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

dynamic response of a rotating timoshenko shaft subject to axial forces and moving loads abstract the dynamic response of a rotating shaft subject to axial force and moving loads is analyzed by using timoshenko beam theory and the assumed mode method the deformations of the shaft are expressed in terms of an inertial reference frame the kinetic and potential energy are then expressed in matrix form by using the assumed mode method the resulting equations of motion in matrix form are derived by using hamilton s principle the influences of the rotational speed of the shaft the axial speed of the loads and the rayleigh coefficient are investigated and compared with the available reported results the effects of compressive axial forces and perturbation of the axial velocity of the moving loads are also included in the analysis

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

information measures for infinite sequences we revisit the notion of computational depth and sophistication for infinite sequences and study the density of the sets of deep and sophisticated infinite sequences koppel defined the sophistication of an object as the length of the shortest total program that given some data as input produces the object and such that the sum of the size of the program with the size of the data is as concise as the smallest description of the object however the notion of sophistication is not properly defined for all sequences we propose a new definition of sophistication for infinite sequences as the limit of the ratio of the sophistication of the initial segments and its length we prove that the set of sequences with sophistication equal to zero has lebesgue measure one and that the set of sophisticated sequences is dense when the sophistication is respectively defined either using lim inf or lim sup antunes fortnow van melkebeek and vinodchandran captured the notion of useful information by computational depth the difference between time bounded and unbounded kolmogorov complexities we show that the set of deep infinite sequences is dense we also prove that sophistication and depth for infinite sequences are distinct information measures

compound poisson approximations for sums of random variables abstract this document shows that a sum of dependent random variables is approximately compound poisson when the variables are rarely nonzero and given they are nonzero their conditional distributions are nearly identical it give several upper bounds on the total variation distance between the distribution of such a sum and a compund poisson distribution included is an example for markovian occurrences of a rare event the bounds are consistent with those that are known for poisson approximations for sums of uniformly small random variables author

the kolmogorov complexity of infinite words we present a brief survey of results on relations between the kolmogorov complexity of infinite strings and several measures of information content dimensions known from dimension theory information theory or fractal geometry special emphasis is placed on bounds on the complexity of strings in constructively given subsets of the cantor space finally we compare the kolmogorov complexity to the subword complexity of infinite strings

a dynamic model for a rotating beam subjected to axially moving forces abstract this paper presents a dynamic model for the vibration of a rotating timoshenko beam subjected to a three directional load moving in the axial direction the model takes into account the axial movement of the axial force component the bending moment produced by this component acting on the surface of the beam is included in the model the two other transverse force components are either of constant magnitude or a linear function of the local deflection lagrange s equations of motion for the modal coordinates are derived based on the assumed mode method these equations are then solved by a fourth order runge kutta algorithm numerical examples are analysed the effects of the axial force component and its induced bending moment and the deflection dependence of the moving forces on the dynamic behaviour of the system at various travelling speeds are investigated it is found that the bending moment induced by the axial force component has a significant influence on the dynamic response of the rotating beam and hence must be considered in such problems as turning operations

dynamic response of a rotating beam subjected to an accelerating distributed surface force abstract a method is given by which the response of a rotating or non rotating timoshenko beam can be determined subjected to an accelerating fixed direction distributed surface force the beam model includes the gyroscopically induced displacement transverse to the direction of the load the solution for pinned supports is set up using multi integral transforms and the inversion is expressed in terms of convulution integrals these are numerically integrated for a uniformly distributed load having an exponentially varying velocity function results are presented for the displacement under the load s center as a function of position and for the displacement of every point on the beam at an instant in time comparisons are made between the beam response to a constant velocity load and its response to a load which accelerates to the same velocity

modal analysis of rotating shafts a body fixed axis formulation approach abstract a spinning timoshenko beam subjected to a constant moving load is analyzed using a modal expansion technique the formulation is based on a body fixed axis reference system and thus should be applicable to handle any general cross sections using this alternative approach dynamic quantities such as natural frequencies mode shapes and system responses are easily computable it is shown that simply supported spinning timoshenko beams posses two pairs of natural frequencies corresponding to each mode shape through appropriate simplifications the coupled eighth order differential equations governing the problem are degenerated into a set of uncoupled fourth order equations with this simplified theory closed form expressions for natural frequencies and system transient response are derived however one pair of the natural frequencies is lost in the process but this is acceptable as the frequencies of this pair are much higher than those of the retained pair in addition a linearized approximation of the natural frequency similar to the one given for euler bernoulli beams is also proposed for a timoshenko shaft numerical simulations are performed to demonstrate the characteristics of the response

an almost machine independent theory of program length complexity sophistication and induction abstract the purpose of this paper is to use a variant of program length complexity to formally define the structure of a binary string where the structure of an object is taken to mean the aggregate of its projectible properties

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

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

compound poisson approximation for nonnegative random variables via stein s method the aim of this paper is to extend stein s method to a compound poisson distribution setting the compound poisson distributions of concern here are those of the form pois nu where nu is a finite positive measure on 0 infty a number of results related to these distributions are established these in turn are used in a number of examples to give bounds for the error in the compound poisson approximation to the distribution of a sum of random variables

computational depth concept and applications we introduce computational depth a measure for the amount of nonrandom or useful information in a string by considering the difference of various kolmogorov complexity measures we investigate three instantiations of computational depth basic computational depth a clean notion capturing the spirit of bennett s logical depth we show that a turing machine m runs in time polynomial on average over the time bounded universal distribution if and only if for all inputs x m uses time exponential in the basic computational depth of x sublinear time computational depth and the resulting concept of shallow sets a generalization of sparse and random sets based on low depth properties of their characteristic sequences we show that every computable set that is reducible to a shallow set has polynomial size circuits distinguishing computational depth measuring when strings are easier to recognize than to produce we show that if a boolean formula has a nonnegligible fraction of its satisfying assignments with low depth then we can find a satisfying assignment efficiently

poisson approximation for excursions of adaptive algorithms with a lattice state space this paper analyzes excursions of adaptive algorithms such as the lms with a lattice state space under certain conditions on the input and disturbance statistics the parameter estimate error forms a markov chain the approximations are valid if this chain has a strong tendency toward an equilibrium point the distribution of the number of excursions in n units of time is approximated by a poisson distribution the mean and distribution of the time of the occurrence of the first excursion are approximated by those of an exponential distribution expressions for the error in the approximations are also derived the approximations are shown to hold asymptotically as the excursion defining set converges to the empty set all the parameters required for the approximations and all expressions for the error in the approximations are calculable in a relatively straightforward manner

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

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

hausdorff dimension and oracle constructions bennett and gill relative to a random oracle a pa npa co npa with probability 1 siam j comput 10 1981 96 113 proved that pa npa relative to a random oracle a or in other words that the set o p np a pa npa has lebesgue measure 0 in contrast we show that o p np has hausdorff dimension 1 this follows from a much more general theorem if there is a relativizable and paddable oracle construction for a complexity theoretic statement then the set of oracles relative to which holds has hausdorff dimension 1 we give several other applications including proofs that the polynomial time hierarchy is infinite relative to a hausdorff dimension 1 set of oracles and that pa npa conpa relative to a hausdorff dimension 1 set of oracles