arbitrary adaptive pole placement for linear multivariable systems this paper presents a methodology for adaptively assigning the closed loop poles of either continuous or discrete time linear mnltivariable systems the scheme is direct in nature in that no parametrized model of the unknown plant needs to be explicitely identified rather the control parameters and an equivalent plant parameterization are simultaneously estimated from input output data using linear parameter estimation procedures implementation of the scheme requires only knowledge of the system controllability indexes and an upper bound on the observabilily index

a survey of the theory of source coding with a fidelity criterion the purpose of this paper is threefold 1 to acquaint the reader with the types of problems that have been considered in the area of source coding with a fidelity criterion 2 to survey results that have been obtained on these problems and 3 to outline future research trends in the area

an algorithm for the computation of eigenvalues spectral zeta functions and zeta determinants on hyperbolic surfaces we present a rigorous scheme that makes it possible to compute eigenvalues of the laplace operator on hyperbolic surfaces within a given precision the method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions it can be applied to check whether or not there is an eigenvalue in an e neighborhood of a given number 0 this makes it possible to find all the eigenvalues in a specified interval up to a given precision with rigorous error estimates the method converges exponentially fast with the number of basis functions used combining the knowledge of the eigenvalues with the selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds as an example we calculate the spectral determinant and the casimir energy of the bolza surface and other surfaces

remarks on persistency of excitation for nonminimal models of systems having purely deterministic disturbances a persistency of excitation result for nonminimal models of systems having purely deterministic disturbances has been given recently in 1 in this note we give a direct and simple proof for the case of bounded signals of the result of 1 and show via a counterexample that an assumption on system signals boundedness which does not appear in 1 is necessary when arbitrary unbounded signals are allowable finally we give a proper persistency of excitation result which does not require signal boundedness for the case when the system input consists of a bounded external signal and of a constant coefficient feedback component

stable hybrid adaptive control this paper examines the asymptotic properties of a hybrid adaptive control scheme while there are many different meanings of the word hybrid control in the literature we are referring to a scheme where the on line parameter estimation and underlying control design are carried out in a discrete time using digital electronics and do not require any analog multiplications or divisions this is of critical importance in practice since analog multipliers dividers are subject to serious drift and offset errors for such hybrid schemes we establish stability and an output tracking property without recourse to persistence of exitation the proof of stability requires a sufficiently fast sampling rate for the discrete time parameter estimation and the underlying control design

an on line universal lossy data compression algorithm via continuous codebook refinement ii optimality for phi mixing source models for pt i see ibid vol 42 no 3 p 803 21 1996 two versions of the gold washing data compression algorithm one with codebook innovation interval and the other with finitely many codebook innovations are considered the version of the gold washing algorithm with codebook innovation interval k is a variant of the gold washing algorithm such that the codebook is innovated once every k 1 source words during the process of encoding the entire source it is demonstrated that when this version of the gold washing algorithm is applied to encode a stationary spl phi mixing source the expected distortion performance converges to the distortion rate function of the source as the codebook length goes to infinity furthermore if the source to be encoded is a markov source or a finite state source then the corresponding sample distortion performance converges almost surely to the distortion rate function the version of the gold washing algorithm with finitely many codebook innovations is a variant of the gold washing algorithm in which after finitely many codebook innovations the codebook is held fixed and reused to encode the forthcoming source sequence block by block similar results are shown for this version of the gold washing algorithm in addition the convergence speed of the algorithm is discussed

extremal properties of the determinant of the laplacian in the bergman metric on the moduli space of genus two riemann surfaces we study extremal properties of the determinant of the laplacian in the bergman metric on the moduli space of compact genus two riemann surfaces by a combination of analytical and numerical methods we identify four non degenerate critical points of this function and compute the signature of the hessian at these points the curve with the maximal number of automorphisms the burnside curve turns out to be the point of the absolute maximum our results agree with the mass formula for virtual euler characteristics of the moduli space a similar analysis is performed for bolza s strata of symmetric riemann surfaces of genus two

stable adaptive regulation of arbitrary nth order plants this paper presents an algorithm for adaptively stabilizing and asymptotically regulating an arbitrary single input single output linear time invariant plant which is controllable and observable of known order n and has unknown parameters no further assumptions are made no external probing signal is required

