subdifferentials faces and dual matrices abstract a characterization of the dual matrices for the unitarily invariant norms is given moreover the connection between the dual matrices the subdifferentials of matrix norms and the faces of the unit ball of matrices is presented

adjustments to the mantel haenszel chi square statistic and odds ratio variance estimator when the data are clustered dependence between observations on a dichotomous variable renders invalid the usual chi square tests of independence and inflates the variances of parameter estimates such a situation occurs for example when subjects consist of members of the same family or with repeated observations on the same person in this paper we describe methods of measuring the extent of correlation or clustering we also develop methods of adjusting the mantel haenszel chi square test statistic and the variance of the mantel haenszel estimate of a common odds ratio in sets of 2 2 contingency tables

comparison of the proportions of affected relatives of cases and controls analysis and minimum sample size formula the problem of familial aggregation has frequently been approached by comparing the proportion of affected relatives of index cases with the proportion of affected relatives of index controls this type of study has an analogy with the case control design and is frequently analysed as if it was one it is however an essentially different design which may involve dependent observations we show that conventional tests for comparing proportions of affected subjects are still valid with this design we give a formula for computing the minimum sample size required to detect a given degree of aggregation measured by the intracluster correlation coefficient which can be estimated very simply by the difference in proportions of affected relatives in the two groups

on minimum fuel control of affine nonlinear systems the minimum fuel control problem is investigated for a class of multiinput affine nonlinear systems whose associated lie algebra is nilpotent after preliminary notions and definitions are presented emphasis is on the solution to the optimal control problem of the lie algebra generated by certain system vector fields is nilpotent consequences of the maximum principle are deduced for such systems

stabilizing controllers for bilinear systems the problem of finding stabilizing feedback controllers for continuous bilinear systems where the controls act additively and multiplicatively simultaneously is treated the applicability of the quadratic control law of jacobson 5 and others is extended to the case when the a matrix has arbitary eigenvalues under certain conditions a class of bilinear systems common among biochemical flow systems is defined the dyadic bilinear systems a control scheme the so called division controller for dyadic bilinear systems is suggested the practicality of the control schemes is demonstrated on the problem of controlling the neutron level in a fission reactor

on the extremal structure of least upper bound norms and their dual abstract given finite dimensional real or complex banach spaces e and f with norms e r and f r we denote by n the least upper bound norm induced on l e f some results are given on the extremal structures of b the unit ball of n of its polar b and of b which is the polar of the unit ball of the least upper bound norm n the exposed faces the extreme points and a large family of other faces of b and b are presented it turns out that b is a subset of b the set of tangency points of the surfaces of b and b is completely determined and represented as the union of the exposed faces of b which are normal to rank one mappings we determine sharp bounds on the ranks of mappings in these exposed faces

on approximation problems with zero trace matrices abstract we consider some approximation problems in the linear space of complex matrices with respect to unitarily invariant norms we deal with special cases of approximation of a matrix by zero trace matrices moreover some characterizations of zero trace matrices are given by means of matrix approximation problems

noncommutative rational functions invariant under the action of a finite solvable group abstract this paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in d generators these invariant skew fields are always finitely generated which contrasts with the free algebra case for abelian groups or solvable groups g with a well behaved representation theory it is shown that the invariant skew fields are free on g d 1 1 generators finally positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented

orthogonality to matrix subspaces and a distance formula abstract we obtain a necessary and sufficient condition for a matrix a to be birkhoff james orthogonal to any subspace w of m n c using this we obtain an expression for the distance of a from any unital c subalgebra of m n c

conservation of generalized momentum maps in mechanical optimal control problems with symmetry summary in this paper a structure preserving direct method for the optimal control of mechanical systems is developed the new method accommodates a large class of one step integrators for the underlying state equations the state equations under consideration govern the motion of affine hamiltonian control systems if the optimal control problem has symmetry associated generalized momentum maps are conserved along an optimal path this is in accordance with an extension of noether s theorem to the realm of optimal control problems in the present work we focus on optimal control problems with rotational symmetries the newly proposed direct approach is capable of exactly conserving generalized momentum maps associated with rotational symmetries of the optimal control problem this is true for a variety of one step integrators used for the discretization of the state equations examples are the one step theta method a partitioned variant of the theta method and energy momentum em consistent integrators numerical investigations confirm the theoretical findings copyright 2016 john wiley sons ltd

