nonlinear fractional dynamics on a lattice with long range interactions abstract a unified approach has been developed to study nonlinear dynamics of a 1d lattice of particles with long range power law interaction a classical case is treated in the framework of the generalization of the well known frenkel kontorova chain model for the non nearest interactions quantum dynamics is considered following davydov s approach for molecular excitons in the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the long range interaction fractional generalizations of the sine gordon nonlinear schrodinger and hilbert schrodinger equations have been found there exists a critical value of the power s of the long range potential below the critical value s 3 s 1 2 we obtain equations with fractional derivatives while for s 3 we have the well known nonlinear dynamical equations with space derivatives of integer order long range interaction impact on the quantum lattice propagator has been studied we have shown that the quantum exciton propagator exhibits transition from the well known gaussian like behavior to a power law decay due to the long range interaction a link between 1d quantum lattice dynamics in the imaginary time domain and a random walk model has been discussed

transition system specifications with negative premises abstract in this article the general approach to plotkin style operational semantics of groote and vaandrager 1989 is extended to transition system specifications tsss with rules that may contain negative premises two problems arise firstly the rules may be inconsistent and secondly it is not obvious how a tss determines a transition relation we present a general method based on the stratification technique in logic programming to prove consistency of a set of rules and we show how a specific transition relation can be associated with a tss in a natural way then a special format for the rules the ntyft ntyxt format is defined it is shown that for this format three important theorems hold the first theorem says that bisimulation is a congruence if all operators are defined using this format the second theorem states that under certain restrictions a tss in ntyft format can be added conservatively to a tss in pure ntyft ntyxt format finally it is shown that the trace congruence for image finite processes induced by the pure ntyft ntyxt format is precisely bisimulation equivalence

fractional variational calculus and the transversality conditions this paper presents the euler lagrange equations and the transversality conditions for fractional variational problems the fractional derivatives are defined in the sense of riemann liouville and caputo the connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined it is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the caputo derivative furthermore both fractional derivatives the riemann liouville and the caputo arise in the formulations even when the fractional variational problem is defined in terms of one fractional derivative only examples are presented to demonstrate the applications of the formulations

ordered sos process languages for branching and eager bisimulation we present a general and uniform method for defining structural operational semantics sos of process operators by traditional plotkin style transition rules equipped with orderings this new feature allows one to control the order of application of rules when deriving transitions of process terms our method is powerful enough to deal with rules with negative premises and copying we show that rules with orderings called ordered sos rules have the same expressive power as gsos rules we identify several classes of process languages with operators defined by rules with and without orderings in the setting with silent actions and divergence we prove that branching bisimulation and eager bisimulation relations are preserved by all operators in process languages in the relevant classes

language preorder as a precongruence groote and vaandrager introduced the tyft format which is a congruence format for strong bisimulation equivalence this article proposes additional syntactic requirements on the tyft format extended with predicates to obtain a precongruence format for language preorder

notions of bisimulation and congruence formats for sos with data while studying the specification of the operational semantics of different programming languages and formalisms one can observe the following three facts first plotkin s style of structural operational semantics has become a standard in defining operational semantics second congruence with respect to some notion of bisimilarity is an interesting property for such languages and it is essential in reasoning third there are numerous languages that contain an explicit data part in the state of the operational semantics the first two facts have resulted in a line of research exploring syntactic formats of operational rules to derive the desired congruence property for free however the third point in combination with the first two is not sufficiently addressed and there is no standard congruence format for operational semantics with an explicit data state in this article we address this problem by studying the implications of the presence of a data state on the notion of bisimilarity furthermore we propose a number of formats for congruence

compositionality of hennessy milner logic by structural operational semantics this paper presents a method for the decomposition of hml formulas it can be used to decide whether a process algebra term satisfies a hml formula by checking whether subterms satisfy certain formulas obtained by decomposing the original formula the method uses the structural operational semantics of the process algebra the main contribution of this paper is the extension of an earlier decomposition method for the de simone format from the ph d thesis of larsen in 1986 to more general formats

