a zero delay sequential scheme for lossy coding of individual sequences we consider adaptive sequential lossy coding of bounded individual sequences when the performance is measured by the sequentially accumulated mean squared distortion the encoder and the decoder are connected via a noiseless channel of capacity r and both are assumed to have zero delay no probabilistic assumptions are made on how the sequence to be encoded is generated for any bounded sequence of length n the distortion redundancy is defined as the normalized cumulative distortion of the sequential scheme minus the normalized cumulative distortion of the best scalar quantizer of rate r which is matched to this particular sequence we demonstrate the existence of a zero delay sequential scheme which uses common randomization in the encoder and the decoder such that the normalized maximum distortion redundancy converges to zero at a rate n sup 1 5 log n as the length of the encoded sequence n increases without bound

suboptimal measures of predictive complexity for absolute loss function the problem of existence of predictive complexity for the absolute loss game is studied predictive complexity is a generalization of kolmogorov complexity which bounds the ability of any algorithm to predict elements of a sequence of outcomes for perfectly mixable loss functions logarithmic and squared difference are among them predictive complexity is defined like kolmogorov complexity to within an additive constant the absolute loss function is not perfectly mixable and the question of existence of the corresponding predictive complexity which is defined to within an additive constant is open we prove that in the case of the absolute loss game the predictive complexity can be defined to within an additive term o n where n is the length of a sequence of outcomes we prove also that in some restricted settings this bound cannot be improved

rates of convergence for minimum contrast estimators we shall present here a general study of minimum contrast estimators in a nonparametric setting although our results are also valid in the classical parametric case for independent observations these estimators include many of the most popular estimators in various situations such as maximum likelihood estimators least squares and other estimators of the regression function estimators for mixture models or deconvolution the main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters optimal rates depending on entropy conditions are already known at least for some of the models involved and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large but under some circumstances large entropies or changes in the entropy structure due to local perturbations the resulting the rates are only suboptimal counterexamples are constructed which show that the phenomenon is real for non parametric maximum likelihood or regression this proves that under purely metric assumptions our theorem is optimal and that minimum contrast estimators happen to be suboptimal

integrators for lie poisson or dynamical systems abstract we reformulate the ge marsden 1988 lie poisson integration algorithm in terms of algebra variables alone and show how to implement it for regular quadratic lie algebras to arbitrary order and for arbitrary lie algebras to first order we also describe two generalizations of ruth s method 1983 that can be used to generate very fast algorithms

on complexity of easy predictable sequences predictive complexity is a generalization of kolmogorov complexity it corresponds to an optimal prediction strategy and gives a natural lower bound to ability of any algorithm to predict elements of a sequence of outcomes a natural question is studied how complex can easy to predict sequences be the standard measure of complexity used in the paper is kolmogorov complexity k which is close to predictive complexity for logarithmic loss function the difficulty of prediction is measured by the notion of predictive complexity kg for bounded loss function of nonlogarithmic type we present an asymptotic relation sup x l x nk x n kg x 1 a log n when n where a is a constant and l x is the length of a sequence x an analogous asymptotic relation holds for relative complexities k x n n and kg x n where n l x to obtain these results we present lower and upper bounds of the cardinality of all sequences of given predictive complexity

a logic of probability with application to the foundations of statistics a simple example shows that the classical theory of probability implies more than one can deduce via kolmogorov s calculus of probability developing dawid s ideas i propose a new calculus of probability which is free from this drawback this calculus naturally leads to a new interpretation of probability i argue that attempts to create a general empirical theory of probability should be abandoned and we should content ourselves with the logic of probability establishing relations between probabilistic theories and observations my approach to the logic of probability is based on a variant of ville s principle of the excluded gambling strategy

regret minimization under partial monitoring we consider repeated games in which the player instead of observing the action chosen by the opponent in each game round receives a feedback generated by the combined choice of the two players we study hannan consistent players for these games that is randomized playing strategies whose per round regret vanishes with probability one as the number n of game rounds goes to infinity we prove a general lower bound of n 1 3 for the convergence rate of the regret and exhibit a specific strategy that attains this rate for any game for which a hannan consistent player exists

