coherent state quantization of paragrassmann algebras by using a coherent state quantization of paragrassmann variables operators are constructed in finite hilbert spaces we thus obtain in a straightforward way a matrix representation of the paragrassmann algebra this algebra of finite matrices realizes a deformed weyl heisenberg algebra the study of mean values in coherent states of some of these operators leads to interesting conclusions

on the representation theory of orthofermions and orthosupersymmetric realization of parasupersymmetry and fractional supersymmetry we construct a canonical irreducible representation for the orthofermion algebra of arbitrary order and show that every representation decomposes into irreducible representations that are isomorphic to either the canonical representation or the trivial representation we use these results to show that every orthosupersymmetric system of order p has a parasupersymmetry of order p and a fractional supersymmetry of order p 1

finite dimensional lie algebras of order f f lie algebras are natural generalizations of lie algebras f 1 and lie superalgebras f 2 when f 2 not many finite dimensional examples are known in this article we construct finite dimensional f lie algebras f 2 by an inductive process starting from lie algebras and lie superalgebras matrix realizations of f lie algebras constructed in this way from su n sp 2n so n and sl n m osp 2 m are given we obtain nontrivial extensions of the poincare algebra by inonu wigner contraction of certain f lie algebras with f 2

tightness and distinguished frechet spaces in this chapter we apply the concept of tightness to study distinguished frechet spaces we show that a frechet space is distinguished if and only if its strong dual has countable tightness this approach to studying distinguished frechet spaces leads to a rich supply of df spaces whose weak duals are quasi suslin but not k analytic the small cardinals mathfrak b and mathfrak d will be used to improve the analysis of kothe s echelon nondistinguished frechet space 1 a

resolutions of topological linear spaces and continuity of linear maps abstract the main result of the paper is the following if an f space x is covered by a family e n n of sets such that e e whenever and f is a linear map from x to a topological linear space y which is continuous on each of the sets e then f is continuous this provides a very strong negative answer to a problem posed recently by j kakol and m lopez pellicer a number of consequences of this result are given some of which are quite curious also inspired by a related question asked by j kakol it is shown that if a linear map is continuous on each member of a sequence of compact sets then it is also continuous on every compact convex set contained in the linear span of the sequence the construction applied to prove this is then used to interpret a natural linear topology associated with the sequence as the inductive limit topology in the sense of ph turpin and thus derive its basic properties

equivalence of variational inequality problems to unconstrained minimization in this paper we propose a class of merit functions for variational inequality problems vi through these merit functions the variational inequality problem is cast as unconstrained minimization problem we estimate the growth rate of these merit functions and give conditions under which the stationary points of these functions are the solutions of vi

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

equivalent unconstrained minimization and global error bounds for variational inequality problems new merit functions for variational inequality problems are constructed through the moreau yosida regularization of some gap functions the proposed merit functions constitute unconstrained optimization problems equivalent to the original variational inequality problem under suitable assumptions conditions are studied for those merit functions to be differentiable and for any stationary point of those functions to be a solution of the original variational inequality problem moreover those functions are shown to provide global error bounds for general variational inequality problems under the strong monotonicity assumption only

regularized gap functions and error bounds for generalized mixed weak vector quasivariational inequality problems in fuzzy environments abstract in this paper we establish gap functions for a new class of generalized mixed weak vector quasivariational inequality problems in fuzzy environments in short mqvipf these functions are constructed through terms of regularized gap functions using the method of nonlinear scalarization functions moreover error bounds are provided for mqvipf via regularized gap functions under suitable assumptions the main results obtained in this paper improve and extend some corresponding known results some examples are given to illustrate our results

filiform lie algebras of order 3 the aim of this work is to generalize a very important type of lie algebras and superalgebras i e filiform lie super algebras into the theory of lie algebras of order f thus the concept of filiform lie algebras of order f is obtained in particular for f 3 it has been proved that by using infinitesimal deformations of the associated model elementary lie algebra it can be obtained families of filiform elementary lie algebras of order 3 analogously as that occurs into the theory of lie algebras m vergne cohomologie des algebres de lie nilpotentes application a l etude de la variete des algebres de lie nilpotentes bull soc math france 98 81 116 1970 also we give the dimension using an adaptation of the sl 2 c module method and a basis of such infinitesimal deformations in some generic cases

