necessary conditions for ternary algebras ternary algebras constructed from ternary commutators or as we call them ternutators defined as the alternating sum of products of three operators have been shown to satisfy cubic identities as necessary conditions for their existence here we examine the situation where we permit identities not solely constructed from ternutators or nested ternutators and we find that in general these impose additional restrictions for example the anti commutators or commutators of the operators must obey some linear relations among themselves

effective critical exponents in two dimensional systems from conformal field theory abstract thermal and field perturbations away from the renormalization group fixed point are considered in 2d conformal field theory general expressions for the effective critical exponents and are obtained in terms of the central charge third moment of energy and spin correlation temperature and external field apparent spin dependence expressions for the effective critical exponents are given for a high spin system with only one fixed point these expression may be described as critical behavior and crossover phenomena for 2d systems away from the critical points

n ary algebras a review with applications this paper reviews the properties and applications of certain n ary generalizations of lie algebras in a self contained and unified way these generalizations are algebraic structures in which the two entries lie bracket has been replaced by a bracket with n entries each type of n ary bracket satisfies a specific characteristic identity which plays the role of the jacobi identity for lie algebras particular attention will be paid to generalized lie algebras which are defined by even multibrackets obtained by antisymmetrizing the associative products of its n components and that satisfy the generalized jacobi identity and to filippov or n lie algebras which are defined by fully antisymmetric n brackets that satisfy the filippov identity three lie algebras have surfaced recently in multi brane theory in the context of the bagger lambert gustavsson model because of this filippov algebras will be discussed at length including the cohomology complexes that govern their central extensions and their deformations it turns out that whitehead s lemma extends to all semisimple n lie algebras when the skewsymmetry of the lie or n lie algebra bracket is relaxed one is led to a more general type of n algebras the n leibniz algebras these will be discussed as well since they underlie the cohomological properties of n lie algebras the standard poisson structure may also be extended to the n ary case we shall review here the even generalized poisson structures whose generalized jacobi identity reproduces the pattern of the generalized lie algebras and the nambu poisson structures which satisfy the filippov identity and determine filippov algebras finally the recent work of bagger lambert and gustavsson on superconformal chern simons theory will be briefly discussed emphasis will be made on the appearance of the 3 lie algebra structure and on why the a4 model may be formulated in terms of an ordinary lie algebra and on its nambu bracket generalization

higher identities for the ternary commutator we use computer algebra to study polynomial identities for the trilinear operation a b c abc acb bac bca cab cba in the free associative algebra it is known that a b c satisfies the alternating property in degree 3 no new identities in degree 5 a multilinear identity in degree 7 which alternates in 6 arguments and no new identities in degree 9 we use the representation theory of the symmetric group to demonstrate the existence of new identities in degree 11 the only irreducible representations of dimension 400 with new identities correspond to partitions 25 1 and 24 13 and have dimensions 132 and 165 we construct an explicit new multilinear identity for partition 25 1 and we demonstrate the existence of a new non multilinear identity in which the underlying variables are permutations of a2b2c2d2e2f

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

approach for transforming linear constraints on petri nets an efficient supervisor synthesis method is proposed based on the constraint transformation since enforcing a logic expression of linear constraints on petri nets essentially involves finding admissible constraints it consists of cooperative algorithms that are used to obtain the logic expression of constraints for any set of forbidden markings to remove redundant constraints using the quantity information of constraints to judge the existence of a permissive supervisor to equivalently reduce multiple constraints into a single one using net structural properties and to approach the admissible constraints from the original ones for ordinary petri nets these algorithms are derived from a series of theorems and lemmas presented to construct the theory of constraint transformation or reduction in addition the linear constraint whose uncontrollable subnet is a forward concurrent free net can be equivalently transformed into an admissible linear constraint using this method

geometry of four vector fields on quaternionic flag manifolds the purpose of this paper is to describe certain natural 4 vector fields on quaternionic flag manifolds which geometrically determine the bruhat cell decomposition these structures naturally descend from the symplectic group sp n and are related to the dressing action given by the iwasawa decomposition of the general linear group over the quaternions gln

