nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

optimal control of systems possessing symmetries it is shown that a symmetry in an optimization problem induces a decomposition of the optimal feedback control law into factors one factor can be calculated algebraically and depends only on the symmetry the other factor corresponds to a lower dimensional optimization problem this gives a priori information about the structure of the optimal feedback control law and indicates a possibly more efficient method for optimizing such systems

25th order high temperature expansion results for three dimensional ising like systems on the simple cubic lattice 25th order high temperature series are computed for a general nearest neighbor three dimensional ising model with arbitrary potential on the simple cubic lattice in particular we consider three improved potentials characterized by suppressed leading scaling corrections critical exponents are extracted from high temperature series specialized to improved potentials obtaining gamma 1 2373 2 nu 0 63012 16 alpha 0 1096 5 eta 0 03639 15 beta 0 32653 10 delta 4 7893 8 moreover biased analyses of the 25th order series of the standard ising model provide the estimate delta 0 52 3 for the exponent associated with the leading scaling corrections by the same technique we study the small magnetization expansion of the helmholtz free energy the results are then applied to the construction of parametric representations of the critical equation of state using a systematic approach based on a global stationarity condition accurate estimates of several universal amplitude ratios are also presented

the structure of nonlinear control systems possessing symmetries a concept of symmetry is defined for general nonlinear control systems it is shown under various technical conditions that nonlinear control systems with symmetries admit local and or global decompositions in terms of lower dimensional subsystems and feedback loops the structure of the individual subsystems is dependent on the structure of the symmetry group for example if the symmetry group is abelian one of the subsystems is a quadrature an additional feature of the decomposition is that the state space dimensions of the subsystems sum to the state space dimension of the original system

monte carlo study of the droplet formation dissolution transition on different two dimensional lattices in 2003 biskup et al commun math phys 242 137 2003 gave a rigorous proof for the behavior of equilibrium droplets in the two dimensional 2d spin 1 2 ising model or equivalently a lattice gas of particles on a finite square lattice of volume v with a given excess ensuremath delta m ensuremath equiv m ensuremath m 0 of magnetization compared to the spontaneous magnetization m 0 m 0 v by identifying a dimensionless parameter ensuremath delta ensuremath delta m and a universal constant ensuremath delta c they showed in the limit of large system sizes that for ensuremath delta l ensuremath delta c the excess is absorbed in the background evaporated system while for ensuremath delta g ensuremath delta c a droplet of the minority phase occurs condensed system by minimizing the free energy of the system they derived an explicit formula for the fraction ensuremath lambda ensuremath delta of excess magnetization forming the droplet to check the applicability of the asymptotic analytical results to much smaller practically accessible system sizes we performed several monte carlo simulations of the 2d ising model with nearest neighbor couplings on a square lattice at fixed magnetization m thereby we measured the largest minority droplet corresponding to the condensed phase in the lattice gas interpretation at various system sizes l 40 80 dots 640 with analytical values for the spontaneous magnetization density m 0 the susceptibility ensuremath chi and the wulff interfacial free energy density ensuremath tau w for the infinite system we were able to determine ensuremath lambda numerically in very good agreement with the theoretical prediction furthermore we did simulations for the spin 1 2 ising model on a triangular lattice and with next nearest neighbor couplings on a square lattice again finding a very good agreement with the analytic formula we demonstrate the universal aspects of the theory with respect to the underlying lattice and type of interaction for the case of the next nearest neighbor model where ensuremath tau w is unknown analytically we present different methods to obtain it numerically by fitting to the distribution of the magnetization density p m

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

symmetry reduction contact geometry and partial feedback linearization let pfaffian system omega define an intrinsically nonlinear control system which is invariant under a lie group of symmetries g using the contact geometry of brunovsky normal forms and symmetry reduction this paper solves the problem of constructing subsystems alpha subset omega such that alpha defines a static feedback linearizable control system a method for representing the trajectories of omega from those of alpha using reduction by a distinguished class g of lie symmetries is described a control system will often have a number of inequivalent linearizable subsystems depending upon the subgroup structure of g this can be used to obtain a variety of representations of the system trajectories in particular if g is solvable the construction of trajectories can be reduced to quadrature it is shown that the identification of linearizable subsystems in any given problem can be carried out algorithmically once the explicit lie algebra of g is known all the const

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

on controlled invariance in nonlinear systems abstract the notion of generalized controlled invariance gci is introduced to allow for the loss of degrees of freedom in control that may be the necessary price for achieving a desired invariance it is shown that subject to some technical rank restrictions controlled invariance implies gci a sequential procedure that definitively establishes for a given system pair whether is a gci of is introduced and through it a verdict o the solvability of the controlled invariance problem cip is reached

