nonlinear kolmogorov equations in infinite dimensional spaces the backward stochastic differential equations approach and applications to optimal control solutions of semilinear parabolic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations parabolic equations are intended in a mild sense that reveals to be suitable also towards applications to optimal control

optimal control problem for nonlinear stochastic difference second kind volterra equations abstract the problems of stability and optimal control for stochastic difference equations are receiving important attention now see for example 1 6 in this paper the necessary optimality condition for nonlinear stochastic difference second kind volterra equation are constructed for stochastic integral functional equations analogous results were obtained in 7

topological properties of eschenburg spaces and 3 sasakian manifolds we examine topological properties of the seven dimensional positively curved eschenburg biquotients and find many examples which are homeomorphic but not diffeomorphic a special subfamily of these manifolds also carries a 3 sasakian metric among these we construct a pair of 3 sasakian spaces which are diffeomorphic to each other thus giving rise to the first example of a manifold which carries two non isometric 3 sasakian metrics

computational methods for discrete boundary value problems on considere des methodes constructives pour les problemes aux valeurs limites discrets y t 1 g t y t t i n f y 0 ou y et g t y sont des vecteurs a n dimensions i n est l ensemble des points discrets 0 1 n f y est un operatuer de b i n dans r n et b i n est l espace de toutes les fonctions n vectorielles reelles definies sur i n

initial value methods for discrete boundary value problems motivated by the work of angel and kalaba s and golberg 8 on twopoint boundary value problems for difference equations in this paper we shall study several initial value methods for multipoint boundary value problems for linear and nonlinear difference systems we generalize method of adjoints goodman and lance 9 method of complementary functions tifford 17 d an method of particular solutions miele 111 proposed for two point continuous problems to linear multipoint problems for difference equations next we show that method of adjoint and method of complementary functions can be used in an iterative way to solve nonlinear problems to justify the results obtained here several examples are also given the methods discussed here are similar to our earlier work l 3 on continuous problems we will take up the initial adjusting method of ojika kasue and welsh 12 15 18 for the discrete case later throughout the paper we shall assume the existence uniqueness and stability properties of solutions of the problems discussed and refer to 4 for some of these questions

forward backward and symmetric stochastic integration we define three types of non causal stochastic integrals forward backward and symmetric our approach consists in approximating the integrator two optics are considered the first one is based on traditional usual stochastic calculus and the second one on wiener distributions

a diffeomorphism classification of 7 dimensional homogeneous einstein manifolds with mathrm su 3 times mathrm su 2 times mathrm u 1 symmetry on donne des exemples qui montrent que des espaces homogenes homeomorphes ne sont pas necessairement diffeomorphes

computational methods for discrete boundary value problems ii abstract for a given nonlinear discrete system together with nonlinear boundary conditions we use newton s iterative scheme and provide sufficient conditions for its convergence to the unique solution then the computational aspects of this iterative method on a floating point system are discussed this includes necessary and sufficient conditions for the convergence of the approximate newton s method an oscillatory state and the stopping criterion to make the analysis widely applicable all the results are proved component wise

surgery and duality surgery as developed by browder kervaire milnor novikov sullivan wall and others is a method for comparing homotopy types of topological spaces with difieomorphism or homeomorphism types of manifolds of dimension 5 in this paper a modiflcation of this theory is presented where instead of flxing a homotopy type one considers a weaker information roughly speaking one compares n dimensional compact manifolds with topological spaces whose k skeletons are flxed where k is at least n 2 a particularly attractive example which illustrates the concept is given by complete intersections by the lefschetz hyperplane theorem a complete intersection of complex dimension n has the same n skeleton as n and one can use the modifled theory to obtain information about their difieomorphism type although the homotopy classiflcation is not known the theory reduces this classiflcation result to the determination of complete intersections in a certain bordism group this was under certain restrictions carried out in tr the restrictions are if d d1 dr is the total degree of a complete intersection x n1 dr of complex dimension n then the assumption is that for all primes p with p pi1 n 1 the total degree d is divisible by p 2n 1 2pi1 1 theorem a two complete intersections x n

