reply to comments on the discrete periodic radon transform this paper presents the reply for the comment made on the discrete periodic radon transform by a m grigoryan this comment presents a series of paper about tensor and paired transform as applied to the fast realization of two dimensional discrete fourier transform the reply indicates that the claim made by a m grigoryan was incorrect the contributions of t hsung and d p k lun are far more than that described by a m grigoryan the dprt is not only a forward transform for computing 2 d dft it is a complete discrete transformation method that includes efficient inverse transform

nonlinear programming algorithms using trust regions and augmented lagrangians with nonmonotone penalty parameters the strategy for obtaining global convergence is based on the trust region approach the merit function is a type of augmented lagrangian a new updating scheme is introduced for the penalty parameter by means of which monotone increase is not necessary global convergence results are proved and numerical experiments are presented

a unified method for evaluating the error correction radius of reliability based soft decision algorithms for linear block codes this paper presents a unified method for evaluating the error correction radii of many reliability based soft decision decoding algorithms for binary linear block codes based on this unified method these decoding algorithms as well as their potential improvements can be compared directly the error correction radius for each of these decoding algorithms is determined by finding the closest point on the boundary of the decision region of the soft decision decoder to the transmitted signal sequence it is shown that this problem can be formulated as a constrained optimization problem although no general closed form expression is possible a simple algorithm that always converges to the optimum solution of this optimization problem is presented based on these results the error correction radii of some well known reliability based soft decision decoding algorithms are revisited and compared

blur identification by the method of generalized cross validation the point spread function psf of a blurred image is often unknown a priori the blur must first be identified from the degraded image data before restoring the image generalized cross validation gcv is introduced to address the blur identification problem the gcv criterion identifies model parameters for the blur the image and the regularization parameter providing all the information necessary to restore the image experiments are presented which show that gvc is capable of yielding good identification results a comparison of the gcv criterion with maximum likelihood ml estimation shows the gcv often outperforms ml in identifying the blur and image model parameters

convergence of the iterates of descent methods for analytic cost functions in the early eighties lojasiewicz in seminari di geometria 1982 1983 universita di bologna istituto di geometria dipartimento di matematica 1984 pp 115 117 proved that a bounded solution of a gradient flow for an analytic cost function converges to a well defined limit point in this paper we show that the iterates of numerical descent algorithms for an analytic cost function share this convergence property if they satisfy certain natural descent conditions the results obtained are applicable to a broad class of optimization schemes and strengthen classical weak convergence results for descent methods to strong limit point convergence for a large class of cost functions of practical interest the result does not require that the cost has isolated critical points and requires no assumptions on the convexity of the cost nor any nondegeneracy conditions on the hessian of the cost at critical points

reliability based soft decision decoding with iterative information set reduction the reliability based decoding approach using the reprocessing of the most reliable information set only is extended into the iterative reprocessing of several information sets at the end of each information set reprocessing some information bits are delivered by the decoder consequently information sets with decreasing cardinality values are considered at each iteration a tight upper bound on the error performance achieved by this new method is derived compared to previously proposed competitive approaches this new method reduces the number of candidate codewords needed to achieve practically optimum decoding importantly it also preserves the very simple structured implementation of the order statistic decoding

on the sequential quadratically constrained quadratic programming methods an iteration of the sequential quadratically constrained quadratic programming method sqcqp consists of minimizing a quadratic approximation of the objective function subject to quadratic approximation of the constraints followed by a line search in the obtained direction methods of this class are receiving attention due to the development of efficient interior point techniques for solving subproblems with this structure via formulating them as second order cone programs recently fukushima et al 2003 proposed a sqcqp method for convex minimization with twice continuously differentiable data their method possesses global and locally quadratic convergence and it is free of the maratos effect the feasibility of subproblems in their method is enforced by switching between the linear and quadratic approximations of the constraints this strategy requires computing a strictly feasible point as well as choosing some further parameters we propose a sqcqp method where feasibility of subproblems is ensured by introducing a slack variable and hence is automatic in addition we do not assume convexity of the objective function or twice differentiability of the problem data while our method has all the desirable convergence properties it is easier to implement among other things it does not require computing a strictly feasible point which is a nontrivial task in addition its global convergence requires weaker assumptions