invariants and numerical methods for odes the preservation of some invariants is important during the numerical integration of odes in some cases failure to maintain certain invariants leads to physically impossible solutions in other cases to instability some authors report that the long term macroscopic characteristics of a solution are better represented if certain invariants are maintained this paper considers two types of invariants point invariants which are functions along a particular trajectory and differential invariants which are invariant relations between neighboring trajectories such as symplectic invariants conventional numerical methods usually introduce o hp 1 errors in an invariant at each numerical step and these will accumulate over the integration interval two approaches to preventing this accumulation are 1 to find methods that maintain the invariants within round off error or 2 to find methods that maintain another invariant which is no more than o hq different from the invariant satisfied by the ode these approaches will be illustrated with some examples the major objective of this paper is to consider the potential of general methods that is methods that do not depend on the specific differential equation to maintain the invariants

new tools for characterizing swarming systems a comparison of minimal models abstract we compare three simple models that reproduce qualitatively the emergent swarming behavior of bird flocks fish schools and other groups of self propelled agents by using a new set of diagnosis tools related to the agents spatial distribution two of these correspond in fact to different implementations of the same model which had been previously confused in the literature all models appear to undergo a very similar order to disorder phase transition as the noise level is increased if we only compare the standard order parameter which measures the degree of agent alignment when considering our novel quantities however their properties are clearly distinguished unveiling previously unreported qualitative characteristics that help determine which model best captures the main features of realistic swarms additionally we analyze the agent clustering in space finding that the distribution of cluster sizes is typically exponential at high noise and approaches a power law as the noise level is reduced this trend is sometimes reversed at noise levels close to the phase transition suggesting a non trivial critical behavior that could be verified experimentally finally we study a bi stable regime that develops under certain conditions in large systems by computing the probability distributions of our new quantities we distinguish the properties of each of the coexisting metastable states our study suggests new experimental analyses that could be carried out to characterize real biological swarms

incompressible fluid turbulence at large reynolds numbers theoretical basis for the t 1 decay law and the form of the longitudinal correlation function approximately valid for large values of the time t a formal solution to the hopf equation is obtained here as an asymptotic power series in t 1 this approximate solution is directly applicable to grid generated isotropic homogeneous turbulence at large reynolds numbers during the initial inertial force dominated period of decay thus the solution accounts for the observed t 1 decay law and the fact that the longitudinal correlation function f is independent of t it is observed that the longitudinal correlation function measured by frenkiel klebanoff and huang is consistent with the theoretical asymptotic behavior f const r 3 as r and fitted by the expression f 1 0 770 r m 3 where m is the grid mesh length and the separation distance r is greater than the taylor microscale 10 t 1 2 interestingly enough this form for the longitudinal correlation function is shown to be derivable from a variational principle

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

online forecast combinations of distributions worst case bounds this paper considers forecasts with distribution functions that may vary through time the forecast is achieved by time varying combinations of individual forecasts we derive theoretical worst case bounds for general algorithms based on multiplicative updates of the combination weights the bounds are useful for studying properties of forecast combinations when data are non stationary and there is no unique best model

toward a classification of finite partial monitoring games partial monitoring games constitute a mathematical framework for sequential decision making problems with imperfect feedback the learner repeatedly chooses an action the opponent responds with an outcome and then the learner suffers a loss and receives a feedback signal both of which are fixed functions of the action and the outcome the goal of the learner is to minimize his total cumulative loss we make progress toward the classification of these games based on their minimax expected regret namely we classify almost all games with two outcomes and a finite number of actions we show that their minimax expected regret is either zero email protected t q t 2 3 or q t and we give a simple and efficiently computable classification of these four classes of games our hope is that the result can serve as a stepping stone toward classifying all finite partial monitoring games

