locally compact convergence groups and n transitive actions all sigma compact locally compact groups acting sharply n transitively and continuously on compact spaces m have been classified except for n 2 3 when m is infinite and disconnected we show that no such actions exist for n 2 and that these actions for n 3 coincide with the action of a hyperbolic group on a space equivariantly homeomorphic to its hyperbolic boundary we further characterize non compact groups acting 3 properly and transitively on infinite compact sets as non elementary boundary transitive hyperbolic groups which in turn were recently studied by caprace de cornulier monod and tessera as an important tool we generalize bowditch s topological characterization of discrete hyperbolic groups to locally compact hyperbolic groups finally we show that if a locally compact group acts continuously 4 properly and 4 cocompactly on a locally connected metrizable compactum m then m has a global cut point which is in sharp contrast to the 3 proper 3 cocompact case due to the solution of bowditch s cut point conjecture

open 3 manifolds whose fundamental groups have infinite center and a torus theorem for 3 orbifolds our main result is a characterization of open seifert fibered 3 manifolds in terms of the fundamental group and large scale geometric properties of a triangulation as an application we extend the seifert fiber space theorem and the torus theorem to a class of 3 orbifolds

homeomorphic conjugates of fuchsian groups by a fuchsian group we mean a discrete subgroup of m it may contain also orientation reversing elements usually a fuchsian group is thought to act on i but this is not a problem since the action on s can be uniquely extended to an action on 1 and later we might also consider a fuchsian group to act on 1 depending on the situation we are here concerned with the problem when is a group g ofhomeornorphisms of s topologically conjugate to a fuchsian group we propose that this happens if and only if g is a convergence group of s which means that it satisfies the convergence property this means that whenever gi g are distinct we can pass to a subsequence in such a way that for some a b c s

a unified approach to some recurrence sequences via faa di bruno s formula abstract the well known faa di bruno formula for higher derivatives of a composite function plays an important role in combinatorics in this paper we obtain a unified approach to some recurrence sequences for the complete bell polynomials by using faa di bruno s formula furthermore we obtain an inverse relation of lah numbers of two types and by using this relation we obtain some new and interesting combinatorial identities

a topological characterisation of hyperbolic groups we characterise word hyperbolic groups as those groups which act properly discontinuously and cocompactly on the space of distinct triples of a compact metrisable space this is in turn equivalent to a convergence group for which every point of the space is a conical limit point

van kampen s embedding obstruction for discrete groups we give a lower bound to the dimension of a contractible mani fold on which a given group can act properly discontinuously in particular we show that the n fold product of nonabelian free groups cannot act prop erly discontinuously on r 2n 1

hadamard products of stable polynomials are stable a real polynomial is asymptotically stable when all of its zeros lie in the open left half of the complex plane we show that the hadamard coefficient wise product of two stable polynomials is again stable improving upon some known results via the associated hurwitz matrices we find another example of a class of totally nonnegative matrices which is closed under hadamard multiplication

expanders and box spaces we consider box spaces of finitely generated residually finite groups g and try to distinguish them up to coarse equivalence we show that for n 2 the group sln z has a continuum of box spaces which are pairwise non coarsely equivalent expanders moreover varying the integer n 3 expanders given as box spaces of sln z are pairwise inequivalent similarly varying the prime p expanders given as box spaces of sl2 z p are pairwise inequivalent a strong form of non expansion for a box space is the existence of 0 1 such that the diameter of each component xn satisfies diam xn xn by bt15 the existence of such a box space implies that g virtually maps onto z we establish the converse for the lamplighter group z 2z z and for a semi direct product z 2 z such box spaces are explicitly constructed using specific congruence subgroups we finally introduce the full box space of g i e the coarse disjoint union of all finite quotients of g we prove that the full box space of a group mapping onto the free group f2 is not coarsely equivalent to the full box space of an s arithmetic group satisfying the congruence subgroup property

