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

uniform partitions of frames of exponentials into riesz sequences abstract the feichtinger conjecture if true would have as a corollary that for each set e 0 1 and z there is a partition 1 n of z such that for each 1 i n exp 2 i x i is a riesz sequence in this paper sufficient conditions on sets e 0 1 and r are given so that exp 2 i x 1 e can be uniformly partitioned into riesz sequences

structured condition numbers and small sample condition estimation of symmetric algebraic riccati equations abstract this paper is devoted to a structured perturbation analysis of the symmetric algebraic riccati equations by exploiting the symmetry structure based on the analysis the upper bounds for the structured normwise mixed and componentwise condition numbers are derived due to the exploitation of the symmetry structure our results are improvements of the previous work on the perturbation analysis and condition numbers of the symmetric algebraic riccati equations our preliminary numerical experiments demonstrate that our condition numbers provide accurate estimates for the change in the solution caused by the perturbations on the data moreover by applying the small sample condition estimation method we propose a statistical algorithm for practically estimating the condition numbers of the symmetric algebraic riccati equations

the matrix sign function a survey of the matrix sign function is presented including some historical background definitions and properties approximation theory and computational methods and condition theory and estimation procedures applications to areas such as control theory eigendecompositions and roots of matrices are outlined and some new theoretical results are also given

new q deformed coherent states with an explicitly known resolution of unity we construct a new family of q deformed coherent states zq where 0 q 1 these states are normalizable on the whole complex plane and continuous in their label z they allow the resolution of unity in the form of an ordinary integral with a positive weight function obtained through the analytic solution of the associated stieltjes power moment problem and expressed in terms of one of the two jacksons s q exponentials they also permit exact evaluation of matrix elements of physically relevant operators we use this to show that the photon number statistics for the states is sub poissonian and that they exhibit quadrature squeezing as well as an enhanced signal to quantum noise ratio over the conventional coherent state value finally we establish that they are the eigenstates of some deformed boson annihilation operator and study some of their characteristics in deformed quantum optics

the effect of problem perturbations on nonlinear dynamical systems and their reduced order models reduced order models are used extensively in many areas of science and engineering for simulation design and control reduction techniques for nonlinear dynamical systems produce models that depend strongly on the nominal set of parameters for which the reduction is carried out in this paper we address the following two questions what is the effect of perturbations in the problem parameters on the output functional of a nonlinear dynamical system and to what extent does the reduced order model capture this effect

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

a flexible condition number for weighted linear least squares problem and its statistical estimation in this paper we mainly focus on the derivation of a flexible condition number for weighted linear least squares problem and its estimation with frechet derivative and kronecker product the explicit expression of the condition number is obtained when the coefficient matrices are large and dense considering the difficulties of explicitly forming the expression of condition number in computer we also present its simplified form through a simple and easy to operate method two different condition estimation methods are introduced and the numerical experiments are also performed to show the efficiency of the condition estimators

a condition number for point matching with application to registration and postregistration error estimation selecting salient points from two or more images for computing correspondence is a well studied problem in image analysis this paper describes a new and effective technique for selecting these tiepoints using condition numbers with application to image registration and mosaicking condition numbers are derived for point matching methods based on minimizing windowed objective functions for 1 translation 2 rotation scaling translation rst and 3 affine transformations our principal result is that the condition numbers satisfy k sub trans spl les k sub rst spl les k sub affine that is if a point is ill conditioned with respect to point matching via translation then it is also unsuited for matching with respect to rst and affine transforms this is fortunate since k sub trans is easily computed whereas k sub rst and k sub affine are not the second half of the paper applies the condition estimation results to the problem of identifying tiepoints in pairs of images for the purpose of registration once these points have been matched after culling outliers using a ransac like procedure the registration parameters are computed the postregistration error between the reference image and the stabilized image is then estimated by evaluating the translation between these images at points exhibiting good conditioning with respect to translation the proposed method of tiepoint selection and matching using condition number provides a reliable basis for registration the method has been tested on a large number of diverse collection of images multidate landsat images aerial images aerial videos and infrared images a web site where the users can try our registration software is available and is being actively used by researchers around the world