robustness of an adaptive pole placement algorithm in the presence of bounded disturbances and slow time variation of parameters the robustness properties of a specific adaptive pole placement scheme in the presence of bounded disturbances and slow time variation of parameters are studied it is shown that the least squares algorithm used in this scheme retains the global stability of the overall system in the presence of bounded disturbances of small magnitude and slow time variation of system parameters within a compact set

an on line universal lossy data compression algorithm via continuous codebook refinement iii redundancy analysis for pt ii see ibid vol 42 p 822 36 1996 the gold washing data compression algorithm is an adaptive vector quantization algorithm with vector dimension n in this paper a redundancy problem of the gold washing data compression algorithm is considered it is demonstrated that for any memoryless source with finite alphabet a and generic distribution p and for any r 0 the redundancy of the gold washing data compression algorithm with dimension n defined as the difference between the average performance of the algorithm and the distortion rate function d p r of p is upper bounded by sub spl delta r sup spl delta d p r a 2 spl xi 4 log n 2n spl sigma logn n where sub spl delta r sup spl delta d p r is the partial derivative of d p r with respect to r a is the cardinality of a and spl xi 0 is a parameter used to control the threshold in the gold washing algorithm in connection with the results of zhang yang and wei see ibid vol 43 no 1 p 71 91 1997 on the redundancy of lossy source coding this shows that the gold washing algorithm has the optimal convergence rate among all adaptive finite state vector quantizers

the performance of universal encoding universal coding theory is surveyed from the viewpoint of the interplay between delay and redundancy the price for universality turns out to be acceptably small

adaptive regulation of mimo linear discrete time systems without requiring a persistent excitation an adaptive regulator which does not require a persistent excitation is developed for multiinput multioutput mimo linear discrete time systems the assumptions on the plant are 1 an upper bound on the system dimension is known 2 the unknown system parameters belong to known bounded intervals 3 the plant is stabilizable for all possible values of the unknown system parameters the stabilizability condition can be tested before the system is put into operation a recursive parameter estimation scheme is given which forces the parameter estimates to converge to the known intervals using this estimator an adaptive regulator is derived which results in a globally stable closed loop system

adaptive stabilization of unstable and nonminimum phase stochastic systems abstract the system under consideration is the discrete time stochastic system a z yn zb z un wn driven by a martingale difference sequence wn where a z are polynomilas in backward shift operator z with unknown coefficients and both a z and b z may be unstable with the purpose of demonstrating theoretical possibility rather than designing a practically applicable control law this paper constructs an adaptive control that stabilizes the system and simultaneously guarantees strong consistency of the least squares estimates for unknown coefficients

a comment on a rate of convergence result for a universal d semifaithful code in the above paper see ibid vol 39 no 3 p 813 20 1993 yu and speed propose a universal pointwise d semifaithful code whose expected compression ratio for discrete memoryless sources approaches the rate distortion function at a rate o n sup 1 log n they also conjecture that this is the fastest achievable convergence rate for pointwise d semifaithful codes in this correspondence we use a simple extension of kraft s inequality and prove that this conjecture is true at least for the hamming distortion measure

global stability of adaptive pole placement algorithms this paper presents direct and indirect adaptive control schemes for assigning the closed loop poles of a single input single output system in both the continuous and discrete time cases the resulting closed loop system is shown to be globally stable when driven by an external reference signal consisting of a sum of sinusoids in particular persistent excitation of the potentially unbounded closed loop input output data and hence convergence of a sequential least squares identification algorithm is proved the results are applicable to standard sequential least squares and least squares with covariance reset

global adaptive pole positioning adaptive pole positioning for linear time invariant discrete time systems is considered under the constraint that the controller consists of an identifier the gains in which do not go to zero an observer and a state feedback law the latter two elements being viewed in transfer function form a globally convergent algorithm is presented for achieving a prescribed set of closed loop poles persistency of excitation of an external input is required together with an underbound on the magnitude of the sylvester resultant of the plant numerator and denominator polynomials this in effect being an underbound on the product of a measure of pole zero separation and a generalized gain

