morse theory for periodic solutions of hamiltonian systems and the maslov index in this paper we prove morse type inequalities for the contractible 1 periodic solutions of time dependent hamiltonian differential equations on those compact symplectic manifolds m for which the symplectic form and the first chern class of the tangent bundle vanish over q m the proof is based on a version of infinite dimensional morse theory which is due to floer the key point is an index theorem for the fredholm operator which plays a central role in floer homology the index formula involves the maslov index of nondegenerate contractible periodic solutions this maslov index plays the same role as the morse index of a nondegenerate critical point does in finite dimensional morse theory we shall use this connection between floer homology and maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every i periodic solution has at least one floquet multiplier which is not equal to 1

rotation numbers eigenvalues and the poincare birkhoff theorem using the relation between rotation numbers and eigenvalues we prove the existence of nontrivial t periodic solutions having precise oscillatory properties for a class of asymptotically linear second order differential equations

multiple periodic solutions for discrete hamiltonian systems abstract several existence results of multiple periodic solutions for discrete hamiltonian system are established where the nonlinearity is either asymptotically linear or superlinear at infinity the approach used here is based on morse theory

fuzzy control on the basis of equality relations with an example from idle speed control the way engineers use fuzzy control in real world applications is often not coherent with an understanding of the control rules as logical statements or implications in most cases fuzzy control can be seen as an interpolation of a partially specified control function in a vague environment which reflects the indistinguishability of measurements or control values in this paper the authors show that equality relations turn out to be the natural way to represent such vague environments and they develop suitable interpolation methods to obtain a control function as a special case of our approach the authors obtain mamdani s model and can justify the inference mechanism in this model and the use of triangular membership functions not only for the reason of simplified computations and they can explain why typical fuzzy partitions are preferred the authors also obtain a criterion for reasonable defuzzification strategies the fuzzy control methodology introduced in this paper has been applied successfully in a case study of engine idle speed control for the volkswagen golf gti

periodic solutions of second order equations via rotation numbers abstract we consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems we show how a rotation number approach together with the poincare birkhoff theorem and the phase plane analysis of the spiral properties allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity these systems may also be resonant in particular our approach can be used to deal with the problems without both global existence of the cauchy problems associated to the equation and the sign assumption on f the typical example is a partially superlinear second order equation our main result generalizes some classical results of jacobowitz and hartman among others

similarities between powersets of terms generalisation of the foundational basis for many valued logic programming builds upon generalised terms in the form of powersets of terms a categorical approach involving set and term functors as monads allows for a study of monad compositions that provide variable substitutions and compositions thereof in this paper substitutions and unifiers appear as constructs in kleisli categories related to particular composed powerset term monads specifically we show that a frequently used similarity based approach to fuzzy unification is compatible with the categorical approach and can be adequately extended in this setting also some examples are included in order to illuminate the definitions

a self organizing fuzzy controller with a fixed maximum number of rules and an adaptive similarity factor abstract this paper proposes a self organizing fuzzy controller with a broad generality for minimum phase and stable systems the controller learns the rules on line with a minimum knowledge about the process the rule base is built and permanenetly actualized from input output real time data and has a fixed maximum number of rules fmnr an on line adaptive similarity factor implements a special efficient inference technique feedforward and predictive effect is introduced in fuzzification and defuzzification stages the defuzzification is carried out in such a way that as the learning process progresses the interval of the control becomes more and more accurate results are shown concerning simulations for non linear siso miso and mimo systems a nd a r eal e xperimental application using a low cost microcomputer

finite tame and wild actions of parabolic subgroups in gl v on certain unipotent subgroups abstract let p be a parabolic subgroup of some general linear group gl v where v is a finite dimensional vector space over an infinite field the group p acts by conjugation on its unipotent radical pu and via the adjoint action on p u the lie algebra of pu more generally we consider the action of p on the lth member of the descending central series of p u denoted by p l u let l p u denote the nilpotency class of pu in our main result we show that p acts on p l u with a finite number of orbits precisely when l p u 4 for l 0 or l p u 5 2l for l 1 moreover in case the field is algebraically closed we consider the modality mod p p l u of the action of p on p l u we show that mod p p l u grows linearly in the minimal cases which admit infinitely many orbits i e l p u 5 for l 0 or l p u 6 2l for l 1 whereas the corresponding modality grows quadratically in all other infinite cases these results are obtained by interpreting the orbits of p on p l u as isomorphism classes of good modules over certain quasi hereditary algebras and by a detailed inspection of the representation types of these algebras

