lifespan theorem for constrained surface diffusion flows we consider closed immersed hypersurfaces in mathbb r 3 and mathbb r 4 evolving by a class of constrained surface diffusion flows our result similar to earlier results for the willmore flow gives both a positive lower bound on the time for which a smooth solution exists and a small upper bound on a power of the total curvature during this time by phrasing the theorem in terms of the concentration of curvature in the initial surface our result holds for very general initial data and has applications to further development in asymptotic analysis for these flows

noncommutative ricci flow in a matrix geometry we study noncommutative ricci flow in a finite dimensional representation of a noncommutative torus it is shown that the flow exists and converges to the flat metric we also consider the evolution of entropy and a definition of scalar curvature in terms of the ricci flow

an improved stable adaptive fuzzy control method stable adaptive fuzzy control is a self tuning concept for fuzzy controllers that uses a lyapunov based learning algorithm thus guaranteeing stability of the system plant controller learning algorithm and convergence of the plant output to a given reference signal in the paper two new methods for stable adaptive fuzzy control are presented the first method is an extension of an existing concept it is shown that a major drawback of that concept the necessity for new adaptation at every change of the reference signal can be avoided by a simple modification the main focus of the paper is on the presentation of a second method which extends the applicability of stable adaptive fuzzy control to a broader class of nonlinear plants this is achieved by an improved controller structure adopted from the neural network domain performance and limitations of the proposed methods as well as some practical design aspects are discussed and illustrated with simulation results

parabolic monge ampere equations on riemannian manifolds on a compact riemannian manifold m g we consider the parabolic monge ampere equation t x t logdet g x hess x t detg x x t f x x 0 phiv 0 x here is a real parameter andf 0 m r are smooth functions we show existence of for all timestindependent of if 0 then t t converges exponentially towards a solution of the stationary problem ast in the special case 0 f 0 one has 0 and we determine the convergence rate t in thel2 norm more precisely

deforming three manifolds with positive scalar curvature in this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact three manifold is path connected the proof uses the ricci flow with surgery the conformal method and the connected sum construction of gromov and lawson the work of perelman on hamilton s ricci flow is fundamental as one of the applications we prove the path connectedness of the space of trace free asymptotically flat solutions to the vacuum einstein constraint equations on r 3

analysis of a parabolic cross diffusion population model without self diffusion abstract the global existence of non negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown the cross diffusion terms are allowed to be arbitrarily large whereas the self diffusion terms are assumed to disappear the last assumption complicates the analysis since these terms usually provide h 1 estimates of the solutions the existence proof is based on a positivity preserving backward euler galerkin approximation discrete entropy estimates and l 1 weak compactness arguments furthermore employing the entropy entropy production method we show for special stationary solutions that the transient solution converges exponentially fast to its steady state as a by product we prove that only constant steady states exist if the inter specific competition parameters disappear no matter how strong the cross diffusion constants are

lifespan theorem for simple constrained surface diffusion flows abstract we consider closed immersed hypersurfaces in r 3 and r 4 evolving by a special class of constrained surface diffusion flows this class of constrained flows includes the classical surface diffusion flow in this paper we present a lifespan theorem for these flows which gives a positive lower bound on the time for which a smooth solution exists and a small upper bound on the total curvature during this time the hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature this turns out to be a deep property of the initial manifold as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in l 2 for m 2 immersed in r 3 and additionally on the concentration in l 3 for m 3 immersed in r 4 this is stronger than a previous result on a different class of constrained surface diffusion flows as here we obtain an improved lower bound on maximal time a better estimate during this period and eliminate any assumption on the area of the evolving hypersurface

smooth solutions to a quasi linear system of diffusion equations for a certain population model abstract the system of diffusion equations proposed by kawasaki shigesada and teramoto describes a population model of two competing species with self and cross population pressures the densities of the two species are denoted by u and v in this paper the author studies the initial boundary value problem associated with the neumann boundary condition corresponds to the case where the flux is zero at the boundary many investigators have considered nonlinear diffusion systems arising from various physical and biological problems these equations however have a special structure the highest order derivatives are not coupled or at least the coefficient matrix for the highest order derivatives is positive definite this is not the case for the system and hence some of the techniques which are effective for the system are no longer applicable to our case nevertheless we can still use sobolevski s method see reference 2 to establish the local existence of solutions under the special assumption c1 c2 in we can also prove the global existence of solutions by energy estimates the unusual structure of seems to make it difficult to settle the question of asymptotic stability of solutions

