ensembles of fuzzy linear model trees for the identification of multioutput systems we address the task of discrete time modeling of nonlinear dynamic systems with multiple outputs using measured data in the area of control engineering this task is typically converted into a set of classical regression problems one for each output which can then be solved with any nonlinear regression approach fuzzy models in the takagi sugeno form are popular in this context we use lolimot which is a tree learning method to build fuzzy linear model trees in this paper we propose implement and empirically evaluate three extensions of fuzzy linear model trees first we consider and evaluate multioutput models second we propose to use ensembles of such models third we investigate the use of a search heuristic based on simulation error as opposed to one step ahead prediction error specific to the context of modeling dynamic systems finally we perform an empirical evaluation and compare these approaches on six multioutput case studies using both measured and simulated data with noise the case studies include modeling of the inverse dynamics of a robot arm as well as five additional process industry systems ensembles improve the performance of both single and multioutput trees while the heuristic specific to modeling dynamic systems only improves performance very slightly multioutput model trees exhibit comparable or worse predictive performance to a set of single output models while providing a more compact model overall we can recommend the use of bagging of single output lolimot models learned by using the simulation error as a search heuristic

almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

li yau gradient bounds on compact manifolds under nearly optimal curvature conditions abstract we prove li yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the ricci flow in the former case the curvature condition is r i c l p for some p n 2 or sup m m r i c 2 y d 2 n x y d y where n is the dimension of the manifold in the later case one only needs scalar curvature being bounded we will explain why the conditions are nearly optimal and give an application the li yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing ricci curvature not bounded from below

on the heat kernel of the bergmann metric on algebraic varieties the restriction of the standard fubini study metric of ycpn l to m m is an incomplete kahler metric g called the bergmann metric let us consider the operators d and 3 defined on c1 functions and c1 1 forms on m m respectively we define the domain 9y d of d to be the set of c1 functions 2 f defined on m m such that both f and df are in l similarly we define the domain 92 6 of to be the set of c1 1 forms co such that both co and 2 6 o are in l we then define the laplacian a with respect to the bergmann metric by a d with domain 9 a given by the set of c2 functions f such that f e 9 d and df e 6 the main purpose of this paper is to show that

remarks on differential harnack inequalities abstract in this paper we give a generalization of global and local differential harnack inequalities for heat equations obtained by li and xu j f li x j xu differential harnack inequalities on riemannian manifolds i linear heat equation adv math 226 5 2011 4456 4491 and baudoin and garofalo f baudoin n garofalo perelman s entropy and doubling property on riemannian manifolds j geom anal 21 2011 1119 1131 from this we can derive new harnack inequalities and new lower bounds for the associated heat kernel also we provide some new entropy formulas with monotonicity

li yau and harnack type inequalities in rcd k n metric measure spaces abstract metric measure spaces satisfying the reduced curvature dimension condition cd k n and where the heat flow is linear are called rcd k n spaces this class of non smooth spaces contains gromov hausdorff limits of riemannian manifolds with ricci curvature bounded below by k and dimension bounded above by n we prove that in rcd k n spaces the following properties of the heat flow hold true a li yau type inequality a bakry qian inequality the harnack inequality

the swift hohenberg equation on conic manifolds abstract we consider the swift hohenberg equation on manifolds with conical singularities and show existence uniqueness and maximal regularity of the short time solution in terms of mellin sobolev spaces moreover we give a necessary and sufficient condition so that the above solution exists for all times space asymptotic expansion of the solution near the singularity is also provided and its relation to the local geometry is shown the same problem is considered on closed manifolds and similar results are obtained by using the above singular analysis theory

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

elliptic gradient estimates for a nonlinear heat equation and applications abstract in this paper we study elliptic gradient estimates for a nonlinear f heat equation which is related to the gradient ricci soliton and the weighted log sobolev constant of smooth metric measure spaces precisely we obtain hamilton s and souplet zhang s gradient estimates for positive solutions to the nonlinear f heat equation only assuming the bakry emery ricci tensor is bounded below as applications we prove parabolic liouville properties for some kind of ancient solutions to the nonlinear f heat equation some special cases are also discussed