hamiltonian and quasi hamiltonian systems nambu poisson structures and symmetries the theory of hamiltonian and quasi hamiltonian systems with respect to nambu poisson structures is studied it is proved that if a dynamical system is endowed with certain properties related to the theory of symmetries then it can be considered as a quasi hamiltonian or hamiltonian system with respect to an appropriate nambu poisson structure several examples of this construction are presented these examples are related to integrability and also to superintegrability

checking cardy s formula for the baxter model a general formula recently obtained by cardy 1988 which relates the second moment of the energy density correlator away from criticality to the value of the conformal anomaly is verified in the baxter model 1971 this result is the first analytical confirmation of cardy s formula in a non trivial interacting system

free akivis algebras primitive elements and hyperalgebras free akivis algebras and primitive elements in their universal enveloping algebras are investigated it is proved that subalgebras of free akivis algebras are free and that finitely generated subalgebras are finitely residual decidability of the word problem for the variety of akivis algebras is also proved the conjecture of k h hofmann and k strambach problem 6 15 in topological and analytic loops in quasigroups and loops theory and applications series in pure mathematics o chein h o pflugfelder and j d h smith eds vol 8 pp 205 262 heldermann verlag berlin 1990 on the structure of primitive elements is proved to be not valid and a full system of primitive elements in free nonassociative algebra is constructed finally it is proved that every algebra b can be considered as a hyperalgebra that is a system with a series of multilinear operations that plays a role of a tangent algebra for a local analytic loop where the hyperalgebra operations on b are interpreted by certain primitive elements

dynamics of generalized poisson and nambu poisson brackets a unified setting for generalized poisson and nambu poisson brackets is discussed it is proved that a nambu poisson bracket of even order is a generalized poisson bracket characterizations of poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and lagrangian submanifolds of product and tangent manifolds respectively

on a class of n leibniz deformations of the simple filippov algebras we study the problem of infinitesimal deformations of all real simple finite dimensional filippov or n lie algebras considered as a class of n leibniz algebras characterized by having an n bracket skewsymmetric in its n 1 first arguments we prove that all n 3 simple finite dimensional filippov algebras fas are rigid as n leibniz algebras of this class this rigidity also holds for the leibniz deformations of the semisimple n 2 filippov i e lie algebras the n 3 simple fas however admit a nontrivial one parameter infinitesimal 3 leibniz algebra deformation we also show that the n 3 simple filippov algebras do not admit nontrivial central extensions as n leibniz algebras of the above class

homology and cohomology on generalized poisson manifolds the canonical homology of a generalized poisson manifold is introduced and the two spectral sequences associated with the periodic double complex are studied the generalized poisson cohomology is also discussed and a natural parity with the canonical homology is established the results hold for nambu poisson manifolds

dirac reduction of dual poisson presymplectic pairs a new notion of a dual poisson presymplectic pair is introduced and its properties are examined the procedure of dirac reduction of poisson operators onto submanifolds proposed by dirac is in this paper embedded in a geometric procedure of reduction of dual poisson presymplectic pairs the method presented generalizes those introduced by j marsden and t ratiu for reductions of poisson manifolds two examples are given

contractions of filippov algebras we introduce in this paper the contractions gc of n lie or filippov algebras g and show that they have a semidirect structure as their n 2 lie algebra counterparts as an example we compute the nontrivial contractions of the simple an 1 filippov algebras by using the inonu wigner and the generalized weimar woods contractions of ordinary lie algebras we compare in the g an 1 simple case the lie algebras liegc the lie algebra of inner endomorphisms of gc with certain contractions lie g iw and lie g w w of the lie algebra lieg associated with g

control of vector discrete event systems ii controller synthesis this paper shows that associated control synthesis problems of ramadge wonham type can be efficiently solved under mild technical restrictions using elementary methods combined with linear integer programming both memoryless and dynamic state feedback are treated structural conditions are provided under which the controller can be computed in closed form namely represented as a vdes or generalized vdes gvdes the results are applied to a model of an agv system adapted from work of holloway and krogh 1990