the topology of certain 3 sasakian 7 manifolds we calculate the integer cohomology ring and stable tangent bundle of a family of compact 3 sasakian 7 manifolds constructed by boyer galicki mann and rees the most important part of the cohomology ring is a previously undetermined torsion group that we describe explicitly and whose order we compute there is a surprising connection with the combinatorics of trees

a homotopy classification of certain 7 manifolds this paper gives a homotopy classification of wallach spaces and a partial homotopy classification of closely related spaces obtained by free sl actions on su 3 and on s3 x s5

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

boundary value problems for general discrete systems on infinite intervals this paper continues the study of boundary value problems for a discrete system which in particular includes the prototype equation x k 1 f k x k equations with finite as well as infinite delays equations of neutral type and the discrete integral equations of volterra type while the finite discrete interval case regular as well as at resonance has been discussed in 1 2 in the present paper the infinite interval case will be addressed

asymptotic behavior of volterra difference equations abstract in this paper we investigate boundedness and stability properties of some classes of discrete volterra equations the main tool in this work is the use of a representation formula which allows us to express the solution of discrete volterra equations in terms of the resolvent matrix of the corresponding system of volterra difference equations

infinite horizon backward stochastic differential equations and elliptic equations in hilbert spaces solutions of semilinear elliptic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations the backward equation is considered on an infinite time horizon and a suitable growth condition replaces the final condition elliptic equations are intended in a mild sense suitable also for applications to optimal control we finally notice that due to the lack of smoothing properties the elliptic partial differential equation considered here could not be treated by analytic methods

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

stochastic optimal control problems and parabolic equations in banach spaces we consider stochastic optimal control problems in banach spaces these problems are related to nonlinear controlled equations with dissipative nonlinearity and are treated via the backward stochastic differential equation approach which also allows us to solve in a mild sense hamilton jacobi bellman equations in banach spaces we apply the results to controlled stochastic heat and wave equations with a cost functional that is well defined on continuous functions and to delay equations in spaces of p integrable functions

the classification of compact simply connected biquotients in dimension 6 and 7 we classify all compact simply connected biquotients of dimension 6 and 7 for each 6 dimensional biquotient all pairs of groups g h and homomorphisms h rightarrow g times g giving rise to it are classified

finite group actions on kervaire manifolds abstract let m k 4 k 2 be the kervaire manifold a closed piecewise linear pl manifold with kervaire invariant 1 and the same homology as the product s 2 k 1 s 2 k 1 of spheres we show that a finite group of odd order acts freely on m k 4 k 2 if and only if it acts freely on s 2 k 1 s 2 k 1 if m k is smoothable then each smooth structure on m k admits a free smooth involution if k 2 j 1 then m k 4 k 2 does not admit any free top involutions free exotic pl involutions are constructed on m k 30 m k 62 and m k 126 each smooth structure on m k 30 admits a free z 2 z 2 action

free actions of finite groups on s n x s n let p be an odd prime we construct a non abelian extension of s by z p z p and prove that any finite subgroup of acts freely and smoothly on s2p 1 s2p 1 in particular for each odd prime p we obtain free smooth actions of infinitely many non metacyclic rank two p groups on s2p 1 s2p 1 these results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres

nonorientable 4 manifolds with fundamental group of order 2 in this paper we classify nonorientable topological closed 4 manifolds with fundamental group z 2 up to homeomorphism our results give a complete list of such manifolds and show how they can be distinguished by explicit invariants including characteristic numbers and the ti invariant associated to a normal pinc structure by the spectral asymmetry of a certain dirac operator in contrast to the oriented case there exist homotopy equivalent nonorientable topological 4 manifolds which are stably homeomorphic after connected sum with s2 x s2 but not homeomorphic