theory of two index bessel functions and applications to physical problems in this article the theory of two index bessel functions is presented their generating function series expansion and integral representations are discussed their usefulness in physical problems is also discussed in the context of analysis of radiation emitted by relativistic electrons in two frequency undulators finally the theoretical analysis proving addition and multiplication theorems for two index bessel functions are completed and their modified forms are introduced

quasi isometry invariance of group splittings we show that a finitely presented one ended group which is not commensurable to a surface group splits over a two ended group if and only if its cayley graph is separated by a quasi line this shows in particular that splittings over two ended groups are preserved by quasi isometries

the generalized touchard polynomials revisited we discuss the generalized touchard polynomials introduced recently by dattoli et al as well as their extension to negative order introduced by the authors with operational methods the connection to generalized stirling and bell numbers is elucidated and analogs to burchnall s identity are derived a recursion relation for the generalized touchard polynomials is established and it is shown that one can interpret some of the resulting formulas as binomial theorems for particular noncommuting variables we suggest to generalize the generalized touchard polynomials still further and introduce so called comtet touchard functions which are associated to the powers of an arbitrary derivation

the spherical bessel and struve functions and operational methods abstract we review some aspects of the theory of spherical bessel functions and struve functions by means of an operational procedure essentially of umbral nature capable of providing the straightforward evaluation of their definite integrals and of successive derivatives the method we propose allows indeed the formal reduction of these family of functions to elementary ones of gaussian type we study the problem in general terms and present a formalism capable of providing a unifying point of view including anger and weber functions too the link to the multi index bessel functions is also briefly discussed

divided differences and a general explicit formula for sequences of mansour mulay shattuck divided differences is a very important tool often used both in theory and in practice for the approximation of functions in this paper we employ divided differences with repetitions to find a general explicit formula for the sequences satisfying mansour mulay shattuck s two term recurrence with arbitrary b i i 0

planar groups and the seifert conjecture we describe a number of characterisations of virtual surface groups which are based on the following result let g be a group and f be a field we show that if g is fp2 over f and if h2 g f thought of as a k vector space contains a 1 dimensional ginvariant subspace then g is a virtual surface group i e contains a subgroup of finite index which is the fundamental group of a closed surface other than the sphere or projective plane in particular this applies to rational poincare duality groups we also conclude that a finitely presented group which is semistable at infinity and with infinite cyclic fundamental group at infinity is a virtual surface group we recover the result of mess which characterises such groups as groups which are quasiisometric to complete riemannian planes we also give a cohomological version of the seifert conjecture from which the topological seifert conjecture proven by tukia mess gabai casson and jungreis can be recovered via work of zieschang and scott

morse theory and finiteness properties of groups we examine the finiteness properties of certain subgroups of right angled artin groups in particular we find an example of a group that is of type fp z but is not finitely presented

touchard like polynomials and generalized stirling numbers abstract the theory of touchard polynomials is generalized using a method based on the definition of exponential operators which extend the notion of the shift operator the proposed technique along with the use of the relevant operational formalism allows the straightforward derivation of properties of this family of polynomials and their relationship to different forms of stirling numbers

symbolic methods for the evaluation of sum rules of bessel functions the use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials we rederive in this way known sum rules and addition theorems for bessel functions furthermore we obtain a set of new closed form sum rules involving various special polynomials and bessel functions the examples we consider are relevant for applications ranging from plasma physics to quantum optics

on a connection between symmetric polynomials generalized stirling numbers and the newton general divided difference interpolation polynomial this paper gives a connection between symmetric polynomials generalized stirling numbers and the newton general divided difference interpolation polynomial the generalized stirling numbers of the first and second kind denoted s n k a and s n k a respectively are given symbolically for the case n 5 as an illustrative example by using maple programming maple v r3 programming reference manual these numbers depend on n k and n distinct values a 1 a 2 a n the ordinary stirling numbers s n k and s n k may be obtained as special cases by taking a i i 1 i 1 1 n

