saturated fusion systems over mathcal a _2 groups in this paper we study the saturated fusion systems mathcal f over mathcal a 2 groups s of order greater than p 4 this study requires the analysis of the possible saturated fusion systems in terms of the mathcal f automorphisms of the possible mathcal f centric and mathcal f radical subgroups also for each case in the classification either s is resistant or mathcal f is constrained the classification identifies a large family of resistant groups complementing the results on resistant groups found by various authors such as green minh stancu and diaz ruiz viruel moreover the classification provides some interesting examples of saturated fusion systems which have an mathcal f essential abelian subgroup of index p

a quantum exactly solvable nonlinear oscillator related to the isotonic oscillator a nonpolynomial one dimensional quantum potential representing an oscillator which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator harmonic oscillator with a centripetal barrier is studied first the general case that depends on a parameter a is considered and then a particular case is studied with great detail it is proven that it is schr dinger solvable and then the wavefunctions n and the energies en of the bound states are explicitly obtained finally it is proven that the solutions determine a family of orthogonal polynomials related to the hermite polynomials and such that i every is a linear combination of three hermite polynomials and ii they are orthogonal with respect to a new measure obtained by modifying the classic hermite measure

saturated fusion systems over 2 groups we develop methods for listing for a given 2 group s all nonconstrained centerfree saturated fusion systems over s these are the saturated fusion systems which could potentially include minimal examples of exotic fusion systems fusion systems not arising from any finite group to test our methods we carry out this program over four concrete examples two of order 2 7 and two of order 2 10 our long term goal is to make a wider more systematic search for exotic fusion systems over 2 groups of small order

reduced tame and exotic fusion systems we define here two new classes of saturated fusion systems reduced fusion systems and tame fusion systems these are motivated by our attempts to better understand and search for exotic fusion systems fusion systems which are not the fusion systems of any finite group our main theorems say that every saturated fusion system reduces to a reduced fusion system which is tame only if the original one is realizable and that every reduced fusion system which is not tame is the reduction of some exotic nonrealizable fusion system

fusion systems and amalgams we study reduced fusion systems from the point of view of their essential subgroups using the classification by goldschmidt and fan of amalgams of prime index to analyze certain pairs of such subgroups our results are applied here to study reduced fusion systems over 2 groups of order at most 64 and also reduced fusion systems over 2 groups having abelian subgroups of index two more applications are given in later papers

saturated fusion systems over a class of finite p groups abstract let p be an odd prime and f be a saturated fusion system over a finite p group s with derived subgroup of prime order excepting the case when s p a where p is a minimal nonabelian p group with p 1 p 1 1 p is cyclic and a is a finite abelian p group in this paper we prove that s f that is s is resistant this generalizes a result of stancu in the odd prime case

master symmetries for volterra equation belov chaltikian and blaszak marciniak lattice equations it is shown how to derive master symmetries for nonlinear lattice equations systematically using the basic principles but without using either their zero curvature equations or the bi hamiltonian structure this has been illustrated for volterra equation two coupled belov chaltikian bc and three coupled blaszak marciniak bm lattice equations the existence of a sequence of master symmetries is one of the characteristics of completely integrable nonlinear partial differential and differential difference equations admitting hamiltonian structure

negatively curved manifolds elliptic operators and the martin boundary consider a riemannian manifold m of dimension n 2 and assume that m is complete simply connected and that its sectional curvatures are bounded between two negative constants in a recent paper 7 m anderson and r schoen have proved that the martin boundary of m with respect to the laplace beltrami operator a is naturally homeomorphic to soo m the sphere at infinity of m here we shall first establish a similar result for a wide class of elliptic operators on m giving at the same time a new proof of the theorem of anderson and schoen our proof relies heavily on several concepts of potential theory but on the other hand avoids the quite difficult calculations of 7 and perhaps makes the result more understandable roughly speaking we show that if y is a second order elliptic operator defined over m and has a uniformly nice behaviour on each ball of radius 1 in m then dq m the y martin boundary of m is s m provided that the following condition holds there is a positive y ci superharmonic function on m for some positive if instead we only assume the existence of the green function for y a minimal requirement when studying the y martin boundary then d m may be very different from so m even for a rotation invariant operator y on the unit ball of rn equipped

normal subsystems of fusion systems the notion of a fusion system was first defined and explored by puig in the context of modular representation theory later broto levi and oliver significantly extended the theory of fusion systems as a tool in homotopy theory in this paper we begin a program to establish a local theory of fusion systems similar to the local theory of finite groups in particular we define the notion of a normal subsystem of a saturated fusion system and prove some basic results about normal subsystems and factor systems

