conditions for the equivalence of internal and external stability for distributed parameter systems conditions are given on a controlled observed distributed parameter system under which external stability also known as input output stability implies internal stability which is the exponential stability of the underlying semigroup generator it is shown that when the system satisfies a general definition of stabilizability and detectability external stability is equivalent to internal stability unlike the case previous definitions of stabilizability and detectability a system which satisfies these does not necessarily have a spectrum decomposition therefore in order to check internal stability it is often sufficient to check the boundedness of the transfer function in the right half complex plane this is illustrated with a simple example

essential spectral radius for markov semigroups i discrete time case using two new measures of non compactness p and w p for a positive kernel p on a polish space e we obtain a new formula of nussbaum gelfand type for the essential spectral radius ress p on b using that formula we show that different known sufficient conditions for geometric ergodicity such as doeblin s condition drift condition by means of lyapunov function geometric recurrence etc lead to variational formulas of the essential spectral radius all those can be easily transported on the weighted space bu some related results on l2 are also obtained especially in the symmetric case moreover we prove that for a strongly feller and topologically transitive markov kernel the large deviation principle of donsker varadhan for occupation measures of the associated markov process holds if and only if the essential spectral radius is zero this result allows us to show that the sufficient condition of donsker varadhan for the large deviation principle is in fact necessary the knowledge of ress p allows us to estimate eigenvalues of p in l2 in the symmetric case and to estimate the geometric convergence rate by means of that in the metric of wasserstein applications to different concrete models are provided for illustrating those general results

optimizability and estimatability for infinite dimensional linear systems an infinite dimensional linear system described by dot x t ax t bu t t geq 0 is said to be optimizable if for every initial state x 0 an input u in l 2 can be found such that x in l 2 here a is the generator of a strongly continuous semigroup on a hilbert space and b is an admissible control operator for this semigroup in this paper we investigate optimizability also known as the finite cost condition and its dual estimatability we explore the connections with stabilizability and detectability we give a very general theorem about the equivalence of input output stability and exponential stability of well posed linear systems the two are equivalent if the system is optimizable and estimatable we conclude that a well posed system is exponentially stable if and only if it is dynamically stabilizable and input output stable we illustrate the theory by two examples based on pdes in two or more space dimensions the wave equation and a structural acoustics model

poincare and transportation inequalities for gibbs measures under the dobrushin uniqueness condition in in this paper we establish an explicit and sharp estimate of the spectral gap poincare inequality and the transportation inequality for gibbs measures under the dobrushin uniqueness condition moreover we give a generalization of the liggett s m e theorem for interacting particle systems

a moderate deviation principle for 2 d stochastic navier stokes equations in this paper we prove a central limit theorem and establish a moderate deviation principle for twodimensional stochastic navier stokes equations with multiplicative noise the weak convergence method plays an important role

moderate deviations of dependent random variables related to clt this paper consists of three parts in the first part we find a common condition the c 2 regularity both for clt and for moderate deviations we show that this condition is verified in two important situations the lee yang theorem case and the fkg system case in the second part we apply the previous results to the additive functionals of a markov process by means of feynman kac formula and kato s analytic perturbation theory we show that the lee yang theorem holds under the assumption that 1 is an isolated simple and the only eigenvalue with modulus 1 of the operator p 1 acting on an appropriate banach space be c b e l 2 the last part is devoted to some applications to statistical mechanical systems where the c 2 regularity becomes a property of the pressure functionals and the two situations presented above become exactly the lee yang theorem case and the fkg system case we shall discuss in detail the ferromagnetic model and give some general remarks on some other models

a comparison of regularizations for an ill posed problem we consider numerical methods for a quasi boundary value regularization of the backward parabolic problem given by u t au 0 0 t t u t f where a is positive self adjoint and unbounded the regularization due to clark and oppenheimer perturbs the final value u t by adding u 0 where is a small parameter we show how this leads very naturally to a reformulation of the problem as a second kind fredholm integral equation which can be very easily approximated using methods previously developed by ames and epperson error estimates and examples are provided we also compare the regularization used here with that from ames and epperson