a fuzzy disturbance observer and its application to control in this paper a fuzzy disturbance observer fdo is developed and its application to the control of a nonlinear system under the internal and external disturbances is presented to construct the fdo two parameter tuning methods are proposed and shown to be useful in adjusting the parameters of the fdo the first tuning method employs the disturbance observation error to guarantee that the fdo monitors the unknown disturbance the next one enlarges the concept of error and introduces augmented error to guarantee that the fdo monitors the disturbance and the control objective is achieved in addition the relationships between the suggested fdo based control and the conventional adaptive fuzzy controls reported in the previous literatures are discussed and it is shown in a rigorous manner that the disturbance observation error or the augmented error converges to a region of which size can be kept arbitrarily small finally some examples and computer simulation results are presented to illustrate the effectiveness and the applicability of the fdo

perelman s entropy functional at type i singularities of the ricci flow we study blow ups around fixed points at type i singularities of the ricci flow on closed manifolds using perelman s w functional first we give an alternative proof of the result obtained by naber 18 and enders muller topping 8 that blow up limits are non flat gradient shrinking ricci solitons our second and main result relates a limit w density at a type i singular point to the entropy of the limit gradient shrinking soliton obtained by blowing up at this point in particular we show that no entropy is lost at infinity during the blow up process

collapsing irreducible 3 manifolds with nontrivial fundamental group let m be a closed orientable irreducible non simply connected 3 manifold we prove that if m admits a sequence of riemannian metrics which volume collapses and whose sectional curvature is locally controlled then m is a graph manifold this is the last step in perelman s proof of thurston s geometrisation conjecture

uniform boundedness and convergence of solutions to cross diffusion systems abstract uniform boundedness and convergence of global solutions are proved for cross diffusion systems in population dynamics gagliardo nirenberg type inequalities are used in the estimates of solutions in order to establish w 2 1 bounds uniform in time in each step of estimates the contribution of the diffusion coefficients are emphasized and it is concluded that the uniform bound remains independent of the growth of the diffusion coefficient in the system hence convergence of solutions are established for systems with large diffusion coefficients

a modified kahler ricci flow in this note we study a kahler ricci flow modified from the classic version in the non degenerate case strong convergence at infinite time is achieved the main focus should be on degenerate case where some partial results are presented

coupling of brownian motions and perelman s l functional abstract we show that on a manifold whose riemannian metric evolves under backwards ricci flow two brownian motions can be coupled in such a way that their normalized l distance is a supermartingale as a corollary we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non decreasing function and the normalized l distance in particular it provides a new proof of a recent result of topping p topping l optimal transportation for ricci flow j reine angew math 636 2009 93 122

nonexistence of nonconstant steady state solutions in a triangular cross diffusion model abstract in this paper we study the shigesada kawasaki teramoto model for two competing species with triangular cross diffusion we determine explicit parameter ranges within which the model exclusively possesses constant steady state solutions

sampled data fuzzy control for a class of nonlinear systems with missing data and disturbances this paper considers the sampled data fuzzy control of nonlinear systems in strict feedback form with disturbances and random missing input data we propose a novel method in which a state observer and a disturbance observer are combined to construct a sampled data fuzzy output feedback controller the stochastic variables with a bernoulli distributed sequence are used to model missing input data fuzzy logic systems are applied to approximate nonlinearities without requiring prior knowledge the relation between observer gain and sampling period is established the output feedback controller designed guarantees that the nonlinear system is globally stable a simulation example of four degrees of freedom robotic arm is used to demonstrate the effectiveness and applicability of the proposed control scheme

curvature contraction of convex hypersurfaces by nonsmooth speeds the research of the second author was supported by an australian postgraduate award the research of the first third fourth and fifth authors was supported by discovery project grant dp120100097 of the australian research council

