remarks on parameter identification ii convergence of the algorithm on demontre un theoreme de convergence pour l algorithme d identification du coefficient a x de l equation parabolique u t a x u x x f x t et on donne des resultats de stabilite

hochschild cohomology and group actions given a finite group action on a suitably enhanced triangulated category linear over a field we establish a formula for the hochschild cohomology of the category of invariants assuming the order of the group is coprime to the characteristic of the base field the formula shows that the cohomology splits canonically with one summand given by the invariant subspace of the hochschild cohomology of the original category we also prove that serre functors act trivially on hochschild cohomology and combine this with our formula to give a useful mechanism for computing the hochschild cohomology of fractional calabi yau categories

quillen stratification for the steenrod algebra let a be the mod 2 steenrod algebra its cohomology h a f2 exta f2 f2 is the e2 term of the adams spectral sequence converging to the 2 component of ir ss the stable homotopy groups of spheres as such this cohomology algebra has been studied extensively for over thirty five years there have been a number of nice structural results and degree by degree computations many of which have implications for rxs0 information about the ring structure of h a f2 though has been harder to come by analogously in the study of the stable homotopy groups of spheres there were many stem by stem calculations before nishida proved in 14 that every element in e3i o i7r so was nilpotent since 7r8s0 z and rs0 o when i 0 in his landmark 1971 paper 18 quillen gave a description of the cohomology algebra h g k modulo nilpotent elements as an inverse limit of the cohomology algebras of the elementary abelian p subgroups of g this has led for instance to the work of benson carlson et al on the theory of varieties for kg modules and in general to deep structural information about modular representations of finite groups in this paper we describe the cohomology of the steenrod algebra modulo nilpotent elements according to the recipe in quillen s result as the inverse limit of the cohomology rings of the elementary abelian sub hopf algebras of a we hope that this leads to further study in several directions first of all according to the point of view of axiomatic stable homotopy theory in 7 the representation theory of any cocommutative hopf algebra has formal similarities to stable homotopy theory viewed this way our main result is an

representation theoretic support spaces for finite group schemes we introduce the space p g of abelian p points of a finite group scheme over an algebraically closed field of characteristic p 0 we construct a homeomorphism g p g proj g from p g to the projectivization of the cohomology variety for any finite group g for an elementary abelian p group respectively an infinitesimal group scheme p g can be identified with the projectivization of the variety of cyclic shifted subgroups resp variety of 1 parameter subgroups for a finite dimensional g module m g restricts to a homeomorphism p g m proj g m thereby giving a representation theoretic interpretation of the cohomological support variety

nilpotence and descent in equivariant stable homotopy theory abstract let g be a finite group and let f be a family of subgroups of g we introduce a class of g equivariant spectra that we call f nilpotent this definition fits into the general theory of torsion complete and nilpotent objects in a symmetric monoidal stable category with which we begin we then develop some of the basic properties of f nilpotent g spectra which are explored further in the sequel to this paper in the rest of the paper we prove several general structure theorems for categories of module spectra over objects such as equivariant real and complex k theory and borel equivariant mu using these structure theorems and a technique with the flag variety dating back to quillen we then show that large classes of equivariant cohomology theories for which a type of complex orientability holds are nilpotent for the family of abelian subgroups in particular we prove that equivariant real and complex k theory as well as the borel equivariant versions of complex oriented theories have this property

existence and multiplicity for positive solutions of a multi point boundary value problem abstract in this paper we study the existence and multiplicity of positive solutions of multi point boundary value problems for systems of nonlinear second order ordinary differential equations

positive solutions for singular systems of three point boundary value problems this paper investigates the following singular systems of nonlinear second order three point boundary value problems u f t v email protected 0 1 v g t u email protected 0 1 u 0 v 0 0 u 1 email protected h v 1 email protected h where email protected 0 1 0 email protected 1 f and g may be singular at t 0 and or t 1 under some weaker conditions the existence of positive solutions is obtained by applying the fixed point theorem of cone expansion and compression two examples are then presented to demonstrate the application of our main results