shannon wavelet regularization methods for a backward heat equation abstract we consider the backward heat equation u x x x t u t x t x 0 t t the solution u x t on the final value t t is an known function g t x this is a typical ill posed problem since the solution if it exists does not depend continuously on the final data in this paper we shall give a shannon wavelet regularization method and obtain some quite sharp error estimates between the exact solution and the approximate solution defined in the scaling space v j

a comparison of some inverse methods for estimating the initial condition of the heat equation in this work we analyze two explicit methods for the solution of an inverse heat conduction problem and we confront them with the least squares method using for the solution of the associated direct problem a classical finite difference method and a method based on an integral formulation finally the tikhonov regularization connected to the least squares criterion is examined we show that the explicit approaches to this inverse heat conduction problem will present disastrous results unless some kind of regularization is used

admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

a regularization parameter for nonsmooth tikhonov regularization in this paper we develop a novel rule for choosing regularization parameters in nonsmooth tikhonov functionals it is solely based on the value function and applicable to a broad range of nonsmooth models and it extends one known criterion a posteriori error estimates of the approximations are derived an efficient numerical algorithm for computing the minimizer is developed and its convergence properties are discussed numerical results for several common nonsmooth models are presented including deblurring natural images the numerical results indicate the rule can yield results comparable with those achieved with the discrepancy principle and the optimal choice and the algorithm merits a fast and steady convergence

fourier regularization for a backward heat equation abstract in this paper a simple and convenient new regularization method for solving backward heat equation fourier regularization method is given meanwhile some quite sharp error estimates between the approximate solution and exact solution are provided a numerical example also shows that the method works effectively

equivalent conditions for stabilizability of infinite dimensional systems with admissible control operators in this paper we study the optimizability of infinite dimensional systems with admissible control operators we show that under a weak condition such a system is optimizable if and only if the system can be split into an exponentially stable subsystem and an unstable subsystem that is exactly controllable in finite time the state space of the unstable subsystem equals the span of all unstable generalized eigenvectors of the original system this subsystem can be infinite dimensional furthermore the unstable poles satisfy a summability condition the state space of the exponentially stable subsystem is given by all vectors for which the action of the original c0 semigroup is stable

the theory of large deviations from boltzmann s 1877 calculation to equilibrium macrostates in 2d turbulence abstract after presenting some basic ideas in the theory of large deviations this paper applies the theory to a number of problems in statistical mechanics these include deriving the form of the gibbs state for a discrete ideal gas describing probabilistically the phase transition in the curie weiss model of a ferromagnet and deriving variational formulas that describe the equilibrium macrostates in models of two dimensional turbulence a general approach to the large deviation analysis of models in statistical mechanics is also formulated

gradient estimates of poisson equations on riemannian manifolds and applications abstract for the reflected diffusion generated by l v on a connected and complete riemannian manifold m with empty or convex boundary we establish some sharp estimates of sup x m g x of the poisson equation l g g in terms of the dimension the diameter and the lower bound of curvature applications to transportation information inequality to cheeger s isoperimetric inequality and to gaussian concentration inequality are given several examples are provided

entropy for translation invariant random cluster measures we study translation invariant random cluster measures with techniques from large deviation theory and convex analysis in particular we prove a large deviation principle with rate function given by a specific entropy and a dobrushinlanfordruelle variational principle that characterizes translation invariant random cluster measures as the solutions of the variational equation for free energy consequences of these theorems include inequalities for edge and cluster densities of translationinvariant random cluster measures 1 introduction the random cluster model was introduced around 1972 by fortuin and kasteleyn 18 as a family of models that contains the bernoulli bond percolation model the ising model and the q state potts models as special cases it has developed into an important and very successful tool for studying the ising and potts models our paper focuses on the properties of the random cluster measures themselves our starting point is grimmett s 23 development of elements of a thermodynamic formalism for this model beginning with a definition of infinite volume random cluster measures in terms of a dlr dobrushinlanfordruelle equation just as for gibbsian systems natural next questions concern the validity of large deviation principles and variational characterizations of infinite volume measures these questions we answer in the present paper and derive some consequences here is a brief overview of the dlr approach of 23 the random cluster measures of interest live on the graph formed by putting edges between nearest neighbor vertices of the d dimensional hypercubic lattice d the edge that connects two vertices x and y that are l 1 distance one apart is