identities for the ternary commutator abstract this paper classifies all identities of degree 7 satisfied by the ternary commutator in an associative ternary algebra seven is the lowest degree for which non trivial identities exist these identities are ternary generalizations of the jacobi identity for lie algebras

multi operator brackets acting thrice we generalize an identity first found by bremner for nambu 3 brackets for odd n brackets built from associative operator products we show that for any fixed a when totally antisymmetrized over all the bs

new generalized poisson structures new generalized poisson structures are introduced by using suitable skew symmetric contravariant tensors of even order the corresponding jacobi identities are provided by conditions on these tensors which may be understood as cocycle conditions as an example we provide the linear generalized poisson structures which can be constructed on the dual spaces of simple lie algebras

a suboptimal deadlock control policy for designing non blocking supervisors in flexible manufacturing systems this work aims to resolve deadlock problems in flexible manufacturing systems a novel deadlock control policy in the frame of petri nets formalism is presented for a class of generalized petri nets namely g systems which can usually model assembly and disassembly operations of multiple resource acquisition based on the concept of elementary siphons all strict minimal siphons sms in an uncontrolled net system are first divided into elementary siphons and dependent ones after that a set of linear inequality constraints expressed by state vectors can be formalized for elementary siphons after being modified by utilizing the conjunctive disjunctive resources upstream neighborhood policy a set of generalized mutual exclusion constraints expressed by marking vectors can be obtained then the additional monitors are only designed for elementary siphons whose control depth variables can be obtained by solving a linear programming problem as a result the controllability of dependent ones can be ensured by properly supervising its elementary siphons finally all strict minimal siphons are max controlled and no insufficiently marked siphon is generated which indicates that the sequential resources are allocated reasonably to guarantee the absence of deadlock states the proposed method can usually lead to a near optimal non blocking supervisor with simple structure a g system example prone to deadlocks is used to illustrate the applicability and the effectiveness of the proposed method

alternating quaternary algebra structures on irreducible representations of sl2 c abstract we determine the multiplicity of the irreducible representation v n of the simple lie algebra sl 2 c as a direct summand of its fourth exterior power 4 v n the multiplicity is 1 resp 2 if and only if n 4 6 resp n 8 10 for these n we determine the multilinear polynomial identities of degree 7 satisfied by the sl 2 c invariant alternating quaternary algebra structures obtained from the projections 4 v n v n we represent the polynomial identities as the nullspace of a large integer matrix and use computational linear algebra to find the canonical basis of the nullspace

quadratic poisson structures and nambu mechanics abstract we study a special class of poisson structures namely quadratic poisson structures on r n which contains the celebrated sklyanin algebras which play the significant role in the theory of exactly solvable models of classical and quantum systems in this paper we study such quadratic poisson structures via nambu mechanics

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

an application of lattice basis reduction to polynomial identities for algebraic structures abstract the authors recent classification of trilinear operations includes among other cases a fourth family of operations with parameter q q and weakly commutative and weakly anticommutative operations these operations satisfy polynomial identities in degree 3 and further identities in degree 5 for each operation using the row canonical form of the expansion matrix e to find the identities in degree 5 gives extremely complicated results we use lattice basis reduction to simplify these identities we compute the hermite normal form h of et obtain a basis of the nullspace lattice from the last rows of a matrix u for which uet h and then use the lll algorithm to reduce the basis

identities for generalized lie and jordan products on totally associative triple systems abstract we determine the identities of degree 9 satisfied by the new ternary operations abc bca cab acb bac cba symmetric sum abc bca cab acb bac cba alternating sum and abc bca cab cyclic sum on every triple system satisfying the total associativity identities abc de a bcd e ab cde these new operations are ternary analogs of the lie and jordan products the identities they satisfy may be regarded as ternary versions of the jacobi and jordan identities

