the bifree locally inverse semigroup on a set abstract a class of regular semigroups closed under taking direct products regular subsemigroups and homomorphic images is an e xistence variety of regular semigroups for an e variety v of locally inverse or e solid regular semigroups the bifree object bf v x on a set x is the natural concept of a free object in v its existence has been proved by y t yeh in this paper the bifree locally inverse semigroup bf l j x is described as a homomorphic image of the absolutely free algebra of type 2 2 generated by x and the set of formal inverses x and equivalently as subsemigroup of a semidirect product of a suitable free semilattice by the bifree completely simple semigroup on x this latter realization is used to show that bf l j x is combinatorial completely semisimple and satisfies several finiteness conditions furthermore the approach of biidentities is used to formulate a birkhoff type theorem for e varieties of locally inverse semigroups and to establish a one one correspondence between locally inverse e varieties and fully invariant congruences on bf l j x for countably infinite x as an application it is shown that in each e variety of locally inverse semigroups all free products exist

general relaxation methods for initial value problems with application to multistep schemes recently an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial value problems using runge kutta methods we generalize this approach to multistep methods including all general linear methods of order two or higher and many other classes of schemes we prove the existence of a valid relaxation parameter and high order accuracy of the resulting method in the context of general equations including but not limited to conservative or dissipative systems the theory is illustrated with several numerical examples

adaptive systems and time varying plants abstract the behaviour of four adaptive algorithms lms equation error output error identification and the goodwin ramadge caines adaptive control algorithm are examined in the face of plant parameter variations the key point to note is that persistency of excitation conditions in the time invariant parameter case guarantees exponential convergence then standard and newly developed lyapunov stability results allow us to guarantee tracking error and parameter error boundedness when the plant parameters are actually time varying

enumeration of idempotents in planar diagram monoids we classify and enumerate the idempotents in several planar diagram monoids namely the motzkin jones a k a temperley lieb and kauffman monoids the classification is in terms of certain vertex and edge coloured graphs associated to motzkin diagrams the enumeration is necessarily algorithmic in nature and is based on parameters associated to cycle components of these graphs we compare our algorithms to existing algorithms for enumerating idempotents in arbitrary regular semigroups and give several tables of calculated values

equational theories of semigroups with involution abstract we employ the techniques developed in an earlier paper to show that involutory semigroups arising in various contexts do not have a finite basis for their identities among these are partition semigroups endowed with their natural inverse involution including the full partition semigroup c n for n 2 the brauer semigroup b n for n 4 and the annular semigroup a n for n 4 n even or a prime power also all of these semigroups as well as the jones semigroup j n for n 4 turn out to be inherently nonfinitely based when equipped with another involution the skew one finally we show that similar techniques apply to the finite basis problem for existence varieties of locally inverse semigroups

analysis of energy and quadratic invariant preserving equip methods abstract in this paper we are concerned with the analysis of a class of geometric integrators at first devised in brugnano et al 2010 and brugnano et al 2012 which can be regarded as an energy conserving variant of gauss collocation methods with these latter they share the property of conserving quadratic first integrals but in addition they also conserve the hamiltonian function itself we here reformulate the methods in a more convenient way and propose a more refined analysis than that given in brugnano et al 2012 also providing as a by product a practical procedure for their implementation a thorough comparison with the original gauss methods is carried out by means of a few numerical tests solving hamiltonian and poisson problems

cellularity of diagram algebras as twisted semigroup algebras abstract the temperley lieb and brauer algebras and their cyclotomic analogues as well as the partition algebra are all examples of twisted semigroup algebras we prove a general theorem about the cellularity of twisted semigroup algebras of regular semigroups which allows us to easily reproduce the cellularity of these algebras this theorem generalizes a result of east about the cellularity of semigroup algebras of inverse semigroups

on groups generated by involutions of a semigroup abstract an involution on a semigroup s or any algebra with an underlying associative binary operation is a function s s that satisfies x y y x and x x for all x y s the set i s of all such involutions on s generates a subgroup c s i s of the symmetric group sym s on the set s we investigate the groups c s for certain classes of semigroups s and also consider the question of which groups are isomorphic to c s for a suitable semigroup s

