a comparison estimate of toponogov type for ricci curvature it seems a natural question to ask to what extent the results and tools for sectional curvature remain valid for ricci curvature there is rapid progress in both positive and negative directions toponogov comparison theorem has been the most powerful tool in the study of sectional curvature underlying the proof of the soul theorem the diameter sphere theorem the uniform estimate of betti numbers and the finiteness theorems toponogov comparison theorem is also the characterizing property of lower or upper sectional curvature bounds which led to generalizations of the concept of sectional curvature bounds to non smooth space thus it could not possibly hold for ricci curvature which makes problems a lot harder for ricci curvature from a geometric point of view it also makes the ricci curvature very different from the sectional curvature as one gradually comes to realize it is interesting then that we find a comparison estimate of toponogov type for ricci curvature a hinge in a complete riemannian manifold consists of two geodesic segments vi y2 such that yi ii 2 0 9 we denote it by 1 2 where z l o we will be using the following modified version of the conjugate radius function pc p

cohomogeneity one manifolds with positive ricci curvature one of the central problems in riemannian geometry is to determine how large the classes of manifolds with positive nonnegative sectional ricci or scalar curvature are see gr for scalar curvature the situation is fairly well understood by comparison special surgery constructions as in sy wr and bundle constructions as in na have resulted in a large number of interesting manifolds with positive ricci curvature so far the only known obstructions to have positive ricci curvature come from obstructions to have positive scalar curvature see li and rs and from the classical bonnet myers theorem which implies that a closed manifold with positive ricci curvature must have finite fundamental group it is well known that among homogeneous manifolds g h this is also a sufficient condition see e g the proof of corollary 3 5 or br in this paper we prove that this is true as well when the manifold admits an action by a compact lie group g with orbits of codimension one

set partition by globally optimized cluster seed points abstract a new method is proposed for solving set partition clustering problems the suggested approach is based on the application of global optimization methodology for selecting best cluster seed points the applicability of the method is illustrated by numerical examples

invariant metrics of positive ricci curvature on principal bundles let y be a compact connected riemannian manifold with a metric of positive ricci curvature let pi p rightarrow y be a principal bundle over y with compact connected structure group g if the fundamental group of p is finite we show that p admits a g invariant metric with positive ricci curvature so that pi is a riemannian submersion

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

generalized liouville property for schrodinger operator on riemannian manifolds in this paper we prove that the dimension of the space of positive bounded respectively mathcal l harmonic functions on a complete riemannian manifold with mathcal l regular ends is equal to the number of ends mathcal l nonparabolic ends respectively this result is a solution of an open problem of grigor yan related to the liouville property for the schrodinger operator mathcal l we also prove that if a given complete riemannian manifold is roughly isometric to a complete riemannian manifold satisfying the volume doubling condition the poincare inequality and the finite covering condition on each end then the dimension of the space of positive bounded respectively solutions for the schrodinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends mathcal l nonparabolic ends respectively this is a partial answer of the question suggested by li related to the regularity of ends of a complete riemannian manifold especially our results directly generalize various earlier results of yau of li and tam of grigor yan and of present authors but with different techniques that the peculiarity of the schrodinger operator demands

a method for globally minimizing concave functions over convex sets a method is described for globally minimizing concave functions over convex sets whose defining constraints may be nonlinear the algorithm generates linear programs whose solutions minimize the convex envelope of the original function over successively tighter polytopes enclosing the feasible region the algorithm does not involve cuts of the feasible region requires only simplex pivot operations and univariate search computations to be performed allows the objective function to be lower semicontinuous and nonseparable and is guaranteed to converge to the global solution computational aspects of the algorithm are discussed

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

subdivision direction selection in interval methods for global optimization the role of the interval subdivision selection rule is investigated in branch and bound algorithms for global optimization the class of rules that allows convergence for the model algorithm is characterized and it is shown that the four rules investigated satisfy the conditions of convergence a numerical study with a wide spectrum of test problems indicates that there are substantial differences between the rules in terms of the required cpu time the number of function and derivative evaluations and space complexity and two rules can provide substantial improvements in efficiency

numerical infinitesimals in a variable metric method for convex nonsmooth optimization abstract the objective of the paper is to evaluate the impact of the infinity computing paradigm on practical solution of nonsmooth unconstrained optimization problems where the objective function is assumed to be convex and not necessarily differentiable for such family of problems the occurrence of discontinuities in the derivatives may result in failures of the algorithms suited for smooth problems we focus on a family of nonsmooth optimization methods based on a variable metric approach and we use the infinity computing techniques for numerically dealing with some quantities which can assume values arbitrarily small or large as a consequence of nonsmoothness in particular we consider the case treated in the literature where the metric is defined via a diagonal matrix with positive entries we provide the computational results of our implementation on a set of benchmark test problems from scientific literature

