complex network dimension and path counts large complex networks occur in many applications of computer science the complex network zeta function and the graph surface function have been used to characterize these networks and to define a dimension for complex networks in this work we present three new results related to the complex network dimension first we show the relationship of the concept to kolmogorov complexity second we show how the definition of complex network dimension can be reformulated by defining the concept for a single node and then defining the complex network dimension as the supremum over all nodes this makes the concept work better for formally infinite graphs third we study interesting parallels to zeta dimension a notion originally from number theory which has found connections to theoretical computer science these parallels lead to a deeper insight into the complex network dimension e g the formulation in terms of the entropy and a theorem relating dimension to connectivity

generic density and small span theorem we refine the genericity concept of ambos spies by assigning a real number in 0 1 to every generic set called its generic density we construct sets of generic density any e computable real in 0 1 and show a relationship between generic density and lutz resource bounded dimension we also introduce strong generic density and show that it is related to packing dimension we show that all four notions are different we show that whereas dimension notions depend on the underlying probability measure generic density does not which implies that every dimension result proved by generic density arguments simultaneously holds under any biased coin based probability measure we prove such a result we improve the small span theorem of juedes and lutz to the packing dimension setting for k bounded truth table reductions under any biased coin probability measure

bounding the dimension of points on a line abstract we use kolmogorov complexity methods to give a lower bound on the effective hausdorff dimension of the point x a x b given real numbers a b and x we apply our main theorem to a problem in fractal geometry giving an improved lower bound on the classical hausdorff dimension of generalized sets of furstenberg type

generalization of automatic sequences for numeration systems on a regular language let l be an infinite regular language on a totally ordered alphabet feeding a finite deterministic automaton with output with the words of l enumerated lexicographically with respect to leads to an infinite sequence over the output alphabet of the automaton this process generalizes the concept of k automatic sequence for abstract numeration systems on a regular language instead of systems in base k here we study the first properties of these sequences and their relations with numeration systems

critical properties of directed self avoiding walks the generating functions and mean displacements of various two dimensional directed self avoiding walk models are calculated exactly by a simple transfer matrix approach asymptotically the authors find r sub n approximately rn2 1 2 approximately n and rperpendicular to n2 1 approximately n1 2 where n is the number of steps in the walk and and perpendicular to refer to projections of the displacement parallel and perpendicular to the preferred axis of the walk respectively some general properties of directed self avoiding walks for arbitrary dimensions are discussed as well

finite state incompressible infinite sequences in this paper we define and study finite state complexity of finite strings and infinite sequences as well as connections between these complexity notions to randomness and normality we show that the finite state complexity does not only depend on the codes for finite transducers but also on how the codes are mapped to transducers as a consequence we relate the finite state complexity to the plain kolmogorov complexity to the process complexity and to prefix free complexity working with prefix free sets of codes we characterise martin lof random sequences in terms of finite state complexity the weak power of finite transducers is compensated by the high complexity of enumeration of finite transducers we also prove that every finite state incompressible sequence is normal but the converse implication is not true these results also show that our definition of finite state incompressibility is stronger than all other known forms of finite automata based incompressibility in particular the notion related to finite automaton based betting systems introduced by schnorr and stimm the paper concludes with a discussion of open questions

nilpotent orbits and commutative elements abstract let w be a simply laced coxeter group with generating set s and let w c denote the subset consisting of those elements whose reduced expressions have no substrings of the form sts for any non commuting s t s we give a root system characterization of w c and in the case where w corresponds to a finite weyl group show that w c is a union of spaltenstein springer steinberg cells the latter is valid also for affine weyl groups of type a but not for type d or e