local parabolic and elliptic gradient estimates for a generalized heat type equation under the yamabe flow abstract in this paper we investigate local parabolic li yau s type and j li s type and local elliptic hamilton s type shouplet zhang s type gradient estimates for positive solutions to a generalized heat type equation of t u h u q under the yamabe flow as applications of local parabolic gradient estimates we establish associated harnack inequalities

hamilton s gradient estimates for fast diffusion equations under the ricci flow abstract let m be a complete noncompact riemannian manifold of dimension n we derive a hamilton s gradient estimate for positive solutions of the fast diffusion equations u t u m 1 4 n 8 m 1 on m 0 under the ricci flow

on the equivalence of the entropic curvature dimension condition and bochner s inequality on metric measure spaces we prove the equivalence of the curvature dimension bounds of lott sturm villani via entropy and optimal transport and of bakry emery via energy and gamma 2 calculus in complete generality for infinitesimally hilbertian metric measure spaces in particular we establish the full bochner inequality on such metric measure spaces moreover we deduce new contraction bounds for the heat flow on riemannian manifolds and on mms in terms of the l 2 wasserstein distance

differential harnack estimates for a nonlinear heat equation abstract we consider positive solutions to the semilinear heat equation w t w a w log w a 0 on complete riemannian manifolds without boundary this equation has applications to studying ricci flow and gradient ricci solitons we derive several differential harnack inequalities which improve on those of y yang 2008 13 we use these inequalities to derive bounds on gradient ricci solitons

gradient estimates and harnack inequalities of a parabolic equation under geometric flow abstract in this paper we consider a manifold evolving by a general geometric flow and study parabolic equation q x t t u x t a u x t x t m 0 t we establish space time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this equation by integrating the gradient estimates we derive the corresponding harnack inequalities finally as applications we give gradient estimates of some specific parabolic equations

gradient estimates and liouville theorems for nonlinear parabolic equations on noncompact riemannian manifolds abstract let m be a complete noncompact riemannian manifold in this paper we derive a local gradient estimate for positive solutions of the nonlinear parabolic equation t u x t x t u x t 0 on m 0 we also obtain a theorem of liouville type for positive solutions of this nonlinear equation this paper extends the result of souplet and zhang p souplet qi s zhang sharp gradient estimate and yau s liouville theorem for the heat equation on noncompact manifolds bull lond math soc 38 2006 1045 1053

sequential decomposition of the matrix riccati equation and its application to the linear quadratic regulator problem the differential matrix riccati equation for the multi input multi output linear quadratic optimal regulator problem is considered two methods are presented successive and parallel that decompose this equation into a set of riccati equations that correspond to optimal regulator problems of possibly reduced dimensions the additivity of the solutions to the equations obtained by the successive sequential decomposition method sdm and the additivity of the inverses of the solutions to the equations obtained by the parallel sdm are established some duality relations between the successive and the parallel methods are presented via the use of the adjoint riccati equation the theory developed is extended to the algebraic matrix riccati equation as a limiting case the application of the sdms in the infinite time linear quadratic regulator problem is investigated special attention is paid to the partially cheap problem where the cost of some of the regulator controls or of their combinations tends asym

hierarchical mixtures of experts and the em algorithm we present a tree structured architecture for supervised learning the statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models glim s learning is treated as a maximum likelihood problem in particular we present an expectation maximization em algorithm for adjusting the parameters of the architecture we also develop an on line learning algorithm in which the parameters are updated incrementally comparative simulation results are presented in the robot dynamics domain

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

large time behavior of schrodinger heat kernels and applications we obtain global in time and qualitatively sharp bounds for the heat kernel g of the schrodinger operator v the potential v satisfies v x c d x b near infinity with b 0 2 when v 0 our result can be described as follows g is bounded from above and below by the multiples of standard gaussians with a weight function if b 2 then the weight is bounded between two positive it constants if b 2 the weight is bounded between two positive functions of t d x and d y which have polynomial decay if b 2 or b 0 an application to a semilinear elliptic problem is also given