randomized residual based error estimators for the proper generalized decomposition approximation of parametrized problems this paper introduces a novel error estimator for the proper generalized decomposition pgd approximation of parametrized equations the estimator is intrinsically random it builds on concentration inequalities of gaussian maps and an adjoint problem with random right hand side which we approximate using the pgd the effectivity of this randomized error estimator can be arbitrarily close to unity with high probability allowing the estimation of the error with respect to any user defined norm as well as the error in some quantity of interest the performance of the error estimator is demonstrated and compared with some existing error estimators for the pgd for a parametrized time harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high dimensional parameter space

one dimensional partially asymmetric simple exclusion process with open boundaries orthogonal polynomials approach the stationary state of the partially asymmetric simple exclusion process with open boundaries is reconsidered the so called matrix product ansatz is employed this enables us to construct the stationary state in the form of the product of the matrices d and e noticing the fact that the matrix c d e for the model is related to certain q orthogonal polynomials the model is analysed for a wide range of parameters the current and the correlation length are evaluated in the thermodynamic limit it turns out that the phase diagram for the correlation length is richer than that for the totally asymmetric case

representations of the quadratic algebra and partially asymmetric diffusion with open boundaries we consider the one dimensional partially asymmetric exclusion model with open boundaries the model describes a system of hard core particles that hop stochastically in both directions with different rates at both boundaries particles are injected and extracted by means of the method of derrida et al the stationary probability measure can be expressed as a matrix product state involving two matrices forming a fock like representation of a general quadratic algebra we obtain the representations of this algebra which were unknown in the mathematical literature and use the two dimensional one to derive exact expressions for the density profile and correlation functions using the correspondence between the stochastic model and a quantum spin chain we obtain exact correlation functions for a spin heisenberg xxz chain with non diagonal boundary terms generalizations to other reaction diffusion models 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

completeness of coherent states associated with self similar potentials and ramanujan s integral extension of the beta function a decomposition of identity is given as a complex integral over the coherent states associated with a class of shape invariant self similar potentials there is a remarkable connection between these coherent states and ramanujan s integral extension of the beta function

many body problems with composite particles and q heisenberg algebras we propose to employ deformed commutation relations to treat many body problems of composite particles the deformation parameter is interpreted as a measure of the effects of the statistics of the internal degrees of freedom of the composite particles a simple application of the method is made for the case of a gas of composite bosons

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

realizability of a model in infinite statistics following greenberg and others we study a space with a collection of operatorsa k satisfying the q mutator relations a l a k a l k l corresponding forq 1 to classical bose and fermi statistics we show that then n matrixa n q representing the scalar products ofn particle states is positive definite for alln ifq lies between 1 and 1 so that the commutator relations have a hilbert space representation in this case this has also been proved by fivel and by bozejko and speicher we also give an explicit factorization ofa n q as a product of matrices of the form 1 q jt 1 with 1 j n andt a permutation matrix in particular a n q is singular if and only ifq m 1 for some integerm of the formk 2 k 2 k n

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

time quantization and q deformations we extend to quantum mechanics the technique of stochastic subordination by means of which one can express any semi martingale as a time changed brownian motion as examples we considered two versions of the q deformed harmonic oscillator in both ordinary and imaginary time and show how these various cases can be understood as different patterns of time quantization rules

nonlinear lie algebra and ladder operators for orbital angular momentum based on the nonlinear lie algebra a new framework is put forward to treat the problem of angular momentum from this framework the ladder operators for the orbital angular momentum can be solved as simultaneous eigenvectors of the lie derivations ad l2 and ad l2 in an operator space the ladder operators for solvable central potentials can also be found in a united way

a macroscopic multifractal analysis of parabolic stochastic pdes it is generally argued that the solution to a stochastic pde with multiplicative noise such as dot u frac12 u u xi where xi denotes space time white noise routinely produces exceptionally large peaks that are macroscopically multifractal see for example gibbon and doering arch ration mech anal 177 115 150 2005 gibbon and titi proc r soc a 461 3089 3097 2005 and zimmermann et al phys rev lett 85 17 3612 3615 2000 a few years ago we proved that the spatial peaks of the solution to the mentioned stochastic pde indeed form a random multifractal in the macroscopic sense of barlow and taylor j phys a 22 13 2621 2626 1989 proc lond math soc 3 64 125 152 1992 the main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio temporal peaks of the solution form infinitely many different multifractals on infinitely many different scales which we sometimes refer to as stretch factors a simpler though still complex such structure is shown to also exist for the constant coefficient version of the said stochastic pde

