the faces of the unit balls of c norms and c spectral norms abstract we give a complete description of the faces of the closed unit ball of an arbitrary c norm on r n and we determine the barycenter the volume and the symmetries of each face as a consequence of a previous result we characterize the faces of the closed unit ball of the corresponding c spectral norm on the space of n m real or complex matrices

isometries for the induced c norm on square matrices and some related results abstract let fn be the linear space of column vectors with n coordinates over f r or c denote by gp n the group of n n generalized permutation matrices i e matrices with exactly one nonzero entry with magnitude 1 in each row and column given a nonzero vector c c1 cn t fn define the c norm of x x1 xn t fn by x c max c t p x p gp n and the induced c norm of a fn n by a c max a y c y c 1 we characterize the isometries for induced c norms to achieve our goal we study the geometric properties of fn n and obtain some inequalities related to the induced c norm as by products these results are of independent interest

asymptotically tight bounds on the depth of estimated context trees we study the maximum markov model order that can be estimated for an individual input sequence x of length n over a finite alphabet of size alpha for popular estimators of tree models where the estimated order is determined by the depth of a context tree estimate and in the special case of plain markov models where the tree is constrained to be perfect with all leaves at the same depth first we consider penalized maximum likelihood estimators where a context tree hat t is obtained by minimizing a cost of the form log hat p t x f n s t where hat p t x is the ml of x under a model with context tree t s t is the set of leaves of t and f n is an increasing penalization function of n the popular bic estimator is a special case with f n frac alpha 1 2 log n for plain markov models a simple argument yields a known upper bound overline k n o log n on the maximum order that can be estimated for x and we show that in fact this simple bound is not far from tight for general context trees we derive an asymptotic upper bound n 1 2 o 1 on the estimated depth and we exhibit explicit input sequences that asymptotically attain the bound up to a multiplicative constant factor we show that a similar upper bound applies also to mdl estimators based on the kt probability assignment and moreover the same example sequences asymptotically approach the upper bound also in this case

the isometries of certain maximum norms abstract let a 1 a m be nonsingular n n matrices over f r or c and let 1 p j for j 1 m we characterize the linear isometries for the norm n on f n defined by n x max l p j a j x 1 j m where l p y denotes the l p norm of y f n as an application we study the conditions on matrices a 1 a m for which n is absolute or a symmetric gauge function

redundancy of the context tree weighting method on renewal and markov renewal processes the context tree weighting method ctw is shown to be almost adaptive on the classes of renewal and markov renewal processes up to logarithmic factor ctw achieves the minimax pointwise redundancy described by i csiszaacuter and p shields in ieee trans inf theory vol 42 no 6 pp 2065 2072 nov 1996 this result not only complements previous results on the adaptivity of the context tree weighting method on the relatively small class of all finite context tree sources which encompasses the class of all finite order markov sources it shows that almost minimax redundancy can be achieved on massive classes of sources classes that cannot be smoothly parameterized by subsets of finite dimensional spaces moreover it shows that almost adaptive compression can be achieved in a computationally efficient way on those massive classes while previous adaptivity results for ctw could rely on the fact that any markov source is a finite context tree source this is no longer the case for renewal sources in order to prove almost adaptivity of ctw over renewal sources it is necessary to establish that ctw carefully balances estimation error and approximation error

lee distance and topological properties of k ary n cubes in this paper we consider various topological properties of a k ary n cube q sub n sup k using lee distance we feel that lee distance is a natural metric for defining and studying a q sub n sup k after defining a q sub n sup k graph using lee distance we show how to find all disjoint paths between any two nodes given a sequence of radix k numbers a function mapping the sequence to a gray code sequence is presented and this function is used to generate a hamiltonian cycle embedding the graph of a mesh and the graph of a binary hypercube into the graph of a q sub n sup k is considered using a k ary gray code we show the embedding of a k n sub 1 spl times k n sub 2 spl times spl times k n sub m dimensional mesh into a q sub n sup k where n spl sigma sub i l sup m n sub i then using a single digit 4 ary reflective gray code we demonstrate embedding a q sub n into q sub n 2 sup 4 we look at how lee distance may be applied to the problem of resource placement in a q sub n sup k by using a lee distance error correcting code although the results in this paper are only preliminary lee distance error correcting codes have not been applied previously to this problem finally we consider how lee distance can be applied to message routing and single node broadcasting in a q sub n sup k in this section we present two single node broadcasting algorithms that are optimal when single port and multi port i o is used