large deviations for the empirical field of a gibbs measure on applique le principe de contraction pour obtenir un principe de grande deviation pour la distribution des distributions empiriques on donne une description detaillee de la fonction de taux resultante

riesz bases and exact controllability of c0 groups with one dimensional input operators abstract this paper considers linear infinite dimensional systems with c 0 group generators and one dimensional admissible input operators the exact controllability and riesz basis generation property are discussed the corresponding results of jacob and zwart advances in mathematical systems theory birkhauser boston ma 2000 under the assumption of algebraic simplicity for eigenvalues of the generator are generalized to the case in which the eigenvalues are allowed to be algebraically multiple but with a uniform bound on the multiplicity

asymptotics of sample entropy production rate for stochastic differential equations using the dimension free harnack inequality and the integration by parts formula for the associated diffusion semigroup we prove the central limit theorem the moderate deviation principle and the logarithmic iteration law for the sample entropy production rate of a family of stochastic differential equations

moderate deviations for empirical measures of markov chains lower bounds we obtain upper bounds for moderate deviations of empirical measures of markov chains with general state space in the topology under the minimal assumption of geometric ergodicity the regeneration split chain method and a result on projective systems are employed in the proof

infinite dimensional continuous time linear systems stabilityand structure analysis the question of exponential and asymptotic stability of infinite dimensional continuous time state space systems is investigated it is shown that every par balanced realization is asymptotically stable conditions are given for par balanced input normal or output normal ealizations to be asymptotically and or exponentially stable the boundedness of the system operators is also studied examples of delay systems are given to illustrate the theory

the regularity of the wave equation with partial dirichlet control and colocated observation in this paper we analyze a multidimensional controlled wave equation on a bounded domain subject to partial dirichlet control and colocated observation by means of a partial fourier transform it is shown that the system is well posed and regular in the sense of d salamon and g weiss the corresponding feedthrough operator is found to be the identity operator on the input space

equivalence between open loop and closed loop invariance for infinite dimensional systems a frequency domain approach in this paper the relation between various system invariance concepts is considered and it is proved that open and closed loop invariance are equivalent concepts using this equivalence the author investigates the existence of the largest closed loop invariant subspace and derives necessary and sufficient conditions such that the largest open loop invariant subspace is equal to the largest closed loop invariant subspace if this equality holds then the solvability of the disturbance decoupling problem ddp is equivalent to the solvability of a meromorphic matrix equation

controllability and stability of a second order hyperbolic system with collocated sensor actuator a second order hyperbolic system with collocated sensor actuator is considered the semigroup generation is shown for the closed loop system under the feedback of a generic unbounded observation operator the equivalence between the exponential stability of the closed loop system and exact controllability of the open loop system is established in the general framework of well posed linear systems finally the conditions are weakened for the diagonal semigroups with finite dimensional inputs example of beam equation is presented to display the application

the boundary element method for the solution of the backward heat conduction equation in this paper we consider the numerical solution of the one dimensional unsteady heat conduction equation in which dirichlet boundary conditions are specified at two space locations and the temperature distribution at a particular time say t0 is given the temperature distribution for all times t t0 is now required and this backward heat conduction problem is a well known improperly posed problem in order to solve this problem the minimal energy technique has been introduced in order to modify the boundary element method and this results in a stable approximation to the solution and the accuracy of the numerical results are very encouraging