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

oscillation criteria for second order nonlinear neutral delay dynamic equations in this paper we will establish some oscillation criteria for the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t here 0 is a quotient of odd positive integers with r t and p t real valued positive functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales so this paper initiates the study

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below ii we prove as our main theorem the finiteness of topological type of a complete open riemannian manifold m with a base point p m whose radial curvature at p is bounded from below by that of a non compact model surface of revolution m which admits a finite total curvature and has no pair of cut points in a sector here a sector is by definition a domain cut off by two meridians emanating from the base point p m notice that our model m does not always satisfy the diameter growth condition introduced by abresch and gromoll in order to prove the main theorem we need a new type of the toponogov comparison theorem as an application of the main theorem we present a partial answer to milnor s open conjecture on the fundamental group of complete open manifolds

on complete manifolds of nonnegative kth ricci curvature in this paper we establish some vanishing and finiteness theorems for the topological type of complete open riemannian manifolds under certain positivity conditions for curvature key tools are comparison techniques and morse theory of busemann and distance functions

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

inclusion functions and global optimization 11 inclusion functions combined with special subdivision strategies are an effective means of solving the global unconstrained optimization problem although these techniques were determined and numerically tested about ten years ago they are nearly unknown and scarcely used in order to make the role of inclusion functions and subdivision strategies more widespread and transparent we will discuss a related simplified basic algorithm it computes approximations of the global minimum and at the same time bounds the absolute approximation error we will show that the algorithm works and converges under more general assumptions than it has been known hitherto that is only appropriate inclusion functions are expected to exist the number of minimal points finite or infinite is not of importance lipschitz conditions or continuity are not assumed as shown in the appendix the required inclusion functions can be constructed and programmed without difficulty in a natural way using interval analysis

stochastic global optimization methods part 1 clustering methods in this stochastic approach to global optimization clustering techniques are applied to identify local minima of a real valued objective function that are potentially global three different methods of this type are described their accuracy and efficiency are analyzed in detail

oscillation criteria for second order quasilinear neutral delay differential equations on time scales in this paper we consider the second order quasilinear neutral dynamic equation r t z d t a 1z d t d q t x d t b 1x d t 0 on a time scale t where z t x t p t x t t a b 0 are constants and obtain oscillation criteria for the equation when b a email protected and b a respectively our results extend some known results when p t 0 in the literature and contain the continuous and discrete cases as special cases

a note on the use of a fuzzy approach in adaptive partitioning algorithms for global optimization in global optimization adaptive partitioning algorithms apa operate on the basis of partitioning the feasible region into subregions sampling and evaluating each subregion and selecting one or more subregions for repartitioning the purpose of the repartitioning process is to locate a narrow neighborhood around the global optimum in this correspondence we propose to use a fuzzy approach in the assessment of subregions using random samples taken from these subregions we discuss different types of uncertainties involved in apa and we conclude that the use of a fuzzy approach in the assessment of subregions is in concurrence with apa s convergence property we provide numerical results for the fuzzy approach on 13 test functions from the literature

optimal search for the global maximum of functions with bounded seminorm this paper deals with methods seeking the global maximum of a real valued function defined over a real interval a b with prescribed seminorm i e the integral of the square of the derivative an optimal one step search algorithm is devised which provides the minimum guaranteed value of the lebesgue measure of the set of possible values of the global maximum after evaluating a given information operator at the next point at each step of the algorithm the value of the function at a new point x n and the se minorm taken over the interval a x n are evaluated in order to obtain the optimal algorithm the interpolatory envelope of a certain class of functions is constructed utilizing the minimum seminorm properties of the first order spline functions a simple search method which is near optimal is also described