linearized stability analysis of discrete volterra equations abstract various different types of stability are defined in a unified framework for discrete volterra equations of the type x n f n n j 0 k n j x n n 0 under appropriate assumptions stability results are obtainable from those valid in the linear case k n j x n b n j x j and a linearized stability theory is studied here by using the fundamental and resolvent matrices several necessary and sufficient conditions for stability are obtained for solutions of the linear equation by considering the equations in various choices of banach space b the elements of which are sequences of vectors x n f n e d b n j e d e d n j 0 etc we show that the theory including a number of new results as well as results already known can be presented in a systematic framework in which results parallel corresponding results for classical volterra integral equations of the second kind

on the number of positive solutions of a nonlinear algebraic system abstract in this paper we study the nonlinear algebraic system of the form e x af x where 0 is a parameter x and f x denote the column vectors col x 1 x 2 x n and col f 1 x 1 f 2 x 2 f n x n respectively with fk r r k 1 2 n 1 n and n is a positive integer a aij n n is an n n matrix and all its entries are positive numbers many problems in various areas such as difference equations boundary value problems dynamical networks stochastic process numerical analysis etc can be converted to system e applying fixed point theorems we prove results on existence uniqueness multiplicity and nonexistence of positive solutions for e

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

real killing spinors and holonomy we give a description of all complete simply connected riemannian manifolds carrying real killing spinors furthermore we present a construction method for manifolds with the exceptional holonomy groupsg2 and spin 7

stability of difference volterra equations direct liapunov method and numerical procedure abstract a procedure to construct liapunov functionals for discrete volterra equations is proposed using this procedure stability conditions are derived for general volterra difference equations some applications of the proposed procedure for obtaining stability conditions for linear multistep methods for volterra integro differential equations are presented

homological stability for moduli spaces of high dimensional manifolds i s galatius was partially supported by nsf grants dms 1105058 and dms 1405001 the european research council erc under the european union s horizon 2020 research and innovation pro gramme grant agreement no 682922 as well as the danish national research foundation through the centre for symmetry and deformation dnrf92 and erc 682992 o randal williams was partially supported by the herchel smith fund and epsrc grant ep m027783 1

optimal pinching constants of odd dimensional homogeneous spaces in this article we compute the pinching constants of all invariant riemannian metrics on the berger space b13 su 5 sp 2 2s1 and of all invariant u 2 biinvariant riemannian metrics on the aloff wallach space w7 1 1 su 3 s1 1 1 we prove that the optimal pinching constants are precisely in both cases so far b13 and w7 1 1 were only known to admit riemannian metrics with pinching constants we also investigate the optimal pinching constants for the invariant metrics on the other aloff wallach spaces w7 k l su 3 s1 k l our computations cover the cone of invariant t2 biinvariant riemannian metrics this cone contains all invariant riemannian metrics unless k l 1 it turns out that the optimal pinching constants are given by a strictly increasing function in k l 0 1 thus all the optimal pinching constants are in order to determine the extremal values of the sectional curvature of an invariant riemannian metric on w7 k l we employ a systematic technique which can be applied to other spaces as well the computation of the pinching constants for b13 is reduced to the curvature computation for two proper totally geodesic submanifolds one of them is diffeomorphic to 3 2 and inherits an sp 2 invariant riemannian metric and the other is w7 1 1 embedded as recently found by taimanov this approach explains in particular the coincidence of the optimal pinching constants for w7 1 1 and the berger space b13

positive ricci curvature on highly connected manifolds for k geq 2 let m 4k 1 be a closed 2k 2 connected manifold if k equiv 1 mod 4 assume further that m is 2k 1 parallelisable then there is a homotopy sphere sigma 4k 1 such that m sharp sigma admits a ricci positive metric this follows from a new description of these manifolds as the boundaries of explicit plumbings

ito formula for c 1 functions of semimartingales we establish an ito formula forc1 functions of processes whose time reversal are semimartingales and forc1 functions whose first derivatives are holder continuous of any parameter and the process comes out from a stochastic flow of homeomorphism

