on the l p solvability of higher order parabolic and elliptic systems with bmo coefficients we prove the solvability in sobolev spaces for both divergence and non divergence form higher order parabolic and elliptic systems in the whole space on a half space and on a bounded domain the leading coefficients are assumed to be merely measurable only in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders for the proof we develop a set of new techniques to produce mean oscillation estimates for systems on a half space

frequency domain design of pid controllers for stable and unstable systems with time delay an approach for tuning pid type controllers is developed for single input single output linear time invariant systems based on an extension to the method of d partition this method permits design for simultaneous minimum gain and phase margin requirements it also allows design for specified maximum gain and phase cross over frequencies of the controlled system the technique can be applied to systems with stable or unstable plants as well as to irrational systems with significant time delay another advantage of the method is that it can be used for various controller configurations including derivative in the feedback path three examples illustrate the tuning method

uniformly nondegenerate elliptic equations with partially bmo coefficients in nonsmooth domains abstract a global estimate in weighted lorentz sobolev spaces is obtained for the weak solutions to divergence form uniformly nondegenerate elliptic equations over a bounded nonsmooth domain here the leading coefficients are assumed to be merely measurable in one variable and have small bmo semi norms in the remaining variables under the assumption that one variable direction is perpendicular to the boundary points which is close to the boundary while a geometric assumption on the boundary is a locally bounded reifenberg flatness in addition we also investigate regularities in lorentz morrey morrey and holder spaces for elliptic equations under the same assumptions on the leading coefficients and the boundary of domain

second order eliptic and parabolic equations with b r2 v mo coefficients the solvability in sobolev spaces w 1 2 p is proved for nondivergence form second order parabolic equations for p 2 close to 2 the leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables and almost vmo vanishing mean oscillation with respect to the other coordinates this implies the w 2 p solvability for the same p of nondivergence form elliptic equations with leading coefficients measurable in two coordinates and vmo in the others under slightly different assumptions we also obtain the solvability results when p 2

brief paper a parameter space approach to constrained variance pid controller design this paper presents a parameter space approach to constrained variance and minimum variance pid controller design for lti models the technique is based on rational transfer functions of the plant and noise models loci corresponding to a fixed variance can be mapped into parameter planes for pid and pi type controllers to graphically display regions which satisfy the constraint thereby guiding a design to appropriate controller gains requirements for tracking regulation can be readily achieved without excessively increasing the output variance the usual advantages of parameter space techniques apply where other design criteria may be superimposed allowing multiple objectives to be achieved non conservatively the superposition of the parameter space boundaries from existing robust control techniques thus allows non conservative robust minimum variance pid design a design example compares the technique to an algebraic minimum variance design using an integrator when tracking is required

elliptic equations with measurable coefficients in reifenberg domains abstract we prove w 1 p estimates for elliptic equations in divergence form under the assumption that for each point and for each sufficiently small scale there is a coordinate system so that the coefficients have small oscillation in n 1 directions we assume the boundary to be reifenberg flat and the coefficients having small oscillation in the flat direction of the boundary

comments on a regulating and tracking pi d controller eitelberg 1987 presents a modification of the classical pid controller which amounts to a two degree of freedom structure beside the fact that this modification is already known from the literature its potentials are not fully exploited by eitelberg based on a closed loop interpretation of the approach a systematic tuning procedure and an additional extension for pi controllers are presented

higher order nondivergence elliptic and parabolic equations in sobolev spaces and orlicz spaces abstract in this paper we obtain the global regularity estimates of the solutions in sobolev spaces and orlicz spaces for higher order elliptic and parabolic equations of nondivergence form in the whole space we only need to focus on the parabolic case since the corresponding result in the elliptic case can be obtained as a corollary

on 1 f noise and other distributions with long tails abstract it is shown following shockley shockley w 1957 proc ire 45 279 290 that when a population is engaged in tasks whose completion requires the successful conclusion of many independent subtasks the distribution function for successes in the primary task is log normal it is also shown that when the dispersion of the log normal distribution is large the distribution is mimicked by a 1 x distribution over a wide range of x this argument provides a generic set of processes that yields the much observed 1 x distribution and will also lead to a 1 f noise spectrum it is commonly found that distributions that seem to be log normal over a broad range say to the 95th percentile of a population change to an inverse fractional power pareto distribution for the last few percentile annual income distributions are examples with this structure the very wealthy generally achieve their superwealth through amplification processes that are not available to most we have introduced a simple amplification model to characterize the transition from a log normal distribution to an inverse power pareto tail