a topological version of the poincare birkhoff theorem with two fixed points the main result of this paper gives a topological property satisfied by any homeomorphism of the annulus mathbb a mathbb s 1 times 1 1 isotopic to the identity and with at most one fixed point this generalizes the classical poincare birkhoff theorem because this property certainly does not hold for an area preserving homeomorphism h of mathbb a with the usual boundary twist condition we also have two corollaries of this result the first one shows in particular that the boundary twist assumption may be weakened by demanding that the homeomorphism h has a lift h to the strip tilde mathbb a mathbb r times 1 1 possessing both a forward orbit unbounded on the right and a forward orbit unbounded on the left as a second corollary we get a new proof of a version of the conley zehnder theorem in mathbb a if a homeomorphism of mathbb a isotopic to the identity preserves the area and has mean rotation zero then it possesses two fixed points

removing index 0 fixed points for area preserving maps of two manifolds using the method of free modifications developed by m brown and extended to area preserving homeomorphisms we prove the following fixed point removal theorem theorem let h m m be an orientation preserving area preserving homeomorphism of an orientable two manifold m having an isolated fixed point p of index 0 given any open neighborhood n of p such that n fix h p there exists an area preserving homeomorphism h such that i h h on m n and ii h is fixed point free on n two applications of this theorem are the second fixed point for the topological version of the conley zehnder theorem on the two torus and a new proof of the second fixed point for the poincare birkhoff fixed point theorem

fuzzy sets and vague environments abstract in this paper we propose a natural approach to handle imprecise numbers as they arise for example from measurements fuzzy sets turn out to be a canonical representation for such imprecise numbers that are induced by taking different tolerance or error bounds into account fuzzy sets are induced by scaling factors that describe the magnitude of the imprecision on the other the scaling factors can be derived from given fuzzy sets so that we have a correspondence between scaling factors and fuzzy sets when these concepts are applied to control problems the max min rule is rediscovered as an interpolation technique viewing fuzzy control as an interpolation technique in vague environments enables us to validate various concepts for the design and tuning of fuzzy controllers and also suggests new methods based on clear semantics

continuation theorems for periodic perturbations of autonomous systems it is first shown in this paper that whenever it exists the coincidence degree of the left hand member of an autonomous differential equation x g x 0 in the space of periodic functions with fixed period omega can be computed in terms of the brouwer degree of g this result provides efficient continuation theorems specially for omega periodic perturbations of autonomous systems extensions to differential equations in flow invariant enr s are also given

fuzzy homomorphisms of algebras in this paper we consider fuzzy relations compatible with algebraic operations which are called fuzzy relational morphisms in particular we aim our attention to those fuzzy relational morphisms which are uniform fuzzy relations called uniform fuzzy relational morphisms and those which are partially uniform f functions called fuzzy homomorphisms both uniform fuzzy relations and partially uniform f functions were introduced in a recent paper by us uniform fuzzy relational morphisms are especially interesting because they can be conceived as fuzzy congruences which relate elements of two possibly different algebras we give various characterizations and constructions of uniform fuzzy relational morphisms and fuzzy homomorphisms we establish certain relationships between them and fuzzy congruences and we prove homomorphism and isomorphism theorems concerning them we also point to some applications of uniform fuzzy relational morphisms

uniform fuzzy relations and fuzzy functions in this paper we introduce and study the concepts of a uniform fuzzy relation and a partially uniform f function we give various characterizations and constructions of uniform fuzzy relations and uniform f functions we show that the usual composition of fuzzy relations is not convenient for f functions so we introduce another kind of composition and we establish a mutual correspondence between uniform f functions and fuzzy equivalences we also give some applications of uniform fuzzy relations in approximate reasoning especially in fuzzy control and we show that uniform fuzzy relations are closely related to the defuzzification problem

resonance and rotation numbers for planar hamiltonian systems multiplicity results via the poincare birkhoff theorem abstract in the general setting of a planar first order system 0 1 u g t u u r 2 with g 0 t r 2 r 2 we study the relationships between some classical nonresonance conditions including the landesman lazer one at infinity and in the unforced case i e g t 0 0 at zero and the rotation numbers of large and small solutions of 0 1 respectively such estimates are then used to establish via the poincare birkhoff fixed point theorem new multiplicity results for t periodic solutions of unforced planar hamiltonian systems j u u h t u and unforced undamped scalar second order equations x g t x 0 in particular by means of the landesman lazer condition we obtain sharp conclusions when the system is resonant at infinity

nonautonomous nonlinear odes nonresonance conditions and rotation numbers abstract we deal with the t periodic problem associated with a nonlinear scalar differential equation like 1 x f t x 0 where for x 0 and x the nonlinearity f is assumed behave linearly with a time dependent coefficient we prove that the landesman lazer conditions at zero and at infinity possess a rotational effect on small and large in the phase plane solutions of 1 as a consequence we are able to generalize previous multiplicity results in the resonant case i e when the linearizations at zero and or at infinity are resonant through the use of the poincare birkhoff fixed point theorem

