on optimal and achievable fix free codes fix free codes are prefix condition codes which can also be decoded in the reverse direction they have attracted attention from several communities and are used in video standards two variations with additional constraints have also been considered for joint source channel coding 1 symmetric codes which require the codewords to be palindromes 2 codes with distance constraints on pairs of codewords approaches to determine the existence of a fix free code with a given set of codeword lengths for each of the three variations are proposed these appear to involve the first use of boolean satisfiability sat for the existence and design of source codes branch and bound algorithms to find the collection of optimal codes for asymmetric and symmetric fix free codes are described the first bound for the performance of optimal symmetric binary fix free codes is provided an earlier conjecture on optimal symmetric binary fix free codes is proven and related results for asymmetric fix free codes are presented a variation of the 3 4 conjecture for fix free codes is introduced a key idea in the first conjecture and the new one is a definition of how one sequence of nondecreasing natural numbers dominates another

tight bounds on the redundancy of huffman codes a method for deriving optimal upper bounds on the redundancy of binary huffman codes in terms of the probability p sub 1 of the most likely source letter is presented this method will be used to compute bounds for all p sub 1 or 1 127 which were previously known only for a few special cases furthermore the known optimal lower bound for binary huffman codes is generalized to arbitrary code alphabets and some upper bounds for d ary huffman codes 2 or 1 2

sequentially constructible reversible variable length codes dominant codelength sequences for reversible variable length codes rvlcs have been recently introduced and studied as a means to looking into optimal rvlcs however obtaining the dominant sequences for rvlcs is computationally challenging in this paper we consider a special subset of all rvlcs namely the sequentially constructible sc rvlcs for which the dominant sequences can be obtained with less computational complexity of course this time saving is achieved at the cost of losing the optimality however it is shown that the dominant sequences for sc rvlcs provide acceptable performance in terms of the redundancy specifically it is seen that the worst case penalty in using the optimal sc rvlcs with respect to the optimal rvlcs is at most 2 9 bit per symbol for alphabet size of up to 16 while obtaining the dominant sequences of sc rvlcs is relatively faster it will still become challenging as the search space of the relevant branch and bound algorithm gets larger when the source alphabet size increases in order to further reduce the time complexity we propose an alternative approach to a table of sc rvl codelength sequences which avoids the branch and bound algorithm it is shown that the codes obtained by this approach perform almost as well as the sc dominant sequences specifically for an alphabet size of up to 21 the redundancy of this approach is at most 2 19 bit more than the optimal sc rvlcs

conditional statistical inference with multistage testing designs in this paper it is demonstrated how statistical inference from multistage test designs can be made based on the conditional likelihood special attention is given to parameter estimation as well as the evaluation of model fit two reasons are provided why the fit of simple measurement models is expected to be better in adaptive designs compared to linear designs more parameters are available for the same number of observations and undesirable response behavior like slipping and guessing might be avoided owing to a better match between item difficulty and examinee proficiency the results are illustrated with simulated data as well as with real data

architecture constraints and behavior this paper aims to bridge progress in neuroscience involving sophisticated quantitative analysis of behavior including the use of robust control with other relevant conceptual and theoretical frameworks from systems engineering systems biology and mathematics familiar and accessible case studies are used to illustrate concepts of robustness organization and architecture modularity and protocols that are central to understanding complex networks these essential organizational features are hidden during normal function of a system but are fundamental for understanding the nature design and function of complex biologic and technologic systems

tight upper bounds on the redundancy of huffman codes bounds on the redundancy of huffman codes in terms of the probability p sub 1 of the most likely source letter are provided in particular upper bounds are presented that are sharper than the bounds given recently by r g gallager ibid vol it 24 no 6 p 668 74 nov 1978 and by r m capocelli et al ibid vol it 32 no 6 p 854 857 nov 1986 for an interval 2 2 sup l 1 1 or 2 it is shown that the new bounds are the tightest possible for these intervals