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

essential spectrum and heat kernel abstract upper bounds are derived for the large time behavior of the heat kernel on riemannian manifolds whose essential spectrum has a positive lower bound applications include a relative signature theorem and the construction of bounded harmonic functions

superlinear elliptic inequalities on manifolds abstract let m be a complete non compact riemannian manifold and let be a radon measure on m we study the problem of existence or non existence of positive solutions to a semilinear elliptic inequality u u q in m where q 1 we obtain necessary and sufficient criteria for existence of positive solutions in terms of green function of in particular explicit necessary and sufficient conditions are given when m has nonnegative ricci curvature everywhere in m or more generally when green s function satisfies the 3g inequality

blow up and global existence of solutions to an inhomogeneous parabolic system abstract we study the critical phenomenon of a weakly coupled inhomogeneous parabolic system which is largely unexplored even though the homogeneous problem has been widely studied the conditions on the powers of the nonlinear terms are new compared with the well known homogeneous case the method appears to be different from the standard ones in the literature an application to the lame emden system is also given

a cameron martin type quasi invariance theorem for pinned brownian motion on a compact riemannian manifold the results in driver 13 for quasi invariance of wiener measure on the path space of a compact riemannian manifold m are extended to the case of pinned wiener measure to be more explicit let h 0 1 tom be a c1 function where m is a compact riemannian manifold o e m is a base point and tom is the tangent space to m at o e m let w m be the space of continuous paths from 0 1 into m v be wiener measure on w m concentrated on paths starting at o e m and hs w denote the stochastic parallel translation operator along a path to e w m up to time s note hs w is only well defined up to v equivalence for c e w m let xh w denote the vector field along a given by xsh c hs cw h s for each s e 0 1 one should interpret xh as a vector field on w m the vector field xh induces a flow sh t w m w m which leaves wiener measure v quasi invariant see driver 13 it is shown in this paper that the same result is valid if h l 0 and the wiener measure v is replaced by a pinned wiener measure ive the measure ve is proportional to the measure v conditioned on the set of paths which start at 0 e m and end at a fixed end point e e m also as in 13 one gets an integration by parts formula for the vector fields xh defined above

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

online adaptive decision trees pattern classification and function approximation recently we have shown that decision trees can be trained in the online adaptive oadt mode basak 2004 leading to better generalization score oadts were bottlenecked by the fact that they are able to handle only two class classification tasks with a given structure in this article we provide an architecture based on oadt exoadt which can handle multiclass classification tasks and is able to perform function approximation exoadt is structurally similar to oadt extended with a regression layer we also show that exoadt is capable not only of adapting the local decision hyperplanes in the nonterminal nodes but also has the potential of smoothly changing the structure of the tree depending on the data samples we provide the learning rules based on steepest gradient descent for the new model exoadt experimentally we demonstrate the effectiveness of exoadt in the pattern classification and function approximation tasks finally we briefly discuss the relationship of exoadt with other classification models

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

heat kernel on smooth metric measure spaces with nonnegative curvature we derive a local gaussian upper bound for the f heat kernel on complete smooth metric measure space m g e f dv with nonnegative bakry emery ricci curvature as applications we obtain a sharp l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness property for nonnegative solutions of the f heat equation assuming f is of at most quadratic growth in particular any l f 1 integrable f subharmonic function on gradient shrinking and steady ricci solitons must be constant we also provide explicit f heat kernel for gaussian solitons

on the asymptotic behaviour of optimal root loci optimal root loci have received some attention in the recent literature results concerning the state space characterization of optimal root loci however either deal with lower order behaviour only or apply to a restricted class of a b c state space models in the present paper we develop a simple algorithm which deals with the state space uhaialtcn7 ation of all orders of optimal root locus behaviour and which applies to any system a b c

a uniform bound for the solutions to a simple nonlinear equation on riemannian manifolds abstract let m be a complete noncompact manifold with ricci curvature bounded below in this note we derive a uniform bound for the solutions to the nonlinear equation u a u log u 0 on m where a is a real constant our method is based on the refined global gradient estimates for the corresponding evolution equation which is due to yau s t yau harnack inequality for non self adjoint evolution equations math res lett 2 1995 387 399 we partially generalize the result of yang y y yang gradient estimates for a nonlinear parabolic equation on riemannian manifolds proc amer math soc 136 2008 4095 4102 in the particular case of complete riemannian manifolds with nonnegative curvature we get a sharp upper bound for the positive solutions this upper bound is independent of the dimension of the manifolds

harnack estimates for a heat type equation under the ricci flow abstract in this paper we consider the gradient estimates for a positive solution of the nonlinear parabolic equation t u t u h u p on a riemannian manifold whose metrics evolve under the ricci flow two harnack inequalities and other interesting results are obtained