learning from examples in large neural networks a statistical mechanical theory of learning from examples in layered networks at finite temperature is studied when the training error is a smooth function of continuously varying weights the generalization error falls off asymptotically as the inverse number of examples by analytical and numerical studies of single layer perceptrons we show that when the weights are discrete the generalization error can exhibit a discontinuous transition to perfect generalization for intermediate sizes of the example set the state of perfect generalization coexists with a metastable spin glass state

the nonstochastic multiarmed bandit problem in the multiarmed bandit problem a gambler must decide which arm of k nonidentical slot machines to play in a sequence of trials so as to maximize his reward this classical problem has received much attention because of the simple model it provides of the trade off between exploration trying out each arm to find the best one and exploitation playing the arm believed to give the best payoff past solutions for the bandit problem have almost always relied on assumptions about the statistics of the slot machines in this work we make no statistical assumptions whatsoever about the nature of the process generating the payoffs of the slot machines we give a solution to the bandit problem in which an adversary rather than a well behaved stochastic process has complete control over the payoffs in a sequence of t plays we prove that the per round payoff of our algorithm approaches that of the best arm at the rate o t 1 2 we show by a matching lower bound that this is the best possible we also prove that our algorithm approaches the per round payoff of any set of strategies at a similar rate if the best strategy is chosen from a pool of n strategies then our algorithm approaches the per round payoff of the strategy at the rate o log n1 2 t 1 2 finally we apply our results to the problem of playing an unknown repeated matrix game we show that our algorithm approaches the minimax payoff of the unknown game at the rate o t 1 2

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

exactly conservative integrators traditional explicit numerical discretizations of conservative systems generically predict artificial secular drifts of any nonlinear invariants in this work we present a general approach for developing explicit nontraditional algorithms that conserve such invariants exactly we illustrate the method by applying it to the three wave truncation of the euler equations the lotka volterra predator prey model and the kepler problem the ideas are discussed in the context of symplectic phase space conserving integration methods as well as nonsymplectic conservative methods we comment on the application of our method to general conservative systems

tracking the best quantizer an algorithm is presented for online prediction that allows to track the best expert efficiently even when the number of experts is exponentially large provided that the set of experts has a certain additive structure as an example we work out the case where each expert is represented by a path in a directed graph and the loss of each expert is the sum of the weights over the edges in the path these results are then used to construct universal limited delay schemes for lossy coding of individual sequences in particular we consider the problem of tracking the best scalar quantizer that is adaptively matched to the source sequence with piecewise different behavior a randomized algorithm is presented which can perform on any source sequence asymptotically as well as the best scalar quantization algorithm that is matched to the sequence and is allowed to change the employed quantizer for a given number of times the complexity of the algorithm is quadratic in the sequence length but at the price of some deterioration in performance the complexity can be made linear analogous results are obtained for sequential multiresolution and multiple description scalar quantization of individual sequences

new numerical integrator for the stackel system conserving the same number of constants of motion as the degree of freedom a new numerical integrator for a wide class of separable hamiltonian systems called the stackel system is presented this integrator is designed so as to conserve the same number of constants of motion as the degree of freedom of the stackel system separation of variables of the stackel system is most fundamental for the integrator a combination of canonical transformations and an energy preserving method with a variable step size plays a key role to design such an integrator some typical and important examples of the stackel system are then discretized explicitly they are the three dimensional kepler motion the holt system and the integrable henon heiles system in celestial mechanics

the weighted majority algorithm we study the construction of prediction algorithms in a situation in which a learner faces a sequence of trials with a prediction to be made in each and the goal of the learner is to make few mistakes we are interested in the case where the learner has reason to believe that one of some pool of known algorithms will perform well but the learner does not know which one a simple and effective method based on weighted voting is introduced for constructing a compound algorithm in such a circumstance we call this method the weighted majority algorithm we show that this algorithm is robust in the presence of errors in the data we discuss various versions of the weighted majority algorithm and prove mistake bounds for them that are closely related to the mistake bounds of the best algorithms of the pool for example given a sequence of trials if there is an algorithm in the pool a that makes at most m mistakes then the weighted majority algorithm will make at most c log a m mistakes on that sequence where c is fixed constant