characterization of the subdifferential of some matrix norms abstract a characterization is given of the subdifferential of matrix norms from two classes orthogonally invariant norms and operator or subordinate norms specific results are derived for some special cases

symmetry preserving observers this paper presents the theory of invariant observers i e symmetry preserving observers we consider an observer to consist of a copy of the system and a correction term and we propose a constructive method based on the cartan moving frame method to find all the symmetry preserving correction terms the construction relies on an invariant frame a classical notion and on an invariant output error a less standard notion precisely defined here using the theory we build three non linear observers for three examples of engineering interest a non holonomic car a chemical reactor and an inertial navigation system for each example the design is based on physical symmetries and the convergence analysis relies on the use of invariant state errors a symmetry preserving way to define the estimation error

reduction of sufficient conditions for optimal control problems with subgroup symmetry in this paper we apply symmetry reduction techniques from geometric mechanics to sufficient conditions for local optimality in optimal control problems after reinterpreting some previous results for left invariant problems on lie groups we focus on optimal control problems with subgroup symmetry for these problems the necessary conditions for optimality can be simplified by exploiting symmetries so as to reduce the number of variables needed to describe trajectories of the system we show that sufficient conditions for optimality based on the non existence of conjugate points can be simplified in an analogous way to the necessary conditions we demonstrate these simplifications by analyzing an optimal control problem that models a spinning top in a gravitational field and we give particular attention to the example of an axisymmetric sleeping top the results we derive in this paper allow us to determine which trajectories of a sleeping top are locally optimal solutions of the optimal control problem which is a new result that has not appeared in previous literature

on the characterization of the extremal points of the unit sphere of matrices abstract we give a characterization of the extremal points of the unit sphere of matrices for the unitarily invariant norms we also investigate the properties of the dual matrices moreover we show that a result given by lau and riha is not correct

a semismooth newton cg based dual ppa for matrix spectral norm approximation problems we consider a class of matrix spectral norm approximation problems for finding an affine combination of given matrices having the minimal spectral norm subject to some prescribed linear equality and inequality constraints these problems arise often in numerical algebra engineering and other areas such as finding chebyshev polynomials of matrices and fastest mixing markov chain models based on classical analysis of proximal point algorithms ppas and recent developments on semismooth analysis of nonseparable spectral operators we propose a semismooth newton cg based dual ppa for solving the matrix norm approximation problems furthermore when the primal constraint nondegeneracy condition holds for the subproblems our semismooth newton cg method is proven to have at least a superlinear convergence rate we also design efficient implementations for our proposed algorithm to solve a variety of instances and compare its performance with the nowadays popular first order alternating direction method of multipliers admm the results show that our algorithm which is warm started with an initial point obtained from the admm substantially outperforms the pure admm especially for the constrained cases and it is able to solve the problems robustly and efficiently to a relatively high accuracy

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

multi agent compositional stability exploiting system symmetries this paper considers nonlinear symmetric control systems by exploiting the symmetric structure of the system stability results are derived that are independent of the number of components in the system this work contributes to the fields of research directed toward compositionality and composability of large scale system in that a system can be built up by adding components while maintaining system stability the modeling framework developed in this paper is a generalization of many existing results which focus on interconnected systems with specific dynamics the main utility of the stability result is one of scalability or compositionality if the system is stable for a given number of components under appropriate conditions stability is then guaranteed for a larger system composed of the same type of components which are interconnected in a manner consistent with the smaller system the results are general and applicable to a wide class of problems the examples in this paper focus on the formation control problems for multi agent robotic systems

random effects models for serial observations with binary response this paper presents a general mixed model for the analysis of serial dichotomous responses provided by a panel of study participants each subject s serial responses are assumed to arise from a logistic model but with regression coefficients that vary between subjects the logistic regression parameters are assumed to be normally distributed in the population inference is based upon maximum likelihood estimation of fixed effects and variance components and empirical bayes estimation of random effects exact solutions are analytically and computationally infeasible but an approximation based on the mode of the posterior distribution of the random parameters is proposed and is implemented by means of the em algorithm this approximate method is compared with a simpler two step method proposed by korn and whittemore using data from a panel study of asthmatics originally described in that paper one advantage of the estimation strategy described here is the ability to use all of the data including that from subjects with insufficient data to permit fitting of a separate logistic regression model as required by the korn and whittemore method however the new method is computationally intensive

nilpotent approximations of control systems and distributions a constructive method is given to approximate the vector fields in a nonlinear control system which is linear in the controls by a system of similar form and on the same state space with the describing vector fields of the approximating system generating a lie algebra which has a certain relevant sub algebra nilpotent given a k dimensional distribution on r n say locally defined near zero the method leads to an approximating k distribution which has nilpotent basis agrees with the original distribution at zero and each derived distribution of the approximating distribution and original distribution also agree at zero

