the q deformation of quantum mechanics of one degree of freedom the q quantum mechanics of the one degree of freedom is studied among others the holomorphic representation of q deformed heisenberg weyl algebra and its realization by covariant berezin symbols is described

coherent states of the q canonical commutation relations for theq deformed canonical commutation relationsa f a g 1 q f g 1 qa g a f forf g in some hilbert space we consider representations generated from a vector satisfyinga f where we show that such a representation exists if and only if 1 moreover for 1 these representations are unitarily equivalent to the fock representation obtained for 0 on the other hand representations obtained for different unit vectors are disjoint we show that the universal c algebra for the relations has a largest proper closed two sided ideal the quotient by this ideal is a naturalq analogue of the cuntz algebra obtained forq 0 we discuss the conjecture that ford this analogue should in fact be equal to the cuntz algebra itself in the limiting casesq 1 we determine all irreducible representations of the relations and characterize those which can be obtained via coherent states

coherent state maps related to the bounded positive operators we show that for any bounded positive operator h with the simple spectrum one can canonically define two coherent state maps the algebras generated by annihilation operators defined by these coherent state maps are studied we describe also how the toda isospectral deformation of h deforms the corresponding coherent state maps and the related operator algebras

a fast sinc function gridding algorithm for fourier inversion in computer tomography the fourier inversion method for reconstruction of images in computerized tomography has not been widely used owing to the perceived difficulty of interpolating from polar or other measurement grids to the cartesian grid required for fast numerical fourier inversion although the fourier inversion method is recognized as being computationally faster than the back projection method for parallel ray projection data the artifacts resulting from inaccurate interpolation have generally limited application of the method this paper presents a computationally efficient gridding algorithm which can be used with direct fourier transformation to achieve arbitrarily small artifact levels the method has potential for application to other measurement geometries such as fan beam projections and diffraction tomography and nmr imaging

some expansions related to the hubbell rectangular source integral abstract four series expansions are presented and examined for the hubbell rectangular source integral f a b b0 arctan a 1 x 2 1 x 2 1 dx for the extremal situations when one or both of the parameters a b tend to zero or infinity

multivariate hypergeometric functions as τ functions of toda lattice and kadomtsev petviashvili equation we present the q deformed multivariate hypergeometric functions related to schur polynomials as tau functions of the kp and of the two dimensional toda lattice hierarchies the variables of the hypergeometric functions are the higher times of those hierarchies the discrete toda lattice variable shifts parameters of hypergeometric functions the role of additional symmetries in generating hypergeometric tau functions is explained

tau method approximation of some integrals related to radiation field problems abstract a more general and unified form of the hubbell rectangular source integral is defined as h a b p v b d 2 p 1 2 b 2 f beta a 2 2 p d subject to 0 a b p 0 1 special cases of this integral occur in some radiation field problems under certain restrictions h may be expressed in terms of appell s double hypergeometric functions f2 and f4 tau method approximations of the gauss hypergeometric function 2f1 are employed in the evaluation of this integral and upper bound estimates on the tau error are given

w gordon s integral 1929 and its representations by means of appell s functions f2 f1 and f3 exact analytical representations for w gordon s integral by means of appell s functions are given it is shown that they correlate between each other but representation of this integral by means of appell s function f2 is more attractive

c algebraic partial compact quantum groups abstract in this paper we introduce c algebraic partial compact quantum groups which are quantizations of topological groupoids with discrete object set and compact morphism spaces these c algebraic partial compact quantum groups are generalizations of hayashi s compact face algebras to the case where the object set can be infinite they form the c algebraic counterpart of an algebraic theory of partial compact quantum groups developed in an earlier paper by the author and t timmermann the correspondence between which will be dealt with in a separate paper as an interesting example to illustrate the theory we show how the dynamical quantum su 2 group as studied by etingof varchenko and koelink rosengren fits into this framework