transition to chaos in discrete nonlinear schrodinger equation with long range interaction discrete nonlinear schrodinger dnls equation describes a chain of oscillators with nearest neighbor interactions and a specific nonlinear term we consider its modification with long range interaction through a potential proportional to 1 l1 with fractional 2 and l as a distance between oscillators this model is called dnls it exhibits competition between the nonlinearity and a level of correlation between interacting far distanced oscillators that is defined by the value of we consider transition to chaos in this system as a function of and nonlinearity it is shown that decreasing of with respect to nonlinearity stabilize the system connection of the model to the fractional generalization of the nls called fnls in the long wave approximation is also discussed and some of the results obtained for dnls can be correspondingly extended to the fnls

divide and congruence ii from decomposition of modal formulas to preservation of delay and weak bisimilarity abstract earlier we presented a method to decompose modal formulas for processes with the internal action and congruence formats for branching and bisimilarity were derived on the basis of this decomposition method the idea is that a congruence format for a semantics must ensure that the formulas in the modal characterisation of this semantics are always decomposed into formulas that are again in this modal characterisation in this follow up paper the decomposition method is enhanced to deal with modal characterisations that contain a modality a to derive congruence formats for delay and weak bisimilarity

the theory of interactive generalized semi markov processes in this paper we introduce the calculus of interactive generalized semi markov processes igsmps a stochastic process algebra which can express probabilistic timed delays with general distributions and synchronizable actions with zero duration and where choices may be probabilistic non deterministic and prioritized igsmp is equipped with a structural operational semantics which generates semantic models in the form of generalized semi markov processes gsmps i e probabilistic systems with generally distributed time extended with action transitions representing interaction among system components this is obtained by expressing the concurrent execution of delays through a variant of st semantics which is based on dynamic names the fact that names for delays are generated dynamically by the semantics makes it possible to define a notion of observational congruence for igsmp that abstracts from internal actions with zero duration simply as a combination of standard observational congruence and probabilistic bisimulation we also present a complete axiomatization for observational congruence over igsmp finally we show how to derive a gsmp from a given igsmp specification in order to evaluate the system performance and we present a case study

maximum entropy type lyapunov functions for robust stability and performance analysis we present two lyapunov functions that ensure the unconditional stability and robust performance of a modal system with uncertain damped natural frequency each lyapunov function involves the sum of two matrices the first being the solution to the so called maximum entropy equation and the second being a constant auxiliary portion the significant feature of these lyapunov functions is that the guaranteed robust stability region is independent of the weighting matrix while the performance bounds are relatively tight compared to alternative approaches thus these lyapunov functions are less conservative than standard bounds that tend to be highly sensitive to the choice of state space basis kevwords maximum entropy function robust stability robust performance

robust stabilization with positive real uncertainty beyond the small gain theory in many applications of feedback control phase information is available concerning the plant uncertainty for example lightly damped flexible structures with colocated rate sensors and force actuators give rise to positive real transfer functions closed loop stability is thus guaranteed by means of negative feedback with strictly positive real compensators in this paper the properties of positive real transfer functions are used to guarantee robust stability in the presence of positive real but otherwise unknown plant uncertainty these results are then used for controller synthesis to address the problem of robust stabilization in the presence of positive real uncertainty one of the principal motivations for these results is to utilize phase information in guaranteeing robust stability in this these sense results go beyond the usual limitations the small gain theorem and quadratic lyapunov functions which may be conservative when phase information is available the results of this paper are based upon a riccati equation formulation of the positive real lemma and thus are in the spirit of recent riccati based approaches to bounded real h control

the fractional boltzmann transport equation the fractional boltzmann transport equation is derived making use of the fractional hamilton s equations based on the fractional actionlike variational approach by simply defining a distribution function and inspecting its time derivative many important results in statistical physics can be derived

a conservative look at operational semantics with variable binding we set up a formal framework to describe transition system specifications in the style of plotkin this framework has the power to express many sortedness general binding mechanisms and substitutions among other notions such as negative hypotheses and unary predicates on terms the framework is used to present a conservativity format in operational semantics which states sufficient criteria to ensure that the extension of a transition system specification with new transition rules does not affect the semantics of the original terms