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

improved harmonic approximation and the 2d ising model at t tc and h 0 we propose a new method to determine the unknown parameter associated with a self consistent harmonic approximation we check the validity of our technique in the context of the sine gordon model as a non trivial application we consider the scaling regime of the 2d ising model away from the critical point and in the presence of a magnetic field h we derive an expression that relates the approximate correlation length t tc and h

virtual copies of semisimple lie algebras in enveloping algebras of semidirect products and casimir operators given a semidirect product g s r of semisimple lie algebras s and solvable algebras r we construct polynomial operators in the enveloping algebra u g of g that commute with r and transform like the generators of s up to a functional factor that turns out to be a casimir operator of r such operators are said to generate a virtual copy of s in u g and allow to compute the casimir operators of g in closed form using the classical formulae for the invariants of s the behavior of virtual copies with respect to contractions of lie algebras is analyzed applications to the class of hamilton algebras and their inhomogeneous extensions are given

jordan quadruple systems abstract we define jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the jordan tetrad a b c d a b c d d c b a as a quadrilinear operation on associative algebras we find further identities in degree 10 which are not consequences of the defining identities we introduce four infinite families of finite dimensional jordan quadruple systems and construct the universal associative envelope for a small system in each family we obtain analogous results for the anti tetrad a b c d a b c d d c b a our methods rely on computer algebra especially linear algebra on large matrices the lll algorithm for lattice basis reduction representation theory of the symmetric group noncommutative grobner bases and wedderburn decompositions of associative algebras

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

the z2 graded schouten nijenhuis bracket and generalized super poisson structures the super or z2 graded schouten nijenhuis bracket is introduced using it new generalized super poisson structures are found which are given in terms of certain graded skew symmetric contravariant tensors of even order the corresponding super jacobi identities are expressed by stating that these tensors have a zero super schouten nijenhuis bracket with themselves 0 as a particular case we provide the linear generalized super poisson structures which can be constructed on the dual spaces of simple superalgebras with a non degenerate killing metric the su 3 1 superalgebra is given as a representative example

brief paper maximally permissive supervisor synthesis based on a new constraint transformation method the method is proposed to design the maximally permissive and efficient supervisor for enforcing linear constraints in which the weights of places are not negative on ordinary petri nets with uncontrollable transitions first the weakly admissible linear constraint is introduced second a method is proposed to design the monitor place for enforcing a weakly admissible linear constraint on petri nets third a theorem proving that a linear constraint can be equivalently transformed at an uncontrollable transition into a disjunction of new constraints is proposed fourth using this theorem an algorithm is presented to equivalently transform a linear constraint each place weight of which is not negative into a disjunction of weakly admissible ones lastly the supervisor which consists of the plant net and a set of monitor places is designed for the weakly admissible linear constraints calculated by the above algorithm

characterization of admissible marking sets in petri nets with uncontrollable transitions this work studies the equivalent transformation from a legal marking set to its admissible marking set first the concepts of escaping marking set and transforming marking set are defined and two algorithms are provided to compute the transforming marking set and the admissible marking set second the equivalent transformation of a disjunction of linear constraints expressed in terms of generalized mutual exclusion constraints gmecs with non negative weights via uncontrollable transitions is established

universal amplitude in the sizes of rings in two dimensions for self avoiding rings of n steps in two dimensions the limiting value as n to infinity of the combination npn r2 nxcn where pn is the number of distinct rings r2 n is their mean square radius of gyration and xc is the critical fugacity is equal to a calculable lattice dependent number times a universal amplitude this latter quantity is calculated exactly using methods of conformal invariance the value is in good agreement with the results of enumeration studies

an envelope for bol algebras abstract we prove that bol algebras arise as primitive elements of certain bialgebras which generalize the usual universal enveloping algebras of lie and malcev algebras the bol algebra is located inside the generalized left alternative nucleus of the envelope and its binary and ternary products are naturally recovered from the product of the envelope

deformation quantization and nambu mechanics starting from deformation quantization star products the quantization problem of nambu mechanics is investigated after considering some impossibilities and pushing some analogies with field quantization a solution to the quantization problem is presented in the novel approach of zariski quantization of fields observables functions in this case polynomials this quantization is based on the factorization over of polynomials in several real variables we quantize the infinite dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is abelian associative and distributive this procedure is then adapted to derivatives needed for the nambu brackets which ensures the validity of the fundamental identity of nambu mechanics also at the quantum level our construction is in fact more general than the particular case considered here it can be utilized for quite general defining identities and for much more general star products

