cappell shaneson homotopy spheres are standard we show here that an infinite sequence of homotopy 4 spheres constructed by cappell shaneson are all diffeomorphic to s 4 this generalizes previous results of akbulut kirby and gompf

chord arc surfaces with small constant ii good parameterizations abstract in a companion paper we introduced a notion for hypersurfaces in r d 1 that is analogous to the chord arc condition with small constant for planar curves in this paper we address the problem of finding good parameterizations for these surfaces in particular we show that these surfaces must be homeomorphic to r d and we build parameterizations with l p bounds on the gradient and the gradient of the inverse with p as the chord arc constant goes to 0 when d 2 we use the uniformization theorem to find a quasisymmetric parameterization we do not know how to build bilipschitz parameterizations when d 2 or quasisymmetric parameterizations when d 3 the author conjectures that they do not exist in general however we can build bilipschitz reflections in all dimensions and these reflections will satisfy certain extra estimates in terms of carleson measures

a spectral approach to lower bounds with applications to geometric searching we establish a nonlinear lower bound for halfplane range searching over a group specifically we show that summing up the weights of n weighted points within n halfplanes requires omega n log n additions and subtractions this is the first nontrivial lower bound for range searching over a group by contrast range searching over a semigroup which forbids subtractions is almost completely understood our proof has two parts first we develop a general entropy based method for relating the linear circuit complexity of a linear map a to the spectrum of a top a in the second part of the proof we design a high spectrum geometric set system for halfplane range searching and using techniques from discrepancy theory we estimate the median eigenvalue of its associated map interestingly the method also shows that using up to a linear number of help gates cannot help these are gates that can compute any bivariate function

euclidean spaces as weak tangents of infinitesimally hilbertian metric measure spaces with ricci curvature bounded below we show that in any infinitesimally hilbertian cd k n space at almost every point there exists a euclidean weak tangent i e there exists a sequence of dilations of the space that converges to a euclidean space in the pointed measured gromov hausdorff topology the proof follows by considering iterated tangents and the splitting theorem for infinitesimally hilbertian cd 0 n spaces

bifurcations in a delayed fractional complex valued neural network complex valued neural networks cvnns with integer order have attracted much attention and which have been well discussed fractional complex valued neural networks fcvnns are more suitable to describe the dynamical properties of neural networks but have rarely been studied it is the first time that the stability and bifurcation of a class of delayed fcvnn is investigated in this paper the activation function can be expressed by separating into its real and imaginary parts by using time delay as the bifurcation parameter the dynamical behaviors that including local asymptotical stability and hopf bifurcation are discussed the conditions of emergence of bifurcation are obtained furthermore it reveals that the onset of the bifurcation point can be delayed as the order increases finally an illustrative example is provided to verify the correctness of the obtained theoretical results

the covering spectrum of a compact length space we define a new spectrum for compact length spaces and riemannian manifolds called the covering spectrum which roughly measures the size of the one dimensional holes in the space more specifically the covering spectrum is a set of real numbers 0 which identify the distinct covers of the space we investigate the relationship between this covering spectrum the length spectrum the marked length spectrum and the laplace spectrum we analyze the behavior of the covering spectrum under gromov hausdorff convergence and study its gap phenomenon

analysis and geometry on manifolds with integral ricci curvature bounds ii we extend several geometrical results for manifolds with lower ricci curvature bounds to situations where one has integral lower bounds in particular we generalize colding s volume convergence results and extend the cheeger colding splitting theorem

a new notion of angle between three points in a metric space we give a new notion of angle in general metric spaces more precisely given a triple a points p x q in a metric space x d we introduce the notion of angle cone pxq as being an interval pxq pxq pxq where the quantities pxq are defined in terms of the distance functions from p and q via a duality construction of differentials and gradients holding for locally lipschitz functions on a general metric space our definition in the euclidean plane gives the standard angle between three points and in a riemannian manifold coincides with the usual angle between the geodesics if x is not in the cut locus of p or q we show that in general the angle cone is not single valued even in case the metric space is a smooth riemannian manifold if x is in the cut locus of p or q but if we endow the metric space with a positive borel measure m obtaining the metric measure space x d m then under quite general assumptions which include many fundamental examples as riemannian manifolds finite dimensional alexandrov spaces with curvature bounded from below gromov hausdorff limits of riemannian manifolds with ricci curvature bounded from below and normed spaces with strictly convex norm fixed p q x the angle cone at x is single valued for m a e x x we prove some basic properties of the angle cone such as the invariance under homotheties of the space and we analyze in detail the case x d m is a measured gromov hausdorff limit of a sequence of riemannian manifolds with ricci curvature bounded from below showing the consistency of our definition with a recent construction of honda 24

