of thirty internal criterion measures for cluster analysis a monte carlo evaluation of thirty internal criterion measures for cluster analysis was conducted artificial data sets were constructed with clusters which exhibited the properties of internal cohesion and external isolation the data sets were analyzed by four hierarchical clustering methods the resulting values of the internal criteria were compared with two external criterion indices which determined the degree of recovery of correct cluster structure by the algorithms the results indicated that a subset of internal criterion measures could be identified which appear to be valid indices of correct cluster recovery indices from this subset could form the basis of a permutation test for the existence of cluster structure or a clustering algorithm

the global behavior of heat kernels in exterior domains abstract we establish the global boundary behavior of dirichlet heat kernels on some unbounded domains these combined with the interior estimate in the recent paper by grigoryan and saloff coste comm pure appl math 55 1 2002 93 provide a complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on some unbounded c1 1 domains d m here m is certain complete noncompact manifold with nonnegative ricci curvature for example there exist positive constants c1 c2 depending on the unbounded domain d m such that for x dist x d x t 1 1 y t 1 1 c 1 e c 2 d x y 2 t b x t g x t y 0 x t 1 1 y t 1 1 e d x y 2 c 2 t c 1 b x t for all x y d and t 0 existence of these global bounds had been an unresolved problem even in the euclidean case

an examination of procedures for determining the number of clusters in a data set a monte carlo evaluation of 30 procedures for determining the number of clusters was conducted on artificial data sets which contained either 2 3 4 or 5 distinct nonoverlapping clusters to provide a variety of clustering solutions the data sets were analyzed by four hierarchical clustering methods external criterion measures indicated excellent recovery of the true cluster structure by the methods at the correct hierarchy level thus the clustering present in the data was quite strong the simulation results for the stopping rules revealed a wide range in their ability to determine the correct number of clusters in the data several procedures worked fairly well whereas others performed rather poorly thus the latter group of rules would appear to have little validity particularly for data sets containing distinct clusters applied researchers are urged to select one or more of the better criteria however users are cautioned that the performance of some of the criteria may be data dependent

gerber shiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times in this paper we consider a compound poisson risk model perturbed by a brownian motion we construct the bivariate cumulative distribution function of the claim size and interclaim time by farlie gumbel morgenstern copula the integro differential equations and the laplace transforms for the gerber shiu functions are obtained we also show that the gerber shiu functions satisfy some defective renewal equations for exponential claims some explicit expressions are obtained and numerical examples for the ruin probabilities are also given

the effect of cluster size dimensionality and the number of clusters on recovery of true cluster structure an evaluation of four clustering methods and four external criterion measures was conducted with respect to the effect of the number of clusters dimensionality and relative cluster sizes on the recovery of true cluster structure the four methods were the single link complete link group average upgma and ward s minimum variance algorithms the results indicated that the four criterion measures were generally consistent with each other of which two highly similar pairs were identified the tirst pair consisted of the rand and corrected rand statistics and the second pair was the jaccard and the fowlkes and mallows indexes with respect to the methods recovery was found to improve as the number of clusters increased and as the number of dimensions increased the relative cluster size factor produced differential performance effects with ward s procedure providing the best recovery when the clusters were of equal size the group average method gave equivalent or better recovery when the clusters were of unequal size

weighted graph cuts without eigenvectors a multilevel approach a variety of clustering algorithms have recently been proposed to handle data that is not linearly separable spectral clustering and kernel k means are two of the main methods in this paper we discuss an equivalence between the objective functions used in these seemingly different methods in particular a general weighted kernel k means objective is mathematically equivalent to a weighted graph clustering objective we exploit this equivalence to develop a fast high quality multilevel algorithm that directly optimizes various weighted graph clustering objectives such as the popular ratio cut normalized cut and ratio association criteria this eliminates the need for any eigenvector computation for graph clustering problems which can be prohibitive for very large graphs previous multilevel graph partitioning methods such as metis have suffered from the restriction of equal sized clusters our multilevel algorithm removes this restriction by using kernel k means to optimize weighted graph cuts experimental results show that our multilevel algorithm outperforms a state of the art spectral clustering algorithm in terms of speed memory usage and quality we demonstrate that our algorithm is applicable to large scale clustering tasks such as image segmentation social network analysis and gene network analysis

