short communication a note on quasi copulas and signed measures in this note we provide two alternative proofs to that given in nelsen et al quasi copulas and signed measures fuzzy sets syst 161 2010 2328 2336 of the fact that the best possible lower bound for the set of n quasi copulas does not induce a stochastic measure on 0 1 n for na c 3 43 firstly by focusing on its mass distribution and secondly by using concepts of self affinity

transitivity bounds in additive fuzzy preference structures transitivity plays a crucial role in preference modeling and related fields in this paper we discuss this property in the general context of additive fuzzy preference structures of particular interest is the decomposition of a large preference relation r in its symmetric part i indifference relation and its asymmetric part p strict preference relation by means of a so called indifference generator i given the type of transitivity of a large preference relation r w r t a conjunctor and a generator we establish basic lower bounds and general upper bounds on the transitivity of p and i these bounds are due to the careful design of generic counterexamples moreover we identify the situations in which these bounds are effectively reached thereby establishing connections with interesting properties such as dominance bisymmetry the 1 lipschitz property and rotation invariance

meta theorems on inequalities for scalar fuzzy set cardinalities we present meta theorems stating general conditions ensuring that certain inequalities for cardinalities of ordinary sets are preserved under fuzzification when adopting a scalar approach to fuzzy set cardinality the conditions pertain to the commutative conjunctor used for modelling fuzzy set intersection in particular this conjunctor should fulfil a number of bell type inequalities the advantage of these meta theorems is that repetitious calculations can be avoided this is illustrated in the demonstration of the lukasiewicz transitivity of fuzzified versions of the simple matching coefficient and the jaccard coefficient or equivalently the triangle inequality of the corresponding dissimilarity measures

some parseval goldstein type identities involving the fs 2 transform the fc 2 transform and the p4 transform and their applications abstract in the present paper the authors consider several new integral transforms including the f s 2 transform the f c 2 transform and the p 4 transform as generalizations of the classical fourier sine transform the classical fourier cosine transform and the classical stieltjes transform respectively many identities involving these transforms are given by making use of these identities a number of new parseval goldstein type identities are obtained for these and many other well known integral transforms the identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions some examples are also given as illustrations of the results presented here

exact solutions to a class of feedback systems on so n this paper provides a novel approach to the problem of attitude tracking for a class of almost globally asymptotically stable feedback laws on so n the closed loop systems are solved exactly for the rotation matrices as explicit functions of time the initial conditions and the gain parameters of the control laws the exact solutions provide insight into the transient dynamics of the system and can be used to prove almost global attractiveness of the identity matrix applications of these results are found in model predictive control problems where detailed insight into the transient attitude dynamics is utilized to approximately complete a task of secondary importance knowledge of the future trajectory of the states can also be used as an alternative to the zero order hold in systems where the attitude is sampled at discrete time instances

new results on discrete copulas and quasi copulas abstract in this paper we prove that the set of all irreducible discrete quasi copulas coincides with the set of all discrete quasi copulas with minimal range we also provide answers throughout some counterexamples to two questions posed in aguilo et al 2 regarding both discrete quasi copulas with minimal range and irreducible discrete quasi copulas some additional results concerning discrete quasi copulas are also given

some new characterizations and properties of quasi copulas in this paper we characterize the concept of quasi copula in three different ways as a special subclass of aggregation operators in terms of its associated volume and for the bivariate case in terms of non increasing tracks on 0 1 2 we also provide sufficient conditions and new properties for quasi copulas

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

metrics and t equalities abstract the relationship between metrics and t equalities is investigated the latter are a special case of t equivalences a natural generalization of the classical concept of an equivalence relation it is shown that in the construction of metrics from t equalities triangular norms with an additive generator play a key role conversely in the construction of t equalities from metrics this role is played by triangular norms with a continuous additive generator or equivalently by continuous archimedean triangular norms these results are then applied to the biresidual operator e t of a triangular norm t it is shown that e t is a t equality on 0 1 if and only if t is left continuous furthermore it is shown that to any left continuous triangular norm t there correspond two particular t equalities on f x the class of fuzzy sets in a given universe x one of these t equalities is obtained from the biresidual operator e t t by means of a natural extension procedure these t equalities then give rise to interesting metrics on f x