integrable multi boson systems and orthogonal polynomials the strict relation between a certain class of multi boson hamiltonian systems and the corresponding class of orthogonal polynomials is established the correspondence is effectively used to integrate the systems as an explicit example we integrate the class of multi boson systems corresponding to q hahn class polynomials

conjugate phase mri reconstruction with spatially variant sample density correction a new image reconstruction method to correct for the effects of magnetic field inhomogeneity in non cartesian sampled magnetic resonance imaging mri is proposed the conjugate phase reconstruction method which corrects for phase accumulation due to applied gradients and magnetic field inhomogeneity has been commonly used for this case this can lead to incomplete correction in part due to the presence of gradients in the field inhomogeneity function based on local distortions to the k space trajectory from these gradients a spatially variant sample density compensation function is introduced as part of the conjugate phase reconstruction this method was applied to both simulated and experimental spiral imaging data and shown to produce more accurate image reconstructions two approaches for fast implementation that allow the use of fast fourier transforms are also described the proposed method is shown to produce fast and accurate image reconstructions for spiral sampled mri

a class of vector coherent states defined over matrix domains a general scheme is proposed for constructing vector coherent states in analogy with the well known canonical coherent states and their deformed versions when these latter are expressed as infinite series in powers of a complex variable z in the present scheme the variable z is replaced by matrix valued functions over appropriate domains as particular examples we analyze the quaternionic extensions of the canonical coherent states and the gilmore perelomov and barut girardello coherent states arising from representations of su 1 1 possible physical applications are indicated

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

sl 2 r symmetry and solvable multiboson systems the one mode and the two mode multiboson systems with sl 2 r symmetry are investigated hamiltonians of these systems are integrated using the theory of orthogonal polynomials the coherent state representation for these systems is constructed

a note on the epstein hubbell generalized elliptic type integral abstract the main object of this note is to present a remarkably simple derivation of the explicit representation of the epstein hubbell generalized elliptic type integral as a gauss hypergeometric function of argument k4 the method applied here is based upon rather elementary formulas and unlike the earlier derivations it does not make use of complex integration with or without residues it is also pointed out how this method can be applied mutatis mutandis in substantially more general cases

a realization of generalized deformed oscillators with cyclic symmetry the algebra of generalized deformed oscillators that possess cyclic symmetry is studied the general set of multilinear operator relations is presented which characterizes the multilinear part of the algebra of cyclic extended deformed oscillators the realization scheme of generalized deformed oscillators with cyclic symmetry in terms of those without cyclic symmetry is established it is shown that all generalized deformed oscillators with cyclic symmetry constructed by the scheme are actually equivalent in the viewpoint of mathematics though each generalized deformed oscillator admits its own particular cyclic extended counterpart examples are given to illustrate this peculiar realization scheme

the darboux like transform and some integrable cases of the q riccati equation using the q version of the darboux transform we obtain the general solution of q difference riccati equation from a special one by the action of one parameter group this allows us to construct the solutions for the large class of q difference riccati equations as well as q difference schrodinger equations which are different from those obtained by the standard darboux transform

representations of coherent states in non orthogonal bases starting with the canonical coherent states we demonstrate that all the so called nonlinear coherent states used in the physical literature as well as large classes of other generalized coherent states can be obtained by changes of bases in the underlying hilbert space this observation leads to an interesting duality between pairs of generalized coherent states bringing into play a gelfand triple of rigged hilbert spaces moreover it is shown that in each dual pair of families of nonlinear coherent states at least one family is related to a generally non unitary projective representation of the weyl heisenberg group which can then be thought of as characterizing the dual pair

algebraic properties of rogers szego functions i applications in quantum optics by means of a well established algebraic framework rogers szego functions associated with a circular geometry in the complex plane are introduced in the context of q special functions and their properties are discussed in detail the eigenfunctions related to the coherent and phase states emerge from this formalism as infinite expansions of rogers szego functions the coefficients being determined through proper eigenvalue equations in each situation furthermore a complementary study on the robertson schrodinger and symmetrical uncertainty relations for the cosine sine and nondeformed number operators is also conducted corroborating in this way certain features of q deformed coherent states