stable adaptive fuzzy control of nonlinear systems a direct adaptive fuzzy controller that does not require an accurate mathematical model of the system under control is capable of incorporating fuzzy if then control rules directly into the controllers and guarantees the global stability of the resulting closed loop system in the sense that all signals involved are uniformly bounded is developed the specific formula for the bounds is provided so that controller designers can determine the bounds based on their requirements the direct adaptive fuzzy controller is used to regulate an unstable system to the origin and to control the duffing chaotic system to track a trajectory the simulation results show that the controller worked without using any fuzzy control rules and that after fuzzy control rules were incorporated the adaptation speed became much faster it is shown explicitly how the supervisory control forces the state to remain within the constraint set and how the adaptive fuzzy controller learns to regain control

global existence analysis of cross diffusion population systems for multiple species the existence of global in time weak solutions to reaction cross diffusion systems for an arbitrary number of competing population species is proved the equations can be derived from an on lattice random walk model with general transition rates in the case of linear transition rates it extends the two species population model of shigesada kawasaki and teramoto the equations are considered in a bounded domain with homogeneous neumann boundary conditions the existence proof is based on a refined entropy method and a new approximation scheme global existence follows under a detailed balance or weak cross diffusion condition the detailed balance condition is related to the symmetry of the mobility matrix which mirrors onsager s principle in thermodynamics under detailed balance and without reaction the entropy is nonincreasing in time but counter examples show that the entropy may increase initially if detailed balance does not hold

complete gradient shrinking ricci solitons with pinched curvature we prove that any n dimensional complete gradient shrinking ricci soliton with pinched weyl curvature is a finite quotient of mathbb r n mathbb r times mathbb s n 1 or mathbb s n in particular we do not need to assume the metric to be locally conformally flat

scalar curvature estimates by parallel alternating torsion we generalize llarull s scalar curvature comparison to riemann ian manifolds admitting metric connections with parallel and alternating torsion and having a nonnegative curvature operator on 2 tm as a by product we show that the euler number and signature of such manifolds are determined by their global holonomy representation our result holds in particular for all quotients of compact lie groups of equal rank equipped with a normal homogeneous metric we also correct a mistake in the treatment of odd dimensional spaces in our earlier papers

ricci flow and the metric completion of the space of kahler metrics we consider the space of kahler metrics as a riemannian submanifold of the space of riemannian metrics and study the associated submanifold geometry in particular we show that the intrinsic and extrinsic distance functions are equivalent we also determine the metric completion of the space of kahler metrics making contact with recent generalizations of the calabi yau theorem due to dinew and guedj zeriahi as an application we obtain a new analytic stability criterion for the existence of a kahler einstein metric on a fano manifold in terms of the ricci flow and the distance function we also prove that the kahler ricci flow converges as soon as it converges in the metric sense

ricci soliton homogeneous nilmanifolds we study a notion weakening the einstein condition on a left invariant riemannian metric g on a nilpotent lie groupn we consider those metrics satisfying ric g ci d for some c in mathbb r and some derivationd of the lie algebra mathfrak n ofn where ric g denotes the ricci operator of n g this condition is equivalent to the metric g to be a ricci soliton we prove that a ricci soliton left invariant metric on n is unique up to isometry and scaling the following characterization is also given n g is a ricci soliton if and only if n g admits a metric standard solvable extension whose corresponding standard solvmanifold s tilde g is einstein this gives several families of new examples of ricci solitons by a variational approach we furthermore show that the ricci soliton homogeneous nilmanifolds n g are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set

the second twisted betti number and the convergence of collapsing riemannian manifolds let mi overset d gh longrightarrow x denote a sequence of n manifolds converging to a compact metric space x in the gromov hausdorff topology such that the sectional curvature is bounded in absolute value and dim x n we prove the following stability result if the fundamental groups of mi are torsion groups of uniformly bounded exponents and the second twisted betti numbers of mi vanish then there is a manifold m and a sequence of diffeomorphisms from m to a subsequence of mi such that the distance functions of the pullback metrics converge to a pseudo metric in c0 norm furthermore m admits a foliation with leaves diffeomorphic to flat manifolds not necessarily compact such that a vector is tangent to a leaf if and only if its norm converges to zero with respect to the pullback metrics these results lead to a few interesting applications

