curvature diameter homotopy groups and cohomology rings we establish two topological results a if m is a 1 connected compact n manifold and q 2 then the minimal number of generators for the qth homotopy group q m mng q m is bounded above by a number depending only on mng h m and q where h m is the homology group c let mathscr m n y be the collection of compact orientable n manifolds whose oriented bundles admit so n invariant fibrations over y with fiber compact nilpotent manifolds such that the induced so n actions on y are equivalent then q m finitely generated m mathscr m n y contains only finite isomorphism classes depending only on n y q together with the results of cg and gr1 from a we conclude that i if m is a complete n manifold of nonnegative curvature then mng q m is bounded above by a number depending only on n and q 2 together with the results of ch and cfg from c we conclude that ii if m is a compact n manifold whose sectional curvature and diameter satisfy secm and diamm d then q m has a finite number of possible isomorphism classes depending on n d q 2 provided q m is finitely generated we also show that b if m is a compact n manifold with secm and diam m d then the cohomology ring h m may have infinitely many isomorphism classes in particular b answers some questions raised by k grove gro

topological obstructions to nonnegative curvature we find new obstructions to the existence of complete riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups in particular we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metrics

some group actions on homotopy spheres of dimension seven and fifteen 0 introduction this paper is based on the simple observation that every 4 plane bundle over s4 admits a natural action of 50 3 by bundle maps and that every 8 plane bundle over 58 admits a natural action of the compact lie group g2 by bundle maps these actions are easy to see once one remembers that so 3 is the group of automorphisms of the quaternions and that g2 is the group of automorphisms of the cayley numbers the study of these actions is closely connected to several well known phenomena in differential topology and compact transformation groups the most obvious connection is to milnor s original construction of exotic 7 spheres as 3 sphere bundles over 54 26 milnor proved that if the euler class of such a bundle is a generator of h4 s4 z then its total space is homeomorphic to 57 he also defined a numerical invariant of the diffeomorphism type and used it to detect an exotic differential structure on some of these sphere bundles subsequently eells and kuiper 12 introduced a refinement of this invariant called the it invariant using the u invariant they proved that the sphere bundle m7 milnor s notation is a generator for the group of homotopy 7 spheres these constructions also work for 7 sphere bundles over 58 32 and the manifold m15 is a generator for bp16 the group of homotopy 15 spheres which bound wn manifolds it is a routine matter to check that milnor s arguments and their subsequent refinements work g equivariantly where g 5so 3 or g2 and we shall do this in sections 2 and 3 in particular each sphere bundle with the correct euler class is g homeomorphic to an orthogonal action on s2 1 where 2n 1 7 or 15 moreover distinct sphere bundles have distinct oriented g diffeomorphism types the proof of the second fact uses an equivariant version of the it invariant as originally defined this invariant takes values in q z however it is well known that in the presence of a g action with s c g its value in q is well defined using

transitive holonomy group and rigidity in nonnegative curvature in this note we examine the relationship between the twisting of a vector bundle xi over a manifold m and the action of the holonomy group of a riemannian connection on xi for example if there is a holonomy group which does not act transitively on each fiber of the corresponding unit sphere bundle then for any f s n to m the pullback f xi of xi admits a nowhere zero cross section these facts are then used to derive a rigidity result for complete metrics of nonnegative sectional curvature on noncompact manifolds

capacitated facility location separation algorithms and computational experience we consider the polyhedral approach to solving the capacitated facility location problem the valid inequalities considered are the knapsack cover flow cover effective capacity single depot and combinatorial inequalities the flow cover effective capacity and single depot inequalities form subfamilies of the general family of submodular inequalities the separation problem based on the family of submodular inequalities is np hard in general for the well known subclass of flow cover inequalities however we show that if the client set is fixed and if all capacities are equal then the separation problem can be solved in polynomial time for the flow cover inequalities based on an arbitrary client set and general capacities and for the effective capacity and single depot inequalities we develop separation heuristics an important part of these heuristics is based on the result that two specific conditions are necessary for the effective cover inequalities to be facet defining the way these results are stated indicates precisely how structures that violate the two conditions can be modified to produce stronger inequalities the family of combinatorial inequalities was originally developed for the uncapacitated facility location problem but is also valid for the capacitated problem no computational experience using the combinatorial inequalities has been reported so far here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities we report on computational results from solving 60 medium size problems