integral preserving integrators ordinary differential equations having a first integral may be solved numerically using one of several methods with the integral preserved to machine accuracy one such method is the discrete gradient method it is shown here that the order of the method can be bootstrapped repeatedly to higher orders of accuracy the method is illustrated using the henon heiles system

global stability of adaptive pole placement algorithms this paper presents direct and indirect adaptive control schemes for assigning the closed loop poles of a single input single output system in both the continuous and discrete time cases the resulting closed loop system is shown to be globally stable when driven by an external reference signal consisting of a sum of sinusoids in particular persistent excitation of the potentially unbounded closed loop input output data and hence convergence of a sequential least squares identification algorithm is proved the results are applicable to standard sequential least squares and least squares with covariance reset

highly efficient invariant conserving explicit runge kutta schemes for nonlinear hamiltonian differential equations abstract a unified framework of invariant conserving explicit runge kutta schemes for the nonlinear hamiltonian odes and pdes are proposed by utilizing the invariant energy quadratization technique first the nonlinear hamiltonian differential equation is transformed into an equivalent reformulation which admits a quadratic energy invariant for the nonlinear ode the reformulation is then discretized using a class of relaxation runge kutta schemes for the nonlinear pde the reformulation is first discretized in the space direction by adopting the fourier pseudo spectral discretization which preserves the semi discrete quadratic conservation laws then the semi discrete system is integrated in the time direction using the explicit relaxation runge kutta method the obtained method can conserve different quadratic conservation laws to machine accuracy which improves the numerical stability during long time computations besides the proposed method keeps the same convergence rate of the standard runge kutta scheme when computed with time step size rescaling and gives a rate of convergence reduced by at most one when computed without step size rescaling numerical experiments for several odes and pdes are provided to illustrate the advantages of the proposed algorithms over long time and verify the theoretical analysis

on adaptive state regulation an indirect adaptive state regulator consisting of an adaptive observer and an asymptotic feedback matrix synthesis is discussed in a rigorous analysis lyapunov stability of the resulting closed loop system without the need for external excitation possible global instabilities as well as global convergence subject to sufficient excitation are shown moreover the behavior of the closed loop system is analyzed for cases in which the true plant differs from the assumed linear time invariant model in that it is of higher order contains slight nonlinearities and time variation effects and is subject to plant and measurement disturbances

line integral methods which preserve all invariants of conservative problems recently the class of hamiltonian boundary value methods hbvms has been introduced with the aim of preserving the energy associated with polynomial hamiltonian systems and more in general with all suitably regular hamiltonian systems however many interesting problems admit other invariants besides the hamiltonian function it would be therefore useful to have methods able to preserve any number of independent invariants this goal is achieved by generalizing the line integral approach which hbvms rely on thus obtaining a number of generalizations which we collectively name line integral methods in fact it turns out that this approach is quite general so that it can be applied to any numerical method whose discrete solution can be suitably associated with a polynomial such as a collocation method as well as to any conservative problem in particular a completely conservative variant of both hbvms and gauss collocation methods is presented numerical experiments confirm the effectiveness of the proposed methods

local asymptotic stability via averaging for time varying systems with unbounded dynamics with respect to time the purpose of this note is to provide sufficient conditions for asymptotic stability for time varying systems whose dynamics are unbounded with respect to time an averaging approach is established that constitutes a generalization of the well known methodology adopted in the literature for systems whose dynamics are bounded with respect to time

a new adaptive law for robust adaptation without persistent excitation a new adaptive law motivated by the work of loannou and kokotovic 1983 is proposed for the robust adaptive control of plants with unknown parameters in this adaptive law the output error plays a dual role in the adjustment of the control parameter vector the advantages of using the adaptive law over others proposed in the literature are discussed in the ideal case the adaptive system has bounded solutions in addition the origin of the error equations is exponentially stable when the reference input is persistently exciting and has a sufficiently large amplitude the adaptive system is also shown to be robust under bounded external disturbances finally it is shown that by suitably modifying the adaptive law the overall system can be made robust in the presence of a class of unmodeled dynamics of the plant simulation results are presented throughout the paper to complement the theoretical developments