an approximation scheme for a class of risk averse stochastic equilibrium problems we consider two models for stochastic equilibrium one based on the variational equilibrium of a generalized nash game and the other on the mixed complementarity formulation each agent in the market solves a single stage risk averse optimization problem with both here and now investment variables and production wait and see variables a shared constraint couples almost surely the wait and see decisions of all the agents an important characteristic of our approach is that the agents hedge risk in the objective functions on costs or profits of their optimization problems which has a clear economic interpretation this feature is obviously desirable but in the risk averse case it leads to variational inequalities with set valued operators a class of problems for which no established software is currently available to overcome this difficulty we define a sequence of approximating differentiable variational inequalities based on smoothing the nonsmooth risk measure in the agents problems such as average or conditional value at risk the smoothed variational inequalities can be tackled by the path solver for example the approximation scheme is shown to converge including the case when smoothed problems are solved approximately an interesting by product of our proposal is that the smoothing approach allows us to show existence of an equilibrium for the original problem to assess the proposed technique numerical results are presented the first set of experiments is on randomly generated equilibrium problems for which we compare the proposed methodology with the standard smooth reformulation of average value at risk minimization using additional variables to rewrite the associated max function the second set of experiments deals with a part of the real life european gas network for which dantzig wolfe decomposition can be combined with the smoothing approach

merit functions for semi definite complementarity problems merit functions such as the gap function the regularized gap function the implicit lagrangian and the norm squared of the fischer burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors we study the extension of these merit functions to complementarity problems defined over the cone of block diagonal symmetric positive semi definite real matrices the extension suggests new solution methods for the latter problems

on the construction of generalized grassmann representatives of state vectors generalized zk graded grassmann variables are used to label coherent states related to the nilpotent representation of the q oscillator of biedenharn and macfarlane when the deformation parameter is a root of unity these states are then used to construct generalized grassmann representatives of state vectors

a general methodology for designing globally convergent optimization neural networks we present a general methodology for designing optimization neural networks we prove that the neural networks constructed by using the proposed method are guaranteed to be globally convergent to solutions of problems with bounded or unbounded solution sets in contrast with the gradient methods whose convergence is not guaranteed we show that the proposed method contains both the gradient methods and nongradient methods employed in existing optimization neural networks as special cases based on the theoretical results of the proposed method we study the convergence and stability of general gradient models in the case of unisolated solutions using the proposed method we derive some new neural network models for a very large class of optimization problems in which the equilibrium points correspond to exact solutions and there is no variable parameter finally some numerical examples show the effectiveness of the method

weight of precompact subsets and tightness pfister 1976 and cascales and orihuela 1986 proved that precompact sets in df and lm spaces have countable weight i e are metrizable improvements by valdivia 1982 cascales and orihuela 1987 and kakol and saxon preprint have varying methods of proof for these and other improvements a refined method of upper semi continuous compact valued maps applied to uniform spaces will suffice at the same time this method allows us to dramatically improve kaplansky s theorem that the weak topology of metrizable spaces has countable tightness extending it to include all lm spaces and all quasi barrelled df spaces both in the weak and original topologies one key is showing that for a large class g including all df and lm spaces countable tightness of the weak topology of e in g is equivalent to realcompactness of the weak topology of the dual of e

spaces of continuous maps on a class of noncompact metric spaces for a metric space x let c x be the family of hypographs of all con tinuous maps from x to i 0 1 endowed with the topology induced by the hausdorff metric of the metric space x i in the present paper the following result is proved if x i s a noncompact locally compact totally bounded separable metric space then c x i s home omorphic to c0 if and only if the set of all isolated points of x is not dense in x w here c0 xn n n 1 1 s up xn 1 and limn xn 0 specially for the open interval 0 1 with the usual metric c 0 1 is homeomorphic to c0

coercivity conditions and variational inequalities various coercivity conditions appear in the literature in order to guarantee solutions for the variational inequality problem we show that these conditions are equivalent to each other and that they are not only sufficient but also necessary for the set of solutions to be non empty and bounded

noncommutative differential geometry of matrix algebras the noncommutative differential geometry of the algebra mn c of complex n n matrices is investigated the role of the algebra of differential forms is played by the graded differential algebra c sl n c mn c mn c sl n c sl n c acting by inner derivations on mn c a canonical symplectic structure is exhibited for mn c for which the poisson bracket is to within a factor i the commutator also a canonical riemannian structure is described for mn c finally the analog of the maxwell potential is constructed and it is pointed out that there is a potential with a vanishing curvature that is not a pure gauge