levy stable probability densities and mechanical relaxation in solid polymers early investigations by weber r and f kohlrausch maxwell and boltzmann of relaxation in viscoelastic solids are reviewed a two state model stress tensor describing strain coupling to internal conformations of a polymer chain is used to derive a linear response version of the boltzmann superposition principle for shear stress relaxation the relaxation function of kohlrausch t exp t is identical to the williams watts empirical dielectric relaxation function and in the model corresponds to the autocorrelation function of a segment s differential shape anisotropy tensor by analogy with the dielectric problem exp t is interpreted as the survival probability of a frozen segment in a swarm of hopping defects with a stable waiting time distributionat for defect motion the exponent a is the fractal dimension of a hierarchical scaling set of defect hopping times integral transforms of t needed for data analysis are evaluated the cosine and inverse laplace transforms are stable probability densities the reciprocal kernel for short time compliance is discussed

gradient estimates for higher order elliptic equations on nonsmooth domains abstract we establish optimal gradient estimates in orlicz space for a nonhomogeneous elliptic equation of higher order with discontinuous coefficients on a nonsmooth domain our assumption is that for each point and for each sufficiently small scale the coefficients have small mean oscillation and the boundary of the domain is sufficiently close to a hyperplane as a consequence we prove the classical w m p m 1 2 1 p estimates for such a higher order equation our results easily extend to higher order elliptic and parabolic systems

cauchy noise and affiliated stochastic processes by departing from the previous attempt phys rev e 51 4114 1995 we give a detailed construction of conditional and perturbed markov processes under the assumption that the cauchy law of probability replaces the gaussian law appropriate for the wiener process as the model of primordial noise all considered processes are regarded as probabilistic solutions of the so called schrodinger interpolation problem whose validity is thus extended to the jump type processes and their step process approximants

applications of extreme value statistics in physics we present a descriptive review of physical problems dealing with extreme values in several fields of physics we consider different physical situations involving random variables that are correlated or not and study the statistics of extremal variables which is relevant for situations where height fluctuations catastrophic events such as material failure or power outrage occur we describe the general theory and relate the cumulative limit distributions that can be accessible in experiments to microscopic models in many cases however the random variables are correlated in interface problems for example and the characteristics of the interaction are revealed in the asymptotic behavior of the limit distribution

exact propagator and eigenfunctions for multistable models with arbitrarily prescribed n lowest eigenvalues the author presents a method to construct potentials for schrodinger equations with some prescribed features for which all eigenfunctions and the time dependent propagator can be explicitly calculated the prescribed features can be formulated by choosing arbitrarily the n lowest eigenvalues alternatively one can prescribe some qualitative behaviour for the potential like the number and relative depths of wells and barriers the results can also be applied to the construction of fokker planck models with prescribed properties and explicitly calculable transition probability density the method is based on ideas of supersymmetric quantum mechanics and the theory of solitons that can be traced back to the work of darboux 1894 and crum 1955

weighted lq estimates for stationary stokes system with partially bmo coefficients abstract we prove the unique solvability of solutions in sobolev spaces to the stationary stokes system on a bounded reifenberg flat domain when the coefficients are partially bmo functions i e locally they are merely measurable in one direction and have small mean oscillations in the other directions using this result we establish the unique solvability in muckenhoupt type weighted sobolev spaces for the system with partially bmo coefficients on a reifenberg flat domain we also present weighted a priori l q estimates for the system when the domain is the whole euclidean space or a half space

supersymmetry and potentials with bound states at arbitrary energies ii for pt i see ibid vol 19 p 2297 1986 it has been shown previously that a potential v0 chi in one dimension which supports no bound states may be used as a reference potential from which by successive applications of the concept of a supersymmetric partner to a given hamiltonian it is possible to find a potential vn chi which supports any specified number n of bound states at any chosen energies ej j 1 n the reflection coefficient of vn is related to the reflection coefficient of v0 various alternative representations of the potentials constructed by this procedure are presented an illustrative example in which vn is constructed by using a sech2 chi barrier as the reference potential is discussed