fixed rate universal lossy source coding and rates of convergence for memoryless sources a fixed rate universal lossy coding scheme is introduced for independent and identically distributed i i d sources it is shown for finite alphabet sources and arbitrary single letter distortion measures that as the sample size n grows the expected distortion obtained using this universal scheme converges to shannon s distortion rate function d r at a rate o log n n the scheme can be extended to universal quantization of real i i d sources subject to a squared error criterion it is shown in this case that the per letter distortion converges to d r at a rate o spl radic log n n both in expectation and almost surely for any real valued bounded i i d source

stability analysis of elliott s direct adaptive pole placement abstract the stability of elliott s direct adaptive pole placement algorithm is examined without the persistency of excitation assumption the conditions for the sequence of estimated parameters that are sufficient for the plant input and the plant output to be bounded are given it is also shown that the modifications proposed in many papers for ensuring the global stability of the indirect adaptive pole placement algorithm can be also applied to the direct one if an auxiliary estimation algorithm for identifying the plant parameters is applied

indirect adaptive pole placement control of mimo stochastic systems self tuning results in this paper we consider indirect adaptive pole placement control appc of linear multivariable stochastic systems instead of the canonical representation often used in the literature we propose using a non minimal but otherwise uniquely identifiable pseudo canonical parameterization that is more suitable for multivariable armax model identification to identify the plant we use the weighted extended least squares wels algorithm a least squares method with slowly decreasing weights which was introduced in bercu 1995 the pole placement controller parameters are then calculated by using a certain perturbation of the parameter estimates such that the linear models corresponding to the perturbed estimates are uniformly controllable and observable we prove that with a reasonable amount of prior information the resulting appc scheme is globally stabilizing and asymptotically self tuning regardless of the degree of persistency of external excitation these results represent the most complete study of stochastic multivariable appc systems to this date

redundancy of the lempel ziv incremental parsing rule the lempel ziv codes are universal variable to fixed length codes that have become virtually standard in practical lossless data compression for any given source output string from a markov or unifilar source we upper bound the difference between the number of binary digits needed to encode the string and the self information of the string we use this result to demonstrate that for unifilar or markov sources the redundancy of encoding the first n letters of the source output with the lempel ziv incremental parsing rule lz 78 the welch modification lzw or a new variant is o ln n sup 1 and we upper bound the exact form of convergence we conclude by considering the relationship between the code length and the empirical entropy associated with a string

robust adaptive control of discrete nominally stabilizable plants this paper presents an indirect pole placement based adaptive control scheme for discrete linear time invariant non necessarily inversely stable and stabilizable systems the scheme is a pole placement type without requiring the plant inverse stability assumption the control objective is that the plant output should asymptotically track in the absence of unmodeled dynamics and bounded disturbances a reference signal given by an arbitrary stable filter under a bounded tracking error while ensuring robust closed loop stability the adaptive stabilizability and the robustness of this system under the presence of unmodeled dynamics and possibly bounded disturbances are proved without assuming the controllability of the modeled plant a normalized least squares algorithm with a relative adaptation dead zone and a posteriori modification of the parameter estimates is used to ensure the controllability of the modified estimation plant model at all sampling instants and at the limit such a property is crucial to solve the stable pole placement problem on the other hand the relative adaptation dead zone is included for robustness purposes under unmodeled dynamics and possibly bounded disturbances

on the parameter convergence properties of a combined vs adaptive control scheme during sliding motion the effects of the use of a variable structure control component within an adaptive control scheme previously proposed by the authors 1992 1995 are investigated in this paper the discontinuous control action does not allow the explicit identification of the coefficients of the filters but only the identification of the so called equivalent control coinciding with the ideal control by coupling the variable structure control strategy with a continuous parameter adjustment mechanism it is instead possible to obtain an equivalent control which during the sliding motion turns out to be equal to a suitably constructed prediction error as long as the parameter adaptation is driven by both the tracking error which vanishes in finite time and the prediction error then the controller parameters prove to be exponentially convergent to the ideal values provided that the reference signal is sufficiently persistently exciting

on the performance of data compression algorithms based upon string matching lossless and lossy data compression algorithms based on string matching are considered in the lossless case a result of wyner and ziv 1989 is extended in the lossy case a data compression algorithm based on approximate string matching is analyzed in the following two frameworks 1 the database and the source together form a markov chain of finite order 2 the database and the source are independent with the database coming from a markov model and the source from a general stationary ergodic model in either framework it is shown that the resulting compression rate converges with probability one to a quantity computable as the infimum of an information theoretic functional over a set of auxiliary random variables the quantity is strictly greater than the rate distortion function of the source except in some symmetric cases in particular this result implies that the lossy algorithm proposed by steinberg and gutman 1993 is not optimal even for memoryless or markov sources