conservation laws and the numerical solution of odes ii abstract many physical systems described by an initial value problem for a system of ordinary differential equations odes conserve physical quantities such as the net charge or total energy as the system evolves typical codes for the numerical solution of the odes will not conserve these quantities and this can lead to solutions which are not even qualitatively correct one way to impose conservation laws is to perturb the numerical solution at each step of the integration a simple theory is developed in this paper which tells how this should be done so as to guarantee convergence of codes based on one step methods it is also easy to interpret the effect on the accuracy of the perturbations

conserving first integrals under discretization with variable step size integration procedures abstract it is well known that the application of one step or linear multistep methods to an ordinary differential equation with first integrals will destroy the conserving quantities with the use of stabilization techniques similar to ascher chin reich numer math 67 1997 131 149 we derive stabilized variants of our original problem we show that variable step size one step and linear multistep methods applied to the stabilized equation will reproduce that phase portrait correctly in particular this technique will conserve first integrals over an infinite time interval within the local error of the used method

new embedded explicit pairs of exponentially fitted runge kutta methods two new embedded pairs of exponentially fitted explicit runge kutta methods with four and five stages for the numerical integration of initial value problems with oscillatory or periodic solutions are developed in these methods for a given fixed w the coefficients of the formulae of the pair are selected so that they integrate exactly systems with solutions in the linear space generated by sinh wt cosh wt the estimate of the local error behaves as o h 4 and the high order formula has fourth order accuracy when the stepsize h 0 these new pairs are compared with another one proposed by franco j m franco an embedded pair of exponentially fitted explicit runge kutta methods j comput appl math 149 2002 407 414 on several problems to test the efficiency of the new methods

persistence of excitation conditions and the convergence of adaptive schemes in the study of the behavior of adaptive filtering algorithms persistence of excitation of the input process arises as a sufficient condition for convergence and perhaps more importantly for convergence rate of the parameter estimates in this paper the underlying nature of the persistence requirement is presented and discussed relating its normal specification in terms of moment conditions with covariance decays etc to sample path properties deterministic and stochastic persistence conditions and persistence measures are treated as well as persistence conditions for output error equation error and adaptive control schemes

fast extremum seeking on hammerstein plants partial plant knowledge may be used to develop model based extremum seekers however existing results rely on a type of time scale separation which leads to slow optimization relative to the plant dynamics in this work a fast model based extremum seeking scheme is proposed for a hammerstein plant and semi global stability results are provided the structure of a hammerstein plant is used to advantage in designing filters that enable the extremum seeker to act on a faster time scale than the plant dynamics this leads to fast convergence while maintaining semi global stability

adaptive observer for continuous descriptor systems we design an adaptive observer for a stable single input and single output continuous descriptor generalized state space or singular system under the assumption that the descriptor system is observable it is transformed into what we call descriptor observer canonical form this form also allows the descriptor variable to be represented by a linear combination of the 2n unknown parameters and the filtered input and output measurement we also show that the estimates converge to the true values when the bounded input satisfies the richness condition under the assumption that the system is controllable

persistent excitation in adaptive systems the importance of the concept of persistent excitation pe in adaptive identification and control has been recognized for some time recently it has become evident that it also plays a central role in many questions related to the robustness of adaptive systems there is every reason to believe that arguments involving this concept will continue to feature prominently in the analysis of most of the important problems of adaptive control hence there is a real need for a deeper understanding of the concept the paper is written with three objectives the first which is tutorial in nature is to provide a general framework for the discussion of persistent excitation and to collect results in the area which are scattered throughout the adaptive literature the second objective is to present some new results related to the uniform asymptotic stability u a s and robustness of adaptive systems and the relation of pe to the stability properties of a class of non linear systems the final objective is to di

