adaptive steganalysis of least significant bit replacement in grayscale natural images this paper deals with the detection of hidden bits in the least significant bit lsb plane of a natural image the mean level and the covariance matrix of the image considered as a quantized gaussian random matrix are unknown an adaptive statistical test is designed such that its probability distribution is always independent of the unknown image parameters while ensuring a high probability of hidden bits detection this test is based on the likelihood ratio test except that the unknown parameters are replaced by estimates based on a local linear regression model it is shown that this test maximizes the probability of detection as the image size becomes arbitrarily large and the quantization step vanishes this provides an asymptotic upper bound for the detection of hidden bits based on the lsb replacement mechanism numerical results on real natural images show the relevance of the method and the sharpness of the asymptotic expression for the probability of detection

on the support of mse optimal fixed rate scalar quantizers this paper determines how the support regions of optimal and asymptotically optimal fixed rate scalar quantizers with respect to mean squared error depend on the number of quantization points n and the probability density of the variable being quantized it shows that for asymptotic optimality it is necessary and sufficient that the support region grow fast enough that the outer or overload distortion decreases as o 1 n sup 2 formulas are derived for the minimal support of asymptotically optimal quantizers for generalized gamma densities including gaussian and laplacian interestingly these turn out to be essentially the same as for the support of optimal fixed rate uniform scalar quantizers heuristic arguments are then used to find closed form estimates for the support of truly optimal quantizers for generalized gamma densities these are found to be more accurate than the best prior estimates as computed by numerical algorithms they demonstrate that the support of an optimal quantizer is larger than the minimal asymptotically optimal support by a factor depending on the density but not n and that the outer distortion of optimal quantizers decreases as 1 n sup 3

automorphisms of b h with respect to minus partial order abstract let h be an infinite dimensional hilbert space and b h the algebra of all bounded linear operators on h we discuss possible extensions of the concept of the minus partial order from matrix to operator algebras in particular we show that mitra s unified theory of matrix partial orders based on generalized inverses can be modified in such a way that we get a unified approach to partial orders on b h or even more general algebras and rings then we choose the most natural among possible definitions of minus partial order on b h and describe the structure of corresponding automorphisms

a new asymptotically optimal code for the positive integers a new universal binary code for the positive integers is proposed as a modified version of m wang s see ibid vol 34 p 324 6 mar 1988 flag encoding scheme the codeword length of the new scheme is shorter than wang s on an average for large initial segments of the positive integers the performance of the new scheme is also compared with that of other universal schemes furthermore it is shown that an asymptotically optimal code can be achieved by modifying the new flag scheme such that the flag length varies dynamically

variance mismatched fixed rate scalar quantization of laplacian sources the paper investigates the mean squared error mse distortion of a variance mismatched fixed rate scalar quantizer that is optimal or asymptotically optimal in the minimum mse sense for a laplacian density with standard deviation q but is applied to another with standard deviation p a nitadori like formula is discovered for an optimal quantizer when p q 1 2 also it is shown that the distortion expressions derived rigorously for asymptotically optimal quantile quantizers essentially confirm the heuristically obtained previous result that the distortion decreases as 1 n3 for the heavy mismatch of 3 2 as lnn n2 for the critical mismatch of 3 2 and as 1 n2 for the mild mismatch of 1 and the resulting distortion decreases as in n n2 1 p

comments on a source matching approach to finding minimax codes by davisson l d and leon garcia a in the above paper 1 davisson and leon garcia prove a result equating the minimax redundancy for universal codes to a channel capacity that is computable by standard techniques they point out that the result had been developed earlier in unpublished work by gallager i would like to point out an alternative development of this result and to compare this work with that of davisson and leon garcia

nonnegative idempotent matrices and the minus partial order abstract we describe the structure of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus order

