on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

linear discrete time h sub spl infin optimal tracking with preview the problem of finite time h sub spl infin tracking for linear discrete time varying systems is considered no a prior knowledge of the dynamics of the reference signal is assumed a distinction between three cases is made depending on whether the reference signal is perfectly known in advance measured online or previewed in a fixed interval of time ahead the tracking problem is formulated as a game where the controller plays against nature which may choose the initial condition or the system and any energy bounded driving disturbance and measurement noise inputs necessary and sufficient conditions are derived for the existence of saddle point equilibrium solutions to the three different information structures of the reference and the corresponding tracking controllers are derived

integration with respect to fractal functions and stochastic calculus i the classical lebesgue stieltjes integral b a fdg of real or complex valued functions on a finite interval a b is extended to a large class of integrands f and integrators g of unbounded variation the key is to use composition formulas and integration by part rules for fractional integrals and weyl derivatives in the special case of holder continuous functions f and g of summed order greater than 1 convergence of the corresponding riemann stieltjes sums is proved the results are applied to stochastic integrals where g is replaced by the wiener process and f by adapted as well as anticipating random functions in the anticipating case we work within slobodeckij spaces and introduce a stochastic integral for which the classical ito formula remains valid moreover this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional brownian motion

stochastic differential games in a non markovian setting stochastic differential games are considered in a non markovian setting typically in stochastic differential games the modulating process of the diffusion equation describing the state flow is taken to be markovian then nash equilibria or other types of solutions such as pareto equilibria are constructed using hamilton jacobi bellman hjb equations but in a non markovian setting the hjb method is not applicable to examine the non markovian case this paper considers the situation in which the modulating process is a fractional brownian motion fractional noise calculus is used for such models to find the nash equilibria explicitly although fractional brownian motion is taken as the modulating process because of its versatility in modeling in the fields of finance and networks the approach in this paper has the merit of being applicable to more general gaussian stochastic differential games with only slight conceptual modifications this work has applications in finance to stock price modeling which incorporates the effect of institutional investors and to stochastic differential portfolio games in markets in which the stock prices follow diffusions modulated with fractional brownian motion

brief h performance of preview control systems performance of the preview control is investigated in terms of h criterion in the infinite horizon time setting the limitation of h performance in preview control action is characterized with the hamiltonian matrix by employing the property of hamiltonian matrix the solvability and analytic solution for the preview full information control problem are clarified based on an auxiliary introduced matrix riccati equation the property of resulting control law is illustrated with a numerical example

signal detection in fractional gaussian noise several results related to the reproducing kernel hilbert space of fractional brownian motion are presented to facilitate the study of signal detection in additive fractional gaussian noise this hilbert space is completely characterized and an alternative characterization for the restriction of this class of functions to a compact interval 0 t is given infinite and finite interval whitening filters for fractional brownian motion are also developed the application of these results to the signal detection problem yields necessary and sufficient conditions for a deterministic or stochastic signal to produce a nonsingular shift when embedded in additive fractional gaussian noise a formula for the likelihood ratio corresponding to any deterministic nonsingular shift is developed

a limit theorem for financial markets with inert investors we study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time it turns out that semi markov processes are tailor made for modelling inert investors with a suitable scaling we show that when the price is driven by the market imbalance the log price process is approximated by a process with long range dependence and non gaussian returns distributions driven by a fractional brownian motion consequently investor inertia may lead to arbitrage opportunities for sophisticated market participants the mathematical contributions are a functional central limit theorem for stationary semi markov processes and approximation results for stochastic integrals of continuous semimartingales with respect to fractional brownian motion

intervalwise receding horizon h sub spl infin tracking control for discrete linear periodic systems in this paper a fixed horizon h sub spl infin tracking control htc for discrete time varying systems is obtained in a state feedback form from the htc an intervalwise receding horizon htc ihtc is suggested for discrete periodic systems conditions are proposed under which ihtc can stabilize a closed loop system it is shown that under these conditions ihtc guarantees an h sub spl infin norm bound and that ihtc with integral action provides zero offset for a constant command signal the performance of ihtc is compared with those of existing controls via an example the results in this paper are also applicable to time invariant systems which belong to the class of periodic systems