robust gamma stability analysis in a plant parameter space abstract given a characteristic polynomial whose coefficients depend polynomially on l uncertain parameters the following robustness problem arises determine whether all the roots of the polynomial are located in a prescribed region in the complex plane for all admissible parameter values to this end the boundary of is mapped into the parameter space a necessary and sufficient condition for stability of an operating domain in parameter space is that it contains at least one stable point and is not intersected by the image of this condition may be tested graphically by gridding l 2 parameters and projecting all boundaries into a two dimensional subspace of the parameter space finally the method is applied to a track guided bus with uncertain mass and velocity

solvability of parabolic equations in divergence form with partially bmo coefficients abstract we prove the h p 1 solvability of second order parabolic equations in divergence form with leading coefficients a i j measurable in t x 1 and having small bmo bounded mean oscillation semi norms in the other variables additionally we assume a 11 is measurable in x 1 and has small bmo semi norms in the other variables the corresponding results for the cauchy problem are also established parabolic equations in sobolev spaces h q p 1 with mixed norms are also considered under the same conditions of the coefficients

properties of stabilizing pid gain set in parameter space the properties of the pid gain set for which all the closed loop poles lie in the left half of the complex plane are examined on the parameter plane a new method of determining the increase in the number of unstable poles when the gain crosses the boundary of stability regions is proposed this method is useful for determining the number of unstable poles in each region these results are extended to gamma stability based on conformal mapping

global gradient estimates for general nonlinear parabolic equations in nonsmooth domains we establish the natural calderon zygmund theory for a nonlinear parabolic equation of p laplacian type in divergence form 0 1 ut diva du x t div f p 2f in t by essentially proving that 0 2 f p lq t du p lq t for every q 1 the equation under consideration is of general type and not necessarily of variation form the involved nonlinearity a a x t is assumed to have a small bmo semi norm with respect to x t variables and the lateral boundary of the domain is assumed to be reifenberg flat as a consequence we are able to not only relax the known regularity requirements on the nonlinearity for such a regularity theory but also extend local results to a global one in a nonsmooth domain whose boundary has a fractal property we also find an optimal regularity estimate in orlicz sobolev spaces for such nonlinear parabolic problems

second order elliptic equations with variably partially vmo coefficients abstract the solvability in w p 2 r d spaces is proved for second order elliptic equations with coefficients which are measurable in one direction and vmo in the orthogonal directions in each small ball with the direction depending on the ball this generalizes to a very large extent the case of equations with continuous or vmo coefficients

brief fast calculation of stabilizing pid controllers a fast method for the computation of stabilizing pid controllers is proposed it is shown in particular that for a fixed value of the proportional term k p the resulting stabilizing pid compensators form a finite set of disjoint polyhedral sets in the parameter space a complete characterization and a practical way of finding such stabilizing regions are given moreover the bounds on stabilizing k p parameters are derived the method therefore has several advantages over recent results and is very promising for real time and or robust control applications

fox function representation of non debye relaxation processes applying the liouville riemann fractional calculus we derive and solve a fractional operator relaxation equation we demonstrate how the exponent of the asymptotic power law decay t relates to the order of the fractional operatordv dtv 0 1 continuous time random walk ctrw models offer a physical interpretation of fractional order equations and thus we point out a connection between a special type of ctrw and our fractional relaxation model exact analytical solutions of the fractional relaxation equation are obtained in terms of fox functions by using laplace and mellin transforms apart from fractional relaxation fox functions are further used to calculate fourier integrals of kohlrausch williams watts type relaxation functions because of its close connection to integral transforms the rich class of fox functions forms a suitable framework for discussing slow relaxation phenomena

global regularity for higher order divergence elliptic and parabolic equations abstract in this paper we obtain the global regularity estimates of the weak solutions in sobolev spaces and orlicz spaces for higher order elliptic and parabolic equations of divergence form with small bmo coefficients in the whole space we only focus on the parabolic case while the corresponding result in the elliptic case can be obtained as a corollary

tuning of pid type controllers for stable and unstable systems with time delay abstract a graphical technique for tuning pid type controllers based on the method of d partition is proposed for single input single output linear time invariant systems the tuning method shifts all the roots of the characteristic equation of the controlled system to a certain desirable region in the left half of the complex plane lhp to achieve a specified least absolute and least relative stability margin the method can be used to tune a precompensator in a two degree of freedom structure so as to improve tracking behaviour the method can be used for both stable and unstable systems for systems with significant time delay and for various controller configurations including derivative in the feedback path