path embedding in star graphs abstract the star graph interconnection network has been introduced as an attractive alternative to the hypercube network in this paper we consider the path embedding problem in star graphs assume that n 4 we prove that paths of all even lengths from d x y to n 2 can be embedded between two arbitrary vertices x and y from the same partite set in the n dimensional star graph in addition paths of all odd lengths from d x y to n 1 can be embedded between two arbitrary vertices x and y from different partite sets in the n dimensional star graph except that if x and y are adjacent there is no path of length 3 between them the result is optimal in the sense that paths of all possible lengths are found in star graphs

certain isometries on rn abstract let r n be the linear space of all real column vectors with n coordinates given x x 1 x n t r n its p k norm is defined by x p k max x i 1 p x i k p 1 p 1 i 1 k n where k is a positive integer satisfying 1 k n and p satisfies 1 p whereas its c norm is defined by x c max c t px p is a generalized permutation matrix for any nonzero c r n this paper characterizes the isometries for c norms and p k norms some related results are also considered

variable length markov chains we study estimation in the class of stationary variable length markov chains vlmc on a finite space the processes in this class are still markovian of high order but with memory of variable length yielding a much bigger and structurally richer class of models than ordinary high order markov chains from an algorithmic view the vlmc model class has attracted interest in information theory and machine learning but statistical properties have not yet been explored provided that good estimation is available the additional structural richness of the model class enhances predictive power by finding a better tradeoff between model bias and variance and allowing better structural description which can be of specific interest the latter is exemplified with some dna data a version of the tree structured context algorithm proposed by rissanen in an information theoretical setup is shown to have new good asymptotic properties for estimation in the class of vlmcs this remains true even when the underlying model increases in dimensionality fur thermore consistent estimation of minimal state spaces and mixing properties of fitted models are given we also propose a new bootstrap scheme based on fitted vlmcs we show its validity for quite general stationary categorical time series and for a broad range of statistical procedures

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

g invariant norms and g c radii abstract let v be a finite dimensional inner product space over f r or c and let g be a closed subgroup of the group of unitary operators on v a norm or a seminorm on v is said to be g invariant if g x x for all g g x v the concept of g invariant norm specializes to many interesting particular cases such as the absolute norms on f n symmetric gauge functions on r n unitarily invariant norms on f m n etc which are of wide research interest in this paper we study the general properties of g invariant norms our main strategy is to study g invariant norms via the g c radius rg c on v defined by r g c x max x g c g g where c v it is shown that the g c radii are very important g invariant seminorms because every g invariant norm or seminorm admits a representation in terms of them as a result one may focus attention on g c radii in order to get results on g invariant norms we study the norm properties of g c radii and obtain various results relating g invariant norms and g c radii the linear operators on v that preserve g invariant norms g invariant seminorms or g c radii are also investigated several open questions are mentioned

bic context tree estimation for stationary ergodic processes context trees of arbitrary stationary ergodic processes with finite alphabets are considered such a process is not necessarily a markov chain so the context tree may be of infinite depth calculated from a sample of size n the bayesian information criterion bic is shown to provide a strongly consistent estimator of the context tree of the process via minimization over hypothetical context trees without any restriction on the hypothetical context trees strong consistency means that the estimated context tree recovers the true one up to a level k eventually almost surely as n tends to infinity under some conditions on the process it is shown that the recovery level k can grow with n at a specific rate determined by the distribution of the process thus the bic estimator can recover the true context tree to larger and larger depths the results include for the special case of k being an arbitrary constant that the strong consistency is satisfied without any assumption on the stationary ergodic process which itself improves the existing results where either the true context tree was assumed to be of finite depth or the depth of the hypothetical context trees was bounded by o log n

