distribution of periodic torus orbits and duke s theorem for cubic fields we study periodic torus orbits on spaces of lattices using the action of the group of adelic points of the underlying tori we define a natural equivalence relation on these orbits and show that the equivalence classes become uniformly distributed this is a cubic analogue of duke s theorem about the distribution of closed geodesics on the modular surface suitably interpreted the ideal classes of a cubic totally real field are equidistributed in the modular 5 fold sl3 z sl3 r so3 in particular this proves a stronger form of the folklore conjecture that the collection of maximal compact flats in sl3 z sl3 r so3 of volume infinity

limiting distribution of translates of the orbit of a maximal pmb mathbb q q torus from identity on mathrm sl _n pmb mathbb r mathrm sl _n pmb mathbb z sl n r sl n z consider the orbit of real points of a maximal mathbb q torus from identity coset on mathrm sl n mathbb r mathrm sl n mathbb z it naturally carries a possibly infinite haar measure we classify all possible limit measures of it when translated by a sequence of elements from mathrm sl n mathbb r this is a natural extension of shapira and zheng s work where only mathbb q split tori are considered

on manipulator control by exact linearization comments on the application to rigid link manipulators of geometric control theory resolved acceleration control operational space control and nonlinear decoupling theory are given and the essential unity of these techniques for externally linearizing and decoupling end effector dynamics is discussed exploiting the fact that the mass matrix of a rigid link manipulator is positive definite and the fact that there is an independent input for each degree of freedom it is shown that a necessary and sufficient condition for a locally externally linearizing and output decoupling feedback law to exist is that the end effector jacobian matrix be nonsingular

exponential control of a rotational motion of a rigid body using quaternions a new developed technique is used to study the problem of the exponential stabilization of the rotational motion of a rigid body about a fixed axis in the body with the help of rotors in this technique the exponential asymptotic stability of this motion is proved the control moments leading to the rotational motion are obtained the conditions that ensure the exponential asymptotic stabilization of this motion are used to obtain the stabilizing control law as non linear functions of the phase coordinates of the system finally many special cases and analyzing the results of the studied problem are investigated

integer points on spheres and their orthogonal lattices linnik proved in the late 1950 s the equidistribution of integer points on large spheres under a congruence condition the congruence condition was lifted in 1988 by duke building on a break through by iwaniec using completely different techniques we conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement we use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition

on the existence of solutions to the quadratic mixed integer mean variance portfolio selection problem abstract in the standard mean variance portfolio selection approach several operative features are not taken into account among these neglected aspects one of particular interest is the finite divisibility of the stock assets i e the obligation to buy sell only integer quantities of asset lots whose number is pre established in order to consider such a feature we deal with a suitably defined quadratic mixed integer programming problem in particular we formulate this problem in terms of quantities of asset lots instead of as usual in terms of capital per cent quotas secondly we provide necessary and sufficient conditions for the existence of a non empty mixed integer feasible set of the considered programming problem thirdly we present some rounding procedures for finding in a finite number of steps a feasible mixed integer solution which is better than the one detected by the necessary and sufficient conditions in terms of the value assumed by the portfolio variance finally we perform an extensive computational experiment by means of which we verify the goodness of our approach

representations of integers by an invariant polynomial and unipotent flows we study a refined version of linnik s problem on the asymptotic behavior of the number of representations of integers m by an integral polynomial as m tends to infinity assuming that the polynomials arise from invariant theory we reduce the question to the study of limiting behavior of measures invariant under unipotent flows our main tool is then ratner s theorem on the uniform distribution of unipotent flows in a form refined by dani and margulis dm2

spacecraft attitude control using explicit model predictive control in this paper an explicit model predictive controller for the attitude of a satellite is designed explicit solutions to constrained linear mpc problems can be computed by solving multi parametric quadratic programs mpqp where the parameters are the components of the state vector the solution to the mpqp is a piecewise affine pwa function which can be evaluated at each sample to obtain the optimal control law the on line computation effort is restricted to a table lookup and the controller can be implemented on inexpensive hardware as fixed point arithmetics can be used this is useful for systems with limited power and cpu resources an example of such systems is micro satellites which is the focus of this paper in particular the explicit mpc empc approach is applied to the sseti eseo micro satellite initiated by the european space agency esa the theoretical results are supported by simulations

