diffusion in poro elastic media existence uniqueness and regularity theory is developed for a general initial boundary value problem for a system of partial differential equations which describes the biot consolidation model in poro elasticity as well as a coupled quasi static problem in thermoelasticity additional effects of secondary consolidation and pore fluid exposure on the boundary are included this quasi static system is resolved as an application of the theory of linear degenerate evolution equations in hilbert space and this leads to a precise description of the dynamics of the system

the covariance matrix of the potts model a random cluster analysis we consider the covariance matrix g mm q2 of thed dimensionalq states potts model rewriting it in the random cluster representation of fortuin and kasteleyn in any of theq ordered phases we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances thereby attributing algebraic significance to these stochastic geometric quantities we also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers for dimensiond 2 we show that this correlation length and the correlation length of the two point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two for systems with first order transitions this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self dual point o for systems with second order transitions this relation implies the equality of the correlation length exponents from above and below threshold as well as an amplitude ratio of two in the course of proving the above results we establish several properties of independent interest including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions and left continuity of the two dimensional free boundary condition percolation probability at o we also introduce dlr equations for the random cluster model and use them to establish ergodicity of the free measure in order to prove these results we introduce a new class of events which we call decoupling events and two inequalities for these events the first is similar to the fkg inequality but holds for events which are neither increasing nor decreasing the second is similar to the van den berg kesten inequality in standard percolation both inequalities hold for an arbitrary fkg measure

cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

on discrete time linear systems with almost periodic kernels the paper considers several different stability notions for discrete time linear time varying systems specified by almost periodic kernels various input output stability notions are analysed including those in which the signal spaces consist of bounded power and asymptotically almost periodic functions further it is proved that every stable time varying discrete time system commuting with a polynomial in the shift operator is periodic an example shows that this result does not hold for unstable time varying systems

finite size scaling for potts models recently borgs and koteck developed a rigorous theory of finite size effects near first order phase transitions here we apply this theory to the ferromagneticq state potts model which forq large andd 2 undergoes a first order phase transition as the inverse temperature is varied we prove a formula for the internal energy in a periodic cube of side lengthl which describes the rounding of the infinite volume jump e in terms of a hyperbolic tangent and show that the position of the maximum of the specific heat is shifted by m l inq e l d o l 2d with respect to the infinite volume transition point t we also propose an alternative definition of the finite volume transition temperature t l which might be useful for numerical calculations because it differs only by exponentially small corrections from t

interfaces for random cluster models a random cluster measure on mathbb z d that is not translationally invariant is constructed for d 3 the critical density pc and sufficiently large q the resulting measure is proven to be a gibbs state satisfying cluster model dlr equations

graphical representations for ising and potts models in general external fields this work is concerned with the theory of graphical representation for the ising and potts models over general lattices with non translation invariant external field we explicitly describe in terms of the random cluster representation the distribution function and consequently the expected value of a single spin for the ising and q state potts models with general external fields we also consider the gibbs states for the edwards sokal representation of the potts model with non translation invariant magnetic field and prove a version of the fkg inequality for the so called general random cluster model grc model with free and wired boundary conditions in the non translation invariant case adding the amenability hypothesis on the lattice we obtain the uniqueness of the infinite connected component and the almost sure quasilocality of the gibbs measures for the grc model with such general magnetic fields as a final application of the theory developed we show the uniqueness of the gibbs measures for the ferromagnetic ising model with a positive power law decay magnetic field with small enough power as conjectured in bissacot et al commun math phys 337 41 53 2015

local global double algebras for slow h sup infinity adaptation i inversion and stability for pt ii see ibid vol 36 no 2 p 143 51 1991 in this two part work a common algebraic framework is introduced for the frozen time analysis of stability and h sup infinity optimization in slowly time varying systems based on the notion of a normed algebra on which local and global products are defined relations between local stability local near optimality local coprime factorization and global versions of these properties are sought the framework is valid for time domain disturbances in l sup infinity h sup infinity behavior is related to l sup infinity input output behavior via the device of an approximate isometry between frequency and time domain norms the authors presently elaborate on the double algebra concept for volterra operators which approximately commute with the shift and summarize the main algebraic properties and norm inequalities local conditions for global invertibility are also obtained classical frozen time stability conditions are incorporated in relations between local and global spectra

gibbs states of graphical representations of the potts model with external fields we consider the ferromagnetic q state potts model with each of the q spin values coupled to an external field we also introduce a generalized random cluster model which includes both the potts model in arbitrary homogeneous external fields and the noninteger q random cluster model as special cases we establish the fkg property the finite energy condition uniqueness of the infinite cluster and gibbsianness of limit states for this generalized model furthermore we develop the theory of gibbs states for the edwards sokal representation of the potts model in a field and relate the phase structure in this representation to those in the spin and random cluster representations finally we characterize the possible color s of the infinite cluster s and show that the correspondence between edwards sokal gibbs states and their random cluster marginals is bijective once the color of the infinite cluster is fixed