effective strong dimension in algorithmic information and computational complexity the two most important notions of fractal dimension are hausdorff dimension developed by hausdorff math ann 79 1919 pp 157 179 and packing dimension developed independently by tricot math proc cambridge philos soc 91 1982 pp 57 74 and sullivan acta math 153 1984 pp 259 277 both dimensions have the mathematical advantage of being defined from measures and both have yielded extensive applications in fractal geometry and dynamical systems lutz proceedings of the 15th ieee conference on computational complexity florence italy 2000 ieee computer society press piscataway nj 2000 pp 158 169 has recently proven a simple characterization of hausdorff dimension in terms of gales which are betting strategies that generalize martingales imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of hausdorff dimension including constructive computable polynomial space polynomial time and finite state dimensions work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science in this paper we show that packing dimension can also be characterized in terms of gales moreover even though the usual definition of packing dimension is considerably more complex than that of hausdorff dimension our gale characterization of packing dimension is an exact dual of and every bit as simple as the gale characterization of hausdorff dimension effectivizing our gale characterization of packing dimension produces a variety of effective strong dimensions which are exact duals of the effective dimensions mentioned above in general and in analogy with the classical fractal dimensions the effective strong dimension of a set or sequence is at least as great as its effective dimension with equality for sets or sequences that are sufficiently regular we develop the basic properties of effective strong dimensions and prove a number of results relating them to fundamental aspects of randomness kolmogorov complexity prediction boolean circuit size complexity polynomial time degrees and data compression aside from the above characterization of packing dimension our two main theorems are the following 1 if vec beta beta 0 beta 1 ldots is a computable sequence of biases that are bounded away from 0 and r is random with respect to vec beta then the dimension and strong dimension of r are the lower and upper average entropies respectively of vec beta 2 for each pair of delta 0 2 computable real numbers 0 alpha le beta le 1 there exists a in rm e such that the polynomial time many one degree of a has dimension alpha in e and strong dimension beta in e our proofs of these theorems use a new large deviation theorem for self information with respect to a bias sequence vec beta that need not be convergent

normal numbers and finite automata we give an elementary and direct proof of the following theorem a real number is normal to a given integer base if and only if its expansion in that base is incompressible by lossless finite state compressors these are finite automata augmented with an output transition function such that the automata input output behaviour is injective they are also known as injective finite state transducers as a corollary we obtain v n agafonov s theorem on the preservation of normality on subsequences selected by finite automata

finite state dimension classical hausdorff dimension sometimes called fractal dimension was recently effectivized using gales betting strategies that generalize martingales thereby endowing various complexity classes with dimension structure and also defining the constructive dimensions of individual binary infinite sequences in this paper we use gales computed by multi account finite state gamblers to develop the finite state dimensions of sets of binary sequences and individual binary sequences the theorem of eggleston quart j math oxford ser 20 1949 31 36 relating hausdorff dimension to entropy is shown to hold for finite state dimension both in the space of all sequences and in the space of all rational sequences binary expansions of rational numbers every rational sequence has finite state dimension 0 but every rational number in 0 1 is the finite state dimension of a sequence in the low level complexity class ac0 our main theorem shows that the finite state dimension of a sequence is precisely the infimum of all compression ratios achievable on the sequence by information lossless finite state compressors

entropy rates and finite state dimension the effective fractal dimensions at the polynomial space level and above can all be equivalently defined as the c entropy rate where c is the class of languages corresponding to the level of effectivization for example pspace dimension is equivalent to the pspace entropy rate at lower levels of complexity the equivalence proofs break down in the polynomial time case the p entropy rate is a lower bound on the p dimension equality seems unlikely but separating the p entropy rate from p dimension would require proving p np we show that at the finite state level the opposite of the polynomial time case happens the reg entropy rate is an upper bound on the finite state dimension we also use the finite state genericity of ambos spies and busse automatic forcing and genericity on the diagonalization strength of finit automata in proc fourth int conf on discrete mathematics and theoretical computer science 2003 springer berlin pp 97 108 to separate finite state dimension from the reg entropy rate however we point out that a block entropy rate characterization of finite state dimension follows from the work of ziv and lempel compression of individual sequences via variable rate coding ieee trans inform theory 24 1978 530 536 on finite state compressibility and the compressibility characterization of finite state dimension by dai et al finite state dimension theoret comput sci 310 1 3 2004 1 33 as applications of the reg entropy rate upper bound and the block entropy rate characterization we prove that every regular language has finite state dimension 0 and that normality is equivalent to finite state dimension 1