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

generalization of n ary nambu algebras and beyond the aim of this paper is to introduce n ary hom algebra structures generalizing the n ary algebras of lie type including n ary nambu algebras n ary nambu lie algebras and n ary lie algebras and n ary algebras of associative type including n ary totally associative and n ary partially associative algebras we provide examples of the new structures and present some properties and construction theorems we describe the general method allowing one to obtain an n ary hom algebra structure starting from an n ary algebra and an n ary algebra endomorphism several examples are derived using this process also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three dimensional hom nambu lie structures with diagonal homomorphism

poisson jacobi reduction of homogeneous tensors the notion of homogeneous tensors is discussed we show that there is a one to one correspondence between multivector fields on a manifold m homogeneous with respect to a vector field on m and first order polydifferential operators on a closed submanifold n of codimension 1 such that is transversal to n this correspondence relates the schouten nijenhuis bracket of multivector fields on m to the schouten jacobi bracket of first order polydifferential operators on n and generalizes the poissonization of jacobi manifolds actually it can be viewed as a super poissonization this procedure of passing from a homogeneous multivector field to a first order polydifferential operator can also be understood as a sort of reduction in the standard case a half of a poisson reduction a dual version of the above correspondence yields in particular the correspondence between homogeneous symplectic structures on m and contact structures on n

higher order simple lie algebras it is shown that the non trivial cocycles on simple lie algebras may be used to introduce antisymmetric multibrackets which lead to higher order lie algebras the definition of which is given their generalised jacobi identities turn out to be satisfied by the antisymmetric tensors or higher order structure constants which characterise the lie algebra cocycles this analysis allows us to present a classification of the higher order simple lie algebras as well as a constructive procedure for them our results are synthesised by the introduction of a single complete brst operator associated with each simple algebra

nonhomogeneous subalgebras of lie and special jordan superalgebras abstract we consider polynomial identities satisfied by nonhomogeneous subalgebras of lie and special jordan superalgebras we ignore the grading and regard the superalgebra as an ordinary algebra the lie case has been studied by volichenko and baranov they found identities in degrees 3 4 and 5 which imply all the identities in degrees 6 we simplify their identities in degree 5 and show that there are no new identities in degree 7 the jordan case has not previously been studied we find identities in degrees 3 4 5 and 6 which imply all the identities in degrees 6 and demonstrate the existence of further new identities in degree 7 our proofs depend on computer algebra we use the representation theory of the symmetric group the hermite normal form of an integer matrix the lll algorithm for lattice basis reduction and the chinese remainder theorem

on lie k algebras abstract we define the notion of a lie k algebra to be a k 1 ary skew symmetric operation on a bigraded vector space which satisfies a certain relation of degree 2 k 1 the notion of lie 1 algebra coincides with the notion of lie superalgebra an ordinary lie algebra is precisely a lie 1 algebra with odd elements we show first that the boundary map in the koszul complex constructed as the koszul complex for ordinary lie algebras squares to zero we then show that the 1 nk 1 homogeneous part of the free lie k algebra with nk 1 even generators is isomorphic as an s nk 1 module to the cohomology of 1 nk 1 the poset of all partitions of nk 1 in which every block size is congruent to 1 mod k this result is analogous to a classical result relating the free lie algebra with n generators to the cohomology of the partition lattice we also construct an explicit basis for the 1 nk 1 homogeneous part of the free lie k algebra with nk 1 even generators and for the cohomology of 1 nk 1 lastly we compute the lie k algebra homology of the free lie k algebra