convergence of kahler ricci flow in this paper we prove a theorem on convergence of kahler ricci flow on a compact kahler manifold which admits a kahler ricci soliton

wave character of metrics and hyperbolic geometric flow in this paper we illustrate the wave character of the metrics and curvatures of manifolds and introduce a new understanding tool the hyperbolic geometric flow this kind of flow is new and very natural to understand certain wave phenomena in the nature as well as the geometry of manifolds it possesses many interesting properties from both mathematics and physics several applications of this method have been found

generation and propagation of interfaces for reaction diffusion equations abstract this paper is concerned with the asymptotic behavior as e 0 of the solution ue of the reaction diffusion equation in r n r u t 1 2 0 where is the derivative of a bistable potential we show that if the initial data u 0 has values in both domains of attraction of the potential then an interface will develop in a short time o e 2 ln e we also show that if the wells of the potential are of equal depth then this interface will propagate with normal velocity equal to its mean curvature our result is valid as long as the interface remains smooth if the initial interface is compact and n 2 then the interface will disappear in a finite time but not if n 1 in case the depths of the wells are not equal then in order to obtain get reasonable results we must work on the scaled time s t e slower time scale in this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells along the normal towards the domain of the deeper well this result is valid for all s 0 and does not actually depend on the regularity of the interface we also extend the above results to the homogeneous second initial boundary value problem in case the depths of the wells are equal and the initial interface is orthogonal to the boundary we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally

entropic structure and duality for multiple species cross diffusion systems this paper deals with the existence of global weak solutions for a wide class of multiple species cross diffusions systems the existence is based on two different ingredients an entropy estimate giving some gradient control and a duality estimate that gives naturally l 2 control the proof relies on a semi implicit scheme tailored for cross diffusion systems firstly defined by the two authors and collaborators these results are applied to models having an entropy relying on the detailed balance condition recently exhibited by chen et al

plurisubharmonic functions and the structure of complete kahler manifolds with nonnegative curvature in this paper we study global properties of continuous plurisubharmonic functions on complete noncompact kahler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds we prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow deformation optimal liouville type theorem for the plurisubharmonic functions as well as a splitting theorem in terms of harmonic functions and holomorphic functions are established the results are then applied to prove several structure theorems on complete noncompact kahler manifolds with nonnegative bisectional or sectional curvature

adaptive control of a class of nonlinear systems with fuzzy logic an adaptive tracking control architecture is proposed for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture employs fuzzy systems which are expressed as a series expansion of basis functions to adaptively compensate for the plant nonlinearities global asymptotic stability of the algorithm is established in the lyapunov sense with tracking errors converging to a neighborhood of zero simulation results for an unstable nonlinear plant are included to demonstrate that incorporating the linguistic fuzzy information from human experts results in superior tracking performance

conformal deformations of integral pinched 3 manifolds abstract in this paper we prove that under an explicit integral pinching assumption between the l 2 norm of the ricci curvature and the l 2 norm of the scalar curvature a closed 3 manifold with positive scalar curvature admits a conformal metric of positive ricci curvature in particular using a result of hamilton this implies that the manifold is diffeomorphic to a quotient of s 3 the proof of the main result of the paper is based on ideas developed in an article by m gursky and j viaclovsky

flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space abstract this paper concerns closed h convex hypersurfaces of dimension n 2 in the hyperbolic space h n 1 of constant sectional curvature evolving in direction of its normal vector where the speed equals a power 1 of a curvature function f which is monotone symmetric homogeneous of degree 1 it is shown that if the initial h convex hypersurface is pinched then this is maintained under the flow and the hypersurfaces shrink to a round point in h n 1 in finite time as a consequence when rescaling appropriately the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of r n 1

boundedness in a quasilinear parabolic parabolic keller segel system with subcritical sensitivity abstract we consider the quasilinear parabolic parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a convex smooth bounded domain r n with n 1 it is proved that if s u d u c u with 2 n and some constant c 0 for all u 1 then the classical solutions to the above system are uniformly in time bounded provided that d u satisfies some technical conditions such as algebraic upper and lower growth resp decay estimates as u this boundedness result is optimal according to a recent result by the second author winkler 2010 27 which says that if s u d u c u for u 1 with c 0 and some 2 n n 2 then for each mass m 0 there exist blow up solutions with mass u 0 m in addition this paper also proves a general boundedness result for quasilinear non uniformly parabolic equations by modifying the iterative technique of moser alikakos alikakos 1979 1