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

geometric approach to non linear optimal control problems minimum energy control problems are investigated for affine non linear systems evolving on manifolds in the framework of the geometric theory of hamiltonian systems some insights into the problems are gained via this machinery

reduction of polysymplectic manifolds the aim of this paper is to generalize the classical marsden weinstein reduction procedure for symplectic manifolds to polysymplectic manifolds in order to obtain quotient manifolds which inherit the polysymplectic structure this generalization allows us to reduce polysymplectic hamiltonian systems with symmetries such as those appearing in certain kinds of classical field theories as an application of this technique an analogue to the kirillov kostant souriau theorem for polysymplectic manifolds is obtained and some other mathematical examples are also analyzed our procedure corrects some mistakes and inaccuracies in previous papers gunther 1987 j differ geom 25 23 53 munteanu et al 2004 j math phys 45 1730 51 on this subject

symmetries in optimal control it is argued that the existence of symmetries may simplify as in classical mechanics the solution of optimal control problems a procedure for obtaining symmetries for the optimal hamiltonian resulting from the maximum principle is given this avoids the actual calculation of the optimal hamiltonian this procedure is based upon the notion of symmetry for the hamiltonian system with inputs and outputs associated with an optimal control problem

explicitly solvable non linear optimal control problems the method of collective hamiltonians is applied to the study of optimal control problems on manifolds this approach enables us to single out exactly solvable situations various examples are considered necessary results from hamiltonian formalism are given including an industrial technique for the construction of completely integrable systems

nonlinearizable single input control systems do not admit stationary symmetries the main result of the paper states that almost any analytic single input control system which is truly nonlinear that is not feedback linearizable with controllable linearization at an equilibrium point does not admit any symmetry preserving that point by almost any system we mean that we exclude a small class of odd systems that admit just one nontrivial symmetry conjugated to minus identity the obtained results are based on a recent classification of nonlinear single input systems under formal feedback we also describe symmetries of feedback linearizable systems

orthogonality of matrices abstract let a and b be rectangular matrices then a is orthogonal to b if a b a for every scalar some approximation theory and convexity results on matrices are used to study orthogonality of matrices and answer an open problem of bhatia and semrl

optimal feedback control of non linear systems abstract a method is proposed for computing optimal nonlinear feedback control laws it is shown that the feedback loop satisfies a system of quasi linear partial differential equations this system is of first order when the dimensions of the state and the control vector are the same hereby it degenerates into algebraic equations when the performance index does not depend on the control these important results offer new ways for determining optimal feedback laws connections with the volterra series and the hamilton jacobi bellman equation are treated some examples are given to illustrate the advantages of this method

on zero trace matrices let a be an n n complex matrix and let 1 p it is shown that tr a 0 if and only if i za p nsol1p for all z c some related characterizations are also given

isoholonomic problems and some applications we study the problem of finding the shortest loops with a given holonomy we show that the solutions are the trajectories of particles in yang mills potentials theorem 4 or equivalently the projections of kaluza klein geodesics theorem 2 applications to quantum mechanics berry s phase sect 3 and the optimal control of deformable bodies sect 6 are touched upon

on a class of suboptimal controls for infinite dimensional bilinear systems the suboptimal control of a bilinear system is considered with respect to a quadratic cost criterion the feedback control is in the space of formal power series on a hilbert space

exposed faces and duality for symmetric and unitarily invariant norms abstract let be a unitarily invariant norm on the space of real or complex n m matrices and g the associated symmetric gauge function thus a g s a where s a is the decreasing sequence of singular values of a denote by b and bg the closed unit balls of and g in a previous paper we showed a close relationship between the faces of b and those of bg in particular to each face of b we associated a standard face of bg and we used this association to completely describe the matrices that are members of an individual face of in the present paper we consider the duality operator that transforms each exposed face of b into an exposed face of the unit ball of s dual and the duality operator that does the same with exposed faces of bg we show that these two operators are very nicely related among other results we prove the following given an exposed face e of b the standard face associated with the dual of e is precisely the dual of the standard face of bg associated with e e is an exposed face of b if and only if its associated standard face is an exposed face of bg as a by product we completely determine the subdifferential of in terms of the subdifferential of g and we completely characterize the matrices that are dual to a given matrix a with respect to