a qp free algorithm without a penalty function or a filter for nonlinear general constrained optimization abstract in this paper we present a qp free algorithm without a penalty function or a filter for nonlinear general constrained optimization at each iteration three systems of linear equations with the same coefficient matrix are solved to yield search direction the nonmonotone line search ensures that the objective function or constraint violation function is sufficiently reduced there is no feasibility restoration phase in our algorithm which is necessary for filter methods the algorithm possesses global convergence as well as superlinear convergence under some mild conditions including a weaker assumption of positive definiteness finally some preliminary numerical results are reported

feasible methods for nonconvex nonsmooth problems with applications in green communications we propose a general feasible method for nonsmooth nonconvex constrained optimization problems the algorithm is based on the inexact solution of a sequence of strongly convex optimization subproblems followed by a step size procedure key features of the scheme are i it preserves feasibility of the iterates for nonconvex problems with nonconvex constraints ii it can handle nonsmooth problems and iii it naturally leads to parallel distributed implementations we illustrate the application of the method to an open problem in green communications whereby the energy consumption in mimo multiuser interference networks is minimized subject to nonconvex quality of service constraints

reliability based code search algorithms for maximum likelihood decoding of block codes efficient code search maximum likelihood decoding algorithms based on reliability information are presented for binary linear block codes the codewords examined are obtained via encoding the information set utilized for encoding comprises the positions of those columns of a generator matrix g of the code which for a given received sequence constitute the most reliable basis for the column space of g substantially reduced computational complexity of decoding is achieved by exploiting the ordering of the positions within this information set the search procedures do not require memory the codeword to be examined is constructed from the previously examined codeword according to a fixed rule consequently the search algorithms are applicable to codes of relatively large size they are also conveniently modifiable to achieve efficient nearly optimum decoding of particularly large codes

active sets nonsmoothness and sensitivity nonsmoothness pervades optimization but the way it typically arises is highly structured nonsmooth behavior of an objective function is usually associated locally with an active manifold on this manifold the function is smooth whereas in normal directions it is vee shaped active set ideas in optimization depend heavily on this structure important examples of such functions include the pointwise maximum of some smooth functions and the maximum eigenvalue of a parametrized symmetric matrix among possible foundations for practical nonsmooth optimization this broad class of partly smooth functions seems a promising candidate enjoying a powerful calculus and sensitivity theory in particular we show under a natural regularity condition that critical points of partly smooth functions are stable small perturbations to the function cause small movements of the critical point on the active manifold

determination of the ma order of an arma process using sample correlations it is shown that autoregressive information and moving average information are implicitly contained in two different correlation matrices an algorithm for ma order determination which adopts only autocorrelations and the ar order without requiring ar coefficients is proposed numerical simulations are presented to show the practical value of the proposed singular value decomposition svd based algorithm

a robust sequential quadratic programming method the sequential quadratic programming method developed by wilson han and powell may fail if the quadratic programming subproblems become infeasible or if the associated sequence of search directions is unbounded this paper considers techniques which circumvent these difficulties by modifying the structure of the constraint region in the quadratic programming subproblems furthermore questions concerning the occurrence of an unbounded sequence of multipliers and problem feasibility are also addressed

on fienup methods for sparse phase retrieval alternating minimization or fienup methods have a long history in phase retrieval we provide new insights related to the empirical and theoretical analysis of these algorithms when used with fourier measurements and combined with convex priors in particular we show that fienup methods can be viewed as performing alternating minimization on a regularized nonconvex least squares problem with respect to amplitude measurements furthermore we prove that under mild additional structural assumptions on the prior semialgebraicity the sequence of signal estimates has a smooth convergent behavior toward a critical point of the nonconvex regularized least squares objective finally we propose an extension to fienup techniques based on a projected gradient descent interpretation and acceleration using inertial terms we demonstrate experimentally that this modification combined with an ell 1 prior constitutes a competitive approach for sparse phase retrieval