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

some generalised radiation field integrals in this paper two families of integrals that are generalisations of the radiation field integrals are considered i a b 0b 0ao 2 2 1 2 2 1 d d and y a b 0b 0ao 1 lambda 1 d d where and are arbitrary constants except that 0 1 2 and o x is the confluent hypergeometric function of the first kind they are expressed in the form of an infinite series and the corresponding coefficients are evaluated some particular cases are mentioned algorithms for the tabulation of these integrals are developed and tables with better precision than those available in the literature for numerical values have been obtained for some parameter values

on a modified beta function and some applications abstract in this work we study the modified beta function b 1 2 x a b 1 2 x 1 v a 1 v b 1 dv x 0 1 that appeared in marus ic and bajzer s solution m marus ic z bajzer generalized two parameter equation of growth j math anal appl 179 1993 446 462 of the generalized two parameter equation of tumor growth y ay by in particular we provide methods for computing this function for special values of the function s parameters some analytic solutions of the generalized two parameter growth equation are obtained

formulae concerning the computation of the clausen integral cl2 θ the main problem under study concerns the expression of the clausen integral cl2 in closed form in terms of known constants and special functions when is equal to a rational multiple of belonging to 0 2 a general formula giving cl2 p q in terms of the derivative of the di gamma function and the sine function is deduced from an appropriate fourier series expansion some variants of this formula are obtained in further sections the formulae expressing cl2 2 and more generally cl2 m m 2 3 4 as linear combinations of terms of the form cl2 const are established the various results are illustrated by means of typical examples of practical application the last section contains two simple approximations enabling the computation of cl for any in 0 with a relative error smaller than 0 63 and 0 003 resp the paper ends with an appendix in which among other things a peculiar trigonometric identity is established as a by product

on reproducing kernels and quantization of states quantization of a mechanical system with the phase space a kahler manifold is studied it is shown that the calculation of the feynman path integral for such a system is equivalent to finding the reproducing kernel function the proposed approach is applied to a scalar massive conformal particle interacting with an external field which is described by deformation of a hermitian line bundle structure

contiguous relations for the fox wright function abstract contiguous relations are a powerful tool for dealing with hypergeometric functions in this paper four different types of contiguous relations for the fox wright function are given we demonstrate the usefulness of our formulas by showing how they can be used to recover the famous hagen rothe convolution

quantum riemann surfaces i the unit disc we construct a non commutative algebra c mu bar u which is a quantum deformation of the algebra of continuous functions on the closed unit disc bar u c mu bar u is generated by the toeplitz operators on a suitable hilbert space of holomorphic functions onu

some reduction and transformation formulas for the appell hypergeometric function f2 abstract an integral representation of the appell series f 2 1 2 1 2 x y is used here to obtain several finite sum expansions in terms of the less cumbersome hypergeometric functions 2f1 and 3f2 in the case when the parameters 1 2 1 and 2 are all positive integers some of our results may be seen as a generalization of the finite sum expansions of the appell series f 1 obtained by cuyt et al j comput appl math 5 1999 213 219 other important consequences as well as potential applications of our results are discussed and a listing of some useful reduction and transformation formulas is provided

derivatives of horn hypergeometric functions with respect to their parameters the derivatives of eight horn hypergeometric functions four appell f1 f2 f3 and f4 and four degenerate confluent 1 2 1 and 1 with respect to their parameters are studied the first derivatives are expressed systematically as triple infinite summations or alternatively as single summations of two variable kampe de feriet functions taking advantage of previously established expressions for the derivative of the confluent or gaussian hypergeometric functions the generalization to the nth derivative of horn s functions with respect to their parameters is rather straightforward in most cases the results are expressed in terms of n 2 infinite summations following a similar procedure mixed derivatives are also treated an illustration of the usefulness of the derivatives of f1 with respect to the first and third parameters is given with the study of autoionization of atoms occurring as part of a post collisional process their evaluation setting the coulomb charge to zero provides the

