thermoacoustic tomography for an integro differential wave equation modeling attenuation abstract in this article we study the inverse problem of thermoacoustic tomography tat on a medium with attenuation represented by a time convolution or memory term and whose consideration is motivated by the modeling of ultrasound waves in heterogeneous tissue via fractional derivatives with spatially dependent parameters under the assumption of being able to measure data on the whole boundary we prove uniqueness and stability and propose a convergent reconstruction method for a class of smooth variable sound speeds by a suitable modification of the time reversal technique we obtain a neumann series reconstruction formula

decentralized supervision of petri nets this note extends previous results on the supervision of petri nets pns to the decentralized setting while focusing on the extension of supervision based on place invariants sbpi the proposed approach is more general and could be applied to other types of supervision as well we begin by introducing d admissibility as an extension to the decentralized setting of the centralized admissibility concept we define also structural d admissibility as the counterpart of the simple sufficient conditions for centralized admissibility in the context of the sbpi note that structural d admissibility is only sufficient for a specification to be enforcible with the same permissiveness as in the centralized setting with full controllability and observability however structural d admissibility can be checked with low polynomial complexity based on the d admissibility concept we propose two suboptimal methods to design decentralized supervisors the first method is to find a centralized solution and then distribute the centralized supervisory policy by means of communication the amount of communication can be minimized by means of an integer linear program ilp the second method is to transform the specification to a more restrictive d admissible specification by means of an ilp in the case of decentralized supervision with communication the ilp can be used to minimize the amount of communication required by the solution

multidimensional solutions of space fractional diffusion equations green s functions and propagators for the multidimensional anisotropic spacetime fractional diffusion equation are obtained in integral form uniqueness of the solutions is studied in the framework

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

modular feedback logic for discrete event systems we examine a modular approach to the synthesis of state feedback controls for the problem of maintaining a predicate on the state set of a discrete dynamic system invariant dynamical systems are modeled by automata together with a mechanism for enabling and disabling a subset of state transitions the basic problem of interest is to ensure by appropriate control action that a given predicate on the state set of the process remains invariantly true whenever it is initially satisfied assuming the predicate can be decomposed into the conjunction or disjunction of component predicates we determine conditions under which it is possible to synthesize the appropriate control in a modular fashion

observability of place transition nets we discuss the problem of estimating the marking of a place transition p t net based on event observation we assume that the net structure is known while the initial marking is totally or partially unknown we give algorithms to compute a marking estimate that is a lower bound of the actual marking the special structure of petri nets allows us to use a simple linear algebraic formalism for estimate and error computation the error between actual marking and estimate is a monotonically nonincreasing function of the observed word length and words that lead to null error are said to be complete we define several observability properties related to the existence of complete words and show how they can be proved to prove some of them we also introduce a useful tool the observer coverability graph i e the usual coverability graph of a p t net augmented with a vector that keeps track of the estimation error on each place of the net finally we show how the estimate generated by the observer may be used to design a state feedback controller for forbidden marking specifications

existence results for an evolution problem with fractional nonlocal conditions a second order abstract problem involving derivatives of non integer order in the nonlinearity as well as in the nonlocal conditions is considered this problem covers many situations arising in real world phenomena and extends the case of integer order to the case of non integer order we establish the existence of mild solutions under extra appropriate sufficient assumptions on the nonlinearity and on the initial data these mild solutions are shown to be classical

anomalous diffusion with ballistic scaling a new fractional derivative abstract anomalous diffusion with ballistic scaling is characterized by a linear spreading rate with respect to time that scales like pure advection ballistic scaling may be modeled with a symmetric riesz derivative if the spreading is symmetric however ballistic scaling coupled with a skewness is observed in many applications including hydrology nuclear physics viscoelasticity and acoustics the goal of this paper is to find a governing equation for anomalous diffusion with ballistic scaling and arbitrary skewness to address this problem we propose a new operator called the zolotarev derivative which is valid for all orders 0 2 the fourier symbol of this operator is related to the characteristic function of a stable random variable in the zolotarev m parameterization in the symmetric case the zolotarev derivative reduces to the well known riesz derivative for 1 the zolotarev derivative is a linear combination of riemann liouville fractional derivatives and a first derivative for 1 the zolotarev derivative is a non local operator that models ballistic anomalous diffusion we prove that this operator is continuous with respect to we derive generator caputo and riemann liouville forms of this operator and provide two examples the solutions of diffusion equations utilizing the zolotarev derivative with an impulse initial condition are shifted and scaled stable densities in the m parameterization