z3 graded differential geometry of the quantum plane in this work the z3 graded differential geometry of the quantum plane is constructed the corresponding quantum lie algebra and its hopf algebra structure are obtained the dual algebra i e the universal enveloping algebra of the quantum plane is explicitly constructed

a new unconstrained differentiable merit function for box constrained variational inequality problems and a damped gauss newton method in this paper we propose a new unconstrained differentiable merit function f for box constrained variational inequality problems vip l u f we study various desirable properties of this new merit function f and propose a gauss newton method in which each step requires only the solution of a system of linear equations global and superlinear convergence results for vip l u f are obtained key results are the boundedness of the level sets of the merit function for any uniform p function and the superlinear convergence of the algorithm without a nondegeneracy assumption numerical experiments confirm the good theoretical properties of the method

a new algorithm for solving variational inequalities with application to the traffic assignment problem the variational inequality problem in euclidian space is formulated as a nonconvex nondifferentiable optimization problem we show that any stationary point is optimal and we propose a solution algorithm that decreases the nondifferential objective monotonically application to the asymmetric traffic assignment problem is considered

asymmetric variational inequality problems over product sets applications and iterative methods in this paper we i describe how several equilibrium problems can be uniformly modelled by a finite dimensional asymmetric variational inequality defined over a cartesian product of sets and ii investigate the local and global convergence of various iterative methods for solving such a variational inequality problem because of the special cartesian product structure these iterative methods decompose the original variational inequality problem into a sequence of simpler variational inequality subproblems in lower dimensions the resulting decomposition schemes often have a natural interpretation as some adjustment processes

sets of filter convergence of sequences of continuous functions abstract we consider sets of filter convergence and divergence to infinity of sequences of continuous functions for borel filters we characterize the sets of filter convergence of sequences of continuous functions on the real line for borel filters we also give sufficient and necessary conditions for results involving sets of filter divergence to infinity in particular we give the full description of such sets for the statistical convergence for metric spaces

a posteriori error bounds for the linearly constrained varitional inequality problem in this paper we derive a posteriori error bound of an approximate solution to a linearly constrained variational inequality problem three measures of error are used with each depending on a particular characterization of an exact solution and related to a certain class of iterative methods

numerical comparison between prediction correction methods for general variational inequalities in this paper we propose two prediction correction methods for solving general variational inequalities the first method makes use of a descent direction to produce the new iterate while the second method generates the corrector by the improved extragradient method under certain conditions the global convergence of both methods is proved it is proved theoretically that the lower bound of the progress obtained by the second method is greater than that by the first one numerical results are given to verify and compare the numerical efficiency of these two predictor corrector methods

a general descent framework for the monotone variational inequality problem we present a framework for descent algorithms that solve the monotone variational inequality problem vip v which consists in finding a solutionv v satisfyings v t v v 0 for allv v this unified framework includes as special cases some well known iterative methods and equivalent optimization formulations a descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given based on this unified logarithmic framework we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions

a globally convergent newton method for solving strongly monotone variational inequalities variational inequality problems have been used to formulate and study equilibrium problems which arise in many fields including economics operations research and regional sciences for solving variational inequality problems various iterative methods such as projection methods and the nonlinear jacobi method have been developed these methods are convergent to a solution under certain conditions but their rates of convergence are typically linear in this paper we propose to modify the newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length the purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate it is then shown that under the strong monotonicity assumption the method is globally convergent and under some additional assumptions the rate of convergence is quadratic limited computational experience indicates the high efficiency of the proposed method

merit functions for general variational inequalities in this paper we consider some classes of merit functions for general variational inequalities using these functions we obtain error bounds for the solution of general variational inequalities under some mild conditions since the general variational inequalities include variational inequalities quasivariational inequalities and complementarity problems as special cases results proved in this paper hold for these problems in this respect results obtained in this paper represent a refinement of previously known results for classical variational inequalities

lagrange multipliers exact regularization and error bounds for monotone variational inequalities we examine two central regularization strategies for monotone variational inequalities the first a direct regularization of the operative monotone mapping and the second via regularization of the associated dual gap function a key link in the relationship between the solution sets to these various regularized problems is the idea of exact regularization which in turn is fundamentally associated with the existence of lagrange multipliers for the regularized variational inequality a regularization is said to be exact if a solution to the regularized problem is a solution to the unregularized problem for all parameters beyond a certain value the lagrange multipliers corresponding to a particular regularization of a variational inequality on the other hand are defined via the dual gap function our analysis suggests various conceptual iteratively regularized numerical schemes for which we provide error bounds and hence stopping criteria under the additional assumption that the solution set to the unregularized problem is what we call weakly sharp of order greater than one

fractional superlie algebras and groups the nth root of a lie algebra and its dual that is the fractional supergroup based on the permutation group sn invariant forms is formulated in the hopf algebra formalism detailed discussion of s3 graded sl 2 algebras is performed