on the average redundancy rate of the lempel ziv code in this paper we settle a long standing open problem concerning the average redundancy r sub n of the lempel ziv 78 lz78 code we prove that for a memoryless source the average redundancy rate attains asymptotically er sub n a spl delta n log n o log log n log sup 2 n where a is an explicitly given constant that depends on source characteristics and spl delta x is a fluctuating function with a small amplitude we also derive the leading term for the kth moment of the number of phrases we conclude by conjecturing a precise formula on the expected redundancy for a markovian source the main result of this paper is a consequence of the second order properties of the lempel ziv algorithm obtained by jacquet and szpankowski 1995 these findings have been established by analytical techniques of the precise analysis of algorithms we give a brief survey of these results since they are interesting in their own right and shed some light on the probabilistic behavior of pattern matching based data compression

exponential convergence of adaptive identification and control algorithms output and equation error adaptive identification algorithms are shown to be exponentially convergent under a deterministic or stochastic persistently exciting or spanning condition on the system inputs together with several other standard conditions an adaptive control algorithm is shown to be exponentially convergent under a deterministic or stochastic persistently exciting condition on the reference trajectory together with some standard conditions

the redundancy of source coding with a fidelity criterion ii coding at a fixed rate level with unknown statistics the redundancy problem of universal lossy source coding at a fixed rate level is considered under some condition on the single letter distortion measure which implies that the cardinality k of the reproduction alphabet is not greater than the cardinality j of the source alphabet it is shown that the redundancy of universally coding memoryless sources p by nth order block codes of rate r goes like spl part spl part r d p r kln n 2n o ln n n for all memoryless sources p except a set whose volume goes to 0 as the block length n goes to infinity where d p r denotes the distortion rate function of p specifically for any sequence c sub n sub n 1 sup spl infin of block codes where c sub n is an nth order block code at the fixed rate r and any spl epsiv 0 the redundancy d sub n c sub n p of c sub n for p is greater than or equal to spl part spl part r d p r k spl epsiv ln n 2n for all p satisfying some regular conditions except a set whose volume goes to 0 as n spl rarr spl infin on the other hand there exists a sequence c sub n sub n 1 sup spl infin of block codes at the rate r such that for any p satisfying some regular conditions the super limit of d sub n c sub n p ln n n is less than or equal to spl part spl part r d p r k 2

explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions abstract we consider transfer operators acting on spaces of holomorphic functions and provide explicit bounds for their eigenvalues more precisely if is any open set in c d and l is a suitable transfer operator acting on bergman space a 2 its eigenvalue sequence n l is bounded by n l a exp a n 1 d where a a 0 are explicitly given

extremal metrics for spectral functions of dirac operators in even and odd dimensions abstract let m n g be a closed smooth riemannian spin manifold and denote by its atiyah singer dirac operator we study the variation of riemannian metrics for the zeta function and functional determinant of 2 and prove finiteness of the morse index at stationary metrics and local extremality at such metrics under general i e not only conformal change of metrics in even dimensions which is also a new case for the conformal laplacian the relevant stability operator is of log polyhomogeneous pseudodifferential type and we prove new results of independent interest on the spectrum for such operators we use this to prove local extremality under variation of the riemannian metric which in the important example when m n g is the round n sphere gives a partial verification of branson s conjecture on the pattern of extremals thus det 2 has a local max max min min when the dimension is 4 k 4 k 1 4 k 2 4 k 3 respectively

direct model reference adaptive pole placement control with exponential weighting properties a parameterization for a linear system is presented to design a direct model reference adaptive pole placement controller this parameterized model is one of the structured nonminimal models the exponentially weighted least squares algorithm is employed to estimate the control parameters the direct adaptive controller has the exponential weighting properties by the proposed method of selecting the characteristic polynomials of the sensitivity function filters in connection with the reference model