nonparametric direct mapping of rainfall runoff relationships an alternative approach to data analysis and modeling we present a new approach for the analysis and modeling of catchment rainfall runoff relationships that uses as predictor variables input history summary variables only the latter are defined as linear combinations of inputs at a given number of previous time steps this transforms the dynamic identification problem into a static one as the identification algorithm we use regression trees which act as a nonlinear nonparametric model the original algorithm is adapted to account for serial correlation in variables the new method is applied to two subcatchments of the u s department of agriculture forest service andrews experimental forest watershed oregon united states simple and interpretable tree models explain more than 80 of the initial deviance of the observations in both calibration and validation this suggests that the selected variables have a good predictive power and that further modeling attempts using them are warranted the models show a distinct pattern of the selected explanatory variables applications of the method include data quality control comparative analysis assessment of hydrological change and multicriterion evaluation of parametric hydrological models

geno fuzzy classification trees making the non terminal nodes of a binary tree classifier fuzzy can mitigate tree brittleness using a genetic algorithm two optimization techniques are explored in one case each generation minimizes classification error by optimizing a common fuzzy percent pt used to determine parameters at every node in the other case each generation yields a sequence of minimized node specific parameters the output value is determined through defuzzification after input vectors in general take both paths at each node with a weighting factor determined by the node membership functions experiments conducted using this geno fuzzy approach yield an improvement compared with other classical algorithms

path integral formulae for heat kernels and their derivatives the heat kernel and its derivatives of a vector laplacian on the sections of a bundle over a compact riemannian manifold are expressed as products of the scalar heat kernel of the manifold and path integrals over the brownian bridge the small time asymptotics of these integrals are computed

decision tree based fuzzy classifier of h1 magnetic resonance spectra from cerebrospinal fluid samples this paper presents a method for classifying cerebrospinal fluid csf samples studied by proton magnetic resonance spectroscopy h1 mrs into clinical subgroups by means of a fuzzy classifier the method focuses on the analysis of a low signal to noise region of the spectra and is designed to use a small number of samples because sampling can only be done through an invasive technique the proposed method involves the fusion of classifiers based on decision trees designed using fuzzy techniques the fusion step was carried out by ordered weighted averaging owa operators the quality of the proposed classifier was evaluated by efficiency and robustness quality indexes using a method based on a cross validation technique results show excellent classification levels and satisfactory robustness in both training and test sets

globally optimal fuzzy decision trees for classification and regression a fuzzy decision tree is constructed by allowing the possibility of partial membership of a point in the nodes that make up the tree structure this extension of its expressive capabilities transforms the decision tree into a powerful functional approximant that incorporates features of connectionist methods while remaining easily interpretable fuzzification is achieved by superimposing a fuzzy structure over the skeleton of a cart decision tree a training rule for fuzzy trees similar to backpropagation in neural networks is designed this rule corresponds to a global optimization algorithm that fixes the parameters of the fuzzy splits the method developed for the automatic generation of fuzzy decision trees is applied to both classification and regression problems in regression problems it is seen that the continuity constraint imposed by the function representation of the fuzzy tree leads to substantial improvements in the quality of the regression and limits the tendency to overfitting in classification fuzzification provides a means of uncovering the structure of the probability distribution for the classification errors in attribute space this allows the identification of regions for which the error rate of the tree is significantly lower than the average error rate sometimes even below the bayes misclassification rate

a harnack type inequality for non symmetric markov semigroups abstract for a strong feller and irreducible markov semigroup on a locally compact polish space the harnack type inequality 1 1 holds if and only if the semigroup has a unique invariant probability measure and is ultracontractive moreover new sufficient conditions for this inequality to hold as well as upper bound estimates of the underlying constant are presented for diffusion semigroups on riemannian manifolds

a neural network of smooth hinge functions smooth hinging hyperplane shh has been proposed as an improvement over the well known hinging hyperplane hh by the fact that it retains the useful features of hh while overcoming hh s drawback of nondifferentiability this paper introduces a formal characterization of smooth hinge function shf which can be used to generate shh as a neural network a method for the general construction of shf is also given furthermore the work proves that shh is better than hh in functional approximation i e the optimal error of shh approximating a general function is always smaller or equal to that of hh particularly in the case that the shf is generated via the integration of a class of sigmoidal functions it is further proven that the corresponding shh of the 2m shfs would outperform a neural network with m of the sigmoidal function from which the shf is derived any upper bound established on the approximation error of a neural network of m sigmoidal activation functions can hence be translated to the shh of m shfs by replacing m with m 2 the work also includes an algorithm for the identification of shh making use of its differentiability property simulation experiments are presented to validate the theoretical conclusions to possible extent