selection of a convolution function for fourier inversion using gridding computerised tomography application in the technique known as gridding the data samples are weighted for sampling density and convolved with a finite kernel then resampled on a grid preparatory to a fast fourier transform the authors compare the artifact introduced into the image for various convolving functions of different sizes including the kaiser bessel window and the zero order prolate spheroidal wave function pswf they also show a convolving function that improves upon the pswf in some circumstances

coherent states and bayesian duality we demonstrate how large classes of discrete and continuous statistical distributions can be incorporated into coherent states using the concept of a reproducing kernel hilbert space each family of coherent states is shown to contain in a sort of duality which resembles an analogous duality in bayesian statistics a discrete probability distribution and a discretely parametrized family of continuous distributions it turns out that nonlinear coherent states of the type widely studied in quantum optics are a particularly useful class of coherent states from this point of view in that they contain many of the standard statistical distributions we also look at vector coherent states and multidimensional coherent states as carriers of mixtures of probability distributions and joint probability distributions

characterization of mathcal r p q deformed rogers szego polynomials associated quantum algebras deformed hermite polynomials and relevant properties this paper addresses a new characterization of deformed rogers szego polynomials by providing their three term recurrence relation and the associated quantum algebra built with corresponding creation and annihilation operators the whole construction is performed in a unified way generalizing all known relevant results which are straightforwardly derived as particular cases continuous deformed hermite polynomials and their recurrence relation are also deduced novel relations are provided and discussed

vector coherent states from plancherel s theorem clifford algebras and matrix domains as a substantial generalization of the technique for constructing canonical and the related nonlinear and q deformed coherent states we present here a method for constructing vector coherent states vcs in the same spirit these vcs may have a finite or an infinite number of components the resulting formalism which involves an assumption on the existence of a resolution of the identity is broad enough to include all the definitions of coherent states existing in the current literature subject to this restriction as examples we first apply the technique to construct vcs using the plancherel isometry for groups and vcs associated with clifford algebras in particular quaternions as physical examples we discuss vcs for a quantum optical model and finally apply the general technique to build vcs over certain matrix domains

gauss legendre approximations for the hubbell rectangular source integral gauss legendre quadrature is used to develop a family of approximation formulas for the accurate numerical computation of the hubbell rectangular source integral

coherent states for systems with discrete and continuous spectrum for a system with one degree of freedom coherent states that are parametrized by classical canonical action angle variables are introduced these states also possess continuity of labelling a resolution of unity and temporal stability the insistence on canonical action angle variables strongly restricts any remaining arbitrariness in the coherent state definition such states are introduced for semibounded hamiltonian operators having either a discrete or a continuous spectrum hamiltonians that have both discrete and continuous parts in their spectrum are also discussed

the construction of some important classes of generalized coherent states the nonlinear coherent states method considering some important classes of generalized coherent states known in the literature we demonstrated that all of them can be created via conventional methods i e the lowering operator eigenstate and the displacement operator techniques using the nonlinear coherent states approach as a result we obtained a unified method to construct a large class of coherent states which have already been introduced by different prescriptions

q ν deformation of generalized basic hypergeometric states we provide with a q deformation of the generalized hypergeometric coherent states defined such that the normalization function is given by generalized basic hypergeometric functions these states are eigenstates of suitably defined deformed lowering operators we study the domain of convergence of the corresponding normalization function on the basis of these states we investigate generalized basic hypergeometric husimi distributions and corresponding phase distributions as well as new analytic basis representations of arbitrary quantum states in bargmann and hardy spaces the quantum statistical properties of the states such as photon counting statistics and quadrature squeezing are analytically and numerically discussed in the framework of conventional quantum optics