new volume comparison results and applications to degeneration of riemannian metrics abstract we consider a condition on the ricci curvature involving vector fields which is broader than the bakry emery ricci condition under this condition volume comparison laplacian comparison isoperimetric inequality and gradient bounds are proven on the manifold specializing to the bakry emery ricci curvature condition we initiate a short cut approach to work on the original manifold which yields under a weaker than usual assumption the results mentioned above for the original manifold these results are different from most well known ones in the literature where the conclusions are made on the weighted manifold instead applications on convergence and degeneration of riemannian metrics under this curvature condition are given to this effect in particular for the bakry emery ricci curvature condition the gradient of the potential function is allowed to have singularity of order close to 1 which is broader than those allowed by the traditional weighted method in general coupling with some new techniques in constructing model functions this approach enables us to extend some of the results in the papers 14 11 36 29 and 31 the condition also covers general ricci solitons instead of just gradient ricci solitons

poincare type inequalities and finding good parameterizations a very important question in geometric measure theory is how geometric features of a set translate into analytic information about it reifenberg bull am math soc 66 312 313 1960 proved that if a set is well approximated by planes at every point and at every scale then the set is a bi holder image of a plane it is known today that carleson type conditions on these approximating planes guarantee a bi lipschitz parameterization of the set in this paper we consider an n ahlfors regular rectifiable set m subset mathbb r n d that satisfies a poincare type inequality involving lipschitz functions and their tangential derivatives then we show that a carleson type condition on the oscillations of the tangent planes of m guarantees that m is contained in a bi lipschitz image of an n plane we also explore the poincare type inequality considered here and show that it is in fact equivalent to other poincare type inequalities considered on general metric measure spaces

hopf bifurcation in numerical approximation of a class delay differential equations abstract in this paper we consider the numerical solution of a delay differential equation undergoing a hopf bifurcation we prove that if the delay differential equation has a hopf bifurcation point at then the numerical solution of the equation also has a hopf bifurcation point at h o h

volume comparison and its applications in riemann finsler geometry abstract we introduce a new geometric quantity the mean covariation for finsler metrics and establish a volume comparison theorem as an application we obtain some precompactness and finiteness theorems for finsler manifolds

on eigenvalue pinching in positive ricci curvature we shall show that for manifolds with ric n 1 the radius is close to iff the n 1 st eigenvalue is close to n this extends results of cheng and croke which show that the diameter is close to iff the first eigenvalue is close to n we shall also give a new proof of an important theorem of colding to the effect that if the radius is close to then the volume is close to that of the sphere and the manifold is gromov hausdorff close to the sphere from work of cheeger and colding these conditions imply that the manifold is diffeomorphic to a sphere

discrete time analogues of integrodifferential equations modelling bidirectional neural networks we formulate discrete time analogues of integrodifferential equations modelling bidirectional neural networks studied by gopalsamy and he the discrete time analogues are considered to be numerical discretizations of the continuous time networks and we study their dynamical characteristics it is shown that the discrete time analogues preserve the equilibria of the continuous time networks by constructing a lyapunov type sequence we obtain easily verifiable sufficient conditions under which every solution of the discrete time analogue converges exponentially to the unique equilibrium the sufficient conditions are identical to those obtained by gopalsamy and he for the uniqueness and global asymptotic stability of the equilibrium of the continuous time network by constructing discrete time versions of halanay type inequalities we obtain another set of easily verifiable sufficient conditions for the global exponential stability of the unique equilibrium of the discrete time analogue the latter sufficient conditions have not been obtained in the literature of continuous time bidirectional neural networks several computer simulations are provided to illustrate the advantages of our discrete time analogue in numerically simulating the continuous time network with distributed delays over finite intervals

three circles theorems for harmonic functions we proved two three circles theorems for harmonic functions on manifolds in integral sense as one application on manifold with nonnegative ricci curvature whose tangent cone at infinity is the unique metric cone with unique conic measure we showed the existence of nonconstant harmonic functions with polynomial growth this existence result recovered and generalized the former result of ding and led to a complete answer of ni s conjecture furthermore in similar context combining the techniques of estimating the frequency of harmonic functions with polynomial growth which were developed by colding and minicozzi we confirmed their conjecture about the uniform bound of frequency