a loss bound model for on line stochastic prediction algorithms abstract in this paper we consider the problem of on line prediction in which at each time an unlabeled instance is given and then a prediction algorithm outputs a probability distribution over the set of labels rather than 0 1 values before it sees the correct label for this setting we propose a weighted average type on line stochastic prediction algorithm wa which can be regarded as a hybrid of the bayes algorithm and a sequential real valued parameter estimation method we derive upper bounds on the instantaneous logarithmic loss and cumulative logarithmic loss for wa in both the example dependent form and the expected form the expectation is taken with respect to the fixed target distribution of sequences specifically under some specific parametric assumptions for target rules we prove that wa is optimal in the sense that the upper bound on the expected cumulative logarithmic loss for wa asymptotically matches rissanen s coding theoretic lower bound further we derive an upper bound on the expected cumulative quadratic loss by making use of relationships between the quadratic loss and the logarithmic loss throughout the paper we relate computational learning theory to information theory most specifically rissanen s predictive minimum description length principle by giving noiseless coding theoretic interpretations to the loss bounds

exact theory for the self similarity and decay of homogeneous turbulence it is shown that space time dilatation invariance x 1x t 2t with concomitant transformations for dependent variables and linearity of the equation engender an exact time explicit generic form for the solution applicable to freely decaying homogeneous incompressible fluid turbulence this solution features a summation over mutually independent dynamical modes labeled by the dilatation scaling index n 1 without the assumption of isotropy nor introduction of a closure approximation procedure the theory provides an explanation for the experimentally observed self similarity of the correlation tensors and the decay laws u x t 2 t n for the different types and decay stages of homogeneous turbulence

sequential prediction of individual sequences under general loss functions we consider adaptive sequential prediction of arbitrary binary sequences when the performance is evaluated using a general loss function the goal is to predict on each individual sequence nearly as well as the best prediction strategy in a given comparison class of possibly adaptive prediction strategies called experts by using a general loss function we generalize previous work on universal prediction forecasting and data compression however here we restrict ourselves to the case when the comparison class is finite for a given sequence we define the regret as the total loss on the entire sequence suffered by the adaptive sequential predictor minus the total loss suffered by the predictor in the comparison class that performs best on that particular sequence we show that for a large class of loss functions the minimax regret is either spl theta log n or spl omega spl radic spl lscr log n depending on the loss function where n is the number of predictors in the comparison class and spl lscr is the length of the sequence to be predicted the former case was shown previously by vovk 1990 we give a simplified analysis with an explicit closed form for the constant in the minimax regret formula and give a probabilistic argument that shows this constant is the best possible some weak regularity conditions are imposed on the loss function in obtaining these results we also extend our analysis to the case of predicting arbitrary sequences that take real values in the interval 0 1

gains and losses are fundamentally different in regret minimization the sparse case we demonstrate that in the classical non stochastic regret minimization problem with d decisions gains and losses to be respectively maximized or minimized are fundamentally different indeed by considering the additional sparsity assumption at each stage at most s decisions incur a nonzero outcome we derive optimal regret bounds of different orders specifically with gains we obtain an optimal regret guarantee after t stages of order t log s so the classical dependency in the dimension is replaced by the sparsity size with losses we provide matching upper and lower bounds of order ts log d d which is decreasing in d eventually we also study the bandit setting and obtain an upper bound of order ts log d s when outcomes are losses this bound is proven to be optimal up to the logarithmic factor log d s

unitary integration with operator splitting for weakly dissipative systems unitary integration is a numerical method that preserves the structure of the quantum liouville equation by evolving the density via unitary transformations unitary integrators preserve the kinematic invariants cj tr j j 1 n to all orders in the time step here we extend unitary integration to weakly dissipative systems we apply the technique of operator splitting using a unitary integrator for the hamiltonian evolution and a conventional integrator for the dissipative piece in this way we guarantee that all dissipation and decoherence variation of the cj is due to the new non hamiltonian terms and not to any numerical artifacts we illustrate the method with examples