optimal codes with limited kraft sum the well known huffman algorithm is an elegant approach to solve the basic problem of finding the optimal code among those with kraft sums smaller than or equal to 1 in this paper an extended problem is investigated where the kraft sum of the usable codes is limited to a given real number gamma 0 unlike the case gamma 1 i e huffman coding for gamma the kraft sum of the optimal code is not necessarily equal to gamma however we show that the kraft sum of the optimal code lies in a finite set of rational numbers which depend on the value of gamma this motivates us to first devise a recursive algorithm to obtain the optimal codelengths with kraft sum equal to gamma where gamma has a finite binary representation and 0 to do so the set of n tuple codelength vectors with a specific kraft sum gamma is characterized the binary representation of gamma plays a fundamental role in all problems that arise for gamma

a rasch model for partial credit scoring a unidimensional latent trait model for responses scored in two or more ordered categories is developed this partial credit model is a member of the family of latent trait models which share the property of parameter separability and so permit specifically objective comparisons of persons and items the model can be viewed as an extension of andrich s rating scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item the difference between the parameters in this model and the category boundaries in samejima s graded response model is demonstrated an unconditional maximum likelihood procedure for estimating the model parameters is developed

on constructing the huffman code based reversible variable length codes in this letter we propose a generic and efficient algorithm that can construct both asymmetrical and symmetrical reversible variable length codes rvlcs starting from a given huffman code the construction is based on two developed codeword selection mechanisms for the symmetrical case and the asymmetrical case respectively it is shown that the two mechanisms possess simple features and can generate efficient rvlcs easily in addition two new asymmetrical rvlcs are constructed and shown to be very efficient for further reducing the coding overheads in mpeg 4 when operating in the reversible decoding mode

an a based algorithm for constructing reversible variable length codes with minimum average codeword length variable length codes vlcs are widely adopted in many compression standards due to their good coding efficiency on average codeword length however an inherent problem with a vlc is that an error of even one bit can cause serious error propagation and thus loss of synchronization at the receiver which would lead to a series of non correctly decoded symbols reversible variable length codes rvlcs were introduced to significantly mitigate this phenomenon in this work a method to find an optimal rvlc in terms of the minimum average codeword length is first formulated as a tree searching problem and then instead of performing an exhaustive search an a based construction algorithm is proposed to find an optimal rvlc the proposed algorithm has been applied to several benchmarks for sources and has found respective optimal symmetric and asymmetric rvlcs

marginal maximum likelihood estimation of item parameters application of an em algorithm maximum likelihood estimation of item parameters in the marginal distribution integrating over the distribution of ability becomes practical when computing procedures based on an em algorithm are used by characterizing the ability distribution empirically arbitrary assumptions about its form are avoided the em procedure is shown to apply to general item response models lacking simple sufficient statistics for ability this includes models with more than one latent dimension

bounds on the redundancy of huffman codes corresp new upper bounds on the redundancy of huffman codes are provided a bound that for 2 9 leq p 1 leq 0 4 is sharper than the bound of gallager when the probability of the most likely source letter p 1 is the only known probability is presented the improved bound is the tightest possible for 1 3 leq p 1 leq 0 4 upper bounds are presented on the redundancy of huffman codes when the extreme probabilities p 1 and p n are known

optimal binary one ended codes binary prefix free codes in which all codewords end with a 1 have been introduced by berger and yeung 1990 a recursive method is given here for the construction of all optimal 1 ended codes with n codewords it is shown that the set of codes obtained by the construction contains only optimal codes we also compute recursively the number of essentially different optimal 1 ended codes with n codewords and show that their number grows faster than any polynomial in n

on the penalty of optimal fix free codes in this paper the difference between the redundancy of the optimal asymmetric symmetric fix free code and that of the optimal prefix free code is considered as the penalty of benefiting from the desired properties of fix free codes this penalty is studied from different perspectives in particular it is shown that the average penalty of asymmetric fix free codes is less than 0 21 bit per symbol moreover it is proved that when the source alphabet size is sufficiently large for almost all sources the penalty is less than or equal to 0 182 bit per symbol regarding symmetric fix free codes it is shown that the average penalty tends to infinity as the source alphabet size increases

a lower bound on the redundancy of d ary huffman codes corresp a necessary and sufficient condition for the most likely letter of any discrete source to be coded by a single symbol with a d ary huffman code 2 leq d is derived as a consequence a lower bound on the redundancy of a d ary huffman code is established