brief paper stability and resonance conditions of elementary fractional transfer functions elementary fractional transfer functions are studied in this paper some basic properties of elementary transfer functions of the first kind are recalled then two main results are presented regarding elementary fractional transfer functions of the second kind written in a canonical form and characterized by a commensurate order a pseudo damping factor and a natural frequency first stability conditions are established in terms of the pseudo damping factor and the commensurate order as a corollary to matignon s stability theorem they extend the previous result into conditions that are simpler to check then resonance conditions are established numerically in terms of the commensurate order and the pseudo damping factor and give interesting information on the frequency behavior of fractional systems it is shown that elementary transfer functions of the second kind might have up to two resonant frequencies moreover three abaci are given allowing to determine the pseudo damping factor and the commensurate order for respectively a desired normalized gain at each resonance a desired phase at each resonance and a desired normalized first or second resonant frequency

a general sos theory for the specification of probabilistic transition systems this article focuses on the formalization of the structured operational semantics approach for languages with primitives that introduce probabilistic and non deterministic behavior we define a general theoretic framework and present the rule format that guarantees that bisimulation equivalence in the probabilistic setting is a congruence for any operator defined in this format we show that the bisimulation is fully abstract w r t the format and possibilistic trace equivalence in the sense that bisimulation is the coarsest congruence included in trace equivalence for any operator definable within the format in other words bisimulation is the smallest congruence relation guaranteed by the format we also provide a conservative extension theorem and show that languages that include primitives for exponentially distributed time behavior such as imc and markov automata based language fit naturally within our framework

discrete time generative reactive probabilistic processes with different advancing speeds we present a process algebra expressing probabilistic external internal choices multi way synchronizations and processes with different advancing speeds in the context of discrete time i e where time is not continuous but is represented by a sequence of discrete steps as in discrete time markov chains dtmcs to this end we introduce a variant of csp that employs a probabilistic asynchronous parallel operator whose synchronization mechanism is based on a mixture of the classical generative and reactive models of probability in particular differently from existing discrete time process algebras where parallel processes are executed in synchronous locksteps the parallel operator that we adopt allows processes with different probabilistic advancing speeds mean number of actions executed per time unit to be modeled moreover our generative reactive synchronization mechanism makes it possible to always derive dtmcs in the case of fully specified systems we then present a sound and complete axiomatization of probabilistic bisimulation over finite processes of our calculus that is a smooth extension of the axiom system for a standard process algebra thus solving the open problem of cleanly axiomatizing action restriction in the generative model as a further result we show that when evaluating steady state based performance measures which are expressible by attaching rewards to actions our approach provides an exact solution even if the advancing speeds are considered not to be probabilistic without incurring the state space explosion problem that arises with standard synchronous approaches we finally present a case study on multi path routing showing the expressiveness of our calculus and that it makes it particularly easy to produce scalable specifications

multiobjective fractional variational calculus in terms of a combined caputo derivative abstract the study of fractional variational problems in terms of a combined fractional caputo derivative is introduced necessary optimality conditions of euler lagrange type for the basic isoperimetric and lagrange variational problems are proved as well as transversality and sufficient optimality conditions this allows to obtain necessary and sufficient pareto optimality conditions for multiobjective fractional variational problems

global renaming operators in concrete process algebra renaming operators are introduced in concrete process algebra concrete means that abstraction and silent moves are not considered examples of renaming operators are given encapsulation pre abstraction and localization we show that renamings enhance the defining power of concrete process algebra by using the example of a queue we give a definition of the trace set of a process see when equality of trace sets implies equality of processes and use trace sets to define the restriction of a process finally we describe processes with actions that have a side effect on a state space and show how to use this for a translation of computer programs into process algebra

a general conservative extension theorem in process algebras with inequalities we prove a general conservative extension theorem for transition system based process theories with easy to check and reasonable conditions the core of this result is another general theorem which gives sufficient conditions for a system of operational rules and an extension of it in order to ensure conservativity that is provable transitions from an original term in the extension are the same as in the original system as a simple corollary of the conservative extension theorem we prove a completeness theorem we also prove a general theorem giving sufficient conditions to reduce the question of ground confluence modulo some equations for a large term rewriting system associated with an equational process theory to a small term rewriting system under the condition that the large system is a conservative extension of the small one we provide many applications to show that our results are useful the applications include but are not limited to various real and discrete time settings in acp atp and ccs and the notions projection renaming stage operator priority recursion the silent step autonomous actions the empty process divergence etc