stationary and transition probabilities in slow mixing long memory markov processes we observe a length n sample generated by an unknown stationary ergodic markov process model over a finite alphabet a given any string w of symbols from a we want estimates of the conditional probability distribution of symbols following w as well as the stationary probability of w two distinct problems that complicate estimation in this setting are i long memory and ii slow mixing which could happen even with only one bit of memory for our problem any consistent estimator can only converge pointwise over the class of all ergodic markov models namely given any estimator and any sample size n the underlying model could be such that the estimator performs poorly on a sample of size n with high probability but can we look at a length n sample and identify if an estimate is likely to be accurate since the memory is unknown a priori a natural approach is to estimate a potentially coarser model with memory kn o log n 1 as n grows estimates get refined and this approach is consistent with the above scaling of kn also known to be essentially optimal but while effective asymptotically the situation is quite different when we want the best answers possible with a length n sample rather than just consistency combining results in universal compression with aldous coupling arguments we obtain sufficient conditions on the length n sample even for slow mixing models to identify when naive i estimates of the model parameters and ii estimates related to the stationary probabilities are accurate and also bound the deviations of the naive estimates from true values

faces of the unit ball of a unitarily invariant norm abstract let be a unitarily invariant norm on the space of all n m complex matrices and let g r n r be the symmetric gauge function associated with that is to say we have a g s a for any a where s a is the nonincreasing sequence of singular values of a in this article we consider the relationship between the facial structures of the closed unit balls of g and which we denote by bg and b our main result gives a complete characterization of the faces of b in terms of the faces of bg for with each face e of b we associate a standard face f e of bg i e f e is a face of bg whose barycenter is a nonnegative nonincreasing n vector conversely each standard face f of bg originates a set of n m matrices say the set m f which is proven to be a face of b our main result also asserts that any face e of b is of the form q m fr where q and r are unitary matrices of appropriate orders and f f e we then explore the main result and draw some of its consequences for example we show that the singular values of the barycenter of a face e of b have the same multiplicities as the barycenter of f e formulas are given which relate the dimensions of e and f e the relative interior and the relative boundary of e are characterized in terms of the relative interior and the relative boundary of f e we completely describe the face of b generated by a matrix a our results also cover the case of real matrices

context tree estimation in variable length hidden markov models we address the issue of context tree estimation in variable length hidden markov models we propose an estimator of the context tree of the hidden markov process which needs no prior upper bound on the depth of the context tree we prove that the estimator is strongly consistent this uses information theoretic mixture inequalities in the spirit of the literatures of finesso and gassiat and boucheron we propose an algorithm to efficiently compute the estimator and provide simulation studies to support our result

stochastic complexity and modeling on demontre un theoreme fondamental qui donne une borne inferieure pour la longueur de code et donc pour les erreurs de prediction on definit les notions d information a priori et d information utile dans les donnees

rvccc a new variational network of cube connected cycles and its topological properties abstract the ccc r n network is an extension of the hypercube which replaces each vertex with a cycle of length n providing that the hypercube is r dimensional when n r the ccc r n network contains more vertices than that of ccc r r which makes it more useful in the construction of a large scale interconnection network however the ccc r n is irregular when n r which makes their properties difficult to be analyzed in this paper we propose a new variational network of the cube connected cycles rvccc the rvccc networks have the properties of regularity vertex symmetry and low diameter compared with the general ccc networks the rvccc networks are more suitable for constructing a large scale interconnection network after the shortest internode distance between any two vertices in the rvccc was determined the exact diameter of this network was calculated and the communication algorithms including the routing algorithm and the broadcasting algorithm were also developed

space efficient representation of truncated suffix trees with applications to markov order estimation suffix trees st are useful in many text processing applications for example to determine the number of occurrences of patterns of arbitrary length in an input string x if the length n of x is large the memory required to represent the st may become a practical performance bottleneck this problem can be alleviated in cases where a nontrivial upper bound is known on the lengths of the patterns of interest by using a truncated st tst however conventional tst implementations still require n bits of memory since they store x we describe a new tst representation that avoids this limitation by storing all the information necessary to reconstruct the tst edge labels in a string y that is often much shorter than x we apply tsts to the implementation of markov order estimators where an upper bound k n on the estimated order can be derived or it is imposed for consistency for example the new representation allows for estimator implementations with sublinear space complexity in some cases of interest in other cases we show experimentally that even when the new representation does not have an asymptotic advantage it still achieves very significant memory savings in practice

