fluctuation theory of connectivities for subcritical random cluster models we develop a fluctuation theory of connectivities for subcritical random cluster models the theory is based on a comprehensive nonperturbative probabilistic description of long connected clusters in terms of essentially one dimensional chains of irreducible objects statistics of local observables for example displacement over such chains obey classical limit laws and our construction leads to an effective random walk representation of percolation clusters the results include a derivation of a sharp ornstein zernike type asymptotic formula for two point functions a proof of analyticity and strict convexity of inverse correlation length and a proof of an invariance principle for connected clusters under diffusive scaling in two dimensions duality considerations enable a reformulation of these results for supercritical nearest neighbor random cluster measures in particular for nearest neighbor potts models in the phase transition regime accordingly we prove that in two dimensions potts equilibrium crystal shapes are always analytic and strictly convex and that the interfaces between different phases are always diffusive thus no roughening transition is possible in the whole regime where our results apply our results hold under an assumption of exponential decay of finite volume wired connectivities assumption 1 2 below in rectangular domains that is conjectured to hold in the whole subcritical regime the latter is known to be true in any dimensions when q 1 q 2 and when q is sufficiently large in two dimensions assumption 1 2 holds whenever there is an exponential decay of connectivities in the infinite volume measure by duality this includes all supercritical nearest neighbor potts models with positive surface tension between ordered phases

harmonic crystal on the wall a microscopic approach a three dimensional winterbottom type construction in the regime of partial wetting is derived in a scaling limit of a gas of microscopic gaussian sos droplets under the fixed volume constraint the proof is based on a coarse graining of the random microscopic region wetted by the crystal random walk representations of various quantities related to free massless fields and a stability analysis of the torsional rigidity problem

a subrate weighted minimum variance control law the paper describes the development of a weighted minimum variance control law with application to single input single output systems in which the control is applied at a slower rate than the available measurements of the plant output signal the theory is extended to incorporate dynamic compensation in the controller and to encompass generalized system models a time domain modelling approach is employed whereby the system dynamics and sampling operations are defined as a series of cyclically time varying difference equations the technique in which the cost functions penalize both the control and a set of output signals generates straightforward expressions for the closed loop characteristic polynomial and is accordant with pole placement synthesis methods the control law suffers from similar difficulties with respect to non minimum phase systems as its single rate counterpart

convexity properties of the surface tension and equilibrium crystals we study the thermodynamic limit of the orientation dependent surface tension under general conditions which we show to hold true for a large class of lattice systems we prove that the limit exists and that it satisfies some convexity properties related to the pyramidal inequality introduced by r l dobrushin and s b shlosman 1 we discuss some consequences of these results for the equilibrium crystal shape

equilibrium shapes of crystals attached to walls we discuss equilibrium shapes of crystals attached to walls optimal shapes for different configurations of walls are found and the minimality of the overall surface tension is proven with the help of a simple geometrical argument based on the isoperimetric inequality and monotonicity stability results in the form of bonnesen inequalities are obtained in the two dimensional case

sos approximants for potts crystal shapes abstract an approximation made when replacing the orientation dependent potts model surface tension by the sos one is rigorously estimated as a result an sos model may be used for an explicit and precise construction of crystal and meniscus shapes for potts models at low temperatures

gmv control of non linear continuous time systems including common delays and state space models a non linear generalized minimum variance control law is proposed for the control of non linear continuous time multivariable systems with common delays on input and output channels the quadratic cost index involves both error and control signal costing terms the solution for the control law is obtained using a non linear operator representation of the plant and a linear state equation model for the disturbance and reference models the reference and disturbance models are represented by linear subsystems however the plant model can be in a very general non linear operator form which could involve state space transfer operators or non linear function look up tables the structure of the system and criterion is chosen so that a simple controller structure and solution is obtained the controller obtained is simple to implement particularly in one form which might be considered to be a state space version of a non linear smith predictor the results are related to those for discrete time systems but

a control weighted minimum variance controller for non minimum phase systems a now minimum variance type of control law is derived for use in self tuning control schemes where the plant is non minimum phase the controller has some of the features of the minimum variance controller and is related to the generalized or minimum variance controller which includes control weighting terms the controller has the advantage that when the control weighting scalar tends to zero the system is assured of being asymptotically stable the generalized minimum variance controller employed in the self tuning control law results in an unstable closed loop system in this situation the controller may also be used to stabilize plants which are both unstable and non minimum phase the closed loop system corresponding to the generalized minimum variance controller may be unstable for all values of the weighting scalar in this situation

