existence of positive solutions to some singular and nonsingular second order boundary value problems abstract the topological transversality theorem is employed to study existence of solutions to a variety of singular and nonsingular second order boundary value problems of the form y f t y y 0 0 t t 0 t 1 y 0 and or y 0 the principal tools are the arzela ascoli theorem essential maps and a priori bounds on solutions

approachability and fixed points for non convex set valued maps this paper studies the approachability of upper semicontinuous nonconvex set valued maps by single valued continuous functions and its applications to fixed point theory the use of approximate selections was initiated by j von neumann 41 in the proof of his well known minimax principle this remarkable property was extended to the normed space setting by cellina s and cellina and lasota 9 in the context of a degree theory for upper semicontinuous convex compact valued maps it also holds for maps with compact contractible values defined on finite polyhedra of iw ma cole11 40 and more generally defined on compact anrs mclennan 39 it was recently taken up by gbrniewicz granas and kryszewski 22 241 in the context of an index theory for non convex maps defined on compact anrs the content of this paper is divided into three parts the first part is devoted to background material concerning the types of spaces and set valued maps studied herein in the second part we define the abstract class d of approachable set valued maps that is the class of maps a x y

prinjective modules reflection functors quadratic forms and auslander reiten sequences let a b be artinian rings and let a m b be an a b bimodule which is a finitely generated left a module and a finitely generated right b module a right a m b prinjective module is a finitely genrated module x r x a x b x a a m b x b over the triangular matrix ring r a o a m b b such that x a is a projective a module x b is an injective b module and is a b homomorphism we study the category prin r b a of right a m b prinjective modules it is an additive krull schmidt subcategory of mod r closed under extensions

a reduction functor tameness and tits form for a class of orders abstract let d be a complete discrete valuation domain which is an algebra over an algebraically closed field k we study in the paper a class of suborders of tiled d orders by means of a rational quadratic form q associated to and a finite poset i having exactly two maximal elements and criteria for finite lattice type and for tame lattice type of are given in terms of the form q and of the category i spr of socle projective k linear representations of the poset i the shape of auslander reiten quiver latt is described in corollary 3 2 a reduction functor h latt i spr preserving representation types is constructed it is shown in corollary 2 10 that for any poset i having exactly two maximal elements there exists a d order in our class an additive functor latt i spr preserving representation types and a poset isomorphism i i

internal duality for resolution of rings the following example provided the motivation for this paper let f be 8 the field with 8 elements and let f be its group of units which acts on f 8 8 u u via left multiplication the cohomology ring h f i f f 8 8 2 w xzr7 w x f x x x is of much interest to topologists a cs m it is a 2 1 2 3 straightforward exercise to construct minimal sets of homogeneous generators and relations for this ring there are 13 generators and 54 relations the surprising fact is that the minimal relations exhibit a certain internal duality

indecomposablea module summands inh v which are unstable algebras letp be a prime and denote bya the modp steenrod algebra we determine the indecomposablea module summands ofh p d f p which admit the structure of an unstablea algebra in fact it turns out that this is equivalent to the problem of determining those indecomposablea module summands which arise as the modp cohomology of a space or even a classifying space of a finite group we reduce this problem to one in modular representation theory namely for whichd andp is the projective cover of the trivial one dimensional gl d f p representationf p a permutation module our solution of this latter problem makes use of the classification of subgroups of gl d f p acting transitively on f p d 0 and hence depends on the classification of finite simple groups on feit thompson s odd order theorem ifp 2

note to the asymptotic behaviour of solutions of damped pendulum equations under forcing although the mathematical or the physical pendulum seems to be one of the most popular objects in modelling see the nice exposition lo and therefore no wonder that for the related autonomous equations studied by means of the planar geometrical methods a vast literature exists see the monographs 2 3 151 or more recently the beautiful modern article 4 there are still several white places of its study in the case of forcing recently the big activity is devoted for proving the existence of harmonics to its equation for the survey of results in this field see s lo or chaotic behaviour of the dynamics see e g 6 or the different approaches in 12 is nevertheless we have found only one reference 16 on the asymptotic behaviour of solutions of the pendulum type equations under forcing which is already 30 years old hence our aim is to extend the results in this field for the equation

using solution sets for solving boundary value problems for ordinary differential equations where f e c i x r r s c c z r consists of finding fixed points of a solution map e for a suitable linearized problem associated to bv see e g 2 5 6 71 in these problems the map x is either continuous and single valued or upper semicontinuous set valued with convex values these are rather strong restrictions because the set of solutions of a general boundary value problem may well be neither a singleton nor a convex set the case in which the solution map is upper semicontinuous but with nonnecessarily convex values is considered in this paper more precisely we want to consider the case when the values of x form an acyclic set and the case when the values of c are sets with a finite number of elements an application to the existence of periodic solutions for a licnard type equation is also given moreover an existence theorem for a boundary value problem of the following form f l x x x teicr x e r

prevalence of stable periodic solutions for duffing equations abstract we study the prevalence of stable periodic solutions of the duffing equations for external force which guarantees the existence of periodic solutions both the dissipative case and the conservative case are considered the nonlinearity may be periodic or satisfies the condition of landesman lazer type