brief paper suboptimal supervisory control of petri nets in presence of uncontrollable transitions via monitor places this paper deals with the problem of enforcing generalized mutual exclusion constraints gmec on place transition nets with uncontrollable transitions an efficient control synthesis technique which has been proposed in the literature enforces gmec constraints by introducing monitor places to create suitable place invariants the method has been shown to be maximally permissive and to give a unique control structure in the case that the set of legal markings is controllable this paper investigates on and formally shows that the class of controllers obtained by this technique may not have a supremal element for uncontrollable specifications moreover it is shown that the family of monitor places enforcing an uncontrollable specification can be parameterized with respect to the solution of a linear system of equation an algorithm to obtain such parameterization is presented here

on special classes of n algebras we define n algebras as linear spaces on which the internal composition law involves n elements m v n v it is known that such algebraic structures are interesting for their applications to problems of modern mathematical physics using the notion of a commutant of two subalgebras of an n algebra we distinguish certain classes of n algebras with reasonable properties semisimple abelian nilpotent solvable we also consider a few examples of n algebras of different types and show their properties

on the geometry of lie algebras and poisson tensors a geometric programme to analyse the structure of lie algebras is presented with special emphasis on the geometry of linear poisson tensors the notion of decomposable poisson tensors is introduced and an algorithm to construct all solvable lie algebras is presented poisson liouville structures are also introduced to discuss a new class of lie algebras which include as a subclass semi simple lie algebras a decomposition theorem for poisson tensors is proved for a class of poisson manifolds including linear ones simple lie algebras are also discussed from this viewpoint and lower dimensional real lie algebras are analysed

remarks on nambu poisson and nambu jacobi brackets we show that nambu poisson and nambu jacobi brackets can be defined inductively an n bracket n 2 is nambu poisson respectively nambu jacobi if and only if on fixing an argument we obtain an n 1 nambu poisson respectively nambu jacobi bracket as a by product we obtain relatively simple proofs of darboux type theorems for these structures

varieties of anticommutativen ary algebras abstract a fundamental problem in the theory of n ary algebras is to determine the correct generalization of the jacobi identity this paper describes some computational results on this problem using representations of the symmetric group it is well known that over a field of characteristic 0 any variety of n ary algebras can be defined by multilinear identities in the anticommutative case it is shown that for n 8 the formula dimensional s 2 n 1 module of multilinear identities in which each term involves two n ary products i e two pairs of n ary anticommutative brackets decomposes as the direct sum of the n distinct simple modules labelled by the n partitions of 2 n 1 in which only 1 and 2 occur as parts in the cases n 3 resp n 4 the kernel of the commutator expansion map and a generator for each of the 7 resp 15 nonzero submodules are determined the paper concludes with some conjectures for n 5

on one class of exact poisson structures we discuss some properties of a natural class of poisson structures on euclidean spaces and abstract manifolds in particular it is proved that such structures are always exact and may be reconstructed from their casimir functions it is shown that in low dimensions they give the whole class of exact poisson structures the dimension of poisson homology of these structures is computed in terms of the milnor number of their casimir functions we also analyze some concrete examples of such structures in low dimensions and show that their centers are generated by casimir functions

simultaneous reduction of petri nets and linear constraints for efficient supervisor synthesis due to state space explosion and uncontrollable events in discrete event systems it is very difficult to design supervisors to enforce user defined linear constraints and ensure the liveness of their petri net pn models with complex structures different from all the existing methods which are to transform original constraints into admissible or weakly admissible ones the method proposed in this work aims to reduce linear constraints and pn models simultaneously as a result an original pn control problem is equivalently reduced to a simpler one i e the optimal supervisors for them make the same restriction on the behavior of a discrete event system moreover it can be guaranteed that the original pn system is live if and only if the reduced one is so since the state space of a pn may grow exponentially with its size and the sizes of real discrete event systems are often too large to handle the proposed method is useful to greatly reduce the computational complexity of both property analysis and supervisor synthesis of discrete event systems