on a jump type stochastic fractional partial differential equation with fractional noises abstract we study a class of stochastic fractional partial differential equations of order 1 driven by a pure jump levy space time white noise and a fractional noise we prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions

absolute versus relative time in process algebras abstract timed process algebras are useful tools for the specification and verification of real time systems we study the relationships between two of these algebras c i pa closed interval process algebra and tccs temporal ccs which deal with temporal aspects of concurrent systems by following very different interpretations durational actions versus durationless actions absolute time versus relative time timed functional behavior versus time and functional behavior local clocks versus global clocks we show that these different choices are not irreconcilable by presenting simple mappings from c i pa to tccs which preserve the behavioral equivalences over the two timed calculi these results hold whenever basic actions are interpreted as either eager or lazy whenever the starting time of action execution is observed rather than their completion time a study on the size of the labelled transition systems describing the transitional semantics of cipa processes and those describing the transitional semantics of their translated versions is also presented

on a stochastic fractional partial differential equation driven by a levy space time white noise we study the existence and uniqueness of the global mild solution for a stochastic fractional partial differential equation driven by a levy space time white noise moreover the flow property for the solution is also studied

divide and congruence from decomposition of modal formulas to preservation of branching and η bisimilarity we present a method for decomposing modal formulas for processes with the internal action t to decide whether a process algebra term satisfies a modal formula one can check whether its subterms satisfy formulas that are obtained by decomposing the original formula the decomposition uses the structural operational semantics that underlies the process algebra we use this decomposition method to derive congruence formats for two weak and rooted weak semantics branching and h bisimilarity

quantales finite observations and strong bisimulation abstract it has often been claimed that bisimulation makes distinctions that cannot be observed in practice abramsky and vickers proposed an algebraic framework based on quantales for describing observations on concurrent processes without hidden transitions and used it in order to provide an observational explanation of several process equivalences ranging from hoare trace equivalence to ready simulation we follow their approach and argue that strong bisimulation can be explained in the same way at least in the case of image computable labelled transition systems

cpo models for compact gsos languages in this paper we present a general way of giving denotational semantics to a class of languages equipped with an operational semantics that fits the gsos format of bloom istrail and meyer the canonical model used for this purpose will be abramsky s domain of synchronization trees and the denotational semantics automatically generated by our methods will be guaranteed to be fully abstract with respect to the finitely observable part of the bisimulation preorder in the process of establishing the full abstraction result we also obtain several general results on the bisimulation preorder including a complete axiomatization for it and give a novel operational interpretation of gsos languages

observation equivalence as a testing equivalence a notion of testing is developed for transition systems with divergence the forms of testing include traces refusals copying and global testing both denotational and operational formulations of testing are given the equivalence based on this notion of testing is shown to coincide with observation equivalence

ntyft ntyxt rules reduce to ntree rules groote and vaandrager introduced thetyft tyxt formatfor transition system specifications tsss and established that for each tss in this format that iswell founded the bisimulation equivalence it induces is a congruence in this paper we construct for each tss in tyft tyxt format an equivalent tss that consists oftree rulesonly as a corollary we can give an affirmative answer to an open question namely whether the well foundedness condition in the congruence theorem for tyft tyxt can be dropped these results extend to tyft tyxt with negative premises and predicates

stability robustness bounds for linear state space models with structured uncertainty in this note we consider the robust stability analysis problem in linear state space models we consider systems with structured uncertainty some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived these bounds are shown to be less conservative than the existing ones