the context tree weighting method basic properties describes a sequential universal data compression procedure for binary tree sources that performs the double mixture using a context tree this method weights in an efficient recursive way the coding distributions corresponding to all bounded memory tree sources and achieves a desirable coding distribution for tree sources with an unknown model and unknown parameters computational and storage complexity of the proposed procedure are both linear in the source sequence length the authors derive a natural upper bound on the cumulative redundancy of the method for individual sequences the three terms in this bound can be identified as coding parameter and model redundancy the bound holds for all source sequence lengths not only for asymptotically large lengths the analysis that leads to this bound is based on standard techniques and turns out to be extremely simple the upper bound on the redundancy shows that the proposed context tree weighting procedure is optimal in the sense that it achieves the rissanen 1984 lower bound

the context tree weighting method extensions first we modify the basic binary context tree weighting method such that the past symbols x sub 1 d x sub 2 d x sub 0 are not needed by the encoder and the decoder then we describe how to make the context tree depth d infinite which results in optimal redundancy behavior for all tree sources while the number of records in the context tree is not larger than 2t 1 here t is the length of the source sequence for this extended context tree weighting algorithm we show that with probability one the compression ratio is not larger than the source entropy for source sequence length t spl rarr spl infin for stationary and ergodic sources

some aspects of the theory of norms abstract let v be a finite dimensional vector space motivated by theory or applications one might want to consider different kinds of norms on v in this paper we discuss some results and problems involving different classes of norms on a vector space studied by this author in the past few years the paper consists of five sections section 1 concerns the conditions on two vectors x y v satisfying x y for all in a certain class of norms section 2 concerns the isometry groups of g invariant norms i e norms that satisfy g x x for all x v and for all g g where g is a group of unitary orthogonal operators on v section 3 concerns g invariant norms that satisfy some special properties section 4 concerns the best approximation s x 0 t of y where y t v with respect to different kinds of norms additional open problems topics and references are mentioned in section 5

consistency of the unlimited bic context tree estimator the bayesian information criterion bic and krichevsky trofimov kt version of minimum description length mdl principle are popular in the study of model selection for order estimation of markov chains both are known to be strongly consistent when there is an upper bound on the order in the unbounded case the bic is also known to be consistent but the kt estimator is consistent only with a bound o log n on the order for context trees a flexible generalization of markov models widely used in data processing the problem is more complicated both in theory and practice given the substantially higher number of possible candidate models imre csiszar and zsolt talata proved the consistency of bic and kt when the hypothetical tree depths are allowed to grow as o log n this correspondence proves that such a restriction is not necessary for finite context sources the bic context tree estimator is strongly consistent even if there is no constraint at all on the size of the chosen tree moreover an algorithm computing the tree minimizing the bic criterion among all context trees in linear time is provided

embedding meshes into twisted cubes the n dimensional twisted cube tn n is a variation of the hypercube in this paper we study embedding of meshes into tn n we prove three major results in this paper 1 for any integer n 1 a 2x2 n 1 mesh can be embedded into tn n with dilation 1 and expansion 1 2 for any integer n 4 an mxk m 3 k 3 mesh cannot be embedded into tn n with dilation 1 3 for any integer n 4 two node disjoint 4x2 n 3 meshes can be embedded into tn n with dilation 2 and expansion 1

the isometries and the g invariance of certain seminorms abstract let f be the field of real or complex numbers and let g be a subgroup of the general linear group gl n f if a is an m n matrix over f let a be the seminorm on f n defined by x a ax for all x f n in this paper we characterize the linear isometries for the seminorm a and study the conditions on a for which a is g invariant that is sx a x a for all x f n and all s g as a special case we describe all matrices a for which a is absolute or a symmetric gauge function

the consistency of the bic markov order estimator the bayesian information criterion bic estimates the order of a markov chain with finite alphabet a from observation of a sample path x 1 x 2 x n as that value k k that minimizes the sum of the negative logarithm of the kth order maximum likelihood and the penalty term a k a 1 2 log n we show that k equals the correct order of the chain eventually almost surely as n oo thereby strengthening earlier consistency results that assumed an apriori bound on the order a key tool is a strong ratio typicality result for markov sample paths we also show that the bayesian estimator or minimum description length estimator of which the bic estimator is regarded as an approximation fails to be consistent for the uniformly distributed i i d process