fuzzy unit hydrograph 1 the unit hydrograph method frequently used for representing the rainfall runoff transformation is subject to a number of uncertainties related both to the model assumption and data acquisition despite them usually being disregarded a methodology to account for these uncertainties based on the fuzzy approach is proposed and illustrated it is assumed that the unit hydrograph can be represented through a fuzzy schematization and effective precipitation is also considered a fuzzy series by applying the extension principle to the convolution integral a fuzzy runoff is obtained in particular the nash cascade parametric form of the unit hydrograph is used and its parameters are considered fuzzy numbers the general methodology and the assessment of the parameters are explained using a case study relevant to the subcatchments of the neckar catchment southwest germany the proposed procedure enables bands to be calculated that describe the range of possible hydrographs for predefined h levels the uncertainty in the runoff calculation is thus represented in terms of fuzzy vagueness by the width of these bands the bands decrease with the increase of the h level the calibrated parameters are then used with the extension principle for calculating the vagueness of different runoff characteristics such as flood peak discharge and time to peak as fuzzy numbers

steady state groundwater flow simulation with imprecise parameters a methodology based on fuzzy set theory is developed to incorporate imprecise parameters into steady state groundwater flow models in this case fuzzy numbers are used to represent parameter imprecision as such they are also used as a measure for the uncertainty associated with the hydraulic heads due to the imprecision in the input parameters the imprecise input parameters may come from indirect measurements subjective interpretation and expert judgment of available information in the methodology a finite difference method is combined with level set operations to formulate the fuzzy groundwater flow model this fuzzy modeling technique can handle imprecise parameters in a direct way without generating a large number of realizations two numerical solution methods are used to solve the fuzzy groundwater flow model the groundwater model operator method proposed in this methodology and the iterative algorithm based on conventional interval arithmetics the iterative method is simple but may overestimate the uncertainty of hydraulic heads the groundwater model operator method not only provides the hull of the solution set for the hydraulic heads but also considers the dependence of hydraulic head coefficients which are functions of imprecise parameters sensitivity analysis shows that the dependence of hydraulic head coefficients has a critical impact on the model results and neglecting this dependence may result in significant overestimation of the uncertainty of hydraulic heads a numerical model based on the methodology is tested by comparing it with the analytical solution for a homogeneous radial flow problem it is also applied to a simplified two dimensional heterogeneous flow case to demonstrate the methodology

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

concavity and rigidity in non negative curvature we show that for a manifold with non negative curvature one obtains a collection of concave functions special cases of which are the concavity of the length of a jacobi field in dimension 2 and the concavity of the volume in general we use these functions to show that there are many cohomogeneity one manifolds which do not carry an analytic invariant metric with non negative curvature this implies in particular that one of the candidates in positively curved cohomogeneity one manifolds and 3 sasakian geometry does not carry an invariant metric with positive curvature

cohomogeneity one topological manifolds revisited we prove a structure theorem for closed topological manifolds of cohomogeneity one this result corrects an oversight in the literature we complete the equivariant classification of closed simply connected cohomogeneity one topological manifolds in dimensions 5 6 and 7 and obtain topological characterizations of these spaces in these dimensions these manifolds are homeomorphic to smooth manifolds

non negative pinching moduli spaces and bundles with infinitely many souls we show that in each dimension n 10 there exist infinite sequences of homotopy equivalent but mutually non homeomorphic closed simply connected riemannian n manifolds with 0 sec 1 positive ricci curvature and uniformly bounded diameter we also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually non homeomorphic finally we construct examples of noncompact manifolds whose moduli spaces of complete metrics with sec 0 have infinitely many connected components

local rigidity of symmetric spaces we show that on a symmetric space of noncompact or compact type the metric is locally rigid in the sense that if one changes the metric locally but preserves the curvature bounds then the new metric is isometric to the old one we also prove an analytic continuation property for symmetric spaces of rank 3 the aim of this paper is to show that a symmetric space is rigid under a local modification of the metric which preserves suitable curvature bounds our results were motivated by the following theorem of gromov bgs 5 theorem 1 gromov let x be a simply connected symmetric space of rank 2 and with nonpositive sectional curvature and q the complement of a compact set in x if f q y is a totally geodesic isometric embedding where y is a complete simply connected riemannian manifold with nonpositive sectional curvature then there exists a unique extension of f to a totally geodesic isometric embedding tf x y we will give a slightly different proof of this theorem which will then enable us to also prove the following dual result theorem 2 let x be a simply connected symmetric space of rank 2 and with sectional curvature 0 0 and dim y dim x and if q is the complement of a ball of radius 7r 30 in x then any isometric embedding f q y extends uniquely to an isometry 7 x y if in the above two theorems we choose y x and f id we obtain the desired local rigidity of symmetric spaces theorem 2 applies in particular to s2 x s and shows that the hopf conjecture is at least locally correct the optimal radius for the ball in theorem 2 would be the radius of a convex ball 7r 2 under those assumptions indeed in the case of nonpositive curvature gromov bgs 5 proved the following local version of theorem 1 received by the editors november 4 1987 and in revised form september 12 1988 1980 mathematics subject classification 1985 revision primary 53c35 secondary 53c20 the first author is supported by the schweizerischer nationalfond the second author is partially supported by a grant from the national science foundation and would like to thank the ihes and the max planck institut for its hospitality 1990 american mathematical society 0002 9947 90 1 00 25 per page