multi parametric deformed heisenberg algebras a route to complexity we introduce a generalization of the heisenberg algebra which is written in terms of a functional of one generator of the algebra f j0 that can be any analytical function when f is linear with slope we show that the algebra in this case corresponds to q oscillators for q2 tan the case where f is a polynomial of order n in j0 corresponds to an n parameter deformed heisenberg algebra the representations of the algebra when f is any analytical function are shown to be obtained through the study of the stability of the fixed points of f and their composed functions the case when f is a quadratic polynomial in j0 the simplest nonlinear scheme which is able to create chaotic behaviour is analysed in detail and special regions in the parameter space give representations that cannot be continuously deformed to representations of heisenberg algebra

three dimensional quantum groups from contractions of su 2 q contractions of lie algebras and of their representations are generalized to define new quantum groups an explicit and complete exposition is made for the one dimensional heisenberg h 1 q and the two dimensional euclidean quantum group e 2 q obtained by contracting su 2 q

quasi additive estimates on the hamiltonian for the one dimensional long range ising model in this work we study the problem of getting quasi additive bounds for the hamiltonian of the long range ising model when the two body interaction term decays proportionally to 1d2 0 1 we revisit the paper by cassandro et al j math phys 46 053305 2005 where they extend to the case 0 ln3ln2 1 the result of the existence of a phase transition by using a peierls argument given by frohlich and spencer commun math phys 84 87 101 1982 for 0 the main arguments of cassandro et al j math phys 46 053305 2005 are based in a quasi additive decomposition of the hamiltonian in terms of hierarchical structures called triangles and contours which are related to the original definition of contours introduced by frohlich and spencer commun math phys 84 87 101 1982 in this work we study the existence of a quasi additive decomposition of the hamiltonian in terms of the contours defined in the work of cassandro et al j math phys 46 053305 2005 the most relevant r

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

deformed systems at finite temperature abstract we apply the formalism of thermo field dynamics to two kinds of deformed hamiltonians these hamiltonians are made up of operators which satisfy deformed heisenberg algebras the entropy and the specific heat are computed for small q 1 where q is the deformation parameter

fractional dimension of sets in discrete spaces the authors give a new definition dimh a for the dimension of an arbitrary subset of the lattice d they establish elementary properties and calculate the dimension for some examples finally they announce a result which states that if dimh a d 2 then a is recurrent

relations for clebsch gordan and racah coefficients in suq 2 and yang baxter equations relations are exploited among clebsch gordan cg and racah coefficients in the algebra suq 2 known as a deformation of su 2 these are used to show that the yang baxter yb relation for the irf interaction round a face model results from one of the symmetry relations for the 9 j symbol specific to suq 2 and that in an asymptotic limit this yb relation becomes the yb relation for the two dimensional vertex model the racah coefficient which has a particularly simple dependence on q is efficiently used such that an asymptotic limit of the racah coefficient is the cg coefficient and another limit gives the factorized s matrix of the vertex model

brst cohomology for uq sl 2 representations a nilpotent brst operator qq is constructed for the q deformation of the universal enveloping algebra of sl 2 an associated operator rq such that qq rq cq is introduced where cq is the q deformed casimir invariant cq fe 1 2h q 1 2h 1 q the related operator qq conjugate to qq is also considered the resulting brst cohomology of uq sl 2 representations is discussed

small sample statistical condition estimation of large scale generalized eigenvalue problems we consider the evaluation of the sensitivity or condition number of generalized eigenvalue problems for a large and sparse real matrix or matrix pair in r n n through some coupled sylvester equation using newton s method the technique of the statistical condition estimation has been adapted to the sensitivity of symmetric matrices as well as general matrices with special structures under some assumptions on various types of perturbations

error estimation for reduced order models of dynamical systems the use of reduced order models to describe a dynamical system is pervasive in science and engineering often these models are used without an estimate of their error or range of validity in this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition we show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method most important the proposed approach allows the assessment of regions of validity for reduced models i e ranges of perturbations in the original system over which the reduced model is still appropriate numerical examples validate our approach the error norm estimates approximate well the forward error while the derived bounds are within an order of magnitude