calculation of the generalized hubbell rectangular source integrals using binomial coefficients the series expansion formulas in terms of binomial coefficients are derived for the family of generalized source integrals hrsis appearing in the evaluation of radiation field from a plane isotropic rectangular source the formulas given in this study for the evaluation of radiation integrals hrsis show good rate of convergence and great numerical stability the results obtained by the present approach are found to be in excellent agreement with other theoretical studies

recursion formulas for appell s hypergeometric function f2 with some applications to radiation field problems abstract using some contiguous function relations for the classical gauss hypergeometric series 2 f 1 we establish a number of new recursion formulas for the appell functions f 2 1 2 n 1 2 x y and f 2 1 2 1 2 n x y where n n 0 0 1 2 n 0 some important applications of our results to the evaluation of radiation field integrals are discussed and a listing of some useful reduction and transformation formulas is provided

superfast tikhonov regularization of toeplitz systems toeplitz structured linear systems arise often in practical engineering problems correspondingly a number of algorithms have been developed that exploit toeplitz structure to gain computational efficiency when solving these systems the earliest fast algorithms for toeplitz systems required o n 2 operations while more recent superfast algorithms reduce the cost to o n log n 2 or below in this work we present a superfast algorithm for tikhonov regularization of toeplitz systems using an extension and transformation technique our algorithm translates a tikhonov regularized toeplitz system into a type of specialized polynomial problem known as tangential interpolation under this formulation we can compute the solution in only o n log n 2 operations we use numerical simulations to demonstrate our algorithm s complexity and verify that it returns stable solutions

the gridding method for image reconstruction by fourier transformation the authors explore a computational method for reconstructing an n dimensional signal f from a sampled version of its fourier transform f spl circ the method involves a window function w spl circ and proceeds in three steps first the convolution g spl circ w spl circ f spl circ is computed numerically on a cartesian grid using the available samples of f spl circ then g wf is computed via the inverse discrete fourier transform and finally f is obtained as g w due to the smoothing effect of the convolution evaluating w spl circ f spl circ is much less error prone than merely interpolating f spl circ the method was originally devised for image reconstruction in radio astronomy but is actually applicable to a broad range of reconstructive imaging methods including magnetic resonance imaging and computed tomography in particular it provides a fast and accurate alternative to the filtered backprojection the basic method has several variants with other applications such as the equidistant resampling of arbitrarily sampled signals or the fast computation of the radon hough transform

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

resampling of data between arbitrary grids using convolution interpolation for certain medical applications resampling of data is required in magnetic resonance tomography mrt or computer tomography ct e g data may be sampled on nonrectilinear grids in the fourier domain for the image reconstruction a convolution interpolation algorithm often called gridding can be applied for resampling of the data onto a rectilinear grid resampling of data from a rectilinear onto a nonrectilinear grid are needed e g if projections of a given rectilinear data set are to be obtained in this paper the authors introduce the application of the convolution interpolation for resampling of data from one arbitrary grid onto another the basic algorithm can be split into two steps first the data are resampled from the arbitrary input grid onto a rectilinear grid and second the rectilinear data is resampled onto the arbitrary output grid furthermore the authors like to introduce a new technique to derive the sampling density function needed for the first step of their algorithm for fast sampling pattern independent determination of the sampling density function the voronoi diagram of the sample distribution is calculated the volume of the voronoi cell around each sample is used as a measure for the sampling density it is shown that the introduced resampling technique allows fast resampling of data between arbitrary grids furthermore it is shown that the suggested approach to derive the sampling density function is suitable even for arbitrary sampling patterns examples are given in which the proposed technique has been applied for the reconstruction of data acquired along spiral radial and arbitrary trajectories and for the fast calculation of projections of a given rectilinearly sampled image