hybrid global exponential stabilization on so 3 we propose a central synergistic hybrid approach for global exponential stabilization on the special orthogonal group so 3 so 3 we introduce a new switching concept relying on a central family of possibly non differentiable potential functions that enjoy the following properties 1 being quadratic as well as their gradients with respect to the euclidean attitude distance and 2 being synergistic with respect to the gradient s singular and or critical points the proposed approach is used to solve the attitude tracking problem leading to global exponential stability results

attitude synchronization of multiple spacecraft with cone avoidance constraints abstract in this paper we consider a team of spacecraft which requires changing its orientation to a common attitude using a decentralized control scheme under a connected communication topology while satisfying cone avoidance constraints due to blind celestial objects plume impingement and so on for this purpose we first combine consensus theory and optimization theory to develop a quaternion based attitude consensus protocol based on the communication graph at each time step each spacecraft generates a guidance command or reference attitude trajectory by synthesizing a series of laplacian like matrix p t using semidefinite programming sdp which involves linear matrix inequalities lmis it is analytically shown that this series of matrices p t is capable of collectively driving the initial attitudes to a common consensus attitude for satisfying cone avoidance constraints exclusion zones are then identified and expressed as lmis this identification of the exclusion zones gives rise to selecting safe waypoints from the reference attitude trajectory and then to passing through the selected waypoints while avoiding the exclusion zones via proper control inputs this solution procedure is demonstrated via numerical simulations of coordinated attitude rendezvous and attitude formation acquisition of multiple spacecraft with cone avoidance manoeuvres

a non linear spacecraft attitude tracking controller for large non constant rate commands a new non linear tracking control algorithm based on an attitude error quaternion is studied in this paper the control law developed here uses the commanded attitude rate without transformation into the body frame the direct use of the commanded attitude rate simplifies the calculation of its derivative which is used in the control law the solutions and the equilibrium points of the closed loop system which is a time varying non linear system are obtained in different scenarios in order to analyse the stability of the system and the tracking performance two different forms of perturbation dynamics with seven state variables are introduced local stability and performance analysis shows that the eigenvalues of the linearized perturbation dynamics are determined only by the gain matrices in the control algorithm and the inertia matrix the existence of globally stable tracking control is proved using a lyapunov function simulation results show that the spacecraft can track the commanded attitude an

on path lifting mechanisms and unwinding in quaternion based attitude control the unit quaternion is a pervasive representation of rigid body attitude used for the design and analysis of feedback control laws because the space of unit quaternions constitutes a double cover of the rigid body attitude space quaternion based control laws are often by design inconsistent i e they do not have a unique value for each rigid body attitude inconsistent quaternion based control laws require an additional mechanism that uniquely converts an attitude estimate into its quaternion representation however conversion mechanisms that are memoryless e g selecting the quaternion having positive scalar component have a limited domain where they remain injective and when used globally introduce discontinuities into the closed loop system we show through an explicit construction and lyapunov analysis that such discontinuities can be hijacked by arbitrarily small measurement disturbances to stabilize attitudes far from the desired attitude to remedy this limitation we propose a hybrid dynamic algorithm for smoothly lifting an attitude path to the unit quaternion space we show that this hybrid dynamic mechanism allows us to directly translate quaternion based controllers and their asymptotic stability properties obtained in the unit quaternion space to the actual rigid body attitude space we also show that when quaternion based controllers are not designed to account for the double covering of the rigid body attitude space by a unit quaternion parameterization they can give rise to the unwinding phenomenon which we characterize in terms of the projection of asymptotically stable sets finally we employ the main results to show that certain hybrid feedbacks can globally asymptotically stabilize the attitude of a rigid body

numerical experience with lower bounds for miqp branch and bound the solution of convex mixed integer quadratic programming miqp problems with a general branch and bound framework is considered it is shown how lower bounds can be computed efficiently during the branch and bound process improved lower bounds such as the ones derived in this paper can reduce the number of quadratic programming qp problems that have to be solved the branch and bound approach is also shown to be superior to other approaches in solving miqp problems numerical experience is presented which supports these conclusions

a topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon we show that a continuous dynamical system on a state space that has the structure of a vector bundle on a compact manifold possesses no globally asymptotically stable equilibrium this result is directly applicable to mechanical systems having rotational degrees of freedom in particular the result applies to the attitude motion of a rigid body in light of this result we explain how attitude stabilizing controllers obtained using local coordinates lead to unwinding instead of global asymptotic stability

bounds for rankin selberg integrals and quantum unique ergodicity for powerful levels reference epfl article 202508doi 10 1090 s0894 0347 2013 00779 1view record in web of science record created on 2014 10 23 modified on 2017 12 10

tracking for fully actuated mechanical systems a geometric framework we present a general framework for the control of lagrangian systems with as many inputs as degrees of freedom relying on the geometry of mechanical systems on manifolds we propose a design algorithm for the tracking problem the notions of error function and transport map lead to a proper definition of configuration and velocity error these are the crucial ingredients in designing a proportional derivative feedback and feedforward controller the proposed approach includes as special cases a variety of results on control of manipulators pointing devices and autonomous vehicles our design provides particular insight into both aerospace and underwater applications where the configuration manifold is a lie group

local global principles for representations of quadratic forms we prove a local global principle for the problem of representations of quadratic forms by quadratic forms over in codimension 5 the proof uses the ergodic theory of p adic groups together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety

stabilization by smooth feedback of the angular velocity of a rigid body abstract stabilization by smooth feedback of the null solution of euler s angular velocity equations is investigated assume there is one external torque aligned with a principal axis the main result is that there exists a stabilizing smooth feedback law if the moment of inertia of the rigid body along that principal axis is either larger or smaller than the remaining two the feedback law is robust with respect to changes in the parameters defining the control law

on measures invariant under tori on quotients of semisimple groups we classify invariant and ergodic probability measures on arithmetic homogeneous quotients of semisimple s algebraic groups invariant under a maximal split torus in at least one simple local factor and show that the algebraic support of such a measure splits into the product of four homogeneous spaces a torus a homogeneous space on which the measure is up to nite index the haar measure a product of homogeneous spaces on each of which the action degenerates to a rank one action and a homogeneous space in which every element of the action acts with zero entropy

heuristic algorithms for the cardinality constrained efficient frontier abstract this paper examines the application of genetic algorithm tabu search and simulated annealing metaheuristic approaches to finding the cardinality constrained efficient frontier that arises in financial portfolio optimisation we consider the mean variance model of markowitz as extended to include the discrete restrictions of buy in thresholds and cardinality constraints computational results are reported for publicly available data sets drawn from seven major market indices involving up to 1318 assets our results are compared with previous results given in the literature illustrating the effectiveness of the proposed metaheuristics in terms of solution quality and computation time

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

an algebraic approach to integer portfolio problems abstract integer variables allow the treatment of some portfolio optimization problems in a more realistic way and introduce the possibility of adding some natural features to the model we propose an algebraic approach to maximize the expected return under a given admissible level of risk measured by the covariance matrix to reach an optimal portfolio it is an essential ingredient the computation of different test sets via grobner basis of linear subproblems that are used in a dual search strategy

robust attitude stabilization of spacecraft using nonlinear quaternion feedback this paper considers the problem of three axis attitude stabilization of a rigid spacecraft a nonlinear control law which uses the feedback of the unit quaternion and the measured angular velocities is proposed and is shown to provide global asymptotic stability the control law does not require the knowledge of the system parameters and is therefore robust to modeling errors the significance of the control law is that it can be used for large angle maneuvers with guaranteed stability

ensemble control of linear systems with parameter uncertainties abstractin this paper we study the optimal control problem for a class of four dimensional linear systems based on quaternionic and fourier analysis when the control is unconstrained the optimal ensemble controller for this linear ensemble control systems is given in terms of prolate spheroidal wave functions for the constrained convex optimisation problem of such systems the quadratic programming is presented to obtain the optimal control laws simulations are given to verity the effectiveness of the proposed theory