volumes of small balls on open manifolds lower bounds and examples question under what curvature assumptions on a complete open manifold is the volume of balls of a fixed radius bounded below independent of the center point xx two theorems establish such assumptions and two examples sharply limit their weakening in particular we give an example of a metric on r4 extending to higher dimensions of positive ricci curvature whose sectional curvatures decay to 0 and such that the volume of balls goes uniformly to 0 as the center goes to infinity 0 introduction the following question came up in discussions of the first author with peter li under what curvature assumptions on a complete open manifold m can one find a constant c m r such that the volume of balls b x r can be bounded from below independent of the midpoint x 0 1 vol b x r c m r note that in the presence of a lower bound on the ricci curvature one needs to prove the above inequality only for some r since the bishop gromov volume comparison theorem gr p 124 then gives a bound c m r for r c n rn in dimension 2 theorem a below is a satisfyingpositive answer since the assumptions do not imply a lower bound on the injectivity radius while any weakening of the assumptions leads to immediate counterexamples theorem a if m2 is complete with curvature k o then there exist c m such that for radii r c m r2 in higher dimensions however we have example 1 there are convex hypersurfaces md in rd 1 such that for d 3 no lower bounds as in 0 1 exist for md in dimensions 2 we therefore assume an upper curvature bound but then one gets uniform control even on the injectivity radius the following theorem has been rediscovered several times received by the editors august 10 1987 1980 mathematics subject classification 1985 revision primary 53c20 secondary 53a07 research supported by nsf grant mcs 7g01780 and the sloan foundation both authors were supported by invitations to the i h e s and the m s r i r 1988 american mathematical society 0002 9947 88 1 00 25 per page

stochastic subsurface hydrology from theory to applications research on stochastic analysis of subsurface flow has developed rapidly in the last decade but applications of this approach have been very limited the purpose of this paper is to illustrate how currently available techniques and results can be used to answer important questions about the large scale behavior of naturally heterogeneous aquifers perturbation based spectral theory which presumes local statistical homogeneity provides generic theoretical results for the head variance effective conductivity tensor and macrodispersivity tensor in a field these results emphasize the key role of the variance and spatial correlation scales of the log hydraulic conductivity field field information of variances and correlation scales of natural materials is summarized the validity of some of the generic stochastic results is evaluated through comparisons with monte carlo simulations and field observations a specific field application example is developed to illustrate how the stochastic results are used to estimate large scale parameters and determine the reliability of three dimensional numerical simulations using typical log conductivity covariance parameters the effective hydraulic conductivity tensor and the macrodispersivity tensor are estimated the calculated head variance based on the simulated mean hydraulic gradient is used as a measure of the adequacy of the calculation of the steady state flow model discussion emphasizes limitations and extensions of this approach and ongoing field evaluations of the results

eulerian lagrangian theory of transport in space time nonstationary velocity fields exact nonlocal formalism by conditional moments and weak approximation a unified eulerian lagrangian theory is presented for the transport of a conservative solute in a random velocity field that satisfies locally v x t f x t where f x t is a random function including sources and or the time derivative of head solute concentration satisfies locally the eulerian equation c x t t j x t g x t where j x t is advective solute flux and g x t is a random source independent of f x t we consider the prediction of c x t and j x t by means of their unbiased ensemble moments c x t and j x t conditioned as implied by the subscript on local hydraulic measurements through the use of the latter in obtaining a relatively smooth unbiased estimate x t of v x t these predictors satisfy c x t v t j x t g x t where j x t x t c x t q x t and q x t is a dispersive flux we show that q is given exactly by three space time convolution integrals of conditional lagrangian kernels with q with c and with c for a broad class of v x t fields including fractals this implies that q x t is generally nonlocal and non fickian rendering c x t non gaussian the direct contribution of random variations in f to q depends on c rather than on c we elucidate the nature of the above kernels discuss conditions under which the convolution of and c becomes pseudo fickian with a lagrangian dispersion tensor similar to that derived in 1921 by taylor recall a 1952 result by batchelor which yields an exact expression for c at early time use the latter to conclude that linearizations which predict that c bifurcates at early time when the probability density function of v is unimodal cannot be correct propose instead a weak approximation which leads to a nonlinear integro differential equation for c due to an instantaneous point source and which improves with the quantity and quality of hydraulic data demonstrate that the weak approximation is analogous to the direct interaction closure of turbulence theory offer non fickian and pseudo fickian weak approximations for the second conditional moment of the concentration prediction error demonstrate that it yields the so called two particle covariance as a special case conclude that the conditional variance of c does not become infinite merely as a consequence of disregarding local dispersion and discuss how to estimate explicitly the cumulative release of a contaminant across a compliance surface together with the associated estimation error