weighted likelihood estimation of ability in item response theory applications of item response theory which depend upon its parameter invariance property require that parameter estimates be unbiased a new method weighted likelihood estimation wle is derived and proved to be less biased than maximum likelihood estimation mle with the same asymptotic variance and normal distribution wle removes the first order bias term from mle two monte carlo studies compare wle with mle and bayesian modal estimation bme of ability in conventional tests and tailored tests assuming the item parameters are known constants the monte carlo studies favor wle over mle and bme on several criteria over a wide range of the ability scale

some basic properties of fix free codes a variable length code is a fix free code if no codeword is a prefix or a suffix of any other codeword this class of codes is applied to speed up the decoding process for the decoder can decode from both sides of the compressed file simultaneously we study some basic properties of fix free codes we prove a sufficient and a necessary condition for the existence of fix free codes and we obtain some new upper bounds on the redundancy of optimal fix free codes

a kraft type sufficient condition for the existence of d ary fix free codes a greedy scheme called greedy codeword assignment scheme gcas is proposed to assign d ary codewords to the given code lengths 1 2 n so that they satisfy the fix free property this scheme guarantees that a d ary fix free code can be obtained whenever i 1 n d d where d is equal to 5 8 for d even and very close to 5 8 for d odd this result can be regarded as an extension of yekhanin s theorem on the existence of binary fix free codes in the special case d 2 the greediness of gcas enables us to prove that if mini i 2 the inequality i 1 n 2 i 21 32 implies the existence of a binary fix free code with code lengths 1 2 n

binary prefix codes ending in a 1 binary prefix codes with the constraint that each codeword must end with a 1 have been recently introduced by berger and yeung 1990 we analyze the performance of such codes by investigating their average codeword length in particular we show that a very simple strategy permits the construction of a 1 ended binary prefix code whose average codeword length is less than h 1 for any discrete source with entropy h we also prove a tight lower bound on the optimal average codeword length in terms of h and of the minimum letter probability of the source finally we discuss the problem of finding an optimum feasible code

sufficient conditions for existence of binary fix free codes two sufficient conditions are given for the existence of binary fix free codes i e both prefix free and suffix free let l be a finite multiset of positive integers whose kraft sum is at most 3 4 it is shown that there exists a fix free code whose codeword lengths are the elements of l if either of the following two conditions holds i the smallest integer in l is at least 2 and no integer in l except possibly the largest one occurs more than 2 sup min l 2 times ii no integer in l except possibly the largest one occurs more than twice the results move closer to the ahlswede balkenhol khachatrian conjecture that kraft sums of at most 3 4 suffice for the existence of fix free codes

the most likely optimal symmetric rvlc for each alphabet size up to 30 a symmetric reversible variable length code rvlc is determined which is most likely to be optimal in average codeword length it is shown that these codes are optimal for more than half of all the possible sources

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

a note on d ary huffman codes an upper bound on the redundancy of d ary huffman codes in terms of the probability p sub 1 of the most likely source letter is provided for large values of p sub 1 the bound improves the one given by r g gallager 1978 additionally some results known for the binary case d 2 are extended to arbitrary d ary huffman codes as a consequence a tight lower bound that corrects a bound recently proposed by j d golic and m m obradovic 1987 is derived

the number of huffman codes compact trees and sums of unit fractions the number of nonequivalent compact huffman codes of length r over an alphabet of size t has been studied frequently equivalently the number of nonequivalent complete t ary trees has been examined we first survey the literature unifying several independent approaches to the problem then improving on earlier work we prove a very precise asymptotic result on the counting function consisting of two main terms and an error term

modified symmetrical reversible variable length code and its theoretical bounds reversible variable length codes rvlcs have been adopted in emerging video coding standards h 263 and mpeg 4 to enhance their error resilience capabilities which are important and essential in error prone environments this study proposes an efficient algorithm to construct a symmetrical rvlc from a given huffman code in addition theoretical bounds on the maximum codeword length for fixed length huffman codes and on the optimal average codeword lengths for sources with exponential distribution are provided

robust stabilization subject to structured uncertainties and mean power constraint abstract this paper deals with a robust stabilization problem for discrete time systems subject to multiple disturbances occurring in controller and actuating channel where both linear structured uncertainties and white gaussian noises are included the desired control law is aimed to robustly stabilize the system and to satisfy some pre specified mean power constraint simultaneously by the philosophy of the mixed h 2 h control a solvability condition is first derived for single input systems that reveals the intrinsic relation between the unstable poles of the plant and the disturbance parameters together with two cross coupled algebraic riccati equations the result is further generalized to multiple input systems with a sufficient condition given again by the unstable poles of the plant an example is included to illustrate the current results

