the quantum heisenberg group h 1 q the structure of the quantum heisenberg group is studied in the two different frameworks of the lie algebra deformations and of the quantum matrix pseudogroups the r matrix connecting the two approaches together with its classical limit r are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed some unusual properties of the quantum enveloping heisenberg algebra are shown

idempotency of linear combinations of an idempotent matrix and a tripotent matrix abstract the problem of characterizing situations in which a linear combination c c 1 a c 2 b of an idempotent matrix a and a tripotent matrix b is an idempotent matrix is thoroughly studied in two particular cases of this problem when either b or b is an idempotent matrix a complete solution follows from the main result in linear algebra appl 321 2000 3 in the present paper a complete solution is established in all the remaining cases when b is an essentially tripotent matrix in the sense that both idempotent matrices b 1 and b 2 constituting its unique decomposition b b 1 b 2 are nonzero the problem is considered also under the additional assumption that the differences a b 1 and a b 2 are hermitian matrices this obviously covers the case when a b 1 and b 2 are hermitian themselves when the problem can be interpreted from a statistical point of view

getting a stochastic process from a conservative lagrangian a first approach abstract the transition probability p v for a stochastic process generated by a conservative lagrangian l l 0 e v is obtained at first order from a perturbation series found using a path integral this p v corresponds to the transition probability for a random walk with a probability density given by the sum of a normal distribution and a perturbation which may be understood as the contribution of the interaction of the random walk with the external field it is also found that the moment generating function for p v can be expressed as the generating function of a normal distribution modified by a perturbation applications of these results to a linear potential a harmonic oscillator potential and an exponentially decaying potential are shown

differentiability of t functionals of location and scatter the paper aims at finding widely and smoothly defined nonparametric location and scatter functionals as a convenient vehicle maximum likelihood estimation of the location vector and scatter matrix e of an elliptically symmetric t distribution on r d with degrees of freedom v i extends to an m functional defined on all probability distributions p in a weakly open weakly dense domain u here u consists of p putting not too much mass in hyperplanes of dimension d as shown for empirical measures by kent and tyler ann statist 19 1991 2102 2119 it will be seen here that e is analytic on u for the bounded lipschitz norm or for d 1 for the sup norm on distribution functions for k 1 2 and other norms depending on k and more directly adapted to t functionals one has continuous differentiability of order k allowing the delta method to be applied to for any p in u which can be arbitrarily heavy tailed these results imply asymptotic normality of the corresponding m estimators n n in dimension d 1 only the t v functional a extends to be defined and weakly continuous at all p

uniform central limit theorems for kernel density estimators let mathbb p n ast k h n x n 1 h n d sum i 1 n k left x x i h n right be the classical kernel density estimator based on a kernel k and n independent random vectors x i each distributed according to an absolutely continuous law mathbb p on mathbb r d it is shown that the processes f longmapsto sqrt n int fd mathbb p n ast k h n mathbb p f in mathcal f converge in law in the banach space ell infty mathcal f for many interesting classes mathcal f of functions or sets some mathbb p donsker some just mathbb p pregaussian the conditions allow for the classical bandwidths h n that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error thus showing that subject to some natural conditions kernel density estimators are plug in estimators in the sense of bickel and ritov ann statist 31 1033 1053 2003 some new results on the uniform central limit theorem for smoothed empirical processes needed in the proofs are also included

graded contractions of representations of orthogonal and symplectic lie algebras with respect to their maximal parabolic subalgebras parabolic gradings of the classical simple lie algebras o n c n or 5 and sp 2n c n or 2 with complex parameters are described for all maximal parabolic subalgebras all contractions which leave a maximal parabolic subalgebra intact and which preserve a parabolic grading parabolic contractions of lie algebras are found contractions of the irreducible representations for each parabolic contraction of the lie algebra are the main results of the article