admissibility in input elements for diagonal semigroups on l 2 abstract we characterize the admissible unbounded input elements for diagonal semigroups on l 2 the result verifies the necessity of a condition whose sufficience was previously established by l f ho and d l russell

moderate deviations and erdos renyi shepp laws for weighted sums let x0 x1 be i i d random variables with e x0 0 e x20 1 and e exp tx0 for any t t0 we prove that the weighted sums v n j 0 aj n xj n 1 obey a moderately large deviation principle if the weights satisfy certain regularity conditions then we prove a new version of the erdos renyi shepp laws for the weighted sums

stabilization and controllability of first order integro differential hyperbolic equations in the present article we study the stabilization of first order linear integro differential hyperbolic equations for such equations we prove that the stabilization in finite time is equivalent to the exact controllability property the proof relies on a fredholm transformation that maps the original system into a finite time stable target system the controllability assumption is used to prove the invertibility of such a transformation finally using the method of moments we show in a particular case that the controllability is reduced to the criterion of fattorini

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

universal prediction of individual binary sequences in the presence of noise the problem of predicting the next outcome of an individual binary sequence based on noisy observations of the past is considered the goal of the predictor is to perform for each individual sequence almost as well as the best in a set of experts where performance is evaluated using a general loss function a comprehensive approach to prediction in this noisy setting is presented and proven generally efficient under appropriate conditions as an illustration of the applicability of the approach suggested for concrete situations two important special cases are explicitly treated the first is the case where the data corrupting noise process is binary valued where the observed bit is the bitwise xor of the clean bit and the noise bit the second case is that of real valued additive noise it is shown that even in this more challenging situation where the information available to the predictor regarding the past sequence is incomplete a predictor can be guaranteed to successfully compete with a whole set of experts in considerably strong senses

self similarity of brownian motion and a large deviation principle for random fields on a binary tree using self similarity of brownian motion and its representation as a product measure on a binary tree we construct a random sequence of probability measures which converges to the distribution of the brownian bridge we establish a large deviation principle for random fields on a binary tree this leads to a class of probability measures with a certain self similarity property the same construction can be carried out forc 0 1 valued processes and we can describe for instance ac 0 1 valued ornstein uhlenbeck process as a large deviation of brownian sheet

entropy limit theorems and variational principles for disordered lattice systems we study infinite volume limits and gibbs states of disordered lattice systems with bounded and continuous potentials our main tools are a generalization of relative entropy for random reference measures and a large deviation theory for nonstationary independent processes we find that many familiar results of invariant potentials such as large deviation theorems variational principles and equivalence of ensembles continue to hold for disordered models with suitably modified statements

asymptotics of the entropy production rate for d dimensional ornstein uhlenbeck processes in the context of non equilibrium statistical physics the entropy production rate is an important concept to describe how far a specific state of a system is from its equilibrium state in this paper we establish a central limit theorem and a moderate deviation principle for the entropy production rate of d dimensional ornstein uhlenbeck processes by the techniques of functional inequalities such as poincare inequality and log sobolev inequality as an application we obtain a law of iterated logarithm for the entropy production rate

spectral regularization method for a cauchy problem of the time fractional advection dispersion equation in this paper a cauchy problem for the time fractional advection dispersion equation tfade is investigated such a problem is obtained from the classical advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order a 0 a 1 we show that the cauchy problem of tfade is severely ill posed and further apply a spectral regularization method to solve it based on the solution given by the fourier method the convergence estimate is obtained under a priori bound assumptions for the exact solution numerical examples are given to show the effectiveness of the proposed numerical method

system identification under regular binary and quantized observations moderate deviations error bounds this technical note presents new results on probabilistic characterization of identification errors in their relationships to data sizes and accuracy requirements employing the moderate deviations principle this technical note shows that if the identification accuracy progressively increases with a suitable rate the probability of an estimate going outside the precision bounds decays exponentially with the data size the precise rate of the decaying probability is obtained system identification under regular binary and quantized observations are considered impact of unmodeled dynamics is also investigated