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

generalized descent methods for asymmetric systems of equations we consider generalizations of the steepest descent algorithm for solving asymmetric systems of equations we first show that if the system is linear and is defined by the matrix m then the method converges if m2 is positive definite we also establish easy to verify conditions on the matrix m that ensure that m2 is positive definite and develop a scaling procedure that extends the class of matrices that satisfy the convergence conditions in addition we establish a local convergence result for nonlinear systems defined by uniformly monotone maps and discuss a class of general descent methods all of the methods that we consider reduce to standard nonlinear programming algorithms for equivalent optimization problems when the jacobian of the underlying problem map is symmetric we interpret the convergence conditions for the generalized steepest descent algorithms as restricting the degree of asymmetry of the problem map

generalized grassmann variables for quantum kit k level systems and barut girardello coherent states for su r 1 algebras this paper concerns the construction of su r 1 barut girardello coherent states in terms of generalized grassmann variables we first introduce a generalized weyl heisenberg algebra a r r 1 generated by r pairs of creation and annihilation operators this algebra provides a useful framework to describe qubit and qukit k level systems it includes the usual weyl heisenberg and su 2 algebras we investigate the corresponding fock representation space the generalized grassmann variables are introduced as variables spanning the fock bargmann space associated with the algebra a r the barut girardello coherent states for su r 1 algebras are explicitly derived and their over completion properties are discussed

on linear differential equations with variable coefficients involving a para grassmann variable linear differential equations with constant coefficients involving a para grassmann variable have been considered recently in the work of mansour and schork symmetry integr geom methods appl 5 73 2009 in the present paper this treatment is extended to linear differential equations with variable coefficients for the equation of first order an explicit formula for the solution is given for the equations of higher order it is shown how the solutions may be determined in terms of the solutions of ordinary differential equations i e involving only bosonic variables some examples of these differential equations are discussed and analogs for the trigonometric functions are introduced

graded gauge theory the mathematical background for a graded extension of gauge theories is investigated after discussing the general properties of graded lie algebras and what may serve as a model for a graded lie group the graded fiber bundle is constructed its basis manifold is supposed to be the so called superspace i e the product of the minkowskian space time with the grassmann algebra spanned by the anticommuting lorentz spinors the vertical subspaces tangent to the fibers are isomorphic with the graded extension of the su n lie algebra the connection and curvature are defined then on this bundle the two different gradings are either independent of each other or may be unified in one common grading which is equivalent to the choice of the spin statistics dependence the yang mills lagrangian is investigated in the simplified case the conformal symmetry breaking is discussed as well as some other physical consequences of the model

image analysis by krawtchouk moments a new set of orthogonal moments based on the discrete classical krawtchouk polynomials is introduced the krawtchouk polynomials are scaled to ensure numerical stability thus creating a set of weighted krawtchouk polynomials the set of proposed krawtchouk moments is then derived from the weighted krawtchouk polynomials the orthogonality of the proposed moments ensures minimal information redundancy no numerical approximation is involved in deriving the moments since the weighted krawtchouk polynomials are discrete these properties make the krawtchouk moments well suited as pattern features in the analysis of two dimensional images it is shown that the krawtchouk moments can be employed to extract local features of an image unlike other orthogonal moments which generally capture the global features the computational aspects of the moments using the recursive and symmetry properties are discussed the theoretical framework is validated by an experiment on image reconstruction using krawtchouk moments and the results are compared to that of zernike pseudo zernike legendre and tchebyscheff moments krawtchouk moment invariants are constructed using a linear combination of geometric moment invariants an object recognition experiment shows krawtchouk moment invariants perform significantly better than hu s moment invariants in both noise free and noisy conditions

on the resolution of the generalized nonlinear complementarity problem minimization of a differentiable function subject to box constraints is proposed as a strategy to solve the generalized nonlinear complementarity problem gncp defined on a polyhedral cone it is not necessary to calculate projections that complicate and sometimes even disable the implementation of algorithms for solving these kinds of problems theoretical results that relate stationary points of the function that is minimized to the solutions of the gncp are presented perturbations of the gncp are also considered and results are obtained related to the resolution of gncps with very general assumptions on the data these theoretical results show that local methods for box constrained optimization applied to the associated problem are efficient tools for solving the gncp numerical experiments are presented that encourage the use of this approach