a matrix li yau hamilton estimate for kahler ricci flow in this paper we prove a new matrix li yau hamilton lyh estimate for kahler ricci flow on manifolds with nonnegative bisectional curvature the form of this new lyh estimate is obtained by the interpolation consideration originated in ch by chow this new inequality is shown to be connected with perelman s entropy formula through a family of differential equalities in the rest of the paper we show several applications of this new estimate and its corresponding estimate for linear heat equation these include a sharp heat kernel comparison theorem generalizing the earlier result of li and tian lt a manifold version of stoll s theorem st on the characterization of algebraic divisors and a localized monotonicity formula for analytic subvarieties which sharpens the bishop volume comparison theorem motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of perelman p we prove a sharp lower bound heat kernel estimate for the time dependent heat equation which is in a certain sense dual to perelman s monotonicity of the reduced volume as an application of this new monotonicity formula we show that the blow down limit of a certain type long time solution is a gradient expanding soliton we also illustrate the connection between the new lyh estimate and the hessian comparison theorem of fin on the forward reduced distance local monotonicity formulae on entropy and forward reduced volume are also derived

self interacting diffusions this paper is concerned with a general class of self interacting diffusions x t t 0 living on a compact riemannian manifold m these are solutions to stochastic differential equations of the form dx t brownian increments drift term depending on x t and t the normalized occupation measure of the process it is proved that the asymptotic behavior of t can be precisely related to the asymptotic behavior of a deterministic dynamical semi flow t t 0 defined on the space of the borel probability measures on m in particular the limit sets of t are proved to be almost surely attractor free sets for these results are applied to several examples of self attracting repelling diffusions on the n sphere for instance in the case of self attracting diffusions our results apply to prove that t can either converge toward the normalized riemannian measure or to a gaussian measure depending on the value of a parameter measuring the strength of the attraction

gradient estimates for the heat equation under the ricci flow the paper considers a manifold m evolving under the ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on m

mixed mode solution to the partially singular discrete time filtering problem by sequential decomposition a sequential decomposition method is developed for solving the stationary discrete time algebraic matrix riccati equation this decomposition is applied to the partially singular stationary filtering problem where some of the system outputs are free of measurement noise explicit expressions are obtained for the minimum error covariance matrices in the case where the number of the noise free measurements is equal to the number of independent inputs the method is generalized also to the case where the singular subsystem is non square

gradient estimates for a simple elliptic equation on complete non compact riemannian manifolds in this paper we study the local gradient estimate for the positive solution to the following equation u aulog u bu 0i nm where a 0 b are real constants m is a complete non compact riemannian manifold our result is optimal in the sense when m g is a complete non compact expanding gradient ricci soliton by definition m g is called an expanding gradient ricci soliton if for some constant c 0 it satisfies that rc cg d 2 f

sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature in this paper we consider complete noncompact riemannian manifolds m g with nonnegative ricci curvature and euclidean volume growth of dimension n ge 3 for every bounded open subset omega subset m with smooth boundary we prove that begin aligned int limits partial omega left frac mathrm h n 1 right n 1 mathrm d sigma ge mathrm avr g big mathbb s n 1 big end aligned where mathrm h is the mean curvature of partial omega and mathrm avr g is the asymptotic volume ratio of m g moreover the equality holds true if and only if m setminus omega g is isometric to a truncated cone over partial omega an optimal version of huisken s isoperimetric inequality for 3 manifolds is obtained using this result finally exploiting a natural extension of our techniques to the case of parabolic manifolds we also deduce an enhanced version of kasue s non existence result for closed minimal hypersurfaces in manifolds with nonnegative ricci curvature

recurrence and transience of branching diffusion processes on riemannian manifolds we relate the recurrence and transience of a branching diffusion process on a riemannian manifold m to some properties of a linear elliptic operator onm including spectral properties there is a trade off between the tendency of the transient brownian motion to escape and the birth process of the new particles if the latter has a high enough intensity then it may override the transience of the brownian motion leading to the recurrence of the branching process and vice versa in the case of a spherically symmetric manifold the critical intensity of the population growth can be found explicitly