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

global minimization of a linearly constrained concave function by partition of feasible domain the problem of minimizing a concave function subject to linear inequality constraints may have many local solutions therefore finding the global constrained minimum is a computationally difficult problem a computational method is described which finds the global minimum of a smooth concave function over a polyhedron in rn the feasible domain is partitioned into a rectangular domain which can be excluded from further consideration and r 2n subdomains at least one of which contains the global minimum a known algorithm can be applied sequentially or in parallel to each of these r subdomains to compute the global minimum a method is also presented appendix b for the construction of nontrivial test problems for which the global minimum point is known given an arbitrary polyhedron and a selected vertex it is shown how to determine a concave quadratic function generally with many local minima with its global minimum at the selected vertex

oscillation criteria of second order half linear dynamic equations on time scales in this paper by using the riccati transformation technique chain rule and inequalitya l lab l 1 l 1 b l 0 l 1 where a and b are positive constants we will establish some oscillation criteria for the second order half linear dynamic equation p t x d t c d q t x c t 0 t a b on time scales where c 1 is an odd positive integer our results not only unify the oscillation of half linear differential and half linear difference equations but can be applied on different types of time scales and improve some well known results in the difference equation case some examples are considered here to illustrate our main results

stochastic global optimization methods part 11 multi level methods in part ii of our paper two stochastic methods for global optimization are described that with probability 1 find all relevant local minima of the objective function with the smallest possible number of local searches the computational performance of these methods is examined both analytically and empirically

local tuning and partition strategies for diagonal go methods in this paper global optimization go lipschitz problems are considered where the multi dimensional multiextremal objective function is determined over a hyperinterval an efficient one dimensional go method using local tuning on the behavior of the objective function is generalized to the multi dimensional case by the diagonal approach using two partition strategies global convergence conditions are established for the obtained diagonal geometric methods results of a wide numerical comparison show a strong acceleration reached by the new methods working with estimates of the local lipschitz constants over different subregions of the search domain in comparison with the traditional approach

triopt a triangulation based partitioning algorithm for global optimization we propose a triangulation based partitioning algorithm triopt for solving low dimensional bound constrained black box global optimization problems the method starts by forming a delaunay triangulation of a given set of samples in the feasible domain and then it assesses the simplices partitions obtained for re partitioning function values at the vertices of each partition are mapped into the zero one interval by a nonlinear transformation function and their aggregate entropy is calculated based on this entropy partitions that hold a promise of containing the global optimum are re partitioned according to different triangular splitting strategies forming new partitions these strategies are efficient in terms of the number of new function evaluations required per new partition a novelty in the search scheme proposed here is that once a partition narrows down to a small size its vertices are eliminated from the available sample set this changes global information on the best solution and triggers a re calculation of transformed values hence revised entropies change the direction of the search to new areas the latter scheme leads to a dynamic parallel search policy which is based on an entropy cut the tree adopts flexible breadth depending on the status of the search in the experimental results it is demonstrated that triopts performance is compatible and often better than that of a well known response surface methodology and two other efficient black box partitioning approaches proposed for global optimization

lexicographic multi objective linear programming using grossone methodology theory and algorithm abstract numerous problems arising in engineering applications can have several objectives to be satisfied an important class of problems of this kind is lexicographic multi objective problems where the first objective is incomparably more important than the second one which in its turn is incomparably more important than the third one etc in this paper lexicographic multi objective linear programming lmolp problems are considered to tackle them traditional approaches either require solution of a series of linear programming problems or apply a scalarization of weighted multiple objectives into a single objective function the latter approach requires finding a set of weights that guarantees the equivalence of the original problem and the single objective one and the search of correct weights can be very time consuming in this work a new approach for solving lmolp problems using a recently introduced computational methodology allowing one to work numerically with infinities and infinitesimals is proposed it is shown that a smart application of infinitesimal weights allows one to construct a single objective problem avoiding the necessity to determine finite weights the equivalence between the original multi objective problem and the new single objective one is proved a simplex based algorithm working with finite and infinitesimal numbers is proposed implemented and discussed results of some numerical experiments are provided