robust noiseless source coding through a game theoretical approach noiseless coding of a discrete source with partially known statistics is formulated as a two person game the payoff is the average codeword length using shannon codes the code designer picks a source probability distribution for the design of the code while an opponent picks the actual source probability distribution it is shown that if the class of probability distributions allowed is convex then there is a saddle point solution which is determined by the maximum entropy distribution of the convex class the maximum entropy element is derived for three families of source probability mass functions pmf a the class of c contaminated pmf s b the class of pmf s for which each probability is known only through an upper and lower bound c the class of pmf s which is a convex hull of a finite number of known pmf s an extension of the robust noiseless source coding problem for families of sources modeled as first order markov chains is discussed

asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization two results are given first using a result of csiszar 1973 the asymptotic i e high resolution low distortion performance for entropy constrained tessellating vector quantization heuristically derived by gersho 1979 is proven for all sources with finite differential entropy this implies using gersho s conjecture and zador s formula that tessellating vector quantizers are asymptotically optimal for this broad class of sources and generalizes a rigorous result of gish and pierce 1968 from the scalar to the vector case second the asymptotic performance is established for zamir and feder s 1992 randomized lattice quantization with the only assumption that the source has finite differential entropy it is proven that the low distortion performance of the zamir feder universal vector quantizer is asympotically the same as that of the deterministic lattice quantizer

rate bounds on ssim index of quantized images in this paper we derive bounds on the structural similarity ssim index as a function of quantization rate for fixed rate uniform quantization of image discrete cosine transform dct coefficients under the high rate assumption the space domain ssim index is first expressed in terms of the dct coefficients of the space domain vectors the transform domain ssim index is then used to derive bounds on the average ssim index as a function of quantization rate for uniform gaussian and laplacian sources as an illustrative example uniform quantization of the dct coefficients of natural images is considered we show that the ssim index between the reference and quantized images fall within the bounds for a large set of natural images further we show using a simple example that the proposed bounds could be very useful for rate allocation problems in practical image and video coding applications

optimal binary search trees abstract we consider the problem of building optimal binary search trees the binary search tree is a widely used data structure for information storage and retrieval a binary search tree t for a set of keys from a total order is a binary tree in which each node has a key value and all the keys of the left subtree are less than the key at the root and all the keys of the right subtree are greater than the key at the root this property holding recursively for the left and right subtrees of the tree t suppose we are given n keys and the probabilities of accessing each key and those occurring in the gap between two successive keys the optimal binary search tree problem is to construct a binary search tree on these n keys that minimizes the expected access time one variant of this problem is when only the gaps have nonzero access probabilities and is called the optimal alphabetic tree problem another related problem is when there is no order between the keys and there are probabilities associated only with the gaps and the objective is to build a binary tree with minimum expected weighted path length from the root this is called the huffman tree problem in this survey we assess known results on the structural properties of the optimal trees algorithms and lower bounds to construct and to verify optimal trees and heuristics to construct nearly optimal trees and other related results

automorphisms of the poset of skew projections abstract any automorphism of the poset b of all idempotents of the algebra b h of all bounded linear operators in a hilbert space h dim h 3 is shown to be of the form p apa 1 or p ap a 1 with a c where c is the set of all semi linear bijections a h h if h is finite dimensional and the set of all continuous invertible linear or conjugate linear operators in h if h is infinite dimensional further applications of the method are outlined

quantizers for the gamma distribution and other symmetrical distributions this paper discusses minimum mean square error quantization for symmetric distributions if the distribution satisfies a log concavity condition the optimal quantizer is itself symmetric for the gamma distribution often used to model speech signals the log concavity condition is not satisfied it is shown that for this distribution both the uniformly spaced and the nonuniformly spaced optimal quantizers are not symmetrical for even numbers of quantizer levels new quantization tables giving the optimal levels for quantizers for the gamma distribution are presented a simple family of symmetric distributions is also examined this family shows that as the distribution gets concentrated near the point of symmetry nonsymmetric solutions become optimal