on the mixing time of the 2d stochastic ising model with plus boundary conditions at low temperature we consider the glauber dynamics for the 2d ising model in a box of side l at inverse temperature and random boundary conditions whose distribution p either stochastically dominates the extremal plus phase hence the quotation marks in the title or is stochastically dominated by the extremal minus phase a particular case is when p is concentrated on the homogeneous configuration identically equal to equal to for large enough we show that for any varepsilon 0 there exists c c beta varepsilon such that the corresponding mixing time tmix satisfies rm lim l to infty bf p left t rm mix ge rm exp cl varepsilon right 0 in the non random case or this implies that t rm mix le rm exp cl varepsilon the same bound holds when the boundary conditions are all on three sides and all on the remaining one the result although still very far from the expected lifshitz behavior tmix o l2 considerably improves upon the previous known estimates of the form t rm mix le rm exp c l frac 12 varepsilon the techniques are based on induction over length scales combined with a judicious use of the so called censoring inequality of y peres and p winkler which in a sense allows us to guide the dynamics to its equilibrium measure

on the microscopic validity of the wulff construction and of the generalized young equation for a large class of 1 1 dimensional interfaces of the solid on solid type we prove on a microscopic basis the validity of the wulff construction and of the generalized young equation which gives the contact angle of a sessile drop on a wall our proof relies on a new method to treat random walks with a finite number of global constraints

motion by mean curvature from the ginzburg landau interface model we consider the scalar field t with a reversible stochastic dynamics which is defined by the standard dirichlet form relative to the gibbs measure with formal energy the potential v is even and strictly convex we prove that under a suitable large scale limit the t field becomes deterministic such that locally its normal velocity is proportional to its mean curvature except for some anisotropy effects as an essential input we prove that for every tilt there is a unique shift invariant ergodic gibbs measure for the field

multivariable weighted minimum variance self tuning controllers abstract the weighted minimum variance wmv controller is employed in a self tuning control system for plants which can be multivariate and non square both explicit and implicit self tuning strategies are discussed existing strategies based upon k steps ahead cost functions have stability problems on systems whose open loop dynamics are both unstable and non minimum phase the wmv controller has the advantage that when plant parameters are known and the control weighting is zero the optimal system is guaranteed to be stable

rigorous probabilistic analysis of equilibrium crystal shapes the rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade we review here the main results that have been obtained both in two and higher dimensions in particular we describe how the phenomenological wulff and winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases we focus on the main conceptual issues and describe the central ideas of the existing approaches

rigorous generalization of young s law for heterogeneous and rough substrates we consider a sos type model of interfaces on a substrate which is both heterogeneous and rough we first show that for appropriate values of the parameters the differential wall tension that governs wetting on such a substrate satisfies a generalized law which combines both cassie and wenzel laws then in the case of an homogeneous substrate we show that this differential wall tension satisfies either the wenzel s law or the cassie s law according to the values of the parameters

paper self tuning control of nonminimum phase systems the instability that arises from the pole zero cancellation of elementary self tuners and most mrac algorithms has given rise to a misapprehension that nonminimum phase systems can pose insoluble problems with the application of adaptive control since most practical processes exhibit some form of nonminimum phase behaviour particularly when controlled digitally this appeared to restrict the usefulness of self tuning methods however during the last few years new algorithms have been suggested which partly overcome these difficulties and have been shown to be effective in industrial trials the paper describes the practical features that a self tuning controller must possess and discusses how plant dead time and excess continuous time poles can lead to discrete time nonminimum phase zeros the source of instability of elementary self tuners is analysed and various suggested approaches for overcoming this problem are reviewed these include the use of a generalized minimum variance cost function incorporating control weighting factorization methods which avoid cancellation of the offending zeros adaptive control based on state space linear quadratic gaussian lqg theory and explicit pole placement via solution of a diophantine or bezoutian equation gawthrop s hybrid controller which avoids those nonminimum phase zeros due to continuous time pole excess is briefly discussed a series of simulations in which a nonminimum phase plant is subjected to a pattern of set point changes is presented to show some of the features of the algorithms in applications the order of the process is unlikely to be known or indeed be finite the pole placement algorithm which is particularly effective in the known order case is shown to lack robustness to underspecified order the use of a large control weighting however restores stability to the generalized minimum variance controller the conclusions are that with appropriate design and engineering judgement adaptive control is viable even for nonminimum phase processes