comparison of partition evaluation measures in an adaptive partitioning algorithm for global optimization abstract an adaptive partitioning algorithm with random search is proposed to locate the global optimum of multimodal functions partitioning algorithms divide the feasible region into nonoverlapping partitions in order to restrict and direct the search to the most promising region expected to contain the global optimum in such a scheme a partition evaluation measure is required to assess sub regions in order to re partition the most promising sub region and intensify the search within that area this study provides computational results on several classes of partition evaluation measures used in the assessment of samples taken from all partitions among the partition evaluation classes used in our comparison are fuzzy statistical and deterministic interval estimation measures performance in terms of solution quality is reported on an extensive set of 77 test functions collected from the literature

on local search in d c optimization problems first we consider a d c minimization problem with a simple feasible set and develop a special method based on the linearization with respect to the basic nonconvexity the convergence of the methods is analyzed and compared with published results theoretical and practical stopping criteria are proposed second we consider a problem with d c constraint and study the properties of special local search method for this problem finally we consider a variant of local search for a general d c optimization problem and investigate its convergence

global optimization of univariate lipschitz functions i survey and properties we consider the following global optimization problems for a univariate lipschitz functionf defined on an interval a b problem p find a globally optimal value off and a corresponding point problem p find a globally optimal value off and a corresponding point problem q localize all globally optimal points problem q find a set of disjoint subintervals of small length whose union contains all globally optimal points problem q find a set of disjoint subintervals containing only points with a globally optimal value and whose union contains all globally optimal points we present necessary conditions onf for finite convergence in problem p and problem q recall the concepts necessary for a worst case and an empirical study of algorithms i e those ofpassive and ofbest possible algorithms summarize and discuss algorithms of evtushenko piyavskii shubert timonov schoen galperin shen and zhu presenting them in a simplified and uniform way in a high level computer language we address in particular the problems of using an approximation for the lipschitz constant reducing as much as possible the expected length of the region of indeterminacy which contains all globally optimal points and avoiding remaining subintervals without points with a globally optimal value new algorithms for problems p and q and an extensive computational comparison of algorithms are presented in a companion paper

oscillation criteria for half linear dynamic equations on time scales this paper is concerned with oscillation of the second order half linear delay dynamic equation r t x p t x t 0 on a time scale t where 1 is the quotient of odd positive integers p t and t t are positive rd continuous functions on t r t is positive and delta differentiable t t and limt t we establish some new sufficient conditions which ensure that every solution oscillates or converges to zero our results in the special cases when t r and t n involve and improve some oscillation results for second order differential and difference equations and when t hn t q n 0 and t n 2 our oscillation results are essentially new some examples illustrating the importance of our results are also included

nonlinear programming and grossone quadratic programing and the role of constraint qualifications abstract a novel and interesting approach to infinite and infinitesimal numbers was recently proposed in a series of papers and a book by sergeyev this novel numeral system is based on the use of a new infinite unit of measure the number grossone indicated by the numeral the number of elements of the set in of natural numbers based on the use of de cosmis and de leone 2012 have then proposed a new exact differentiable penalty function for constrained optimization problems in this paper these results are specialized to the important case of quadratic problems with linear constraints moreover the crucial role of constraint qualification conditions well know in constraint minimization literature is also discussed with reference to the new proposed penalty function

a stochastic method for global optimization a stochastic method for global optimization is described and evaluated the method involves a combination of sampling clustering and local search and terminates with a range of confidence intervals on the value of the global optimum computational results on standard test functions are included as well

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

geometric curvature bounds in riemannian manifolds with boundary an alexandrov upper bound on curvature for a riemannian manifold with boundary is proved to be the same as an upper bound on sectional curvature of interior sections and of sections of the boundary which bend away from the interior as corollaries those same sectional curvatures are related to estimates for convexity and conjugate radii the hadamard cartan theorem and yau s isoperimetric inequality for spaces with negative curvature are generalized

on strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions abstract the implementation of global optimization algorithms using the arithmetic of infinity is considered a relatively simple version of implementation is proposed for the algorithms that possess the introduced property of strong homogeneity it is shown that the p algorithm and the one step bayesian algorithm are strongly homogeneous

axiomatic approach to statistical models and their use in multimodal optimization theory this paper summarizes the results of axiomatic constructing statistical models of complicated multimodal functions it is shown that an optimization algorithm may be constructed on the basis of a statistical model and some ideas of the rational choice theory a brief review of related algorithms and reports on investigations of their efficiency is given