invariant measures for frequently hypercyclic operators abstract we investigate frequently hypercyclic and chaotic linear operators from a measure theoretic point of view among other things we show that any frequently hypercyclic operator t acting on a reflexive banach space admits an invariant probability measure with full support which may be required to vanish on the set of all periodic vectors for t that there exist frequently hypercyclic operators on the sequence space c 0 admitting no ergodic measure with full support and that if an operator admits an ergodic measure with full support then it has a comeager set of distributionally irregular vectors we also give some necessary and sufficient conditions which are satisfied by all the known chaotic operators for an operator t to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures finally we give a baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support

mobius like groups of homeomorphisms of the circle a group g homeo s 1 is a mobius like group if every element of g is conjugate in homeo s 1 to a mobius transformation our main result is given a mobus like like group g which has at least one global fixed point g is conjugate in homeo s 1 to a mobius group if and only if the limit set of g is all of s 1 moreover we prove that if the limit set of g is not si then after identifying some closed subintervals of s 1 to points the induced action of g is conjugate to an action of a mobius group we also show that the above result does not hold in the case when g has no global fixed points namely we construct examples of mobius like groups with limit set equal to s 1 but these groups cannot be conjugated to mobius groups

invariant gaussian measures for operators on banach spaces and linear dynamics we give conditions for an operator t on a complex separable banach space x with sufficiently many eigenvectors associated to eigenvalues of modulus 1 to admit a non degenerate invariant gaussian measure with respect to which it is weak mixing the existence of such a measure depends on the geometry of the banach space and on the possibility of parametrizing the mathbb t eigenvector fields of t in a regular way we also investigate the connection with frequent hypercyclicity

a universal hypercyclic representation abstract for any countable group and also for any locally compact second countable compactly generated topological group g we show the existence of a universal hypercyclic i e topologically transitive representation on a hilbert space in the sense that it simultaneously models every possible ergodic probability measure preserving free action of g

generalized leibniz functional matrices and divided difference form of the lagrange burmann formula abstract in this paper we derive two versions of divided difference form of the lagrange burmann expansion formula the first version expresses all coefficients by a form of a determinant by means of the factorizations of the generalized leibniz functional matrices with respect to a sequence we obtain the second version with coefficients represented by a finite sum some new matrix inversions are also investigated

repeated derivatives of composite functions and generalizations of the leibniz rule we use the properties of hermite and kampe de feriet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher order polynomial these results are extended to product of functions of the above argument thus giving rise to expressions which can formally be interpreted as generalizations of the familiar leibniz rule finally examples of practical interest are discussed

gromov s measure equivalence and rigidity of higher rank lattices in this paper the notion of measure equivalence me of countable groups is studied me was introduced by gromov as a measure theoretic analog of quasi isometries all lattices in the same locally compact group are measure equivalent this is one of the motivations for this notion the main result of this paper is me rigidity of higher rank lattices any countable group which is me to a lattice in a simple lie group g of higher rank is commensurable to a lattice in g

splittings of finitely generated groups over two ended subgroups we describe a means of constructing splittings of a one ended finitely generated group over two ended subgroups starting with a finite collection of codimension one two ended subgroups in the case where all the two ended subgroups have two ended commensurators we obtain an annulus theorem and a form of jsj splitting of rips and sela the construction uses ideas from the work of dunwoody sageev and swenson we use a particular kind of order structure which combines cyclic orders and treelike structures in the special case of hyperbolic groups this provides a link between combinarorial constructions and constructions arising from the topological structure of the boundary in this context we recover the annulus theorem of scott and swarup we also show that a one ended finitely generated groups which contains an infinite order element and such that every infinite cyclic subgroup is virtually codimension one is a virtual surface group