idempotency of linear combinations of three idempotent matrices two of which are commuting abstract the considerations of the present paper were inspired by baksalary o m baksalary idempotency of linear combinations of three idempotent matrices two of which are disjoint linear algebra appl 388 2004 67 78 who characterized all situations in which a linear combination p c 1 p 1 c 2 p 2 c 3 p 3 with c i i 1 2 3 being nonzero complex scalars and p i i 1 2 3 being nonzero complex idempotent matrices such that two of them p 1 and p 2 say are disjoint i e satisfy condition p 1 p 2 0 p 2 p 1 is an idempotent matrix in the present paper by utilizing different formalism than the one used by baksalary the results given in the above mentioned paper are generalized by weakening the assumption expressing the disjointness of p 1 and p 2 to the commutativity condition p 1 p 2 p 2 p 1

graded contractions of the affine lie algebra a1 1 its representations and tensor products and an application to the branching rule a1 1 contains implies a1 1 we describe all contractions of the affine kac moody algebra a1 1 which preserve various gradings by the cyclic groups z2 and z3 as well as a grading by z2 z3 together with the contractions of simultaneously graded representations of a1 1 and their tensor products an application is given to the branching rules for a1 1 contains implies a1 1 for a number of distinct embeddings each associated with a particular grading

idempotency of linear combinations of an idempotent matrix and a t potent matrix that commute abstract this paper deals with idempotent matrices i e a 2 a and t potent matrices i e b t b when both matrices commute we derive a list of all complex numbers c 1 and c 2 such that c 1 a c 2 b is an idempotent matrix in addition the real case is also analyzed

new donsker classes several classes of functions are shown to be donsker by an argument based on partitioning the sample space one example is the class of all nondecreasing functions f r r such that 0 f f for a given function f with f 2 dp p

quantitative concentration inequalities for empirical measures on non compact spaces we establish quantitative concentration estimates for the empirical measure of many independent variables in transportation distances as an application we provide some error bounds for particle simulations in a model mean field problem the tools include coupling arguments as well as regularity and moment estimates for solutions of certain diffusive partial differential equations

donsker type theorems for nonparametric maximum likelihood estimators let mathcal p be a nonparametric probability model consisting of smooth probability densities and let hat p n be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law mathbb p with hat mathbb p n denoting the measure induced by the density hat p n define the stochastic process hat nu n f longmapsto sqrt n int fd hat mathbb p n mathbb p where f ranges over some function class mathcal f we give a general condition for donsker classes mathcal f implying that the stochastic process hat nu n is asymptotically equivalent to the empirical process in the space ell infty mathcal f of bounded functions on mathcal f this implies in particular that hat nu n converges in law in ell infty mathcal f to a mean zero gaussian process we verify the general condition for a large family of donsker classes mathcal f we give a number of applications convergence of the probability measure hat mathbb p n to mathbb p at rate sqrt n in certain metrics metrizing the topology of weak star convergence a unified treatment of convergence rates of the mle in a continuous scale of sobolev norms sqrt n efficient estimation of nonlinear functionals defined on mathcal p limit theorems at rate sqrt n for the maximum likelihood estimator of the convolution product mathbb p ast p

locally operating realizations of transformation lie groups using the mackey theory of induced representations a systematic study of the locally operating multiplier realizations of a connected lie group g that acts transitively on a space time manifold is presented we obtain a mathematical characterization of the locally operating multiplier realizations and a reduction of the problem of multiplier locally operating realizations to linear ones via a splitting group g m for g in this way the locally operating multiplier realizations are obtained by induction from finite dimensional linear representations of a well determined subgroup of g some examples such as the two dimensional euclidean group the galilei group and the one dimensional newton hooke group are given

the gel fand tsetlin representations of the unitary cayley klein algebras the explicit expressions of the operators of the irreducible representations of the unitary cayley klein algebras un j are obtained from the well known gel fand tsetlin representations of un the contractions of the representations of u2 j1 and u3 j1 j2 are regarded as examples this approach gives the representations of the contracted algebras in different bases for example in discrete and continuous ones possible contractions of the irreducible representations are discussed

