an extended kuhn tucker approach for linear bilevel programming kuhn tucker approach has been applied with remarkable success in linear bilevel programming blp however it still has some extent unsatisfactory and incomplete one principle challenges is that it could not well handle a linear blp problem when the constraint functions at the upper level are of arbitrary linear form this paper describes theoretical foundation of kuhn tucker approach and proposes an extended kuhn tucker approach to deal with the problem the results have demonstrated that the extended kuhn tucker approach can solve a wider class of linear blp problems can than current capabilities permit

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self contained and unified way these generalizations are algebraic structures in which the two entries lie bracket has been replaced by a bracket with n entries each type of n ary bracket satisfies a specific characteristic identity which plays the role of the jacobi identity for lie algebras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

adaptive stochastic cellular automata theory abstract the mathematical concept of cellular automata has been generalized to allow for the possibility that the uniform local interaction rules that govern conventional cellular automata are replaced by nonuniform local interaction rules which are drawn from the same probability distribution function in order to guarantee the statistical homogeneity of the cellular automata system adaptation and learning in such a system can be accomplished by evolving the probability distribution function along the steepest descent direction of some objective function in a statistically unbiased way to ensure that the cellular automata s dynamical behavior approaches the desired behavior asymptotically the proposed ca model has been shown mathematically to possess the requisite convergence property under general conditions

solving quadratic convex bilevel programming problems using a smoothing method abstract in this paper we present a smoothing sequential quadratic programming to compute a solution of a quadratic convex bilevel programming problem we use the karush kuhn tucker optimality conditions of the lower level problem to obtain a nonsmooth optimization problem known to be a mathematical program with equilibrium constraints the complementary conditions of the lower level problem are then appended to the upper level objective function with a classical penalty these complementarity conditions are not relaxed from the constraints and they are reformulated as a system of smooth equations by mean of semismooth equations using fisher burmeister functional then using a quadratic sequential programming method we solve a series of smooth regular problems that progressively approximate the nonsmooth problem some preliminary computational results are reported showing that our approach is efficient

an extended kth best approach for linear bilevel programming kth best approach is one of the three popular and workable approaches for linear bilevel programming however it could not well deal with a linear bilevel programming problem when the constraint functions at the upper level are of arbitrary linear form based on our previous work this paper proposes theoretical properties of linear bilevel programming and presents an extended kth best approach for linear bilevel programming which can effectively solve above problem

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

kinematical superalgebras and lie algebras of order 3 we study and classify kinematical algebras which appear in the framework of lie superalgebras or lie algebras of order 3 all these algebras are related through generalized inonu wigner contractions from either the orthosymplectic superalgebra or the de sitter lie algebra of order 3

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

on the definitions of nucleus for dialgebras abstract malcev dialgebras were introduced recently by bremner peresi and sanchez ortega in the present paper we continue their study by introducing the notion of the generalized alternative dinucleus of a 0 dialgebra a general conjecture about the speciality of malcev dialgebras in terms of this dinucleus is formulated in the last section we introduce the appropriate generalization of the associative nucleus for dialgebras and prove an analogue of kleinfeld s theorem for the setting of dialgebras

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

on the definition of linear bilevel programming solution linear bilevel programming theory has been studied for many years by a number of researchers from different aspects yet it still remains to some extent unsatisfactory and incomplete the main challenge is how to solve a linear bilevel programming problem when the upper level s constraint functions are of arbitrary linear form this paper proposes a definition for linear bilevel programming solution the performance comparisons have demonstrated that the new model can solve a wider class of problems than current capabilities permit

hypersymmetry a z3 graded generalization of supersymmetry we propose a generalization of non commutative geometry and gauge theories based on ternary z3 graded structures in the new algebraic structures we define all products of two entities are left free the only constraining relations being imposed on ternary products these relations reflect the action of the z3 group which may be either trivial i e abc bca cab generalizing the usual commutativity or non trivial i e abc jbca with j e 2 i 3 the usual z2 graded structures such as grassmann lie and clifford algebras are generalized to the z3 graded case certain suggestions concerning the eventual use of these new structures in physics of elementary particles and fields are exposed