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

lifschitz tail on hyperbolic space neumann conditions on considere un espace hyperbolique h de dimension n et l operateur laplacien hyperbolique h etant donne une distribution aleatoire de points a l aide d un processus ponctuel de poisson tel que chaque point se situe dans une boule hyperbolique de rayons on etudie la question suivante si est un petit nombre reel combien y a t il de valeurs propres inferieures a pour l operateur 1 2 h dans une boule b de centre o avec des conditions de neumann sur la frontiere et avec des conditions aux limites de dirichlet sur les boules aleatoires formant des intersections avec b si le rayon de b est suffisamment grand

the ricci flow on the 2 sphere the classical uniformization theorem interpreted differential geomet rically states that any riemannian metric on a 2 dimensional surface ispointwise conformal to a constant curvature metric thus one can con sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met ric the primary interest in this question is not so much to give a newproof of the uniformization theorem but rather to understand nonlinearparabolic equations better especially those arising in differential geome try a sufficiently deep understanding of parabolic equations should yieldimportant new results in riemannian geometry the question in the preceding paragraph has been studied by richardhamilton 3 and brad osgood ralph phillips and peter sarnak 6 in 3 hamilton studied the following equation which we refer to as

surfaces with boundary their uniformizations determinants of laplacians and isospectrality let be a compact surface of type g n n 0 obtained by re moving n disjoint disks from a closed surface of genus g assuming 0 while osgood phillips and sarnak o p s3 showed the properness when g 0

extremal of log sobolev inequality and w entropy on noncompact manifolds abstract let m be a complete connected noncompact manifold with bounded geometry under a condition near infinity we prove that the log sobolev functional 1 1 has an extremal function decaying exponentially near infinity we also prove that an extremal function may not exist if the condition is violated this result has the following consequences 1 it seems to give the first example of connected complete manifolds with bounded geometry where a standard log sobolev inequality does not have an extremal 2 it gives a negative answer to the open question on the existence of extremal of perelman s w entropy in the noncompact case which was stipulated by perelman 2002 22 p 9 3 2 remark 3 it helps to prove in some cases that noncompact shrinking breathers of ricci flow are gradient shrinking solitons

global boundedness in a quasilinear attraction repulsion chemotaxis model with nonlinear sensitivity abstract this paper considers the attraction repulsion chemotaxis model with homogeneous neumann boundary conditions in a smooth bounded convex domain in this model when the scaling constant is zero and the chemotactic sensitivity functions are nonlinear we prove that this system possesses a unique global classical solution that is uniformly bounded under some assumptions when the scaling constant is one and one of the chemotactic sensitivity functions is nonlinear we also obtain a unique bounded global classical solution under some assumptions

harnack estimates for nonlinear backward heat equations in geometric flows let m be a closed riemannian manifold with a family of riemannian metrics gij t evolving by a geometric flow tgij 2sij where sij t is a family of smooth symmetric two tensors we derive several differential harnack estimates for positive solutions to the nonlinear backward heat type equation f t f flog f asf where a and are constants and s gijsij is the trace of sij our abstract formulation provides a unified framework for some known results proved by various authors and moreover leads to new harnack inequalities for a variety of geometric flows

compact manifolds with a little negative curvature on etend le theoreme de bochner et le theoreme de myers pour des p formes en donnant des obstructions topologiques pour des varietes admettant des metriques avec une petite courbure negative

a mass for alf manifolds we prove a positive mass theorem on alf manifolds i e complete noncompact manifolds that are asymptotic to a circle fibration over a euclidean base with fibers of asymptotically constant length

fuzzy classification trees for data analysis overly generalized predictions are a serious problem in concept classification in particular the boundaries among classes are not always clearly defined for example there are usually uncertainties in diagnoses based on data from biochemical laboratory examinations such uncertainties make the prediction be more difficult than noise free data to avoid such problems the idea of fuzzy classification is proposed this paper presents the basic definition of fuzzy classification trees along with their construction algorithm fuzzy classification trees is a new model that integrates the fuzzy classifiers with decision trees that can work well in classifying the data with noise instead of determining a single class for any given instance fuzzy classification predicts the degree of possibility for every class some empirical results the dataset from uci repository are given for comparing fct and c4 5 generally speaking fct can obtain better results than c4 5