on some characterizations of pairwise star orthogonality using rank and dagger additivity and subtractivity abstract the main results are that pairwise star orthogonality a i a j 0 and a i a j 0 for all i j where a 1 a k are complex m n matrices is equivalent to i a i a j and to ii a i rs a j and a j a j where i 1 k here the superscript denotes the moore penrose inverse while and rs denote respectively the star and rank subtractivity or minus partial orderings five more characterizations of the pairwise star orthogonality of k complex m n matrices are also presented our characterizations extend earlier results by hartwig and styan 1986 for k 2

preservers of the left star and right star partial orders abstract let m n f be the set of all n n matrices over the field f where f r or f c we characterize maps on m n f which are surjective and preserve either the left star or the right star order in both directions but are not necessarily additive

universal source coding for monotonic and fast decaying monotonic distributions we study universal compression of sequences generated by monotonic distributions we show that for a monotonic distribution over an alphabet of size k each probability parameter costs essentially 0 5log n k3 bits where n is the coded sequence length as long as k o n1 3 otherwise for k o n the total average sequence redundancy is o n1 3 e bits overall we then show that there exists a sub class of monotonic distributions over infinite alphabets for which redundancy of o n1 3 e bits overall is still achievable this class contains fast decaying distributions including many distributions over the integers such as the family of zipf distributions and geometric distributions for some slower decays including other distributions over the integers redundancy of o n bits overall is achievable a method to compute specific redundancy rates for such distributions is derived the results are specifically true for finite entropy monotonic distributions finally we study individual sequence redundancy behavior assuming a sequence is governed by a monotonic distribution we show that for sequences whose empirical distributions are monotonic individual redundancy bounds even tighter than those in the average case can be obtained the relation of universal compression with monotonic distributions to universal compression of patterns of sequences is demonstrated

a new code for encoding all monotone sources with a fixed large alphabet size the problem of designing a fixed code for encoding all monotone sources with a large alphabet size n is studied it is proved that if the codelengths of a code sequence are of order o log n then the redundancy for almost all monotone sources with n symbols is very close to that for a specific distribution so called average distribution noting this point for any large alphabet size n a new code with a very simple structure is proposed whose redundancy is close to that of the huffman code for almost all monotone sources with alphabet size 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

precise minimax redundancy and regret recent years have seen a resurgence of interest in redundancy of lossless coding the redundancy regret of universal fixed to variable length coding for a class of sources determines by how much the actual code length exceeds the optimal ideal over the class code length in a minimax scenario one finds the best code for the worst source either in the worst case called also maximal minimax or on average we first study the worst case minimax redundancy over a class of stationary ergodic sources and replace shtarkov s bound by an exact formula among others we prove that a generalized shannon code minimizes the worst case redundancy derive asymptotically its redundancy and establish some general properties this allows us to obtain precise redundancy for memoryless markov and renewal sources for example we present the exact constant of the redundancy for memoryless and markov sources by showing that the integer nature of coding contributes log logm m 1 logm o 1 where m is the size of the alphabet then we deal with the average minimax redundancy and regret our approach here is orthogonal to most recent research in this area since we aspire to show that asymptotically the average minimax redundancy is equivalent to the worst case minimax redundancy for some classes of sources after formulating some general bounds relating these two redundancies we prove our assertion for memoryless and markov sources nevertheless we provide evidence that maximal redundancy of renewal processes does not have the same leading term as the average minimax redundancy however our general results show that maximal and average regrets are asymptotically equivalent

maps on matrix spaces it is well known that every automorphism of the full matrix algebra is inner we give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras characterizations of monotone and orthogonality preserving maps on idempotent matrices some nonlinear preserver results and some recent theorems concerning geometry of matrices we show that all these topics are closely related and point out the connections with physics and geometry several open problems are posed

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

linear preservers for matrix inequalities and partial orderings we characterize the linear preservers of minus partial order on matrix algebras the developed approach allows us to classify linear transformations that live fixed several matrix relations arising as extremal cases in some classical matrix inequalities including rank additivity relation and related properties applications to the determinant preservers are considered

some orders for operators on hilbert spaces in this paper we present some interesting properties of the diamond left right star and sharp orders for operators on hilbert spaces