entropy and tikhonov based regularization techniques applied to the backwards heat equation abstract the goal of this paper is to analyze the performance of different regularization techniques for an inverse heat conduction problem ihcp the estimation of the initial condition the inverse problem is formulated as a nonlinear constrained optimization problem and a regularization term is added to the objective function with the help of a regularization parameter three classes of regularization methods have been considered tikhonov regularization maximum entropy principle and truncated singular value decomposition concerning the entropic methodology two new techniques are introduced and good results were obtained using synthetic data corrupted with noise the morozov s discrepancy principle is used to find out the regularization parameter

necessary conditions for exponential stabilizability of distributed parameter systems with infinite dimensional unbounded feedback abstract in this paper we consider distributed parameter control systems of the form x t ax t bu t where x t is in the state space x u t is in an infinite dimensional control space u and b is an unbounded input operator which is admissible in a standard sense we find necessary conditions for there to exist a compact operator k x u such that a bk generates a c 0 semigroup which is of exponential growth type in particular for every a must have at most finitely many eigenvalues of finite multiplicity in re z furthermore the part of a corresponding to the remaining spectrum must generate a semigroup with type at most we apply this to a model for a clamped euler bernoulli plate with dirichlet control applied to the entire boundary we show that a compact state feedback cannot exponentially stabilize the plate in the state space of maximum regularity

projections of gibbs measures may be non gibbsian consider the phase of the two dimensional nearest neighbor ferromagnetic ising model at a temperature belowt c let be the restriction of this measure to a coordinate axis we prove that there is no one dimensional translation invariant summable interaction for which is a gibbs measure this is proven by showing that if such an interaction existed would have large deviation properties different from those it actually has percolation methods are used in the proof

complete analyticity for 2d ising completed we study the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh we extend to every subcritical value of the temperature a result previously proven by martirosyan at low enough temperature and which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the bulk if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the bulk as a consequence we are able to complete the proof that complete analyticity for nice sets holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model the main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field and recently extended to all temperatures below the critical one by ioffe

necessary conditions for exact controllability with a finite dimensional input space we derive necessary conditions for exact controllability of infinite dimensional systems described by x ax bu where the generator a has a riesz basis of eigenvectors and the input space is finite dimensional these conditions are in terms of the eigenvalues of a and the degree of unboundedness of b but no further information about b is needed our results easily imply lack of exact controllability for a wide range of distributed parameter systems it is known from partial differential equation pde examples that the addition of damping into an exactly controllable second order system can destroy exact controllability we discuss the effect of damping on such systems using our general conditions we also give simple new proofs for results about the lack of exact controllability of hyperbolic systems in more than one space dimension

simplified tikhonov and fourier regularization methods on a general sideways parabolic equation the inverse heat conduction problem ihcp can be considered to be a sideways parabolic equation in the quarter plane and now the results available in the literature on ihcp mainly devoted to the standard sideways heat equation numerical methods have been developed also for more general equations but in most cases the stability theory and convergence proofs have not been generalized accordingly this paper remedies this by a simplified tikhonov and a new fourier regularization methods on a general sideways parabolic equation some known results for sideways heat equation are only the special case of the conclusions in this paper the numerical example shows that the computation effect is satisfactory

uniqueness of minimizer for countable markov shifts and equidistribution of periodic points for a finitely irreducible countable markov shift and a potential with summable variations we provide a condition on the associated pressure function which ensures that bowen s gibbs state the equilibrium state and the minimizer of the level 2 large deviations rate function are all unique and they coincide from this we deduce that the set of periodic points weighted with the potential equidistributes with respect to the gibbs equilibrium state as the periods tend to infinity applications are given to the gauss map and the bowen series map associated with a finitely generated free fuchsian group with parabolic elements

microcanonical distributions for lattice gases in this article a large deviation principle cf theorem 1 3 for the empirical distribution functional is applied to prove a rather general version of boltzmann s principle cf theorem 3 5 for models with shift invariant finite range potentials the final section contains an application of these considerations to the two dimensional ising model at sub critical temperature

