forward backward and symmetric stochastic integration we define three types of non causal stochastic integrals forward backward and symmetric our approach consists in approximating the integrator two optics are considered the first one is based on traditional usual stochastic calculus and the second one on wiener distributions

discrete time adaptive control in the presence of input constraints the problem of ensuring that a control input remains within certain bounds is ubiquitous in all control applications this problem becomes even more difficult in the context of adaptive control since the plant to be controlled has unknown parameters in this paper we establish conditions under which minimum phase discrete time plants can be adaptively controlled in a stable manner in the presence of amplitude constraints on the input all open loop stable plants are shown to result in global stability with the proposed adaptive controller and open loop unstable plants are shown to result in local stability when the initial conditions of the closed loop system lie within a compact set it is also shown that when initial conditions exceed a certain value open loop unstable plants can become unstable it is furthermore shown that the adaptive algorithms can be appropriately modified in the presence of exogeneous disturbances and certain modeling errors to yield boundedness simulations are presented to support the theoretical derivations

the malliavin calculus a functional analytic approach in his groundbreaking articles s 91 malliavin introduced a technique for obtaining elliptic regularity results using function space calculus in his formulation the function space calculus on which his theory rests is the martingale calculus coming from the ornstein uhlenbeck process on wiener space cf 121 for a detailed discussion of malliavin s theory from this point of view although this formulation has a great deal to recommend it and is particularly pleasing to afficionados of the so called brownian sheet most analysts including those who are reasonably facile with the machinery of probability theory are unlikely to find the technical difficulties inherent in this approach outweighed by the eventual rewards shigekawa 111 discovered that there is an alternative formulation of malliavin s theory in shigekawa s formulation no mention need be made of the ornstein uhlenbeck process instead what he relies on is a sobolev type extension of the frechet dervative with wiener measure playing the role played by lebesgue measure in the finite dimensional context a third formulation of malliavin s theory was recently provided by bismut l bismut s idea is to exploit the quasi invariance of wiener measure as it is manifested in a beautiful relation due to haussmann 4 the present article has two aims in the first place section 1 2 and 3 are devoted to yet another formulation of malliavin s calculus although the end results are precisely the same as those obtained in 121 no reference is made to the ornstein uhlenbeck process on which the original theory depended instead the machinery is entirely that of elementary functional analysis and is devoid of any sophisticated martingale calculus it has been my intent that these three sections should be readily accessible to any analyst who has had some acquintance with wiener measure

backward doubly stochastic differential equations and systems of quasilinear spdes we introduce a new class of backward stochastic differential equations which allows us to produce a probabilistic representation of certain quasilinear stochastic partial differential equations thus extending the feynman kac formula for linear spde s

complexity of graph self assembly in accretive systems and self destructible systems self assembly is a process in which small objects autonomously associate with each other to form larger complexes it is ubiquitous in biological constructions at the cellular and molecular scale and has also been identified by nanoscientists as a fundamental method for building nano scale structures recent years have seen convergent interest and efforts in studying self assembly from mathematicians computer scientists physicists chemists and biologists however most complexity theoretical studies of self assembly utilize mathematical models with two limitations 1 only attraction while no repulsion is studied 2 only assembled structures of two dimensional square grids are studied in this paper we study the complexity of the assemblies resulting from the cooperative effect of repulsion and attraction in a more general setting of graphs this allows for the study of a more general class of self assembled structures than the previous tiling model we define two novel assembly models namely the accretive graph assembly model and the self destructible graph assembly model and identify a fundamental problem in them the sequential construction of a given graph we refer to it as the accretive graph assembly problem agap and the self destructible graph assembly problem dgap in the respective models our main results are i agap is np complete even if the maximum degree of the graph is restricted to 4 or the graph is restricted to be planar with maximum degree 5 ii counting the number of sequential assembly orderings that result in a target graph agap is p complete and iii dgap is pspace complete even if the maximum degree of the graph is restricted to 6 this is the first pspace complete result in self assembly we also extend the accretive graph assembly model to a stochastic model and prove that determining the probability of a given assembly in this model is p complete

stochastic delay equations with hereditary drift estimates of the density abstract we consider a family of stochastic differential equations with a drift depending on the past history and a delayed diffusion term perturbed by a small parameter e 0 we establish the asymptotic behaviour as e 0 for the logarithm of the corresponding family of densities at a fixed time t 0 the proof needs large deviation estimates and malliavin calculus