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 decomposition of nonlinear control systems in this paper the problem is studied when the system x i k 1 u t g t x x 0 x 0 defined on an n dimensional analytic manifold can be decomposed into either independent subsystems or a cascade of subsystems necessary and sufficient conditions for the existence of such local decompositions are given in terms of lie algebras generated by the system

cluster monte carlo algorithm with a conserved order parameter we propose a cluster simulation algorithm for statistical ensembles with fixed order parameter we use the tethered ensemble which features helmholtz s effective potential rather than gibbs s free energy and in which canonical averages are recovered with arbitrary accuracy for the d 2 3 ising model our method s critical slowing down is comparable to that of canonical cluster algorithms yet we can do more than merely reproduce canonical values as an example we obtain a competitive value for the 3d ising anomalous dimension from the maxima of the effective potential

conservation laws in hamiitonian control systems under unmodelled dynamics utilisation pour la definition des lois de conservation de commande d un concept plus general qui comprend la consideration des quantites a la fois conservees et non conservees etude de l influence des dynamiques lineaires non modelisees sur les lois de conservation

synchronisation of complex dynamical networks with different dynamics of nodes via decentralised dynamical compensation controllers this paper investigates the synchronisation of nonlinear coupled complex dynamical networks with different nonlinear nodes and different orders by using decentralised dynamical compensation controllers we propose a dynamical network mathematical model with similar nonlinear nodes whose dimensions are different for this kind of network model the decentralised dynamical compensation controllers are designed for the state synchronisation of the coupled nodes in addition the synchronisation manifold is defined as an invariant manifold which is regarded as the generalised case of dynamical networks with the same nodes dynamics furthermore some stability criteria for the synchronisation are derived by means of rigorous theoretical analysis finally numerical examples are presented to verify the effectiveness of the obtained theoretical results

controlled invariance for nonlinear systems necessary and sufficient conditions are derived for a b invariance called here controlled invariance for nonlinear systems dot x f x u the obtained results generalize and elucidate already known results about systems dot x a x sum min i 1 max m u i b i x a new and direct differential geometric interpretation of the concept of controlled invariance and the derived conditions is given

dynamic critical behavior of the worm algorithm for the ising model we study the dynamic critical behavior of the worm algorithm for the two and three dimensional ising models by monte carlo simulation the autocorrelation functions exhibit an unusual three time scale behavior as a practical matter the worm algorithm is slightly more efficient than the swendsen wang algorithm for simulating the two point function of the three dimensional ising model

the analysis of zero dynamics of nonlinear control systems with symmetries zero dynamics of nonlinear control systems with symmetries are studied in this paper it is shown that the zero dynamics of a symmetric system is also symmetric and admits a special form of cascade decomposition a local zero dynamics is always possible to be extended to the semiglobal case the relation between the zero dynamics and the quotient system is also discussed

on attribute reduction in concept lattices the polynomial time discernibility matrix based method becomes the cr method abstract we eliminate unnecessary computations of attribute discernibility sets in discernibility matrix based methods dm methods of attribute reduction in concept lattices we obtain a polynomial time algorithm we call a skim dm method the skim dm method however is merely slightly different from the standard clarification and reduction cr method this result enables us to express a relationship of the cr method and the dm methods the dm methods are basically the cr method with up to an exponential amount of unnecessary computations additionally we experimentally evaluate the skim dm method in comparison with the cr method

attribute reduction in generalized one sided formal contexts in this paper we present a new pair of adjoint mappings between a power set and the direct product of complete lattices the proposed pair of adjoint mappings form a galois connection and the corresponding concept lattice is constructed from a generalized one sided formal context we also propose a lattice keep based attribute reduction approach for generalized one sided formal contexts specifically we present concrete judgment theorems and an algorithm to calculate the attribute reducts in generalized one sided formal contexts furthermore according to the importance of attributes we discuss attributive characteristics for the proposed generalized one sided concept lattice

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

reducing concept lattices by means of a weaker notion of congruence abstract attribute and size reductions are key issues in formal concept analysis in this paper we consider a special kind of equivalence relation to reduce concept lattices which will be called local congruence this equivalence relation is based on the notion of congruence on lattices with the goal of losing as less information as possible and being suitable for the reduction of concept lattices we analyze how the equivalence classes obtained from a local congruence can be ordered moreover different properties related to the algebraic structure of the whole set of local congruences are also presented finally a procedure to reduce concept lattices by the new weaker notion of congruence is introduced this procedure can be applied to the classical and fuzzy formal concept analysis frameworks