the low temperature expansion of the wulff crystal in the 3d ising model we compute the expansion of the surface tension of the 3d random cluster model for q 1 in the limit where p goes to 1 we also compute the asymptotic shape of a plane partition of n as n goes to this same shape determines the wulff crystal to order o in the 3d ising model and more generally in the 3d random cluster model for q 1 at temperature

on the wulff crystal in the ising model we study the phase separation phenomenon in the ising model in dimensions d 3 to this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization m we confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase moreover the rescaled droplet is asymptotically close to a definite deterministic shape the wulff crystal which minimizes the surface free energy in the course of the proof we establish a surface order large deviation principle for the magnetization our results are valid for temperatures t below a limit of slab thresholds t c conjectured to agree with the critical point t c moreover t should be such that there exist only two extremal translation invariant gibbs states at that temperature a property which can fail for at most countably many values and which is conjectured to be true for every t the proofs are based on the fortuin kasteleyn representation of the ising model along with coarse graining techniques to handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis finally we propose a heuristic picture that for subcritical temperatures close enough to t c the dominant minus spin cluster of the wulff droplet permeates the entire box and has a strictly positive local density everywhere

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

weighted minimum variance self tuning control a weighted minimum variance controller has recently been defined which is based upon a k steps ahead control law and which may be used for unstable and non mimimum phase plants the controller includes control weighting and this results in relatively smooth control signals the controller is also stable for ranges of control weighting where other control laws are unstable both explicit and implicit self tuning algorithms are proposed based upon this controller some of the algorithms use bilinear identification methods developed by astrom for use in his pole zero placement algorithms the self tuning controller is related to that developed by clarke and gawthrop when the system is minimum phase

triangular sos models and cubic crystal shapes a solid on solid sos model in a field h conjugate to the orientation of the surface is exactly solved with the aid of pfaffians the free energy h directly gives the equilibrium shape of a finite crystal the phase diagram exhibits rough and smooth phases corresponding to rounded and flat portions of the crystal surface the solid on solid model undergoes transitions of the pokrovsky talapov type 1979 characterised by a specific heat exponent alpha 1 2 one special point of the phase diagram corresponds to the appearance of a facet via an alpha 0 transition height height correlations are derived along a special line in the phase diagram with the aid of the known equivalence of this sos model with an ising model several exponents can be translated from one model to the other this enables one to derive the topology of the phase diagram of the antiferromagnetic triangular ising model with first and second neighbour couplings in a field

global structure stability of riemann solutions of quasilinear hyperbolic systems of conservation laws shocks and contact discontinuities abstract in this paper the author proves the global structure stability of the lax s riemann solution u u x t containing only shocks and contact discontinuities of general n n quasilinear hyperbolic system of conservation laws more precisely the author proves the global existence and uniqueness of the piecewise c1 solution u u t x of a class of generalized riemann problem which can be regarded as a perturbation of the corresponding riemann problem for the quasilinear hyperbolic system of conservation laws moreover this solution has a global structure similar to that of the solution u u x t combining the results in kong global structure instability of riemann solutions of quasilinear hyperbolic systems of conservation laws rarefaction waves to appear the author proves that the lax s riemann solution of general n n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non degenerate shocks and contact discontinuities but no rarefaction waves and other weak discontinuities

a generalized lqg approach to self tuning control part i aspects of design implicit self tuners based on single stage cost function minimization schemes require exact knowledge of the dead time of the controlled plant explicit pole placement self tuners on the other hand though robust against variations of the time delay are very sensitive to the assumptions made about the model order of the plant wrong assumptions may result in very poor performance an explicit method based on minimization of a multistage cost function and a carima plant model formulated in state space is presented the method extends the design features of the generalized minimum variance scheme to the multistage case these can be used to tailor the closed loop response of the system the approach is shown to be robust against wrong a priori assumptions made about the plant dead time or order thus maintaining good servo and disturbance rejection properties see part ii

paper self tuning leader follower games the optimization of a class of linear discrete time stochastic systems with unknown parameters and multiple decision makers or controllers each having his own single stage quadratic objective is considered based on a centralized information pattern a steady state solution is obtained for the stochastic adaptive leader follower game problem it is shown that this class of leader follower game problems can be transformed to an equivalent class of multiple input multiple output self tuning control problems with a single decision maker the convergence of the multiple input multiple output self tuning controller is established and applied to the leader follower game a numerical simulation illustrates the proposed self tuning algorithm for an adaptive leader follower strategy