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

matrix partial orders through generalized inverses unified theory the unified theory presented here covers as special cases the star order of drazin the minus order of hartwig and nambooripad the sharp order and other partial orders introduced by the author

characterizations of the best linear unbiased estimator in the general gauss markov model with the use of matrix partial orderings abstract under the general gauss markov model y x 2 v two new characterizations of blue x are derived involving the lowner and rank subtractivity partial orderings between the dispersion matrix of blue x and the dispersion matrix of y as particular cases of these characterizations three new criteria for the equality between olse x and blue x are given

on the convexity of the mse distortion of symmetric uniform scalar quantization this paper investigates the convexity of the mean squared error distortion of symmetric uniform scalar quantization with respect to step size the principal results include proofs for odd numbers of levels that distortion is not convex for any symmetric density and that it is convex for even numbers of levels for densities such as gaussian laplacian and gamma but is not in general for two sided rayleigh for the latter case an interval is derived that includes the optimal step size and over which the distortion is convex the proofs of convexity use the euler maclaurin formula applied to the second derivative of distortion with upper bounds on the remainder term these results imply that a zero of the derivative of the distortion for these densities which has been previously conjectured optimal is indeed the optimal step size because the distortion is convex either globally or locally over a sufficiently wide interval to ensure a global minimizer

huffman redundancy for large alphabet sources the performance of optimal prefix free encoding for memoryless sources with a large alphabet size is studied it is shown that the redundancy of the huffman code for almost all sources with a large alphabet size n is very close to that of the average distribution of the monotone sources with n symbols this value lies between 0 02873 and 0 02877 bit for sufficiently large n

asymptotic analysis of optimal fixed rate uniform scalar quantization studies the asymptotic characteristics of uniform scalar quantizers that are optimal with respect to mean squared error mse when a symmetric source density with infinite support is sufficiently well behaved the optimal step size spl delta sub n for symmetric uniform scalar quantization decreases as 2 spl sigma n sup 1 v sup 1 1 6n sup 2 where n is the number of quantization levels spl sigma sup 2 is the source variance and v sup 1 spl middot is the inverse of v y y sup 1 spl int sub y sup spl infin p spl sigma sup 1 x x dx equivalently the optimal support length n spl delta sub n increases as 2 spl sigma v sup 1 1 6n sup 2 granular distortion is asymptotically well approximated by spl delta sub n sup 2 12 and the ratio of overload to granular distortion converges to a function of the limit spl tau spl equiv lim sub y spl rarr spl infin y sup 1 e x x y provided as usually happens that spl tau exists when it does its value is related to the number of finite moments of the source density an asymptotic formula for the overall distortion d sub n is obtained and spl tau 1 is both necessary and sufficient for the overall distortion to be asymptotically well approximated by spl delta sub n sup 2 12 applying these results to the class of two sided densities of the form b x sup spl beta e spl alpha x sup spl alpha which includes gaussian laplacian gamma and generalized gaussian it is found that spl tau 1 that spl delta sub n decreases as ln n sup 1 spl alpha n that d sub n is asymptotically well approximated by spl delta sub n sup 2 12 and decreases as ln n sup 2 spl alpha n sup 2 and that more accurate approximations to spl delta sub n are possible the results also apply to densities with one sided infinite support such as rayleigh and weibull and to densities whose tails are asymptotically similar to those previously mentioned

automorphisms of mn partially ordered by rank subtractivity ordering let mn be the space of all n n matrices with coefficients in r or c where n 3 the rank subtractivity or minus partial order on mn is defined by a rsb iff rankb ranka rank b a we characterize bijective mappings on mn such that a rsb iff a rs b a b mn

on optimal perfect reconstruction feedback quantizers this paper presents novel results on perfect reconstruction feedback quantizers prfqs i e noise shaping predictive and sigma delta a d converters whose signal transfer function is unity our analysis of this class of converters is based upon an additive white noise model of quantization errors our key result is a formula that relates the minimum achievable mse of such converters to the signal to noise ratio snr of the scalar quantizer embedded in the feedback loop this result allows us to obtain analytical expressions that characterize the corresponding optimal filters we also show that for a fixed snr of the scalar quantizer the end to end mse of an optimal prfq which uses the optimal filters which for this case turn out to be iir decreases exponentially with increasing oversampling ratio key departures from earlier work include the fact that fed back quantization noise is explicitly taken into account and that the order of the converter filters is not a priori restricted