linearly coupled quantum oscillators with levy stable noise abstract the influence of an external noise on two linearly coupled quantum oscillators is investigated the noise is described by the probability measure corresponding to a symmetric levy stable distribution in calculations the path integral technique is used this paper is a continuation of investigations started by bose datta and feinsilver

elliptic equations in divergence form with partially bmo coefficients the solvability in sobolev spaces is proved for divergence form second order elliptic equations in the whole space a half space and a bounded lipschitz domain for equations in the whole space or a half space the leading coefficients aij are assumed to be only measurable in one direction and have locally small bmo semi norms in the other directions for equations in a bounded domain additionally we assume that aij have small bmo semi norms in a neighborhood of the boundary of the domain we give a unified approach of both the dirichlet boundary problem and the conormal derivative problem we also investigate elliptic equations in sobolev spaces with mixed norms under the same assumptions on the coefficients

weighted w1 p estimates for weak solutions of degenerate elliptic equations with coefficients degenerate in one variable abstract this paper studies sobolev regularity of weak solution of degenerate elliptic equations in divergence form div a x u div f x where x x y r n r the coefficient matrix a x is a symmetric measurable n 1 n 1 matrix and it could be degenerate or singular in the one dimensional y variable as a weight function in the a 2 muckenhoupt class our results give weighted sobolev regularity estimates of calderon zygmund type for weak solutions of this class of degenerate singular equations as an application of these estimates we establish global fractional sobolev regularity estimates for solutions of the spectral fractional elliptic equation with measurable coefficients this result can be considered as the sobolev counterpart of the recently established schauder regularity theory of fractional elliptic equations

interior w2 p x regularity for elliptic equations of nondivergence form abstract in this paper we consider the following elliptic equations of nondivergence form i j 1 n a i j 2 u x i x j f in r n where the coefficients a i j are real valued bounded measurable functions defined in r n belong to vmo loc see definition 2 5 and satisfy the uniformly elliptic condition under the assumption that p satisfies the log holder continuity condition we will establish the interior w 2 p x regularity for strong solutions to the elliptic equations of nondivergence form

a regulating and tracking pi d controller traditionally pi controllers operate solely on the reference error signal modified reference error in the case of pid controllers in the forward path of the classical feedback loop they are designed either as regulating e g to reduce process disturbances or as tracking controllers e g to follow reference changes if both characteristics are important then a compromise design has to be found here a special connection of the classical pi d controller is proposed which allows a greater degree of independence in the design and practical tuning of the regulating and tracking features

master equations for subordinated processes we study markov jump processes constructed by subordination of diffusion processes the procedure can be viewed as a randomization or a coarse graining of time we construct the master equation for the cases of finite and infinite total jump rates and give a collection of explicitly solvable examples

on levy or stable distributions and the williams watts model of dielectric relaxation this paper is concerned with the levy or stable distribution function defined by the fourier transform q alpha left z right frac 1 2 pi int infty infty exp left izu left u right alpha right du with 0 alpha leqslant 2 when 2 it becomes the gauss distribution function and when 1 the cauchy distribution when 2 the distribution has a long inverse power tail q alpha left z right sim frac gamma left 1 alpha right sin tfrac 1 2 pi alpha pi left z right 1 alpha in the regime of small if logz 1 the distribution is mimicked by a log normal distribution we have derived rapidly converging algorithms for the numerical calculation ofq z for various in the range 0 1 the functionq z appears naturally in the williams watts model of dielectric relaxation in that model one expresses the normalized dielectric parameter as in n left omega right equiv in n left omega right i in n left omega right int 0 infty e i omega t left d phi left t right mathord left vphantom d phi left t right dt right kern nulldelimiterspace dt right dt with phi left t right exp left t mathord left vphantom t tau right kern nulldelimiterspace tau right alpha it has been found empirically by various authors that observed dielectric parameters of a wide variety of materials of a broad range of frequencies are fitted remarkably accurately by using this form of t e n is shown to be directly related toq z it is also shown that if the williams watts exponential is expressed as a weighted average of exponential relaxation functions exp left t mathord left vphantom t tau right kern nulldelimiterspace tau right alpha int 0 infty g left lambda alpha right e lambda t dt the weight functiong is expressible as a stable distribution some suggestions are made about physical models that might lead to the williams watts form of t