quaternion based adaptive output feedback attitude control of spacecraft using chebyshev neural networks this paper investigates the problem of output feedback attitude control of an uncertain spacecraft two robust adaptive output feedback controllers based on chebyshev neural networks cnn termed adaptive neural networks nn controller i and adaptive nn controller ii are proposed for the attitude tracking control of spacecraft the four parameter representations quaternion are employed to describe the spacecraft attitude for global representation without singularities the nonlinear reduced order observer is used to estimate the derivative of the spacecraft output and the cnn is introduced to further improve the control performance through approximating the spacecraft attitude motion the implementation of the basis functions of the cnn used in the proposed controllers depends only on the desired signals and the smooth robust compensator using the hyperbolic tangent function is employed to counteract the cnn approximation errors and external disturbances the adaptive nn controller ii can efficiently avoid the over estimation problem i e the bound of the cnns output is much larger than that of the approximated unknown function and hence the control input may be very large existing in the adaptive nn controller i both adaptive output feedback controllers using cnn can guarantee that all signals in the resulting closed loop system are uniformly ultimately bounded for performance comparisons the standard adaptive controller using the linear parameterization of spacecraft attitude motion is also developed simulation studies are presented to show the advantages of the proposed cnn based output feedback approach over the standard adaptive output feedback approach

equidistribution of heegner points and ternary quadratic forms we prove new equidistribution results for galois orbits of heegner points with respect to single reduction maps at inert primes the arguments are based on two different techniques primitive representations of integers by quadratic forms and distribution relations for heegner points our results generalize an equidistribution result with respect to a single reduction map established by cornut and vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow moreover for fixed fundamental discriminant and variable conductor we deduce an effective surjectivity theorem for the reduction map from heegner points to supersingular points at a fixed inert prime our results are applicable to the setting considered by kolyvagin in the construction of the heegner points euler system

heuristic algorithms for the portfolio selection problem with minimum transaction lots the problem of selecting a portfolio has been largely faced in terms of minimizing the risk given the return while the complexity of the quadratic programming model due to markowitz has been overcome by the recent progress in algorithmic research the introduction of linear risk functions has given rise to the interest in solving portfolio selection problems with real constraints in this paper we deal with the portfolio problem with minimum transaction lots we show that in this case the problem of finding a feasible solution is independently of the risk function np complete moreover given the mixed integer linear model new heuristics are proposed which starting from the solution of the relaxed problem allow to find a solution close to the optimal one the algorithms are based on the construction of mixed integer subproblems using only a part of the securities available formulated using the information obtained from the solution of the relaxed problem the heuristics have been tested with respect to two disjoint time periods using real data from the milan stock exchange

mean absolute deviation portfolio optimization model and its applications to tokyo stock market the purpose of this paper is to demonstrate that a portfolio optimization model using the l1 risk mean absolute deviation risk function can remove most of the difficulties associated with the classical markowitz s model while maintaining its advantages over equilibrium models in particular the l1 risk model leads to a linear program instead of a quadratic program so that a large scale optimization problem consisting of more than 1 000 stocks may be solved on a real time basis numerical experiments using the historical data of nikkei 225 stocks show that the l1 risk model generates a portfolio quite similar to that of the markowitz s model within a fraction of time required to solve the latter

invariant measures and the set of exceptions to littlewood s conjecture we classify the measures on sl k r sl k z which are invariant and ergodic under the action of the group a of positive diagonal matrices with positive entropy we apply this to prove that the set of exceptions to littlewood s conjecture has hausdorff dimension zero

uniform subconvexity and symmetry breaking reciprocity abstract a non symmetric reciprocity formula is established that expresses the fourth moment of automorphic l functions of level q and primitive central character twisted by the l th hecke eigenvalue as a twisted mixed moment of automorphic l functions of level l and trivial central character as an application uniform subconvexity bounds for l functions in the level and the eigenvalue aspect are derived

subconvex equidistribution of cusp forms reduction to eisenstein observables let traverse a sequence of cuspidal automorphic representations of gl2 with large prime level unramified central character and bounded infinity type for g gl1 pgl2 let h g denote the assertion that subconvexity holds for g twists of the adjoint l function of with polynomial dependence upon the conductor of the twist we show that h gl1 implies h pgl2 in geometric terms h pgl2 corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term h gl1 to the special case in which the relevant sequence of measures is tested against an eisenstein series