the shortest common superstring problem average case analysis for both exact and approximate matching the shortest common superstring problem and its extension to approximate matching are considered in the probability model where each string in a given set has the same length and letters of strings are drawn independently from a finite set in the exact matching case several algorithms proposed in the literature are shown to be asymptotically optimal in the sense that the ratio of the savings resulting from the superstring constructed by each of these algorithms that is the difference between the total length of the strings in the given set and the length of the superstring to the optimal savings from the shortest superstring approaches in probability to 1 as the number of strings in the given set increases in the approximate matching case a modified version of the shortest common approximate matching superstring problem is analyzed it is demonstrated that the optimal savings in this case is given approximately by nlogn i sub l q q 2d where n is the number of strings in the given set q is the probability distribution governing the selection of letters of strings i sub l q q 2d is the lower mutual information between q and q with respect to 2d and d spl ges 0 is the distortion allowed in approximate matching in addition an approximation algorithm is proposed and proved asymptotically optimal

on the redundancy of lossy source coding with abstract alphabets the redundancy problem of lossy source coding with abstract source and reproduction alphabets is considered for coding at a fixed rate level it is shown that for any fixed rate r 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the rate r such that the distortion redundancy of c sub n defined as the difference between the performance of c sub n and the distortion rate function d p r of p is upper bounded by spl part d p r spl part r ln n 2n o ln n n for coding at a fixed distortion level it is demonstrated that for any d 0 and any memoryless abstract alphabet source p satisfying some mild conditions there exists a sequence c sub n sub n 1 sup spl infin of block codes at the fixed distortion d such that the rate redundancy of c sub n defined as the difference between the performance of c sub n and the rate distortion function r p d of p is upper bounded by 7ln n 6n o ln n n these results strengthen the traditional berger s 1968 1971 abstract alphabet source coding theorem and extend the positive redundancy results of zhang yang and wei see ibid vol 43 no 1 p 71 91 1997 and ibid vol 42 p 803 21 1996 on lossy source coding with finite alphabets and the redundancy result of wyner see ibid vol 43 p 1452 64 1997 on block coding of memoryless gaussian sources

sequential codes lossless compression of individual sequences and kolmogorov complexity a general class of sequential codes for lossless compression of individual sequences on a finite alphabet is defined including many types of codes that one would want to implement the principal requirement for membership in the class is that the encoding and decoding operations be performable on a computer the opta function for the class of codes is then considered which is the function that assigns to each individual sequence the infimum of the rates at which the sequence can be compressed over this class of sequential codes two results about the opta function are obtained 1 it is shown that any sequential code in the class compresses some individual sequence at a rate strictly greater than the rate for that sequence given by the opta function and 2 it is shown that the opta function takes a value strictly greater than that of the kolmogorov 1965 complexity rate function for some individual sequences

an on line universal lossy data compression algorithm via continuous codebook refinement i basic results a new on line universal lossy data compression algorithm is presented for finite memoryless sources with unknown statistics its performance asymptotically approaches the fundamental rate distortion limit the codebook is generated on the fly and continuously adapted by simple rules there is no separate codebook training or codebook transmission candidate codewords are randomly generated according to an arbitrary and possibly suboptimal distribution through a carefully designed gold washing or information theoretic sieve mechanism good codewords and only good codewords are promoted to permanent status with high probability we also determine the rate at which our algorithm approaches the fundamental limit

a rate of convergence result for a universal d semifaithful code the problem of optimal rate universal coding is considered in the context of rate distortion theory a d semifaithful universal coding scheme for discrete memoryless sources is given the main result is a refined covering lemma based on the random coding argument and the method of types the average codelength of the code is shown to approach its lower bound the rate distortion function at a rate o n sup 1 log n and this is conjectured to be optimal based on a result of a j pilc 1968 issues of constructiveness and universality are addressed

global adaptive pole placement detailed analysis of a first order system this paper studies a deterministic adaptive pole positioning problem for a linear first order system the principal contribution is to specify an adaptive control law which secures global convergence the key conditions required are that an externally available plant input be persistently exciting and that an underbound not necessarily tight be known on the magnitude of the input coupling gain the pattern of the solution to the problem can carry over to high order systems

analytic torsion and closed geodesics on hyperbolic manifolds pour une variete hyperbolique orientee fermee x et une representation orthogonale de 1 x on definit une fonction zeta de type ruelle r s pour re s grand comme le produit sur les geodesiques closes premieres des facteurs det i exp sl ou l est la longueur de on continue analytiquement r s et on calcule son terme principal en s 0

