n dimensional superintegrable systems from symplectic realizations of lie coalgebras the construction of n dimensional nd integrable systems from coalgebras is reviewed in the case of poisson coalgebras a necessary condition for the integrability of the nd coalgebra hamiltonian coming from a given coalgebra is obtained in terms of the dimension of the symplectic realization and the number of nonlinear casimir functions from this viewpoint the full set of three four and five dimensional lie poisson coalgebras is analysed and many new families of multiparametric nd integrable systems coming from the cases that fulfil the integrability condition are obtained including the explicit form of the integrals of the motion the superintegrability of these hamiltonians is also emphasized and the generalization of the whole construction to the quantum mechanical case is straightforward

a statistical decision rule with incomplete knowledge about classes abstract a statistical decision rule among m classes is presented when the a priori knowledge about classes is not complete either the number of classes is not the true one or it is not possible to obtain samples from all the possible classes the reject option proposed by chow is extended by defining an ambiguity reject option and a distance reject option these two types of reject can be defined in a parametric as well as in a non parametric way an example is given in r in order to illustrate this rule this method has been developed essentially to solve diagnostic problems

scheduling to minimize average completion time off line and on line approximation algorithms in this paper we introduce two general techniques for the design and analysis of approximation algorithms for np hard scheduling problems in which the objective is to minimize the weighted sum of the job completion times for a variety of scheduling models these techniques yield the first algorithms that are guaranteed to find schedules that have objective function value within a constant factor of the optimum in the first approach we use an optimal solution to a linear programming relaxation in order to guide a simple list scheduling rule consequently we also obtain results about the strength of the relaxation our second approach yields on line algorithms for these problems in this setting we are scheduling jobs that continually arrive to be processed and for each time t we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards our on line technique yields constant performance guarantees for a variety of scheduling environments and in some cases essentially matches the performance of our off line lp based algorithms

integrable hamiltonian systems with vector potentials we investigate integrable two dimensional hamiltonian systems with scalar and vector potentials admitting second invariants which are linear or quadratic in the momenta in the case of a linear second invariant we provide some examples of weakly integrable systems in the case of a quadratic second invariant we recover the classical strongly integrable systems in cartesian and polar coordinates and provide some new examples of integrable systems in parabolic and elliptical coordinates

on the approximability of single machine scheduling with precedence constraints we consider the single machine scheduling problem to minimize the weighted sum of completion times under precedence constraints in a series of recent papers it was established that this scheduling problem is a special case of minimum weighted vertex cover in this paper we show that the vertex cover graph associated with the scheduling problem is exactly the graph of incomparable pairs defined in the dimension theory of partial orders exploiting this relationship allows us to present a framework for obtaining 2 2 f approximation algorithms provided that the set of precedence constraints has fractional dimension of at most f our approach yields the best known approximation ratios for all previously considered special classes of precedence constraints and it provides the first results for bounded degree and orders of interval dimension 2 on the negative side we show that the addressed problem remains np hard even when restricted to the special case of interval orders furthermore we prove that the general problem if a fixed cost present in all feasible schedules is ignored becomes as hard to approximate as vertex cover we conclude by giving the first inapproximability result for this problem showing under a widely believed assumption that it does not admit a polynomial time approximation scheme

a new superintegrable hamiltonian we identify a new superintegrable hamiltonian in three degrees of freedom obtained as a reduction of pure keplerian motion in six dimensions the new hamiltonian is a generalization of the keplerian one and has the familiar 1 r potential with three barrier terms preventing the particle crossing the principal planes in three degrees of freedom there are five functionally independent integrals of motion and all bound classical trajectories are closed and strictly periodic the generalization of the laplace runge lenz vector is identified and shown to provide functionally independent isolating integrals they are quartic in the momenta and do not arise from separability of the hamilton jacobi equation a formulation of the system in action angle variables is presented

lie hamilton systems on the plane properties classification and applications we study lie hamilton systems on the plane i e systems of first order differential equations describing the integral curves of a t dependent vector field taking values in a finite dimensional real lie algebra of planar hamiltonian vector fields with respect to a poisson structure we start with the local classification of finite dimensional real lie algebras of vector fields on the plane obtained in a gonzalez lopez n kamran and p j olver proc london math soc 64 339 1992 and we interpret their results as a local classification of lie systems moreover by determining which of these real lie algebras consist of hamiltonian vector fields with respect to a poisson structure we provide the complete local classification of lie hamilton systems on the plane we present and study through our results new lie hamilton systems of interest which are used to investigate relevant non autonomous differential equations e g we get explicit local diffeomorphisms between such systems in particular the milne pinney second order kummer schwarz complex riccati and buchdahl equations as well as some lotka volterra and nonlinear biomathematical models are analysed from this lie hamilton approach

