equivalence of q bosons using the exponential phase operator various forms of the q boson are explained and their hidden symmetry revealed by transformations using the exponential phase operator both the one component and the multicomponent q bosons are discussed as a byproduct we obtain a new boson algebra having a shifted vacuum structure and define a global operator u 1 gauge transformation

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

complementarity of suq 3 and uq 2 and q boson realization of the suq 3 irreducible representations the complementarity relationship also termed duality that arises between the irreps of the su 3 and u 2 lie algebras when their direct sum su 3 u 2 is embedded into a larger u 6 algebra and single row irreps of the latter are considered is extended to the corresponding q algebras suq 3 and uq 2 it is demonstrated by explicitly constructing the unique q boson state that is simultaneously maximal in suq 3 and uq 2 for a given u q 2 weight in addition the relations between the suq 3 and uq 2 casimir operators resulting from their complementarity are explicitly found together with the q bargmann representation of q boson operators the complementarity relationship is then used to construct a gel fand tseitlin basis for arbitrary suq 3 irreps in terms of q boson operators

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

asymptotics and uniform asymptotics for finite time and infinite time absolute ruin probabilities in a dependent compound renewal risk model abstract in this paper we consider a dependent compound renewal risk model with constant premium rate and interest rate where the individual claim sizes are widely orthant dependent and the claim number has a distribution belonging to the intersection among the maximum domain of attraction of the gumbel distribution the subexponential class and the rapidly varying class in such a dependent compound renewal or poisson risk model we obtain the asymptotics and uniform asymptotics for the finite time and infinite time absolute ruin probabilities to this end we investigate the tail behavior of the random sum with some widely orthant dependent summands and the random number in the maximum domain of attraction of the gumbel distribution

tail behavior of random sums of negatively associated increments abstract let x x k k 1 be a sequence of negatively associated random variables with a common distribution function and finite expectation and let be a nonnegative integer valued random variable independent of x k k 1 in this paper we give unified form for the asymptotic behavior of the random sums s k 1 x k in the case of lim x p x x p x c 0 the results extend the earlier results of aleskeviciene et al 2008 2

on the aubry mather theory in statistical mechanics we generalize aubry mather theory for configurations on the line to general sets with a group action cocycles on the group play the role of rotation numbers the notion of birkhoff configuration can be generalized to this setting under mild conditions on the group we show how to find birkhoff ground states for many body interactions which are ferromagnetic invariant under the group action and having periodic phase space

the poisson bracket for q deformed systems it is shown that the quantum system of q deformed oscillators with the uq n group symmetry can be obtained by the canonical quantization of a classical system with a modified poisson bracket the modification of the poisson bracket is connected with a non canonical transformation of phase space variables in this approach the deformation parameter turns out to be a function of the planck constant and some dimensional parameters characterizing the classical system

real de sitter and conformal algebras in terms of q oscillators we express the cartan weyl basis of all quantum real d 4 de sitter and d 4 conformal algebras in terms of q deformed oscillators

continuity of depinning force abstract monotone recurrence relations give rise to a class of dynamical systems on the high dimensional cylinder which generalizes the class of monotone twist maps a solution of a monotone recurrence relation corresponds to an equilibrium of the generalized frenkel kontorova fk model the aubry mather theory for monotone recurrence relations says that for each r there is a birkhoff minimizer with rotation number whether all birkhoff minimizers of rotation number form a foliation is a question like whether there is an invariant circle with rotation number for a monotone twist map in this paper we give a criterion for the existence of minimal foliations and study its continuity the depinning force f d depending on rotation numbers is a critical value of the driving force for the fk model under which there are birkhoff equilibria and hence the system is pinned and above which there are no birkhoff equilibria of rotation number and the system is sliding we show that all birkhoff minimizers with irrational rotation number form a foliation if and only if f d 0 if p q is rational then f d p q 0 if and only if all p q periodic birkhoff minimizers constitute a foliation moreover we show that f d is continuous at irrational and holder continuous at diophantine points and it also depends continuously on system parameters

negative association of random variables with applications random variables x 1 cdots x k are said to be negatively associated na if for every pair of disjoint subsets a 1 a 2 of 1 2 cdots k operatorname cov lbrack f x 1 i in a 1 g x j j in a 2 rbrack leq 0 for all nondecreasing functions f g the basic properties of negative association are derived especially useful is the property that nondecreasing functions of mutually exclusive subsets of na random variables are na this property is shown not to hold for several other types of negative dependence recently proposed one consequence is the inequality p x i leq x i i 1 cdots k leq prod k 1p x i leq x i for na random variables x 1 cdots x k and the dual inequality resulting from reversing the inequalities inside the square brackets in another application it is shown that negatively correlated normal random variables are na other na distributions are the a multinomial b convolution of unlike multinomials c multivariate hypergeometric d dirichlet and e dirichlet compound multinomial negative association is shown to arise in situations where the probability measure is permutation invariant applications of this are considered for sampling without replacement as well as for certain multiple ranking and selection procedures in a somewhat striking example na and positive association representing quite strong opposing types of dependence are shown to exist side by side in models of categorical data analysis

generalized q oscillators and their hopf structures we study the relationships among the various forms of the q oscillator algebra and consider the conditions under which it supports a hopf structure we also present a generalization of this algebra together with its corresponding hopf structure its multimode extensions are also considered