causal theories and data for acoustic attenuation obeying a frequency power law this study compares causal theories based on kramers kronig relations fractional calculus and on those derived from new time domain causal relationships to diverse data all these theories are based on the assumptions that the functional form of the attenuation persists beyond the measurement range and that attenuation is much smaller than the wave number the data for lossy media with attenuation having a power law frequency dependence with an exponent y include cases for both liquids and solids ranging from acoustic to ultrasound frequencies data are in closer correspondence with the new theory which predicts decreasing dispersion as the power exponent y approaches zero or an even integer experimental results and supporting evidence show that the classical case of frequency squared attenuation is dispersionless an approximate nearly local kramers kronig theory is in agreement with the time causal theory when the exponent is close to one but deviates for other values the comprehensive time causal theory is shown to be equivalent to two other theories derived from exact kramers kronig relations and from fractional calculus and it covers the y odd integer cases which are missing or incomplete in these approaches attenuation dispersion relations are presented in two forms one for a general frequency range and another for a finite range it is demonstrated that complete velocity dispersion within a signal bandwidth can be predicted from knowledge of the attenuation data and velocity at a single frequency including the velocity at either zero frequency y 1 and or at a high frequency limit 0 y 1 in addition attenuation data can be predicted from velocity data

modified szabo s wave equation models for lossy media obeying frequency power law szabo s models of acoustic attenuation szabo j acoust soc am 96 1 491 500 1994 comply well with the empirical frequency power law involving noninteger and odd integer exponent coefficients while guaranteeing causality but nevertheless encounter the troublesome issues of hypersingular improper integral and obscurity in implementing initial conditions the purpose of this paper is to ease or remove these drawbacks of the szabo s models via the caputo fractional derivative concept the positive time fractional derivative is also introduced to include the positivity of the attenuation processes

detectability of breast lesions with cari ultrasonography using a bioacoustic computational approach compared to other medical breast imaging techniques ultrasonic imaging is more attractive due to its relatively low cost and wide clinical use this study is concerned with clinical amplitude velocity reconstruction imaging cari ultrasonography for accurately detecting breast cancer using two acoustic properties the sound speed and the attenuation bioacoustic modeling is used to describe ultrasound wave propagation in the breast tumor system the feasibility and the sensitivity of the cari device are investigated using a finite element time domain approximation in tissue mimicking breast 2d and 3d simulation experiments show the detectability of small ellipse ellipsoid shaped tumors in terms of the increase in the sound speed in the tumor region our study confirms also that the fetd finite element time domain method is a simple but robust tool to simulate the cari device and other included clinical effects such as focusing and scanning of the beams

boundedness of singular radon transforms on l p spaces under a finite type condition in this paper a new method is presented for proving boundedness of singular radon transforms on l p for 1 p two closely related results generalizing the corresponding theorem of christ nagel stein and wainger are proven 1 introduction in this paper a new method is presented for proving boundedness of singular radon transforms on l p for 1 p two closely related results generalizing the corresponding theorem of christ nagel stein and wainger in 3 are proved singular radon transforms are a type of singular integral operator a prototype for which is the hilbert transform along curves let 1 1 r n be a curve with 0 0 and let f be a schwarz function on r n 1

linking multiple relaxation power law attenuation and fractional wave equations the acoustic wave attenuation is described by an experimentally established frequency power law in a variety of complex media e g biological tissue polymers rocks and rubber recent papers present a variety of acoustical fractional derivative wave equations that have the ability to model power law attenuation on the other hand a multiple relaxation model is widely recognized as a physically based description of the acoustic loss mechanisms as developed by nachman et al j acoust soc am 88 1584 1595 1990 through assumption of a continuum of relaxation mechanisms each with an effective compressibility described by a distribution related to the mittag leffler function this paper shows that the wave equation corresponding to the multiple relaxation approach is identical to a given fractional derivative wave equation this work therefore provides a physically based motivation for use of fractional wave equations in acoustic modeling

