sub quadratic convergence of a smoothing newton algorithm for the p 0 and monotone lcp given equation here the linear complementarity problem lcp is to find equation here such that x s 0 s mx q xts 0 by using the chen harker kanzow smale chks smoothing function the lcp is reformulated as a system of parameterized smooth nonsmooth equations as a result a smoothing newton algorithm which is a modified version of the qi sun zhou algorithm mathematical programming vol 87 2000 pp 1 35 is proposed to solve the lcp with m being assumed to be a p0 matrix p0 lcp the proposed algorithm needs only to solve one system of linear equations and to do one line search at each iteration it is proved in this paper that the proposed algorithm has the following convergence properties i it is well defined and any accumulation point of the iteration sequence is a solution of the p0 lcp ii it generates a bounded sequence if the p0 lcp has a nonempty and bounded solution set iii if an accumulation point of the iteration sequence satisfies a nonsingularity condition which implies the p0 lcp has a unique solution then the whole iteration sequence converges to this accumulation point sub quadratically with a q rate 2 t where t 0 1 is a parameter and iv if m is positive semidefinite and an accumulation point of the iteration sequence satisfies a strict complementarity condition then the whole sequence converges to the accumulation point quadratically

on almost smooth functions and piecewise smooth functions abstract piecewise smooth ps functions are perhaps the best known examples of semismooth functions which play key roles in the solution of nonsmooth equations and nonsmooth optimization recently there have emerged other examples of semismooth functions including the p norm function 1 p defined on r n with n 2 ncp functions smoothing penalty functions and integral functions these semismooth functions share the special property that their smooth point sets are locally connected around their nonsmooth points by extending a result of rockafellar we show that the smooth point set of a ps function cannot have such a property this shows that the above functions though semismooth are not ps we call such functions almost smooth as we show that the b subdifferential of an as function at a point has either one or infinitely many elements which contrasts with ps functions whose b subdifferential at a point has only a finite number of elements we derive other useful properties of as functions and sufficient conditions for a function to be as these results are then applied to various smoothing penalty functions and integral functions

analysis on the smoothing method for the p linear complementarity systems abstract a smoothing method for solving the linear complementarity systems lcs has been proposed by zheng in his thesis in zheng s algorithm the lcs is reformulated as a parameterized differential algebraic equation dae however the existence of the solution of the parameterized dae is not proved the aim of this paper is to establish the existence of the solution of the parameterized dae and to demonstrate the continuous dependence of the solution on the parameter and the initial value of the parameterized dae

generalized heisenberg algebras and fibonacci series we have constructed a heisenberg type algebra generated by the hamiltonian the step operators and an auxiliary operator this algebra describes quantum systems having eigenvalues of the hamiltonian depending on the eigenvalues of the two previous levels this happens for example for systems having the energy spectrum given by a fibonacci sequence moreover the algebraic structure depends on the two functions f x and g x when these two functions are linear we classify analysing the stability of the fixed points of the functions the possible representations for this algebra

global convergence of a class of non interior point algorithms using chen harker kanzow smale functions for nonlinear complementarity problems we propose a class of non interior point algorithms for solving the complementarity problems cp find a nonnegative pair x y 2n satisfying y f x and xiyi 0 for every i 1 2 n where f is a continuous mapping from n to n the algorithms are based on the chen harker kanzow smale smoothing functions for the cp and have the following features a it traces a trajectory in 3n which consists of solutions of a family of systems of equations with a parameter b it can be started from an arbitrary not necessarily positive point in 2n in contrast to most of interior point methods and c its global convergence is ensured for a class of problems including not strongly monotone complementarity problems having a feasible interior point to construct the algorithms we give a homotopy and show the existence of a trajectory leading to a solution under a relatively mild condition and propose a class of algorithms involving suitable neighborhoods of the trajectory we also give a sufficient condition on the neighborhoods for global convergence and two examples satisfying it

a class of algebras similar to the enveloping algebra of sl 2 fix f e c x define r c a b h subject to the relations ha ah a hb bh b ab ba f h we study these algebras for different f and in particular show how they are similar to and different from u sl 2 the enveloping algebra of sl 2 c there is a notion of highest weight modules and a category a for such r for each n 0 if f x x 1 n l xn 1 then r has precisely n simple modules in each finite dimension and every finite dimensional r module is semisimple