a locking free numerical approximation for quasilinear poroelasticity problems abstract this paper is devoted to a new formulation which couples the weak galerkin method and the finite element method for approximating solutions of the equations of quasi static poroelasticity which model flow through elastic porous media it is assumed that the permeability of the elastic matrix depends nonlinearly on the dilatation of the porous medium the steady state version of the system is recast in terms of displacement pressure and volumetric stress and the well posedness of both the continuous and discrete three field formulations is proved error estimates for the proposed numerical method are obtained these show that the method is locking free numerical experiments presented further demonstrate the accuracy and the locking free characteristic of the proposed numerical method

dobrushin interfaces via reflection positivity we study the interfaces separating different phases of 3d systems by means of the reflection positivity method we treat discrete non linear sigma models which exhibit power law decay of correlations at low temperatures and we prove the rigidity property of the interface our method is applicable to the ising and potts models where it simplifies the derivation of some known results the method also works for large entropy systems of continuous spins

robust identification and galois sequences abstract worst case l1 identification is studied for bibo stable linear shift invariant systems it is shown that the chebyshev identification method when used with galois input designs satisfies a certain robust convergence property and provides l1 model error bounds in worst case identification of bibo stable systems with a uniformly bounded noise set up the robust identification methodology developed is compatible with the modelling requirements of modern robust control design

model validation a connection between robust control and identification the gap between the models used in control synthesis and those obtained from identification experiments is considered by investigating the connection between uncertain models and data the model validation problem addressed is given experimental data and a model with both additive noise and norm bounded perturbations is it possible that the model could produce the observed input output data this problem is studied for the standard h sub infinity mu framework models a necessary condition for such a model to describe an experimental datum is obtained for a large class of models in the robust control framework this condition is computable as the solution of a quadratic optimization problem

brief on robustness in system identification this paper studies robustness issues in system identification specifically a general framework for robust convergence analysis is given for identification of stable linear time invariant systems this is used to derive a rather complete set of results on robust convergence under average l p norms orlicz norms and cross correlation type noise semi norms the new theory is used to discuss and to study robustness issues in identification of linear time invariant models when the true plant is not exactly linear robustness issues in system identification against unmodelled nonlinear dynamics are important as real plants exhibit typically at best nearly linear dynamics the results show clearly that the effects of unmodelled dynamics are typically more serious than those of random noise

h optimization and optimal rejection of persistent disturbances abstract in this note it is shown that in a certain well known normed space of persistent signals a natural optimal disturbance rejection problem is equivalent to h optimization thus clarifying certain misconceptions on h optimization in the literature these conclusions are further strengthened using techniques of wiener s generalized harmonic analysis

rigidity of the interface in percolation and random cluster models we study conditioned random cluster measures with edge parameter p and cluster weighting factor q satisfying q 1 the conditioning corresponds to mixed boundary conditions for a spin model interfaces may be defined in the sense of dobrushin and these are proved to be rigid in the thermodynamic limit in three dimensions and for sufficiently large values of p this implies the existence of non translation invariant conditioned random cluster measures in three dimensions the results are valid in the special case q 1 thus indicating a property of three dimensional percolation not previously noted

time domain identification for robust control abstract in this paper we are concerned with a problem of robust control oriented system identification in the time domain based on the well known schur takagi aak theorem we propose a linear algorithm to obtain the nominal model of the plant to be identified and the minimal bound of the uncertainty of the nominal model error which is measured by h norm it is also shown that in the model set defined by the nominal model and the uncertainty bound there exists at least one model which matches the prescribed input output data given in the time domain

the necessity of the small gain theorem for time varying and nonlinear systems it is shown that an appropriately modified version of the small gain theorem which uses the asymptotic gain of a fading memory system leads to a necessary and sufficient condition for the stable invertibility of certain feedback operators this modified small gain theorem is then used to obtain conditions for robust finite energy input output stability of 1 nonlinear plants subject to nonlinear time varying perturbations and 2 linear time varying plants subject to linear time varying perturbations these conditions are shown to be necessary as well as sufficient

an efficient numerical scheme for the biharmonic equation by weak galerkin finite element methods on polygonal or polyhedral meshes this paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak galerkin wg finite element methods the wg finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual h 2 semi norm weak partial derivatives and their approximations called discrete weak partial derivatives are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra the discrete weak partial derivatives serve as building blocks for the wg finite element method the resulting matrix from the wg method is symmetric positive definite and parameter free an error estimate of optimal order is derived in an h 2 equivalent norm for the wg finite element solutions error estimates in the usual l 2 norm are established yielding optimal order of convergence for all the wg finite element algorithms except the one corresponding to the lowest order i e piecewise quadratic elements some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme

modeling of linear fading memory systems motivated by questions of approximate modeling and identification we consider various classes of linear time varying bounded input bounded output bibo stable fading memory systems and the characterizations are proved these include fading memory systems in general almost periodic systems and asymptotically periodic systems we also show that the norm and strong convergence coincide for bibo stable causal fading memory systems

interfaces in the potts model ii antonov s rule and rigidity of the order disorder interface within the ferromagneticq state potts model we discuss the wetting of the interface between two ordered phasesa andb by the disordered phasef at the transition temperature in two or more dimensions and forq large we establish the validity of the antonov s rule ab af fb where denotes the surface tension between the considered phases we also prove that at this temperature in three or more dimensions the interface between any ordered phase and the disordered one is rigid

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

a time domain approach to model validation in this paper we offer a novel approach to control oriented model validation problems the problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input output data our approach directly uses time domain input output data to validate uncertainty models the algorithms we develop are computationally tractable and reduce to generally nondifferentiable convex feasibility problems or to linear programming problems in special cases we give analytical solutions to these problems

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

estimation of model quality this paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems this is a very large field we therefore concentrate on approaches that have been motivated by the need for reliable models for control system design this will involve a discussion of efforts that go under the titles of estimation in th worst case estimation estimation in l1 and stochastic embedding of undermodelling a central theme of this survey is to examine these new methods with reference to the classic bias variance tradeoff in model structure selection

fading memory feedback systems and robust stability abstract this paper considers fading memory for nonlinear time varying systems and associated problems of robust stability we define two notions of fading memory for stable dynamical systems uniform and pointwise we then provide conditions under which stable linear or nonlinear systems exhibit uniform or pointwise fading memory in particular we show that 1 all stable discrete time linear time varying ltv systems have uniform fading memory 2 all stable continuous time ltv systems have pointwise fading memory and 3 stable finite dimensional continuous time ltv systems have uniform fading memory we then show that a version of the small gain theorem which employs the asymptotic gain of a fading memory system is necessary for the stable invertibility of certain feedback operators these results are presented for both continuous time and discrete time systems using general lp or l p notions of input output stability and generalize existing results for l2 stability we further investigate fading memory in a closed loop context for linear plants we parametrize all nonlinear controllers which lead to closed loop pointwise fading memory

lethargy results in lti system modelling modelling of linear time invariant lti systems is studied in an approximate modelling framework motivated by modern robust control theory as convolution type lti systems are represented by their unit impulse responses i e through a sequence of real complex numbers this paper provides also system theoretic results on modelling of sequences and their associated transfer functions the modelling results derived in this paper are mostly of lethargy type i e they establish limitations in approximate modelling of lti systems non existence of coprime factorizations is established for a number of different types of lti systems this leads to the somewhat surprising conclusion that many lti systems are not even stabilizable and that an even larger class of lti systems cannot be well approximated by rational transfer function models in addition it is discussed that often the rate of rational approximation of lti systems is not essentially better than the rate of approximation in some basis such as finite impulse response fir models furthermore different definitions of the important finite power signal setup are studied and new results given

interfaces in the potts model i pirogov sinai theory of the fortuin kasteleyn representation we develop a new analysis of the order disorder transition in ferromagnetic potts models for large numberq of spin states we use the pirogov sinai theory which we adapt to the fortuin kasteleyn representation of the models this theory applies in a rather direct way in our approach and leads to a system of non interacting contours with small activities as a consequence simpler and more natural techniques are found allowing us to recover previous results on the bulk properties of the model which then extend to non integer values ofq and to deal with non translation invariant boundary conditions this will be applied in a second part of this work to study the behaviour of the interfaces at the transition point

almost sure quasilocality in the random cluster model keywords random cluster model non gibbs states quasilocality of conditional distributions reference gr pf article 1995 005doi 10 1007 bf02184883 record created on 2008 11 27 modified on 2017 05 12

robust approximate modelling of stable linear systems robust approximation and worst case approximate modelling of stable shift invariant systems from corrupted transfer function estimates are studied in the h sense connections between the problem formulations of the present work and certain problems of worst case system identification notably the helmicki jacobson nett problem formulation for identification in h are established issues of model set selection are addressed using the n width concept a concrete result establishes a priori knowledge for which a certain rational model set is optimal in then width sense a notion of robust convergence is defined so that any untuned approximation method satisfying it has a generic well posedness property for systems in the disk algebra the existence of robustly convergent approximation methods based on any complete model set in the disk algebra is shown in a constructive way a framework is given in which approximate models can be obtained as stable perturbations of the true system these can be combined

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large