zero sets of copulas abstract we study conditions on sets in order to be zero sets of semi copulas quasi copulas and with special attention of copulas we find necessary and sufficient conditions for characterizing the zero sets of absolutely continuous copulas and copulas whose support coincides with the closure of the complementary of the zero set moreover we study several topological properties and the lattice theoretic structure and characterize the zero sets of the class of archimedean copulas

cayley kinematics and the cayley form of dynamic equations the cayley transform and the cayleytransform kinematic relationships are an important and fascinating set of results that have relevance in ndimensional orientations and rotations in this paper th

invariant higher order variational problems we investigate higher order geometric k splines for template matching on lie groups this is motivated by the need to apply diffeomorphic template matching to a series of images e g in longitudinal studies of computational anatomy our approach formulates euler poincar theory in higher order tangent spaces on lie groups in particular we develop the euler poincar formalism for higher order variational problems that are invariant under lie group transformations the theory is then applied to higher order template matching and the corresponding curves on the lie group of transformations are shown to satisfy higher order euler poincar equations the example of so 3 for template matching on the sphere is presented explicitly various cotangent bundle momentum maps emerge naturally that help organize the formulas we also present hamiltonian and hamilton ostrogradsky lie poisson formulations of the higher order euler poincar theory for applications on the hamiltonian side

best possible bounds on the set of copulas with a given value of spearman s footrule abstract in this paper we find pointwise best possible bounds on the set of copulas with a given value of the spearman s footrule coefficient we show that the lower bound is always a copula but unlike the bounds on sets of copulas with a given value of other measures such as kendall s tau spearman s rho and blonqvist s beta the upper bound can be a copula or a proper quasi copula we characterised both of these cases

some results on a transformation of copulas and quasi copulas this paper provides some results including invariance properties dependence measures convexity properties and tail dependence on a transformation of copulas and quasi copulas

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

homography based visual servo control with imperfect camera calibration in this technical note a robust adaptive uncalibrated visual servo controller is proposed to asymptotically regulate a robot end effector to a desired pose a homography based visual servo control approach is used to address the six degrees of freedom regulation problem a high gain robust controller is developed to asymptotically stabilize the rotation error and an adaptive controller is developed to stabilize the translation error while compensating for the unknown depth information and intrinsic camera calibration parameters a lyapunov based analysis is used to examine the stability of the developed controller

identities for the e2 1 transform and their applications in the present paper the authors introduce the e 2 1 transform with kernel the exponential integral function it is shown that the third iterate of the l 2 transform is the exponential integral transform and some identities involving the new transform the l 2 transform and the widder potential transform are given using the identities parseval goldstein type results involving these transforms are proved some illustrative examples are also given

equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data in this paper we discuss the necessary and sufficient conditions for the equivalence of the dynamics of nonholonomic mechanics and variational nonholonomic vakonomic dynamics for certain initial conditions we derive a priori results for identifying equivalence and specializing to abelian chaplygin systems prove that equivalence results if and only if the constrained nonholonomic equations are lagrangian we eliminate the need to solve the variational nonholonomic problem when checking equivalence by obtaining explicit formulae for the system s multipliers and then derive conditions under which the multiplier free lagrangian gives equivalence of the dynamics we consider nonholonomic systems possessing invariant measures showing when equivalence and hamiltonization are the same we also derive conditions under which measure preserving systems exhibit equivalence we apply the results to many of the known nonholonomic systems

exponential stability of an attitude tracking control system on so 3 for large angle rotational maneuvers abstract this paper studies a tracking control system for the attitude dynamics of a rigid body by selecting an attitude error function carefully we show that the proposed control system guarantees a desirable tracking performance uniformly for rotational maneuvers involving a large initial attitude error a strict lyapunov analysis is presented to show exponential stability and a sufficient condition to avoid non differentiable points of the attitude error function is also shown the proposed control system is directly developed on the special orthogonal group to avoid complexities and ambiguities associated with other attitude representations such as euler angles or quaternions these are illustrated by numerical examples

