a deterministic analysis of stochastic approximation with randomized directions we study the convergence of two stochastic approximation algorithms with randomized directions the simultaneous perturbation stochastic approximation algorithm and the random direction kiefer wolfowitz algorithm we establish deterministic necessary and sufficient conditions on the random directions and noise sequences for both algorithms and these conditions demonstrate the effect of the random directions on the sample path behavior of the algorithms studied we discuss ideas for further research in the analysis and design of these algorithms

generalized simulated annealing we discuss and illustrate a new stochastic algorithm generalized simulated annealing for computationally finding the global minimum of a given not necessarily convex energy cost function defined in a continuous d dimensional space this algorithm recovers as particular cases the so called classical boltzmann machine and fast cauchy machine simulated annealings and turns out to be quicker than both

bubbling bifurcation loss of synchronization and shadowing breakdown in complex systems abstract complex dynamical systems with many degrees of freedom may exhibit a wealth of collective phenomena related to high dimensional chaos this paper focuses on a lattice of coupled logistic maps to investigate the relationship between the loss of chaos synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems in the neighborhood of the critical transition to strongly non hyperbolic behavior the system undergoes on off intermittency with respect to the synchronization manifold this has been confirmed by numerical diagnostics of synchronization and non hyperbolic behavior the latter using the statistical properties of finite time lyapunov exponents

hopf algebras of gk dimension two with vanishing ext group abstract we construct and study a family of finitely generated hopf algebra domains h of gelfand kirillov dimension two such that ext h 1 k k 0 consequently we answer a question of goodearl and the second named author

recurrence plots revisited abstract we show that recurrence plots rps give detailed characterizations of time series generated by dynamical systems driven by slowly varying external forces for deterministic systems we show that rps of the time series can be used to reconstruct the rp of the driving force if it varies sufficiently slowly if the driving force is one dimensional its functional form can then be inferred up to an invertible coordinate transformation the same results hold for stochastic systems if the rp of the time series is suitably averaged and transformed these results are used to investigate the nonlinear prediction of time series generated by dynamical systems driven by slowly varying external forces we also consider the problem of detecting a small change in the driving force and propose a surrogate data technique for assessing statistical significance numerically simulated time series and a time series of respiration rates recorded from a subject with sleep apnea are used as illustrative examples

cooling schedules for optimal annealing a monte carlo optimization technique called simulated annealing is a descent algorithm modified by random ascent moves in order to escape local minima which are not global minima the level of randomization is determined by a control parameter t called temperature which tends to zero according to a deterministic cooling schedule we give a simple necessary and sufficient condition on the cooling schedule for the algorithm state to converge in probability to the set of globally minimum cost states in the special case that the cooling schedule has parametric form tt c log1 t the condition for convergence is that c be greater than or equal to the depth suitably defined of the deepest local minimum which is not a global minimum state

universal enveloping algebras of poisson hopf algebras abstract for a poisson algebra a by exploring its relation with lie rinehart algebras we prove a poincare birkhoff witt theorem for its universal enveloping algebra a e some general properties of the universal enveloping algebras of poisson hopf algebras are studied given a poisson hopf algebra b we give the necessary and sufficient conditions for a poisson polynomial algebra b x p to be a poisson hopf algebra we also prove a structure theorem for b e when b is a pointed poisson hopf algebra namely b e is isomorphic to b h b the crossed product of b and h b where h b is the quotient hopf algebra b e b e b

stochastic discrete optimization in this paper a stochastic search method is proposed for finding a global solution to the stochastic discrete optimization problem in which the objective function must be estimated by monte carlo simulation although there are many practical problems of this type in the fields of manufacturing engineering operations research and management science there have not been any nonheuristic methods proposed for such discrete problems with stochastic infrastructure the proposed method is very simple yet it finds a global optimum solution the method exploits the randomness of monte carlo simulation and generates a sequence of solution estimates this generated sequence turns out to be a nonstationary markov chain and it is shown under mild conditions that the markov chain is strongly ergodic and that the probability that the current solution estimate is global optimum converges to one furthermore the speed of convergence is also analyzed

stochastic optimization on continuous domains with finite time guarantees by markov chain monte carlo methods we introduce bounds on the finite time performance of markov chain monte carlo mcmc algorithms in solving global stochastic optimization problems defined over continuous domains it is shown that mcmc algorithms with finite time guarantees can be developed with a proper choice of the target distribution and by studying their convergence in total variation norm this work is inspired by the concept of finite time learning with known accuracy and confidence developed in statistical learning theory