efficient two dimensional simulations of the fractional szabo equation with different time stepping schemes the modified szabo wave equation is one of the various models that have been developed to model the power law frequency dependent attenuation phenomena in lossy media the purpose of this study is to develop two different efficient numerical methods for the two dimensional szabo equation and to compare the relative merits of each method in both methods we employ the adi scheme to split directions however we use different time discretization specifically in the first adi method adi i we include a third order correction term to achieve second order convergence for smooth solutions hence extending the work of sun and wu 2006 in the second adi method adi ii we employ the scheme in zeng et al submitted for publication to two dimensional fractional wave equation using multiple correction terms to enhance accuracy for non smooth solutions our simulation results show that both methods are computationally efficient for the fractional wave equation but have different advantages in terms of accuracy specifically adi ii seems to produce more accurate results than adi i for non smooth solutions however for smooth solutions and fractional order close to two adi i seems to outperform adi ii

fractional biharmonic operator equation model for arbitrary frequency dependent scattering attenuation in acoustic wave propagation this paper proposes a fractional biharmonic operator equation model in the time space domain to describe scattering attenuation of acoustic waves in heterogeneous media compared with the existing models the proposed fractional model is able to describe arbitrary frequency dependent scattering attenuation which typically obeys an empirical power law with an exponent ranging from 0 to 4 in stark contrast to an extensive and rapidly increasing application of the fractional derivative models for wave absorption attenuation in the literature little has been reported on frequency dependent scattering attenuation this is largely because the order of the fractional laplacian is from 0 to 2 and is infeasible for scattering attenuation in this study the definition of the fractional biharmonic operator in space with an order varying from 0 to 4 is proposed as well as a fractional biharmonic operator equation model of scattering attenuation which is consistent with arbitrary frequency power law dependency an

on deadlock free modular supervisory control of discrete event systems in this paper the problem of synthesizing deadlock free modular supervisors for discrete event systems is discussed by introducing the d invariant relation between a pair of supervisors it is shown that when the control objective is described in terms of intersection of two languages a necessary and sufficient condition for the modular supervisor to be deadlock free is that this pair of subsupervisors satisfies a d invariant relation a procedure for synthesizing the deadlock free modular supervisor is presented some issues concerning deadlock free modular supervisory control are also discussed

full wave modeling of therapeutic ultrasound efficient time domain implementation of the frequency power law attenuation for the simulation of therapeutic ultrasound applications a method including frequency dependent attenuation effects directly in the time domain is highly desirable this paper describes an efficient numerical time domain implementation of the power law attenuation model presented by szabo szabo j acoust soc am 96 491 500 1994 simulations of therapeutic ultrasound applications are feasible in conjunction with a previously presented finite differences time domain fdtd algorithm for nonlinear ultrasound propagation ginter et al j acoust soc am 111 2049 2059 2002 szabo implemented the empirical frequency power law attenuation using a causal convolutional operator directly in the time domain equation though a variety of time domain models has been published in recent years no efficient numerical implementation has been presented so far for frequency power law attenuation models solving a convolutional integral with standard time domain techniques requires enormous computational ef

a generalized fractional order elastodynamic theory for non local attenuating media this study presents a generalized elastodynamic theory based on fractional order operators capable of modelling the propagation of elastic waves in non local attenuating solids and across complex

classes of singular integrals along curves and surfaces this paper is concerned with singular convolution operators in rd d 2 with convolution kernels supported on radial surfaces yd r ly l we show that if r s log s then lp boundedness holds if and only if p 2 this statement can be reduced to a similar statement about the multiplier m r 71 lrl i 7 in r2 we also construct smooth r for which the corresponding operators are bounded for po p 2 but unbounded for p po for given po e 1 2 finally we discuss some examples of singular integrals along convex curves in the plane with odd extensions

characterization of admissible marking sets in petri nets with conflicts and synchronizations in this paper we study the problem of constraint transformation for petri nets with uncontrollable transitions and containing both conflicts and synchronizations we show that given an arbitrary net and a set of legal markings the admissible marking set cannot always be represented by a finite number of disjunctions of gmecs moreover we characterize the gmec inflation phenomenon that is the case in which the representation of the admissible marking set may be too complex to be efficiently implemented in a closed loop net to rule out the possibility of gmec inflation we consider a subclass of constraints called singular gmecs with an acyclic backward conflict free uncontrollable subnet by these assumptions we propose an algorithm to transform a given singular gmec into a controllable or gmec which precisely characterizes its admissible marking set