a general finite element formulation for fractional variational problems abstract this paper presents a general finite element formulation for a class of fractional variational problems fvps the fractional derivative is defined in the riemann liouville sense for fvps the euler lagrange and the transversality conditions are developed in the fractional finite element formulation ffef presented here the domain of the equations is divided into several elements and the functional is approximated in terms of nodal variables minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme three examples are considered to show the performance of the algorithm results show that as the number of discretization is increased the numerical solutions approach the analytical solutions and as the order of the derivative approaches an integer value the solution for the integer order system is recovered for unspecified boundary conditions the numerical solutions satisfy the transversality conditions this indicates that for the class of problems considered the numerical solutions can be obtained directly from the functional and there is no need to solve the fractional euler lagrange equations thus the formulation extends the traditional finite element approach to fvps

the meaning of negative premises in transition system specifications we present a general theory for the use of negative premises in the rules of transition system specifications tsss we formulate a criterion that should be satisfied by a tss in order to be meaningful that is to unequivocally define a transition relation we also provide powerful techniques for proving that a tss satisfies this criterion meanwhile constructing this transition relation both the criterion and the techniques originate from logic programming van gelder et al 1988 gelfond and lifschitz 1988 to which tsss are close in an appendix we provide an extensive comparison between them as in groote 1993 we show that the bisimulation relation induced by a tss is a congruence provided that it is in ntyft ntyxt format and can be proved meaningful using our techniques we also considerably extend the conservativity theorems of groote 1993 and groote and vaandrager 1992 as a running example we study the combined addition of priorities and abstraction to basic process algebra bpa under some reasonable conditions we show that this tss is indeed meaningful which could not be shown by other methods bloom et al 1995 groote 1993 finally we provide a sound and complete axiomatization for this example

incomplete normalization of probability on multifractals abstract this work is an extension of the incomplete probability theory from the simple case of monofractals previously studied to the more general case of multifractals that can occur in the phase space without equiprobable partition

divide and congruence iii from decomposition of modal formulas to preservation of stability and divergence abstract in two earlier papers we derived congruence formats with regard to transition system specifications for weak semantics on the basis of a decomposition method for modal formulas the idea is that a congruence format for a semantics must ensure that the formulas in the modal characterisation of this semantics are always decomposed into formulas that are again in this modal characterisation the stability and divergence requirements that are imposed on many of the known weak semantics have so far been outside the realm of this method stability refers to the absence of a transition we show using the decomposition method how congruence formats can be relaxed for weak semantics that are stability respecting this relaxation for instance brings the priority operator within the range of the stability respecting branching bisimulation format divergence which refers to the presence of an infinite sequence of transitions escapes the inductive decomposition method we circumvent this problem by proving that a congruence format for a stability respecting weak semantics is also a congruence format for its divergence preserving counterpart

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

algebraic laws for nondeterminism and concurrency since a nondeterministic and concurrent program may in general communicate repeatedly with its environment its meaning cannot be presented naturally as an input output function as is often done in the denotational approach to semantics in this paper an alternative is put forth first a definition is given of what it is for two programs or program parts to be equivalent for all observers then two program parts are said to be observation congruent if they are in all program contexts equivalent the behavior of a program part that is its meaning is defined to be its observation congruence class the paper demonstrates for a sequence of simple languages expressing finite terminating behaviors that in each case observation congruence can be axiomatized algebraically moreover with the addition of recursion and another simple extension the algebraic language described here becomes a calculus for writing and specifying concurrent programs and for proving their properties

robust controller synthesis using the maximum entropy design equations this note presents an application of the optimality conditions obtained in 1 for dynamic compensation in the presence of state control and measurement dependent noise by solving these equations which represent a fundamental generalization of standard steady state lqg theory a series of increasingly robust control designs is obtained for the example considered in 2

calculi for synchrony and asynchrony abstract a calculus for distributed computation is studied based upon four combinators a central idea is an abelian group of actions which models the interfaces between components of a distributed computing agent using a notion of bisimulation congruence relations are defined over computing agents and thence an algebraic theory is derived the calculus models both synchronous and asynchronous computation in particular it is shown that the author s calculus of communicating systems 1980 which is an asynchronous model is derivable from the calculus presented here