universal forecasting algorithms abstract we consider forecasting systems which when given an initial segment of a binary string guess the next bit deterministic forecasting or estimate the probability of the next bit being 1 probabilistic forecasting the quality of forecasting is measured by the number of errors in the deterministic case or by the sum of distances between the forecasts and the true probabilities in the probabilistic case a forecasting system is said to be simple if it has a short description e g a simple formula which admits fast computation of the forecasts there is a universal forecasting algorithm which for any given bound on time of forecasting computes forecasts within time and the quality of these forecasts is not much lower than that of any simple forecasting system the complexity of the rival forecasting system may increase as increases the aim of the paper is to study the possibilities and limitations of universal forecasting

equivalence of models for polynomial learnability abstract in this paper we consider several variants of valiant s learnability model that have appeared in the literature we give conditions under which these models are equivalent in terms of the polynomially learnable concept classes they define these equivalences allow comparisons of most of the existing theorems in valiant style learnability and show that several simplifying assumptions on polynomial learning algorithms can be made without loss of generality we also give a useful reduction of learning problems to the problem of finding consistent hypotheses and give comparisons and equivalences between valiant s model and the prediction learning models of haussler littlestone and warmuth in 29th annual ieee symposium on foundations of computer science 1988

an exactly conservative integrator for the n body problem the two dimensional n body problem of classical mechanics is a non integrable hamiltonian system for n 2 traditional numerical integration algorithms which are polynomials in the time step typically lead to systematic drifts in the computed value of the total energy and angular momentum even symplectic integration schemes exactly conserve only an approximate hamiltonian we present an algorithm that conserves the true hamiltonian and the total angular momentum to machine precision it is derived by applying conventional discretizations in a new space obtained by transformation of the dependent variables we develop the method first for the restricted circular three body problem then for the general two dimensional three body problem and finally for the planar n body problem jacobi coordinates are used to reduce the two dimensional n body problem to an n 1 body problem that incorporates the constant linear momentum and centre of mass constraints for a four body choreography we find that a larger time step can be used with our conservative algorithm than with symplectic and conventional integrators

universal modeling and coding the problems arising in the modeling and coding of strings for compression purposes are discussed the notion of an information source that simplifies and sharpens the traditional one is axiomatized and adaptive and nonadaptive models are defined with a measure of complexity assigned to the models a fundamental theorem is proved which states that models that use any kind of alphabet extension are inferior to the best models using no alphabet extensions at all a general class of so called first in first out fifo arithmetic codes is described which require no alphabet extension devices and which therefore can be used in conjunction with the best models because the coding parameters are the probabilities that define the model their design is easy and the application of the code is straightforward even with adaptively changing source models

step decision rules for multistage stochastic programming a heuristic approach stochastic programming with step decision rules spsdr aims to produce efficient solutions to multistage stochastic optimization problems spsdr like plain multistage stochastic programming sp operates on a monte carlo computing sample of moderate size that approximates the stochastic process unlike sp spsdr does not strive to build a balanced event tree out of that sample rather it defines a solution as a special type of decision rule with the property that the decisions at each stage are piecewise constant functions on the sample of scenarios those pieces define a partition of the set of scenarios at each stage t but the partition at t 1 need not be refinement of the partition at t however the rule is constructed so that the non anticipativity condition is met a necessary condition to make the rules operational to validate the method we show how to extend a non anticipatory decision rule to arbitrary scenarios within a very large validation sample of scenarios we apply three methods spsdr sp and robust optimization to the same 12 stage problem in supply chain management and compare them relatively to different objectives and performance criteria it appears that spsdr performs better than sp in that it produces a more accurate estimate prediction of the value achieved by its solution on the validation sample and also that the achieved value is better

online markov decision processes we consider a markov decision process mdp setting in which the reward function is allowed to change after each time step possibly in an adversarial manner yet the dynamics remain fixed similar to the experts setting we address the question of how well an agent can do when compared to the reward achieved under the best stationary policy over time we provide efficient algorithms which have regret bounds with no dependence on the size of state space instead these bounds depend only on a certain horizon time of the process and logarithmically on the number of actions