hierarchical dobinski type relations via substitution and the moment problem we consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson a a 1 monomials of the form exp a ra r 1 2 under the composition of their exponential generating functions they turn out to be of sheffer type we demonstrate that two key properties of these sequences remain preserved under substitutional composition a the property of being the solution of the stieltjes moment problem and b the representation of these sequences through infinite series dobinski type relations we present a number of examples of such composition satisfying properties a and b we obtain new dobinski type formulae and solve the associated moment problem for several hierarchically defined combinatorial families of sequences

the ramanujan master theorem and its implications for special functions abstract we study a number of possible extensions of the ramanujan master theorem which is formulated here by using methods of umbral nature we discuss the implications of the procedure for the theory of special functions like the derivation of formulae concerning the integrals of products of families of bessel functions and the successive derivatives of bessel type functions we stress also that the procedure we propose allows a unified treatment of many problems appearing in applications which can formally be reduced to the evaluation of exponential or gaussian like integrals

the boundary of negatively curved groups gromov s article gr contains fundamental properties of negatively curved groups several sets of seminar notes are available frn swn usn that contain more detailed accounts of and further expand on gromov s article a sequence xi e r i 1 2 is convergent at infinity provided xi x t 00 as i j t 00 two sequences xi yj convergent at infinityare equivalent if xi yi t 00 as i t 00 a point in the boundary ar of r is an equivalence class of sequences convergent at infinity now r u ar has a natural topology in which r is a discrete subspace and a typical not necessarily open neighborhood of a point a e ar represented by a sequence xj is given by y e r u arj a y n where a y lim xi y if y e r and a y lim xi yi if year is represented by a sequence yj ar is a compact metrizable finite dimensional space gr swn the important tool that relates ar with the cohomological properties of r is the rips complex pd r for every d 0 pd r is the simplicial complex whose vertices are elements of r and a collection xi xk e r spans a simplex if d xi x d for all i j the natural group action of r on itself by left translations gives rise to an action on pd r the key observation of rips is that when d is sufficiently large pd r is contractible and therefore when r is torsion free provides a model for er more precisely we have proposition 1 1 gr lemma 1 7 a swn 4 2 proposition 9 suppose that r is a negatively curved group and let t5 be as in the definition choose an integer

new sum rule for products of bessel functions with application to plasma physics in our investigations of the linear theory of the stability of relativistic beam plasma systems immersed in a magnetic field we have been led to consider sum rules for an infinite series of products of bessel functions of the form j n n jj2n n in this work we report on the sum of this series treated as a special case of a more general infinite series we also mention the extension of the results beyond the range of the parameters for which formulae are explicitly given and indicate how intermediate results obtained may be useful in their own right finally an additional application of our result is indicated

products of bessel functions and associated polynomials symbolic methods of umbral nature are exploited to derive series expansion for the products of bessel functions it is shown that the product of two cylindrical bessel functions can be written in terms of jacobi polynomials the procedure is extended to products of an arbitrary number of functions and the link with previous researchers is discussed we show that the technique we propose and the use of the ramanujan master theorem allow the derivation of integrals of practical interest

frequently hypercyclic operators with irregularly visiting orbits abstract we prove that a bounded operator t on a separable banach space x satisfying a strong form of the frequent hypercyclicity criterion which implies in particular that the operator is universal in the sense of glasner and weiss admits frequently hypercyclic vectors with irregularly visiting orbits i e vectors x x such that the set n t x u n 1 t n x u of return times of x into u under the action of t has positive lower density for every non empty open set u x but there exists a non empty open set u 0 x such that n t x u 0 has no density

quasi flats and rigidity in higher rank symmetric spaces in this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces such as sl n 2 n 3 theorem 8 1 is a major step towards the proof of quasiisometric rigidity of such lattices e we also give a different and effective proof of the theorem of kleiner leeb on the quasi isometric rigidity of higher rank symmetric spaces kl