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

adaptive one step ahead control subject to an input amplitude constraint this paper investigates the adaptive control of a single input single output process when there exists a constraint on the amplitude of the applied control signal in particular an adaptive control scheme is proposed and is shown to be globally convergent in the sense that the control asymptotically matches the amplitude constrained one step ahead optimal control some simulation results are presented to demonstrate the performance of the scheme

generalized stochastic integrals and the malliavin calculus the paper first reviews the skorohod generalized stochastic integral with respect to the wiener process over some general parameter space t and it s relation to the malliavin calculus as the adjoint of the malliavin derivative some new results are derived and it is shown that every sufficiently smooth process ut t t can be decomposed into the sum of a malliavin derivative of a wiener functional and a process whose generalized integral over t vanishes using the results on the generalized integral the bismut approach to the malliavin calculus is generalized by allowing non adapted variations of the wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions

estimation de varadhan pour des diffusions a deux parametres sous des conditions de hormander generalisees la loi d une diffusion a parametre planaire possede une densite lisse en dehors des axes dans le cas elliptique nous etablissons une estimation de cette densite pur des petites valeurs du parametre celle ci constitue l analogue de l estimation de varadhan objet qui est bien connu dans le cas des diffusions ordinaires

direct adaptive control subject to input amplitude constraint this paper deals with the problem of controlling discrete time linear systems with unknown dynamics and a saturation type constraint on the control input amplitude we propose a direct adaptive regulator that globally stabilizes any system with one pole at x 1 and all others inside the unit disk the regulator design is carried out using the pole placement technique of particular interest the input saturation vanishes after a finite transient period

regularity of skorohod integral processes based on integrands in a finite wiener chaos letf be a square integrable kernel on them dimensional unit cube u the skorohod integral process in themth wiener chaos associated with it isoperimetric inequalities for functions on wiener space yield the exponential integrability of the increments ofu to this result we apply the majorizing measure technique to show thatu possesses a continuous version and give an upper bound of its modulus of continuity

linear constructions for dna codes in this paper we translate in terms of coding theory constraints that are used in designing dna codes for use in dna computing or as bar codes in chemical libraries we propose new constructions for dna codes satisfying either a reverse complement constraint a gc content constraint or both that are derived from additive and linear codes over four letter alphabets we focus in particular on codes over gf 4 and we construct new dna codes that are in many cases better sometimes far better than previously known codes we provide updated tables up to length 20 that include these codes as well as new codes constructed using a combination of lexicographic techniques and stochastic search

discrete time approximations of stochastic delay equations the milstein scheme in this paper we develop a strong milstein approximation scheme for solving stochastic delay differential equations sddes the scheme has convergence order 1 in order to establish the scheme we prove an infinite dimensional ito formula for tame functions acting on the segment process of the solution of an sdde it is interesting to note that the presence of the memory in the sdde requires the use of the malliavin calculus and the anticipating stochastic analysis of nualart and pardoux given the nonanticipating nature of the sdde the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel

second order taylor expansion for backward doubly stochastic control system we consider the second order taylor expansion for backward doubly stochastic control system the results are obtained under no restriction on the convexity of control domain moreover the control variable is allowed in the drift coefficient and the diffusion coefficient