control of vector discrete event systems i the base model a vector discrete event system vdes is a discrete event system model in which the system state is represented by a vector with integer components and state transitions by integer vector addition the authors investigate such structures from the viewpoint of control theory the objective is to specialize to vdes the general automation based control theory of p j ramadge and w m wonham 1987 and w m wonham and p j ramadge 1988 it is shown that within the vdes framework a useful class of control systems can be compactly modeled and some of the associated control problems of ramadge and wonham can be more efficiently solved

output stabilizability of discrete event dynamic systems the authors investigate the problem of designing stabilizing feedback compensators for discrete event dynamic systems deds modeled as finite state automata in which some transition events are controllable and some events are observed the problem of output stabilization is defined as the construction of a compensator such that all state trajectories in the closed loop system go through a given set e infinitely often the authors also define a stronger notion of output stabilizability which requires that the state not only pass through e infinitely often but that the set of instants when the state is in e and one knows it is in e is also infinite necessary and sufficient conditions are presented for both notions the authors also introduce and characterize a notion of resiliency that corresponds to the system being able to recover from observation errors in addition they provide some general bounds for the algorithms considered and discuss several conditions under which far smaller bounds can be achieved

the existence of mild and classical solutions for a second order abstract fractional problem in this paper we consider a second order abstract problem involving derivatives of non integer order the existence and uniqueness of mild and classical solutions are established in appropriate spaces under weak assumptions

representation of the absorption of nonlinear waves by fractional derivatives fractional derivatives of arbitrary order are incorporated into a model nonlinear equation in order to simulate the absorption mechanism in a finite frequency domain that makes it possible to take into account a power dependence of the absorption coefficient on the frequency at any real degree the model gives a mathematical description of a large set of well known absorption mechanisms the method is applied to calculate weakly nonlinear oscillations in a nonrigid tube as a result the value of the absorption coefficient and its dependence on the frequency are found based on experimental data

time domain wave equations for lossy media obeying a frequency power law for attenuation described by a slowly varying power law function of frequency 0 y classical lossy time domain wave equations exist only for the restricted cases where y 0 or y 2 for the frequently occurring practical situation in which attenuation is much smaller than the wave number a lossy dispersion characteristic is derived that has the desired attenuation general power law dependence in order to obtain the corresponding time domain lossy wave equation time domain loss operators similar in function to existing derivative operators are developed through the use of generalized functions three forms of lossy wave equations are found depending on whether y is an even or odd integer or a noninteger a time domain expression of causality analogous in function to the kramers kronig relations in the frequency domain is used to derive the causal wave equations final causal versions of the time domain wave equations are obtained even for the cases where y 1 which according to the paley wiener th

singular integrals associated to hypersurfaces l2 theory abstract we consider singular integrals associated to a classical calderon zygmund kernel k and a hypersurface given by the graph of t where is an arbitrary c 1 function and is a smooth convex function of finite type we give a characterization of those calderon zygmund kernels k and convex functions so that the associated singular integral operator is bounded on l 2 for all c 1 functions

propagation speed of the maximum of the fundamental solution to the fractional diffusion wave equation in this paper the one dimensional time fractional diffusion wave equation with the fractional derivative of order a 1 a 2 is revisited this equation interpolates between the diffusion and the wave equations that behave quite differently regarding their response to a localized disturbance whereas the diffusion equation describes a process where a disturbance spreads infinitely fast the propagation speed of the disturbance is a constant for the wave equation for the time fractional diffusion wave equation the propagation speed of a disturbance is infinite but its fundamental solution possesses a maximum that disperses with a finite speed in this paper the fundamental solution of the cauchy problem for the time fractional diffusion wave equation its maximum location maximum value and other important characteristics are investigated in detail to illustrate analytical formulas results of numerical calculations and plots are presented numerical algorithms and programs used to produce plots are discussed

modeling power law absorption and dispersion for acoustic propagation using the fractional laplacian the efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics previous wave equations which use time domain fractional operators require the storage of the complete pressure field at previous time steps such operators are convolution based this makes them unsuitable for many three dimensional problems of interest here a wave equation that utilizes two lossy derivative operators based on the fractional laplacian is derived these operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into fourier based pseudospectral and k space methods without the increase in memory required by their time domain fractional counterparts a framework for encoding the developed wave equation using three coupled first order constitutive equations is discussed and the model is demonstrated through several one two and three dimensional simulations