valued preference structures abstract an approach to the axiomatics of valued preference modelling is suggested in the framework of this approach general definitions of strict preference indifference and incomparability relations associated with a valued preference relation are established via solving a system of functional equations any solution of this system satisfies most of the classical properties characterizations of important particular solutions are also proved

lie hamilton systems on the plane applications and superposition rules a lie hamilton lh system is a nonautonomous system of first order ordinary differential equations describing the integral curves of a t dependent vector field taking values in a finite dimensional real lie algebra of hamiltonian vector fields with respect to a poisson structure we provide new algebraic geometric techniques to easily determine the properties of such lie algebras on the plane e g their associated poisson bivectors we study new and known lh systems on with physical biological and mathematical applications new results cover cayley klein riccati equations the here defined planar diffusion riccati systems complex bernoulli differential equations and projective schrodinger equations constants of motion for planar lh systems are explicitly obtained which in turn allow us to derive superposition rules through a coalgebra approach

a maximally superintegrable system on an n dimensional space of nonconstant curvature abstract we present a novel hamiltonian system in n dimensions which admits the maximal number 2 n 1 of functionally independent quadratic first integrals this system turns out to be the first example of a maximally superintegrable hamiltonian on an n dimensional riemannian space of nonconstant curvature and it can be interpreted as the intrinsic smorodinsky winternitz system on such a space moreover we provide three different complete sets of integrals in involution and solve the equations of motion in closed form

min max regret version of a scheduling problem with outsourcing decisions under processing time uncertainty abstract we consider the min max regret version of a single machine scheduling problem to determine which jobs are processed by outsourcing under processing time uncertainty the performance measure is expressed as the total cost for processing some jobs in house and outsourcing the rest processing time uncertainty is described through two types of scenarios either an interval scenario or a discrete scenario the objective is to minimize the maximum deviation from optimality over all scenarios we show that when the cost for in house jobs is expressed as the makespan the problem with an interval scenario is polynomially solvable while the one with a discrete scenario is np hard thus for the discrete scenario case we develop a 2 approximation algorithm and investigate when the problem is polynomially solvable since the problem minimizing the total completion time as a performance measure for in house jobs is known to be np hard for both scenarios we consider the problem with a special structure for the processing time uncertainty and develop a polynomial time algorithm for both scenarios

a systematic construction of completely integrable hamiltonians from coalgebras a universal algorithm to construct n particle classical and quantum completely integrable hamiltonian systems from representations of coalgebras with casimir elements is presented in particular this construction shows that quantum deformations can be interpreted as generating structures for integrable deformations of hamiltonian systems with coalgebra symmetry in order to illustrate this general method the algebra and the oscillator algebra are used to derive new classical integrable systems including a generalization of gaudin calogero systems and oscillator chains quantum deformations are then used to obtain some explicit integrable deformations of the previous long range interacting systems and a non coboundary deformation of the 1 1 poincare algebra is shown to provide a new ruijsenaars schneider like hamiltonian

single machine scheduling with scenarios in the field of robust optimization the goal is to provide solutions to combinatorial problems that hedge against variations of the numerical parameters this constitutes an effort to design algorithms that are applicable in the presence of uncertainty in the definition of the instance we study the single machine scheduling problem with the objective of minimizing the weighted sum of completion times we model uncertainty by replacing the vector of numerical values in the description of the instance by a set of possible vectors called scenarios the goal is to find a schedule of minimum value in the worst case scenario we first show that the general problem cannot be approximated within o log 1 en for any e 0 unless np has quasi polynomial algorithms we then study more tractable special cases and obtain a linear program lp based 2 approximation algorithm for the unweighted case we show that our analysis is tight by providing a matching lower bound on the integrality gap of the lp moreover we prove that the unweighted version is np hard to approximate within a factor less than 6 5 we conclude by presenting a polynomial time algorithm based on dynamic programming for the case when the number of scenarios and the values of the instance are bounded by some constant

n dimensional integrability from two photon coalgebra symmetry a wide class of hamiltonian systems with n degrees of freedom and endowed with at least n 2 functionally independent integrals of motion in involution is constructed by making use of the two photon lie poisson coalgebra h6 the set of n 2 constants of the motion is shown to be a universal one for all these hamiltonians irrespective of the dependence of the latter on several arbitrary functions and n free parameters within this large class of quasi integrable n dimensional hamiltonians new families of completely integrable systems are identified by finding explicitly a new independent integral through the analysis of the sub coalgebra structure of h6 in particular new completely integrable n dimensional hamiltonians describing natural systems geodesic flows and static electromagnetic hamiltonians are presented