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

on linear combinations of two tripotent idempotent and involutive matrices abstract let a c 1 a 1 c 2 a 2 where c 1 c 2 are nonzero complex numbers and a 1 a 2 is a pair of two n n nonzero matrices we consider the problem of characterizing all situations where a linear combination of the form a c 1 a 1 c 2 a 2 is i a tripotent or an involutive matrix when a 1 and a 2 are commuting involutive or commuting tripotent matrices respectively ii an idempotent matrix when a 1 and a 2 are involutive matrices and iii an involutive matrix when a 1 and a 2 are involutive or idempotent matrices

on some linear combinations of commuting involutive and idempotent matrices abstract we consider the problem of characterizing situations where a linear combination of commuting involutive matrices is a k potent matrix where k is a positive integer such that k 2 also we consider the problem of characterizing situations where a linear combination of the form c 1 i n c 2 a c 3 b is a nonsingular or an involutive matrix when idempotent matrices a b c n n satisfy different conditions and c 1 c 2 c 3 c 0

the gel fand tsetlin representations of the orthogonal cayley klein algebras the explicit expressions of the operators of the irreducible representations of the orthogonal cayley klein algebras son 1 j are obtained from the well known gel fand tsetlin representations of son 1 the contractions and the analytic continuations of the representations of so3 j1 j2 so4 j1 j2 j3 are regarded as examples this approach gives the representations of the contracted and the analytically continued algebras in different bases for example in discrete and continuous ones possible contractions of the irreducible representations are discussed

mean square displacement analysis of single particle trajectories with localization error brownian motion in an isotropic medium we examine the capability of mean square displacement msd analysis to extract reliable values of the diffusion coefficient d of a single particle undergoing brownian motion in an isotropic medium in the presence of localization uncertainty the theoretical results supported by simulations show that a simple unweighted least squares fit of the msd curve can provide the best estimate of d provided an optimal number of msd points are used for the fit we discuss the practical implications of these results for data analysis in single particle tracking experiments

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

empirical processes indexed by lipschitz functions on obtient des conditions necessaires et suffisantes sur p pour que les boules unites de bl r et lip r soient des classes de p donsker fonctionnelles

the three dimensional euclidean quantum group e 3 q and its r matrix a contraction procedure starting from so 4 q is used to determine the quantum analog e 3 q of the three dimensional euclidean group and the structure of its representations a detailed analysis of the contraction of the r matrix is then performed and its explicit expression has been found the classical limit of r is shown to produce an integrable dynamical system by means of the r matrix the pseudogroup of the noncommutative representative functions is considered it will finally be shown that a further contraction made on e 3 q produces the two dimensional galilei quantum group and this in turn can be used to give a new realization of e 3 q and e 2 1 q

on a disjoint idempotent decomposition for linear combinations produced from n commutative tripotent matrices abstract it has been established a 3 n term disjoint idempotent decomposition did for the linear combinations produced from n 2 commutative tripotent matrices their products and their products of power 2 at most the results obtained in this way generalize those in y tian a disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications linear multilinear algebra 59 2011 1237 1246 moreover an algorithm to get a did has been provided finally a numerical example has been given to exemplify the results

a note on linear combinations of commuting tripotent matrices abstract the purpose of this note is to characterize all situations in which a linear combination of two commuting tripotent matrices is also a tripotent matrix in the case of real scalars and real symmetric matrices this problem admits an interesting statistical interpretation namely it is equivalent to the question of when a linear combination of two quadratic forms in normal variables each distributed as a difference of two independent 2 variables is also distributed as such a difference

the central limit theorem for empirical processes under local conditions the case of radon infinitely divisible limits without gaussian component on obtient des resultats de convergence faible pour des processus empiriques indices par des classes f de fonctions dans le cas de limites de radon purement de poisson infiniment divisibles