study of the hopf functional equation for turbulence duhamel principle and dynamical scaling we consider a formulation for the hopf functional differential equation which governs statistical solutions of the navier stokes equations by introducing an exponential operator with a functional derivative we recast the hopf equation as an integro differential functional equation by the duhamel principle on this basis we introduce a successive approximation to the hopf equation as an illustration we take the burgers equation and carry out the approximations to the leading order scale invariance of the statistical navier stokes equations in d dimensions is formulated and contrasted with that of the deterministic navier stokes equations for the statistical navier stokes equations critical scale invariance is achieved for the characteristic functional of the d th derivative of the vector potential in d dimensions the deterministic equations corresponding to this choice of the dependent variable acquire the linear fokker planck operator under dynamic scaling in three dimensions it is the vorticity gradient that behaves like a fundamental solution more precisely source type solution of deterministic navier stokes equations in the long time limit physical applications of these ideas include study of a self similar decaying profile of fluid flows moreover we reveal typical physical properties in the late stage evolution by combining statistical scale invariance and the source type solution this yields an asymptotic form of the hopf functional in the long time limit improving the well known hopf titt solution in particular we present analyses for the burgers equations to illustrate the main ideas and indicate a similar analysis for the navier stokes equations

regret bounds and minimax policies under partial monitoring this work deals with four classical prediction settings namely full information bandit label efficient and bandit label efficient as well as four different notions of regret pseudo regret expected regret high probability regret and tracking the best expert regret we introduce a new forecaster inf implicitly normalized forecaster based on an arbitrary function for which we propose a unified analysis of its pseudo regret in the four games we consider in particular for x exp x k inf reduces to the classical exponentially weighted average forecaster and our analysis of the pseudo regret recovers known results while for the expected regret we slightly tighten the bounds on the other hand with x x q k which defines a new forecaster we are able to remove the extraneous logarithmic factor in the pseudo regret bounds for bandits games and thus fill in a long open gap in the characterization of the minimax rate for the pseudo regret in the bandit game we also provide high probability bounds depending on the cumulative reward of the optimal action finally we consider the stochastic bandit game and prove that an appropriate modification of the upper confidence bound policy ucb1 auer et al 2002a achieves the distribution free optimal rate while still having a distribution dependent rate logarithmic in the number of plays

how many strings are easy to predict it is well known in the theory of kolmogorov complexity that most strings cannot be compressed more precisely only exponentially few 2n m binary strings of length n can be compressed by m bits this paper extends the incompressibility property of kolmogorov complexity to the unpredictability property of predictive complexity the unpredictability property states that predictive complexity defined as the loss suffered by a universal prediction algorithm working infinitely long of most strings is close to a trivial upper bound the loss suffered by a trivial minimax constant prediction strategy we show that only exponentially few strings can be successfully predicted and find the base of the exponent

generalised entropies and asymptotic complexities of languages the paper explores connections between asymptotic complexity and generalised entropy asymptotic complexity of a language a language is a set of finite or infinite strings is a way of formalising the complexity of predicting the next element in a sequence it is the loss per element of a strategy asymptotically optimal for that language generalised entropy extends shannon entropy to arbitrary loss functions it is the optimal expected loss given a distribution on possible outcomes it turns out that the set of tuples of asymptotic complexities of a language w r t different loss functions can be described by means of the generalised entropies corresponding to the loss functions

how to filter an individual sequence with feedback we consider causally estimating filtering the components of a noise corrupted sequence relative to a reference class of filters the noiseless sequence to be filtered is designed by a ldquowell informed antagonist rdquo meaning it may evolve according to an arbitrary law unknown to the filter based on past noiseless and noisy sequence components we show that this setting is more challenging than that of an individual noiseless sequence a k a the ldquosemi stochasticrdquo setting in the sense that any deterministic filter even one guaranteed to do well on every noiseless individual sequence fails under some well informed antagonist on the other hand we constructively establish the existence of a randomized filter which successfully competes with an arbitrary given finite reference class of filters under every antagonist thus unlike in the semi stochastic setting randomization is crucial in the antagonist framework our noise model allows for channels whose noisy output depends on the l past channel outputs in addition to the noiseless channel input symbol memoryless channels are obtained as a special case of our model by taking i 0 in this case our scheme coincides with one that was recently shown to compete with an arbitrary reference class when the underlying noiseless sequence is an individual sequence hence our results show that the latter scheme is universal not only for the semi stochastic setting in which it was originally proposed but also under the well informed antagonist