approximations of lipschitz maps via immersions and differentiable exotic sphere theorems abstract as our main theorem we prove that a lipschitz map from a compact riemannian manifold m into a riemannian manifold n admits a smooth approximation via immersions if the map has no singular points on m in the sense of f h clarke where dim m dim n as its corollary we have that if a bi lipschitz homeomorphism between compact manifolds and its inverse map have no singular points in the same sense then they are diffeomorphic we have three applications of the main theorem the first two of them are two differentiable sphere theorems for a pair of topological spheres including that of exotic ones the third one is that a compact n manifold m is a twisted sphere and there exists a bi lipschitz homeomorphism between m and the unit n sphere s n 1 which is a diffeomorphism except for a single point if m satisfies certain two conditions with respect to critical points of its distance function in the clarke sense moreover we have three corollaries from the third theorem the first one is that for any twisted sphere n of general dimension n there exists a bi lipschitz homeomorphism between n and s n 1 which is a diffeomorphism except for a single point in particular there exists such a map between an exotic n sphere n of dimension n 4 and s n 1 the second one is that if an exotic 4 sphere 4 exists then 4 does not satisfy one of the two conditions above the third one is that for any grove shiohama type n sphere n there exists a bi lipschitz homeomorphism between n and s n 1 which is a diffeomorphism except for one of points that attain their diameters

stability hopf bifurcations and spatial patterns in a delayed diffusive predator prey model with herd behavior in this paper we consider a delayed diffusive predator prey model with herd behavior firstly by choosing the appropriate bifurcation parameter the stability of the positive equilibria and the existence of hopf bifurcations induced by diffusion and delay respectively are investigated by analyzing the corresponding characteristic equation then applying the normal form theory and the center manifold argument for partial functional differential equations the formula determining the properties of the hopf bifurcation are obtained furthermore the instability of the hopf bifurcation leads to the emergence of spatial patterns finally some numerical simulations are also carried out to illustrate and expand the theoretical results

a generalized free matrix based integral inequality for stability analysis of time varying delay systems abstract this paper focuses on the delay dependent stability problem of time varying delay systems a generalized free matrix based integral inequality gfmbii is presented this inequality is able to deal with time varying delay systems without using the reciprocal convexity lemma it overcomes the drawback that the bessel legendre inequality is inconvenient to cope with a time varying delay system as the resultant bound contains a reciprocal convexity through the use of the derived inequality and by constructing a suitable lyapunov krasovskii function lkf improved stability criteria are presented in the form of linear matrix inequalities lmis two numerical examples are carried out to demonstrate that the results outperform the state of the art in the literature

convergence of calabi yau manifolds abstract in this paper we study the convergence of calabi yau manifolds under kahler degeneration to orbifold singularities and complex degeneration to canonical singularities including the conifold singularities and the collapsing of a family of calabi yau manifolds

lower bounds on the complexity of some optimal data structures a technique is presented for deriving lower bounds on the complexity of optimal data structures which permit insertions and deletions of records and queries of the form text query region quad text return quad sum begin subarray l text key r in text region end subarray text value r quad text region in gamma where value r the value associated with a record r lies in a commutative semi group s and gamma denotes a set of regions of the space of possible keys this technique is illustrated with several examples

embeddings of riemannian manifolds with heat kernels and eigenfunctions we show that any closed n dimensional riemannian manifold can be embedded by a map constructed from heat kernels at a certain time from a finite number of points both this time and this number can be bounded in terms of the dimension a lower bound on the ricci curvature the injectivity radius and the volume it follows that the manifold can be embedded by a finite number of eigenfunctions of the laplace operator again this number only depends on the geometric bounds and the dimension in addition both maps can be made arbitrarily close to an isometry in the appendix we derive quantitative estimates of the harmonic radius so that the estimates on the number of eigenfunctions or heat kernels needed can be made quantitative as well 2016 wiley periodicals inc