graded contractions of representations of special linear lie algebras with respect to their maximal parabolic subalgebras parabolic gradings of the classical simple lie algebras sl n c n or 3 are described for all maximal parabolic subalgebras all contractions are found which leave a maximal parabolic subalgebra intact and which preserve a parabolic grading parabolic contractions of lie algebras contractions of the irreducible representations of sl n c for each parabolic contraction of the lie algebra are the main results of the article

the jordan schwinger representations of cayley klein groups i the orthogonal groups the cayley klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the orthogonal groups the jordan schwinger representations of cayley klein groups are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type the matrix elements of finite group transformations are obtained in the bases of coherent and fock states

on idempotency and tripotency of linear combinations of two commuting tripotent matrices abstract let t 1 and t 2 be two nonzero commuting n n tripotent matrices and c 1 c 2 two nonzero complex numbers necessary and sufficient conditions for the tripotency and the idempotency of c 1 t 1 c 2 t 2 are obtained the problems considered here have also statistical importance when c 1 c 2 are real scalars and t 1 t 2 are real symmetric matrices

determinantal correlations of brownian paths in the plane with nonintersection condition on their loop erased parts as an image of the many to one map of loop erasing operation l of random walks a self avoiding walk saw is obtained the loop erased random walk lerw model is the statistical ensemble of saws such that the weight of each saw is given by the total weight of all random walks that are inverse images of l we regard the brownian paths as the continuum limits of random walks and consider the statistical ensemble of loop erased brownian paths lebps as the continuum limits of the lerw model following the theory of fomin on nonintersecting lerws we introduce a nonintersecting system of n tuples of lebps in a domain d in the complex plane where the total weight of nonintersecting lebps is given by fomin s determinant of an n n matrix whose entries are boundary poisson kernels in d we set a sequence of chambers in a planar domain and observe the first passage points at which n brownian paths 1 n first enter each chamber under the condition that the loop erased parts l 1 l n make a system of nonintersecting lebps in the domain in the sense of fomin we prove that the correlation functions of first passage points of the brownian paths of the present system are generally given by determinants specified by a continuous function called the correlation kernel the correlation kernel is of eynard mehta type which has appeared in two matrix models and time dependent matrix models studied in random matrix theory conformal covariance of correlation functions is demonstrated

quantum structure of the motion groups of the two dimensional cayley klein geometries a simultaneous and global scheme of quantum deformation is defined for the set of algebras corresponding to the groups of motions of the two dimensional cayley klein geometries their central extensions are also considered under this unified pattern in both cases some fundamental properties characterizing the classical ck geometries as the existence of a set of commuting involutions contractions and dualities relationships remain in the quantum version

non standard quantum so 2 2 and beyond a new non standard quantization of the universal enveloping algebra of the split natural real form so 2 2 of d2 is presented some classical graded contractions of so 2 2 associated to a z2 z2 grading are studied and the automorphisms defining this grading are generalized to the quantum case thus providing quantum contractions of this algebra this produces a new family of non standard quantum algebras some of these algebras can be realized as 2 1 kinematical algebras we explicitly introduce a new deformation of poincare algebra which is naturally linked to the null plane basis another realization of these quantum algebras as deformations of the conformal algebras for the two dimensional euclidean galilei and minkowski spaces is given and its new properties are emphasized

properties of the combinations of commutative idempotents this paper deals with idempotent operators i e p2 p and n potent operators i e pn p we characterize the invertibility the group invertibility and idempotency of the linear combinations of idempotent operators or n potent operators under certain commutativity property imposed on them

the matrix quantum unitary cayley klein groups the author defines the matrix quantum unitary cayley klein groups of arbitrary dimensions by generalizing the r matrix theory of the quantum groups to the case of degenerate hermitic forms the corresponding quantum groups are obtained from the appropriate classical quantum groups by contractions