soft decision decoding of linear block codes using preprocessing and diversification order w reprocessing is a suboptimal soft decision decoding approach for binary linear block codes in which up to w bits are systematically flipped on the so called most reliable information basis mrb this correspondence first incorporates two preprocessing rules into order w reprocessing and shows that with appropriate choice of parameters the proposed order w reprocessing with preprocessing requires comparable complexity to order w reprocessing but achieves asymptotically the performance of order w 2 reprocessing to complement the mrb a second basis is employed for practical snrs and this approach is systematically extended to a multibasis order w reprocessing scheme for high signal to noise ratios snrs it is shown that the proposed multibasis scheme significantly enlarges the error correction radius a commonly used measure of performance at high snrs over the original single basis order w reprocessing as a by product this approach also precisely characterizes the asymptotic performance of the well known chase and generalized minimum distance gmd decoding algorithms the proposed algorithm successfully decodes the 192 96 reed solomon concatenated code and the 256 147 extended bch code in near optimal manner within 0 01 db at a block error rate of 10 5 with affordable computational cost

asymptotic optimality of the gmd and chase decoding algorithms the generalized minimum distance gmd and chase 1972 decoding algorithms are some of the most important suboptimum bounded distance decoding algorithms for binary linear block codes over an additive white gaussian noise awgn channel we compute the limitation of the ratio between the probability of decoding error for the gmd or any one of the chase decoding algorithms and that of the maximum likelihood ml decoding when the signal to noise ratio snr approaches infinity if the minimum hamming distance of the code is greater than 2 the limitation is shown to be equal to 1 and thus the gmd and chase decoding algorithms are asymptotically optimum

identification and restoration of noisy blurred images using the expectation maximization algorithm a maximum likelihood approach to the blur identification problem is presented the expectation maximization algorithm is proposed to optimize the nonlinear likelihood function in an efficient way in order to improve the performance of the identification algorithm low order parametric image and blur models are incorporated into the identification method the resulting iterative technique simultaneously identifies and restores noisy blurred images

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

efficient blind image restoration using discrete periodic radon transform restoring an image from its convolution with an unknown blur function is a well known ill posed problem in image processing many approaches have been proposed to solve the problem and they have shown to have good performance in identifying the blur function and restoring the original image however in actual implementation various problems incurred due to the large data size and long computational time of these approaches are undesirable even with the current computing machines in this paper an efficient algorithm is proposed for blind image restoration based on the discrete periodic radon transform dprt with dprt the original two dimensional blind image restoration problem is converted into one dimensional ones which greatly reduces the memory size and computational time required experimental results show that the resulting approach is faster in almost an order of magnitude as compared with the traditional approach while the quality of the restored image is similar

reliability based soft decision decoding with multiple biases in this paper a new reliability based soft decision decoding algorithm is presented this algorithm repeatedly uses biased reliability values to construct the most reliable basis mrb as a result this new method makes use of multiple information sets in a stochastic way compared to previously proposed competitive approaches this new method produces a more efficient mrb reprocessing type algorithm to achieve near maximum likelihood decoding mld performance with a proper choice of the bias value it can be combined with any mrb reprocessing type algorithm and in each case a tight performance analysis can be derived

a class of globally convergent optimization methods based on conservative convex separable approximations this paper deals with a certain class of optimization methods based on conservative convex separable approximations ccsa for solving inequality constrained nonlinear programming problems each generated iteration point is a feasible solution with lower objective value than the previous one and it is proved that the sequence of iteration points converges toward the set of karush kuhn tucker points a major advantage of ccsa methods is that they can be applied to problems with a very large number of variables say 104 105 even if the hessian matrices of the objective and constraint functions are dense

dual subgradient algorithms for large scale nonsmooth learning problems classical first order fo algorithms of convex optimization such as mirror descent algorithm or nesterov s optimal algorithm of smooth convex optimization are well known to have optimal theoretical complexity estimates which do not depend on the problem dimension however to attain the optimality the domain of the problem should admit a good proximal setup the latter essentially means that 1 the problem domain should satisfy certain geometric conditions of favorable geometry and 2 the practical use of these methods is conditioned by our ability to compute at a moderate cost proximal transformation at each iteration more often than not these two conditions are satisfied in optimization problems arising in computational learning what explains why proximal type fo methods recently became methods of choice when solving various learning problems yet they meet their limits in several important problems such as multi task learning with large number of tasks where the problem domain does not exhibit favorable geometry and learning and matrix completion problems with nuclear norm constraint when the numerical cost of computing proximal transformation becomes prohibitive in large scale problems we propose a novel approach to solving nonsmooth optimization problems arising in learning applications where fenchel type representation of the objective function is available the approach is based on applying fo algorithms to the dual problem and using the accuracy certificates supplied by the method to recover the primal solution while suboptimal in terms of accuracy guaranties the proposed approach does not rely upon good proximal setup for the primal problem but requires the problem domain to admit a linear optimization oracle the ability to efficiently maximize a linear form on the domain of the primal problem