the global extension problem crossed products and co flag non commutative poisson algebras let p be a poisson algebra e a vector space and e p an epimorphism of vector spaces with v ker the global extension problem asks for the classification of all poisson algebra structures that can be defined on e such that e p becomes a morphism of poisson algebras from a geometrical point of view it means to decompose this groupoid into connected components and to indicate a point in each such component all such poisson algebra structures on e are classified by an explicitly constructed classifying set gph2 p v which is the coproduct of all non abelian cohomological objects ph2 p v v v which are the classifying sets for all extensions of p by v v v the second classical poisson cohomology group h2 p v appears as the most elementary piece among all components of gph2 p v several examples are provided in the case of metabelian poisson algebras or co flag poisson algebras over p the latter being poisson algebras q which admit a finite chain of epimorphisms of poisson algebras pn q npn 1 p1 1p0 p such that dim ker i 1 for all i 1 n

noise conditions for prespecified convergence rates of stochastic approximation algorithms we develop deterministic necessary and sufficient conditions on individual noise sequences of a stochastic approximation algorithm for the error of the iterates to converge at a given rate specifically suppose spl rho sub n is a given positive sequence converging monotonically to zero consider a stochastic approximation algorithm x sub n 1 x sub n a sub n a sub n x sub n b sub n a sub n e sub n where x sub n is the iterate sequence a sub n is the step size sequence e sub n is the noise sequence and x is the desired zero of the function f x ax b then under appropriate assumptions we show that x sub n x o spl rho sub n if and only if the sequence e sub n satisfies one of five equivalent conditions these conditions are based on well known formulas for noise sequences kushner and clark s 1978 condition chen s see proc ifac world congr p 375 80 1996 condition kulkarni and horn s see ieee trails automat contr vol 41 p 419 24 1996 condition a decomposition condition and a weighted averaging condition our necessary and sufficient condition on e sub n to achieve a convergence rate of spl rho sub n is basically that the sequence e sub n spl rho sub n satisfies any one of the above five well known conditions we provide examples to illustrate our result in particular we easily recover the familiar result that if a sub n a n and e sub n is a martingale difference process with bounded variance then x sub n x o n sup 1 2 log n sup spl beta for any spl beta 1 2

prime regular hopf algebras of gk dimension one this paper constitutes the first part of a program to classify all affine prime regular hopf algebras h of gelfand kirillov dimension one over an algebraically closed field of characteristic zero we prove a number of properties of such an algebra list some classes of examples and then prove that when the pi degree of h is prime our list contains all such algebras

simulated annealing type markov chains and their order balance equations generalized simulated annealing type markov chains where the transition probabilities are proportional to powers of a vanishing small parameter are considered an order of recurrence which quantifies the asymptotic behavior of the state occupation probability is associated with each state these orders of recurrence satisfy a fundamental balance equation across every edge cut in the graph of the markov chain moreover the markov chain converges in a cesaro sense to the set of states having the largest recurrence orders these results convert the analytic problem of determining the asymptotic properties of the time inhomogeneous stochastic process into a purely algebraic problem of solving the balance equations to determine the recurrence orders graph theoretic algorithms are provided to determine the solutions of the balance equations by applying these results to the problem of optimization by simulated annealing it is shown that the sum of the recurrence order and the cost is a constant for all st

universal enveloping algebras and universal derivations of poisson algebras abstract let k be an arbitrary field of characteristic 0 it is shown that for any n 1 the universal enveloping algebras of the poisson symplectic algebra p n k and the weyl algebra a n k are isomorphic and the canonical isomorphism between them easily leads to the moyal product a basis of the universal enveloping algebra p e of a free poisson algebra p k x 1 x n is constructed and it is proved that the left dependence of a finite number of elements of p e over p e is algorithmically recognizable we describe the poisson dependence of any two elements of a free poisson algebra in characteristic 0 in the language of universal derivatives the fox derivatives on free poisson algebras are defined and it is proved that an analogue of the jacobian conjecture for two generated free poisson algebras is equivalent to the two dimensional classical jacobian conjecture a new proof of the tameness of automorphisms of two generated free poisson algebras is also given