a suboptimal lossy data compression based on approximate pattern matching a practical suboptimal variable source coding algorithm for lossy data compression is presented this scheme is based on approximate string matching and it naturally extends the lossless lempel ziv 1977 data compression scheme among others we consider the typical length of an approximately repeated pattern within the first n positions of a stationary mixing sequence where d percent of mismatches is allowed we prove that there exists a constant r sub 0 d such that the length of such an approximately repeated pattern converges in probability to 1 r sub 0 d log n pr but it almost surely oscillates between 1 r sub spl infin d log n and 2 r sub 1 d log n where r sub spl infin d r sub 0 d r sub 1 d 2 are some constants these constants are natural generalizations of renyi entropies to the lossy environment more importantly we show that the compression ratio of a lossy data compression scheme based on such an approximate pattern matching is asymptotically equal to r sub 0 d we also establish the asymptotic behavior of the so called approximate waiting time n sub l which is defined as the time until a pattern of length c repeats approximately for the first time we prove that log n sub l l spl rarr r sub 0 d pr as l spl rarr spl infin in general r sub 0 d r d where r d is the rate distortion function thus for stationary mixing sequences we settle in the negative the problem investigated by steinberg and gutman by showing that a lossy extension of the wyner ziv 1989 scheme cannot be optimal

the selberg zeta function for convex co compact schottky groups we give a new upper bound on the selberg zeta function for a convex co compact schottky group acting on the hyperbolic space n 1 in strips parallel to the imaginary axis the zeta function is bounded by exp c s where is the dimension of the limit set of the group this bound is more precise than the optimal global bound exp c s n 1 and it gives new bounds on the number of resonances scattering poles of n 1 the proof of this result is based on the application of holomorphic l2 techniques to the study of the determinants of the ruelle transfer operators and on the quasi self similarity of limit sets we also study this problem numerically and provide evidence that the bound may be optimal our motivation comes from molecular dynamics and we consider n 1 as the simplest model of quantum chaotic scattering

complexity of strings in the class of markov sources shannon s self information of a string is generalized to its complexity relative to the class of finite state machine fsm defined sources unlike an earlier generalization the new one is valid for both short and long strings the definition is justified in part by a theorem stating that asymptotically the mean complexity provides a tight lower bound for the mean length of all so called regular codes this also generalizes shannon s noiseless coding theorem for a large subclass of fsm sources a simple algorithm is described for computing the complexity

the redundancy of source coding with a fidelity criterion 1 known statistics the problem of redundancy of source coding with respect to a fidelity criterion is considered for any fixed rate r 0 and any memoryless source with finite source and reproduction alphabets and a common distribution p the nth order distortion redundancy d sub n r of fixed rate coding is defined as the minimum of the difference between the expected distortion per symbol of any block code with length n and rate r and the distortion rate function d p r of the source p it is demonstrated that for sufficiently large n d sub n r is equal to spl part spl part r d p r ln n 2n o ln n n where spl part spl part r d p r is the partial derivative of d p r evaluated at r and assumed to exist for any fixed distortion level d 0 and any memoryless source p the nth order rate redundancy r sub n d of coding at fixed distortion level d or by using d semifaithful codes is defined as the minimum of the difference between the expected rate per symbol of any d semifaithful code of length n and the rate distortion function r p d of p evaluated at d it is proved that for sufficiently large n r sub n d is upper bounded by ln n n o ln n n and lower bounded by in n 2n o in n n as a by product the lower bound of r sub n d derived in this paper gives a positive answer to a conjecture proposed by yu and speed 1993

on the redundancy of the fixed database lempel ziv algorithm for spl phi mixing sources the redundancy problem of the fixed database lempel ziv 1977 algorithm is considered it is demonstrated that for a class of spl phi mixing sources which includes markov sources unifilar sources and finite state sources as special cases the redundancy of the fixed database lempel ziv algorithm with database size n is lower bounded by h loglogn logn o logloglogn logn and upper bounded by 2h loglogn logn o logloglogn logn where h is the entropy rate of the source the method of proof is new and uses the concept of variable length to variable length codes