the critical exponent of a reaction diffusion equation on some lie groups we shall study the global existence and blow up of the following semilinear parabolic cauchy problem 1 1 hu h0u up x i aijxju tu up 0 in rn 0 u x 0 u0 x 0 in rn where aij aij x t are bounded measurable functions and the matrix a aij is uniformly elliptic xj j 1 k are smooth vector fields in rn and x j is the formal adjoint of xj basic assumptions i we assume that xj satisfy the hormander s condition for hypoellipticity i e the rank of the lie algebra generated by x1 xp equals n given two points x y rn d x y will be the x1 xp control distance x denotes the euclidean distance d x d x 0 b x r will represent the metric ball y rn d x y 6 r properties of the distance metric balls and the fundamental solutions of h0 have been studied extensively in nsw fes ks1 2 and sa ii there are positive constants q b c and b such that 1 2 b x r br lim x d x lim d x x

heat kernel bounds and desingularizing weights abstract we study the parabolic operator t x v t x in r 1 r d d 1 with a potential v v v v 0 assumed to be from a parabolic kato class and obtain two sided gaussian bounds on the associated heat kernel the constraints on the kato norms of v and v are completely asymmetric as they should be further improvements to our heat kernel bounds are obtained in the case of time independent potentials if v has singularities of the type c x 2 with a suitably small constant c we obtain new lower and sharp upper weighted heat kernel bounds the rate of growth of the weights depends explicitly on the constant c the standard bounds and methods estimates in lp spaces without desingularizing weights fail for singular potentials

harnack inequality and continuity of solutions to quasi linear degenerate parabolic equations with coefficients from kato type classes abstract for a general class of divergence type quasi linear degenerate parabolic equations with measurable coefficients and lower order terms from nonlinear kato type classes we prove local boundedness and continuity of solutions and the intrinsic harnack inequality for positive solutions

a criterion for determining the number of groups in a data set using sum of squares clustering marriott 1971 biometrics 27 501 514 used a heuristic argument to derive the criterion g2 l w i for determining the number of groups in a data set when the clustering objective function is the withingroup determinant i w 1 an analogous argument is employed to derive a criterion for use with the within group sum of squares objective function trace w the behaviour of both marriott s criterion and the new criterion is investigated by monte carlo methods for homogeneous data based on uniform and independent variables the performance of the new criterion is close to expectation while marriott s criterion shows much more extreme behaviour for grouped data the new criterion correctly identifies the number of groups in 85 of data sets under a wide range of conditions while marriott s criterion shows a success rate of less than 40 the new criterion is illustrated on the wellknown iris data and some cautionary comments are made about its use

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

a nonlinear parabolic problem in an exterior domain abstract we consider the problem formula herep 1 n 2 0 is a finite union of disjoint open sets anduo x is a continuous nonnegative and bounded function such that u o x a x as x 0 4 for somea 0 and 0 in this paper we discuss the asymptotic behaviour of solutions to 0 1 0 4 in terms of the various values of the parametersp a n and a common pattern that emerges from our analysis is the existence of an external zone whereu x t uo x and one or several internal regions where the influence of the set as well as that of diffusion and absorption is most strongly felt we present a complete classification of the stabilization profiles in terms of these parameters