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

generalized grassmannian coherent states for pseudo hermitian n level systems the purpose of this paper is to generalize fermionic coherent states for two level systems described by pseudo hermitian hamiltonian cherbal et al 2007 j phys a math theor 40 1835 to n level systems central to this task is the expression of the coherent states in terms of generalized grassmann variables these kinds of grassmann coherent states satisfy the bi overcompleteness condition instead of the overcompleteness one as it is reasonably expected because of the biorthonormality of the system choosing an appropriate grassmann weight function the resolution of identity is examined moreover grassmannian coherent and squeezed states of the deformed group suq 2 for the three level pseudo hermitian system are presented

on p 2 generalized deformed parafermions and exotic statistics generalized deformed parafermions of order two are analysed according to the lie super algebras they generate a classification of the subtended exotic statistics is then given and for each of them remarkable embedded lie structures are pointed out

quantum z3 graded space in this paper the quantum z3 graded space is discussed and its differential calculus is investigated and the quantum matrices in quantum z3 graded space are obtained

poincare and sl 2 algebras of order 3 in this paper we initiate a general classification for lie algebras of order 3 and we give all lie algebras of order 3 based on sl 2 c and iso 1 3 the poincare algebra in four dimensions we then set the basis of the theory of the deformations in the gerstenhaber sense and contractions for lie algebras of order 3

finite dimensional calculus we discuss topics related to finite dimensional calculus in the context of finite dimensional quantum mechanics the truncated heisenberg weyl algebra is called a taa algebra after tekin aydin and arik who formulated it in terms of orthofermions it is shown how to use a matrix approach to implement analytic representations of the heisenberg weyl algebra in univariate and multivariate settings we provide examples for the univariate case krawtchouk polynomials are presented in detail including a review of krawtchouk polynomials that illustrates some curious properties of the heisenberg weyl algebra as well as presenting an approach to computing krawtchouk expansions from a mathematical perspective we are providing indications as to how to implement infinite terms rota s finite operator calculus

weakly k analytic spaces and the three space property for analyticity abstract let e e be a dual pair of vector spaces the paper studies general conditions which allow to lift analyticity or k analyticity from the weak topology e e to stronger ones in the frame of e e first we show that the mackey dual of a space c p x is analytic iff the space x is countable this yields that for uncountable analytic spaces x the mackey dual of c p x is weakly analytic but not analytic we show that the mackey dual e of an lf space of a sequence of reflexive separable frechet spaces with the heinrich density condition is analytic i e e is a continuous image of the polish space n n this extends a result of valdivia we show also that weakly quasi suslin locally convex baire spaces are metrizable and complete this extends a result of de wilde and sunyach we provide however a large class of weakly analytic but not analytic metrizable separable baire topological vector spaces not locally convex this will be used to prove that analyticity is not a three space property but we show that a metrizable topological vector space e is analytic if e contains a complete locally convex analytic subspace f such that the quotient e f is analytic several questions remarks and examples are included

regularized gap function as penalty term for constrained minimization problems abstract by using the regularized gap function for variational inequalities li and peng introduced a new penalty function p x for the problem of minimizing a twice continuously differentiable function in closed convex subset of the n dimensional space r n under certain assumptions they proved that the original constrained minimization problem is equivalent to unconstrained minimization of p x the main purpose of this paper is to give an in depth study of those properties of the objective function that can be extended from the feasible set to the whole r n by p x for example it is proved that the objective function has bounded level sets or is strongly coercive on the feasible set if and only if p x has bounded level sets or is strongly coercive on r n however the convexity of the objective function does not imply the convexity of p x when the objective function is not quadratic no matter how small is instead the convexity of the objective function on the feasible set only implies the invexity of p x on r n moreover a characterization for the invexity of p x is also given

gap functions and error bounds for inverse quasi variational inequality problems abstract this paper is dedicated to inverse quasi variational inequalities which correspond to a mixture of quasi variational inequalities and inverse variational inequalities our aim is to obtain local global error bounds for inverse quasi variational inequality problems in terms of different gap functions merit functions i e the residual gap function the regularized gap function and the d gap function these bounds provide effective estimated distances between a specific point and the exact solution of the inverse quasi variational inequality problem since the class of inverse quasi variational inequalities includes the classes of general quasi variational inequalities and variational inequalities our results cover and extend similar results for these problems