a comparison of gcv and gml for choosing the smoothing parameter in the generalized spline smoothing problem the partially improper prior behind the smoothing spline model is used to obtain a generalization of the maximum likelihood gml estimate for the smoothing parameter then this estimate is compared with the generalized cross validation gcv estimate both analytically and by monte carlo methods the comparison is based on a predictive mean square error criteria it is shown that if the true unknown function being estimated is smooth in a sense to be defined then the gml estimate undersmooths relative to the gcv estimate and the predictive mean square error using the gml estimate goes to zero at a slower rate than the mean square error using the gcv estimate if the true function is rough then the gcv and gml estimates have asymptotically similar behavior a monte carlo experiment was designed to see if the asymptotic results in the smooth case were evident in small sample sizes mixed results were obtained for n 32 gcv was somewhat better than gml for n 64 and gcv was decidedly superior for n 128 in the n 32 case gcv was better for smaller sigma 2 and the comparison close for larger sigma 2 the theoretical results are shown to extend to the generalized spline smoothing model which includes the estimate of functions given noisy values of various integrals of them

on the complexity of finding first order critical points in constrained nonlinear optimization the complexity of finding epsilon approximate first order critical points for the general smooth constrained optimization problem is shown to be no worse that o epsilon 2 in terms of function and constraints evaluations this result is obtained by analyzing the worst case behaviour of a first order short step homotopy algorithm consisting of a feasibility phase followed by an optimization phase and requires minimal assumptions on the objective function since a bound of the same order is known to be valid for the unconstrained case this leads to the conclusion that the presence of possibly nonlinear nonconvex inequality equality constraints is irrelevant for this bound to apply

discrete radon transform this paper describes the discrete radon transform drt and the exact inversion algorithm for it similar to the discrete fourier transform dft the drt is defined for periodic vector sequences and studied as a transform in its own right casting the forward transform as a matrix vector multiplication the key observation is that the matrix although very large has a block circulant structure this observation allows construction of fast direct and inverse transforms moreover we show that the drt can be used to compute various generalizations of the classical radon transform rt and in particular the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves in fact we describe not a single transform but a class of transforms representatives of which correspond in one way or another to discrete versions of the rt and its generalizations an interesting observation is that the exact inversion algorithm cannot be obtained directly from radon s inversion formula given the fact that the rt has no nontrivial one dimensional analog exact invertibility makes the drt a useful tool geared specifically for multidimensional digital signal processing exact invertibility of the drt flexibility in its definition and fast computational algorithm affect present applications and open possibilities for new ones some of these applications are discussed in the paper

the discrete periodic radon transform in this correspondence a discrete periodic radon transform and its inversion are developed the new discrete periodic radon transform possesses many properties similar to the continuous radon transform such as the fourier slice theorem and the convolution property etc with the convolution property a 2 d circular convolution can be decomposed into 1 d circular convolutions hence improving the computational efficiency based on the proposed discrete periodic radon transform we further develop the inversion formula using the discrete fourier slice theorem it is interesting to note that the inverse transform is multiplication free this important characteristic not only enables fast inversion but also eliminates the finite word length error that may be generated in performing the multiplications

a moving balls approximation method for a class of smooth constrained minimization problems we introduce a new algorithm for a class of smooth constrained minimization problems which is an iterative scheme that generates a sequence of feasible points that approximates the constraints set by a sequence of balls and is accordingly called the moving balls approximation algorithm mba the computational simplicity of mba which uses first order data information makes it suitable for large scale problems theoretical and computational properties of mba in its primal and dual forms are studied and convergence and global rate of convergence results are established for nonconvex and convex problems we then introduce a variant of mba that includes an active set technique and is particularly suitable for problems with a large number of constraints this variant is as simple as mba and is proven to preserve the convergence properties of mba extension of mba is also developed for a class of variational inequalities initial numerical experiments on quadratically constrained problems demonstrate the viability and performance of our methods when compared to some existing state of the art optimization methods software such as a sequential quadratic programming solver from the imsl library and the cvxopt software package for convex optimization