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

jordan triple disystems we take an algorithmic and computational approach to a basic problem in abstract algebra determining the correct generalization to dialgebras of a given variety of nonassociative algebras we give a simplified statement of the kp algorithm introduced by kolesnikov and pozhidaev for extending polynomial identities for algebras to corresponding identities for dialgebras we apply the kp algorithm to the defining identities for jordan triple systems to obtain a new variety of nonassociative triple systems called jordan triple disystems we give a generalized statement of the bso algorithm introduced by bremner and sanchez ortega for extending multilinear operations in an associative algebra to corresponding operations in an associative dialgebra we apply the bso algorithm to the jordan triple product and use computer algebra to verify that the polynomial identities satisfied by the resulting operations coincide with the results of the kp algorithm this provides a large class of examples of jordan triple disystems we formulate a general conjecture expressed by a commutative diagram relating the output of the kp and bso algorithms we conclude by generalizing the jordan triple product in a jordan algebra to operations in a jordan dialgebra we use computer algebra to verify that resulting structures provide further examples of jordan triple disystems for this last result we also provide an independent theoretical proof using jordan structure theory

special identities for the pre jordan product in the free dendriform algebra abstract pre jordan algebras were introduced recently in analogy with pre lie algebras a pre jordan algebra is a vector space a with a bilinear multiplication x y such that the product x y x y y x endows a with the structure of a jordan algebra and the left multiplications l x y x y define a representation of this jordan algebra on a equivalently x y satisfies these multilinear identities x y z u y z x u z x y u z x y u x y z u y z x u x y z u z y x u x z y u z x y u x y z u y z x u the pre jordan product x y x y y x in any dendriform algebra also satisfies these identities we use computational linear algebra based on the representation theory of the symmetric group to show that every identity of degree 7 for this product is implied by the identities of degree 4 but that there exist new identities of degree 8 which do not follow from those of lower degree there is an isomorphism of s 8 modules between these new identities and the special identities for the jordan diproduct in an associative dialgebra

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

a fast parallel horner algorithm the simple homer algorithm solves the problem of evaluating a polynomial of degree d with n indeterminates in this paper it is shown that its implementation on a parallel computer with o d processors can achieve a complexity of 2 lceil log 2 d 1 rceil cdot lceil log 2 n rceil 1 if in addition the evaluation of all partial derivatives is also sought then the full homer algorithm solves this problem on a parallel computer with o d n 1 processors achieving a parallel complexity of 2 lceil log 2 d 1 rceil cdot lceil log 2 d 1 rceil lceil log 2 n rceil 1

jordan trialgebras and post jordan algebras abstract we compute minimal sets of generators for the s n modules n 4 of multilinear polynomial identities of arity n satisfied by the jordan product and the jordan diproduct resp pre jordan product in every triassociative resp tridendriform algebra these identities define jordan trialgebras and post jordan algebras jordan analogues of the lie trialgebras and post lie algebras introduced by dotsenko et al pei et al vallette loday we include an extensive review of analogous structures existing in the literature and their interrelations in order to identify the gaps filled by our two new varieties of algebras we use computer algebra linear algebra over finite fields representation theory of symmetric groups to verify in both cases that every polynomial identity of arity 6 is a consequence of those of arity 4 we conjecture that in both cases the next independent identities have arity 8 imitating the glennie identities for jordan algebras we formulate our results as a commutative square of operad morphisms which leads to the conjecture that the squares in a much more general class are also commutative

parafermions for higher order extensions of the poincare algebra and their associated superspace parafermions of orders 2 and 3 are shown to be the fundamental tool to construct superspaces related to cubic and quartic extensions of the poincare algebra the corresponding superfields are constructed and some of their main properties are analyzed in detail in this context the existence problem of operators acting like covariant derivatives is analyzed and the associated operators are explicitly constructed

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

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

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

a note on the definition of a linear bilevel programming solution an alternative definition of the linear bilevel programming problem blp has recently been proposed by lu shi and zhang this note shows that the proposed definition is a restriction of blp indeed the new definition is equivalent to transferring the first level constraints involving second level variables into the second level resulting in a special case of blp in which there are no first level constraint involving second level variables thus contrary to what is stated by the authors who suggested the new definition this does not allow to solve a wider class of problems but rather relaxes the feasible region allowing for infeasible points to be considered as feasible

multiplication free evaluation of polynomials via a stochastic bernstein representation a new multiplication free method for the evaluation of multidimensional polynomials is proposed the method is based on a stochastic bernstein representation sbr and utilizes a random number generator a locally encoded data structure and a system table the sbr is shown to be capable of representing any polynomial function and of approximating any continuous function arbitrarily closely an error bound analysis is performed using a large deviations technique a variant of the sbr which has been used by others to balance an inverted pendulum is also analyzed