solving ordinary differential equations on the infinity computer by working with infinitesimals numerically there exists a huge number of numerical methods that iteratively construct approximations to the solution y x of an ordinary differential equation ode y x f x y starting from an initial value y 0 y x 0 and using a finite approximation step h that influences the accuracy of the obtained approximation in this paper a new framework for solving odes is presented for a new kind of a computer the infinity computer it has been patented and its working prototype exists the new computer is able to work numerically with finite infinite and infinitesimal numbers giving so the possibility to use different infinitesimals numerically and in particular to take advantage of infinitesimal values of h to show the potential of the new framework a number of results is established it is proved that the infinity computer is able to calculate derivatives of the solution y x and to reconstruct its taylor expansion of a desired order numerically without finding the respective derivatives analytically or symbolically by the successive derivation of the ode as it is usually done when the taylor method is applied methods using approximations of derivatives obtained thanks to infinitesimals are discussed and a technique for an automatic control of rounding errors is introduced numerical examples are given

oscillation of second order nonlinear neutral dynamic equations on time scales abstract this paper is concerned with the oscillation of second order nonlinear neutral dynamic equations of the form r t y t p t y t f t y t 0 on a time scale t by using a generalized riccati technique and integral averaging techniques we establish new oscillation results which handle some cases not covered by known criteria

convergence qualification of adaptive partition algorithms in global optimization following the presentation of a general partition algorithm scheme for seeking the globally best solution in multiextremal optimization problems necessary and sufficient convergence conditions are formulated in terms of respectively implied or postulated properties of the partition operator the convergence results obtained are pertinent to a number of deterministic algorithms in global optimization permitting their diverse modifications and generalizations

examples of manifolds of positive ricci curvature on presente de nouveaux exemples de varietes de riemann simplement connexes de dimension 7 a courbure de ricci positive qui n admettent pas de metrique de courbure sectionnelle non negative dans les cas compacts et non compacts

an algorithm for finding the global maximum of a multimodal multivariate function this algorithm for global optimization uses an arbitrary starting point requires no derivatives uses comparatively few function evaluations and is not side tracked by nearby relative optima the algorithm builds a gradually closer piecewise differentiable approximation to the objective function the computer program exhibits a theoretically expected strong tendency to cluster around relative optima close to the global results of testing with several standard functions are given

global one dimensional optimization using smooth auxiliary functions in this paper new global optimization algorithms are proposed for solving problems where the objective function is univariate and has lipschitzean first derivatives to solve this problem smooth auxiliary functions which are adaptively improved during the course of the search are constructed three new algorithms are introduced the first used the exact a priori known lipschitz constant for derivatives the second when this constant is unknown estimates it during the course of the search and finally the last method uses neither the exact global lipschitz constant nor its estimate but instead adaptively estimates the local lipschitz constants in different sectors of the search region during the course of optimization convergence conditions of the methods are investigated from a general viewpoint and some numerical results are also given 1998 the mathematical programming society inc published by elsevier science b v

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below i we investigate the finiteness structure of a complete non compact n dimensional riemannian manifold m whose radial curvature at a base point of m is bounded from below by that of a non compact von mangoldt surface of revolution with its total curvature greater than we show as our main theorem that all busemann functions on m are exhaustions and that there exists a compact subset of m such that the compact set contains all critical points for any busemann function on m as corollaries by the main theorem m has finite topological type and the isometry group of m is compact

complete manifolds with nonnegative ricci curvature and quadratically nonnegatively curved infinity in this paper we study complete riemannian manifolds with nonnegative ricci curvature and quadratically nonnegatively curved infinity we show that m is of finite topological type if it has either maximal volume growth or minimal volume growth

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

minimization by random search techniques we give two general convergence proofs for random search algorithms we review the literature and show how our results extend those available for specific variants of the conceptual algorithm studied here we then exploit the convergence results to examine convergence rates and to actually design implementable methods finally we report on some computational experience

a volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications where dpo is the distance to po this class was first studied by u abresch abi ab2 because of the decay condition on the curvature faster than quadratic we expect the class of manifolds of asymptotically nonnegative curvature behaves similarly at infinity to manifolds of nonnegative curvature for the case of sectional curvature in abl and ab2 abresch showed that the complexity of topology of such manifolds is similar to that of nonnegative curvature namely they are all of finite topological type with uniform bound on the betti numbers in ka a kasue investigated the theory of harmonic functions for such manifolds again he demonstrated the expected similarity for ricci curvature the examples of j p sha and d g yang showed that we can not hope to control the topology of such manifolds as we do for sectional curvature very little is known for the case of asymptotically nonnegative ricci curvature except the work of abresch and gromoll ab which studied such manifolds under further conditions on the sectional curvature and diameter growth a question of recent interest stimulated by the works of p li and l f tam lt1 lt2 is whether open manifolds of asymptotically nonnegative ricci curvature has finitely many ends for the special case when a has compact support m l cai cai and z d liu liu recently gave a bound on the number of ends in lt2 under the stronger condition of f0o tn a t dt o0 li and tam also gave such a bound

large scale behavior of kernels of schrodinger operators in this paper we study the asymptotic behavior of solutions to schrodinger equations on manifolds with nonnegative ricci curvature and euclidean volume growth as a first application we get a sharp asymptotic estimate on the green s function for the laplacian which sharpens results of li and yau next we establish a definite gap in the possible rates of decay of exterior harmonic functions on these manifolds when the manifold also has quadratic curvature decay we get supremum estimates on the c 2 norm of the green s function this has function theoretic implications for these manifolds we then give a generalization to manifolds with nonnegative ricci curvature outside a compact set finally we study schrodinger operators where the potential decays quadratically at infinity in this setting we obtain a geometric version at infinity of a classical theorem of bers

metrics of positive ricci curvature on quotient spaces one of the classical problems in differential geometry is the investigation of closed manifolds which admit riemannian metrics with given lower bounds for the sectional or the ricci curvature and the study of relations between the existence of such metrics and the topology and geometry of the underlying manifold despite many efforts during the past decades this problem is still far from being understood for example so far the only obstructions to the existence of a metric with positive ricci curvature come from the obstructions to the existence of metrics with positive scalar curvature li hi ro schy ta and the bonnet myers theorem which implies that the fundamental group of a closed manifold with positive ricci curvature must be finite fruitful constructions of metrics with positive ricci curvature on closed manifolds have so far been established by techniques that include deformation of metrics au eh we kahler geometry yau1 yau2 bundles and warping po na bb gpt special kinds of surgery sy wr metrical glueing gz2 and sasakian geometry bgn particularly large classes of examples of manifolds with positive ricci curvature are given by all compact homogeneous spaces with finite fundamental group na and all closed cohomogeneity one manifolds with finite fundamental group gz2 in this article we present several new classes of manifolds which admit metrics of positive or nonnegative ricci curvature the idea is to consider quotients of manifolds m g of positive or nonnegative ricci curvature by a free isometric action while taking such a quotient non decreases the sectional curvature it may well happen in general that the ricci curvature of the quotient is inferior to the one of m our first results however state that the quotient of m does admit metrics of positive ricci curvature if m belongs to one of the aforementioned classes of homogeneous spaces or spaces with a cohomogeneity one action more precisely we prove

the sierpinski curve viewed by numerical computations with infinities and infinitesimals abstract the sierpinski curve is one of the most known space filling curves and one with the highest number of applications we present a recently proposed computational methodology based on the infinite quantity called grossone to investigate the behavior of two different constructions of the sierpinski curve we emphasize that adopting this point of view we have infinitely many sierpinski curves depending contrarily to traditional analysis on each specific starting configuration of particular interest are some power series expansions in the new infinitesimal quantities emerging from the study of the considered curves

oscillation of second order nonlinear neutral delay dynamic equations on time scales in this paper some sufficient conditions for oscillation of the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t are established here 1 is an odd positive integer with r t and p t are rd continuous functions defined on t our results as a special case when t r and t n involve and improve some well known oscillation results for second order neutral delay differential and difference equations when t hn and t qn t t qk k n q 1 i e for generalized neutral delay difference and q neutral delay difference equations our results are essentially new and also can be applied on different types of time scales e g t n2 t2 t n and t tn tn n n0 where tn is the set of harmonic numbers some examples illustrating our main results are given

iterative methods for the localization of the global maximum methods for the search of a maximum utilizing a sequence of majorants to generate a sequence of localizations are analyzed necessary and sufficient conditions for the convergence of this sequence are given the case of lipschitz continuous functions is studied in detail and an analogy with iterative and relaxation methods for the solution of a system of linear equations is made methods of jacobi gauss seidel and chaotic type are proposed