the drazin inverse of the linear combinations of two idempotents in the banach algebra abstract in this paper some drazin inverse representations of the linear combinations of two idempotents in a banach algebra are obtained moreover we present counter examples to and establish the corrected versions of two theorems by cvetkovic ilic and deng

green function estimates for subordinate brownian motions stable and beyond a subordinate brownian motion x is a l evy process which can be obtained by replacing the time of the brownian motion by an independent subordinator in this paper when the laplace exponent phi of the corresponding subordinator satisfies some mild conditions we first prove the scale invariant boundary harnack inequality for x on arbitrary open sets then we give an explicit form of sharp two sided estimates of the green functions of these subordinate brownian motions in any bounded c 1 1 open set as a consequence we prove the boundary harnack inequality for x on any c 1 1 open set with explicit decay rate unlike cite ksv2 ksv4 our results cover geometric stable processes and relativistic geometric stable process i e the cases when the subordinator has the laplace exponent phi lambda log 1 lambda alpha 2 0 alpha and phi lambda log 1 lambda m alpha 2 2 alpha m 0 0 d 2

on idempotency of linear combinations of idempotent matrices p 1 p 2 and p 3 being any three different nonzero mutually commutative nxn idempotent matrices and c 1 c 2 and c 3 being nonzero scalars the problem of characterizing some situations where a linear combination of the form p c 1p 1 c 2p 2 or p c 1p 1 c 2p 2 c 3p 3 is also an idempotent matrix has been considered moreover two interesting results about the idempotency of linear combinations of 2x2 idempotent matrices and 3x3 idempotent matrices have been given a statistical interpretation of the idempotency problem considered in this study has also been pointed out

so 5 q and contraction chevalley basis representations for q generic and root of unity representations of so 5 q are constructed explicitly on the chevalley basis the classical representations once obtained on this basis are very simply deformed for arbitrary q generic and root of unity this is in sharp contrast to the classical canonical gelfand zetlin gz representations which lead to problems studied elsewhere in q deformations for orthogonal though not for unitary algebras our results are limited to irreducible representations labeled by three variable parameters the maximal number being four within this restriction it is shown how to construct all representations for all q s the contraction of the chevalley generators e2 f2 leads to an easily obtained complete hopf algebra the structure of the contracted algebra is quite different even at the classical level from the familiar e 4 the four dimensional euclidean algebra obtained on contracting the generator j45 in the gz formalism the contracted representations are given the so 5 q representations have their own

efficient maximum likelihood estimation in semiparametric mixture models we consider maximum likelihood estimation in several examples of semiparametric mixture models including the exponential frailty model and the errors in variables model the observations consist of a sample of size n from the mixture density int p theta x z d eta z the mixing distribution is completely unknown we show that the first component hat theta n of the joint maximum likelihood estimator hat theta n hat eta n is asymptotically normal and asymptotically efficient in the semiparametric sense

on invertibility of combinations of k potent operators abstract some properties of combinations c 1 p 1 c 2 p 2 c 3 p 1 s p 2 k 1 s where p 1 and p 2 are two different nonzero k potents c 1 c 2 and c 3 are three nonzero complex numbers and positive integers k 2 and s k 1 are obtained furthermore the invertibility the group invertibility and the k potency of the linear combinations of k potents are investigated under certain commutativity properties imposed on them

contractions of the irreducible representations of the quantum algebras suq 2 and soq 3 the contractions of the irreducible representations of the unitary quantum algebra suq 2 and the orthogonal quantum algebra soq 3 in the gel fand tsetlin basis are regarded in detail with the help of the dual numbers

tripotency of a linear combination of two involutory matrices and a tripotent matrix that mutually commute this paper concerns with the tripotency of a linear combination of three matrices which has a background in statistical theory we demonstrate all the possible cases that lead to the tripotency of a linear combination of two involutory matrices and a tripotent matrix that mutually commute by utilizing block technique and returning the partitioned matrices to their original forms we derive the sufficient and necessary conditions such that a linear combination of three mutually commuting involutory matrices is tripotent