learning valued preference structures for solving classification problems this paper introduces a new approach to classification which combines pairwise decomposition techniques with ideas and tools from fuzzy preference modeling more specifically our approach first decomposes a polychotomous classification problem involving m classes into an ensemble of binary problems one for each ordered pair of classes the corresponding classifiers are trained on the relevant subsets of the transformed original training data in the classification phase a new query is submitted to every binary learner the output of each classifier is interpreted as a fuzzy degree of preference for the first in comparison with the second class by combining the outputs of all classifiers one thus obtains a fuzzy preference relation which is taken as a point of departure for the final classification decision this way the problem of classification is effectively reduced to a problem of decision making based on a fuzzy preference relation corresponding techniques which have been investigated quite intensively in the field of fuzzy set theory hence become amenable to the task of classification in particular by decomposing a preference relation into a strict preference an indifference and an incomparability relation this approach allows one to quantify different types of uncertainty in classification and thereby supports sophisticated classification and postprocessing strategies

quantum 1 1 extended galilei algebras from lie bialgebras to quantum r matrices and integrable systems the lie bialgebras of the 1 1 extended galilei algebra are obtained and classified into four multiparametric families their quantum deformations are obtained together with the corresponding deformed casimir operators for the coboundary cases quantum universal r matrices are also given applications of the quantum extended galilei algebras to classical integrable systems are explicitly developed

loop coproducts gaudin models and poisson coalgebras in this paper we show that if a is a poisson algebra equipped with a set of maps i a a n satisfying suitable conditions then the images of the casimir functions of a under the maps i that we call loop coproducts are in involution rational trigonometric and elliptic gaudin models can be recovered as particular cases of this construction and we show that the same happens for the integrable or partially integrable models that can be obtained through the so called coproduct method on the other hand we show that the loop coproduct approach provides a natural generalization of the gaudin algebras from the lie poisson to the generic poisson algebra context and hopefully can lead to the definition of new integrable models

1 1 schrodinger lie bialgebras and their poisson lie groups all lie bialgebra structures for the 1 1 dimensional centrally extended schrodinger algebra are explicitly derived and proved to be of coboundary type therefore since all of them come from a classical r matrix the complete family of schrodinger poisson lie groups can be deduced by means of the sklyanin bracket all possible embeddings of the harmonic oscillator extended galilei and gl 2 lie bialgebras within the schrodinger classification are studied as an application new quantum hopf algebra deformations of the schrodinger algebra including their corresponding quantum universal r matrices are constructed

reducing multiclass to binary a unifying approach for margin classifiers we present a unifying framework for studying the solution of multiclass categorization problems by reducing them to multiple binary problems that are then solved using a margin based binary learning algorithm the proposed framework unifies some of the most popular approaches in which each class is compared against all others or in which all pairs of classes are compared to each other or in which output codes with error correcting properties are used we propose a general method for combining the classifiers generated on the binary problems and we prove a general empirical multiclass loss bound given the empirical loss of the individual binary learning algorithms the scheme and the corresponding bounds apply to many popular classification learning algorithms including support vector machines adaboost regression logistic regression and decision tree algorithms we also give a multiclass generalization error analysis for general output codes with adaboost as the binary learner experimental results with svm and adaboost show that our scheme provides a viable alternative to the most commonly used multiclass algorithms

robust discrete optimization and network flows we propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution and is computationally tractable both practically and theoretically in particular when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation when only the cost coefficients are subject to uncertainty and the problem is a 0 1 discrete optimization problem on n variables then we solve the robust counterpart by solving at most n 1 instances of the original problem thus the robust counterpart of a polynomially solvable 0 1 discrete optimization problem remains polynomially solvable in particular robust matching spanning tree shortest path matroid intersection etc are polynomially solvable we also show that the robust counterpart of an np hard approximable 0 1 discrete optimization problem remains approximable finally we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network

using fuzzy binary relations for identifying noninterior decision alternatives abstract a fuzzy noninferiority relation is introduced based on this relation fuzzy relations of preference strict preference indifference and incomparability are specified and their properties are investigated a fuzzy noninferior subset is defined and its basic characteristic properties are established an algorithm is presented for identifying a fuzzy set of noninferior alternatives in decision making situations that require the selection of a certain number of noninferior alternatives for more detailed study

the use of fuzzy outranking relations in preference modelling this paper deals with preference modelling in the context of decision aid in this framework conflicting systems of logic uncertain knowledge ambiguous positions are always present in order to tackle this problem a multiple criteria methodology is proposed mainly based on fuzzy outranking relations introduced both at one dimensional and multi dimensional levels some properties of outranking relations are investigated such relations are then combined using fuzzy logical connectives to generate relational systems of fuzzy preferences that are shown to be very useful to reflect the vagueness of information in the various preference situations that may be considered in the modelling process

solvable lie algebras with an mathbb n graded nilradical of maximal nilpotency degree and their invariants the class of solvable lie algebras with an graded nilradical of maximal nilpotency index is classified it is shown that such solvable extensions are unique up to isomorphism the generalized casimir invariants for the graded nilradicals and their associated solvable extensions are computed by the method of moving frames

maximal superintegrability of benenti systems for a class of hamiltonian systems naturally arising in the modern theory of separation of variables we establish their maximal superintegrability by explicitly constructing the additional integrals of motion

integrable hamiltonian systems with velocity dependent potentials the integrability of a two dimensional hamiltonian in which the potential depends explicitly on the momenta is investigated hamiltonians of this kind are encountered in the description of the motion of a particle in a magnetic field two integrable classes of potentials are identified and the second integral of motion is constructed for each of them the singularity analysis of the equations of motion is also performed confirming once more the relation between the weak painleve property and integrability

how to construct integrable fokker planck and electromagnetic hamiltonians from ordinary integrable hamiltonians integrable hamiltonians with velocity dependent potentials including those of fokker planck type h 1 2 p2x p2y kxpx kypy are constructed from integrable hamiltonians of type h 1 2 p2x p2y v x y using certain canonical and noncanonical transformations some of the hamiltonians obtained this way are integrable only for zero energy candidates for the potential which is of interest for fokker planck models are constructed in several cases

universal integrals for superintegrable systems on n dimensional spaces of constant curvature an infinite family of classical superintegrable hamiltonians defined on the n dimensional spherical euclidean and hyperbolic spaces are shown to have a common set of 2n 3 functionally independent constants of the motion among them two different subsets of n integrals in involution including the hamiltonian can always be explicitly identified as particular cases we recover in a straightforward way most of the superintegrability properties of the smorodinsky winternitz and generalized kepler coulomb systems on spaces of constant curvature and we introduce as well new classes of quasi maximally superintegrable potentials on these spaces results presented here are a consequence of the poisson coalgebra symmetry of all the hamiltonians together with an appropriate use of the phase spaces associated with poincare and beltrami coordinates

a jordanian quantum two photon schrodinger algebra a non standard quantum deformation of the two photon algebra is constructed and its quantum universal r matrix is given representations of this new quantum algebra are studied on the fock space and translated into fock bargmann realizations that provide a direct formalism for the definition of deformed states of light finally the isomorphism between and the 1 1 schrodinger algebra is used to introduce a new non standard hopf algebra deformation of this latter symmetry algebra

quantum superintegrability and exact solvability in n dimensions a family of maximally superintegrable systems containing the coulomb atom as a special case is constructed in n dimensional euclidean space two different sets of n commuting second order operators are found overlapping in the hamiltonian alone the system is separable in several coordinate systems and is shown to be exactly solvable it is solved in terms of classical orthogonal polynomials the hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine lie algebra

on superintegrable symmetry breaking potentials in n dimensional euclidean space we give a graphical prescription for obtaining and characterizing all separable coordinates for which the schr dinger equation admits separable solutions for one of the superintegrable potentials here xn 1 is a distinguished cartesian variable the algebra of second order symmetries of the resulting schr dinger equation is given and for the first potential the closure relations of the corresponding quadratic algebra these potentials are particularly interesting because they occur in all dimensions n 1 the separation of variables problem is highly nontrivial for them and many other potentials are limiting cases

decompositions network flows and a precedence constrained single machine scheduling problem we present an in depth theoretical algorithmic and computational study of a linear programming lp relaxation to the precedence constrained single machine scheduling problem 1 prec s jw jc jto minimize a weighted sum of job completion times on the theoretical side we study the structure of tight parallel inequalities in thelp relaxation and show thatevery permutation schedule that is consistent with sidney s decomposition has total cost no more than twice the optimum on the algorithmic side we provide a parametric extension to sidney s decomposition and show that a finest decomposition can be obtained by essentially solving a parametric minimum cut problem finally we report results obtained by an algorithm based on these developments on randomly generated instances with up to 2 000 jobs