diffusion maps and coarse graining a unified framework for dimensionality reduction graph partitioning and data set parameterization we provide evidence that nonlinear dimensionality reduction clustering and data set parameterization can be solved within one and the same framework the main idea is to define a system of coordinates with an explicit metric that reflects the connectivity of a given data set and that is robust to noise our construction which is based on a markov random walk on the data offers a general scheme of simultaneously reorganizing and subsampling graphs and arbitrarily shaped data sets in high dimensions using intrinsic geometry we show that clustering in embedding spaces is equivalent to compressing operators the objective of data partitioning and clustering is to coarse grain the random walk on the data while at the same time preserving a diffusion operator for the intrinsic geometry or connectivity of the data set up to some accuracy we show that the quantization distortion in diffusion space bounds the error of compression of the operator thus giving a rigorous justification for k means clustering in diffusion space and a precise measure of the performance of general clustering algorithms

existence and asymptotic behavior of positive continuous solutions for some nonlinear parabolic systems abstract we show that positive continuous solutions exist for the nonlinear parabolic system u u t p x t g v v v t q x t f u on r n 0 n 3 with boundary conditions u x 0 x v x 0 x for nonnegative constants and provided that the functions f g are nonnegative continuous and nondecreasing on 0 also the potentials p q are nonnegative and are required to satisfy some hypotheses related to the parabolic kato class p r n

global existence and asymptotic behavior of solutions for nonlinear parabolic equations on unbounded domains we study the existence and the asymptotic behavior of positive solutions for the parabolic equation a u tu vup 0 on d 0 where a 0 d is a some unbounded domain in rn n 3 and v belongs to a new parabolic class j of singular potentials generalizing the well known parabolic kato class at infinity p introduced recently by zhang we also show that the choice of this class is essentially optimal

on the heat equation with a time dependent singular potential abstract we introduce various classes of potentials on 0 t r n and study the corresponding nonautonomous heat equation given by u t 1 2 u v t x u we prove the existence of the feynman kac propagators for the nonautonomous heat equation and the l p l q smoothing theorem for the propagators

nonautonomous kato classes of measures and feynman kac propagators the behavior of the feynman kac propagator corresponding to a time dependent measure on r n is studied we prove the boundedness of the propagator in various function spaces on r n and obtain a uniqueness theorem for an exponentially bounded distributional solution to a nonautonomous heat equation

on a parabolic equation with a singular lower order term abstract we obtain the existence of the weak green s functions of parabolic equations with lower order coefficients in the so called parabolic kato class which is being proposed as a natural generalization of the kato class in the study of elliptic equations as a consequence we are able to prove the existence of solutions of some initial boundary value problems moreover based on a lower and an upper bound of the green s function we prove a harnack inequality for the non negative weak solutions

existence of positive bounded solutions for some nonlinear elliptic systems abstract we prove some existence results of positive bounded continuous solutions to the semilinear elliptic system u p x g v v q x f u in domains d with compact boundary subject to some dirichlet conditions where and are nonnegative parameters the functions f g are nonnegative continuous monotone on 0 and the potentials p q are nonnegative and satisfy some hypotheses related to the kato class k d

on the gerber shiu discounted penalty function in a risk model with two types of delayed claims and random income abstract this paper considers a risk model with random premium income and two types of by claims which is an extension to the general delayed claims process where there is only one type of by claim and the premium income is a constant assume that each main claim induces one of the two by claims the by claim and its associated main claim occur at the same time when the main claim amount is less than a threshold variable otherwise the occurrence of the by claim will be delayed an integral equations system for the gerber shiu discounted penalty functions is presented by using auxiliary risk models given that the premium size is exponentially distributed the explicit expression for the laplace transform of the gerber shiu penalty function is derived by applying rouche theorem the gerber shiu discounted penalty functions with reciprocal of the mean of premium surplus are obtained according to lagrange interpolation theorem we prove that the gerber shiu discounted penalty function satisfies a defective renewal equation finally under the assumption of the claim sizes satisfying exponential distribution the explicit formula of the ruin probability is derived when the discounted factor equals zero and the penalty function equals one

the gerber shiu discounted penalty function in the delayed renewal risk process with random income abstract this paper considers the delayed renewal risk model with random premium income the gerber shiu discounted penalty function in the delayed renewal model is expressed in terms of the corresponding gerber shiu function in the ordinary renewal model the obtained results can be viewed as the discrete analogy of the classical sparre anderson risk model