on the h 2 two sided model matching problem with preview the h2 optimization problem with preview and asymptotic behavior constraints is considered in a general two sided model matching setting the solution is obtained in terms of two constrained sylvester equations associated with asymptotic behavior and stabilizing solutions of two algebraic riccati equations the riccati equations do not depend on the preview length yet are affected by asymptotic behavior constraints and are thus different from the standard h2 riccati equations arising in problems with no steady state requirements or in one sided problems

a game theoretic approach to a finite time disturbance attenuation problem a disturbance attenuation problem over a finite time interval is considered by a game theoretic approach where the control restricted to a function of the measurement history plays against adversaries composed of the process and measurement disturbances and the initial state a zero sum game formulated as a quadratic cost criterion subject to linear time varying dynamics and measurements is solved by a calculus of variation technique by first maximizing the quadratic cost criterion with respect to the process disturbance and initial state a full information game between the control and the measurement residual subject to the estimator dynamics results the resulting solution produces an n dimensional compensator which expresses the controller as a linear combination of the measurement history a disturbance attenuation problem is solved based on the results of the game problem for time invariant systems it is shown that under certain conditions the time varying controller becomes time invariant on the infinite time interval the resulting controller satisfies an h sub infinity norm bound

feedforward control a full information approach the design of the feedforward part of a two degree of freedom controller is recast as a full information synthesis problem

optimal tracking performance preview control and exponential signals we study tracking performance limitation problems two issues are addressed concerning how earlier results developed elsewhere may be extended to more general classes of reference signals and how tracking performance may be further improved beyond that offered by feedback control toward these issues we consider exponentially increasing reference inputs and examine the use of preview control for tracking using an optimal interpolation approach we develop analytical expressions and a conceptual insight which will aid in the understanding of these issues to this effect we derive explicit expressions for the optimal tracking error either as exact solutions or bounds it is found that for the exponential signals the earlier results can be directly extended and similar conclusive statements can be drawn it is also shown that in general preview can be used to advantage for improving tracking performance especially in dealing with the effect resulted from plant nonminimum phase zeros

continuous time tracking problems in an h sub spl infin setting a game theory approach this paper investigates the problem of finite horizon h sub spl infin tracking for linear time varying systems three tracking problems are considered depending on whether the reference signal is perfectly known in advance measured on line or previewed in fixed interval of time ahead no a priori knowledge of a dynamic model for the reference signal is assumed a game theory approach to the latter tracking problems is presented where given a specific reference signal the controllers play against nature which can choose any initial state and any bounded energy disturbance input and measurement noise necessary and sufficient conditions for the existence of a saddle point equilibrium are determined and h sub spl infin tracking controllers for both the cases of state and output feedback are derived tracking problems for time invariant systems on infinite horizon are also analyzed and time invariant tracking controllers are obtained for zero initial conditions

some discretizations of geometric evolution equations and the ricci iteration on the space of kahler metrics we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite dimensional spaces we illustrate the fruitfulness of this approach in the context of the ricci flow as well as another flow in kahler geometry we introduce and study dynamical systems related to the ricci operator on the space of kahler metrics that arise as discretizations of these flows we pose some problems regarding their dynamics we point out a number of applications to well studied objects in kahler and conformal geometry such as constant scalar curvature metrics kahler ricci solitons nadel type multiplier ideal sheaves balanced metrics the moser trudinger onofri inequality energy functionals and the geometry and structure of the space of kahler metrics e g we obtain a new sharp inequality strengthening the classical moser trudinger onofri inequality on the two sphere

a feedback theory of two degree of freedom optimal wiener hopf design the design of linear two degree of freedom stabilizing controllers is treated in a quadratic cost setting the class of all such controllers which give finite cost is established and the tradeoff possible between optimum performance tracking cost sensitivity and stability margins is discussed