stable adaptive control using fuzzy systems and neural networks stable direct and indirect adaptive controllers are presented which use takagi sugeno fuzzy systems conventional fuzzy systems or a class of neural networks to provide asymptotic tracking of a reference signal for a class of continuous time nonlinear plants with poorly understood dynamics the indirect adaptive scheme allows for the inclusion of a priori knowledge about the plant dynamics in terms of exact mathematical equations or linguistics while the direct adaptive scheme allows for the incorporation of such a priori knowledge in specifying the controller we prove that with or without such knowledge both adaptive schemes can learn how to control the plant provide for bounded internal signals and achieve asymptotically stable tracking of a reference input in addition for the direct adaptive scheme a technique is presented in which linguistic knowledge of the inverse dynamics of the plant may be used to accelerate adaptation the performance of the indirect and direct adaptive schemes is demonstrated through the longitudinal control of an automobile within an automated lane

stable multi input multi output adaptive fuzzy neural control in this letter stable direct and indirect adaptive controllers are presented that use takagi sugeno t s fuzzy systems 1985 conventional fuzzy systems or a class of neural networks to provide asymptotic tracking of a reference signal vector for a class of continuous time multi input multi output mimo square nonlinear plants with poorly understood dynamics the direct adaptive scheme allows for the inclusion of a priori knowledge about the control input in terms of exact mathematical equations or linguistics while the indirect adaptive controller permits the explicit use of equations to represent portions of the plant dynamics we prove that with or without such knowledge the adaptive schemes can learn how to control the plant provide for bounded internal signals and achieve asymptotically stable tracking of the reference inputs we do not impose any initialization conditions on the controllers and guarantee convergence of the tracking error to zero

ricci flow on quasi projective manifolds we consider the kahler ricci flow on complete finite volume metrics that live on the complement of a divisor in a compact kahler manifold x assuming certain spatial asymptotics on the initial metric we compute the singularity time in terms of cohomological data on x we also give a sufficient condition for the singularity if there is one to be type ii

positively curved manifolds with maximal discrete symmetry rank let m be a closed simply connected n manifold of positive sectional curvature we determine its homeomorphism or homotopic type if m also admits an isometric elementary p group action of large rank our main results are there exists a constant p n 0 such that 1 if m 2 n admits an effective isometric inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i action for a prime p p n then k n and implies that m 2 n is homeomorphic to a sphere or a complex projective space 2 if m 2 n 1 admits an isometric s 1 inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i action for a prime p p n then k n and implies that m is homeomorphic to a sphere 3 for m in 1 or 2 if n 7 and k 3 n 4 2 then m is homeomorphic to a sphere or homotopic to a complex projective space

on global existence of solutions to a cross diffusion system abstract we consider a strongly coupled nonlinear parabolic system which arises in population dynamics in n dimensional domains n 1 we prove the global existence of classical solutions to the system for n 10

takagi sugeno fuzzy model based indirect adaptive fuzzy observer and controller design this paper proposes the design scheme of the alternative adaptive observer and controller based on the takagi sugeno t s fuzzy model the t s fuzzy modeling and the state feedback control technique are adopted for the simple structure the proposed method maintains consistent performance in the presence of parameter uncertainties and incorporates linguistic fuzzy information from human operators in addition with the simple adaptive state feedback controller it solves the singularity problem which occurs in the inverse dynamics based on the feedback linearization method using lyapunov theory and lipschitz condition the stability analysis is conducted and the adaptive law is derived the proposed method is applied to the stabilization problem of a flexible joint manipulator in order to guarantee its performance

the kahler ricci flow through singularities we prove the existence and uniqueness of the weak kahler ricci flow on projective varieties with log terminal singularities we also show that the weak kahler ricci flow can be uniquely continued through divisorial contractions and flips if they exist finally we propose an analytic version of the minimal model program with ricci flow

inverse mean curvature flow with forced term abstract in this paper we study the evolution of convex hypersurfaces by inverse mean curvature minus an external force field c we prove that the flow will preserve the convexity for any c if c 1 h on the initial surface we prove that the flow will expand the hypersurface for all time and after rescaling the hypersurface will converge to a sphere if c 1 h on the initial surface we show that the maximum existence time of the flow is finite and the hypersurface will contract to a point when approaching the final time