dimension in complexity classes a theory of resource bounded dimension is developed using gales which are natural generalizations of martingales when the resource bound delta a parameter of the theory is unrestricted the resulting dimension is precisely the classical hausdorff dimension sometimes called fractal dimension other choices of the parameter delta yield internal dimension theories in e e2 espace and other complexity classes and in the class of all decidable problems in general if mathcal c is such a class then every set x of languages has a dimension in mathcal c which is a real number dim x mid mathcal c in 0 1 along with the elements of this theory two preliminary applications are presented for every real number 0 le alpha le frac 1 2 the set rm freq le alpha consisting of all languages that asymptotically contain at most alpha of all strings has dimension mathcal h alpha the binary entropy of alpha in e and in e2 for every real number 0 le alpha le 1 the set rm size alpha frac 2 n n consisting of all languages decidable by boolean circuits of at most alpha frac 2 n n gates has dimension alpha in espace

comments on real space renormalisation group for directed lattice animals the authors carefully analyse the rg results of directed site lattice animals on a square lattice obtained from a new rsrg method proposed by them in 1984 here asymptotic forms for b 1 b and b 1 b the values obtained from cell to cell transformation of linear size b 1 to b are proposed in the large b limit by fitting the data they find 0 796 and 0 507 these results are in very good agreement with the known values

the dimensions of individual strings and sequences a constructive version of hausdorff dimension is developed using constructive supergales which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences this constructive dimension is used to assign every individual infinite binary sequence s a dimension which is a real number dim s in the interval 0 1 sequences that are random in the sense of martin lof have dimension 1 while sequences that are decidable 10 or 10 have dimension 0 it is shown that for every 20 computable real number in 0 1 there is a 20 sequence s such that dim s a discrete version of constructive dimension is also developed using termgales which are supergale like functions that bet on the terminations of finite binary strings as well as on their successive bits this discrete dimension is used to assign each individual string w a dimension which is a nonnegative real number dim w the dimension of a sequence is shown to be the limit inferior of the dimensions of its prefixes the kolmogorov complexity of a string is proven to be the product of its length and its dimension this gives a new characterization of algorithmic information and a new proof of mayordomo s recent theorem stating that the dimension of a sequence is the limit inferior of the average kolmogorov complexity of its first n bits every sequence that is random relative to any computable sequence of coin toss biases that converge to a real number in 0 1 is shown to have dimension h the binary entropy of

minuscule elements of weyl groups the numbers game andd complete posets abstract certain posets associated to a restricted version of the numbers game of mozes are shown to be distributive lattices the posets of join irreducibles of these distributive lattices are characterized by a collection of local structural properties which form the definition of d complete poset in representation theoretic language the top minuscule portions of weight diagrams for integrable representations of simply laced kac moody algebras are shown to be distributive lattices these lattices form a certain family of intervals of weak bruhat orders these bruhat lattices are useful in studying reduced decompositions of minuscule elements of weyl groups and their associated schubert varieties the d complete posets have recently been proven to possess both the hook length and the jeu de taquin properties

directed lattice animals and the lee yang edge singularity the exponents nu perpendicular to gamma of directed animals in d dimensions are shown to equal those for the lee yang problem in d 1 dimensions for d 2 the author obtains the exact results nu perpendicular to theta 1 2 and for d or 7 the scaling relation theta d 1 nu perpendicular to

pushdown dimension resource bounded dimension is a notion of computational information density of infinite sequences based on computationally bounded gamblers this paper develops the theory of pushdown dimension and explores its relationship with finite state dimension the pushdown dimension of any sequence is trivially bounded above by its finite state dimension since a pushdown gambler can simulate any finite state gambler we show that for every rational 0

minuscule elements of weyl groups abstract we provide several characterizations of the minuscule elements of weyl groups studied by d peterson and r a proctor and extend proctor s classification from the simply laced case to the general case

dimensions of copeland erdos sequences the base k copeland erdos sequence given by an infinite set a of positive integers is the infinite sequence cemk a formed by concatenating the base k representations of the elements of a in numerical order this paper concerns the following four quantities the finite state dimension dimfs cek a a finite state version of classical hausdorff dimension introduced in 2001 the finite state strong dimension dimfs cek a a finite state version of classical packing dimension introduced in 2004 this is a dual of dimfs cek a satisfying dimfs cek a dimfs ce k a the zeta dimension dim a a kind of discrete fractal dimension discovered many times over the past few decades the lower zeta dimension dim a a dual of dim a satisfying dim a dim a we prove the following dimfs cek a dim a this extends the 1946 proof by copeland and erdos that the sequence cek primes is borel normal dimfs cek a dim a these bounds are tight in the strong sense that these four quantities can have simultaneously any four values in 0 1 satisfying the four above mentioned inequalities