on the reliability order based decoding algorithms for binary linear block codes in this correspondence we consider the decoding of binary block codes over the additive white gaussian noise awgn channel with binary phase shift keying bpsk signaling by a reliability order based decoding algorithm robda we mean a soft decision decoding algorithm which decodes to the best most likely codeword of the form that is the sum of the hard decision tuple and an error pattern in a set determined only by the order of the reliabilities of the hard decisions examples of robdas include many well known decoding algorithms such as the generalized minimum distance gmd decoding algorithm chase decoding algorithms and the reliability based decoding algorithms proposed by fossorier and lin it is known that the squared error correction radii of robdas can be computed from the minimal squared euclidean distances mseds between the all one sequence and the polyhedra corresponding to the error patterns for the computation of such mseds we give a new method which is more compact than the one proposed by fossorier and lin these results are further used to show that any bounded distance robda is asymptotically optimal the ratio between the probability of decoding error of a bounded distance robda and that of the maximum likelihood ml decoding approaches 1 when the signal to noise ratio snr approaches infinity provided that the minimum hamming distance of the code is greater than 2

on the convergence of successive linear quadratic programming algorithms the global convergence properties of a class of penalty methods for nonlinear programming are analyzed these methods include successive linear programming approaches and more specifically the successive linear quadratic programming approach presented by byrd et al math program 100 2004 pp 27 48 every iteration requires the solution of two trust region subproblems involving piecewise linear and quadratic models respectively it is shown that for a fixed penalty parameter the sequence of iterates approaches stationarity of the penalty function a procedure for dynamically adjusting the penalty parameter is described and global convergence results for it are established

a superlinearly convergent sqp method without boundedness assumptions on any of the iterative sequences this paper is aimed to present a new sequential quadratic programming sqp algorithm for finding a solution to nonlinear constrained programming problems with weak conditions where the improved direction can be yielded by solving one quadratic programming qp and the correction direction can be obtained by solving another qp the main characters of the proposed algorithm are as follows first by limiting infeasibility of sqp iterates the boundedness of the iteration sequence can be obtained in the case of the feasible set being nonempty and bounded as well as the constraint functions being convex second global convergence can be proved under slater constraint qualification cq furthermore superlinear convergence can be ensured under suitable conditions third the proposed algorithm is further improved with a bidirectional line search technique finally some numerical experiments are operated to test the proposed algorithms and the results demonstrate that they are promising

global convergence of an sqp method without boundedness assumptions on any of the iterative sequences usual global convergence results for sequential quadratic programming sqp algorithms with linesearch rely on some a priori assumptions about the generated sequences such as boundedness of the primal sequence and or of the dual sequence and or of the sequence of values of a penalty function used in the linesearch procedure different convergence statements use different combinations of assumptions but they all assume boundedness of at least one of the sequences mentioned above in the given context boundedness assumptions are particularly undesirable because even for non pathological and well behaved problems the associated penalty functions whose descent is used to produce primal iterates may not be bounded below for any value of the penalty parameter consequently boundedness assumptions on the iterates are not easily justifiable by introducing a very simple and computationally cheap safeguard in the linesearch procedure we prove boundedness of the primal sequence in the case when the feasible set is nonempty convex and bounded if in addition the slater condition holds we obtain a complete global convergence result without any a priori assumptions on the iterative sequences the safeguard consists of not accepting a further increase of constraints violation at iterates which are infeasible beyond a chosen threshold which can always be ensured by the proposed modified sqp linesearch criterion