superintegrable three body systems on the line we consider classical three body interactions on a euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momentum first integrals these systems are multiseparable superintegrable and equivalent up to rescalings to a one particle system in the three dimensional euclidean space common features of the dynamics are discussed we show how to determine quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics the conformal multiseparability is discussed and examples of conformal first integrals are given the systems considered here in generality include the calogero wolfes and other three body interactions widely studied in mathematical physics

reduced fusion systems over p groups with abelian subgroup of index p ii abstract for an odd prime p we look at simple fusion systems over a finite nonabelian p group s which has an abelian subgroup a of index p when s has more than one such subgroup we reduce this to a case already studied by ruiz and viruel when a is the unique abelian subgroup of index p in s and is not essential equivalently is not radical in the fusion system we give a complete list of all possibilities which can occur this includes several families of exotic fusion systems including some which have proper strongly closed subgroups

fusion systems over non resistant a2 groups abstract in this paper we study the saturated fusion systems f over potential non resistant a 2 groups of order greater than p 4 for each case in the classification f is constrained and realized by a finite group

almost all generalized extraspecial p groups are resistant a p group p is called resistant if for any finite group g having p as a sylow p subgroup the normalizer ng p controls p fusion in g the aim of this paper is to prove that any generalized extraspecial p group p is resistant excepting the case when p e a where a is elementary abelian and e is dihedral of order 8 when p 2 or extraspecial of order p3 and exponent p when p is odd this generalizes a result of green and minh

all p local finite groups of rank two for odd prime p in this paper we give a classification of the rank two p local finite groups for odd p this study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible f radical subgroups also for each case in the classification either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p local finite group getting some new examples of these for p 3

subgroup families controlling p local finite groups a p local finite group consists of a finite p group s together with a pair of categories which encode conjugacy relations among subgroups of s and which are modelled on the fusion in a sylow p subgroup of a finite group it contains enough information to define a classifying space which has many of the same properties as p completed classifying spaces of finite groups in this paper we examine which subgroups control this structure more precisely we prove that the question of whether an abstract fusion system mathcal f over a finite p group s is saturated can be determined by just looking at smaller classes of subgroups of s we also prove that the homotopy type of the classifying space of a given p local finite group is independent of the family of subgroups used to define it in the sense that it remains unchanged when that family ranges from the set of mathcal f centric mathcal f radical subgroups at a minimum to the set of mathcal f quasicentric subgroups at a maximum finally we look at constrained fusion systems analogous to p constrained finite groups and prove that they in fact all arise from groups

the homotopy theory of fusion systems the main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p completed classifying spaces of finite groups these spaces occur as the classifying spaces of certain algebraic objects which we call p local finite groups a p local finite group consists roughly speaking of a finite p group s and fusion data on subgroups of s encoded in a way explained below our starting point is our earlier paper blo on p completed classifying spaces of finite groups together with the axiomatic treatment by lluis puig pu pu2 of systems of fusion among subgroups of a given p group the p completion of a space x is a space xp which isolates the properties of x at the prime p and more precisely the properties which determine its mod p cohomology for example a map of spaces x f y induces a homotopy equivalence xp y p if and only if f induces an isomorphism in mod p cohomology and h xp fp h x fp in favorable cases if x is p good when g is a finite group the p completion bgp of its classifying space encodes many of the properties of g at p for example not only the mod p cohomology of bg but also the sylow p subgroup of g together with all fusion among its subgroups are determined up to isomorphism by the homotopy type of bgp our goal here is to give a direct link between p local structures and homotopy types which arise from them this theory tries to make explicit the essence of what it means to be the p completed classifying space of a finite group and at the same time yields new spaces which are not of this type but which still enjoy most of the properties a space of the form bgp would have we hope that the ideas presented here will have further applications and generalizations in algebraic topology but this theory also fits well with certain aspects of modular representation theory in particular it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p a saturated fusion system f over a p group s consists of a set homf p q of monomorphisms for each pair of subgroups p q s which form a category under composition include all monomorphisms induced by conjugation in s and satisfy certain other axioms formulated by puig definitions 1 1 and 1 2 below in particular these axioms are satisfied by the conjugacy homomorphisms in a finite group we refer to pu and pu2 for more details of puig s work on saturated

master symmetries non hamiltonian symmetries and superintegrability of the generalized smorodinsky winternitz system the first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach it is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries in the second part the properties of dynamical but non hamiltonian symmetries are studied it is proved that the higher order superintegrability of the generalized smorodinsky winternitz system is a consequence of the existence of symplectic symmetries not preserving the hamiltonian function

the classification of p local finite groups over the extraspecial group of order p 3 and exponent p in this paper we classify the p local finite groups over p1 2 the extraspecial group of order p3 and exponent p for odd p this study reduces to the classification of the saturated fusion systems over p1 2 which will be characterized by the outer automorphism group the number of open image in new window radical subgroups and the automorphism group of each nontrivial open image in new window radical subgroup as part of this classification we obtain three new exotic 7 local finite groups