93 of the 3 over 4 conjecture is already verified one of the most important challenges in the context of fix free codes is proving the 3 4 conjecture which guarantees the existence of fix free codewords of lengths l1 l2 ln if i 1n2 li 3 4 although this conjecture has not become a theorem yet some researchers have proved the problem with some extra constraints on the codelengths one of those we call it yekhanin s constraint is i li 1 2 li 1 2 where minklk in this paper it is shown that such a constraint is not so restrictive we prove that almost 93 8 of the n tuple codelength vectors with kraft sum 3 4 do satisfy yekhanin s constraint one can optimistically interpret this result as almost 93 8 of the road of proving the 3 4 conjecture is paved

the redundancy of an optimal binary fix free code is not greater than 1 bit in the context of fix free codes the most important and immediate consequence of the 3 4 conjecture if it is proven is that the redundancy of an optimal binary fix free code never exceeds 1 bit as with the optimal prefix free codes i e huffman codes in this paper this bound on the redundancy is proven without requiring the conjecture to be true to do so we use two known sufficient conditions for the existence of binary fix free codes to derive an improved upper bound on the redundancy of an optimal fix free code in terms of the largest symbol probability

on the average codeword length of optimal binary codes for extended sources corresp although optimal binary source coding using symbol blocks of increasing length must eventually yield a code having an average codeword length arbitrarily close to the source entropy it is known that the sequence of average codeword lengths need not be nonincreasing the sequence is however bounded above by the average codeword length of the source and certain subsequences must be nondecreasing sufficient conditions are obtained describing sources for which a decrease in average codeword length is achieved when coding pairs of symbols a sufficient condition specifying sources for which no such decrease is possible is also obtained

some tight lower bounds on the redundancy of optimal binary prefix free and fix free codes the tight lower bound on the redundancy of optimal prefix free codes in terms of a given symbol probability not necessarily the largest or the smallest one has been derived in the literature the first goal of this paper is to derive the tight lower bounds in terms of j j 1 known symbol probabilities of the source using some properties of kullback leibler distance since fix free codes are special prefix free codes for which no codeword is a suffix of the other codewords it is clear that all lower bounds on the redundancy of optimal prefix free codes are also valid for fix free ones accordingly the second question of the paper is on the tightness of the derived lower bounds for optimal fix free codes it is proven that these lower bounds are tight for optimal fix free codes if j leq 3 and are not tight for j geq 4 also it is shown that the tight lower bound in terms of the probability of the most likely symbol is the same for optimal prefix free and optimal fix free codes

some upper bounds on the redundancy of optimal binary fix free codes in order to investigate the cost of employing fix free codes instead of the well established prefix free codes some upper bounds on the redundancy of the optimal fix free codes and their difference with the corresponding huffman codes are presented unlike the conventional approach which is based on some shannon like suboptimal codes we examine the redundancy of a better huffman like code an algorithm is proposed for deriving optimal binary codeword lengths with kraft sum 5 8 for a given probability distribution the existence of a fix free code with such codelengths is guaranteed by yekhanin s theorem it is shown that the redundancy of this fix free code is at most 0 8 bit greater than that of the huffman code and does not exceed 8 3 log3 1 0817 bits this upper bound is 4 3 log 5 3 0 5964 bit less than the best known one if the 3 4 conjecture is proved sometime in the future the presented upper bound on the redundancy of the optimal fix free codes i e 1 0817 will be improved by only 5 3 log3 0 0817 in addition to these general bounds all known upper bounds in terms of the largest the smallest and an arbitrary given symbol probability are substantially improved