nonconnected moduli spaces of positive sectional curvature metrics for a closed manifold m let 9 m resp 9 ic m be the space of riemannian metrics on m with positive sectional resp ricci cur vature and let diff m be the diffeomorphism group of m which acts on these spaces we construct examples of 7 dimensional manifolds for which the moduli space 9 m diff m is not connected and others for which 9 c m diff m has infinitely many connected components the examples are obtained by analyzing a family of positive sectional curvature metrics on homogeneous spaces constructed by aloff and wallach on which su 3 acts transitively respectively a family of positive einstein metrics constructed by wang and ziller on homogeneous spaces on which su 3 x su 2 x u i acts transitively max planck institut fur mathematik gottfried claren strasse 26 5300 bonn 3 germany and fachbereich mathematik universitat mainz 6500 mainz germany current address johannes gutenberg universitat mainz fachbereich mathematik staudinger weg 9 6500 mainz germany e mail address kreck topologie mathematik uni mainz de department of mathematics university of notre dame notre dame indiana 46556 e mail address stolz l nd edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

aleatoric and epistemic uncertainty in groundwater flow and transport simulation 1 the characterization of aleatory hydrogeological parameter uncertainty has traditionally been accomplished using probability theory however when consideration is given to epistemic as well as aleatory uncertainty probability theory is not necessarily appropriate this is especially the case where expert opinion is regarded as a suitable source of information when experts opine upon the uncertainty of a parameter value both aleatoric and epistemic uncertainties are introduced and must be modeled appropriately a novel approach to expert provided parameter uncertainty characterization can be defined that bridges an historical gap between probability theory and fuzzy set theory herein a random set a generalization of a random variable is employed to formalize expert knowledge and fuzzy sets are used to propagate this uncertainty to model estimates of contaminant transport the resultant random set based concentration estimates are shown to be more general than the corresponding random variable estimates in some cases the random set based results are shown as upper and lower probabilities that bound the corresponding random variable s cumulative distribution function

compact differentiable transformation groups on exotic spheres 0 introduction 355 1 statement of the main results 357 2 higher dimensional subgroups of o n 1 358 3 higher dimensional transformation groups on homology spheres geometric weight systems and orbit structures 360 4 higher dimensional transformation groups on homotopy spheres and the proof of theorem b 375 5 degree of symmetry of exotic spheres revisited 381 6 large transformation groups on exotic spheres in bp 1 384

exotic spheres with positive ricci curvature we show that a certain class of manifolds admit metrics of positive ricci curvature this class includes many exotic spheres including all homotopy spheres which represent elements of bp2n

the almost cyclicity of the fundamental groups of positively curved manifolds recall that a pure f structure is a kind of generalized torus action the main result asserts that if a compact positively curved manifold mn admits an invariant pure f structure such that each orbit has positive dimension then the fundamental group has a finite cyclic subgroup with index less than wn a constant depending only on n as an application we conclude that for all 0 1 the fundamental group of a pinched n manifold either has a cyclic subgroup with index less than wn or has order less than w n a constant depending only on n and in particular this substantially improves the main result in ro1

twisting and nonnegative curvature metrics on vector bundles over the round sphere a complete noncompact manifold m with nonnegative sectional curvature is diffeomorphic to the normal bundle of a compact submanifold s called the soul of m when s is a round sphere we show that the clutching map of this bundle is restricted this is used to deduce that there are at most finitely many isomorphism types of such bundles with sectional curvature lying in a fixed interval 0 we also examine the opposite question of how the twisting of the bundle limits the type of possible nonnegative curvature metrics on the bundle it turns out that if the bundle does not admit a nowhere zero section then the normal exponential map is necessarily a diffeomorphism onto m and the ideal boundary of m consists of a single point in their paper cg cheeger and gromoll raised the question of which vector bundles over the round sphere admit complete metrics with nonnegative sectional curvature the significance of this problem is that it attempts to determine to what extent a converse to the soul theorem holds recall that this theorem states that every open i e complete noncompact manifold m with nonnegative curvature km is diffeomorphic to a vector bundle over a compact totally geodesic submanifold s called a soul a natural question then is whether all such vector bundles admit complete metrics with km 0 in ow it was shown that when the soul is a bieberbach manifold nonnegative curvature metrics force the vanishing of the euler class of the vector bundle it follows that among oriented plane bundles over the torus only the trivial one admits such a metric the above case is fairly rigid a priori however since for example any bundle with curvature 0 over such a manifold must also admit a flat metric and the corresponding question for a simply connected base remains open an answer could provide insights on the topological structure of compact manifolds with nonnegative curvature since any open manifold with nonnegative curvature can after modifying its metric be isometrically 1991 mathematics subject classification primary 53c20