bivariate quasi copulas and doubly stochastic signed measures we show that there exist bivariate proper quasi copulas that do not induce a doubly stochastic signed measure on 0 1 2 we construct these quasi copulas from the so called proper quasi transformation square matrices

a hitchhiker s guide to quasi copulas abstract this paper presents a review of the concept of an n quasi copula starting from the characterization of functions that can be derived from operations on random variables we recall several other characterizations and properties that have been proven in the literature highlighting the differences between the bivariate case and the higher dimensional case additionally we recall several applications of n quasi copulas such as their role in the study of bounds on sets of n copulas we also recall various classes of n quasi copulas that have been studied in the literature finally we present some questions that have not yet been answered in the literature

almost global tracking of simple mechanical systems on a general class of lie groups we present a general intrinsic tracking controller design for fully actuated simple mechanical systems when the configuration space is one of a general class of lie groups we first express a state feedback controller in terms of a function the error function satisfying certain regularity conditions if an error function can be found then a general smooth and bounded reference trajectory may be tracked asymptotically from almost every initial condition with locally exponential convergence asymptotic convergence from almost every initial condition is referred to as almost global asymptotic stability error functions may be shown to exist on any compact lie group or any lie group diffeomorphic to the product of a compact lie group and r sup n this covers many cases of practical interest such as so n se n their subgroups and direct products we show here that for compact lie groups the dynamic configuration feedback controller obtained by composing the full state feedback law with an exponentially convergent velocity observer is also almost globally asymptotically stable with respect to the tracking error we emphasize that no invariance is needed for these results however for the special case where the kinetic energy is left invariant we show that the explicit expression of these controllers does not require coordinates on the lie group the controller constructions are demonstrated on so 3 and simulated for the axi symmetric top results show excellent performance

eigenfactor solution of the matrix riccati equation a continuous square root algorithm this paper introduces a new algorithm for solving the matrix riccati equation differential equations for the eigenvalues and eigenvectors of the solution matrix are developed in which their derivatives are expressed in terms of the eigenvalues and eigenvectors themselves and not as functions of the solution matrix the solution of these equations yields then the time behavior of the eigenvalues and eigenvectors of the solution matrix a reconstruction of the matrix itself at any desired time is immediately obtained through a trivial similarity transformation this algorithm serves two purposes first being a square root solution it entails all the advantages of square root algorithms such as nonnegative definiteness and accuracy secondly it furnishes the eigenvalues and eigenvectors of the solution matrix continuously without resorting to the complicated route of solving the equation directly and then decomposing the solution matrix into its eigenvalues and eigenvectors the algorithm which handles cases of distinct as well as multiple eigenvalues is tested on several examples through these examples it is seen that the algorithm is indeed more accurate than the ordinary one moreover it is seen that the algorithm works in cases where the ordinary algorithm fails and even in cases where the closed form solution cannot be computed as a result of numerical difficulties

bell type inequalities for quasi copulas we present all bell type inequalities regarding at most four random events of which not more than two are intersected at the same time reformulating these inequalities in the context of fuzzy probability calculus leads to related inequalities on commutative conjunctors we prove that some of these inequalities hold for quasi copulas or copulas additionally for the important parametric family of frank t norms which are also copulas we identify for each of the inequalities the parameter values for which it is fulfilled thereby completing the results of pykacz and d hooghe

a note on an idempotent transformation of absolutely continuous archimedean copulas abstract in de baets et al 6 the authors define for a copula c an idempotent transformation d c which in some cases generates new copulas in this note we give a negative answer to a question posed in 6 showing that d c is not necessarily a copula when c is an absolutely continuous archimedean copula

identities and parseval type relations for the l2 transform abstract in the present paper the authors introduce identities for the l 2 transform which yield a parseval type relationship for the l 2 transform the parseval type relationship proven in this paper give rise to useful corollaries for evaluating indefinite integrals of special functions some examples are also given as illustrations of the results presented here

double bracket equations and geodesic flows on symmetric spaces in this paper we consider the geometry of hamiltonian flows on the cotangent bundle of coadjoint orbits of compact lie groups and on symmetric spaces a key idea here is the use of the normal metric to define the kinetic energy this leads to hamiltonian flows of the double bracket type we analyze the integrability of geodesic flows according to the method of thimm we obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity we demonstrate for example integrability of the geodesic flow on the real and complex grassmannians we also consider right invariant systems and the generalized rigid body equations in this setting