some new examples of universal hypercyclic operators in the sense of glasner and weiss a bounded operator on a real or complex separable infinite dimensional banach space z is universal in the sense of glasner and weiss if for every invertible ergodic measure preserving transformation t of a standard lebesgue probability space x b there exists an a invariant probability measure on h with full support such that the two dynamical systems x b t and h bh a are isomorphic we present a general and simple criterion for an operator to be universal which allows us to characterize universal operators among unilateral or bilateral weighted shifts on p or c0 show the existence of universal operators on a large class of banach spaces and give a criterion for universality in terms of unimodular eigenvectors we also obtain similar results for operators which are universal for all ergodic systems not only for invertible ones and study necessary conditions for an operator on a hilbert space to be universal

some generalizations of the apostol genocchi polynomials and the stirling numbers of the second kind abstract recently the authors introduced some generalizations of the apostol bernoulli polynomials and the apostol euler polynomials see q m luo h m srivastava j math anal appl 308 2005 290 302 and q m luo taiwanese j math 10 2006 917 925 the main object of this paper is to investigate an analogous generalization of the genocchi polynomials of higher order that is the so called apostol genocchi polynomials of higher order for these generalized apostol genocchi polynomials we establish several elementary properties provide some explicit relationships with the apostol bernoulli polynomials and the apostol euler polynomials and derive various explicit series representations in terms of the gaussian hypergeometric function and the hurwitz or generalized zeta function we also deduce their special cases and applications which are shown here to lead to the corresponding results for the genocchi and euler polynomials of higher order by introducing an analogue of the stirling numbers of the second kind that is the so called stirling numbers of the second kind we derive some basic properties and formulas and consider some interesting applications to the family of the apostol type polynomials furthermore we also correct an error in a previous paper q m luo h m srivastava comput math appl 51 2006 631 642 and pose two open problems on the subject of our investigation

frequently hypercyclic operators we investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators t on separable complex f spaces t is frequently hypercyclic if there exists a vector x such that for every nonempty open subset u of x the set of integers n such that t n x belongs to u has positive lower density we give several criteria for frequent hypercyclicity and this leads us in particular to study linear transformations from the point of view of ergodic theory several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting

definite integrals and operational methods abstract an operational method already employed to formulate a generalization of the ramanujan master theorem is applied to the evaluation of integrals of various types this technique provides a very flexible and powerful tool yielding new results encompassing different aspects of the special function theory

on the real rootedness of generalized touchard polynomials we consider the real rootedness of generalized touchard polynomials recently revisited by mansour and schork 2013 towards this end we first describe the normal form of the generalized touchard polynomials by which recurrence relations for the polynomial part are derived by using the recurrence relations we prove the real rootedness of the generalized touchard polynomials for the parameter m 1 k k 1 k n

the algebraic torus theorem a structure theorem is proved for a finitely generated group with a finitely generated virtually polycyclic codimension one subgroup

connectivity at infinity for right angled artin groups we establish sufficient conditions implying semistability and connectivity at infinity properties for cat 0 cubical complexes we use this along with the geometry of cubical k 1 s to give a complete description of the higher connectivity at infinity properties of right angled artin groups among other things this determines which right angled artin groups are duality groups applications to group extensions are also included

quasi actions on trees i bounded valence given a bounded valence bushy tree t we prove that any cobounded quasi action of a group g on t is quasiconjugate to an action of g on another bounded valence bushy tree t this theorem has many applications quasi isometric rigidity for fundamental groups of finite bushy graphs of coarse pd n groups for each fixed n a generalization to actions on cantor sets of sullivan s theorem about uniformly quasiconformal actions on the 2 sphere and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice finally we give the first examples of two finitely generated groups which are quasi isometric and yet which cannot act on the same proper geodesic metric space properly discontinuously and cocompactly by isometries