quasi linear pdes and forward backward stochastic differential equations weak solutions abstract in this paper we study the existence uniqueness and the probabilistic representation of the weak solutions of quasi linear parabolic and elliptic partial differential equations pdes in the sobolev space h 1 r d for this we study first the solutions of forward backward stochastic differential equations fbsdes with smooth coefficients regularity of solutions and their connection with classical solutions of quasi linear parabolic pdes then using the approximation procedure we establish their convergence in the sobolev space to the solutions of the fbsdes in the space l 2 r d r d l 2 r d r k l 2 r d r k d this gives a connection with the weak solutions of quasi linear parabolic pdes finally we study the unique weak solutions of quasi linear elliptic pdes using the solutions of the fbsdes on infinite horizon

anticipated backward doubly stochastic differential equations in this paper we deal with a new type of differential equations called anticipated backward doubly stochastic differential equations anticipated bdsdes the coefficients of these bdsdes depend on the future value of the solution y z we obtain the existence and uniqueness theorem and a comparison theorem for the solutions of these equations besides as an application we also establish a duality between the anticipated bdsdes and the delayed doubly stochastic differential equations delayed dsdes

amplitude constrained adaptive control this paper extends the theoretical results of amplitude constrained adaptive control to a class of type 1 possibly non minimum phase plants the one step ahead adaptive control when subject to a saturation constraint results in a system which is bibo stable in the sense that the output is bounded and the control signal converges to the corresponding non adaptive control the results are readily extended to the model reference adaptive control case

stable adaptation in the presence of input constraints abstract for a class of linear dynamical systems with unknown parameters a direct model reference adaptive control framework is developed that provides stable adaptation in the presence of input constraints the proposed design methodology termed positive modification has the ability to protect the control law from actuator position saturation the reference system is adaptively modified to ensure global asymptotic tracking for open loop stable systems for unstable systems an estimate for the domain of attraction is derived based on the input saturation magnitude and system parameters simulation of a benchmark example verifies the theoretical statements

stochastic calculus with respect to gaussian processes in this paper we develop a stochastic calculus with respect to a gaussian process of the form b t int t 0 k t s dw s where w is a wiener process and k t s is a square integrable kernel using the techniques of the stochastic calculus of variations we deduce change of variable formulas for the indefinite integrals and we study the approximation by riemann sums the particular case of the fractional brownian motion is discussed

covariance identities and inequalities for functionals on wiener and poisson spaces we present covariance identities and inequalities for functionals of the wiener and the poisson processes using malliavin calculus techniques an expansion with a remainder term is obtained for the covariance of such functionals our results extend known identities and inequalities for functions of multivariate random vectors

a change of variables formula for stratonovich integrals and existence of solutions for two point stochastic boundary value problems existence results for a class of two point non adaptive stochastic boundary value problems are derived by combining a new change of variables formula with optimal control techniques the class of equations treated is significantly wider than that for which previously known existence results apply

adaptive control in the presence of input constraints this paper deals with the problem of adaptively controlling a linear time invariant plant in the presence of constraints on the input amplitude we introduce a new adaptive algorithm which ensures that the plant can be locally stabilized in addition to the standard assumptions which are required for adaptive control in the ideal case an upper bound on the plant control parameter is required to be known the results are evaluated by simulation studies

euler maruyama approximations for sdes with non lipschitz coefficients and applications abstract in this paper euler maruyama approximation for sde with non lipschitz coefficients is proved to converge uniformly to the solution in l p space with respect to the time and starting points as an application we also study the existence of solution and large deviation principle for anticipative sde with random initial condition

adaptive tracking with saturating input and controller integral action the authors consider the problem of controlling stable linear plants subject to input saturation constraints and parametric uncertainties a globally stable adaptive controller which performs well in the presence of such constraints is designed using a specific pole placement that accounts for the saturation constraint an integral action to deal with the tracking problem and an adaptive control law with an ad hoc excitation capability together with a parameter adaptation to cope with the parametric uncertainty the proposed controller allows one to track slowly varying reference sequences particularly in the case of a constant reference the output tracking error vanishes asymptotically and the control input does stop saturating after a finite time