how suboptimal is the shannon code in order to determine how suboptimal the shannon code is one should compare its performance with that of the optimal code i e the corresponding huffman code in some sense it is well known that in the worst case the redundancy of both the shannon and huffman codes can be arbitrarily close to 1 beyond this worst case viewpoint very little is known in this paper we compare the performance of these codes from an average point of view the redundancy is considered as a random variable on the set of all sources with n symbols and its average is evaluated it is shown that the average redundancy of the shannon code is very close to 0 5 bits whereas the average redundancy of the huffman code is less than n 1 1 ln n 0 086 bits it is also proven that the variance of the redundancy of the shannon code tends to zero as n increases therefore for sources with alphabet size n the redundancy of the shannon code is approximately 0 5 bits with probability approaching 1 as n

the minimum average code for finite memoryless monotone sources the problem of selecting a code for finite monotone sources with n symbols is considered the selection criterion is based on minimizing the average redundancy called minave criterion instead of its maximum i e minimax criterion the average probability distribution pnmacr whose associated huffman code has the minimum average redundancy is derived the entropy of the average distribution i e h pnmacr and the average entropy of the monotone distributions i e h pnmacr are studied it is shown that both logn h pnmacr and logn h pnmacr are asymptotically equal to a constant sime0 61 therefore there is only a negligible penalty at most 1 61 bits symbol in using a simple fixed length code with respect to the optimal code an efficient near optimal encoding technique is also proposed the consequences of the two approaches i e minave and minimax are compared in terms of their associated distributions and associated codes in order to evaluate the average performance of the minimax code we prove that the informational divergence of the average distribution and minimax distribution asymptotically grows as 2 275 loglogn

monotonicity of step sizes of mse optimal symmetric uniform scalar quantizers for generalized gamma probability densities this paper studies the monotonicity of step sizes of optimal symmetric uniform scalar quantizers with respect to mean squared error distortion the principal results are that for the special cases of gaussian laplacian two sided rayleigh and gamma densities optimal step size monotonically decreases when the number of levels n increases by two and that for any generalized gamma density and all sufficiently large n optimal step size again decreases when n increases by two also it is shown that for a laplacian density and sufficiently large n optimal step size decreases when n increases by just one

relative uniformity of sources and the comparison of optimal code costs conditions on the probability distributions of two sources that describe their relative uniformity are shown to be sufficient for inequalities on their minimum average codeword lengths and entropies the huffman coding problem and variants involving renyi entropy are addressed

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

kalman filtering with relays over wireless fading channels this note studies the use of relays to improve the performance of kalman filtering over packet dropping links packet reception probabilities are governed by time varying fading channel gains and the sensor and relay transmit powers we consider situations with multiple sensors and relays where each relay can either forward one of the sensors measurements to the gateway fusion center or perform a simple linear network coding operation on some of the sensor measurements using an expected error covariance performance measure we consider optimal and suboptimal methods for finding the best relay configuration and power control problems for optimizing the kalman filter performance our methods show that significant performance gains can be obtained through the use of relays network coding and power control with at least 30 40 less power consumption for a given expected error covariance specification