finite state dimension and real arithmetic we use entropy rates and schur concavity to prove that for every integer k 2 every nonzero rational number q and every real number a the base k expansions of a q a and q a all have the same finite state dimension and the same finite state strong dimension this extends and gives a new proof of wall s 1949 theorem stating that the sum or product of a nonzero rational number and a borel normal number is always borel normal

real space renormalisation group for directed systems a new real space renormalisation group method is developed to treat systems with directionally dependent critical behaviour here two effective lengths s sub b and sperpendicular to b are defined for a given cell of linear size b the renormalised lattice is constructed from these effective lengths and is deformed from the original lattice it is the anisotropic rescaling of these effective lengths which give the anisotropic exponents nu sub and nu perpendicular to this method gives the exact result for the case of fully directed saws on a square lattice in the b to infinity limit for directed bond and directed site lattice animals the results obtained here are also in good agreement with the other known values

functions that preserve p randomness we show that polynomial time randomness p randomness is preserved under a variety of familiar operations including addition and multiplication by a nonzero polynomial time computable real number these results follow from a general theorem if email protected r is an open interval f i r is a function and email protected i is p random then f r is p random provided1 f is p computable on the dyadic rational points in i and 2 f varies sufficiently at r i e there exists a real constant c 0 such that either email protected i r f x f r x r c or email protected i r f x f r x r c our theorem implies in particular that any analytic function about a p computable point whose power series has uniformly p computable coefficients preserves p randomness in its open interval of absolute convergence such functions include all the familiar functions from first year calculus

finite state complexity in this paper we develop a version of algorithmic information theory ait based on finite transducers instead of turing machines the complexity induced is called finite state complexity in spite of the fact that the universality theorem true for turing machines is false for finite transducers the invariance theorem holds true for finite state complexity we construct a class of finite state complexities based on various enumerations of the set of finite transducers in contrast with descriptional complexities plain prefix free from ait finite state complexity is computable and there is no a priori upper bound for the number of states used for minimal descriptions of arbitrary strings upper and lower bounds for the finite state complexity of arbitrary strings and for strings of particular types are given and incompressible strings are studied

fractal dimension and logarithmic loss unpredictability we show that the hausdorff dimension equals the logarithmic loss unpredictability for any set of infinite sequences over a finite alphabet using computable feasible and finite state predictors this equivalence also holds for the computable feasible and finite state dimensions combining this with recent results of fortnow and lutz proc 15th ann conf on comput learning theory 2002 380 we have a tight relationship between prediction with respect to logarithmic loss and prediction with respect to absolute loss

a grobner basis for kazhdan lusztig ideals it kazhdan lusztig ideals a family of generalized determinantal ideals investigated in woo yong 08 provide an explicit choice of coordinates and equations encoding a neighborhood of a torus fixed point of a schubert variety on a type a flag variety our main result is a gr o bner basis for these ideals this provides a single geometric setting to transparently explain the naturality of pipe dreams on the it rothe diagram of a permutation and their appearance in begin itemize item combinatorial formulas fomin kirillov 94 for schubert and grothendieck polynomialsof lascoux sch u tzenberger 82 item the equivariant k theory specialization formula of buch rim a nyi 04 and item a positive combinatorial formula for multiplicities of schubert varieties in good cases including those for which the associated kazhdan lusztig ideal is homogeneous under the standard grading end itemize our results generalize with alternate proofs knutson miller 05 s gr o bner basis theorem for schubert determinantal ideals and their geometric interpretation of the monomial positivity of schubert polynomials we also complement recent work of knutson 08 09 on degenerations of kazhdan lusztig varieties in general lie type as well as work of goldin 01 on equivariant localization and of lakshmibai weyman 90 rosenthal zelevinsky 01 and krattenthaler 01 on grassmannian multiplicity formulas

kolmogorov complexity and hausdorff dimension abstract in this paper various relationships between the kolmogorov complexity of infinite strings and measures of information content are given the general approach taken here is to bound the complexity of a maximally complex string in a given set of strings by the hausdorff dimension or the entropy of that set it turns out that hausdorff dimension yields lower bounds to the kolmogorov complexity whereas under certain recursiveness constraints on the structure of the respective sets their entropy yields upper bounds more detailed investigations result in a generalization of p martin lof s theorems on the kolmogorov complexity of random strings to maximally complex strings in regularly structured sets of infinite strings