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

entropy production and information gain in axiom a systems anosov systems are mathematical models for chaotic systems in statistical mechanics and fluid dynamics most of these systems enjoy the property of positive entropy production we introduce the concept of specific information gain or specific relative entropy h in anosov systems and prove that it is identical to the entropy production rate e p defined by ruelle and gallavotti in anosov systems from this point of view the entropy production rate e p characterizes the degree of macroscopic irreversibility of the system

exact observability of diagonal systems with a finite dimensional output operator in this paper equivalent conditions for exact observability of diagonal systems with a finite dimensional output operator are given one of these equivalent conditions is the conjecture of russell and weiss siam j control opt 32 1 1994 1 23 the other conditions are in terms of the eigenvalues and the lyapunov solutions of finite dimensional subsystems

an a posteriori parameter choice rule for the truncation regularization method for solving backward parabolic problems the goal of this paper is to investigate a backward parabolic equation with time dependent coefficient by the truncation method tm the key point to our paper is that we give an a posteriori choice of regularization parameter for the tm with which the convergence estimates between the exact solution and the regularized approximation are obtained numerical implementation sheds light on the accuracy and efficiency of the proposed method

large deviations and concentration properties for ϕ interface models we consider the massless field with zero boundary conditions outside d n d d n n d a suitable subset of d i e the continuous spin gibbs measure n on d n with hamiltonian given by h x y x y 1 v x y and x 0 for x d n c the interaction v is taken to be strictly convex and with bounded second derivative this is a standard effective model for a d 1 dimensional interface represents the height of the interface over the base d n due to the choice of scaling of the base we scale the height with the same factor by setting n n we study various concentration and relaxation properties of the family of random surfaces n and of the induced family of gradient fields n n as the discretization step 1 n tends to zero n in particular we prove a large deviation principle for n and show that the corresponding rate function is given by d u x dx where is the surface tension of the model this is a multidimensional version of the sample path large deviation principle we use this result to study the concentration properties of n under the volume constraint i e the constraint that 1 n d x dn n x stays in a neighborhood of a fixed volume v 0 and the hard wall constraint i e n x 0 for all x this is therefore a model for a droplet of volume v lying above a hard wall we prove that under these constraints the field n of rescaled heights concentrates around the solution of a variational problem involving the surface tension as it would be predicted by the phenomenological theory of phase boundaries our principal result however asserts local relaxation properties of the gradient field n n to the corresponding extremal gibbs states thus our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments the proofs have both probabilistic and analytic aspects essential analytic tools are p estimates for elliptic equations and the theory of young measures on the side of probability tools a central role is played by the helffer sjostrand 31 pde representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of t funaki and h spohn 25 on the structure of gradient fields

large deviation principle for empirical fields of log and riesz gases we study a system of n particles with logarithmic coulomb or riesz pairwise interactions confined by an external potential we examine a microscopic quantity the tagged empirical field for which we prove a large deviation principle at speed n the rate function is the sum of an entropy term the specific relative entropy and an energy term the renormalized energy introduced in previous works coupled by the temperature we deduce a variational property of the sine beta processes which arise in random matrix theory we also give a next to leading order expansion of the free energy of the system proving the existence of the thermodynamic limit

optimal error bound and fourier regularization for identifying an unknown source in the heat equation in this paper we investigate a problem of the identification of an unknown source from one supplementary temperature measurement at a given instant of time for the transient heat equation under an a priori condition we answer the question concerning the best possible accuracy for the problem the fourier regularization method is utilized for solving the problem and its convergent rate is analyzed numerical results are presented to illustrate the accuracy and efficiency of the method