three space economical algorithms for calculating minimum redundancy prefix codes the minimum redundancy prefix code problem is to determine for a given list w spl omega sub 1 spl omega sub n of n positive symbol weights a list l l sub 1 l sub n of n corresponding integer codeword lengths such that spl sigma sub i 1 sup n 2 sup li spl les 1 and spl sigma sub i 1 sup n spl omega sub i l sub i is minimized let us consider the case where w is already sorted in this case the output list l can be represented by a list m m sub 1 m sub h where m sub l for l 1 h denotes the multiplicity of the codeword length l in l and h is the length of the greatest codeword fortunately h is proved to be o min log 1 p sub 1 n where p sub 1 is the smallest symbol probability given by spl omega sub 1 spl sigma sub i 1 sup n spl omega sub i we present the fast lazyhuff f lazyhuff the economical lazyhuff e lazyhuff and the best lazyhuff b lazyhuff algorithms f lazyhuff runs in o n time but requires o min h sup 2 n additional space on the other hand e lazyhuff runs in o n nlog n h time requiring only o h additional space finally b lazyhuff asymptotically overcomes the previous algorithms requiring only o n time and o h additional space moreover our three algorithms have the advantage of not writing over the input buffer during code calculation a feature that is very useful in some applications

asymptotic formulas for variance mismatched fixed rate scalar quantization of a gaussian source asymptotic formulas are derived for the mean squared error mse distortion and the snr of a variance mismatched fixed rate scalar quantizer designed mse optimally for a gaussian source it is discovered that in the case that the standard deviation ratio of the applied to density to the designed for density is greater than 3 2 the snr in db increases at large rate r as 9 03r 2 15 1 1 2 log10r plus a constant that depends only on

on transmit beamforming for miso ofdm channels with finite rate feedback with finite rate feedback we propose two feedback methods for transmit beamforming in a point to point miso ofdm channel for the first method a receiver with perfect channel information quantizes and feeds back the optimal transmit beamforming vectors of a few selected subcarriers which are equally spaced based on those quantized vectors the transmitter applies either constant linear or higher order interpolation with the remaining beamforming vectors with constant interpolation we derive the approximate sum achievable rate and the optimal cluster size that maximizes the approximate rate for linear interpolation we derive a closed form expression for the phase rotation by utilizing the correlation between ofdm subcarriers we also propose a higher order interpolation that requires more than two quantized vectors to interpolate transmit beamformers and is based on existing channel estimation methods numerical results show that interpolation with the optimized cluster size can perform significantly better than that with an arbitrary cluster size for the second proposed method a channel impulse response is quantized with a uniform scalar quantizer with channel quantization we also derive the approximate sum achievable rate we show that switching between the two methods for different feedback rate requirements can perform better than the existing schemes

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

hua s fundamental theorem of the geometry of matrices abstract let m n f be the space of all n n matrices over the field f n 2 two matrices a b m n f are adjacent if rank a b 1 hua s fundamental theorem of the geometry of square matrices characterizes bijective maps on m n f that preserve adjacency in both directions in this paper we treat a long standing open problem whether the result of hua holds true under the weaker assumption of preserving the adjacency in one direction only we answer this question in the affirmative in the case that every nonzero homomorphism f f f is surjective for example the field of real numbers has this property in order to prove this result we have to improve ovchinnikov s characterization of automorphisms of the poset of idempotent matrices

some properties of matrix partial orderings abstract the matrix partial orderings considered are 1 the star ordering and 2 the minus ordering or rank subtractivity both in the set of m n complex matrices and 3 the lowner ordering in the set of m m matrices the problems discussed are 1 inheriting certain properties under a given ordering 2 preserving an ordering under some matrix multiplications 3 relationships between an ordering among direct or kronecker and hadamard products and the corresponding orderings between the factors involved 4 orderings between generalized inverses of a given matrix and 5 preserving or reversing a given ordering under generalized inversions several generalizations of results known in the literature and a number of new results are derived

some remarks on partial orderings of nonnegative definite matrices abstract baksalary and pukelsheim 1990 investigated partial orderings of nonnegative definite matrices some additional remarks are given in this note

a new recursive universal code of the positive integers a new recursive universal code of the positive integers is proposed in which any given sequence can be used as a delimiter of codeword while bit 0 is used as a delimiter in known universal codes e g levenshtein code elias spl omega code even rodeh code stout code bentley yao code etc the codeword length of the proposed code is shorter than log sub 2 sup n n in almost all of sufficiently large positive integers although the known codes are longer than log sub 2 sup n n for any positive integer n