bounded solutions for a forced bounded oscillator without friction under the validity of a landesman lazer type condition we prove the existence of solutions bounded on the real line together with their first derivatives for some second order nonlinear differential equation of the form u g u p t where the reaction term g is bounded the proof is variational and relies on a dual version of the nehari method for the existence of oscillating solutions to superlinear equations

a variational method for the existence of bounded solutions of a sublinear forced oscillator we prove that for every bounded and measurable forcing p t the differential equation ddot u u 1 3 p t has bounded solutions with arbitrarily large amplitude in general it is not possible to say that all solutions are bounded as shown by an example due to littlewood the proof is based on a variational method which can be seen as a dual version of nehari s method for boundary value problems on compact intervals

tame prinjective type and tits form of two peak posets ii throughout this paper we use the terminology and notation introduced w x in 5 7 11 13 in particular k is an algebraically closed field and we denote by i u a finite poset i e partially ordered set with respect to the partial order u we shall write i j if i u j and i j for the sake of simplicity we write i instead of i u we denote by max i the set of all maximal elements of i and i will be called an n peak poset if max i s n let ki be the incidence k algebra of the poset i that is ki is the k subalgebra of the full i i matrix algebra m k consisting of all i w x matrices l in m k such that l s 0 if i uu j in i u we recall from i j i i j w x 13 that the socle soc ki of the right ki module ki is projective and the global dimension of the algebra ki is finite we denote by mod ki the category of finitely generated right ki mod ules and by prin ki the full subcategory of mod ki consisting of all prinjective modules that is modules x in mod ki having a projective resolution of the form

a peak reduction functor for socle projective representations throughout this paper k is a fixed algebraically closed field given a finite poset i e partially ordered set j with the partial order u we denote by max j the set of all maximal elements of j with respect to the order u and we set jys j max j the poset j is said to be an r peak poset provided max j s r throughout this paper we suppose that j s 4 4 1 n p p and max j s p p without loss of generality 1 r 1 r we can assume that the partial order u in j is such that i u j implies 4 i f j in the natural order for all i j g 1 n w x we recall from 18 that the category j spr of peak j spaces or socle projective representations of j over the field k is defined as follows the objects of j spr are systems m s m of finite dimensional k vector j jg j spaces m such that j

on the qualitative theory of lower semicontinuous differential inclusions on etudie les proprietes qualitatives de l ensemble des solutions de caratheodory pour une inclusion differentielle de la forme x t f t x t t 0 t x 0 r n

a reduced tits quadratic form and tameness of three partite subamalgams of tiled orders let d be a complete discrete valuation domain with the unique maximal ideal p we suppose that d is an algebra over an algebraically closed field k and d p k subamalgam d suborders of a tiled d order are studied in the paper by means of the integral tits quadratic form q zn1 2n3 2 z a criterion for a subamalgam d order to be of tame lattice type is given in terms of the tits quadratic form q and a forbidden list 1 17 of minor d suborders of presented in the tables

on weakly non negative unit forms and tame algebras in this paper we consider integral quadratic forms q ip z of the form q x 2i x i jqijx xj and call them unit forms such a unit form q is said to be weakly non negative if q x q whenever jt2 0 that is whenever x o for i 1 n our main purpose is to formulate a criterion for weak non negativity of unit forms this question arises in various areas of representation theory in connection with the characterization of tame representation type in the representation theory of quivers 17 6 partially ordered sets 18 19 certain classes of bimodules 8 23 1 3 and certain classes of algebras 20 14 24 16 for each of these structures q there is a unit form qn naturally attached to q such that the weak non negativity of qn is an equivalent or at least a necessary condition for q to be of tame representation type a class of algebras for which tame representation type is equivalent to the weak non negativity of the associated unit form is described in the last section of this paper a general criterion for weak non negativity using determinants was given by zeldich 28 see also 21 another criterion was obtained by de la pena 21 our approach is similar to that in 11 where we gave a characterization of weakly positive unit forms based on the classification of the critical forms let us recall some notions given a unit form q ip tl and a non empty subset ix im of 1 n we denote by qt the quadratic form qd where d z it maps the cth natural basis vector ek onto eik such a unit form q is called the restriction of q to we call a unit form q on ip a hypercritical form if q is weakly non negative for every proper subset 1 n though q itself is not in these terms a unit form q is weakly non negative if and only if q admits no hypercritical restriction so we have to classify the hypercritical forms in 1 we show that they can be obtained by certain elementary transformations from the unit forms naturally attached to the hyperbolic graphs of fig 1 on p 50 in particular there are only finitely many hypercritical forms however their number is large by a simple process we reduce the problem to the classification of the stretched hypercritical forms in 2 we show that except for a few of them these are of the shape q x q x xn xn ixn where q q n d is critical in a similar way the unit forms attached to the hyperbolic graphs am dm em correspond to those

positive solutions with a complex behavior for superlinear indefinite odes on the real line abstract we show the existence of infinitely many positive solutions defined on the real line for the nonlinear scalar ode u a t a t u 3 0 where a is a periodic sign changing function and the parameter 0 is large such solutions are characterized by the fact of being either small or large in each interval of positivity of a in this way we find periodic solutions having minimal period arbitrarily large and bounded non periodic solutions exhibiting a complex behavior the proof is variational exploiting suitable natural constraints of nehari type