large deviations for lattice systems i parametrized independent fields we prove large deviation theorems for empirical measures of independent random fields whose distributions depend measurably on an auxiliary parameter this dependence respects the action of the shift group and a large deviation principle holds whenever a certain ergodicity condition is satisfied we also investigate the entropy functions for these processes especially in relation to the usual relative entropy

a mollification method for ill posed problems a mollification method for a class of ill posed problems is suggested the idea of the method is very simple and natural if the data are given inexactly then we try to find a sequence of mollification operators which map the improper data into well posedness classes of the problem mollify the improper data within these mollified data our problem becomes well posed and when these facts are in hand we try to obtain error estimates and optimal or quasi optimal mollification parameters the method is working not only for problems in hilbert spaces but also for problems in banach spaces applications of the method to concrete problems like numerical differentiation parabolic equations backwards in time the cauchy problem for the laplace equation one and multidimensional non characteristic cauchy problems for parabolic equations in infinite or finite domains give us very sharp stability estimates of h older continuous type in these cases the method is optimal in the sense that it gives the same order of h older continuous dependence on the data as for the regularized problems furthermore the method may be implemented numerically using fast fourier transforms for the first time a uniform stability estimate of h older continuous type of the solution of the heat equation backwards in time in the space l p bbb r for all p in 1 infty could be established by our mollification method a new simple sharp pointwise estimate of h older type for the weak solution of a non characteristic cauchy problem for parabolic equations in a finite domain is established

stability and stabilizability of infinite dimensional systems in this paper we are concerned with stability problems for infinite dimensional systems first we review the theory for linear systems where the dynamics are governed by strongly continuous semigro

linear state space systems in infinite dimensional space the role and characterization of joint stabilizability detectability several fundamental results from the theory of linear state space systems in finite dimensional space are extended to encompass a class of linear state space systems in infinite dimensional space the results treated are those pertaining to the relationship between input output and internal stability the problem of dynamic output feedback stabilization and the concept of joint stabilizability detectability a complete structural characterization of jointly stabilizable detectable systems is obtained the generalized theory applies to a large class of linear state space systems assuming only that i the evolution of the state is governed by a strongly continuous semigroup of bounded linear operators ii the state space is hilbert space iii the input and output spaces are finite dimensional and iv the sensing and control operators are bounded general conclusions regarding the fundamental structure of control theoretic problems in infinite dimensional space can be drawn from these results

a modified method for a backward heat conduction problem we consider a backward heat conduction problem in a strip where data is given at the final time t t t 0 and a solution for 0 t t is sought the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data in order to numerically solve the problem we study a modification of the equation where a third order mixed derivative term is added error estimates for this problem are given which show that the modified problem is stable and its solution is an approximation of the backward heat conduction problem some numerical tests illustrate that the proposed method is feasible and effective c 2006 elsevier inc all rights reserved

an approximate solution for nonlinear backward parabolic equations we consider the backward parabolic equation ut au f t u t 0 t t u t g where a is a positive unbounded operator and f is a nonlinear function satisfying a lipschitz condition with an approximate datum g the problem is severely ill posed using the truncation method we propose a regularized solution which is the solution of a system of differential equations in finite dimensional subspaces according to some a priori assumptions on the regularity of the exact solution we obtain several explicit error estimates including an error estimate of holder type for all t 0 t an example on heat equations and numerical experiments are given

a numerical technique for backward inverse heat conduction problems in one dimensional space in this article we consider an inverse heat conduction problem in one dimensional space with two unknown terms in this problem the initial temperature and histories of temperature in the finite interval of time may be determined then by using a semi implicit finite difference method an estimation of these unknown terms will be found obviously this problem is ill posed but by applying a suitable parameter we can obtain the best of these estimations finally some numerical experiments are given in the final section of this paper