regularity of einstein manifolds and the codimension 4 conjecture in this paper we are concerned with the regularity of noncollapsed riemannian manifolds m n g with bounded ricci curvature as well as their gromov hausdor limit spaces m n dj dgh x d where dj denotes the riemannian distance our main result is a solution to the codimension 4 conjecture namely thatx is smooth away from a closed subset of codimension 4 we combine this result with the ideas of quantitative stratication to prove a priori l q estimates on the full curvaturejrmj for all q v 0 and diam m d contains at most a nite number of dieomorphism classes a local version is used to show that noncollapsed 4 manifolds with bounded ricci curvature have a priori l 2 riemannian curvature estimates

sobolev spaces laplacian and heat kernel on alexandrov spaces we prove the compactness of the imbedding of the sobolev space w 1 2 0 omega into l 2 omega for any relatively compact open subset omega of an alexandrov space as a corollary the generator induced from the dirichlet energy form has discrete spectrum the generator can be approximated by the laplacian induced from the dc structure on the alexandrov space we also prove the existence of the locally holder continuous heat kernel

local spectral convergence in rcd k n spaces abstract in this paper we give necessary and sufficient conditions for the validity of the local spectral convergence in balls in the setting of rcd spaces

discrete homotopies and the fundamental group abstract we generalize and strengthen the theorem of gromov that the fundamental group of any compact riemannian manifold of diameter at most d has a set of generators g 1 g k of length at most 2 d and relators of the form g i g m g j in particular we obtain an explicit bound for the number k of generators in terms of the number of short loops at every point and the number of balls required to cover a given semi locally simply connected geodesic space as a corollary we obtain a fundamental group finiteness theorem new even for riemannian manifolds that replaces the curvature and volume conditions of anderson and the 1 systole bound of shen wei by more general geometric hypothesis implied by these conditions this theorem in turn is a special case of a theorem for arbitrary compact geodesic spaces proved using the method of discrete homotopies introduced by the first author and v n berestovskii central to the proof is the notion of homotopy critical spectrum introduced in this paper as a natural consequence of discrete homotopy methods this spectrum is closely related to the sormani wei covering spectrum which is a subset of the classical length spectrum studied by de verdiere and duistermaat guillemin it is completely determined including multiplicity by special closed geodesics called essential circles

universal covers for hausdorff limits of noncompact spaces we prove that if y is the gromov hausdorff limit of a sequence of complete manifolds m n i with a uniform lower bound on ricci curvature then y has a universal cover

stability and hopf bifurcation in a delayed ratio dependent holling tanner type model in this study a delayed ratio dependent holling tanner type predator prey model is investigated first the local stability of a positive equilibrium is studied and then the existence of hopf bifurcations is established by using the normal form theory and center manifold theorem the explicit algorithm determining the stability direction of the bifurcating periodic solutions are derived finally we perform the numerical simulations for justifying the theoretical results

collapsing of riemannian manifolds and eigenvalues of laplace operator on etudie la perturbation des valeurs propres de l operateur de laplace sous des deformations des varietes qui ne conservent pas les dimensions

on hopf bifurcation and control for a delay systems abstract hopf bifurcation of a delay differential equation modeling two enterprise interaction mechanism is investigated for the demands of computer simulation applications the discrete time two enterprise interaction model is proposed by using the forward euler scheme by discussing the characteristic equation of linearized part the local stability criterion of the positive fixed point is presented time lag is selected as the bifurcation parameter then it is demonstrated that variation of time lag may result in hopf bifurcation furthermore derivations for the directions and stability of hopf bifurcation are also given for the purpose of controlling hopf bifurcation a time delayed feedback control scheme is proposed finally we simulate the main results

delay variation dependent stability of delayed discrete time systems this note is concerned with the stability analysis of linear discrete time system with a time varying delay a generalized free weighting matrix gfwm approach is proposed to estimate summation terms in the forward difference of lyapunov functional and theoretical study shows that the gfwm approach encompasses several frequently used estimation approaches as special cases moreover an augmented lyapunov functional with a delay product type term is constructed to take into account delay changing information as a result the proposed gfwm approach together with the augmented lyapunov functional leads to a less conservative delay variation dependent stability criterion finally numerical examples are given to illustrate the advantages of the proposed criterion

the space of embedded minimal surfaces of fixed genus in a 3 manifold iv locally simply connected this paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3manifold the key is to understand the structure of an embedded minimal disk in a ball in r3 this was undertaken in cm3 cm4 and the global version of it will be completed here see the discussion around figure 12 for the local case and cm15 for some more details our main results are theorem 0 1 the lamination theorem and theorem 0 2 the one sided curvature estimate the lamination theorem is stated in the global case where the lamination is in fact a foliation the first four papers of this series show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multivalued graph this is done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase like the helicoid to prove that such a disk is a double spiral staircase we showed in the first three papers of the series that it is built out of n valued graphs where n is a fixed number in this paper we will deal with how the multi valued graphs fit together and in particular prove regularity of the set of points of large curvature the axis of the double spiral staircase the first theorem is the global version of the statement that every embedded minimal disk is a double spiral staircase