on a compound poisson risk model with delayed claims and random incomes abstract the main focus of this paper is to analyze the gerber shiu penalty function of a compound poisson risk model with delayed claims and random incomes it is assumed that every main claim will produce a by claim which can be delayed with a certain probability we derive the integral equation satisfied by the gerber shiu penalty function given that the premium size is exponentially distributed the explicit expression for the laplace transform of the gerber shiu penalty function is derived finally when the premium sizes have rational laplace transforms we also obtain the laplace transform of the gerber shiu penalty function

estimating the number of clusters in a data set via the gap statistic we propose a method the gap statistic for estimating the number of clusters groups in a set of data the technique uses the output of any clustering algorithm e g k means or hierarchical comparing the change in within cluster dispersion with that expected under an appropriate reference null distribution some theory is developed for the proposal and a simulation study shows that the gap statistic usually outperforms other methods that have been proposed in the literature

on a gerber shiu type function and its applications in a dual semi markovian risk model in this paper we consider a dual risk process which can be used to model the surplus of a business that invests money constantly and earns gains randomly in both time and amount the occurrences of the gains and their amounts are assumed follow a semi markovian structure e g reinhard 1984 we analyze a quantity resembling the gerber shiu expected discounted penalty function gerber and shiu 1998 that incorporates random variables defined before and after the time of ruin such as the minimum surplus level before ruin and the time of the first gain after ruin general properties of the function are studied and some exact results are derived upon exponential distributional assumptions on either the inter arrival times or the gain amounts applications in a perpetual insurance and the last inter arrival time containing the time of ruin are given along with some numerical examples

the gerber shiu function and the generalized cramer lundberg model abstract in this paper we extend some results in cramer 7 by considering the expected discounted penalty function as a generalization of the infinite time ruin probability we consider his ruin theory model that allows the claim sizes to take positive as well as negative values depending on the sign of these amounts they are interpreted either as claims made by insureds or as income from deceased annuitants respectively we then demonstrate that when the events arrival process is a renewal process the gerber shiu function satisfies a defective renewal equation subsequently we consider some special cases such as when claims have exponential distribution or the arrival process is a compound poisson process and annuity related income has erlang n distribution we are then able to specify the parameter and the functions involved in the above mentioned defective renewal equation

classes of time dependent measures non homogeneous markov processes and feynman kac propagators we study the inheritance of properties of free backward propagators associated with transition probability functions by backward feynman kac propagators corresponding to functions and time dependent measures from non autonomous kato classes the inheritance of the following properties is discussed the strong continuity of backward propagators on the space l r the l r l q smoothing property of backward propagators and various generalizations of the feller property we also prove that a propagator on a banach space is strongly continuous if and only if it is separately strongly continuous and locally uniformly bounded

joint moments of the total discounted gains and losses in the renewal risk model with two sided jumps abstract this paper considers a renewal insurance risk model with two sided jumps e g labbe et al 2011 where downward and upward jumps typically represent claim amounts and random gains respectively a generalization of the gerber shiu expected discounted penalty function gerber and shiu 1998 is proposed and analyzed for sample paths leading to ruin in particular we shall incorporate the joint moments of the total discounted costs associated with claims and gains until ruin into the gerber shiu function because ruin may not occur the joint moments of the total discounted claim costs and gain costs are also studied upon ultimate survival of the process general recursive integral equations satisfied by these functions are derived and our analysis relies on the concept of moment based discounted densities introduced by cheung 2013 some explicit solutions are obtained in two examples under different cost functions when the distribution of each claim is exponential or a combination of exponentials while keeping the distributions of the gains and the inter arrival times between successive jumps arbitrary the first example looks at the joint moments of the total discounted amounts of claims and gains whereas the second focuses on the joint moments of the numbers of downward and upward jumps until ruin numerical examples including the calculations of covariances between the afore mentioned quantities are given at the end along with some interpretations