on the complexity of algebraic numbers i expansions in integer bases let b 2 be an integer we prove that the b ary expansion of every irrational algebraic number cannot have low complexity furthermore we establish that irrational morphic numbers are transcendental for a wide class of morphisms in particular irrational automatic numbers are transcendental our main tool is a new combinatorial transcendence criterion

discontinuous perturbation analysis of discrete event dynamic systems the discontinuous perturbation analysis dpa method is presented to deal with discontinuous sample performance functions this method is applicable to derivative estimation with respect to most threshold type problems in discrete event dynamic systems deds through modeling of discontinuities by step function we are able to provide a unified framework for constructing the derivative estimation of a deds as a result we offer an alternate approach to derive derivative estimation in deds s finally some relationships with other pa techniques are discussed and a numerical example is presented

a sample performance function of closed jackson queueing networks a stochastic system such as a queueing network can be specified by system parameters and a set of sequences of random variables that represents the randomness in the system a sample performance function is a measure of system performance as a function of system parameters for each realization of the random sequences although the average of n sample performance functions converges to the expected value of the performance with probability one when n goes to infinity the average of the derivatives of these n sample performance functions with respect to a parameter may not converge to the derivative of the expected value in this paper we study a sample performance function of a closed jackson queueing network specifically the time required by a server to serve a finite number of customers we show that this sample performance function is a continuous piecewise linear function of the mean service time we prove that the average of derivatives of this sample performance function with a given initial

aggregation of particles which move on deterministic trajectories with fractal dimension two i a simple and new model for dla the author simulates a new type of aggregation process particles follow deterministic trajectories the trajectories have a fractal dimension dw of two dw is defined as s r approximately rdw where s r is the total number of sites a particle visits while travelling a distance r when the growth process is started with a single stationary particle on a square lattice a large aggregate looks like diffusion limited aggregation dla the fractal dimension df of the aggregate is nearly equal to 1 7 that of dla is 1 67 or 0 05 moreover the aggregate has axial anisotropy although it includes less than one thousand particles therefore the aggregation belongs to the same universality class of dla and is supposed to be the highly noise reduced model for dla because the laplace equation has no relationship to the aggregation process the laplace equation does not seem to be a necessary condition for producing aggregates like dla

sample path derivatives for s s inventory systems for s s inventory systems we derive sample path derivatives of performance measures with respect to the two parameters s and s these derivatives yield derivative estimators which can be estimated from a single sample path or simulation of the inventory system in some cases not even requiring actual knowledge of the underlying demand distribution such derivative estimates would be useful in sensitivity analysis or in gradient based optimization techniques we consider the nondiscounted periodic review system with general independent and identically distributed i i d continuous demands full backlogging and general holding and shortage costs for the infinite horizon model consistency proofs are given for some special cases although we argue why the estimators should be correct for the more general case

first and second derivative estimators for cyclic closed queueing networks we consider a cyclic closed queueing network with arbitrary service time distributions and derive first and second derivative estimators of some finite horizon performance metrics with respect to a parameter of any one of the service distributions our approach is based on observing a single sample path of this system and evaluating first and second order effects on departure times as a result of the parameter perturbation we then define an estimator as a conditional expectation over appropriate observable quantities using smoothed perturbation analysis techniques this process recovers the first derivative estimator along the way and gives new insights into event order change phenomena which are of higher order despite the complexity of the analysis the final algorithms we obtain are relatively simple further we show that our estimators are unbiased and include some numerical examples we also show the use of our estimators in obtaining approximations of the entire system response surface as a function of system parameters

perturbation analysis of discrete event systems concepts algorithms and applications abstract perturbation analysis of discrete event systems is a technique that provides estimates of performance sensitivities with respect to system parameters by analyzing a single sample path of a discrete event system this technique saves a great amount of computation in simulation and can be applied to on line performance optimization of real world systems where changing the values of parameters is infeasible this paper gives an overview of the main concepts general methodologies applications and further directions of the area