group splittings and asymptotic topology it is a consequence of the theorem of stallings on groups with many ends that splittings over finite groups are preserved by quasi isometries in this paper we use asymptotic topology to show that group splittings are preserved by quasi isometries in many cases roughly speaking we show that splittings are preserved under quasi isometries when the vertex groups are fundamental groups of aspherical manifolds or more generally coarse pd n groups and the edge groups are smaller than the vertex groups

coarse differentiation of quasi isometries i spaces not quasi isometric to cayley graphs in this paper we prove that certain spaces are not quasi isometric to cayley graphs of nitely generated groups in particular we answer a question of woess and prove a conjecture of diestel and leader by showing that certain homogeneous graphs are not quasi isometric to a cayley graph of a nitely generated group this paper is the rst in a sequence of papers proving results announced in our 2007 article quasi isometries and rigidity of solvable groups in particular this paper contains many steps in the proofs of quasi isometric rigidity of lattices in sol and of the quasi isometry classication of lamplighter groups the proofs of those results are completed in coarse dierentiation of quasi isometries ii rigidity for lattices in sol and lamplighter groups the method used here is based on the idea of coarse dierentiation introduced in our 2007 article

some extensions of faa di bruno s formula with divided differences the well known formula of faa di bruno s for higher derivatives of a composite function has played an important role in combinatorics in this paper we generalize the divided difference form of faa di bruno s formula and give an explicit formula for the n th divided difference of a multicomposite function more generally we establish the relationships of the bell polynomials with respect to multicomposite functions applying these to multicomposite functions we obtain some extensions of faa di bruno s formula

on the asymptotic geometry of abelian by cyclic groups gromov s polynomial growth theorem gro81 states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups gromov s theorem inspired the more general problem see e g gdlh91 of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure in short are solvable groups quasi isometrically rigid in general they aren t very recently a dioubina dio99 has found a solvable group which is quasi isometric to a group which is not virtually solvable these groups are finitely generated but not finitely presentable in the opposite direction first steps in identifying quasi isometrically rigid solvable groups which are not virtually nilpotent were taken for a special class of examples the solvable baumslagsolitar groups in fm98 and fm99b the goal of the present paper is to show that a much broader class of solvable groups the class of finitely presented nonpolycyclic abelian bycyclic groups is characterized among all finitely generated groups by its quasi isometry type we also give a complete quasi isometry classification of the groups in this class such a classification for nilpotent groups remains a major open question motivated by these results we offer a conjectural picture of quasi isometric classification and rigidity for polycyclic abelianby cyclic groups in 10 1

a sierpinski carpet with the co hopfian property motivated by questions in geometric group theory we define a quasisymmetric co hopfian property for metric spaces and provide an example of a metric sierpinski carpet with this property as an application we obtain a quasi isometrically co hopfian gromov hyperbolic space with a sierpinski carpet boundary at infinity in addition we give a complete description of the quasisymmetry group of the constructed sierpinski carpet this group is uncountable and coincides with the group of bi lipschitz transformations

quasi isometries preserve the geometric decomposition of haken manifolds we prove quasi isometry invariance of the canonical decomposition for fundamental groups of haken 3 manifolds with zero euler characteristic we show that groups quasi isometric to haken manifold groups with nontrivial canonical decomposition are finite extensions of haken orbifold groups as a by product we describe all 2 dimensional quasi flats in the universal covers of non geometric haken manifolds

a new class of sum rules for products of bessel functions we derive a new class of sum rules for products of bessel functions of the first kind using standard algebraic manipulations we extend some of the well known properties of jn some physical applications of the results are also discussed a comparison with the newberger b s newberger j math phys 52 1278 1982 sum rules is performed on a typical example

generalized shift operators and pseudo polynomials of fractional order we introduce families of pseudo kampe de feriet polynomials which can be viewed as the natural complement for the theory of fractional derivatives and partial fractional differential equations of evolutive type we show that they allow the possibility of treating a large variety of exponential operators providing generalized fractional forms of shift operators the link of these pseudo polynomials with wright type functions is also discussed