brief adaptive control of input constrained type 1 plants stabilization and tracking we consider the problem of controlling non minimum phase type 1 plants in presence of model uncertainty and a saturation constraint imposed on the control input the control objectives are the following i robust global asymptotic stabilization of the controlled plant with respect to a multiplicative modeling error ii output regulation in presence of external perturbations and iii tracking of output reference trajectories the problem is dealt with using a saturated indirect adaptive regulator based on a specific pole placement of the closed loop poles the slower the reference trajectory and the perturbation sequence the better the tracking and regulation quality

the onsager machlup functional for a class of anticipating processes consider the solution xt 0 t 1 of the equation x t x 0 w t mathop smallint limits 0 t b x s d s wherew wt 0 t 1 is ad dimensional wiener process b is a lipschitz function from d into d andx0 is ad dimensional functional of the wiener process

markov field property of stochastic differential equations the purpose of this paper is to prove a characterization of the conditional independence of two independent random variables given a particular functional of them in terms of a factorization property as an application we discuss the markov field property for solutions of stochastic differential equations with a boundary condition involving the values of the process at times t 0 and t 1

stochastic equivariant cohomologies and cyclic cohomology we give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the chen souriau sense and to establish a link with cyclic cohomology with the second one we can establish a stochastic fixed point theorem

the malliavin calculus and stochastic delay equations we consider stochastic delay systems dx t g x t r dw t driven by multi dimensional brownian motion w the diffusion coefficient g is smooth with a possible degeneracy at 0 for a large class of deterministic initial paths we show that the solution x t admits a smooth density with respect to lebesgue measure the proof is based on malliavin calculus together with new probabilistic lower bounds on the solution x

a two sided stochastic integral and its calculus let x be a forward diffusion and y a backward diffusion both defined on 0 1 x t and y t being respectively adapted to the past of a wiener process w and to its future increments we construct a two sided stochastic integral of the form mathop smallint limits 0 t phi u x u y u dw u which generalizes the backward and forward ito integrals simultaneously our construction is quite intuitive and leads to a generalized stochastic calculus it is also shown that for each fixed t our integral coincides with that defined by skorohod in 18

smooth densities for degenerate stochastic delay equations with hereditary drift we establish the existence of smooth densities for solutions of r d valued stochastic hereditary differential systems of the form dx t h t x dt g t x t r dw t in the above equation w is an n dimensional wiener process r is a positive time delay h is a nonanticipating functional defined on the space of paths in r d and g is an n x d matrix valued function defined on 0 x r d such that gg has degeneracies of polynomial order on a hypersurface in r d in the course of proving this result we establish a very general criterion for the hypoellipticity of a class of degenerate parabolic second order time dependent differential operators with space independent principal part

integration by parts on wiener space and the existence of occupation densities we present a general criterion for the existence of an occupation density which is based on the integration by parts formula on wiener space this criterion is applied to two particular examples of anticipating processes first we discuss the case of brownian motion plus a nonadapted absolutely continuous drift and second we consider the case of a skorohod integral process finally a version of tanaka s formula is proved for the skorohod integral process 1 introduction an extension of the ito stochastic integral for processes which are not necessarily adapted to the brownian motion was introduced by skorohod in 12 the skorohod integral shares some of the probabilistic properties of the ito integral and on the other hand it has a more analytic flavor because it can be regarded as the adjoint of the derivative operator recently a stochastic calculus for skorohod integral processes has been developed in 9 and a version of the change of variable formula ito s formula for skorohod integral processes has been established a crucial point in the derivation of this formula is the fact that skorohod integral processes possess quadratic variations which are given in the same way as for their it6 counterparts and consequently are nontrivial that is the trajectories of these processes are continuous but highly irregular this observation leads us to investigate their occupation densities or local times different methods have been used so far to study the occupation densities of anticipating processes berman s idea to employ fourier analysis see 2 has been applied by imkeller in 6 7 to find integral criteria for the existence of a square integrable local time for a skorohod integral process which belongs to the second wiener chaos a more stochastic approach has been presented in 8 again for processes living in the second wiener chaos although this method can be generalized to deal with more general processes its main idea is to represent the skorohod integral process we are dealing with as the composition of a gaussian semimartingale depending on an infinite dimensional parameter with a gaussian vector and to use an infinite dimensional generalization of kolmogorov s continuity criterion in order to find the local time of this composition