computational applications of nonextensive statistical mechanics computational applications of the nonextensive entropy s q and nonextensive statistical mechanics a current generalization of the boltzmann gibbs bg theory are briefly reviewed the corresponding bibliography is provided as well

noetherian hopf algebra domains of gelfand kirillov dimension two abstract we classify all noetherian hopf algebras h over an algebraically closed field k of characteristic zero which are integral domains of gelfand kirillov dimension two and satisfy the condition ext h 1 k k 0 the latter condition is conjecturally redundant as no examples are known among noetherian hopf algebra domains of gk dimension two where it fails

an alternative proof for convergence of stochastic approximation algorithms an alternative proof for convergence of stochastic approximation algorithms is provided the proof is completely deterministic very elementary involving only basic notions of convergence and direct in that it remains in a discrete setting an alternative form of the kushner clark condition is introduced and utilized and the results are the first to prove necessity for general gain sequences in a hilbert space setting

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

relative frequencies of generalized simulated annealing we consider a class of nonhomogeneous markov chains arising in simulated annealing and related stochastic search algorithms using only elementary first principles we analyze the convergence and rate of convergence of the relative frequencies of visits to states in the markov chain we describe in detail three examples including the standard simulated annealing algorithm to show how our framework applies to specific stochastic search algorithmsthese examples have not previously been recognized to be sufficiently similar to share common analytical grounds our analysis though elementary provides the strongest sample path convergence results to date for simulated annealing type markov chains our results serve to illustrate that by taking a purely sample path view surprisingly strong statements can be made using only relatively elementary tools

threshold accepting a general purpose optimization algorithm appearing superior to simulated annealing abstract a new general purpose algorithm for the solution of combinatorial optimization problems is presented the new threshold accepting method is even simpler structured than the wellknown simulated annealing approach the power of the new algorithm is demonstrated by computational results concerning the traveling salesman problem and the problem of the construction of error correcting codes moreover deterministic versions of the new heuristic turn out to perform nearly equally well consuming only a fraction of the computing time of the stochastic versions as an example the deterministic threshold accepting method yields very near to optimum tours for the famous 442 cities traveling salesman problem of grotschel within 1 to 2 s of cpu time

homological integral of hopf algebras the left and right homological integrals are introduced for a large class of infinite dimensional hopf algebras using the homological integrals we prove a version of maschke s theorem for infinite dimensional hopf algebras the generalization of maschke s theorem and homological integrals are the keys to studying noetherian regular hopf algebras of gelfand kirillov dimension one

classification of connected hopf algebras of dimension p 3 i abstract let p be a prime and k be an algebraically closed field of characteristic p in this paper we provide the classification of connected hopf algebras of dimension p 3 except for the case when the primitive space of the hopf algebra is a two dimensional abelian restricted lie algebra each isomorphism class is presented by generators x y z with relations and hopf algebra structures let be the multiplicative group of p 2 p 1 th roots of unity when the primitive space is one dimensional and p is odd there is an infinite family of isomorphism classes which is naturally parameterized by a k 1

using recurrences to characterize the hyperchaos chaos transition hyperchaos occurs in a dynamical system with more than one positive lyapunov exponent when the equations governing the time evolution of the dynamical system are known the transition from chaos to hyperchaos can be readily obtained when the second largest lyapunov exponent crosses zero if the only information available on the system is a time series however such method is difficult to apply we propose the use of recurrence quantification analysis of a time series to characterize the chaos hyperchaos transition we present results obtained from recurrence plots of coupled chaotic piecewise linear maps and chua matsumoto circuits but the method can be applied as well to other systems even when one does not know their dynamical equations

regularity of involutory pi hopf algebras abstract let h be an involutory hopf algebra over a field of characteristic 0 if h is either a finite module over its affine center or a semiprime noetherian affine pi algebra then it is auslander regular and cohen macaulay and is a finite direct sum of prime rings

homological aspects of noetherian pi hopf algebras and irreducible modules of maximal dimension abstract we prove that a noetherian hopf algebra of finite global dimension possesses further attractive homological properties at least when it satisfies a polynomial identity this applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity as well as to classical enveloping algebras in positive characteristic in all three cases we show that these algebras are auslander regular and macaulay we derive representation theoretic consequences concerning the coincidence of the non azumaya and singular loci for each of the above three classes of algebras