normal form theory and spectral sequences abstract the concept of unique normal form is formulated in terms of a spectral sequence as an illustration of this technique some results of baider and churchill concerning the normal form of the anharmonic oscillator are reproduced the aim of this paper is to show that spectral sequences give us a natural framework in which to formulate normal form theory

quasiconformal 4 manifolds for any pseudo group of homeomorphism of euclidean space one can define the corresponding category of manifolds the most familiar examples in topology are the full pseudo group of homeomorphisms giving rise to the theory of topological manifolds and the subgroup of smooth differomorphisms giving rise to the theory of c manifolds in this paper we discuss an intermediate category quasiconformal homeomorphisms and manifolds recall that a homeomorphism 9 d r n from a domain d in r n to its image 9 d is k quasiconformal if for all x in d

geometry and analysis of dirichlet forms ii abstract given a regular strongly local dirichlet form e under assumption that the lower bound of the ricci curvature of bakry emery the local doubling and local poincare inequalities are satisfied we obtain that i the intrinsic differential and distance structures of e coincide ii the cheeger energy functional ch d e is a quadratic norm this shows that ii is necessary for the riemannian ricci curvature defined by ambrosio gigli savare to be bounded from below this together with some recent results of ambrosio gigli savare yields that the heat flow gives a gradient flow of boltzman shannon entropy under the above assumptions we also obtain an improvement on kuwada s duality theorem for dirichlet forms under the assumptions of doubling and poincare inequalities finally dirichlet forms are constructed to show that doubling and poincare inequalities are not enough to obtain either i or ii above that is the lower bound of the bakry emery curvature condition is essential

lower bounds on the complexity of polytope range searching orthogonal range searching and simplex range searching have received much attention recently in the computational geometry literature whereas the former problem is nearing a definitive solution however the complexity of simplex range searching has long remained elusive to state the problem simply suppose that we are given n points in euclidean d space fixed once and for all and m units of computer memory we wish to organize the memory to be in a position to answer the following type of queries efficiently given an arbitrary simplex q how many of the n points lie inside q a natural variant of the problem calls for reporting the points in question and not simply counting them more generally it is customary to weight the points ahead of time and then ask for the cumulative weight of the subset of points that fall within the query there is abundant practical application to motivate research on this problem 5 6 7 10 11 15 18 20 22 for example clipping and removing hidden surfaces in computer graphics are fundamental tasks whose computational bottlenecks are instances of simplex range searching also of great interest is the central theoretical question lying underneath what is the most efficient way of organizing information to support a given class of queries what takes this question apart from the classical problem of searching a linear list is the power of redundancy while oversupply of memory space is usually of marginal interest when searching a linear list it is often the key to efficiency in multidimensional searching for this reason the principal research activity in that area has been the investigation of space time trade offs our main result is a family of lower bounds on the space time complexity of simplex range searching we prove that the worst case query time is q n l hi in the euclidean plane and more generally q n log n m l d in d space for d 3 where n is the number of points and m is the amount of storage available l these bounds hold with high probability for a random point set

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

lower bounds on the bounded coefficient complexity of bilinear maps we prove lower bounds of order n log n for both the problem of multiplying polynomials of degree n and of dividing polynomials with remainder in the model of bounded coefficient arithmetic circuits over the complex numbers these lower bounds are optimal up to order of magnitude the proof uses a recent idea of r raz proc 34th stoc 2002 proposed for matrix multiplication it reduces the linear problem of multiplying a random circulant matrix with a vector to the bilinear problem of cyclic convolution we treat the arising linear problem by extending j morgenstern s bound j acm 20 pp 305 306 1973 in a unitarily invariant way this establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix in addition we extend these lower bounds for linear and bilinear maps to a model of circuits that allows a restricted number of unbounded scalar multiplications