petri net controller synthesis for partially controllable and observable discrete event systems to enforce linear constraints on petri nets that are partially controllable and observable this work proposes an approach based on constraint transformation first a state space equation of a petri net control system based on event feedback is obtained by expressing a control action as a matrix and the optimal control policy is designed however this policy needs to solve a nonlinear program on line second pre transition gain transformation is proposed to equivalently transform a constraint into a disjunction of new ones for an uncontrollable transition and similarly post transition gain transformation to transform a constraint into a disjunction of new ones for an unobservable transition an algorithm is then given to transform a constraint into a disjunction of admissible ones and consequently an efficient policy which may not be optimal can be designed third in order to guarantee that the policy be both efficient and optimal a dynamic linear constraint is introduced further observing transformation is proposed to simplify a dynamic constraint for an unobservable transition and an algorithm is given to equivalently transform a class of linear constraints into admissible dynamic ones as a result an optimal controller requiring little online computation can be designed accordingly for some class of petri nets finally a maze system is used to illustrate the theoretical results

frequency domain wave equation and its time domain solutions in attenuating media our purpose in this paper is to describe the wave propagation in media whose attenuation obeys a frequency power law to achieve this a frequency domain wave equation was developed using previously derived causal dispersion relations an inverse space and time fourier transform of the solution to this algebraic equation results in a time domain solution it is shown that this solution satisfies the convolutional time domain wave equation proposed by szabo j acoust soc am 96 491 500 1994 the form of the convolutional loss operator contained in this wave equation is obtained solutions representing the propagation of both plane sinusoidal and transient waves propagating in media with specific power law attenuation coefficients are investigated as special cases of our solution using our solution comparisons are made for transient one dimensional propagation in a medium whose attenuation is proportional to frequency with recently obtained numerical solutions of szabo s equation these show good agreement

think globally act locally decentralized supervisory control decentralized supervisory control is investigated by considering problem formulations that model systems whose specifications are given as global constraints but whose solution is described by local controllers a necessary and sufficient condition is given for the existence of a solution to the problem of finding decentralized supervisors that ensure that the behavior of the closed loop system lies in a given range where the range of behavior can be described by regular languages it can be effectively tested whether the decentralized control problem is solvable in this case a procedure is given to compute the associated supervisors

some inequalities for singular convolution operators in spaces suppose that a bounded function m satisfies a localized multiplier condition supt o ii0m tp imp 0 the purpose of this paper is to find easily verified conditions that 1 implies me mp the condition llmll 0 then we ask under which conditions supt 0 qm tp 1mp oc implies m c mp throughout this paper x will always be chosen as in the following definition x c co rn satisfies a tauber condition with respect to the dilations tp if for every x 0 there is a tx such that q tpx 0 sometimes we need special bump functions of the following kind let p e c rn be a p homogeneous distance function this means that p tpx tp x received by the editors august 21 1986 1980 mathermatics subject lassijication 1985 revision primary 42b15 42b20 42b25

deriving fractional acoustic wave equations from mechanical and thermal constitutive equations it is argued that fractional acoustic wave equations come in two kinds the first kind is constructed ad hoc to have loss operators that fit power law measurements the second kind is more fundamental as they in addition are based on underlying physical equations here that means constitutive equations these equations are the fractional kelvin voigt and the more general fractional zener stress strain relationships as well as a fractional version of the fourier heat law the properties of the wave equations are given in terms of attenuation and phase group velocities for low intermediate and high frequency regions in the most general case the attenuation exhibits power law behavior in all frequency ranges while the phase and group velocities increase sharply in the intermediate frequency range and converge to a constant finite value for high frequencies it is also shown that the fractional zener wave equation is equivalent to the multiple relaxation model for attenuation

a k space green s function solution for acoustic initial value problems in homogeneous media with power law absorption an efficient green s function solution for acoustic initial value problems in homogeneous media with power law absorption is derived the solution is based on the homogeneous wave equation for lossless media with two additional terms these terms are dependent on the fractional laplacian and separately account for power law absorption and dispersion given initial conditions for the pressure and its temporal derivative the solution allows the pressure field for any time t 0 to be calculated in a single step using the fourier transform and an exact k space time propagator for regularly spaced cartesian grids the former can be computed efficiently using the fast fourier transform because no time stepping is required the solution facilitates the efficient computation of the pressure field in one two or three dimensions without stability constraints several computational aspects of the solution are discussed including the effect of using a truncated fourier series to represent discrete initial condit