unit quaternion based output feedback for the attitude tracking problem in this note we propose a quaternion based dynamic output feedback for the attitude tracking problem of a rigid body without velocity measurement our approach consists of introducing an auxiliary dynamical system whose output which is also a unit quaternion is used in the control law together with the unit quaternion representing the attitude tracking error roughly speaking the necessary damping that would have been achieved by the direct use of the angular velocity can be achieved in our approach by the vector part mathtilde q of the error signal between the output of the auxiliary system and the unit quaternion tracking error the resulting velocity free control scheme guarantees almost global asymptotic stability which is as strong as the topology of the motion space can permit in the regulation case our control law is a pure quaternion feedback i e consisting of two terms that are vector parts of unit quaternion and hence the control torques are naturally bounded by the control gains simulation results are provided to show the effectiveness of the proposed control scheme

uniform distribution of heegner points let e be a modular elliptic curve over q of conductor n let k denote an imaginary quadratic field of discriminant d with n d 1 if p is a prime then there exists a unique zp extension k k such that gal k q acts nontrivially on gal k k the field k is called the anticyclotomic zp extension of k let e k denote the mordell weil group of e over k then a fundamental conjecture of mazur maz predicts that the size of e k is controlled by the prime factorization of n in k equivalently mazur s conjecture relates the size of the mordell weil group to the sign in the functional equation of certain l series the conjecture was verified by greenberg rohrlich and rubin in what mazur calls the exceptional case when e has complex multiplication by k more generally they settled the conjecture for certain abelian varieties with complex multiplication for a discussion of this cm case we refer the reader to gre roh and rub our goal in this paper is to treat the generic case which occurs either when e has no cm or when the field of complexmultiplications is distinct from k under certain conditions on e and k mazur s conjecture predicts that the group e k is finitely generated our main result asserts that this is in fact the case at least when p is an ordinary prime for e or when the class number of k is prime to p the main new ingredient we introduce is that of equidistribution following ideas used by ferrero andwashington to study the cyclotomic invariant more precisely we show that theheegner points associated to definite quaternion algebras are uniformly distributed on the components of a certain curve x and that the elements of a certain galois group act independently in a suitable sense this combined with a special value formula due to gross allows us to conclude that the special values of anticyclotomic l functions are almost always nonzero so that the statement about the mordell weil groups follows from the machinery of euler systems as developed by bertolini and darmon bd

computational study of a family of mixed integer quadratic programming problems we present computational experience with a branch and cut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables such problems arise in financial applications the algorithm solves the largest real life problems in a few minutes of run time

global set stabilisation of the spacecraft attitude using finite time control technique in this article we revisit the classical problem of attitude stabilisation for a rigid spacecraft with external disturbances a global set stabilisation method using finite time control technique is proposed in the absence of disturbances the states of the closed loop system will be stabilised in finite time to a set consisting of two equilibria in the presence of disturbances the states will be stabilised to a neighbourhood of this set by constructing a particular lyapunov function it is proved that the closed loop system satisfies global set stability the control method in this article is based on set control idea and thus is more natural and energy efficient numerical simulation results show the effectiveness of the method

exact nonlinear control of large angle rotational maneuvers the rigid body attitude control problem with external torques is transformed into an equivalent linear form implementable by three double integrators the linearizing transformations themselves are formulated in vector algebra requiring no integrators for implementation it is thereby shown that optimal command generation for fast slewing maneuvers can be carried out exactly in the transformed system together with regulator design without gain scheduling for correction of unmodeled disturbances in particular a general rest to rest maneuver is computed in closed form and coupled with a previously obtained exact detumbling maneuver so that arbitrary initial conditions can be accommodated an illustrative simulation is appended

a note on the least prime that splits completely in a nonabelian galois number field we prove a nontrivial estimate for the size of the least rational prime that splits completely in the ring of integers of certain families of nonabelian galois number fields our result complements results of linnik and vinogradov and of pollack who studied this problem in the quadratic and abelian number field settings respectively