exotic g_2 g2 manifolds we exhibit the first examples of closed 7 dimensional riemannian manifolds with holonomy g 2 that are homeomorphic but not diffeomorphic these are also the first examples of closed ricci flat manifolds that are homeomorphic but not diffeomorphic the examples are generated by applying the twisted connected sum construction to fano 3 folds of picard rank 1 and 2 the smooth structures are distinguished by the generalised eells kuiper invariant introduced by the authors in a previous paper

optimal control problems for stochastic delay evolution equations in banach spaces we consider the optimal control for a banach space valued stochastic delay evolution equation the existence and uniqueness of the mild solution for the associated hamilton jacobi bellman equations are obtained by means of backward stochastic differential equations an application to optimal control of stochastic delay partial differential equations is also given

monotone iterative methods for a general class of discrete boundary value problems abstract in this paper we shall offer comparison results as well as monotone iterative schemes for the construction of solutions to a very general class of discrete boundary value problems the discrete system we consider includes in particular the n th order prototype systems finite as well as infinite discrete delay equations and discrete integral equations further the boundary conditions we consider include the initial terminal periodic and transport type problems numerical examples illustrating the usefulness of the proposed schemes to a variety of boundary value problems are also included

parallel monotone iterative relaxation methods for a class of two dimensional discrete boundary value problems abstract a parallel monotone iterative relaxation method for a class of two dimensional discrete boundary value problems is established and the sequence of iterations is shown to converge monotonically either from above or below to a solution of the problem this monotone convergence results yields a parallel computational algorithm as well as an existence comparison result for the solutions to compute the sequence of iterations the thomas algorithm can be used in the same fashion as for one dimensional problem the existence and comparison results of the upper and lower solutions are given the local as well as global existence uniqueness of the solution are obtained the global convergence of the iterations is investigated and the influence of the parameters on the rate of convergence of the iterations is analyzed numerical results are given to corroborate the analytical results

on nonlinear discrete boundary value problems mo x 1 x n 0 1 4 where n is a fixed positive integer 8 represent n by n constant matrices a t is an n by n matrix for each te 0 l h t belongs to ir for each nonnegative integer t u is in r 1 is a real parameter f and g are continuously differentiable f r i ir and g r i r for systems 1 1 1 2 we prove that if the linear system x t l a l x t h t 1 5 subject to

discrete systems on infinite intervals abstract we study superlinear boundary value problems for a discrete system which in particular includes the prototype equation x k 1 f k x k equations with finite as well as infinite delays equations of neutral type and the discrete integral equations of volterra type

diffeomorphism type of the berger space so 5 so 3 we compute the eells kuiper invariant of the berger manifold so 5 so 3 and determine that it is diffeomorphic to the total space of an s 3 bundle over s 4 this answers a question raised by k grove and w ziller

the stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls this paper is concerned with a pontryagin s maximum principle for stochastic optimal control problems of delay systems with random coefficients involving both continuous and impulse controls this kind of control problems is motivated by some interesting phenomena arising from economics and finance we establish a necessary maximum principle and a sufficient verification theorem by virtue of the duality and the convex analysis to explain the theoretical results we apply them to a production and consumption choice problem

stochastic systems with memory and jumps abstract stochastic systems with memory naturally appear in life science economy and finance we take the modelling point of view of stochastic functional delay equations and we study these structures when the driving noises admit jumps our results concern existence and uniqueness of strong solutions estimates for the moments and the fundamental tools of calculus such as the ito formula we study the robustness of the solution to the change of noises specifically we consider the noises with infinite activity jumps versus an adequately corrected gaussian noise the study is presented in two different frameworks we work with random variables in infinite dimensions where the values are considered either in an appropriate l p type space or in the space of cadlag paths the choice of the value space is crucial from the modelling point of view as the different settings allow for the treatment of different models of memory or delay our techniques involve tools of infinite dimensional calculus and the stochastic calculus via regularisation

stability of a general class of difference systems abstract in this paper we shall investigate several notions of stability of a general ordinary difference system using the prolific second method of lyapunov