on global error estimation and control for initial value problems this paper addresses global error estimation and control for initial value problems for ordinary differential equations the focus lies on a comparison between a novel approach based on the adjoint method combined with a small sample statistical initialization and the classical approach based on the first variational equation control is achieved through tolerance proportionality both approaches are found to work well and to enable estimation and control in a reliable manner

zeta functions of discrete self similar sets abstract in this paper we study a class of countable and discrete subsets of a euclidean space that are self similar with respect to a finite set of affine similarities any such set can be interpreted as having a fractal structure we introduce a zeta function for these sets and derive basic analytic properties of this fractal zeta function motivating examples that come from combinatorial geometry and arithmetic are given particular attention

metric properties of fractal lattices the authors study the connectivity dimension d of fractal lattices viewed as networks graphs of sites and constant length bond two examples are investigated in detail the 2d sierpinski gasket and 2d infinite percolation clusters on square and triangular lattices in the first case d is shown to coincide with the fractal dimension whereas d 1 72 or 0 02 appears as a universal exponent for percolation clusters in two dimensions considered as an intrinsic parameter the connectivity dimension d is compared with other intrinsic and extrinsic characteristic parameters of fractal lattices in particular the authors argue that d or d or d holds on fractal lattices in general d fractal dimension d spectral dimension

approach of spherical harmonics to the representation of the deformed su 1 1 algebra the m shifting generators of su 2 algebra together with a pair of l shifting ladder symmetry operators have been used in the space of all spherical harmonics ylm in order to introduce a new set of operators expressing the transitions between them it is shown that the space of spherical harmonics whose l 2m or l 2m is given presents negative and positive irreducible representations of a deformed su 1 1 algebra respectively these internal symmetries also suggest new algebraic methods to construct the spherical harmonics in the framework of the spectrum generating algebras

the distribution of zeros of general q polynomials a general system of q orthogonal polynomials is defined by means of its three term recurrence relation this system encompasses many of the known families of q polynomials among them the q analogue of the classical orthogonal polynomials the asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behaviour of the coefficients of the recurrence relation this result is applied to specific classes of polynomials known by the names q hahn q kravchuk q racah q askey and wilson al salam carlitz and the celebrated little and big q jacobi

a study of representations of the algebra of functions on the quantum group glq n a q analogue of the root system is constructed for this algebra which is similar to the root system of classical lie algebras it is then used to construct in detail a class of representations of the algebra of functions on the quantum group glq n and a q boson realization of the generators of glq n is given i also construct infinite dimensional hilbert space representations of this algebra the main result of this paper is stated in proposition 8

a simple difference realization of the heisenberg q algebra a realization of the heisenberg q algebra whose generators are first order difference operators on the full real line is discussed herein the eigenfunctions of the corresponding q oscillator hamiltonian are given explicitly in terms of the q 1 hermite polynomials the nonuniqueness of the measure for these q oscillator states is also studied

suq h cross to 0 2 and suq h cross 2 the classical and quantum q deformations of the su 2 algebra ii the hopf algebra the yang baxter equation and multi deformed algebraic structures for pt i see ibid vol 23 p 4185 1990 the suq 2 algebra is realized by means of both the poisson brackets in classical mechanics and commutators in quantum mechanics in a system with q deformed oscillators of two different types the structures of the hopf algebra and the quantum yang baxter equation are also discussed on a quantum level a set of j representations of the quantum algebra suq 2 is constructed based on the type ii q oscillators multi deformations of the oscillators of the two types and multi deformed algebras expressed in poisson brackets as well as in lie brackets are proposed

structured condition numbers of structured tikhonov regularization problem and their estimations both structured componentwise and structured normwise perturbation analysis of the tikhonov regularization are presented the structured matrices under consideration include toeplitz hankel vandermonde and cauchy matrices structured normwise mixed and componentwise condition numbers for the tikhonov regularization are introduced and their explicit expressions are derived for the general linear structure based on the derived expressions we prove structured condition numbers are smaller than their corresponding unstructured counterparts by means of the power method and small sample statistical condition estimation sce fast condition estimation algorithms are proposed our estimation methods can be integrated into tikhonov regularization algorithms that use the generalized singular value decomposition gsvd for large scale linear structured tikhonov regularization problems we show how to incorporate the sce into the preconditioned conjugate gradient pcg method to get the posterior error estimations the structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones and the proposed condition estimation algorithms are reliable also an image restoration example is tested to show the effectiveness of the sce for large scale linear structured tikhonov regularization problems

dimension of discrete fractal spaces a new definition of fractal dimension in the case of discrete metric spaces is given it defines the dimension of an arbitrary unbounded subset x of the nu dimensional lattice zv