topology of riemannian submanifolds with prescribed boundary we prove that a smooth compact submanifold of codimension 2 immersed in r n 3 bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces this settles a question of guan and spruck related to a problem of yau analogous results for complete fillings of arbitrary riemannian submanifolds are obtained as well on the other hand we show that these finiteness theorems may not hold if the codimension is too high or the prescribed boundary is not sufficiently regular our proofs employ among other methods a relative version of nash s isometric embedding theorem and the theory of alexandrov spaces with curvature bounded below including the compactness and stability theorems of gromov and perelman

the metric projection onto the soul we study geometric properties of the metric projection r m s of an open manifold m with nonnegative sectional curvature onto a soul s ir is shown to be c up to codimension 3 in arbitrary codimensions small metric balls around a soul turn out to be convex so that the unit normal bundle of s also admits a metric of nonnegative curvature next we examine how the horizontal curvatures at infinity determine the geometry of m and stu dy the structure of sharafutdinov lines we conclude with regularity properties of the cut and conjugate loci of m the resolution of the soul conjecture of cheeger and gromoll by perelinan showed that the structure of open manifolds with nonnegative sectional curvature is more rigid than expected orne of the key results in 14 is that the metric projection wf m s which maps a point p in m to the point 7r p in s that is closest to p is a riemannian submersioni perelman also observed that this map is at least of class c1 and later it was shown in 9 that 7w is at least c2 and c at almnost every point the existence of such a riemannian submersion combined with the restriction on the sign of the curvature suggests that the geometry of m inust be special the purpose of this paper is to illustrate this in several different directions in fact we show that most natural geometric objects classically used in the study of these spaces are intrinsically related to the structure of the map w the paper is essentially structured as follows after introducing notation and recalling some basic results in sectioni 1 we establish in section 2 the existence of maps similar to w for any submanifold of m homologous and isometric to the soul and use this to prove the existence of convex tubular neighborhoods for them in particular this also proves theorem 2 5 the unit normal bundle vti s admits a metric with nonnegative sectional curvature as a consequence open manifolds with nonnegative curvature provide examples of compact manifolds with the same lower curvatuire bound see also 81 theorem 2 5 also implies that nontrivial plane bundles over a bieberbach maanifold do not admit nonnegatively curved metrics thereby providing a shorter proof of the main result in 13 we are grateful to the referee for pointing out this fact to us properties such as these are established via stanidard submersion techniques but the fact that 7r is not known to be smooth everywhere prevents us fromi using these techniques to study the geometry of m far away from the soul one way around received by the editors august 18 1997 1991 mathematics subject classification primary 53c20

pontryagin numbers and nonnegative curvature we prove that any rational linear combination of pontryagin numbers that is not a multiple of the signature is unbounded on connected closed oriented manifolds of nonnegative sectional curvature combining our result with gromov s boundedness result for the signature yields a new characterization of the l genus

finiteness theorems for nonnegatively curved vector bundles we prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds more generally we obtain finiteness results for open riemannian manifolds whose topology is concentrated on compact domains of bounded geometry

a perturbation expansion for diffusion in a random velocity field we present a formal perturbation expansion for the large scale effective drift velocity and diffusion matrix of a medium with stationary random velocity v x and constant nonrandom diffusion matrix a on a small scale if mu denotes the mean of v x and we express v x as mu varepsilon u x then the expansion is in powers of varepsilon with mu u and a fixed we obtain explicit expressions up to second order in varepsilon which generalize the standard formulae to the case a ne 0

obstructions to nonnegative curvature and rational homotopy theory we establish a link between rational homotopy theory and the problem which vector bundles admit a complete riemannian metric of nonnegative sectional curvature as an application we show for a large class of simply connected nonnegatively curved manifolds that if c lies in the class and t is a torus of positive dimension then most vector bundles over c x t admit no complete nonnegatively curved metrics

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

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

ricci curvature bounds and einstein metrics on compact manifolds where f mo m1 is a homeomorphism and dil f is the dilatation of f given by dil f supx x2 dist f x f x2 dist x1 x2 if mo and m1 are not homeomorphic define dl mo mi oo gromov 20 proves the remarkable result that the space of compact riemannian manifolds f a 3 d of sectional curvature iki 3 0 and diameter dm v and diameter dm c iki dm vm1 in particular gromov s compactness theorem may be strengthened to the statement that f a v d is c1 l compact in the lipschitz topology in this paper we study the question of lipschitz convergence of compact riemannian manifolds with bounds imposed on the ricci curvature ric in

seven dimensional cohomogeneity one manifolds with nonnegative curvature we show that a certain family of cohomogeneity one manifolds does not admit an invariant metric of nonnegative sectional curvature unless it admits one with positive curvature as a consequence the classification of nonnegatively curved cohomogeneity one manifolds in dimension seven is reduced to only one further family of candidates