an iterative scheme for variational inequalities in this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational inequalities this iterative scheme not only contains as special cases the projection linear approximation and relaxation methods but also induces new algorithms then we show that under appropriate assumptions the proposed iterative scheme converges by establishing contraction estimates involving a sequence of norms in en induced by symmetric positive definite matrices gm thus in contrast to the above mentioned methods this technique allows the possibility of adjusting the norm at each step of the algorithm this flexibility will generally yield convergence under weaker assumptions

fractional supersymmetry and fth roots of representations a generalization of super lie algebras is presented it is then shown that all known examples of fractional supersymmetry can be understood in this formulation however the incorporation of three dimensional fractional supersymmetry in this framework needs some care the proposed solutions lead naturally to a formulation of a fractional supersymmetry starting from any representation d of any lie algebra g this involves taking the fth roots of d in an appropriate sense a fractional supersymmetry in any space time dimension is then possible this formalism finally leads to an infinite dimensional extension of g reducing to the centerless virasoro algebra when g sl 2 r

on cauchy euler s differential equation involving a para grassmann variable in this paper we consider the mth order cauchy euler s differential equation involving a para grassmann variable of order p in the grassmann case i e p 1 we determine the solution for arbitrary order m in the case of arbitrary order p we give a solution for the cases m 1 2

a z3 graded generalization of supermatrices we introduce z3 graded objects which are the generalization of the more familiar z2 graded objects that are used in supersymmetric theories and in many models of non commutative geometry first we introduce the z3 graded grassmann algebra and we use this object to construct the z3 matrices which are the generalizations of the supermatrices then we generalize the concepts of supertrace and superdeterminant

iterative methods for variational and complementarity problems in this paper we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems included among such methods are the newton and several successive overrelaxation algorithms for the most part the study is concerned with the family of linear approximation methods these are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches the paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria finally several convergence results are obtained for some nonlinear approximation methods

non baire hyperplanes in non separable baire spaces abstract given an infinite dimensional banach space e one may ask does e have 1 a properly separable quotient 2 a dense non baire hyperplane every closed hyperplane in a baire space is baire the famous separable quotient problem 1 remains unsolved and question 2 is question 13 1 1 of p perez carreras and j bonet north holland math stud vol 131 north holland amsterdam 1987 in 1966 wilansky klee conjectured the answer to 2 is always no j arias de reyna denied the conjecture math ann 249 1980 111 114 proving the answer to 2 is yes whenever the answer to 1 is yes under assumption of c additivity c a a condition weaker than martin s axiom m valdivia extended the result to e a baire 2nd category in itself topological vector space collect math 34 1983 287 296 in general 2 is non trivial only in case e is baire with e e an extension of valdivia s extension yields the complete answer for 2 in the general locally convex setting affirming question 13 1 1 of perez carreras and bonet referenced above in particular a not necessarily hausdorff locally convex space e has a dense non baire hyperplane if and only if dim e and e e even for e non locally convex dim e and e e suffices but is not necessary whereas e e is obviously always necessary in order that e have a dense non baire hyperplane whether e e suffices in the hausdorff non locally convex case and whether the assumption of c a can be omitted we do not know contrastingly any infinite dimensional e is the union of a sequence of hyperplanes and if e is baire one of the hyperplanes must be dense and baire and every hyperplane of e must be unordered baire like if e is

z3 graded algebras and the cubic root of the supersymmetry translations a generalization of supersymmetry is proposed based on z3 graded algebras introducing the objects whose ternary commutation relations contain the cubic roots of unity e2 i 3 e4 i 3 and 1 the operators whose trilinear combinations yield the supersymmetric translation generators can be constructed cubic matrices forming a ternary algebra are the generalization of pauli s matrices the general properties of z3 grading some other representations of such algebras and their possible pertinence with regard to the quark model are briefly discussed

a new interpretation of the orthofermion algebra we present particle algebras whose representations correspond to states having at most p particles for p 1 the algebra corresponds to fermions for p 2 3 4 the algebra corresponds to the orthofermion algebra with a new interpretation

vector fields and their duals abstract a standard way of realizing a lie algebra is as a family of vector fields closed under commutation using the action on the universal enveloping algebra one finds a realization in a dual form the double dual this is an algebraic fourier transform of a vector fields realization of the lie algebra on the other hand in the subject of umbral calculus canonical boson calculus the duals to vector fields play a primary role it is shown that the double dual realizations of lie algebras provide a rich source of examples for the umbral calculus which complementarily provides a canonical construction of polynomial systems associated to the lie algebra for any finite dimensional lie algebra take an element in the local lie group it generates then there is an abelian family of operators such that acting on a canonical vacuum state the abelian group gives the same result as the group element constructed via the given lie algebra in other words they yield the same coherent states