stabilization of discrete event processes discrete event processes are modelled by state machines in the ramadge wonham framework with control by a feedback event disablement mechanism in this paper concepts of stabilization of discrete event processes are defined and investigated we examine the possibility of driving a process under control from arbitrary initial states to a prescribed subset of the state set and then keeping it there indefinitely this stabilization property is studied also with respect to open loop processes i e uncontrolled processes and their asymptotic behaviour is characterized to this end such well known classical concepts of dynamics as invariant sets and attractors are redefined and characterized in the discrete event control framework we provide polynomial time algorithms for verifying various types of attraction and for the synthesis of attractors

on the supermal controllable sublanguage of a given language the concept of controllable language has been shown to play a basic role in the existence theory of supervisory controls for discrete event processes in this paper the supremal controllable sublanguage s of a given language l is characterized as the largest fixpoint of a monotone operator omega in the case where the languages involved are regular it is shown that the fixpoint s can be computed as the limit of the finite sequence k j given by k j 1 omega k j k 0 l an effective computational algorithm is developed and three examples are provided for illustration

gauss jacobi type quadrature rules for fractional directional integrals fractional directional integrals are the extensions of the riemann liouville fractional integrals from one to multi dimensional spaces and play an important role in extending the fractional differentiation to diverse applications in numerical evaluation of these integrals the weakly singular kernels often fail the conventional quadrature rules such as newton cotes and gauss legendre rules it is noted that these kernels after simple transforms can be taken as the jacobi weight functions which are related to the weight factors of gauss jacobi and gauss jacobi lobatto rules these rules can evaluate the fractional integrals at high accuracy comparisons with the three typical adaptive quadrature rules are presented to illustrate the efficacy of the gauss jacobi type rules in handling weakly singular kernels of different strengths potential applications of the proposed rules in formulating and benchmarking new numerical schemes for generalized fractional diffusion problems are briefly discussed in the final remarking section

static state feedback control of discrete event systems under partial observation studies static state feedback control of discrete event systems under partial observation the authors present a necessary and sufficient condition for the existence of a static state feedback controller a static controller is a special case of a dynamic controller studied by kumar et al the authors condition however has computational advantage in contrast to those obtained by kumar et al the authors also present a necessary and sufficient condition under which the modular feedback synthesis is possible

nonlinear hamiltonian equations with fractional damping existence and uniqueness are proved for a class of hamiltonian nonlinear differential equations of second order with a caputo fractional derivative of order 0 2 for noninteger the rate of convergence of solutions to a critical point is algebraic for 0 1 a non negative dissipation function is constructed for 2 the equation is hamiltonian the solution is an analytic function of for 0 1 and for 1 2 numerical studies of nonlinear pendulum and the duffing equation with a fractional derivative term replacing the usual damping show striking differences between ordinary differential and fractional differential equations

compact structures in a class of nonlinearly dispersive equations with time fractional derivatives in this work variants of the kdv equation with time fractional derivatives which exhibits compactons solitons with compact support are investigated new solitary wave solutions with compact support are developed the homotopy perturbation method is employed to derive compact solutions for these variants the chosen initial solution or trial function plays a major role in changing the physical structure of the solution the proposed scheme can be used to construct compact and noncompact solutions for a wide class of nonlinear fractional evolution equations

effective fractional acoustic wave equations in one dimensional random multiscale media this paper considers multiple scattering of waves propagating in a non lossy one dimensional random medium with short or long range correlations using stochastic homogenization theory it is possible to show that pulse propagation is described by an effective deterministic fractional wave equation which corresponds to an effective medium with a frequency dependent attenuation that obeys a power law with an exponent between 0 and 2 the exponent is related to the hurst parameter of the medium which is a characteristic parameter of the correlation properties of the fluctuations of the random medium moreover the frequency dependent attenuation is associated with a special frequency dependent phase which ensures that causality and kramers kronig relations are satisfied in the time domain the effective wave equation has the form of a linear integro differential equation with a fractional derivative

a temporal logic approach to discrete event control for the safety canonical class this paper presents a temporal logic formulation of discrete event control which forms a new theoretical basis for control analysis and synthesis of a class of discrete event systems des based on the formulation a basic supervisory control theory is developed for a control objective specified by an invariance formula belonging to the safety canonical class of manna and pneuli using the safety canonical class as a basis the refinement and generalization of the existing basic predicate framework are demonstrated a simple example illustrates the formal axiomatic means to perform control theoretic analysis and synthesis under the new formulation