abelian ideals in a borel subalgebra of a complex simple lie algebra let mathfrak g be a complex simple lie algebra and mathfrak b a fixed borel subalgebra of mathfrak g we describe the abelian ideals in mathfrak b in a uniform way that is independent of the classification of complex simple lie algebras as an application we derive a formula for the maximal dimension of a commutative lie subalgebra of mathfrak g

a divergence formula for randomness and dimension if s is an infinite sequence over a finite alphabet s and b is a probability measure on s then the dimension of s with respect to b written dim b s is a constructive version of billingsley dimension that coincides with the constructive hausdorff dimension dim s when b is the uniform probability measure this paper shows that dim b s and its dual dim b s the strong dimension of s with respect to b can be used in conjunction with randomness to measure the similarity of two probability measures a and b on s specifically we prove that the divergence formuladim b r dim b r h a h a d email protected email protected holds whenever a and b are computable positive probability measures on s and email protected email protected is random with respect to a in this formula h a is the shannon entropy of a and d email protected email protected is the kullback leibler divergence between a and b we also show that the above formula holds for all sequences r that are a normal in the sense of borel when dim b r and dim b r are replaced by the more effective finite state dimensions dim f s b r and dim f s b r in the course of proving this we also prove finite state compression characterizations of dim f s b s and dim f s b s

directed self avoiding walks on certain directed random fractals using the recently proposed real space renormalisation group method for directed systems the authors study the critical behaviour of directed self avoiding walks dsaw on both directed lattice animals dla and directed percolation clusters at threshold pc dpc in two dimensions the values for the exponent nu perpendicular to are found to be nu perpendicular to dsawdla approximately 0 590 and nu perpendicular to dsawdpc approximately 0 528 both of which are higher than the mean field value 0 5 it is also shown rigorously that nu sub remains unchanged i e nu sub dsaw dla nu sub dsaw dpc 1

size and shape of directed lattice animals uses position space renormalisation and exact enumeration methods to study configurational properties of directed lattice animals clusters of directed bonds in which the tail of a new bond must be added to the head of existing bonds exponents are found for the divergence of correlation lengths parallel and perpendicular to the preferred axis in addition an exponent is found that characterises the singularity of the directed animal generating function and critical points are located

on tables of random numbers the set theoretic axioms of the calculus of probability in formulating which i had the opportunity of playing some part kolmogorov 1950 had solved the majority of formal difficulties in the construction of a mathematical apparatus which is useful for a very large number of applications of probabilistic methods so successfully that the problem of finding the basis of real applications of the results of the mathematical theory of probability became rather secondary to many investigators i have already expressed the view see kolmogorov 1950 chapter i that the basis for the applicability of the results of the mathematical theory of probabi lity to real random phenomena must depend on some form of the frequency concept of probability the unavoidable nature of which has been established by von mises in a spirited manner however for a long time i had the following views 1 the frequency concept based on the notion of limiting frequency as the number of trials increases to infinity does not contribute anything to substantiate the applicability of the results of probability theory to real practical problems where we have always to deal with a finite number of trials 2 the frequency concept applied to a large but finite number of trials does not admit a rigorous formal exposition within the framework of pure mathematics accordingly i have sometimes put forward the frequency concept which involves the conscious use of certain not rigorously formal ideas about practical reliability approximate stability of the frequency in a long series of trials without the precise definition of the series which are sufficiently large etc see foundation of the theory of probability chapter i and for more details great soviet encyclo paedia section on probability and mathematika iou metod i znachenye chapter on probability theory i still maintain the first of the two theses mentioned above as regards the second however i have come to realise that the concept of random distribution of a property in a large finite population can have a strict formal mathematical exposition in fact we can show that in sufficiently large populations the distribution of the pro perty may be such that the frequency of its occurrence will be almost the same for all sufficiently large sub populations when the law of choosing these is sufficiently simple such a conception in its full development requires the introduction of a measure of the complexity of the algorithm i propose to discuss this question in another article in the present article however i shall use the fact that there cannot be a very large number of simple algorithms 369 3