on complete manifolds with nonnegative ricci curvature complete open riemannian manifolds mn g with nonnegative sectional curvature are well understood the basic results are toponogov s splitting theorem and the soul theorem cg1 the splitting theorem has been extended to manifolds of nonnegative ricci curvature cg2 on the other hand the soul theorem does not extend even topologically according to recent examples in gm2 a different method to construct manifolds which carry a metric with ric 0 but no metric with nonnegative sectional curvature has been given by l berard bergery bb this leads to the question cf also y1 is there any finiteness result for complete riemannian manifolds with ric 0 the answer is certainly affirmative in the low dimensional special cases n 2 where all notions of curvature coincide and n 3 where nonnegative ricci curvature has been studied by means of stable minimal surfaces msy sy on the other hand j p sha and d g yang shy have constructed complete manifolds with strictly positive ricci curvature in higher dimensions for example they can choose the underlying space to be r4 x s3 with infinitely many copies of s3 x cp 2 attached to it by surgery cf also shy 1 it is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions the purpose of this paper is to establish the following main result

exponential stability and periodic oscillatory solution in bam networks with delays both exponential stability and periodic oscillatory solution of bidirectional associative memory bam networks with axonal signal transmission delays are considered by constructing suitable lyapunov functional and some analysis techniques some simple sufficient conditions are given ensuring the global exponential stability and the existence of periodic oscillatory solutions of bam with delays these conditions are presented in terms of system parameters and have important leading significance in the design and applications of globally exponentially stable and periodic oscillatory neural circuits for bam with delays in addition two examples are given to illustrate the results

collapsing and pinching under a lower curvature bound in this paper we are concerned with collapsing phenomena and pinching problems of riemannian manifolds whose sectional curvatures are uniformly bounded from below for a positive integer n and for d 0 let x be the set of compact riemannian n manifolds m with sectional curvatures km 1 and diameters diam m d by the weak compactness theorem of gromov glp x is relatively compact in the set of compact metric spaces with respect to the hausdorff distance furthermore gromov proved in g2 that the sum of betti numbers of any element in x is uniformly bounded in terms of the given constants these results suggest that x is an object of study on which we could develop some geometry and topology in fact when volumes are uniformly bounded away from zero the following results are known let v be the subset of x with volume vol m v grove and petersen gp1 proved that the set of homotopy types of elements in 4 v is finite related results are in y2 for pinching problems otsu shiohama and the author osy y3 obtained some differential sphere theorems in the class x v according to a recent paper gpw if n 3 4 then v contains at most finitely many diffeomorphism types let mi i 1 2 be a convergent sequence in a and x be the limit of them we say that mi collapses to x if the hausdorff dimension of x is less than n by the volume condition no collapsing occurs in 4 v from this point of view it is quite natural to ask what collapsing phenomena occur and what types of pinching theorems hold in the class x in this situation it seems difficult to determine the singularities of x some examples are given in section 1 in this paper we study collapsing phenomena in the case when x is a riemannian manifold and establish a pinching theorem for a riemannian manifold m we denote by inj m the injectivity radius of m when n is written in place of x and the metric of n is normalized our

jost maps ball homogeneous and harmonic manifolds abstract given a real number e 0 small enough an associated jost map j e between two riemannian manifolds is defined then we prove that connected riemannian manifolds for which the center of mass of each small geodesic ball is the center of the ball i e for which the identity is a j e map are ball homogeneous in the analytic case we characterize such manifolds in terms of the euclidean laplacian and we show that they have constant scalar curvature under some restriction on the ricci curvature we prove that riemannian analytic manifolds for which the center of mass of each small geodesic ball is the center of the ball are locally and weakly harmonic

bifurcation and chaos in a discrete physiological control system in this paper we introduce a new method to discrete a physiological control system with delay and investigate the dynamical behavior of the discrete model basic properties of the discrete system are analyzed by means of lyapunov exponent spectrum and bifurcation diagrams analysis results show that this system has complex dynamics with some interesting characteristics which may provide some new knowledge for us in fields of physiology

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

collapsing of abelian fibered calabi yau manifolds we study the collapsing behavior of ricci flat kahler metrics on a projective calabi yau manifold which admits an abelian fibration when the volume of the fibers approaches zero we show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature and along the fibers the rescaled metrics become flat in the limit the limit metric on the base minus the critical locus is locally isometric to an open dense subset of any gromov hausdorff limit space of the ricci flat metrics we then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit in this setting we prove an analogue of a result of gross and wilson for k3 surfaces which is motivated by the strominger yau zaslow picture of mirror symmetry