lyapunov spectrum and synchronization of piecewise linear map lattices with power law coupling we study the synchronization properties of a lattice of chaotic piecewise linear maps the coupling strength decreases with the lattice distance in a power law fashion we obtain the lyapunov spectrum of the coupled map lattice and investigate the relation between spatiotemporal chaos and synchronization of amplitudes and phases using suitable numerical diagnostics

stochastic comparison algorithm for discrete optimization with estimation in this paper we study a class of discrete optimization problems where the objective function for a given configuration can be expressed as the expectation of a random variable in such problems only samples of the random variables are available for the optimization process an iterative algorithm called the stochastic comparison sc algorithm is developed the convergence of the sc algorithm is established based on an examination of the quasi stationary probabilities of a time inhomogeneous markov chain we also present some numerical experiments

the structure of hopf algebras with a projection the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1 5 and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afiine group schemes respectively let

co frobenius hopf algebras and the coradical filtration we prove that a hopf algebra with a finite coradical filtration is co frobenius we also characterize co frobenius hopf algebras with coradical a hopf subalgebra let h be a hopf algebra whose coradical is a hopf algebra let gr h be the associated graded coalgebra and let r be the diagram of h c f 2 then the following are equivalent 1 h is co frobenius 2 gr h is co frobenius 3 r is finite dimensional 4 the coradical filtration of h is finite this theorem allows us to give systematically examples of co frobenius hopf algebras and opens the way to the classification of ample classes of such hopf algebras

noetherian pi hopf algebras are gorenstein we prove that every noetherian affine pi hopf algebra has finite injective dimension which answers a question of brown 1998

on the convergence and applications of generalized simulated annealing the convergence of the generalized simulated annealing with time inhomogeneous communication cost functions is discussed this study is based on the use of log sobolev inequalities and semigroup techniques in the spirit of a previous article by one of the authors we also propose a natural test set approach to study the global minima of the virtual energy the second part of the paper is devoted to the application of these results we propose two general markovian models of genetic algorithms and we give a simple proof of the convergence toward the global minima of the fitness function finally we introduce a stochastic algorithm that converges to the set of the global minima of a given mean cost optimization problem

a p ge semiconductor experiment showing chaos and hyperchaos abstract a p ge semiconductor experiment is investigated by the help of both probabilistic and dynamical characterization methods dimensions lyapunov exponents and the corresponding scaling functions are calculated two exemplary files of data from the p ge semiconductor experiment exhibiting spontaneous i e undriven resistance oscillations in the low temperature avalanche breakdown are shown to be chaotic and hyperchaotic respectively for the first file we obtained a fractal dimension between two and three and one positive lyapunov exponent whereas for the second file we found a fractal dimension between three and four and two positive lyapunov exponents adopting the terminology introduced by rossler the behavior corresponding to the latter file is called hyperchaotic furthermore using the language of the thermodynamical formalism the probabilistic scaling function evaluated for the hyperchaotic state indicates a phase transition like behavior

on the dixmier moeglin equivalence for poisson hopf algebras abstract we prove that the poisson version of the dixmier moeglin equivalence holds for cocommutative affine poisson hopf algebras this is a first step towards understanding the symplectic foliation and the representation theory of cocommutative affine poisson hopf algebras our proof makes substantial use of the model theory of fields equipped with finitely many possibly noncommuting derivations as an application we show that the symmetric algebra of a finite dimensional lie algebra equipped with its natural poisson structure satisfies the poisson dixmier moeglin equivalence

markov chains with rare transitions and simulated annealing we consider approximate stationary markov chains in which the entries of the one step transition probability matrix are known to be of different orders of magnitude and whose structure that is the orders of magnitude of the transition probabilities does not change with time for such markov chains we present a method for generating order of magnitude estimates for the t step transition probabilities for any t we then notice that algorithms of the simulated annealing type may be represented by markov chains which are approximately stationary over fairly long time intervals using our results we obtain a characterization of the convergent cooling schedules for the most general class of algorithms of the simulated annealing type

a convergence analysis of generalized hill climbing algorithms generalized hill climbing ghc algorithms provide a well defined framework for describing the performance of local search algorithms for discrete optimization problems necessary and sufficient convergence conditions for ghc algorithms are presented these convergence conditions are derived using a new iteration classification scheme for ghc algorithms the implications of the necessary and the sufficient convergence conditions fur ghc algorithms with respect to existing convergence theory for simulated annealing are also discussed