the moser veselov equation abstract we study the orthogonal solutions of the matrix equation xj jx t m where j is symmetric positive definite and m is skew symmetric this equation arises in the discrete version of the dynamics of a rigid body investigated by moser and veselov commun math phys 139 1991 217 we show connections between orthogonal solutions of this equation and solutions of a certain algebraic riccati equation this will bring out the symplectic geometry of the moser veselov equation and also reduces most computational issues about solutions to finding invariant subspaces of a certain hamiltonian matrix necessary and sufficient conditions for the existence of orthogonal solutions and methods to compute them are presented our method is contrasted with the moser veselov approach commun math phys 139 1991 217 we also exhibit explicit solutions of a particular case of the moser veselov equation which appears associated with the continuous version of the dynamics of a rigid body

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

lagrangian mechanics and variational integrators on two spheres euler lagrange equations and variational integrators are developed for lagrangian mechanical systems evolving on a product of two spheres the geometric structure of a product of two spheres is carefully considered in order to obtain global equations of motion both continuous equations of motion and variational integrators completely avoid the singularities and complexities introduced by local parameterizations or explicit constraints we derive global expressions for the euler lagrange equations on two spheres which are more compact than existing equations written in terms of angles since the variational integrators are derived from hamilton s principle they preserve the geometric features of the dynamics such as symplecticity momentum maps or total energy as well as the structure of the configuration manifold computational properties of the variational integrators are illustrated for several mechanical systems in addition lie group variational integrators can be used to integrate lagrangian flows on more general homogeneous spaces this is achieved by lifting the discrete hamilton s principle on homogeneous spaces to a discrete variational principle on the lie group that is constrained by a discrete connection copyright 2009 john wiley sons ltd

asymptotic smooth stabilization of the inverted 3 d pendulum the 3 d pendulum consists of a rigid body supported at a fixed pivot with three rotational degrees of freedom it is acted on by gravity and it is fully actuated by control forces the 3 d pendulum has two disjoint equilibrium manifolds namely a hanging equilibrium manifold and an inverted equilibrium manifold the contribution of this paper is that two fundamental stabilization problems for the inverted 3 d pendulum are posed and solved the first problem asymptotic stabilization of a specified equilibrium in the inverted equilibrium manifold is solved using smooth and globally defined feedback of angular velocity and attitude of the 3 d pendulum the second problem asymptotic stabilization of the inverted equilibrium manifold is solved using smooth and globally defined feedback of angular velocity and a reduced attitude vector of the 3 d pendulum these control problems for the 3 d pendulum exemplify attitude stabilization problems on the configuration manifold so 3 in the presence of potential forces lyapunov analysis and nonlinear geometric methods are used to assess global closed loop properties yielding a characterization of the almost global domain of attraction for each case

a new method of solving hermite s differential equation using the l2 transform in the present paper a new method of solving hermite s differential equation is given using the l 2 transform

control of a driftless bilinear vector field on n sphere in this paper we consider a multi input driftless bilinear system evolving on the n dimensional sphere s n we first provide examples drawn from rigid body mechanics that provide the motivation for the control of bilinear systems on s n for the general framework we establish the global controllability on s n and propose two linear control laws on s n that achieve asymptotic stabilization of an equilibrium point with an almost global domain of attraction further the asymptotically stable closed loop system trajectories are shown to be arcs on the geodesics of s n for a particular choice of the equilibrium point next we propose two linear time varying control laws to achieve trajectory tracking on s n and show the asymptotic stability of the tracking error a distributed control is designed for the consensus of multiagent bilinear systems on s n with an undirected tree as the communication graph the consensus manifold is shown to have an almost global domain of attraction

quasi copulas and signed measures we study the relationship between multivariate quasi copulas and measures that they may or may not induce on 0 1 n we first study the mass distribution of the pointwise best possible lower bound for the set of n quasi copulas for n 3 as a consequence we show that not every n quasi copula induces a signed measure on 0 1 n