a comparison of neural net classifiers and linear tree classifiers their similarities and differences abstract both neural net classifiers utilizing multilayer perceptron and linear tree classifiers composed of hierarchically structured linear discriminant functions can form arbitrarily complex decision boundaries in the feature space and have similar decision making processes the structure of the linear tree classifier can be easily mapped to that of the neural nets having two hidden layers by using the hyperplanes produced by the linear tree classifier a new method for mapping the linear tree classifier to the neural nets having one hidden layer is presented with theoretical basis of mapping the convex decision regions produced by the linear tree classifier to the neurons in the neural nets this mapping may be useful for choosing appropriately sized neural nets having one or two hidden layers as well as for initializing the connection weights to speed up the learning rate and avoid the local trapping of the backpropagation algorithm also the internal operations of the hidden layer neurons in the three layer feedforward nets could be well described by this mapping experimental results on both synthetic and real data suggest that this mapping is effective and that there exists no significant difference in the classification accuracy of the neural net classifiers having one and two hidden layers substantial advantages of the neural net classifiers over the linear tree classifier were not found in the experiment while the latter is much faster in both training and classifying stages

a complete fuzzy decision tree technique in this paper a new method of fuzzy decision trees called soft decision trees sdt is presented this method combines tree growing and pruning to determine the structure of the soft decision tree with refitting and backfitting to improve its generalization capabilities the method is explained and motivated and its behavior is first analyzed empirically on 3 large databases in terms of classification error rate model complexity and cpu time a comparative study on 11 standard uci repository databases then shows that the soft decision trees produced by this method are significantly more accurate than standard decision trees moreover a global model variance study shows a much lower variance for soft decision trees than for standard trees as a direct cause of the improved accuracy

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

pattern trees induction a new machine learning method fuzzy classification is one of the most important applications in fuzzy set and fuzzy logic related research its goal is to find a set of fuzzy rules that form a classification model most of the existing fuzzy rule induction methods e g the fuzzy decision trees fdts induction method focus on searching rules consisting of triangular norms t norms i e and only but not triangular conorms t conorms or explicitly this may lead to the omission of generating important rules that involve t conorms explicitly this paper proposes a type of tree termed pattern trees pts that makes use of different aggregations including both t norms and t conorms like decision trees pts are an effective tool for classification applications this paper discusses the difference between decision trees and pts and also shows that the subsethood based method sbm and the weighted subsethood based method wsbm are two specific cases of pt induction a novel pt induction method is proposed using similarity measure and fuzzy aggregations the comparison to other classification methods including sbm wsbm c4 5 nearest neighbor support vector machine and fdt induction shows that 1 pts can obtain high accuracy rates in classifications 2 pts are robust to overfltting and 3 pts especially simple pattern trees spts maintain compact tree structures

harnack inequality heat kernel bounds and eigenvalue estimates under integral ricci curvature bounds abstract let m n g i j be a complete riemannian manifold we prove that for any p n 2 when k p 1 is small enough some parabolic type gradient bounds hold for the positive solutions of a nonlinear parabolic equation u t u a u log u on geodesic balls b o r in m with 0 r 1 we can also derive the gradient estimates for any solutions to the above nonlinear parabolic equation along the ricci flow on a closed manifold without any curvature conditions as its application we derive some parabolic type gradient estimates for a positive solution u x t of the heat equation u t u moreover our estimate is stronger than zhang and zhang s estimate see remark 2 in section 4 those results are generalizations of li yau hamilton li xu type gradient estimates under the integral ricci curvature bounds by utilizing the gradient estimates of the heat equation we obtain harnack inequalities the upper bound and the lower bound for the heat kernel eigenvalue estimate and the lower bound of green s function on riemannian manifolds under the integral ricci curvature bounds

learning pairwise image similarities for multi classification using kernel regression trees we are often faced with the problem of distinguishing between visually similar objects that share the same general appearance characteristics as opposed to object categorization this task is focused on capturing fine image differences in a pose dependent fashion our research addresses this particular family of problems and is centered around the concept of learning from example pairs formally we construct a parameterized visual similarity function optimally separating the pairs of images that depict the objects of the same class or identity from the pairs representing different object classes identities it combines various image distances that are quantified by comparing local descriptor responses at the corresponding locations in both paired images to find the best combinations we train ensembles of so called kernel regression trees which model the desired similarity function as a hierarchy of fuzzy decision stumps the obtained function is then used within a k nn like framework to address complex multi classification problems through the experiments with several image datasets we demonstrate the numerous advantages of the proposed approach high classification accuracy scalability ease of interpretation and the independence of the feature representation