on the complexity of algebraic numbers ii continued fractions the continued fraction expansion of an irrational number alpha is eventually periodic if and only if alpha is a quadratic irrationality however very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three because of some numerical evidence and a belief that these numbers behave like most numbers in this respect it is often conjectured that their partial quotients form an unbounded sequence more modestly we may expect that if the sequence of partial quotients of an irrational number alpha is in some sense simple then alpha is either quadratic or transcendental the term simple can of course lead to many interpretations it may denote real numbers whose continued fraction expansion has some regularity or can be produced by a simple algorithm by a simple turing machine for example or arises from a simple dynamical system the aim of this paper is to present in a unified way several new results on these different approaches of the notion of simplicity complexity for the continued fraction expansion of algebraic real numbers of degree at least three

gradient estimation for multicomponent maintenance systems with age replacement policy we consider multicomponent maintenance systems with an f failure group age replacement policy it keeps failed components idling until f components are failed and then replaces all failed components together with the nonfailed components whose age has passed the critical threshold age n for components of type n with each maintenance action costs are associated we derive various unbiased gradient estimators based on the measure valued differentiation approach for the gradient of the average cost each estimator has its own domain of applicability we also compare the performance of our gradient estimators when applied to stochastic optimization with other general gradient free methods

smoothed conditional perturbation analysis of discrete event dynamical systems a new sample path analysis approach based on the smoothing property of conditional expectation for estimating the performance sensitivity of discrete event dynamical systems is proposed several examples are presented to show how this approach overcomes a difficulty of the ordinary infinitesimal perturbation analysis the basic message is that one can get more knowledge about the system performance by observing and analyzing the sample path than by using the conventional simulation approach it is also pointed out that the classical queueing theory approach for getting the performance sensitivity and the sample path based infinitesimal perturbation analysis approach can be unified in the framework of the new approach the smoothed conditional perturbation analysis

metric properties of fractal lattices the authors study the connectivity dimension d of fractal lattices viewed as networks graphs of sites and constant length bond two examples are investigated in detail the 2d sierpinski gasket and 2d infinite percolation clusters on square and triangular lattices in the first case d is shown to coincide with the fractal dimension whereas d 1 72 or 0 02 appears as a universal exponent for percolation clusters in two dimensions considered as an intrinsic parameter the connectivity dimension d is compared with other intrinsic and extrinsic characteristic parameters of fractal lattices in particular the authors argue that d or d or d holds on fractal lattices in general d fractal dimension d spectral dimension

second derivative sample path estimators for the gi g m queue applying the technique of smoothed perturbation analysis spa to the gi g m queue with first come first served fcfs queue discipline we derive sample path estimators for the second derivative of mean steady state system time with respect to a parameter of the service time distribution such estimators provide a possible means for speeding up the convergence of gradient based stochastic optimization algorithms the derivation of the estimators sheds some new light on the complications encountered in applying the technique of spa the most general cases require the simulation of additional sample subpaths however an approximation procedure is also introduced which eliminates the need for additional simulation simulation results indicate that the approximation procedure is reasonably accurate when the service times are exponential or deterministic the estimator simplifies and the approximation procedure becomes exact for the m m 2 queue the estimator is proved to be strongly consistent

geometry of differential polynomial functions ii algebraic curves introduction the present paper is a direct continuation of b5 we shall freely borrow terminology and notations from that paper which shall be referred to from now on as part i however understanding the introduction of the present paper only requires familiarity with the introduction and first appendix of part i here we investigate 6 polynomial functions of smooth projective curves x defined over a 6 closed field f we will prove the surprising result that any smooth projective curve of genus 2 over f which does not descend to c can be naturally embedded via a 6 regular map as a 6 closed subset of a projective space which lies completely outside a unique hyperplane this provides in particular remarkable new examples of ritt s anomaly on the 6 dimension of intersections r p 133 we will also prove an infinitesimal analogue of serge lang s conjecture la which in special significant cases implies our 6 algebraic analogue of lang s conjecture b4 and hence the geometric analogue of lang s conjecture let f be as in the introduction of part i a 6 closed field with derivation operator 6 and constant field c and let x be a smooth projective f variety then one can always define two sequences of 6 regular maps