on attribute reduction in concept lattices methods based on discernibility matrix are outperformed by basic clarification and reduction abstract several recent papers have studied the problem of attribute reduction in concept lattices by using a discernibility matrix in various extensions of formal concept analysis we recall the clarification and reduction method by ganter wille and show that it is strictly superior to the methods based on discernibility matrix consequently we show how to adapt the clarification and reduction method to the extensions of formal concept analysis considered in the recent papers

cascade decomposition of nonlinear systems foliations and ideals of transitive lie algebras abstract we relate the cascade decomposition of nonlinear systems to the joint notions of foliations and ideals of transitive lie algebras

symmetries and conservation laws for hamiltonian systems with inputs and outputs a generalization of noether s theorem the definition of symmetries and conservation laws for autonomous i e without external cases hamiltonian systems are generalized to hamiltonian systems with inputs and outputs it is shown that a symmetry implies the existence of a conservation law and vice versa thereby generalizing noether s theorem for autonomous hamiltonian systems

symmetric systems structure of the state space it is shown that every module of every finite dimensional associative algebra is associated with the state space of some class of symmetric systems the general structure of symmetric systems is investigated

a methodology for analysis of concept lattice reduction independent methodology for analysis of concept lattice reduction we use sets of proper implications holding in the original and reduced structure highlight the kinds of changes propitiated by the different classes of techniques four reduction techniques were used to illustrate the proposed methodology formal concept analysis fca is a mathematical theory of data analysis with applications in many areas the problem of obtaining a concept lattice of an appropriate size was identified in several applications as one of the most important problems of fca in order to deal with this problem several techniques with different characteristics were proposed for concept lattice reduction however there are currently no adequate methods to assess what types of knowledge transformations can result from a reduction a methodology for analysis of concept lattice reduction is presented here it is based on the use of sets of proper implications holding in the original and reduced formal contexts or concept lattices working with both sets of implications the methodology is able to show what is preserved eliminated inserted or transformed by a reduction technique three classes of reduction techniques are analyzed from the standpoint of the methodology in order to highlight techniques of each class have in common with respect to the transformations performed such analysis is followed by specific examples in each class

large scale dissipative and passive control systems and the role of star and cyclic symmetries in this technical note symmetries in large scale dissipative and passive control systems are considered in the framework of dissipativity and passivity theory stability conditions for large scale systems are derived by categorizing agents into symmetry groups and applying local control laws under limited interconnections with neighbors building upon previous studies on stability of q s r dissipative large scale systems we show that for cyclic and star shaped symmetric systems there exists an upper bound on the number of subsystems that can be added to preserve stability of dissipative systems in cyclic and star shaped symmetric systems the subsystems can be heterogeneous as long as they satisfy the same dissipative inequalities approximate symmetry with respect to interconnections is also considered and the robustness of the results is demonstrated

a mathcal b invariant distributions and disturbance decoupling of nonlinear systems the concept of a b invariant subspaces has resulted in a unified approach to many of the basic structural properties of time invariant linear systems w m wonham lecture notes in economics and mathematical systems vol 101 springer verlag new york 1974 the purpose of this paper is to introduce the more general notion of a mathcal b invariant distributions on differentiable manifolds and to use this idea to study the disturbance decoupling problem for a class of nonlinear systems which evolve on real analytic manifolds

bisimilar control affine systems the notion of bisimulation plays a very important role in theoretical computer science where it provides several notions of equivalence between models of computation these equivalences are in turn used to simplify verification and synthesis for these models as well as to enable compositional reasoning in systems theory a similar notion is also of interest in order to develop modular verification and design tools for purely continuous or hybrid control systems in this paper we introduce two notions of bisimulation for nonlinear systems we present differential geometric characterizations of these notions and show that bisimilar systems of different dimensions are obtained by factoring out certain invariant distributions furthermore we also show that all bisimilar systems of different dimension are of this form

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

optimal planar turns under acceleration constraints this technical note considers the problem of finding optimal trajectories for a particle moving in a 2 d plane from a given initial position and velocity to a specified terminal heading under a magnitude constraint on the acceleration the cost functional to be minimized is the integral over time of a non negative power of the particle s speed special cases of such a cost functional include travel time and path length unlike previous work on related problems variations in the magnitude of the velocity vector are allowed pontryagin s maximum principle is used to show that the optimal paths possess a special property whereby the vector that divides the velocity and acceleration vectors in a specific ratio which depends on the cost functional is a constant this property is used to characterize the optimal acceleration vector and obtain the parametric equations of the corresponding optimal paths sufficiency conditions for optimality are used to show that trajectories satisfying the necessary conditions are actually optimal if the heading change required is sufficiently small solutions of the time optimal and the length optimal problems are obtained as special cases of the general problem