group theoretical foundations of fractional supersymmetry fractional supersymmetry denotes a generalization of supersymmetry which may be constructed using a single real generalized grassmann variable n 0 for arbitrary integer n 2 3 an explicit formula is given in the case of general n for the transformations that leave the theory invariant and it is shown that these transformations possess interesting group properties it is shown also that the two generalized derivatives that enter the theory have a geometric interpretation as generators of left and right transformations of the fractional supersymmetry group careful attention is paid to some technically important issues including differentiation that arise as a result of the peculiar nature of quantities such as

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

a class of gap functions for variational inequalities recently auchmuty 1989 has introduced a new class of merit functions or optimization formulations for variational inequalities in finite dimensional space we develop and generalize auchmuty s results and relate his class of merit functions to other works done in this field especially we investigate differentiability and convexity properties and present characterizations of the set of solutions to variational inequalities we then present new descent algorithms for variational inequalities within this framework including approximate solutions of the direction finding and line search problems the new class of merit functions include the primal and dual gap functions introduced by zuhovickii et al 1969a 1969b and the differentiable merit function recently presented by fukushima 1992 also the descent algorithm proposed by fukushima is a special case from the class of descent methods developed in this paper through a generalization of auchmuty s class of merit functions we extend those inherent in the works of dafermos 1983 cohen 1988 and wu et al 1991 new algorithmic equivalence results relating these algorithm classes to each other and to auchmuty s framework are also given

error bounds of regularized gap functions for nonsmooth variational inequality problems we study the clarke rockafellar directional derivatives of the regularized gap functions and of some modified ones for the variational inequality problem vip defined by a locally lipschitz but not necessarily differentiable function on a closed convex set in an euclidean space as applications we show that under the strong monotonicity assumption the regularized gap functions have fractional exponent error bounds and consequently that the sequences provided by an algorithm of armijo type converge to the solution of the vip

compact coverings for baire locally convex spaces very recently tkachuk has proved that for a completely regular hausdorff space x the space cp x of continuous real valued functions on x with the pointwise topology is metrizable complete and separable iff cp x is baire i e of the second baire category and is covered by a family k n n of compact sets such that k k if our general result which extends some results of de wilde sunyach and valdivia states that a locally convex space e is separable metrizable and complete iff e is baire and is covered by an ordered family k n n of relatively countably compact sets consequently every baire locally convex space which is quasi suslin is separable metrizable and complete

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

an appa based descent method with optimal step sizes for monotone variational inequalities to solve monotone variational inequalities some existing appa based descent methods utilize the iterates generated by the well known approximate proximal point algorithms appa to construct descent directions this paper aims at improving these appa based descent methods by incorporating optimal step sizes in both the extra gradient steps and the descent steps global convergence is proved under mild assumptions the superiority to existing methods is verified both theoretically and computationally

convergence rate analysis of iteractive algorithms for solving variational inequality problems we present a unified convergence rate analysis of iterative methods for solving the variational inequality problem our results are based on certain error bounds they subsume and extend the linear and sublinear rates of convergence established in several previous studies we also derive a new error bound for gamma strictly monotone variational inequalities the class of algorithms covered by our analysis in fairly broad it includes some classical methods for variational inequalities e g the extragradient matrix splitting and proximal point methods for these methods our analysis gives estimates not only for linear convergence which had been studied extensively but also sublinear depending on the properties of the solution in addition our framework includes a number of algorithms to which previous studies are not applicable such as the infeasible projection methods a separation projection method inexact hybrid proximal point methods and some splitting techniques finally our analysis covers certain feasible descent methods of optimization for which similar convergence rate estimates have been recently obtained by luo 14

solving variational inequality and fixed point problems by line searches and potential optimization we introduce a general adaptive line search framework for solving fixed point and variational inequality problems our goals are to develop iterative schemes that i compute solutions when the underlying map satisfies properties weaker than contractiveness for example weaker forms of nonexpansiveness ii are more efficient than the classical methods even when the underlying map is contractive and iii unify and extend several convergence results from the fixed point and variational inequality literatures to achieve these goals we introduce and study joint compatibility conditions imposed upon the underlying map and the iterative step sizes at each iteration and consider line searches that optimize certain potential functions as a special case we introduce a modified steepest descent method for solving systems of equations that does not require a previous condition from the literature the square of the jacobian matrix is positive definite since the line searches we propose might be difficult to perform exactly we also consider inexact line searches