dubins problem on surfaces ii nonpositive curvature let m be a complete connected two dimensional riemannian manifold with nonpositive gaussian curvature k we say that m satisfies the unrestricted complete controllability ucc property for the dubins problem if the following holds given any p 1 v 1 and p 2 v 2 in tm there exists for every eps 0 a curve gamma in m with geodesic curvature smaller than eps such that gamma connects p 1 to p 2 and for i 1 2 dot gamma is equal to v i at p i property ucc is equivalent to the complete controllability of a family of control systems of dubins type parameterized by eps it is well known that the poincare half plane does not verify property ucc in this paper we provide a complete characterization of the two dimensional nonpositively curved manifolds m with either uniformly negative or bounded curvature that satisfy property ucc more precisely if sup mk infty we show that ucc holds if and only if i m is of the first kind or ii the curvature satisfies a suitable integral decay condition at infinity

cheeger manifolds and the classification of biquotients a closed manifold is called a biquotient if it is diffeomorphic to k g h for some compact lie group g with closed subgroups k and h such that k acts freely on g h every biquotient has a riemannian metric of nonnegative sectional curvature in fact almost all known manifolds of nonnegative curvature are biquotients the only known closed manifolds of nonnegative sectional curvature which were not defined in this way are those found by cheeger 6 in 1973 and grove and ziller 16 in 2000 in order to understand these constructions better this paper analyzes which of the cheeger and grove ziller manifolds are actually diffeomorphic to biquotients in the process we develop a general approach to the classification of biquotient manifolds a priori it is hard to determine whether a given manifold is a biquotient because there is no obvious upper bound on the dimension of the groups involved we give a procedure which allows us to reduce the dimension of the groups needed to describe a given manifold as a biquotient a surprising application is that gromoll meyer s example of an exotic 7 sphere which is a biquotient 15 is the only exotic sphere of any dimension which is a biquotient more generally we classify all biquotients which are simply connected rational homology spheres of any dimension theorem 6 1 the classification of biquotients which are simply connected rational homology spheres has been given independently by kapovitch and ziller 17 more generally they classify all simply connected biquotients whose rational cohomology ring is generated by one element another application of our general classification theory for biquotients is the precise determination of which cheeger manifolds the connected sums of two rank one symmetric spaces with any orientations are diffeomorphic to biquotients theorem 2 1 some of the cheeger manifolds were known to be biquotients such as cp2 cp2 the nontrivial s2 bundle over s2 but we find that many of the other cheeger manifolds are also biquotients such as cp2 cp2 on the other hand the cheeger manifold cp4 hp2 is not diffeomorphic or even homotopy equivalent to a biquotient the positive result that some cheeger manifolds such as cp2 cp2 are biquotients implies that they have nonnegatively curved riemannian metrics with various good properties that were not clear from cheeger s construction first the new metrics are real analytic further the new metrics determine a natural complex analytic structure on the whole tangent bundle since aguilar 2 showed that biquotients have this property my paper 26 has a weaker result which applies to all the cheeger manifolds finally the geodesic flow for the new metrics on the cheeger manifolds has zero topological entropy paternain 23 proved the latter property for cheeger s metric on cp2 cp2 but now we know it for some metric on a large class of the cheeger manifolds finally our general classification results imply that there are only finitely many

riemannian submersions from compact four manifolds we show that under certain conditions a nontrivial riemannian submersion from positively curved four manifolds does not exist this gives a partial answer to a conjecture due to fred wilhelm we also prove a rigidity theorem for riemannian submersions with totally geodesic fibers from compact four dimensional einstein manifolds

network formulations of mixed integer programs we consider mixed integer sets described by system of linear inequalities in which the constraint matrix a is totally unimodular the right hand side is arbitrary vector and a subset of the variables is required to be integer we show that the problem of checking nonemptiness of a set of this type is np complete even in the case in which the linear system describes mixed integer network flows with half integral requirement on the nodes this is in contrast to the case in which a is totally unimodular and contains at most two nonzeros per row in this case we provide an extended formulation for the convex hull of solutions whose constraint matrix is a dual network matrix with an integral right hand side vector the size of this formulation depends on the number of distinct fractional parts taken by the continuous variables in the extreme points of the convex hull of the given set when this number is polynomial in the dimension of a the extended formulation is of polynomial size if in addition the corresponding list of fractional parts can be computed efficiently then our result provides a polynomial algorithm for the optimization problem over these sets we show that there are instances for which this list is of exponential size and we also give conditions under which it is short and can be efficiently computed finally we show that these results provide a unified framework leading to polynomial size extended formulations for several generalizations of mixing sets and lot sizing sets studied in the last few years

on the polyhedral structure of a multi item production planning model with setup times we present and study a mixed integer programming model that arises as a substructure in many industrial applications this model generalizes a number of structured mip models previously studied and it provides a relaxation of various capacitated production planning problems and other fixed charge network flow problems we analyze the polyhedral structure of the convex hull of this model as well as of a strengthened lp relaxation among other results we present valid inequalities that induce facets of the convex hull under certain conditions we also discuss how to strengthen these inequalities by using known results for lifting valid inequalities for 0 1 continuous knapsack problems