how to use expert advice we analyze algorithms that predict a binary value by combining the predictions of several prediction strategies called experts our analysis is for worst case situations i e we make no assumptions about the way the sequence of bits to be predicted is generated we measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence where the expectation is taken with respect to the randomization in the predictins we show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert and we give efficient algorithms that achieve this our upper and lower bounds have matching leading constants in most cases we then show how this leads to certain kinds of pattern recognition learning algorithms with performance bounds that improve on the best results currently know in this context we also compare our analysis to the case in which log loss is used instead of the expected number of mistakes

an exact representation of the space time characteristic functional of turbulent navier stokes flows with prescribed random initial states and driving forces by adopting a formal operator viewpoint the space time characteristic functional associated with navier stokes turbulence is expressed in terms of a linear operator acting on the space of functionals obtained by a simple similarity transformation of the local translation operator generated by the nonlinear terms in the navier stokes equation this operator is unitary with respect to the formal scalar product of functionals the equivalence of this operator representation to the functional integral representation of rosen is shown and for gaussian initial velocity and external force fields some consequences of this representation are presented

rigorous shadowing of numerical solutions of ordinary differential equations by containment an exact trajectory of a dynamical system lying close to a numerical trajectory is called a shadow we present a general purpose method for proving the existence of finite time shadows of numerical ode integrations of arbitrary dimension in which some measure of hyperbolicity is present and there are either 0 or 1 expanding modes or 0 or 1 contracting modes much of the rigor is provided automatically by interval arithmetic and validated ode integration software that is freely available the method is a generalization of a previously published containment process that was applicable only to two dimensional maps we extend it to handle maps of arbitrary dimension with the above restrictions and finally to odes the method involves building n cubes around each point of the discrete numerical trajectory through which the shadow is guaranteed to pass at appropriate times the proof consists of two steps first the rigorous computational verification of a simple geometric property which we call the inductive containment property and second a simple geometric argument showing that this property implies the existence of a shadow the computational step is almost entirely automated and easily adaptable to any ode problem the method allows for the rescaling of time which is a necessary ingredient for successfully shadowing odes finally the method is local in the sense that it builds the shadow inductively requiring information only from the most recent integration step rather than more global information typical of several other methods the method produces shadows of comparable length and distance to all currently published results finally we conjecture that the inductive containment property implies the existence of a shadow without restriction on the number of expanding and contracting modes although proof currently eludes us

predicting 0 1 functions on randomly drawn points we consider the problem of predicting 0 1 valued functions on rn and smaller domains based on their values on randomly drawn points our model is related to valiant s pac learning model but does not require the hypotheses used for prediction to be represented in any specified form in our main result we show how to construct prediction strategies that are optimal to within a constant factor for any reasonable class f of target functions this result is based on new combinatorial results about classes of functions of finite vc dimension we also discuss more computationally efficient algorithms for predicting indicator functions of axis parallel rectangles more general intersection closed concept classes and halfspaces in rn these are also optimal to within a constant factor finally we compare the general performance of prediction strategies derived by our method to that of those derived from methods in pac learning theory

universal prediction of individual sequences the problem of predicting the next outcome of an individual binary sequence using finite memory is considered the finite state predictability of an infinite sequence is defined as the minimum fraction of prediction errors that can be made by any finite state fs predictor it is proven that this fs predictability can be achieved by universal sequential prediction schemes an efficient prediction procedure based on the incremental parsing procedure of the lempel ziv data compression algorithm is shown to achieve asymptotically the fs predictability some relations between compressibility and predictability are discussed and the predictability is proposed as an additional measure of the complexity of a sequence

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells