a new matrix method for the casimir operators of the lie algebras w mathfrak s mathfrak p n hbox mathbb r and i mathfrak s mathfrak p 2n hbox mathbb r a method is given to determine the casimir operators of the perfect lie algebras and the inhomogeneous lie algebras in terms of polynomials associated with a parametrized 2n 1 2n 1 matrix for the inhomogeneous symplectic algebras this matrix is shown to be associated to a faithful representation we further analyse the invariants for the extended schrodinger algebra in n 1 dimensions which arises naturally as a subalgebra of the method is extended to other classes of lie algebras and some applications to the missing label problem are given

local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

convergence rates for forward backward dynamical systems associated with strongly monotone inclusions abstract we investigate the convergence rates of the trajectories generated by implicit first and second order dynamical systems associated to the determination of the zeros of the sum of a maximally monotone operator and a monotone and lipschitz continuous one in a real hilbert space we show that these trajectories strongly converge with exponential rate to a zero of the sum provided the latter is strongly monotone we derive from here convergence rates for the trajectories generated by dynamical systems associated to the minimization of the sum of a proper convex and lower semicontinuous function with a smooth convex one provided the objective function fulfills a strong convexity assumption in the particular case of minimizing a smooth and strongly convex function we prove that its values converge along the trajectory to its minimum value with exponential rate too

grading refinements in the contractions of lie algebras and their invariants all mutually inequivalent toroidal gradings of the lie algebra are constructed each of them provides several different graded contractions to other eight dimensional lie algebras these can be either continuous or discrete the continuous ones are related to generalized wigner inonu contractions the behaviour of casimir operators under the graded contractions is studied and the casimir and generalized casimir operators of the contracted lie algebras are presented

prox penalization and splitting methods for constrained variational problems this paper is concerned with the study of a class of prox penalization methods for solving variational inequalities of the form ax n c x ni0 where mathcal h is a real hilbert space a mathcal h rightrightarrows mathcal h is a maximal monotone operator and n c is the outward normal cone to a closed convex set c subset mathcal h given psi mathcal h to mathbb r cup infty which acts as a penalization function with respect to the constraint x in c and a penalization parameter beta n we consider a diagonal proximal algorithm of the form x n i lambda n a beta n partial psi 1 x n 1 and an algorithm which alternates proximal steps with respect to a and penalization steps with respect to c and reads as x n i lambda n beta n partial psi 1 i lambda na 1 x n 1 we obtain weak ergodic convergence for a general maximal monotone operator a and weak convergence of the whole sequence x n when a is the subdifferential of a proper lower semicontinuous convex function mixing with passty s idea we can extend the ergodic convergence theorem thus obtaining the convergence of a prox penalization splitting algorithm for constrained variational inequalities governed by the sum of several maximal monotone operators our results are applied to an optimal control problem where the state variable and the control are coupled by an elliptic equation we also establish robustness and stability results that account for numerical approximation errors

a note on design transformation and binning in nonparametric curve estimation summary methods based on design transformation and equal number binning are shown to overcome problems of design sparseness while retaining excellent theoretical properties in company with local linear smoothing they produce optimal estimators although with optimality defined a little differently from in the nontransformed case when used in conjunction with wavelet techniques they overcome problems of stochastic design but unlike related techniques based on convolution and interpolation do not degrade features of the signal through prior smoothing or inflate the variance

graded contractions of casimir operators we describe graded contractions of casimir operators of lie algebras the formalism applies to any casimir operator given as a symmetric form we deal with the quadratic casimir operator in exhaustive detail and indicate the modifications for operators of higher degree

asymptotic control and stabilization of nonlinear oscillators with non isolated equilibria let h r be a c1 function on a real hilbert space h and let 0 be a positive damping parameter for any control function e r r which tends to zero as t we study the asymptotic behavior of the trajectories of the damped nonlinear oscillator hbfc x t x t x t e t x t 0 we show that if e t does not tend to zero too rapidly as t then the term e t x t asymptotically acts as a tikhonov regularization which forces the trajectories to converge to a particular equilibrium indeed in the main result of this paper it is established that when is convex and s argmin under the key assumption that e is a slow control i e 0 e t dt then each trajectory of the hbfc system strongly converges as t to the element of minimal norm of the closed convex set s as an application we consider the damped wave equation with neumann boundary condition formula

non solvable contractions of semisimple lie algebras in low dimension the problem of non solvable contractions of lie algebras is analysed by means of a stability theorem the problem is shown to be deeply related to the embeddings among semisimple lie algebras and the resulting branching rules for representations with this procedure we determine all deformations of indecomposable lie algebras having a nontrivial levi decomposition onto semisimple lie algebras of dimension n 8 and obtain the non solvable contractions of the latter class of algebras

trigonometry of spacetimes a new self dual approach to a curvature signature in dependent trigonometry a new method to obtain trigonometry for the real spaces of constant curvature and metric of any even degenerate signature is presented the method could be described as curvature signature in dependent trigonometry and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation this brings to its logical end the idea of an absolute trigonometry and provides equations which hold true for the nine two dimensional spaces of constant curvature and any signature this family of spaces includes both relativistic and non relativistic spacetimes therefore a complete discussion of trigonometry in the six de sitter minkowskian newton hooke and galilean spacetimes follow as particular instances of the general approach distinctive traits of the method are universality and self duality every equation is meaningful for the nine spaces at once and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves these basic structural properties allow a complete study of trigonometry and in fact any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes in most cases these equations are new

2 1 kinematical expansions from galilei to de sitter algebras expansion of a lie algebra is the opposite process to contraction starting from a lie algebra the expansion process goes to another algebra which is non isomorphic and less abelian we propose an expansion method based on the casimir invariants of the initial and expanded algebras and where the free parameters involved in the expansion are the curvatures of their associated homogeneous spaces this method is applied for expansions within the family of lie algebras of three dimensional spaces and 2 1 d kinematical algebras we show that these expansions are classed into two types the first type makes the curvature of space or spacetime different from zero i e it introduces a space or universe radius while the other has a similar interpretation for the curvature of the space of worldlines which is non positive and equal to 1 c2 in the kinematical algebras we obtain expansions which go from galilei to either newton hooke or poincarealgebras and from these to de sitter algebras as well as some other examples

interpolation methods for nonlinear wavelet regression with irregularly spaced design we introduce interpolation methods that enable nonlinear wavelet estimators to be employed with stochastic design or nondyadic regular design in problems of nonparametric regression this approach allows relatively rapid computation involving dyadic approximations to wavelet after interpolation techniques new types of interpolation are described enabling first order variance reduction at the expense of second order increases in bias the effect of interpolation on threshold choice is addressed and appropriate thresholds are suggested for error distributions with as few as four finite moments

wavelet based estimation with multiple sampling rates we suggest an adaptive sampling rule for obtaining information from noisy signals using wavelet methods the technique involves increasing the sampling rate when relatively high frequency terms are incorporated into the wavelet estimator and decreasing it when again using thresholded terms as an empirical guide signal complexity is judged to have decreased through sampling in this way the algorithm is able to accurately recover relatively complex signals without increasing the long run average expense of sampling it achieves this level of performance by exploiting the opportunities for near real time sampling that are available if one uses a relatively high primary resolution level when constructing the basic wavelet estimator in the practical problems that motivate the work where signal to noise ratio is particularly high and the long run average sampling rate may be several hundred thousand operations per second high primary resolution levels are quite feasible

determinantal formulae for the casimir operators of inhomogeneous lie algebras the author would like to correct some inaccuracy in formula 6 please see pdf for details

strong asymptotic convergence of evolution equations governed by maximal monotone operators with tikhonov regularization abstract we consider the tikhonov like dynamics u t a u t e t u t where a is a maximal monotone operator on a hilbert space and the parameter function e t tends to 0 as t with 0 e t d t when a is the subdifferential of a closed proper convex function f we establish strong convergence of u t towards the least norm minimizer of f in the general case we prove strong convergence towards the least norm point in a 1 0 provided that the function e t has bounded variation and provide a counterexample when this property fails

asymptotic behavior of coupled dynamical systems with multiscale aspects abstract we study the asymptotic behavior as time variable t goes to of nonautonomous dynamical systems involving multiscale features as a benchmark case given h a general hilbert space h r and h r two closed convex functions and a function of t which tends to as t goes to we consider the differential inclusion x t x t t x t 0 this system models the emergence of various collective behaviors in game theory as well as the asymptotic control of coupled systems we show several results ranging from weak ergodic to strong convergence of the trajectories as a key ingredient we assume that for every p belonging to the range of n c 0 t p t c p t d t where is the fenchel conjugate of c is the support function of c argmin and n c x is the normal cone to c at x as a by product we revisit the system x t t x t x t 0 where t tends to zero as t goes to and 0 t d t whose asymptotic behavior can be derived from the preceding one by time rescaling applications are given in game theory optimal control variational problems and pdes

from constants of motion to superposition rules for lie hamilton systems a lie system is a non autonomous system of first order differential equations possessing a superposition rule i e a map expressing its general solution in terms of a generic finite family of particular solutions and some constants lie hamilton systems form a subclass of lie systems whose dynamics is governed by a curve in a finite dimensional real lie algebra of functions on a poisson manifold it is shown that lie hamilton systems are naturally endowed with a poisson coalgebra structure this allows us to devise methods for deriving in an algebraic way their constants of motion and superposition rules we illustrate our methods by studying kummer schwarz equations riccati equations ermakov systems and smorodinsky winternitz systems with time dependent frequency

generalized two mode harmonic oscillator dynamical group and squeezed states in this paper we generalize our previous results of the generalized one mode harmonic oscillator to the generalized two mode case systematic use is made of the so 3 2 dynamical group and we are able to write a general form for the exact time evolution operator in terms of squeezing operators of one and two modes a complete classification of the exact solutions is made and we derive them explicitly whenever possible the relevant results on algebraic decomposition coherent state generators and classification of the solutions are shown in tables a plethora of soluble hamiltonians already treated in the literature which appear to be particular cases of the general formalism presented herein are analysed as well as new cases which to the authors knowledge have not yet been considered

a new class of alternating proximal minimization algorithms with costs to move given two objective functions f mathcal x mapsto r cup infty and g mathcal y mapsto r cup infty on abstract spaces mathcal x and mathcal y and a coupling function c mathcal x times mathcal y mapsto r we introduce and study alternative minimization algorithms of the following type x 0 y 0 in mathcal x times mathcal y mbox given x n y n rightarrow x n 1 y n rightarrow x n 1 y n 1 mbox as follows left begin array l x n 1 in mbox argmin f xi beta n c xi y n alpha n h x n xi xi in mathcal x y n 1 in mbox argmin g eta mu n c x n 1 eta nu n k y n eta eta in mathcal y end array right their most original feature is the introduction of the terms h mathcal x times mathcal x mapsto r and k mathcal y times mathcal y mapsto r which are costs to change or to move distance like functions relative entropies accounting for various inertial friction or anchoring effects these algorithms are studied in a general abstract framework the introduction of the costs to change h and k leads to proximal minimizations with corresponding dissipative effects as a result the algorithms enjoy nice convergent properties coefficients alpha n beta n mu n nu n are nonnegative parameters when taking alpha n nu n 0 and quadratic costs on a hilbert space one recovers the classical alternating minimization algorithm which itself is a natural extension of the alternating projection algorithm of von neumann a number of new significant results hold in general metric spaces we pay particular attention to the following cases 1 mathcal x d mathcal x and mathcal y d mathcal y are complete metric spaces and h geq d mathcal x k geq d mathcal y high local costs to move the algorithms then provide sequences that converge to nash equilibria 2 mathcal x mathcal y mathcal h is a hilbert space the costs to change are quadratic low local costs to move and the functions f g mathcal h mapsto r cup infty are closed convex proper then some of the classical convergence theorems for alternating convex minimization algorithms including those of acker and prestel are properly extended with original proofs

the general solution of the real graded contractions of the general solution of the graded contraction equations for a grading of the real compact simple lie algebra so n 1 is presented in an explicit way it turns out to depend on independent real parameters the structure of the general graded contractions is displayed for the low dimensional cases and kinematical algebras are shown to appear straightforwardly the geometrical or physical meaning of the contraction parameters as curvatures is also analysed in particular for kinematical algebras these curvatures are directly linked to geometrical properties of possible homogeneous spacetimes

asymptotic control and stabilization of nonlinear oscillators with non isolated equilibria a note from l 1 to non l 1 in h attouch m o czarnecki asymptotic control and stabilization of non linear oscillators with non isolated equilibria j differential equations 179 2002 278 310 we exhibited a sharp condition ensuring the efficiency of the tikhonov like control term in the hbfc system precisely let h r be a c1 function on a real hilbert space h let 0 be a positive damping parameter and let r r be a control function which decreases to zero as t in order to select particular equilibria in the important case where has non isolated equilibria we introduced in h attouch m o czarnecki j differential equations 179 2002 278 310 the following damped nonlinear oscillator and studied its asymptotic behavior hbfc x t x t x t t x t 0 we established that when is convex and s argmin under the key assumption that is a slow control i e 0 t dt then each trajectory of the hbfc system strongly converges when t to the element of minimal norm of the closed convex set s the condition on the control term is sharp indeed when 0 t dt the trajectory weakly converges but it may not strongly converge and we have no information a priori on the weak limit in this note we give an answer to the following question when does an l1 control term becomes or behave non l1 precisely take a control term l1 let x be the solution of the corresponding hbfc system take n to be a non increasing truncation of n t t for t 0 n let x be the solution of the corresponding hbfc system we show that limn xn x 0 in particular the weak limits of the trajectories xn strongly converge when n to the strong limit of the trajectory x in other words there is no loss of the information gained by the slow behavior for t n

asymptotic behavior of nonautonomous monotone and subgradient evolution equations in a hilbert setting h we study the asymptotic behavior of the trajectories of nonautonomous evolution equations dot x t a t x t ni 0 where for each t geq 0 a t h rightrightarrows h denotes a maximal monotone operator we provide general conditions guaranteeing the weak ergodic convergence of each trajectory x cdot to a zero of a limit maximal monotone operator a infty as the time variable t tends to infty the crucial point is to use the brezis haraux function or equivalently the fitzpatrick function to express at which rate the excess of mathrm gph a infty over mathrm gph a t tends to zero this approach gives a sharp and unifying view of this subject in the case of operators a t partial phi t which are subdifferentials of proper closed convex functions phi t we show convergence results for the trajectories then we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions illustrations are given in the field of coupled systems and partial differential equations

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

central extensions of the quasi orthogonal lie algebras we determine the central extensions of a whole family of lie algebras obtained by the method of graded contractions from n arbitrary all the inhomogeneous orthogonal and pseudo orthogonal algebras are members of this family as well as a large number of other non semisimple algebras all of which have at least a semidirect structure in some cases two or more the dimensions of their second cohomology groups and the explicit expression of their central extensions are given

internal labelling operators and contractions of lie algebras we analyse under which conditions the missing label problem associated with a reduction chain of simple lie algebras can be completely solved by means of an inonu wigner contraction naturally related to the embedding this provides a new interpretation of the missing label operators in terms of the casimir operators of the contracted algebra and shows that the available labelling operators are not completely equivalent further the procedure is used to obtain upper bounds for the number of invariants of affine lie algebras arising as contractions of semi simple algebras

approaching the solving of constrained variational inequalities via penalty term based dynamical systems abstract we investigate the existence and uniqueness of locally absolutely continuous trajectories of a penalty term based dynamical system associated to a constrained variational inequality expressed as a monotone inclusion problem relying on lyapunov analysis and on the ergodic continuous version of the celebrated opial lemma we prove weak ergodic convergence of the orbits to a solution of the constrained variational inequality under investigation if one of the operators involved satisfies stronger monotonicity properties then strong convergence of the trajectories can be shown

linear functions of concomitants of order statistics with application to nonparametric estimation of a regression function abstract let xi yi i 1 2 n be independent identically distributed as x y then the rth ordered x variate is denoted by xr n and the associated y variate the concomitant of the rth order statistic by y r n this paper considers statistics of the form and more generally of the form where j is a bounded smooth function and may depend on n under certain regularity conditions the asymptotic normality of these statistics is established these statistics are used to construct consistent estimators of various conditional quantities for example e y x x p y a x x and var y x x

the family of quaternionic quasi unitary lie algebras and their central extensions the family of quaternionic quasi unitary or quaternionic unitary cayley klein algebras is described in a unified setting this family includes the simple algebras sp n 1 and sp p q in the cartan series cn 1 as well as many non semisimple real lie algebras which can be obtained from these simple algebras by particular contractions the algebras in this family are realized here in relation with the groups of isometries of quaternionic hermitian spaces of constant holomorphic curvature this common framework allows one to perform the study of many properties for all these lie algebras simultaneously in this paper the central extensions for all quasi simple lie algebras of the quaternionic unitary cayley klein family are shown to be trivial no matter their dimension

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

on the choice of smoothing parameter threshold and truncation in nonparametric regression by non linear wavelet methods summary concise asymptotic theory is developed for non linear wavelet estimators of regression means in the context of general error distributions general designs general normalizations in the case of stochastic design and non structural assumptions about the mean the influence of the tail weight of the error distribution is addressed in the setting of choosing threshold and truncation parameters mainly the tail weight is described in an extremely simple way by a moment condition previous work on this topic has generally imposed the much more stringent assumption that the error distribution be normal different approaches to correction for stochastic design are suggested these include conventional kernel estimation of the design density in which case the interaction between the smoothing parameters of the non linear wavelet estimator and the linear kernel method is described

a convergence result for nonautonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming abstract we present a new result on the asymptotic behavior of nonautonomous subgradient evolution equations of the form u t t u t where t t 0 is a family of closed proper convex functions the result is used to study the flow generated by the family t x f x r t where f x r ctx r exp aix bi r is the exponential penalty approximation of the linear program min ctx ax b and r t is a positive function tending to 0 when t we prove that the trajectory u t converges to an optimal solution u of the linear program and we give conditions for the convergence of an associated dual trajectory t toward an optimal solution of the dual program

asymptotic behavior of gradient like dynamical systems involving inertia and multiscale aspects abstract in a hilbert space h we study the asymptotic behavior as time variable t goes to of nonautonomous gradient like dynamical systems involving inertia and multiscale features given h r and h r two convex differentiable functions a positive damping parameter and t a function of t which tends to zero as t goes to we consider the second order differential equation x t x t x t t x t 0 this system models the emergence of various collective behaviors in game theory as well as the asymptotic control of coupled nonlinear oscillators assuming that t tends to zero moderately slowly as t goes to infinity we show that the trajectories converge weakly in h the limiting equilibria are solutions of the hierarchical minimization problem which consists in minimizing over the set c of minimizers of as key assumptions we suppose that 0 t d t and that for every p belonging to a convex cone c depending on the data and 0 t p c t p d t where is the fenchel conjugate of and c is the support function of c an application is given to coupled oscillators

graded contractions of bilinear invariant forms of lie algebra we introduce a new construction for bilinear invariant forms on lie algebras based on the method of graded contractions the general method is described and the z2 z3 and z2 x z2 contractions are found the results can be applied to all lie algebras and superalgebras finite or infinite dimensional which admit the chosen gradings we consider some examples contractions of the killing form toroidal contractions of sl 3 c and we briefly discuss the limit to new wess zumino witten actions

a dynamic gradient approach to pareto optimization with nonsmooth convex objective functions in a general hilbert framework we consider continuous gradient like dynamical systems for constrained multiobjective optimization involving non smooth convex objective functions our approach is in the line of a previous work where was considered the case of convex di erentiable objective functions based on the yosida regularization of the subdi erential operators involved in the system we obtain the existence of strong global trajectories we prove a descent property for each objective function and the convergence of trajectories to weak pareto minima this approach provides a dynamical endogenous weighting of the objective functions applications are given to cooperative games inverse problems and numerical multiobjective optimization

cayley klein algebras as graded contractions of so n 1 we study z2 x n graded contractions of the real compact simple lie algebra so n 1 and we identify within them the cayley klein algebras as a naturally distinguished subset

second order forward backward dynamical systems for monotone inclusion problems we begin by considering second order dynamical systems of the from ddot x t gamma t dot x t lambda t b x t 0 where b cal h rightarrow cal h is a cocoercive operator defined on a real hilbert space cal h lambda 0 infty rightarrow 0 infty is a relaxation function and gamma 0 infty rightarrow 0 infty is a damping function both depending on time for the generated trajectories we show existence and uniqueness of the generated trajectories as well as their weak asymptotic convergence to a zero of the operator b the framework allows us to address from similar perspectives second order dynamical systems associated with the problem of finding zeros of the sum of a maximally monotone operator and a cocoercive one this captures as a particular case the minimization of the sum of a nonsmooth convex function with a smooth convex one furthermore we prove that when b is the gradient of a smooth convex function the value of the latter converges along the ergo

versions of kernel type regression estimators abstract we explore the aims of and relationships between various kernel type regression estimators to do so we identify two general types of direct kernel estimators differing in their treatment of the nuisance density function associated with regressor variable design we look at the well known gasser muller nadaraya watson and priestley chao methods in this light in the random design case none of these methods is totally adequate and we mention a novel direct kernel method with appropriate properties disadvantages of even the latter idea are remedied by kernel weighted local linear fitting a well known technique that is currently enjoying renewed popularity we see how to fit this approach into our general framework and hence form a unified understanding of how these kernel type smoothers interrelate though the mission of this article is unificatory and even pedagogical the desire for better understanding of superficially different approaches is motivated by the need to improve prac

contractions of the irreducible representations of the quantum algebras suq 2 and soq 3 the contractions of the irreducible representations of the unitary quantum algebra suq 2 and the orthogonal quantum algebra soq 3 in the gel fand tsetlin basis are regarded in detail with the help of the dual numbers

convergence of damped inertial dynamics governed by regularized maximally monotone operators in a hilbert space setting we study the asymptotic behavior as time t goes to infinity of the trajectories of a second order differential equation governed by the yosida regularization of a maximally monotone operator with time varying positive index lambda t the dissipative and convergence properties are attached to the presence of a viscous damping term with positive coefficient gamma t a suitable tuning of the parameters gamma t and lambda t makes it possible to prove the weak convergence of the trajectories towards zeros of the operator when the operator is the subdifferential of a closed convex proper function we estimate the rate of convergence of the values these results are in line with the recent articles by attouch cabot and attouch peypouquet in this last paper the authors considered the case gamma t frac alpha t which is naturally linked to nesterov s accelerated method we unify and often improve the results already present in the literature

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

four dimensional quantum affine algebras and space time q symmetries a global model of the q deformation for the quasiorthogonal lie algebras generating the groups of motions of the four dimensional affine cayley klein ck geometries is obtained starting from the three dimensional deformations it is shown how the main algebraic classical properties of the ck systems can be implemented in the quantum case quantum deformed versions either of the space time or space symmetry algebras poincare 3 1 galilei 3 1 4 d euclidean as well as others appear in this context as particular cases for some of these classical algebras several q deformations are directly obtained

superintegrability on curved spaces orbits and momentum hodographs revisiting a classical result by hamilton the equation of the orbits in the configuration space and of the hodographs in the momentum plane for the curved kepler and harmonic oscillator systems living in a configuration space of any constant curvature and either signature type are derived by purely algebraic means this result extends to the curved kepler or harmonic oscillator for the classical hamilton derivation of the orbits of the euclidean kepler problem through its hodographs in both cases the fundamental property allowing these derivations to work is the superintegrability of the curved kepler and harmonic oscillator no matter whether the constant curvature of the configuration space is zero or not or whether the configuration space metric is riemannian or lorentzian in the curved case the basic result does not refer to the velocity hodograph but to the momentum hodograph both coincide in a euclidean configuration space but only the latter is unambiguously defined in all curved spaces

anti de sitter poincare symmetries and representations from poincare galilei through a classical deformation approach a classical deformation procedure based on universal enveloping algebras casimirs and curvatures of symmetrical homogeneous spaces is applied to several cases of physical relevance starting from the 3 1 d galilei algebra we describe at the level of representations the process leading to its two physically meaningful deformed neighbours the poincare algebra is obtained by introducing a negative curvature in the flat galilean phase space or space of worldlines while keeping a flat spacetime to be precise starting from a representation of the galilei algebra with both casimirs different from zero we obtain a representation of the poincare algebra with both casimirs necessarily equal to zero the poincare angular momentum pauli lubanski components position and velocity operators etc are expressed in terms of galilean operators through some expressions deforming the proper galilean ones similarly the newton hooke algebras appear by endowing spacetime with a nonzero curvature while keeping a flat phase space the same approach starting from the 3 1 d poincare algebra provides representations of the anti de sitter as poincare deformations

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

casimir invariants for the complete family of quasisimple orthogonal algebras a complete choice of generators of the centre of the enveloping algebras of real quasisimple lie algebras of orthogonal type for arbitrary dimension is obtained in a unified setting the results simultaneously include the well known polynomial invariants of the pseudo orthogonal algebras as well as the casimirs for many non simple algebras such as the inhomogeneous the newton hooke and galilei type etc which are obtained by contraction s starting from the simple algebras the dimension of the centre of the enveloping algebra of a quasisimple orthogonal algebra turns out to be the same as for the simple algebras from which they come by contraction the structure of the higher order invariants is given in a convenient pyramidal manner in terms of certain sets of pauli lubanski elements in the enveloping algebras as an example showing this approach at work the scheme is applied to recovering the casimirs for the 3 1 kinematical algebras some prospects on the relevance of these results for the study of expansions are also given