kreck stolz invariants for quaternionic line bundles we generalise the kreck stolz invariants s2 and s3 by defining a new invariant the t invariant for quaternionic line bundles over closed spin manifolds m of dimension 4k 1 with h 3 m q h 4 m q 0 the t invariant classifies closed smooth oriented 2 connected rational homology 7 spheres up to almost diffeomorphism and detects exotic homeomorphisms between such manifolds the t invariant also provides information about quaternionic line bundles over a fixed manifold and we use it to give a new proof of a theorem of feder and gitler about the values of the second chern classes of quaternionic line bundles over hp k the t invariant for s 4k 1 is closely related to the adams e invariant on the 4k 5 stem

periodic solutions of a single species discrete population model with periodic harvest stock abstract we discuss a discrete population model describing single species growth with periodic harvest stock the theory of coincidence degree is applied to show that the model equation admits two periodic solutions under minor technical assumptions we show that one of these two periodic solutions is positive and attracts almost all positive solutions

optimal control problem for stochastic evolution equations in hilbert spaces in this article we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in hilbert spaces and the cost functional has a quadratic growth the existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value as an application an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

the topology of stein fillable manifolds in high dimensions i we continue our study of contact structures on manifolds of dimension at least five using complex surgery theory we show that in each dimension 2q 1 3 there are maximal almost contact manifolds to which there is a stein cobordism from any other 2q 1 dimensional contact manifold we show that the product m x s 2 admits a weakly fillable contact structure provided m admits a weak symplectic filling w omega with omega pi 2 m 0 we also study the connection between stein fillability and connected sums we give examples of almost contact manifolds for which the connected sum is stein fillable while the components are not concerning obstructions to stein fillings we show that the 8k 1 dimensional sphere has an almost contact structure which is not stein fillable once k 1 as a consequence we deduce that any highly connected almost contact 8k 1 manifold with k 1 admits an almost contact structure which is not stein fillable

general theorems for stability and boundedness for nonlinear functional discrete systems abstract we consider the nonlinear functional discrete system x n 1 g n x s 0 s n def g n x and obtain sufficient conditions for stability properties of the zero solution and for the boundedness of solutions

a diffeomorphism classification of manifolds which are like projective planes we give a complete diffeomorphism classification of 1 connected closed manifolds m with integral homology h m z z z provided that dim m 6 4

g2 manifolds and associative submanifolds via semi fano 3 folds we construct many new topological types of compact g 2 manifolds i e riemannian 7 manifolds with holonomy group g 2 to achieve this we extend the twisted connected sum construction first developed by kovalev and apply it to the large class of asymptotically cylindrical calabi yau 3 folds built from semi fano 3 folds constructed previously by the authors in many cases we determine the diffeomorphism type of the underlying smooth 7 manifolds completely we find that many 2 connected 7 manifolds can be realised as twisted connected sums in a variety of ways raising questions about the global structure of the moduli space of g 2 metrics many of the g 2 manifolds we construct contain compact rigid associative 3 folds which play an important role in the the higher dimensional enumerative geometry gauge theory calibrated submanifolds approach to defining deformation invariants of g 2 metrics by varying the semi fanos used to build different g 2 metrics on the same 7 manifold we can change the number of rigid associative 3 folds we produce

stochastic equations with delay optimal control via bsdes and regular solutions of hamilton jacobi bellman equations we consider an ito stochastic differential equation with delay driven by brownian motion whose solution by an appropriate reformulation defines a markov process x with values in a space of continuous functions mathbf c with generator mathcal l we then consider a backward stochastic differential equation depending on x with unknown processes y z and we study properties of the resulting system in particular we identify the process z as a deterministic functional of x we next prove that the forward backward system provides a suitable solution to a class of parabolic partial differential equations on the space mathbf c driven by mathcal l and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects we also include applications to optimal stochastic control of differential equation with delay in particular we characterize optimal controls as feedback laws in terms of the process x