edge disjoint hamiltonian cycles in k ary n cubes and hypercubes solutions for decomposing a higher dimensional torus to edge disjoint lower dimensional tori in particular edge disjoint hamiltonian cycles are obtained based on the coding theory approach first lee distance gray codes in z sub k sup n are presented and then it is shown how these codes can directly be used to generate edge disjoint hamiltonian cycles in k ary n cubes further some new classes of binary gray codes are designed from these lee distance gray codes and using these new classes of binary gray codes edge disjoint hamiltonian cycles in hypercubes are generated

faces and traces of the unit ball of a symmetric gauge function abstract we investigate the faces of the closed unit ball bg x g x 1 of a symmetric gauge function g r n r firstly we describe the group of symmetries of a face f of bg in terms of its barycenter we introduce the concept of standard face of bg i e a face whose relative interior meets the set d x r n x 1 x n 0 called here the standard cone of r n and the concept of trace of a face f which is the intersection of f with d characterizations of standard faces are given and we show that the trace of a face if nonempty is the trace of a standard face it is proved that a face f and its trace have the same dimension iff f is a standard face a relation is found between the dimension of f and the dimension of its symmetry core i e the set of points of f that are fixed by all symmetries of f other results of this type are given the main result has a long and delicate proof it completely characterizes in several different ways the subsets of bg that are traces of faces of bg

k restricted edge connectivity for some interconnection networks abstract k restricted edge connectivity is an important parameter in measuring the reliability and fault tolerance of large interconnection networks in this paper we present two families of graphs similar with the networks proposed by chen et al y c chen j j m tan l h hsu s s kao super connectivity and super edge connectivity for some interconnection networks applied mathematics and computation 140 2003 245 254 and study the k k 2 3 restricted edge connectivity of these graphs in particular as the applications of our results the k k 2 3 restricted edge connectivity of the recursive circulant graphs and the n ary k cubes is given

complete path embeddings in crossed cubes crossed cubes are popular variants of hypercubes in this paper we study path embeddings between any two distinct nodes in crossed cubes we prove two important results in the n dimensional crossed cube a for any two nodes all paths whose lengths are greater than or equal to the distance between the two nodes plus 2 can be embedded between the two nodes with dilation 1 b for any two integers n 2 and l with 1

the minimum description length principle in coding and modeling we review the principles of minimum description length and stochastic complexity as used in data compression and statistical modeling stochastic complexity is formulated as the solution to optimum universal coding problems extending shannon s basic source coding theorem the normalized maximized likelihood mixture and predictive codings are each shown to achieve the stochastic complexity to within asymptotically vanishing terms we assess the performance of the minimum description length criterion both from the vantage point of quality of data compression and accuracy of statistical inference context tree modeling density estimation and model selection in gaussian linear regression serve as examples

a universal data compression system a universal data compression algorithm is described which is capable of compressing long strings generated by a finitely generated source with a near optimum per symbol length without prior knowledge of the source this class of sources may be viewed as a generalization of markov sources to random fields moreover the algorithm does not require a working storage much larger than that needed to describe the source generating parameters

context tree estimation for not necessarily finite memory processes via bic and mdl the concept of context tree usually defined for finite memory processes is extended to arbitrary stationary ergodic processes with finite alphabet these context trees are not necessarily complete and may be of infinite depth the familiar bayesian information criterion bic and minimum description length mdl principles are shown to provide strongly consistent estimators of the context tree via optimization of a criterion for hypothetical context trees of finite depth allowed to grow with the sample size n as o logn algorithms are provided to compute these estimators in o n time and to compute them on line for all i les n in o nlogn time

generalized hypercubes edge disjoint hamiltonian cycles and gray codes some new classes of hamming metric gray codes over zpn where p is a prime and n is an integer power of 2 are described then how these gray codes can be used to generate the maximum number of edge disjoint hamiltonian cycles in an n dimensional generalized hypercube ghc qpn is shown for qpn the number of edge disjoint hamiltonian cycles generated using these methods is n p 1 2 which is the maximum possible since the degree of each node in qpn is n p 1 in addition for any integers p and n p not necessarily a prime and n not necessarily a power of 2 how to generate the maximum number of edge disjoint hamiltonian cycles in qnp is also described