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

on the optimal control of bilinear systems and its relation to lie algebras the optimal control of bilinear systems is considered and related to the lie algebra generated by the system matrices interesting results obtain when this lie algebra is nilpotent

symmetry reduction of optimal control systems and principal connections this paper explores the role of symmetries and reduction in nonlinear control and optimal control systems the focus of the paper is to give a geometric framework of symmetry reduction of optimal control systems as well as to show how to obtain explicit expressions of the reduced system by exploiting the geometry in particular we show how to obtain a principal connection to be used in the reduction for various choices of symmetry groups as opposed to assuming such a principal connection is given or choosing a particular symmetry group to simplify the setting our result synthesizes some previous works on symmetry reduction of nonlinear control and optimal control systems affine and kinematic optimal control systems are of particular interest we explicitly work out the details for such systems and also show a few examples of symmetry reduction of kinematic optimal control problems

singular value analysis of nonlinear symmetric systems in this paper we introduce the notions of state space symmetry for nonlinear systems and of the cross operator as a nontrivial natural extension of the linear symmetric case in terms of the controllability and observability operators associated to it we give a characterization of a symmetric nonlinear system in terms of the cross operator and a coordinate transformation then we analyze the use of the cross operator for solving the hankel singular value problem of the system the result is a new and simpler characterization of the solutions of this problem in terms of the cross operator and a metric

ck surface approximation from surface patches we study the problem of constructing a smooth approximant from a finite set of patches given on a surface defined by an equation x3 f x1 x2 as an approximant of f a discrete smoothing spline belonging to a suitable piecewise polynomial space is proposed error results and numerical results are given

sample size formula for the proportional hazards regression model a formula is derived for determining the number of observations necessary to test the equality of two survival distributions when concomitant information is incorporated this formula should be useful in designing clinical trials with a heterogeneous patient population schoenfeld 1981 biometrika 68 316 319 derived the asymptotic power of a class of statistics used to test the equality of two survival distributions that result is extended to the case where concomitant information is available for each individual and where the proportional hazards model holds the loss of efficiency caused by ignoring concomitant variables is also computed

characterizations of the trace abstract the paper contains a number of equivalent conditions which characterize the trace among the linear functionals on the matrix algebra some of these results are extended to more general operator algebras

orthogonality of matrices and some distance problems abstract if a and b are matrices such that a zb a for all complex numbers z then a is said to be orthogonal to b we find necessary and sufficient conditions for this to be the case some applications and generalisations are also discussed

local asymptotic approximation of nonlinear control systems let g1 gm be smooth vector fields on rd using a canonical linear rescaling of time and space coordinates the control system is transformed into a system on the same state space which is arbitrarily close to a nilpotent one this approximation takes place in the c topology and is particularly useful for the study of local properties of which depend on the high order variational structure of the vector fields gi

properties of linear approximations of matrices in the spectral norm abstract the properties of linear approximations of a matrix are presented with respect to the spectrum norm we describe the connection between all spectral approximations of a given matrix and we give a sufficient condition under which the spectral approximation is unique