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

single value simulation of fuzzy variables abstract the basic ideas of the present paper arise from relations between random sets and fuzzy ones the suggested approach which may be called fuzzy numerical simulation allows for ascribing a precise numerical value to a fuzzy variable by generating a value of a random variable related in some way to the fuzzy variable the so called generative characteristics of a fuzzy variable being the expected value and variance of the generating random variable are introduced the problem of simultaneous generating values of two fuzzy variables is also discussed finally methods of generating when a fuzzy variable is described with a complex body of evidence are presented

when strictly locally convex hypersurfaces are embedded in this paper we will prove hadamard stoker type theorems in the following ambient spaces mathcal m n times mathbb r where mathcal m n is a 1 4 pinched manifold and certain killing submersions e g berger spheres and heisenberg spaces that is under the condition that the principal curvatures of an immersed hypersurface are greater than some non negative constant depending on the ambient space we prove that such a hypersurface is embedded and we also study its topology

cohomogeneity one disk bundles with normal homogeneous collars we consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved1 invariant metric with normal collar i e such that near the boundary the metric is the product of an interval and a normal homogeneous space if such a bundle is not the quotient of a trivial bundle then we show that its rank has to be in 2 3 4 6 8 moreover we give a complete classification of such bundles of rank 6 and 8 and a partial classification for rank 3

a classification of hyperpolar and cohomogeneity one actions an isometric action of a compact lie group on a riemannian manifold is called hyperpolar if there exists a closed connected submanifold that is flat in the induced metric and meets all orbits orthogonally in this article a classification of hyperpolar actions on the irreducible riemannian symmetric spaces of compact type is given since on these symmetric spaces actions of cohomogeneity one are hyperpolar i e normal geodesics are closed we obtain a classification of the homogeneous hypersurfaces in these spaces by computing the cohomogeneity for all hyperpolar actions this result implies a classification of the cohomogeneity one actions on compact strongly isotropy irreducible homogeneous spaces

curvature and symmetry of milnor spheres since milnor s discovery of exotic spheres mi one of the most intriguing problems in riemannian geometry has been whether there are exotic spheres with positive curvature it is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature hi on the other hand there are many examples of exotic spheres with positive ricci curvature cf ch1 he po and na and this work recently culminated in wr where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive ricci curvature this includes all exotic spheres in dimension 7 so far however no example of an exotic sphere with positive sectional curvature has been found in fact until now only one example of an exotic sphere with nonnegative sectional curvature was known the so called gromoll meyer sphere gm in dimension 7 as one of our main results we prove

stochastic theory of field scale fickian dispersion in anisotropic porous media a three dimensional theory is described for field scale fickian dispersion in anisotropic porous media due to the spatial variability of hydraulic conductivities the study relies partly on earlier work by the authors the attributes of which are briefly reviewed it leads to results which differ in important ways from earlier theoretical conclusions about dispersion in anisotropic media we express the dispersion tensor d as the sum of a local component d and a field scale component the local component is assumed to be independent of velocity which is most appropriate if it represents molecular diffusion and its principal terms are taken to act parallel and normal to the mean velocity vector the field scale component is written as where is a dispersivity tensor and we show that at large peclet numbers p the dispersivity tensor reduces to a single principal component parallel to the mean velocity regardless of how is oriented this result valid for arbitrary covariance functions of log hydraulic conductivity differs from that of l w gelhar and c l axness 1983 according to whom the asymptotic dispersivity tensor may possess more than one nonzero eigen value they calculate the direction of the largest principal dispersivity to be offset from the mean velocity toward the direction of least spatial correlation or away from the stratification in typical layered media we show that this principal dispersivity is offset in the opposite direction at small and intermediate peclet numbers but rotates toward the mean velocity as p increases the largest eigen value is constant and dominated by field scale velocity fluctuations at large p values the other two eigen values diminish asymptotically in proportion to p 1 and are controlled by d as well as by field scale differential convection the range of small peclet numbers has not been previously investigated under anisotropic conditions yet is of much importance for transport in low permeability rocks or soils we show that at low p values all three principal dispersivities are proportional to p and thus is proportional to 2 a phenomenon reminiscent of taylor diffusion when the mean velocity is inclined to the axes of anisotropy the eigen values of are neither parallel nor normal to however since d is dominated by d at small peclet numbers the principal dispersion coefficients are asymptotically as p 0 parallel and normal to the mean velocity just like when p is large their maximum deviation from these directions occurs at intermediate p values