on a new homotopy continuation trajectory for nonlinear complementarity problems most known continuation methods for p 0 complementarity problems require some restrictive assumptions such as the strictly feasible condition and the properness condition to guarantee the existence and the boundedness of certain homotopy continuation trajectory to relax such restrictions we propose a new homotopy formulation for the complementarity problem based on which a new homotopy continuation trajectory is generated for p 0 complementarity problems the most promising feature of this trajectory is the assurance of the existence and the boundedness of the trajectory under a condition that is strictly weaker than the standard ones used widely in the literature of continuation methods particularly the often assumed strictly feasible condition is not required here when applied to p complementarity problems the boundedness of the proposed trajectory turns out to be equivalent to the solvability of the problem and the entire trajectory converges to the unique least element solution provided that it exists moreover for monotone complementarity problems the whole trajectory always converges to a least 2 norm solution provided that the solution set of the problem is nonempty the results presented in this paper can serve as a theoretical basis for constructing a new path following algorithm for solving complementarity problems even for the situations where the solution set is unbounded

improved smoothing type methods for the solution of linear programs we consider a smoothing type method for the solution of linear programs its main idea is to reformulate the primal dual optimality conditions as a nonlinear and nonsmooth system of equations and to apply a newton type method to a smooth approximation of this nonsmooth system the method presented here is a predictor corrector method and is closely related to some methods recently proposed by burke and xu on the one hand and by the authors on the other hand however here we state stronger global and or local convergence properties moreover we present quite promising numerical results for the whole netlib test problem collection

finite dimensional simple modules over generalized heisenberg algebras abstract generalized heisenberg algebras h f for any polynomial f h c h have been used to explain various physical systems and many physical phenomena for the last 20 years in this paper we first obtain the center of h f and the necessary and sufficient conditions on f for two h f to be isomorphic then we determine all finite dimensional simple modules over h f for any polynomial f h c h more precisely there are three classes of them a h f z a b h f z a and c h f z n if f w h c for any c c and n th n 1 primitive root w of unity we actually obtain a complete classification of all irreducible modules over h f

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

quadratic one step smoothing newton method for p0 lcp without strict complementarity in this paper we propose a modified one step smoothing newton method for solving the p 0 linear complementarity problem p 0 lcp based on kanzow s smoothing function our smoothing newton method solves only one linear system of equations and performs only one line search at each iteration it is proved that our proposed algorithm has global convergence and local quadratic convergence in absence of strict complementarity assumption at the p 0 lcp solution under weaker conditions our convergence results are much stronger than many previous literatures

multi parametric deformed heisenberg algebras a route to complexity we introduce a generalization of the heisenberg algebra which is written in terms of a functional of one generator of the algebra f j0 that can be any analytical function when f is linear with slope we show that the algebra in this case corresponds to q oscillators for q2 tan the case where f is a polynomial of order n in j0 corresponds to an n parameter deformed heisenberg algebra the representations of the algebra when f is any analytical function are shown to be obtained through the study of the stability of the fixed points of f and their composed functions the case when f is a quadratic polynomial in j0 the simplest nonlinear scheme which is able to create chaotic behaviour is analysed in detail and special regions in the parameter space give representations that cannot be continuously deformed to representations of heisenberg algebra

a full newton step non interior continuation algorithm for a class of complementarity problems in this paper we investigate a class of nonlinear complementarity problems arising from the discretization of the free boundary problem which was recently studied by sun and zeng z sun j zeng a monotone semismooth newton type method for a class of complementarity problems j comput appl math 235 5 2011 1261 1274 we propose a new non interior continuation algorithm for solving this class of problems where the full newton step is used in each iteration we show that the algorithm is globally convergent where the iteration sequence of the variable converges monotonically we also prove that the algorithm is globally linearly and locally superlinearly convergent without any additional assumption and locally quadratically convergent under suitable assumptions the preliminary numerical results demonstrate the effectiveness of the proposed algorithm

a fixed point method for a class of super large scale nonlinear complementarity problems we consider a class of complementarity problems involving functions which are nonlinear in this paper we reformulate this nonlinear complementarity problem as a system of absolute value equations which is nonsmooth then we propose a fixed point method to solve this nonsmooth system we prove that the proposed method is globally linearly convergent under a mild condition the proposed method is greatly effective not only for small and medium size problems but also for large and super large scale problems especially our method can efficiently solve super large scale problems with a million variables in a few tens of minutes on a pc