on the generalized reflexive and anti reflexive solutions to a system of matrix equations let p and q be two generalized reflection matrices i e p ph p2 i and q qh q2 i an n n matrix a is said to be generalized reflexive generalized anti reflexive with respect to the matrix pair p q if a paq a paq it is obvious that any n m matrix is also a generalized reflexive with respect to the matrix pair in im by extending the conjugate gradient least square cgls approach the present paper treats two iterative algorithms to solve the system of matrix equations f1 x a1 f2 x a2 fm x am including the sylvester and lyapunov matrix equations as special cases over the generalized reflexive and anti reflexive matrices where f1 f2 fm are the linear operators from cn n onto cri si and ai cri si for i 1 2 m when this system is consistent over the generalized reflexive generalized anti reflexive matrix it is proved that the first second iterative algorithm converges to a generalized reflexive generalized anti reflexive solution for any initial generalized reflexive generalized anti reflexive matrix also the first second iterative algorithm can obtain the the least frobenius norm generalized reflexive generalized anti reflexive solution for special initial generalized reflexive generalized anti reflexive matrix furthermore the optimal approximation generalized reflexive generalized anti reflexive solution to a given generalized reflexive generalized anti reflexive matrix can be derived by finding the least frobenius norm generalized reflexive generalized anti reflexive solution of a new system of matrix equations finally we test the proposed iterative algorithms and show their effectiveness using numerical examples

optimality reduction and collective motion the planar self steering particle model of agents in a collective gives rise to dynamics on the n fold direct product of se 2 the rigid motion group in the plane assuming a connected undirected graph of interaction between agents we pose a family of symmetric optimal control problems with a coupling parameter capturing the strength of interactions the hamiltonian system associated with the necessary conditions for optimality is reducible to a lie poisson dynamical system possessing interesting structure in particular the strong coupling limit reveals additional hidden symmetry beyond the manifest one used in reduction this enables explicit integration of the dynamics and demonstrates the presence of a master clock that governs all agents to steer identically for finite coupling strength we show that special solutions exist with steering controls proportional across the collective these results suggest that optimality principles may provide a framework for understanding imitative behaviours observed in certain animal aggregations

polynomial filtering of discrete time stochastic linear systems with multiplicative state noise the problem of finding an optimal polynomial state estimate for the class of stochastic linear models with a multiplicative state noise term is studied for such models a technique of state augmentation is used leading to the definition of a general polynomial filter the theory is developed for time varying systems with nonstationary and non gaussian noises moreover the steady state polynomial filter for stationary systems is also studied numerical simulations show the high performances of the proposed method with respect to the classical linear filtering techniques

control of nonlinear systems in singular problems abstract the minimum singular functional control problem is analyzed for a class of multi input affine nonlinear systems we develop an algorithm for solving the singular problem that is applicable whether or not singular sub arcs exist in the optimal control the optimal control of bilinear systems to obtain by considering the lie algebra generated by the system matrices

score test for familial aggregation in probands studies application to alzheimer s disease when studying familial aggregation of a disease the following two stage design is often used first select index subjects cases and controls then record data on their relatives the likelihood corresponding to this design is derived and a score test of homogeneity is proposed for testing the hypothesis of no aggregation this test takes into account the selection procedure and allows adjustment to be made for explanatory variables it appears as the sum of three terms a pure test of homogeneity a test of comparison of observed minus expected cases in the two groups and a term which adjusts for the possible unequal probabilities of disease of the index subjects asymptotic efficiency and a simulation study show that the proposed test is superior to either the pure homogeneity test or tests based on the comparison of numbers of affected in the two groups the test statistic which has an asymptotically standard normal distribution is applied to a study of familial aggregation of early onset alzheimer s disease for which a highly significant value 9 46 is obtained this is the highest value among the three tests compared in agreement with the simulation study a logistic normal model is fitted to the data taking account of the selection procedure it allows to estimate the regression parameters and the variance of the random effect the likelihood ratio test for familial aggregation seems less powerful than the score test

algebraic riccati equation and symplectic algebra questions of existence uniqueness and the parametric dependence of solutions of the algebraic riccati equation are considered the different criteria for the solubility of this equation are obtained with the help of symplectic algebra

on minimum quadratic functional control of affine nonlinear systems abstract the minimization control problem of the quadratic functionals for the class of affine nonlinear systems in the hypothesis of nilpotent associated lie algebra is analyzed the optimal control correspondent to the first second and third order nilpotent operator is determined h bourdache siguerdinljane has applied the method of lie algebras to the study of optimal control regulation of satellites s banks and m yew have studied the optimal control of energy consumption minimization for a class of bilinear systems and j s liu et al have generalized this result for the class of affine nonlinear systems the optimal control determination of nonlinear system with a nilpotent structure minimizing the quadratic functionals generalizes the results of j s liu k yuan w s lin and s p banks and m k yew regarding the energy minimization of the affine nonlinear and bilinear systems