a compendium of fuzzy weak orders representations and constructions the present paper gives a state of the art overview of representation and construction results for fuzzy weak orders we do not assume that the underlying domain is finite instead we concentrate on results that hold in the most general case that the underlying domain is possibly infinite this paper presents three fundamental representation results each of which also provides a construction method i score function based representations ii inclusion based representations iii representations by decomposition into crisp linear orders and fuzzy equivalence relations which also facilitates a pseudo metric based construction

preservation of the maslov index along bifurcating branches of solutions of first order systems in rn abstract in this paper we study bifurcation from simple eigenvalues for systems of differential equations we prove the existence of global bifurcating branches of solutions on which the maslov index of suitable associated linear systems is preserved

fuzzy control of model car this paper discusses fuzzy control of a model car fuzzy control rules are derived by modelling an expert s driving actions experiments are performed using a model car with a sensing unit and a micro computer

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

on degenerations and extensions of finite dimensional modules abstract we derive a cancellation theorem for degenerations of modules that says in particular that projective or injective common direct summands can always be neglected combining the cancellation result with the existence of almost split sequences we characterize the orbit closure of a module living on preprojective components by the fact that the dimension of the homomorphism space to any other module does not decrease for representation directed algebras whence in particular for path algebras of dynkin quivers we provide an alternative proof which shows in addition that any minimal degenerationnofmcomes from an exact sequence with middle termmwhose end terms add up ton by a careful examination the same is true for degenerations of matrix pencils having used so far the existence of certain extensions to obtain degenerations we then turn the tables and use degenerations to produce a lot of interesting short exact sequences in particular we show that any non simple indecomposable over a tame quiver is an extension of an indecomposable and a simple

half eigenvalues of periodic sturm liouville problems we consider the nonlinear sturm liouville problem 1 pu qu au bu u in 0 2 2 u 0 u 2 pu 0 pu 2 where 1 p q l1 0 2 with p 0 a e on 0 2 a b l1 0 2 is a real parameter and u t max u t 0 for t 0 2 values of for which 1 2 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions the set of half eigenvalues will be denoted by h we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having certain nodal properties and we obtain certain spectral and degree theoretic properties associated with h these properties yield results on the existence and non existence of solutions of the problem 3 pu qu f t u h in 0 2 together with 2 where h l1 0 2 f 0 2 r r is a caratheodory function and the limits a t lim f t b t lim f t exist for a e t 0 2 when the functions a and b are constant the set h is closely related to the fucik spectrum of the problem and equivalent solvability results are obtained using the two approaches however when a and b are not constant the half eigenvalue approach yields stronger results

a note on a modified version of the poincare birkhoff theorem abstract we show by constructing a suitable area preserving homeomorphism without fixed points that the twist condition considered at zero and at infinity in a recent modified version of the poincare birkhoff fixed point theorem cannot be interchanged

positive periodic solutions to an indefinite minkowski curvature equation abstract we investigate the existence non existence multiplicity of positive periodic solutions both harmonic i e t periodic and subharmonic i e kt periodic for some integer k 2 to the equation u 1 u 2 a t g u 0 where 0 is a parameter a t is a t periodic sign changing weight function and g 0 0 is a continuous function having superlinear growth at zero in particular we prove that for both g u u p with p 1 and g u u p 1 u p q with 0 q 1 p the equation has no positive t periodic solutions for close to zero and two positive t periodic solutions a small one and a large one for large enough moreover in both cases the small t periodic solution is surrounded by a family of positive subharmonic solutions with arbitrarily large minimal period the proof of the existence of t periodic solutions relies on a recent extension of mawhin s coincidence degree theory for locally compact operators in product of banach spaces while subharmonic solutions are found by an application of the poincare birkhoff fixed point theorem after a careful asymptotic analysis of the t periodic solutions for

generalizations of the poincare birkhoff theorem in this article we prove some generalizations of the classical poincar birkhoff theorem on area preserving homeomorphisms of the annulus which satisfy a boundary twist condition the work of g d birkhoff on this theorem and its applications can be found in b1 b2 and chapter v of b3 a more modern treatment can be found in b n we prove a theorem for the open annulus a s1 x 0 1 since as we will see the theorem for the closed annulus can easily be obtained from this result for the open annulus however it is not immediately obvious what should be the analogue of the twist condition it turns out that the most general hypothesis and the most natural from the point of view of our proof involves the notion of positively and negatively returning disks for some lift of f to the covering space a r x 0 1 more precisely if f a a is a lift of f a a we will say that the e is a positively returning disk for f if there is an open disk u c a such that f u n u 0 and fp u n u k 0 forsome n k 0 here u k denotes the set x k t i x t e u thus u is disjoint from its image but under iteration by f returns to a positive translate of itself a negatively returning disk is defined similarly but with k 0 such that fl u n u 0 in section 2 we prove the following