diagonal versions and quantum stochastic integrals on the symmetric fock space with nonadapted integrands we develop a quantum stochastic calculus for nonadapted processes and the three standard integrators on the symmetric fock space over an arbitrary polish space the integrable processes are restricted to a class of operators which possess some kernel description our approach is mainly based on the concept of diagonalized versions 16 in the case of the real line we consider connections to the usual stochastic and quantum stochastic calculi and derive ito formulae besides the standard integrals with two integrands and one integrator these formulae lead to a new type of quantum stochastic integrals which has two integrators and three integrands

stable adaptive control for not necessarily stable discrete linear plants with bounded non linear inputs abstract this paper deals with the design of stable adaptive control schemes for discrete plants under non linear bounded inputs two algorithms are studied one of them being a generalization of the d step ahead standard algorithm while the second one involves the use of a time decreasing adaptation gain the class of inputs for which the algorithms perform in a stable fashion include saturations relays and saturated dead zones the plant stability assumption is not requested to prove standard non asymptotic stability results the mechanism to achieve scheme s stability is the use of an extra weighted additive term in the adaptation error which takes into account the output deviation with respect to that associated with linear inputs

complexities for generalized models of self assembly in this paper we study the complexity of self assembly under models that are natural generalizations of the tile self assembly model in particular we extend rothemund and winfree s study of the tile complexity of tile self assembly proceedings of the 32nd annual acm symposium on theory of computing portland or 2000 pp 459 468 they provided a lower bound of omega frac log n log log n on the tile complexity of assembling an n times n square for almost all n adleman et al proceedings of the 33rd annual acm symposium on theory of computing heraklion greece 2001 pp 740 748 gave a construction which achieves this bound we consider whether the tile complexity for self assembly can be reduced through several natural generalizations of the model one of our results is a tile set of size o sqrt log n which assembles an n times n square in a model which allows flexible glue strength between nonequal glues this result is matched for almost all n by a lower bound dictated by kolmogorov complexity for three other generalizations we show that the omega frac log n log log n lower bound applies to n times n squares at the same time we demonstrate that there are some other shapes for which these generalizations allow reduced tile sets specifically for thin rectangles with length n and width k we provide a tighter lower bound of omega frac n 1 k k for the standard model yet we also give a construction which achieves o frac log n log log n complexity in a model in which the temperature of the tile system is adjusted during assembly we also investigate the problem of verifying whether a given tile system uniquely assembles into a given shape we show that this problem is np hard for three of the generalized models

quantum and non causal stochastic calculus the quantum stochastic calculus initiated by hudson and parthasarathy and the non causal stochastic calculus originating with the papers of hitsuda and skorohod are two potent extensions of the ito calculus currently enjoying intensive development the former provides a quantum probabilistic extension of schrodinger s equation enabling the construction of a markov process for a quantum dynamical semigroup the latter allows the treatment of stochastic differential equations which involve terms which anticipate the future in this paper the close relationship between these theories is displayed and a noncausal quantum stochastic calculus already in demand from physics is described

dipole codes attractively encode glue functions dipole words are sequences of magnetic dipoles in which alike elements repel and opposite elements attract magnetic dipoles contrast with more general sets of bonding types called glues in which pairwise bonding strength is specified by a glue function we prove that every glue function g has a set of dipole words called a dipole code that attractively encodes g the pairwise attractions positive or non positive bond strength between the words are identical to those of g moreover we give such word sets of asymptotically optimal length similar results are obtained for a commonly used subclass of glue functions