synthesis of supervisors enforcing general linear constraints in petri nets efficient techniques exist for the design of supervisors enforcing constraints consisting of linear marking inequalities this note shows that without losing the benefits of the prior techniques the class of constraints can be generalized to linear constraints containing marking terms firing vector terms and parikh vector terms we show that this extended class of constraints is more expressive furthermore we show that the extended constraints can describe any supervisor consisting of control places arbitrarily connected to the transitions of a plant petri net pn the supervisor design procedure we propose is as follows for pns without uncontrollable and unobservable transitions a direct method for the design of a pn supervisor that is least restrictive is given for pns with uncontrollable and or unobservable transitions we reduce the problem to the design of supervisors enforcing linear marking inequalities

ternary analogues of lie and malcev algebras abstract we consider two analogues of associativity for ternary algebras total and partial associativity using the corresponding ternary associators we define ternary analogues of alternative and assosymmetric algebras on any ternary algebra the alternating sum a b c abc acb bac bca cab cba the ternary analogue of the lie bracket defines a structure of an anticommutative ternary algebra we determine the polynomial identities of degree 7 satisfied by this operation in totally and partially associative alternative and assosymmetric ternary algebras these identities define varieties of ternary algebras which can be regarded as ternary analogues of lie and malcev algebras our methods involve computational linear algebra based on the representation theory of the symmetric group

the partially alternating ternary sum in an associative dialgebra the alternating ternary sum in an associative algebra gives rise to the partially alternating ternary sum in an associative dialgebra with products and by making the argument a the center of each term we use computer algebra to determine the polynomial identities in degree 9 satisfied by this new trilinear operation in degrees 3 and 5 we obtain these identities define a new variety of partially alternating ternary algebras we show that there is a 49 dimensional space of multilinear identities in degree 7 and we find equivalent nonlinear identities we use the representation theory of the symmetric group to show that there are no new identities in degree 9

simple facts concerning nambu algebras a class of substitution equations arising in the extension of jacobi identity for n gebras is studied and solved graded bracket and cohomology adapted to the study of formal deformations are presented new identities in the case of nambu lie algebras are proved the triviality in the gerstenhaber sense of certain deformed n skew symmetric brackets satisfying the leibniz rule with respect to a star product is shown for n 3

the schouten nijenhuis bracket cohomology and generalized poisson structures newly introduced generalized poisson structures based on suitable skew symmetric contravariant tensors of even order are discussed in terms of the schouten nijenhuis bracket the associated jacobi identities are expressed as conditions on these tensors the cohomological contents of which is given in particular we determine the linear generalized poisson structures which can be constructed on the dual spaces of simple lie algebras

nambu jacobi and generalized jacobi manifolds the geometry of nambu jacobi and generalized jacobi manifolds is studied a large collection of examples is given the characteristic distribution generated by the hamiltonian vector fields on a nambu jacobi manifold is proved to be completely integrable and the induced geometrical structure of the leaves of the corresponding generalized foliation is ellucidated

k leibniz algebras from lower order ones from lie triple to lie l ple systems two types of higher order lie l ple systems are introduced in this paper they are defined by brackets with l 3 arguments satisfying certain conditions and generalize the well known lie triple systems one of the generalizations uses a construction that allows us to associate a 2n 3 leibniz algebra l with a metric n leibniz algebra l by using a 2 n 1 linear kasymov trace form for l some specific types of k leibniz algebras relevant in the construction are introduced as well both higher order lie l ple generalizations reduce to the standard lie triple systems for l 3

on the higher order generalizations of poisson structures the characterization of the nambu poisson n tensors as a subfamily of the generalized poisson ones recently introduced and here extended to the odd order case is discussed the homology and cohomology complexes of both structures are compared and some physical considerations are made

generalized jacobi structures jacobi brackets a generalization of standard poisson brackets in which leibniz s rule is replaced by a weaker condition are extended to brackets involving an arbitrary even number of functions this new structure includes as a particular case the recently introduced generalized poisson structures the linear case on simple group manifolds is also studied and non trivial examples different from those coming from generalized poisson structures of this new construction are found by using the cohomology ring of the given group