representations and constructions of similarity based fuzzy orderings this contribution advocates the dissemination of similarity based fuzzy orderings a generalized concept of fuzzy orderings which appears to be unavoidable as soon as the classical correspondences between preorderings equivalence relations and orderings are taken into account in contrast to earlier papers on the topic this paper is focused on representation and construction results which once more demonstrate the feasibility of the proposed concept

a multiplicity result for a class of strongly indefinite asymptotically linear second order systems we prove a multiplicity result for a class of strongly indefinite nonlinear second order asymptotically linear systems with dirichlet boundary conditions the key idea for the proof is to bring together the classical shooting method and the maslov index of the linear hamiltonian systems associated to the asymptotic limits of the given nonlinearity

the bruhat chevalley order of parabolic group actions in general linear groups and degeneration for δ filtered modules abstract let k be an algebraically closed field and v a finite dimensional k space let gl v be the general linear group of v and p a parabolic subgroup of gl v now p acts on its unipotent radical p u and on p u lie p u the lie algebra of p u via the adjoint action more generally we consider the action of p on the l th member of the descending central series of p u denoted by p l u all instances when p acts on p l u for l 0 with a finite number of orbits are known in this note we give a complete combinatorial description of the closure relation on the set of p orbits on p l u i e the bruhat chevalley order for every finite case there is a canonical bijection between the set of p orbits on p l u and the set of isomorphism classes of filtered modules of a particular dimension vector e of a certain quasi hereditary algebra a t l these isomorphism classes in turn are given by the orbits of a reductive group g e on the variety r e of all a t l modules with filtration and dimension vector e the subcategory of a t l mod of all filtered a t l modules of dimension vector e is denoted by f e in our chief result theorem 1 1 we show that provided there is only a finite number of isomorphism classes of indecomposable modules in f the following three posets coincide 1 the bruhat chevalley order on the set of p orbits on p l u 2 the bruhat chevalley order on the set of g e orbits on r e 3 the poset opposite to the so called hom order on the set of isomorphism classes of f e the advantage of this hom order is that it is given purely by discrete invariants and that it can be computed explicitly for any given finite case theorem 1 1 then in turn allows us to explicitly determine the closure relations for the p orbits on p l u with the aid of this hom order we present some examples of hasse diagrams in an appendix

on the use of morse index and rotation numbers for multiplicity results of resonant bvps abstract motivated by a recent series of papers by k li and co workers 14 15 16 17 18 we consider the problem of the existence and multiplicity of solutions for the neumann or periodic bvps associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behavior of its solutions near zero and near infinity may be compared with the behavior of the solutions of two suitable linear systems one considered near zero and the other near infinity we show how a rotation number approach together with the poincare birkhoff theorem and a recent variant of it allows to obtain multiplicity results in terms of the gap between the morse indexes of the referred linear systems at zero and at infinity these systems may also be resonant when the gaps are sufficiently large our multiplicity results improve the ones obtained by variational methods in the quoted papers also our approach allows a description of the solutions obtained in terms of their nodal properties

an improvement of the poincare birkhoff fixed point theorem if g is a twist homeomorphism of an annulus a in the plane which leaves at most one point in the interior of a fixed then there is an essential simple closed curve in the interior of a which meets its image in at most one point hence the annular region bounded by this simple closed curve and the inside component of the boundary of a is mapped onto either a proper subset or a proper superset of itself

equality relations as a basis for fuzzy control abstract the aim of this paper is to introduce a fuzzy control model with well founded semantics in order to explain the concepts applied in fuzzy control assuming that the domains of the input and output variables for the process are endowed with equality relations that reflect the indistinguishability of values lying closely together the use of triangular and trapezoidal membership functions can be justified and max inference where is a t norm turns out to be a consequence of our model distinguishing between a functional and a relational view of the control rules it is possible to explain when defuzzification strategies like mom or coa are appropriate or lead to undesired results

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

foundations of fuzzy sets abstract this paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice it then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy and connective within these categories weak forms of subobject representations provide an internal second order logic an approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal second order theory

maslov index poincare birkhoff theorem and periodic solutions of asymptotically linear planar hamiltonian systems abstract we study the relationship between the twist condition in the poincare birkhoff fixed point theorem and the assumptions on the maslov index for asymptotically linear planar hamiltonian systems for this aim we develop a variant of the poincare birkhoff theorem which together with its classical version allows to obtain a lower bound for the number of nontrivial periodic solutions in terms of the gap between the maslov indexes associated to the linearizations of the planar system at zero and at infinity