transfinite adams representability abstract we consider the following problems in a well generated triangulated category t let be a regular cardinal and t t the full subcategory of compact objects is every functor h t op ab that preserves products of h t x t for some x in t is every natural transformation t x t t y t of the form t f t for some f x y in t if the answer to both questions is positive we say that t satisfies adams representability a classical result going back to brown and adams shows that the stable homotopy category satisfies 0 adams representability the case 0 is well understood thanks to the work of christensen keller and neeman in this paper we develop an obstruction theory to decide whether t satisfies adams representability we derive necessary and sufficient conditions of homological nature and we compute several examples in particular we show that there are rings satisfying adams representability for all 0 and rings which do not satisfy adams representability for any 0 moreover we exhibit rings for which the answer to both questions is no for all 2

realizability of algebraic galois extensions by strictly commutative ring spectra we discuss some of the basic ideas of galois theory for commutative s algebras originally formulated by john rognes we restrict our attention to the case of finite galois groups and to global galois extensions we describe parts of the general framework developed by rognes central roles are played by the notion of strong duality and a trace mapping constructed by greenlees and may in the context of generalized tate cohomology we give some examples where algebraic data on coefficient rings ensures strong topological consequences we consider the issue of passage from algebraic galois extensions to topological ones by applying obstruction theories of robinson and goerss hopkins to produce topological models for algebraic galois extensions and the necessary morphisms of commutative s algebras examples such as the complex k theory spectrum as a ko algebra indicate that more exotic phenomena occur in the topological setting we show how in certain cases topological abelian galois extensions are classified by the same harrison groups as algebraic ones and this leads to computable harrison groups for such spectra we end by proving an analogue of hilbert s theorem 90 for the units associated with a galois extension

positive solutions of n order and m order multi point singular boundary value system in this paper we study the existence of positive solutions for the nonlinear n order and m order multi point singular boundary value system by using the fixed point index theory the existence of positive solution and many positive solutions for nonlinear singular boundary value system are obtained

chromatic homotopy theory is asymptotically algebraic inspired by the ax kochen isomorphism theorem we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects we employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory more precisely we show that the ultraproduct of the e n p local categories over any non principal ultrafilter on the set of prime numbers is equivalent to the ultraproduct of certain algebraic categories introduced by franke this shows that chromatic homotopy theory at a fixed height is asymptotically algebraic

k theoretic obstructions to bounded t structures schlichting conjectured that the negative k groups of small abelian categories vanish and proved this for noetherian abelian categories and for all abelian categories in degree 1 the main results of this paper are that mathrm k 1 e vanishes when e is a small stable infty category with a bounded t structure and that mathrm k n e vanishes for all n geqslant 1 when additionally the heart of e is noetherian it follows that barwick s theorem of the heart holds for nonconnective k theory spectra when the heart is noetherian we give several applications to non existence results for bounded t structures and stability conditions to possible k theoretic obstructions to the existence of the motivic t structure and to vanishing results for the negative k groups of a large class of dg algebras and ring spectra

positive solutions of three point boundary value problems for systems of nonlinear second order ordinary differential equations abstract in this paper we study the three point boundary value problems for systems of nonlinear second order ordinary differential equations of the form u f t v t 0 1 v g t u t 0 1 u 0 v 0 0 u u 1 v v 1 under some conditions we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones

homotopy coherent adjunctions and the formal theory of monads abstract in this paper we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its n arrows using a graphical calculus that we develop here the hom spaces are appropriately fibrant indeed are nerves of categories which indicates that all of the expected coherence equations in each dimension are present to justify our terminology we prove that any adjunction of quasi categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contractible kan complexes we extract several simplicial functors from the free homotopy coherent adjunction and show that quasi categories are closed under weighted limits with these weights these weighted limits are used to define the homotopy coherent monadic adjunction associated to a homotopy coherent monad we show that each vertex in the quasi category of algebras for a homotopy coherent monad is a codescent object of a canonical diagram of free algebras to conclude we prove the quasi categorical monadicity theorem describing conditions under which the canonical comparison functor from a homotopy coherent adjunction to the associated monadic adjunction is an equivalence of quasi categories our proofs reveal that a mild variant of beck s argument is all in the weights much of it independent of the quasi categorical context

three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second order ordinary differential equations abstract in this paper we consider the existence of at least three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second order ordinary differential equations the associated green s function for the boundary value problems is first given the proofs of our main results are based upon the leggett williams fixed point theorem finally we give an example to demonstrate our result

the galois group of a stable homotopy theory abstract to a stable homotopy theory a presentable symmetric monoidal stable category we naturally associate a category of finite etale algebra objects and using grothendieck s categorical machine a profinite group that we call the galois group we then calculate the galois groups in several examples for instance we show that the galois group of the periodic e algebra of topological modular forms is trivial and that the galois group of k n local stable homotopy theory is an extended version of the morava stabilizer group we also describe the galois group of the stable module category of a finite group a fundamental idea throughout is the purely categorical notion of a descendable algebra object and an associated analog of faithfully flat descent in this context

the cohomology algebras of finite dimensional hopf algebras the cohomology algebra of a finite dimensional graded connected cocommutative biassociative hopf algebra over a field k is shown to be a finitely generated k algebra counterexamples to the analogue of a result of quillen that nonnilpotent cohomology classes should have nonzero restriction to some abelian sub hopf algebra are constructed but an elementary proof of the validity of this detection principle for the special case of finite sub hopf algebras of the mod 2 steenrod algebra is given as an application an explicit formula for the krull dimension of the cohomology algebras of the finite skeletons of the mod 2 steenrod algebra is given if a is an augmented algebra over the field k the cohomology algebra h a is defined as exta k k if a is finite dimensional as a k vector space h a may still fail to be a finitely generated k algebra e g lofwall 12 however if a k g the group algebra of a finite group then h a is finitely generated evens 6 and venkov 21 cocommutative hopf algebras are one generalization of group algebras and connected graded cocommutative hopf algebras are closely analogous to finite p groups it is the intent of this work to push this analogy as far as possible the first positive result is that finite generation holds in this context also all hopf algebras mentioned in this work are biassociative and either commutative or cocommutative theorem a if a is a finite dimensional graded connected cocommutative k hopf algebra then h a exta k k is a finitely generated k algebra the strategy of the proof is essentially that developed by adams 1 and liulevicius 10 for the computation of the cohomology of small sub hopf algebras of the steenrod algebras one resolves a as a sequence of iterated central extensions of hopf algebras each such extension has an associated spectral sequence and some hold on the differentials is provided by the transgression theorem relating steenrod operations on the fiber and base the only philosophical difference between the present plan and that of 1 10 is that precise computational results are sacrificed for the sake of generality received by the editors december 10 1979 ams mos subject classifications 1970 primary 55g10 55f35 55h99 55j99 57f05 18a24 18h15

a note on the cohomology of finite dimensional cocommutative hopf algebras abstract in the context of finite dimensional cocommutative hopf algebras we prove versions of various group cohomology results the quillen venkov theorem on detecting nilpotence in group cohomology chouinard s theorem on determining whether akg module is projective by restricting to elementary abelianp subgroups ofg and quillen s theorem which identifies the cohomology ofg modulo nilpotent elements this last result is only proved for graded connected hopf algebras

positive solvability of systems of nonlinear hammerstein integral equations the paper deals with the existence of positive continuous solutions to systems of nonlinear hammerstein integral equations the main tool used in the proofs is fixed point index theory in a cone the results obtained here are essentially different from existing ones in the literature