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations abstract we prove interior and boundary holder continuity for bounded weak soluttions of a class of quasilinear parabolic equations whose prototype is u t div u m 1 du p 2 du

real analyticity of solutions of hamilton s equation il existe des constantes positives cn et sn dependant seulement de la dimension n de la variete m telles que pour une metrique initiale g 0 avec r g 0 1 l equation de hamilton a une solution jusqu a t sn et l estimation suivante est vraie pour la metrique solution en t o t sn k 0 t k k 1 12 k r 2 c n

the hele shaw problem and area preserving curve shortening motions we prove existence locally in time of a solution of the following hele shaw problem given a simply connected curve contained in a smooth bounded domain find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature we show that this motion is a curve shortening motion which does not change the area of the region enclosed by the curve in case is the whole plane 2 we also show that if the initial curve is close to an equilibrium curve i e to a circle then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity

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

the calabi flow on kahler surfaces with bounded sobolev constant i we consider the formation of singularities along the calabi flow by assuming the uniformly bounded sobolev constants on kahler surfaces we prove that if curvature tensor is not uniformly bounded then one can form a singular model called deepest bubble such deepest bubble has to be a scalar flat ale kahler metric in certain kahler classes on toric fano surfaces the sobolev constants are a priori bounded along the calabi flow with small calabi energy we can also show in certain cases no deepest bubble can form along the flow it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently to illustrate our results more clearly we focus on an example on mathbb cp 2 blown up three points at generic position our result also implies existence of constant scalar curvature metrics on mathbb cp 2 blown up three points at generic position in the kahler classes where the exceptional divisors have the same area

approximation theory of fuzzy systems spl minus siso case in this paper the approximation properties of mimo fuzzy systems generated by the product inference are discussed we first give an analysis of fuzzy basic functions fbf s and present several properties of fbf s based on these properties of fbf s we obtain several basic approximation properties of fuzzy systems 1 basic approximation property which reveals the basic approximation mechanism of fuzzy systems 2 uniform approximation bounds which give the uniform approximation bounds between the desired control or decision functions and fuzzy systems 3 uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions and 4 universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect the similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

on the convergence of a modified kahler ricci flow we study the convergence of a modified kahler ricci flow defined by zhou zhang we show that the modified kahler ricci flow converges to a singular metric when the limit class is degenerate this proves a conjecture of zhang

on the positive mass theorem for manifolds with corners we study the positive mass theorem for certain non smooth metrics following p miao s work our approach is to smooth the metric using the ricci flow as well as improving some previous results on the behaviour of the adm mass under the ricci flow we extend the analysis of the zero mass case to higher dimensions

the uniformization theorem for compact kahler manifolds of nonnegative holomorphic bisectional curvature soit x h une variete de kahler compacte a n dimensions de courbure bisectionnelle holomorphe non negative et soit x h son espace de recouvrement universel alors il existe des entiers non negatifs k n 1 n l et des espaces symetriques hermitiens compacts ineductibles m 1 m k de rang 2 tels que x h est isometriquement bi holomorphe a c k g 0 p nu 1 1 p nu l l m 1 g 1 m p g p ou g 0 est la metrique euclidienne sur c k g 1 g p sont des metriques canoniques sur m 1 m p et i 1 i l est une metrique de kahler sur p nu i portant une courbure bisectionnelle holomorphe non negative

an initial boundary value problem for a certain density dependent diffusion system on considere le systeme parabolique quasilineaire u t x c 1 d 1 v v e 1 a 1 v b 1 u u v t x c 2 d 2 u v e 2 a 2 u b 2 o v dans o t u x s 0 v x s sur o t u x o w 1 x v x o w 2 x sur ou r n est un domaine borne a frontiere lisse