covariance averaging in the analysis of uncertain systems the effects of parametric uncertainty in stable state space systems are analyzed by averaging the state covariance over the statistics of the uncertain parameters for natural frequency uncertainty this computation is related to the fourier transform of the probability density function of the uncertain parameter equipartition and incoherence are illustrated for a single mode oscillator averaging over a discrete uncertainty model yields the bourret design equations while averaging over a cauchy uncertainty distribution yields the maximum entropy covariance equation of d c hyland 1982

a format for semantic equivalence comparison this paper presents a new format for process algebras the extended tyft tyxt format which generalises the tyft tyxt format of groote and vaandrager the format differs from most previous formats in that the labels on transitions are treated as many sorted terms bisimulation is a congruence for all operators defined by extended transition system specifications in this format when one extended transition system specification is summed with another the resulting bisimulation can either identify more terms an abstracting extension up to bisimulation or fewer terms a refining extension up to bisimulation than the original bisimulation on the individual system the notions of abstracting extension and refining extension are defined and two theorems are presented giving conditions required for achieving each type of extension these results provide a way to compare different semantic equivalences defined for different process algebras finally an application of this theory to semantic equivalence comparison is given for a new result relating castellani s pomset equivalence and krishan s multiprocessor equivalence

on the operational semantics of nondeterminism and divergence abstract an operational model of nondeterministic processes coupled with a novel theory of divergence is presented the operational model represents internal nondeterminism without using explicit internal transitions here the notion of internal state effectively replaces the familiar notion of internal transition giving rise to an alternative operational view of processes the weak process roughly a weak process is a collection of stable internal states together with a set of transitions each of which is defined from an internal state to another weak process internal nondeterminism arises from such refinement of processes into multiple internal states a simple extension to the basic weak process model gives rise to an elaborate operational theory of divergence according to this theory the ability of a process to undertake an infinite internal computation which is pathological or persistent is distinguished from its ability to undertake an infinite internal computation which is not although applicable to process algebraic languages with an internal action construct the resulting model is most suitable for supplying operational semantics to process algebras which express internal nondeterminism by an internal choice construct the distinction between the two forms of divergence is in particular taken into account when the hiding construct of such a process algebra is assigned a weak process semantics

a comparison of statecharts step semantics the paper studies some variants of statecharts step semantics in the framework of structural operational semantics the chosen framework allows to study precongruence and congruence properties of behavioral preorders and equivalences and to compare with respect to these properties the different step semantics considered

on the axiomatisability of priority iii priority strikes again abstract aceto et al proved that over the process algebra bccsp with the priority operator of baeten bergstra and klop the equational theory of order insensitive bisimilarity is not finitely based however it was noticed that by substituting the action prefixing operator of bccsp with bpa s sequential composition the infinite family of equations used to show that non finite axiomatisability result could be proved by a finite collection of sound equations that observation left as an open question the existence of a finite axiomatisation for order insensitive bisimilarity over bpa with the priority operator in this paper we provide a negative answer to this question we prove that in the presence of at least two actions order insensitive bisimilarity is not finitely based over bpa with priority

robust static and dynamic output feedback stabilization deterministic and stochastic perspectives three parallel gaps in robust feedback control theory are examined sufficiency versus necessity deterministic versus stochastic uncertainty modeling and stability versus performance deterministic and stochastic output feedback control problems are considered with both static and dynamic controllers the static and dynamic robust stabilization problems involve deterministically modeled bounded but unknown measurable time varying parameter variations while the static and dynamic stochastic optimal control problems feature state control and measurement dependent white noise general sufficiency conditions for the deterministic problems are obtained using lyapunov s direct method while necessary conditions for the stochastic problems are derived as a consequence of minimizing a quadratic performance criterion the sufficiency tests are then applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems as an additional application of the deterministic result the modified riccati equation approach of petersen and hollot is generalized in the static case and extended to dynamic compensation

a generalized fractional kn equation hierarchy and its fractional hamiltonian structure a generalized hamiltonian structure of the fractional soliton equation hierarchy is presented by using differential forms and exterior derivatives of fractional orders we construct the generalized fractional trace identity through the riemann liouville fractional derivative an example of the fractional kn soliton equation hierarchy and hamiltonian structure is presented which is a new integrable hierarchy and possesses hamiltonian structure