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

q deformed oscillator algebra as a quantum group it is proved that the q deformed oscillator algebra is a quantum group the hopf algebraic structure is set up and verified to be consistent and compatible the universal r matrix and yang baxter equation are constructed the q differential operator algebra is briefly discussed

error estimation for reduced order models of dynamical systems the use of reduced order models to describe a dynamical system is pervasive in science and engineering often these models are used without an estimate of their error or range of validity in this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition we show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method most importantly the proposed approach allows the assessment of regions of validity for reduced models i e ranges of perturbations in the original system over which the reduced model is still appropriate numerical examples validate our approach the error norm estimates approximate well the forward error while the derived bounds are within an order of magnitude

reply to a paper by m t barlow and s j taylor fractional dimension of sets in discrete systems it is pointed out that the different definitions of the fractional dimension of an arbitrary subset a of d given by naudts 1988 and by barlow and taylor 1989 are interrelated but obtained using completely different criteria

condition numbers for the nonlinear matrix equation and their statistical estimation abstract in this paper we consider four different types of condition numbers i e two normwise ones mixed and componentwise ones for the nonlinear matrix equation x a f x a q and derive their explicit expressions using the probabilistic spectral norm estimator and the statistical condition estimation method these condition numbers can be estimated with high reliability two numerical examples are provided to illustrate the difference between the normwise condition numbers and the mixed and componentwise condition numbers and the efficiency of the estimators respectively

discrete fractal dimensions of the ranges of random walks in mathbb z d associate with random conductances let x x t t ge 0 be a continuous time random walk in an environment of i i d random conductances mu e in 1 infty e in e d where e d is the set of nonoriented nearest neighbor bonds on the euclidean lattice mathbb z d and d 3 let rm r x in mathbb z d x t x rm for some t ge 0 be the range of x it is proved that for almost every realization of the environment dimh r dimp r 2 almost surely where dimh and dimp denote respectively the discrete hausdorff and packing dimension furthermore given any set a subseteq mathbb z d a criterion for a to be hit by x t for arbitrarily large t 0 is given in terms of dimh a similar results for bouchoud s trap model in mathbb z d d 3 are also proven

verma module of the quantum group gl n q and its q boson and heisenberg weyl relation realizations the structure and verma module of the quantum matrix element algebra a n q of the quantum group gl n q are studied the method used here is similar to that for studying the structure and verma module of semisimple lie algebras the concept of the cartan subalgebra raising and lowering operators and their pairs is defined the q boson and the corresponding heisenberg weyl relation realizations are generally studied and the cyclic representations of a n q are obtained the explicit examples a 2 q and a 3 q are discussed in detail

path integral formalism for suq 2 coherent states a path integral formulation in the representation of coherent states for the quantum group suq 2 is introduced an expression for the transition amplitude connecting two suq 2 coherent states is constructed and the corresponding canonical equations of motion are also derived

a new completeness relation in the q deformed two mode fock space in two mode fock space a new completeness relation for the q analogue of the single complex variable state is proved by virtue of some q deformed combinatorial identities which are also derived here for the first time

the q t gaussian process the q t fock space fq t h introduced in this paper is a deformation of the q fock space of bozejko and speicher the corresponding creation and annihilation operators now satisfy the commutation relation aq t f aq t g qaq t g aq t f f g htn turn mathjax off a defining relation of the chakrabarti jagannathan deformed quantum oscillator algebra the moments of the deformed gaussian element sq t h aq t h aq t h are encoded by the joint statistics of crossings and nestings in pair partitions the q 0 t specialization yields a natural single parameter deformation of the full boltzmann fock space of free probability with the corresponding semicircular measure variously encoded via the rogers ramanujan continued fraction the t airy function the t catalan numbers of carlitz riordan and the first order statistics of the reduced wigner process