simple applications of q bosons a deformation of the harmonic oscillator algebra associated with the morse potential and the su 2 algebra is derived using the quantum analogue of the anharmonic oscillator we use the quantum oscillator algebra or q boson algebra which is a generalization of the heisenberg weyl algebra obtained by introducing a deformation parameter q further we present a new algebraic realization of the q bosons for the case of q being a root of unity which corresponds to a periodic structure described by a finite dimensional representation we show that this structure represents the symmetry of a linear lattice with periodic boundary conditions

smoothing type algorithm for solving linear programs by using an augmented complementarity problem we present a smoothing type algorithm for solving the linear program lp by making use of an augmented system of its optimality conditions the algorithm is shown to be globally convergent without requiring any assumption it only needs to solve one system of linear equations and to perform one line search at each iteration in particular if the lp has a solution and hence it has a strictly complementary solution then the algorithm will generate a strictly complementary solution of the lp and if the lp is infeasible then the algorithm will correctly detect infeasibility of the lp to the best of our knowledge this is the first smoothing type algorithm for solving the lp having the above desired convergence features

chaos control and hybrid projective synchronization for a class of new chaotic systems the problems on chaos control and hybrid projective synchronization for a class of new chaotic systems are considered first new 4d chaotic systems are proposed by introducing an additional state into a 3d quadratic chaotic system and the states of the systems corresponding to the different ranges of parameter b are exhibited second a single scalar adaptive feedback controller for chaos control of the systems is presented third hybrid projective synchronization hps of two of the chaotic systems with parameters in different conditions are investigated by presenting adaptive feedback control strategies with adaptive parameter update laws and considering controller simplification to achieve complete synchronization finally numerical simulations are demonstrated to verify the effectiveness of the strategies

existence and limiting behavior of a non interior point trajectory for nonlinear complementarity problems without strict feasibility condition for p0 complementarity problems most existing non interior point path following methods require the existence of a strictly feasible point for a p complementarity problem the existence of a strictly feasible point is equivalent to the nonemptyness and the boundedness of the solution set in this paper we propose a new homotopy formulation for complementarity problems by which a new non interior point continuation trajectory is generated the existence and the boundedness of this non interior point trajectory for p0 complementarity problems are proved under a very mild condition that is weaker than most conditions used in the literature one prominent feature of this condition is that it may hold even when the often assumed strict feasibility condition fails to hold in particular for a p problem it turns out that the new non interior point trajectory exists and is bounded if and only if the problem has a solution we also study the convergence of this trajectory and characterize its limiting point as the parameter approaches zero

a regularization semismooth newton method based on the generalized fischer burmeister function for p0 ncps we consider a regularization method for nonlinear complementarity problems with f being a p 0 function which replaces the original problem with a sequence of the regularized complementarity problems in this paper this sequence of regularized complementarity problems are solved approximately by applying the generalized newton method for an equivalent augmented system of equations constructed by the generalized fischer burmeister fb ncp functions f p with p 1 we test the performance of the regularization semismooth newton method based on the family of ncp functions through solving all test problems from mcplib numerical experiments indicate that the method associated with a smaller p for example email protected 1 1 2 usually has better numerical performance and the generalized fb functions f p with email protected 1 1 2 can be used as the substitutions for the fb function f 2

a smoothing newton method for minimizing a sum of euclidean norms we consider the problem of minimizing a sum of euclidean norms f x sum i 1 m b i a i tx this problem is a nonsmooth problem because f is not differentiable at a point x when one of the norms is zero in this paper we present a smoothing newton method for this problem by applying the smoothing newton method proposed by qi sun and zhou math programming 87 2000 pp 1 35 directly to a system of strongly semismooth equations derived from primal and dual feasibility and a complementarity condition this method is globally and quadratically convergent as applications to this problem smoothing newton methods are presented for the euclidean facilities location problem and the steiner minimal tree problem under a given topology preliminary numerical results indicate that this method is extremely promising

thermodynamic properties of a solid exhibiting the energy spectrum given by the logistic map we show that the infinite dimensional representation of the recently introduced logistic algebra can be interpreted as a nontrivial generalization of the heisenberg or oscillator algebra this allows us to construct a quantum hamiltonian having the energy spectrum given by the logistic map we analyze the hamiltonian of a solid whose collective modes of vibration are described by this generalized oscillator and compute the thermodynamic properties of the model in the two cycle and r 3 6785 chaotic region of the logistic map