merit functions and error bounds for generalized variational inequalities we consider the generalized variational inequality and construct certain merit functions associated with this problem in particular those merit functions are everywhere nonnegative and their zero sets are precisely solutions of the variational inequality we further use those functions to obtain error bounds i e upper estimates for the distance to solutions of the problem

a note on the meaning of covariant derivatives in supersymmetry we analyze in this paper the group theoretical meaning of the covariant derivatives and show that they are horizontal left invariant vector fields on superspace obtained from a super lie group which at the same time exhibits the structure of a principal bundle with a canonical connection the geometrical construction is general and not restricted to the super poincare group

a relaxed projection method for variational inequalities this paper presents a modification of the projection methods for solving variational inequality problems each iteration of the proposed algorithm consists of projection onto a halfspace containing the given closed convex set rather than the latter set itself the algorithm can thus be implemented very easily and its global convergence to the solution can be established under suitable conditions

a braided interpretation of fractional supersymmetry in higher dimensions a many variable q calculus is introduced using the formalism of braided covector algebras its properties are discussed in detail and related to fractional supersymmetry when certain of its deformation parameters are roots of unity the special cases of two dimensional supersymmetry and fractional supersymmetry are developed in detail

applying coordinate products to the topological identification of normed spaces using the l 2 products we find pre hilbert spaces that are absorbing sets for all borelian classes of order 1 we also show that the following spaces are homeomorphic to the countable product of the space x n r x n is bounded 1 every coordinate product c h n of normed spaces h n in the sense of a banach space c where each h n is an absolute f set and infinitely many of the h n s are z spaces 2 every function space l p p p l p with the l q topology 0 q p 3 every sequence space l p p p l p with the l q topology 0 p q we also note that each additive and multiplicative borelian class of order 2 each projective class and the class of nonprojective spaces contain uncountably many topologically different pre hilbert spaces which are z spaces

a sequential linear programming algorithm for solving monotone variational inequalities applied to strongly monotone variational inequalities newton s algorithm achieves local quadratic convergence in this paper it is shown how the basic newton method can be modified to yield an algorithm whose global convergence can be guaranteed by monitoring the monotone decrease of the gap function associated with the variational inequality each iteration consists in the solution of a linear program in the space of primal dual variables and of a linesearch convergence does not depend on strong monotonicity however under strong monotonicity and geometric stability assumptions the set of active constraints at the solution is implicitly identified and quadratic convergence is achieved

gap functions and global error bounds for set valued variational inequalities the set valued variational inequality problem is very useful in economics theory and nonsmooth optimization in this paper we introduce some gap functions for set valued variational inequality problems under suitable assumptions by using these gap functions we derive global error bounds for the solution of the set valued variational inequality problems our results not only generalize the previously known results for classical variational inequalities from single valued case to set valued but also present a way to construct gap functions and derive global error bounds for set valued variational inequality problems

on topological classification of function spaces _ of low borel complexity we prove that if x is a countable nondiscrete completely regular space such that the function space cp x is an absolute fa set then cp x is homeomorphic to v where a xi e r xi o for all but finitely many i as an application we answer in the negative some problems of a v arhangel skil by giving examples of countable completely regular spaces x and y such that x fails to be a br space and a k space and hence x is not a kc space and not a sequential space and y fails to be an 80 space while the function spaces cp x and cp y are homeomorphic to cp x for the compact metric space 3s 0 u n1 n 1 2

two metric projection methods for constrained optimization this paper is concerned with the problem min f x mid x in x where x is a convex subset of a linear space h and f is a smooth real valued function on h we propose the class of methods x k 1 p x k alpha k g k where p denotes projection on x with respect to a hilbert space norm cdot g k denotes the frechet derivative of f at x k with respect to another hilbert space norm cdot k on h and alpha k is a positive scalar stepsize we thus remove an important restriction in the original proposal of goldstein 1 and levitin and poljak 2 where the norms cdot and cdot k must be the same it is therefore possible to match the norm cdot with the structure of x so that the projection operation is simplified while at the same time reserving the option to choose cdot k on the basis of approximations to the hessian of f so as to attain a typically superlinear rate of convergence the resulting methods are particularly attrac