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

spectral properties of finite dimensional perturbed linear semigroups abstract many interesting cauchy problems arising in applications can be treated as finite dimensional perturbations of relatively simple problems in most applications these perturbations take place via boundary conditions it is the aim of this paper to investigate such cauchy problems and especially the spectral properties of the corresponding generators some general results are obtained that unify several well known results for functional differential equations and age dependent population dynamics applications to control problems as well as some nonlinear generalizations will appear in forthcoming papers

a non local boundary value problem method for parabolic equations backward in time abstract the ill posed parabolic equation backward in time u t a u 0 0 t t u t f subject to the constraint u 0 e with the positive self adjoint unbounded operator a and e 0 being given is regularized by the well posed non local boundary value problem u t a u 0 0 t t u 0 u t f 0 the error estimates of holder type of the regularized solutions are obtained these estimates improve the related results by mel nikova denche and bessila

detection of time varying heat sources using an analytic forward model abstract we present a simple analytic point source model for both static and time varying point like heat sources and the resulting temperature profile that solves the heat equation in dimension three simple algorithms to detect the location and spectral content of these sources are developed and numerically tested using finite element mesh simulations the resulting framework for heat source reconstruction problems which are ill posed inverse problems seems promising and warrants for future research possible fields of application for our work are material testing to detect manufacturing defects and medical imaging to detect abnormal health conditions

a moderate deviation principle for 2 d stochastic navier stokes equations driven by multiplicative levy noises abstract in this paper we establish a moderate deviation principle for two dimensional stochastic navier stokes equations driven by multiplicative levy noises the weak convergence method introduced by budhiraja dupuis and ganguly in 3 plays a key role

a backward parabolic equation with a time dependent coefficient regularization and error estimates we consider the problem of determining the temperature u x t for x t 0 p x 0 t in the parabolic equation with a time dependent coefficient this problem is severely ill posed i e the solution if it exists does not depend continuously on the given data in this paper we use a modified method for regularizing the problem and derive an optimal stability estimation a numerical experiment is presented for illustrating the estimate

numerical solution of forward and backward problem for 2 d heat conduction equation for a two dimensional heat conduction problem we consider its initial boundary value problem and the related inverse problem of determining the initial temperature distribution from transient temperature measurements the conditional stability for this inverse problem and the error analysis for the tikhonov regularization are presented an implicit inversion method which is based on the regularization technique and the successive over relaxation sor iteration process is established due to the explicit difference scheme for a direct heat problem developed in this paper the inversion process is very efficient while the application of sor technique makes our inversion convergent rapidly numerical results illustrating our method are also given

optimal filtering for the backward heat equation for the backward heat equation stabilized by an a priori initial bound an estimator is determined for intermediate values that is optimal with respect to the bound and the observation accuracy it is shown how this may be implemented computationally with error estimates for the computed approximation which can be made arbitrarily close to the uncertainty level induced by the ill posedness of the underlying problem thus the feasibility of this for practical computation inevitably severely limited by that inherent uncertainty is as good as possible

surface tension of binary mixtures an exact relation within sos like models we investigate the statistical mechanics of binary mixtures within the framework of a two component sos model where we obtain an exact formula for the free energy this allows for an explicit expression linking the surface tension of a binary mixture to those of its components which is valid at all temperatures answering an open question asked by de coninck et al 2005 j stat phys 119 597 642

rank dependent moderate deviations of u empirical measures in strong topologies we prove a rank dependent moderate deviation principle for u empirical mea sures where the underlying i i d random variables take values in a measurable not neces sarily polish space s s the result can be formulated on a suitable subset of all signed measures on s m s m we endow this space with a topology which is stronger than the usual topology a moderate deviation principle for banach space valued u statistics is obtained as a particular application the advantage of our result is that we obtain in the degenerate case moderate deviations in non gaussian situations with non convex rate func tions

heuristic parameter choice rules for convex variational regularization based on error estimates in this paper we are interested in heuristic parameter choice rules for general convex variational regularization which are based on error estimates two such rules are derived and generalize those from quadratic regularization namely the hanke raus rule and quasi optimality criterion a posteriori error estimates are shown for the hanke raus rule and convergence for both rules is also discussed numerical results for both rules are presented to illustrate their applicability