dimension dependent hypercontractivity for gaussian kernels we derive sharp local and dimension dependent hypercontractive bounds on the markov kernel of a large class of diffusion semigroups unlike the dimension free ones they capture refined properties of markov kernels such as trace estimates they imply classical bounds on the ornstein uhlenbeck semigroup and a dimensional and refined transportation talagrand inequality when applied to the hamilton jacobi equation hypercontractive bounds on the ornstein uhlenbeck semigroup driven by a non diffusive levy semigroup are also investigated curvature dimension criteria are the main tool in the analysis

an iterative growing and pruning algorithm for classification tree design a critical issue in classification tree design obtaining right sized trees i e trees which neither underfit nor overfit the data is addressed instead of stopping rules to halt partitioning the approach of growing a large tree with pure terminal nodes and selectively pruning it back is used a new efficient iterative method is proposed to grow and prune classification trees this method divides the data sample into two subsets and iteratively grows a tree with one subset and prunes it with the other subset successively interchanging the roles of the two subsets the convergence and other properties of the algorithm are established theoretical and practical considerations suggest that the iterative free growing and pruning algorithm should perform better and require less computation than other widely used tree growing and pruning algorithms numerical results on a waveform recognition problem are presented to support this view

sharp differential estimates of li yau hamilton type for positive p p forms on kahler manifolds in this paper we study the heat equation of hodge laplacian deformation of p p forms on a kahler manifold after identifying the condition and establishing that the positivity of a p p form solution is preserved under such an invariant condition we prove the sharp differential harnack in the sense of liyau hamilton estimates for the positive solutions of the hodge laplacian heat equation we also prove a nonlinear version coupled with the kahler ricci flow and some interpolating matrix differential harnack type estimates for both the kahler ricci flow and the ricci flow 2011 wiley periodicals inc

integrated density of states for random metrics on manifolds we study ergodic random schrodinger operators on a covering manifold where the randomness enters both via the potential and the metric we prove measurability of the random operators almost sure constancy of their spectral properties the existence of a self averaging integrated density of states and a pastur ubin type trace formula

a methodology for automated fuzzy model generation in this paper we propose a generic methodology for the automated generation of fuzzy models the methodology is realized in three stages initially a crisp model is created and in the second stage it is transformed to a fuzzy one in the third stage all parameters entering the fuzzy model are optimized the proposed methodology is novel and generic since it can integrate alternative techniques in each of its stages a specific realization of this methodology is implemented using decision trees for the creation of the crisp model the sigmoid function the min max operators and the maximum defuzzifier for the transformation of the crisp model into a fuzzy one and four different optimization strategies including global and local optimization techniques as well as hybrid approaches the proposed methodology presents several advantages and novelties the transformation of the crisp model to the respective fuzzy one is straightforward ensuring its full automated nature and it introduces a set of parameters expressing the importance of each fuzzy rule the above realization is extensively evaluated using several benchmark data sets from the uci machine learning repository and the obtained results indicate its high efficiency

remarks on li yau inequality on graphs abstract in this paper we study li yau gradient estimates for the solutions u to the heat equation t u u on graphs under the curvature condition c d n k which is introduced by bauer et al in 5 as applications we derive harnack inequalities and heat kernel estimates on graphs also we present a type of hamilton gradient estimates

gradient estimates and harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on riemannian manifolds abstract we derive the gradient estimates and harnack inequalities for positive solutions of nonlinear parabolic and nonlinear elliptic equations t u x t h x t u x t 0 and u b u hu 0 on riemannian manifolds we also obtain a theorem of liouville type for positive solutions of the nonlinear elliptic equation

gradient estimates and harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces abstract let m n g e d v be a smooth metric measure space in this paper we derive a series of gradient estimates and harnack inequalities for positive solutions of a nonlinear parabolic partial differential equation t u p u q u a 1 in m n where p q c 2 1 m n and a is a positive constant