majorization minimization algorithms in signal processing communications and machine learning this paper gives an overview of the majorization minimization mm algorithmic framework which can provide guidance in deriving problem driven algorithms with low computational cost a general introduction of mm is presented including a description of the basic principle and its convergence results the extensions acceleration schemes and connection to other algorithmic frameworks are also covered to bridge the gap between theory and practice upperbounds for a large number of basic functions derived based on the taylor expansion convexity and special inequalities are provided as ingredients for constructing surrogate functions with the pre requisites established the way of applying mm to solving specific problems is elaborated by a wide range of applications in signal processing communications and machine learning

iterative soft input soft output decoding of reed solomon codes by adapting the parity check matrix an iterative algorithm is presented for soft input soft output siso decoding of reed solomon rs codes the proposed iterative algorithm uses the sum product algorithm spa in conjunction with a binary parity check matrix of the rs code the novelty is in reducing a submatrix of the binary parity check matrix that corresponds to less reliable bits to a sparse nature before the spa is applied at each iteration the proposed algorithm can be geometrically interpreted as a two stage gradient descent with an adaptive potential function this adaptive procedure is crucial to the convergence behavior of the gradient descent algorithm and therefore significantly improves the performance simulation results show that the proposed decoding algorithm and its variations provide significant gain over hard decision decoding hdd and compare favorably with other popular soft decision decoding methods

a global convergence theory for dennis el alem and maciel s class of trust region algorithms for constrained optimization without assuming regularity this work presents a convergence theory for dennis el alem and maciel s class of trust region based algorithms for solving the smooth nonlinear programming problem with equality constraints the results are proved under very mild conditions on the quasi normal and tangential components of the trial steps the lagrange multiplier estimates and the hessian estimates are assumed to be bounded no regularity assumption is made in particular linear independence of the gradients of the constraints is not assumed the theory proves global convergence for the class in particular it shows that a subsequence of the iteration sequence satisfies one of four types of mayer bliss stationary conditions in the limit

global and local convergence of a penalty free method for nonlinear programming we present a class of trust region algorithms that do not use any penalty function or a filter for nonlinear equality constrained optimization in each iteration the infeasibility is controlled by a progressively decreasing upper limit and trial steps are computed by a byrd omojokun type trust region strategy measures of optimality and infeasibility are computed whose relationship serves as a criterion on which the algorithm decides which one to focus on improving as a result the algorithm keeps a balance between the improvements on optimality and feasibility even if no restoration phase which is required by filter methods is used the framework of the algorithm ensures the global convergence without assuming regularity or boundedness on the iterative sequence by using a second order correction strategy marato s effect is avoided and fast local convergence is shown the preliminary numerical results are reported

on a ast algorithms for decoding short linear block codes the a ast algorithm for decoding linear block codes is an efficient tree search method for decoding short linear block codes the information contained in the parity bits are potentially useful for reducing the edges needed in the tree search we propose to extract the information embedded in the parity bits by considering sums of parity bits which can be decided by part of the message bits analysis for the number of required message bits for obtaining sums of parity bits is provided the information contained in the parity bits can help to reduce the number of search tree edges

a very simple sqcqp method for a class of smooth convex constrained minimization problems with nice convergence results we introduce a new and very simple algorithm for a class of smooth convex constrained minimization problems which is an iterative scheme related to sequential quadratically constrained quadratic programming methods called sequential simple quadratic method ssqm the computational simplicity of ssqm which uses first order information makes it suitable for large scale problems theoretical results under standard assumptions are given proving that the whole sequence built by the algorithm converges to a solution and becomes feasible after a finite number of iterations when in addition the objective function is strongly convex then asymptotic linear rate of convergence is established

majorization minimization procedures and convergence of sqp methods for semi algebraic and tame programs in view of solving nonsmooth and nonconvex problems involving complex constraints like standard nlp problems we study general maximization minimization procedures produced by families of strongly convex subproblems using techniques from semi algebraic geometry and variational analysis in particular ojasiewicz inequality we establish the convergence of sequences generated by these types of schemes to critical points the broad applicability of this process is illustrated in the context of nlp in that case critical points coincide with kkt points when the data are semi algebraic or real analytic our method applies for instance to the study of various sequential quadratic programming sqp schemes the moving balls method the penalized sqp method and the extended sqp method under standard qualification conditions this provides to the best of our knowledge the first general convergence results for general nonlinear programming problems we emphasize the fact that unlike most works on this subjec