statistical condition estimation for the roots of polynomials this paper presents fast and reliable condition estimates for the roots of a real polynomial based on the method of statistical condition estimation sce by kenney and laub using relative perturbations we provide both a basic implementation of sce and an advanced one via the condition of the eigenvalues of the corresponding companion matrix new results for estimating the condition of eigenvalues are given fast structured calculations of the eigenvalues of the companion matrix are used and fast structured manipulations of the schur decomposition and the invariant subspaces of the companion matrix are presented the overall process is based on fast operations for sequentially semiseparable structures with small off diagonal ranks the cost of obtaining the condition estimates for all of the roots is o n 2 where n is the degree of the polynomial numerical examples are used to demonstrate the accuracy and efficiency of the proposed methods

planck distribution for a q boson gas the energy density distributions for bosons obeying the q deformation of the harmonic oscillator algebra have been studied in order to obtain some physical insight into the parameter q the action of this parameter on the energy distributions resembles the action of the temperature parameter in the classical planck law q 1

on position and momentum operators in the q oscillator algebra the aim of this paper is to study the position and momentum operators in q deformed oscillator algebras the natural form of the position operator is xp qpn a a qpn where p is a real number this operator is an operator representable by a jacobi matrix using the theory of jacobi matrices the theory of classical moment problem and the theory of basic hypergeometric functions it is shown that depending on values of q and p xp can be unbounded symmetric operator which has the deficiency indices 1 1 and hence is not self adjoint but has self adjoint extensions bounded self adjoint operator with continuous simple spectrum or self adjoint operator of trace class therefore with discrete spectrum with zero as the point of accumulation of eigenvalues the connection of the q deformed heisenberg relation px qxp 1 for the position and momentum operators with a q deformation of the quantum harmonic oscillator is also considered

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

small sample statistical condition estimates for general matrix functions a new condition estimation procedure for general matrix functions is presented that accurately gauges sensitivity by measuring the effect of random perturbations at the point of evaluation in this procedure the number of extra function evaluations used to evaluate the condition estimate determines the order of the estimate that is the probability that the estimate is off by a given factor is inversely proportional to the factor raised to the order of the method the transpose free nature of this new method allows it to be applied to a broad range of problems in which the function maps between spaces of different dimensions this is in sharp contrast to the more common power method condition estimation procedure that is limited in the usual case where the frechet derivative is known only implicitly to maps between spaces of equal dimension a group of examples illustrates the flexibility of the new estimation procedure in handling a variety of problems and types of sensitivity estimates such as mix

intermittency and multifractality a case study via parabolic stochastic pdes let denote space time white noise and consider the following stochastic partial dierential equations i u 1 u 00 u started identically at one and ii z 1 z 00 started identically at zero it is well known that the solution to i is intermittent whereas the solution to ii is not and the two equations are known to be in dierent universality classes we prove that the tall peaks of both systems are multifractals in a natural large scale sense some of this work is extended to also establish the multifractal behavior of the peaks of stochastic pdes on r r d with d 2 g lawler has asked us if intermittency is the same as multifractality the present work gives a negative answer to this question as a byproduct of our methods we prove also that the peaks of the brownian motion form a large scale monofractal whereas the peaks of the ornstein uhlenbeck process on r are multifractal throughout we make extensive use of the macroscopic fractal theory of m t barlow and s j taylor 3 4 we expand on aspects of the barlow taylor theory as well

statistical condition estimation for linear systems the standard approach to measuring the condition of a linear system compresses all sensitivity information into one number thus a loss of information can occur in situations in which the standard condition number with respect to inversion does not accurately reflect the actual sensitivity of a solution or particular entries of a solution it is shown that a new method for estimating the sensitivity of linear systems addresses these difficulties the new procedure measures the effects on the solution of small random changes in the input data and by properly scaling the results obtains reliable condition estimates for each entry of the computed solution moreover this approach which is referred to as small sample statistical condition estimation is no more costly than the standard 1 norm or power method 2 norm condition estimates and it has the advantage of considerable flexibility for example it easily accommodates restrictions on or structure associated with allowable perturbations the method also has a rigorous statistical theory available for the probability of accuracy of the condition estimates however it gives no estimate of an approximate null vector for nearly singular systems the theory of this approach is discussed along with several illustrative examples