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

tail probability of a random sum with a heavy tailed random number and dependent summands abstract let k k 1 be a sequence of widely orthant dependent random variables with common distribution f satisfying e 0 let be a nonnegative integer valued random variable in this paper we discuss the tail probabilities of random sums s n 1 n when the random number has a heavier tail than the summands i e p x p x 0 as x under some additional technical conditions we prove that if has a consistently varying tail then s has a consistently varying tail and p s x p x e on the other hand the converse problem is also equally interesting we prove that if s has a consistently varying tail then has a consistently varying tail and that p s x p x e still holds in particular the random number is not necessarily assumed to be independent of the summands k k 1 in theorem 3 1 and theorem 3 3 finally some applications to the asymptotic behavior of the finite time ruin probabilities in some insurance risk models are given

deformations of the fermion realization of the sp 4 algebra and its subalgebras with a view towards future applications in nuclear physics the fermion realization of the compact symplectic sp 4 algebra and its q deformed versions are investigated three important reduction chains of the sp 4 algebra are explored in both the classical and deformed cases the deformed realizations are based on distinct deformations of the fermion creation and annihilation operators for the primary reduction the su 2 substructure can be interpreted as either the spin isospin or angular momentum algebra whereas for the other two reductions su 2 can be associated with pairing between fermions of the same type or pairing between two distinct fermion types each reduction provides for a complete classification of the basis states the deformed induced u 2 representations are reducible in the action spaces of sp 4 and are decomposed into irreducible representations

generalized q deformed symplectic sp 4 algebra for multi shell applications am ulti shell generalization of a fermion representation of the q deformed compact symplectic spq 4 algebra is introduced an analytic form for the action of two or more generators of the spq 4 symmetry on the basis states is determined and the result used to derive formulae for the overlap between number preserving states as well as for matrix elements of a model hamiltonian a second order operator in the generators of spq 4 is identified that is diagonal in the basis set and that reduces to the casimir invariant of the sp 4 algebra in the non deformed limit of the theory the results can be used in nuclear structure applications to calculate decay transition probabilities and to provide for a description of pairing and higher order interactions in systems with nucleons occupying more than a single j orbital

ghost circles in lattice aubry mather theory abstract monotone lattice recurrence relations such as the frenkel kontorova lattice arise in hamiltonian lattice mechanics as models for ferromagnetism and as discretization of elliptic pdes mathematically they are a multi dimensional counterpart of monotone twist maps such recurrence relations often admit a variational structure so that the solutions x z d r are the stationary points of a formal action function w x given any rotation vector r d classical aubry mather theory establishes the existence of a large collection of solutions of w x 0 of rotation vector for irrational this is the well known aubry mather set it consists of global minimizers and it may have gaps in this paper we study the parabolic gradient flow d x d t w x and we will prove that every aubry mather set can be interpolated by a continuous gradient flow invariant family the so called ghost circle the existence of these ghost circles is known in dimension d 1 for rational rotation vectors and morse action functions the main technical result of this paper is therefore a compactness theorem for lattice ghost circles based on a parabolic harnack inequality for the gradient flow this implies the existence of lattice ghost circles of arbitrary rotation vectors and for arbitrary actions as a consequence we can give a simple proof of the fact that when an aubry mather set has a gap then this gap must be filled with minimizers or contain a non minimizing solution

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

convolution equivalence and distributions of random sums a serious gap in the proof of pakes s paper on the convolution equivalence of infinitely divisible distributions on the line is completely closed it completes the real analytic approach to sgibnev s theorem then the convolution equivalence of random sums of iid random variables is discussed some of the results are applied to random walks and levy processes in particular results of bertoin and doney and of korshunov on the distribution tail of the supremum of a random walk are improved finally an extension of rogozin s theorem is proved

q oscillators realization of fq 4 and gq 3 a realization of the quantum superalgebras corresponding to f 4 and g 3 is given in terms of q deformed bosonic and fermionic oscillators

the q boson realizations of the quantum group uq sl n 1 c the authors give an explicit expression of the recurrency formulae of canonical realization for quantum enveloping algebras uq sl n 1 c in these formulae the generators of the algebra uq sl n 1 c are expressed by means of n canonical q boson pairs one auxiliary representation of the algebra uq gl n c

vertex representations of quantum affine algebras we construct vertex representations of quantum affine algebras of ade type which were first introduced in greater generality by drinfeld and jimbo the limiting special case of our construction is the untwisted vertex representation of affine lie algebras of frenkel kac and segal our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisfying the defining relations in the case of the quantum affine algebra of type a we introduce vertex operators corresponding to all the roots and determine their commutation relations this provides an analogue of a chevalley basis of the affine lie algebra unk n in the basic representation

on the destruction of minimal foliations monotone variational recurrence relations arise in solid state physics conservative lattice dynamics and the theory of hamiltonian twist maps an example is the frenkel kontorova lattice for such recurrence relations aubry mather theory guarantees the existence of solutions of every rotation number 2 r when is irrational they are the action minimizers that constitute the aubry mather set this aubry mather set is either connected or a cantor set in the rst case it is called a minimal foliation and otherwise a minimal lamination in this paper we prove that when the rotation number of a minimal foliation is easy to approximate by rational numbers then the foliation can be destroyed into a lamination by an arbitrarily small smooth perturbation of the recurrence relation this generalizes a theorem of mather for twist maps to a large class of monotone variational problems we moreover provide a proof that may allow for further generalizations

on the decoupling of the homogeneous and inhomogeneous parts in inhomogeneous quantum groups we show that if there exists a realization of a hopf algebra h in a h module algebra then one can split their cross product into the tensor product algebra of itself with a subalgebra isomorphic to h and commuting with this result applies in particular to the algebra underlying inhomogeneous quantum groups like the euclidean groups which are obtained as cross products of the quantum euclidean spaces qn with the quantum groups of rotation uq so n of qn for which it has no classical analogue