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

stochastic h sub spl infin tracking with preview for state multiplicative systems the problem of finite horizon h sub spl infin tracking for linear time varying systems with stochastic parameter uncertainties is investigated we consider three tracking patterns depending on the nature of the reference signal i e whether it is perfectly known in advance measured on line or previewed in a fixed time interval ahead the stochastic uncertainties appear in both the dynamic and measurement matrices of the system for each of the above three cases a game theory approach is applied for the state feedback case where given a specific reference signal the controller plays against nature which chooses the initial condition and the energy bounded disturbance the problems are solved using an expected value of the standard performance index over the stochastic parameters where necessary and sufficient conditions are found for the existence of a saddle point equilibrium the infinite horizon time invariant tracking problem is also solved the theory developed is demonstrated by a simple tracking example

the ricci flow on the 2 sphere the classical uniformization theorem interpreted differential geomet rically states that any riemannian metric on a 2 dimensional surface ispointwise conformal to a constant curvature metric thus one can con sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met ric the primary interest in this question is not so much to give a newproof of the uniformization theorem but rather to understand nonlinearparabolic equations better especially those arising in differential geome try a sufficiently deep understanding of parabolic equations should yieldimportant new results in riemannian geometry the question in the preceding paragraph has been studied by richardhamilton 3 and brad osgood ralph phillips and peter sarnak 6 in 3 hamilton studied the following equation which we refer to as

stochastic calculus for fractional brownian motion i theory in this paper a stochastic calculus is given for the fractional brownian motions that have the hurst parameter in 1 2 1 a stochastic integral of ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment this integral uses the wick product and a derivative in the path space some ito formulae or change of variables formulae are given for smooth functions of a fractional brownian motion or some processes related to a fractional brownian motion a stochastic integral of stratonovich type is defined and the two types of stochastic integrals are explicitly related a square integrable functional of a fractional brownian motion is expressed as an infinite series of orthogonal multiple integrals

a game theoretic approach to h control for time varying systems a representation formula for all controllers that satisfy an mathcal l infty type constraint is derived for time varying systems it is now known that a formula based on two indefinite algebraic riccati equations may be found for time invariant systems over an infinite time support see j c doyle et al ieee trans automat control ac 34 1989 pp 831 847 k glover and j c doyle systems control lett 11 1988 pp 167 172 k glover et al siam j control optim 29 1991 pp 283 324 m green et al siam j control optim 28 1990 pp 1350 1371 d j n limebeer et al in proc ieee conf on decision and control austin tx 1988 g tadmor math control systems signal processing 3 1990 pp 301 324 in the time varying case two indefinite riccati differential equations are required a solution to the design problem exists if these equations have a solution on the optimization interval the derivation of the representation formula illustrated in this paper mak

two degree of freedom h sub infinity optimization of multivariable feedback systems a solution to the two degree of freedom h sub infinity minimization problem that arises in the design of multivariable optimal continuous time stochastic control systems is derived a decoupling approach that enables a partially independent design of the prefilter and the feedback controller and yields a simple solution to the optimization problem is applied this solution is obtained by transforming the optimization problem into two standard form four block problems

sharp local embedding inequalities in this paper we establish local versions of the onofri and sharp sobolev inequalities such local inequalities enable us to give a more direct and simpler proof of the onofri inequality on 2 as well as an alternative proof of sharp sobolev inequalities on n for n 3 2005 wiley periodicals inc

integration questions related to fractional brownian motion let b h u u be a fractional brownian motion fbm with index h 0 1 and b h be the closure in l 2 of the span sp b h of the increments of fbm b h it is well known that when b h b 1 2 is the usual brownian motion bm an element x b 1 2 can be characterized by a unique function f x l 2 in which case one writes x in an integral form as x f x u db 1 2 u from a different though equivalent perspective the space l 2 forms a class of integrands for the integral on the real line with respect to bm b 1 2 in this work we explore whether a similar characterization of elements of b h can be obtained when h 0 1 2 or h 1 2 1 since it is natural to define the integral of an elementary function f k 1 n f k 1 uk uk 1 by k 1 n f k b h u k 1 b h u k we want the spaces of integrands to contain elementary functions these classes of integrands are inner product spaces if the space of integrands is not complete then it characterizes only a strict subset of b h when 0 h 1 2 by using the moving average representation of fbm b h we construct a complete space of integrands when 1 2 h 1 however an analogous construction leads to a space of integrands which is not complete when 0 h 1 2 or 1 2 h 1 we also consider a number of other spaces of integrands while smaller and henceincomplete they form a natural choice and are convenient to workwith we compare these spaces of integrands to the reproducing kernel hilbert space of fbm