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

joint propagation and exploitation of probabilistic and possibilistic information in risk assessment random variability and imprecision are two distinct facets of the uncertainty affecting parameters that influence the assessment of risk while random variability can be represented by probability distribution functions imprecision or partial ignorance is better accounted for by possibility distributions or families of probability distributions because practical situations of risk computation often involve both types of uncertainty methods are needed to combine these two modes of uncertainty representation in the propagation step a hybrid method is presented here which jointly propagates probabilistic and possibilistic uncertainty it produces results in the form of a random fuzzy interval this paper focuses on how to properly summarize this kind of information and how to address questions pertaining to the potential violation of some tolerance threshold while exploitation procedures proposed previously entertain a confusion between variability and imprecision thus yielding overly conservative results a new approach is proposed based on the theory of evidence and is illustrated using synthetic examples

fixed charge transportation on a path optimization lp formulations and separation in the fixed charge transportation problem the goal is to optimally transport goods from depots to clients when there is a fixed cost associated to transportation or equivalently to opening an arc in the underlying bipartite graph we further motivate its study by showing that it is both a special case and a strong relaxation of the big bucket multi item lot sizing problem and a generalization of a simple variant of the single node flow set this paper is essentially a polyhedral analysis of the polynomially solvable special case in which the associated bipartite graph is a path we give a mathcal o n 3 time optimization algorithm and a mathcal o n 2 size linear programming extended formulation we describe a new class of inequalities that we call path modular inequalities we give two distinct proofs of their validity the first one is direct and crucially relies on sub and super modularity of an associated set function the second proof is by showing that the projection of the extended linear programming formulations onto the original variable space yields exactly the polyhedron described by the path modular inequalities thus we also show that these inequalities suffice to describe the convex hull of the set of feasible solutions

lifted flow cover inequalities for mixed 0 1 integer programs we investigate strong inequalities for mixed 0 1 integer programs derived from flow cover inequalities flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities however because of the sequential nature of the standard lifting techniques and the complexity of the optimization problems that have to be solved to obtain lifting coefficients lifting of flow cover inequalities is computationally very demanding we present a computationally efficient way to lift flow cover inequalities based on sequence independent lifting techniques and give computational results that show the effectiveness of our lifting procedures

the cut off covering spectrum we introduce the r cut off covering spectrum and the cut off covering spectrum of a metric space or a riemannian manifold the spectra measure the sizes of localized holes in the space and are defined using covering spaces called covers and r cut off covers they are investigated using 6 homotopies which are homotopies via grids whose squares are mapped into balls of radius on locally compact spaces we prove that these new spectra are subsets of the closure of the length spectrum we prove the r cut off covering spectrum is almost continuous with respect to the pointed gromov hausdorff convergence of spaces and that the cut off covering spectrum is also relatively well behaved this is not true of the covering spectrum defined in our earlier work which was shown to be well behaved on compact spaces we close by analyzing these spectra on riemannian manifolds with lower bounds on their sectional and ricci curvature and their limit spaces

nonnegative curvature and cobordism type we show that in each dimension n 4k k 2 there exist infinite sequences of closed simply connected riemannian n manifolds with nonnegative sectional curvature and mutually distinct oriented cobordism type

strongly nonnegative curvature we prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition their curvature operator can be modified with a 4 form to become positive semidefinite

approximate extended formulations mixed integer programming mip formulations are typically tightened through the use of a separation algorithm and the addition of violated cuts using extended formulations involving new variables is a possible alternative but this often results in prohibitively large mips where even the linear programming relaxations are hard or impossible to solve in this paper we demonstrate how in certain cases it is possible and interesting to define approximate extended formulations in all the examples considered our description involves a single control parameter k large values of k result in strong but large formulations in particular when k takes its maximum value the approximate formulation is identical to the complete extended formulation through this approximation parameter the user has control over the tradeoff between the strength and the size of the formulation

conditions for nonnegative curvature on vector bundles and sphere bundles this paper addresses cheeger and gromoll s question of which vector bundles admit a complete metric of nonnegative curvature and relates their question to the issue of which sphere bundles admit a metric of positive curvature we show that any vector bundle which admits a metric of nonnegative curvature must admit a connection a tensor and a metric on the base space which together satisfy a certain differential inequality on the other hand a slight sharpening of this condition is sufficient for the associated sphere bundle to admit a metric of positive curvature our results sharpen and generalize walschap and strake s conditions under which a vector bundle admits a connection metric of nonnegative curvature

projecting an extended formulation for mixed integer covers on bipartite graphs we consider the mixed integer version of bipartite vertex cover this is equivalent to a mixed integer network dual model introduced recently that generalizes several mixed integer sets arising in production planning we derive properties of inequalities that are valid for the convex hull of the mixed integer bipartite covers by projecting an extended formulation onto the space of the original variables this permits us to give a complete description of the facet inducing inequalities of the double mixing set and of the continuous mixing set with flows two mixed integer sets that generalize several models studied in the literature