optimisation of a single component maintenance system a smoothed perturbation analysis approach abstract in this paper we show how the technique of smoothed perturbation analysis spa can be applied to optimize threshold values in a maintenance model we do so for a particular model in which a single component is minimally repaired up to an age threshold t and preventively replaced at age tp where tp t with each maintenance action such as minimal repair replacement after failure or preventive replacement costs are associated these costs may depend on the sample path history of the component we derive an estimator for the derivative of the cost performance with respect to t and tp

addendum to extensions and generalizations of smoothed perturbation analysis in a generalized semi markov process framework under a very general framework both in terms of finite time performance measures and system structure the authors derive smoothed perturbation analysis spa estimators and prove their unbiasedness the commuting condition which has been key in previous work is not required a priori and thus the framework includes such systems as the gi g 1 k queue and multiclass queueing networks which do not satisfy the commuting condition the generality achieved is traded off against the fact that the estimator is not always easily implementable on a single sample path the use of the commuting condition in a local sense is proposed to help simplify the estimators derived queueing and multiclass queueing networks are used as illustrative examples for a simple multiclass closed queueing network some simulation results are provided when the commuting condition is satisfied globally the framework allows the recovery of previous results on ipa and spa estimators as corollaries of the main theorems

semi galois categories ii an arithmetic analogue of christol s theorem abstract in connection with our previous work on semi galois categories 1 2 this paper proves an arithmetic analogue of christol s theorem concerning an automata theoretic characterization of when a formal power series n t n f q t over finite field f q is algebraic over the polynomial ring f q t there are by now several variants of christol s theorem all of which are concerned with rings of positive characteristic this paper provides an arithmetic or f 1 variant of christol s theorem in the sense that it replaces the polynomial ring f q t with the ring o k of integers of a number field k and the ring f q t of formal power series with the ring of witt vectors we also study some related problems

on the convergence rates of ipa and fdc derivative estimators we show that under the sufficient conditions usually given for infinitesimal perturbation analysis ipa to apply for derivative estimation a finite difference scheme with common random numbers fdc has the same order of convergence namely o n 1 2 provided that the size of the finite difference interval converges to zero fast enough this holds for both one and two sided fdc this also holds for different variants of ipa such as some versions of smoothed perturbation analysis spa which is based on conditional expectation finally this also holds for the estimation of steady state performance measures by truncated horizon estimators under some ergodicity assumptions our developments do not involve monotonicity but are based on continuity and smoothness we give examples and numerical illustrations which show that the actual difference in mean square error mse between ipa and fdc is typically negligible we also obtain the order of convergence of that difference which is faster than the c

aggregation by attractive particle cluster interaction a quasideterministic irreversible growth model is studied where aggregates are generated by attractive cluster particle forces proportional ro r alpha it is shown that fractal dimensions and growth site probability measures of the resulting fractal structures strongly depend on the parameter alpha comparison with experimental results indicates that this model represents a further step towards realistic simulations of growth processes

smoothed perturbation analysis for a class of discrete event systems a gradient estimation procedure for a general class of stochastic discrete event systems is developed in contrast to most previous work the authors focus on performance measures whose realizations are inherently discontinuous in fact piecewise constant functions of the parameter of differentiation two broad classes of finite horizon discontinuous performance measures arising naturally in applications are considered because of their discontinuity these important classes of performance measures are not susceptible to infinitesimal perturbation analysis ipa instead the authors apply smoothed perturbation analysis formalizing it and generalizing it in the process smoothed perturbation analysis uses conditional expectations to smooth jumps the resulting gradient estimator involves two factors the conditional rate at which jumps occur and the expected effect of a jump among the types of performance measures to which the methods can be applied are transient state probabilities finite horizon throughputs distributions on arrival and expected terminal cost