quantum 2 1 kinematical algebras a global approach in this paper we give an approach to quantum deformations of the 2 1 kinematical lie algebras within a scheme that simultaneously describes all groups of motions of classical geometries in n 3 dimensions we cover at once all the kinematical geometries including the quantum versions of inonu wigner contractions which are defined in a natural way and relate q deformations as expected we thus obtain some q deformations previously known for the three dimensional euclidean and 2 1 poincare algebras and also some new q deformations for these and other kinematical algebras such as the 2 1 de sitter galilei and newton hooke algebras

idempotency of linear combinations of two idempotent matrices abstract a complete solution is established to the problem of characterizing all situations where a linear combination of two different idempotent matrices p 1 and p 2 is also an idempotent matrix including naturally three such situations known in the literature viz if the combination is either the sum p 1 p 2 or one of the differences p 1 p 2 p 2 p 1 and appropriate additional conditions are fulfilled the solution asserts that in the particular case where p 1 and p 2 are complex matrices such that p 1 p 2 is hermitian these three situations exhaust the list of all possibilities and that this list extends when the above assumption on p 1 and p 2 is violated a statistical interpretation of the idempotency problem considered in this note is also pointed out

four dimensional quantum affine algebras and space time q symmetries a global model of the q deformation for the quasiorthogonal lie algebras generating the groups of motions of the four dimensional affine cayley klein ck geometries is obtained starting from the three dimensional deformations it is shown how the main algebraic classical properties of the ck systems can be implemented in the quantum case quantum deformed versions either of the space time or space symmetry algebras poincare 3 1 galilei 3 1 4 d euclidean as well as others appear in this context as particular cases for some of these classical algebras several q deformations are directly obtained

lie bialgebra contractions and quantum deformations of quasi orthogonal algebras lie bialgebra contractions are introduced and classified a non degenerate coboundary bialgebra structure is implemented into all pseudo orthogonal real algebras so p q starting from the one corresponding to so n 1 it allows us to introduce a set of lie bialgebra contractions which leads to lie bialgebras of quasi orthogonal algebras this construction is explicitly given for n 2 3 4 all lie bialgebra contractions studied in this paper define hopf algebra contractions for the drinfel d jimbo deformations uzso p q they are explicitly used to generate new non semisimple quantum algebras as it is the case for the euclidean poincare and galilean algebras

on the spectra of some combinations of two generalized quadratic matrices let a and b be two generalized quadratic matrices with respect to idempotent matrices p and q respectively such that a p a p 0 a p p a a b q b q 0 b q q b b p q q p ab ba and a b p q p q a b with c let a b be diagonalizable the relations between the spectrum of the matrix a b and the spectra of some matrices produced from a and b are considered moreover some results on the spectrum of the matrix a b are obtained when a b is not diagonalizable finally some results and examples illustrating the applications of the results in the work are given

casimir invariants for the complete family of quasisimple orthogonal algebras a complete choice of generators of the centre of the enveloping algebras of real quasisimple lie algebras of orthogonal type for arbitrary dimension is obtained in a unified setting the results simultaneously include the well known polynomial invariants of the pseudo orthogonal algebras as well as the casimirs for many non simple algebras such as the inhomogeneous the newton hooke and galilei type etc which are obtained by contraction s starting from the simple algebras the dimension of the centre of the enveloping algebra of a quasisimple orthogonal algebra turns out to be the same as for the simple algebras from which they come by contraction the structure of the higher order invariants is given in a convenient pyramidal manner in terms of certain sets of pauli lubanski elements in the enveloping algebras as an example showing this approach at work the scheme is applied to recovering the casimirs for the 3 1 kinematical algebras some prospects on the relevance of these results for the study of expansions are also given