rapid gridding reconstruction with a minimal oversampling ratio reconstruction of magnetic resonance images from data not falling on a cartesian grid is a fourier inversion problem typically solved using convolution interpolation also known as gridding gridding is simple and robust and has parameters the grid oversampling ratio and the kernel width that can be used to trade accuracy for computational memory and time reductions we have found that significant reductions in computation memory and time can be obtained while maintaining high accuracy by using a minimal oversampling ratio from 1 125 to 1 375 instead of the typically employed grid oversampling ratio of two when using a minimal oversampling ratio appropriate design of the convolution kernel is important for maintaining high accuracy we derive a simple equation for choosing the optimal kaiser bessel convolution kernel for a given oversampling ratio and kernel width as well we evaluate the effect of presampling the kernel a common technique used to reduce the computation time and find that using linear interpolation between samples adds negligible error with far less samples than is necessary with nearest neighbor interpolation we also develop a new method for choosing the optimal presampled kernel using a minimal oversampling ratio and presampled kernel we are able to perform a three dimensional 3 d reconstruction in one eighth the time and requiring one third the computer memory versus using an oversampling ratio of two and a kaiser bessel convolution kernel while maintaining the same level of accuracy

vector coherent states with matrix moment problems canonical coherent states can be written as infinite series in powers of a single complex number z and a positive integer m the requirement that these states realize a resolution of the identity typically results in a moment problem where the moments form the positive sequence of real numbers m m 0 in this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number z and the moments m of the canonical coherent states by n n matrices associated oscillator algebras are discussed with the aid of a generalized matrix factorial two physical examples are discussed in the first example coherent states are obtained for the jaynes cummings model in the weak coupling limit and some physical properties are discussed in terms of the constructed coherent states in the second example coherent states are obtained for a conditionally exactly solvable supersymmetric radial harmonic oscillator

coherent states and geometric quantization based on the concept of generalized coherent states a theory of mechanical systems is formulated in a way which naturally exhibits the mutual relation of classical and quantum aspects of physical phenomena

quantum algebras and q special functions related to coherent states maps of the disc the quantum algebras generated by the coherent states maps of the disc are investigated it is shown that the analytic realization of these algebras leads to a generalized analysis which includes standard analysis as well as q analysis the applications of the analysis to star product quantizations and q special functions theory are given among others the meromorphic continuation of the generalized basic hypergeometric series is found and a reproducing measure is constructed when the series is treated as a reproducing kernel

a completeness relation for the coherent states of the p q oscillator by p q integration p q integration is defined for the p q oscillator this is used to prove a completeness relation for the coherent states of the p q oscillator the p q analogue of the bargmann fock representation is also discussed

generalized bivariate beta distributions involving appell s hypergeometric function of the second kind let x 1 x 2 and x 3 be independent random variables x 1 and x 2 having a confluent hypergeometric function kind 1 distribution with probability density function proportional to x i n i 1 1f 1 a i b i x i i 1 2 and x 3 having a standard gamma distribution with shape parameter n 3 define y 1 y 2 x 1 x 3 x 2 x 3 and z 1 z 2 x 1 x 2 x 1 x 2 x 3 in this article we derive probability density functions of y 1 y 2 and z 1 z 2 and study their properties we use the second hypergeometric function of appell to express these density functions

fast higher order mr image reconstruction using singular vector separation medical resonance imaging mri conventionally relies on spatially linear gradient fields for image encoding however in practice various sources of nonlinear fields can perturb the encoding process and give rise to artifacts unless they are suitably addressed at the reconstruction level accounting for field perturbations that are neither linear in space nor constant over time i e dynamic higher order fields is particularly challenging it was previously shown to be feasible with conjugate gradient iteration however so far this approach has been relatively slow due to the need to carry out explicit matrix vector multiplications in each cycle in this work it is proposed to accelerate higher order reconstruction by expanding the encoding matrix such that fast fourier transform can be employed for more efficient matrix vector computation the underlying principle is to represent the perturbing terms as sums of separable functions of space and time compact representations with this property are found by singular vector analysis of the perturbing matrix guidelines for balancing the accuracy and speed of the resulting algorithm are derived by error propagation analysis the proposed technique is demonstrated for the case of higher order field perturbations due to eddy currents caused by diffusion weighting in this example image reconstruction was accelerated by two orders of magnitude