on the equivalence of the parallel channel and the correlated cluster relaxation models the question of the origins of nonexponential relaxation is addressed in terms of the probabilistic approach to relaxation the interconnection between two differently rooted probabilistic models i e between the parallel channel and the correlated cluster models is presented we show that clearly different probabilistic origins yield in both approaches a well defined class of universally valid two power law responses with the stretched exponential and exponential decay laws as special cases the equivalence of both models indicates that variations in the local environment of the relaxing configurational units parallel channel relaxation can provide a basis for self similar relaxation dynamics without the need for hierarchically constrained dynamics correlated clusters relaxation

elliptic and parabolic equations with measurable coefficients in weighted sobolev spaces abstract we consider both divergence and non divergence parabolic equations on a half space in weighted sobolev spaces all the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient which is assumed to be only measurable either in the time or the spatial variable as functions of the other variables the coefficients have small bounded mean oscillation bmo semi norms the lower order coefficients are allowed to blow up near the boundary with a certain optimal growth condition as a corollary we also obtain the corresponding results for elliptic equations

from classical dynamics to continuous time random walks the migration of a classical dynamical system between regions of configuration space can be treated as a continuous time random walk between these regions derivation of a classical analog of the quantum mechanical generalized master equation provides expressions for the waiting time distribution in terms of transition memory functions a short memory approximation to these memory functions is equivalent to the well known transition state method an example is discussed for which this approximation seems reasonable but is entirely wrong

ornstein uhlenbeck cauchy process we combine earlier investigations of linear systems subject to levy fluctuations with recent attempts to give meaning to so called levy flights in external force fields we give a complete construction of the ornstein uhlenbeck cauchy process as a fully computable paradigm example of doob s stable noise supported ornstein uhlenbeck process despite the nonexistence of all moments we determine local characteristics forward drift of the process generators of forward and backward dynamics and relevant pseudodifferential evolution equations the induced nonstationary spatial process is proved to be markovian and quite apart from its inherent discontinuity defines an associated velocity process in a probabilistic sense

a generalized cauchy process and its application to relaxation phenomena we study some of the basic properties of a generalized cauchy process indexed by two parameters the application of the lamperti transformation to the generalized cauchy process leads to a self similar process which preserves the long range dependence the asymptotic properties of spectral density of the process are derived possible application of this process to model relaxation phenomena is considered

controllers of ziegler nichols type for unstable process with time delay abstract proportional p proportional plus integral pi and proportional plus integral plus derivative pid controllers are designed for a time delayed process having latency l and a single right half plane pole at s with l 1 explicit designs are derived from the concept of phase margin and an optimum stability approach for pi controllers is based on a parameter plane study an optimum gain margin design is also given for p control

brief analytical pid parameter expressions for higher order systems most approaches to feedback synthesis are numerical design techniques that produce their results in form of numerical values for the controller parameters in contrast analytical design techniques produce the controller parameters as explicit expressions that are a function of open loop system parameters and the desired closed loop advantages include fast commissioning in terms of physical system parameters gain scheduling without extra design effort and analytical sensitivity insights so far it has not been known how to derive analytical pid expressions for higher than second order systems the paper proposes a technique to address this problem analyses its properties demonstrates computational simplicity and shows that its performance is comparable to known numerical techniques

on elliptic equations in a half space or in convex wedges with irregular coefficients abstract we consider second order elliptic equations in a half space with leading coefficients measurable in a tangential direction we prove the w p 2 estimate and solvability for the dirichlet problem when p 1 2 and for the neumann problem when p 2 we then extend these results to equations with more general coefficients which are measurable in a tangential direction and have small mean oscillations in the other directions as an application we obtain the w p 2 solvability of elliptic equations in convex wedge domains or in convex polygonal domains with discontinuous coefficients

higher order elliptic and parabolic systems with variably partially bmo coefficients in regular and irregular domains abstract the solvability in sobolev spaces is proved for divergence form complex valued higher order parabolic systems in the whole space on a half space and on a reifenberg flat domain the leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder the directions in which the coefficients are only measurable vary depending on each cylinder the corresponding elliptic problem is also considered

global weighted estimates in orlicz spaces for second order nondivergence parabolic equations abstract in this paper we prove the following global weighted estimates in orlicz spaces f l w t u d u d 2 u u t l w t for second order parabolic equations of nondivergence form with small bmo coefficients u t a i j x t u x i x j f in t u x t 0 on p t