branch and bound experiments in convex nonlinear integer programming the branch and bound principle has long been established as an effective computational tool for solving mixed integer linear programming problems this paper investigates the computational feasibility of branch and bound methods in solving convex nonlinear integer programming problems the efficiency of a branch and bound method often depends on the rules used for selecting the branching variables and branching nodes among others the concepts of pseudo costs and estimations are implemented in selecting these parameters since the efficiency of the algorithm also depends on how fast an upper bound on the objective minimum is attained heuristic rules are developed to locate an integer feasible solution to provide an upper bound the different criteria for selecting branching variables branching nodes and heuristics form a total of 27 branch and bound strategies these strategies are computationally tested and the empirical results are presented

weak subconvexity for central values of l functions we describe a general method to obtain weak subconvexity bounds for many classes of l functions we give several examples of our bound and our work has applications to a conjecture of rudnick and sarnak for the mass equidistribution of hecke eigenforms

global finite time attitude tracking via quaternion feedback abstract this paper addresses the attitude tracking of a rigid body using a quaternion description global finite time attitude controllers are designed with three types of measurements namely full states attitude plus constant biased angular velocity and attitude only in all three scenarios hybrid control techniques are utilized to overcome the well known topological constraint on the attitude manifold while coupled nonsmooth feedback inputs are designed via homogeneous theory to achieve finite time stability specially a finite time bias observer is derived in the second scenario and a quaternion filter is constructed to provide damping in the absence of velocity feedback the proposed methods ensure bounded control torques a priori and in particular include several existing attitude controllers as special cases

state feedback spl hscr sub spl infin suboptimal control of a rigid spacecraft a state feedback spl hscr sub spl infin suboptimal control problem on tso 3 for a rigid spacecraft with three control torques and disturbances is addressed the hamilton jacobi inequality associated with a corresponding state feedback spl hscr sub spl infin suboptimal control problem on ts sup 3 is solved globally the spl hscr sub spl infin suboptimal state feedback on ts sup 3 involves the euler parameters and solves the original problem on tso 3 when the initial values of the euler parameters are appropriately chosen

the attitude control problem a general framework for the analysis of the attitude tracking control problem for a rigid body is presented a large family of globally stable control laws is obtained by using the globally nonsingular unit quaternion representation in a lyapunov function candidate whose form is motivated by the consideration of the total energy of the rigid body the controllers share the common structure of a proportional derivative feedback plus some feedforward which can be zero the model independent case the coriolis torque compensation or an adaptive compensation these controller structures are compared in terms of the requirement on the a priori model information guaranteed transient performance and robustness the global stability of the luh walker paul robot end effector controller is also analyzed in this framework

sparse equidistribution problems period bounds and subconvexity we introduce a geometric method to bound periods of automorphic forms the key features of this method are the use of equidistribution results in place of mean value theorems and the systematic use of mixing and the spectral gap applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of cm points to subconvexity of the triple product period in the level aspect over number fields which implies subconvexity for certain standard and rankin selberg l functions and to bounding fourier coefficients of automorphic forms

a comparison between hmlp and hrbf for attitude control in this paper the problem of controlling the attitude of a rigid body such as a spacecraft in three dimensional space is approached by introducing two new control strategies developed in hypercomplex algebra the proposed approaches are based on two parallel controllers both derived in quaternion algebra the first is a feedback controller of the proportional derivative pd type while the second is a feedforward controller which is implemented either by means of a hypercomplex multilayer perceptron hmlp neural network or by means of a hypercomplex radial basis function hrbf neural network several simulations show the performance of the two approaches the results are also compared with a classical pd controller and with an adaptive controller showing the improvements obtained by using neural networks especially when an external disturbance acts on the rigid body in particular the hmlp network gave better results when considering trajectories not presented during the learning phase

the distribution of periodic torus orbits on homogeneous spaces we prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces with particular focus on the case of the diagonal torus acting on quotients of pgln r after attaching to each periodic orbit an integral invariant the discriminant our results have the following flavour certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most o orbits of discriminant the proof relies on the well separatedness of periodic orbits together with measure rigidity for torus actions we also give examples of sequences of periodic orbits of this action that fail to become equidistributed even in higher rank we give an application of our results to sharpen a theorem of minkowski on ideal classes in totally real number fields of cubic and higher degrees

new class of control laws for robotic manipulators part 1 non adaptive case a new class of exponentially stabilizing control laws for joint level control of robot arms is introduced it has recently been recognized that the non linear dynamics associated with robotic manip