stabilization of nonholonomic systems using isospectral flows in this paper we derive and analyze a discontinuous stabilizing feedback for a lie algebraic generalization of a class of kinematic nonholonomic systems introduced by brockett new directions in applied mathematics p hilton and g young eds springer verlag new york 1982 pp 11 27 the algorithm involves discrete switching between isospectral and norm decreasing flows we include a rigorous analysis of the convergence

on the existence of a trivariate copula with given values of a trivariate quasi copula at several points the existence of a multivariate copula with a given value of a multivariate quasi copula at a single point is known in the bivariate case the existence of a copula with given values of a quasi copula at two or three arbitrary points is also known in this paper we give an alternative proof of the existence of a trivariate copula with a given value of a trivariate quasi copula at a single point this proof relies on the reformulation of the existence problem as a linear programming minimization problem and its solution by means of the simplex algorithm the same method is then used to prove the existence of a trivariate copula with given values of a trivariate quasi copula at two arbitrary points in the unit cube we furthermore establish a counter example showing that the existence for given values at three points is not guaranteed this completes the analysis of the trivariate case

orientation control on so 3 with piecewise sinusoids this paper addresses rigid body orientation control for the case that angular velocity is restricted to a class of piecewise sinusoidal signals the orientation kinematics are represented by the group of rotation matrices so 3 we present a new closed form solution of the rotation matrix kinematics and we use this solution to prove controllability we then present kinematic level orientation feedback controllers for setpoint tracking and command following

hamel coefficients for the rotational motion of an n dimensional rigid body many of the kinematic and dynamic concepts relating to rotational motion have been generalized for ndimensional rigid bodies in this paper a new derivation of the generalized euler equations is pr

optimal control and geodesic flows in this paper we analyze and generalize from the point of view of the maximum principle a class of nonlinear optimal control problems originally introduced in brockett 1994 the optimal control problems discussed here lie on the adjoint orbits of compact lie groups or on grassmann manifolds and the structure of the equations that arise depends on the metric used the so called normal metric we consider some special cases of the general problem and their meaning from the point of view of riemannian geometry and hamiltonian mechanics

solving quantum optimal control problems using clebsch variables and lin constraints clebsch variables and lin constraints are applied to the study of a class of optimal control problems for affine controlled quantum systems the optimal control problem will be modelled with controls defined on an auxiliary space where the dynamical group of the system acts freely the reciprocity between both theories the classical theory defined by the objective functional and the quantum system is established by using a suitable version of lagrange s multipliers theorem and a geometrical interpretation of the constraints of the system as defining a subspace of horizontal curves in an associated bundle it is shown how the solutions of the variational problem defined by the objective functional determine solutions of the quantum problem then a new way of obtaining explicit solutions for a family of optimal control problems for affine controlled quantum systems finite or infinite dimensional is obtained one of its main advantages is the the use of clebsch variables allows to compute such solutions from solutions of invariant problems that can often be computed explicitly this procedure can be presented as an algorithm that can be applied to a large class of systems finally some simple examples spin control a simple quantum hamiltonian with an elroy beanie type classical model and a controlled one dimensional quantum harmonic oscillator illustrating the main features of the theory will be discussed

minimal parameter solution of the orthogonal matrix differential equation the straightforward solution of the first order differential equation satisfied by all nth order orthogonal matrices requires n sup 2 integrations to obtain the matrix elements there are however only n n 1 2 independent parameters which determine an orthogonal matrix the questions of choosing them finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered in the present work several possibilities which are based on attitude determination in three dimensions are examined it is concluded that not all 3 d methods have useful extensions to other dimensions and that the 3 d gibbs vector or cayley parameters provide the most useful extension an algorithm is developed using the resulting parameters which are termed extended rodrigues parameters and numerical results are presented of the application of the algorithm to a fourth order matrix

completely integrable gradient flows in this paper we exhibit the toda lattice equations in a double bracket form which shows they are gradient flow equations on their isospectral set on an adjoint orbit of a compact lie group representations for the flows are given and a convexity result associated with a momentum map is proved some general properties of the double bracket equations are demonstrated including a discussion of their invariant subspaces and their function as a lie algebraic sorter