energy preserving methods for poisson systems we present and analyze energy conserving methods for the numerical integration of ivps of poisson type that are able to preserve some casimirs their derivation and analysis is done following the ideas of hamiltonian bvms hbvms see brugnano et al 10 and references therein it is seen that the proposed approach allows us to obtain the methods recently derived in cohen and hairer 2011 17 giving an alternative derivation of such methods and a new proof of their order sufficient conditions that ensure the existence of a unique solution of the implicit equations defining the formulae are given a study of the implementation of the methods is provided in particular order and preservation properties when the involved integrals are approximated by means of a quadrature formula are derived

persistency of excitation in the presence of possibly unbounded signals this note develops an explicit characterization of the concept of persistency of excitation for time invariant linear systems in the presence of possibly unbounded signals this result can be used to establish exponential convergence for a range of adaptive control algorithms a feature of our proof is a particular technique for dealing with the initial condition response

uniform global asymptotic stability of adaptive cascaded nonlinear systems with unknown high frequency gains abstract we study adaptive tracking problems for nonlinear systems with unknown control gains we construct controllers that yield uniform global asymptotic stability for the error dynamics and hence tracking and parameter estimation for the original systems our result is based on a new explicit global strict lyapunov function construction we illustrate our work using a brushless dc motor turning a mechanical load we quantify the effects of time varying uncertainties on the motor electric parameters

preserving algebraic invariants with runge kutta methods abstract we study runge kutta methods for the integration of ordinary differential equations and the retention of algebraic invariants as a general rule we derive two conditions for the retention of such invariants the first is a condition on the coefficients of the methods the second is a pair of partial differential equations that otherwise must be obeyed by the invariant this paper extends previous work on multistep methods in iserles technical report na1997 13 damtp university of cambridge 1997 the cases related to the retention of quadratic and cubic invariants perhaps of greatest relevance in applications are thoroughly discussed we conclude recommending a generalized class of runge kutta schemes namely lie group type runge kutta methods these are schemes for the solution of odes on lie groups but can be employed together with group actions to preserve a larger class of algebraic invariants without restrictions on the coefficients

stable model reference adaptive control in the presence of bounded disturbances the adaptive control of a linear time invariant plant in the presence of bounded disturbances is considered in addition to the usual assumptions made regarding the plant transfer function it is also assumed that the high frequency gain k p of the plant and an upper bound on the magnitude of the controller parameters are known under these conditions the adaptive controller suggested assures the boundedness of all signals in the overall system

explicit methods based on a class of four stage fourth order runge kutta methods for preserving quadratic laws this paper concerns the study of explicit 4 stage fourth order runge kutta methods which preserve quadratic conservation laws when their weights are made solution dependent application to orthogonal and hamiltonian linear systems and numerical tests are also reported

state agreement for continuous time coupled nonlinear systems two related problems are treated in continuous time first the state agreement problem is studied for coupled nonlinear differential equations the vector fields can switch within a finite family associated to each vector field is a directed graph based in a natural way on the interaction structure of the subsystems generalizing the work of moreau under the assumption that the vector fields satisfy a certain subtangentiality condition it is proved that asymptotic state agreement is achieved if and only if the dynamic interaction digraph has the property of being sufficiently connected over time the proof uses nonsmooth analysis second the rendezvous problem for kinematic point mass mobile robots is studied when the robots fields of view have a fixed radius the circumcenter control law of ando e ieee trans robotics automation 15 1999 pp 818 828 is shown to solve the problem the rendezvous problem is a kind of state agreement problem but the interaction structure is state dependent

stability and robustness properties of a simple adaptive controller a low order adaptive tracking controller is proposed for linear time invariant plants with a relative degree not exceeding two the order of the plant is not required to be known a priori the algorithm is robustly stable with respect to small linear time invariant plant perturbations and bounded disturbances the robustness is achieved by using a projection of the parameter estimates in the control law an additional a priori information needed to design the controller is bounds on the plant parameters in the absence of plant perturbations and disturbances perfect tracking is obtained

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

geometric integration on spheres and some interesting applications geometric integration theory can be employed when numerically solving odes or pdes with constraints in this paper we present several one step algorithms of various orders for odes on a collection of spheres to demonstrate the versatility of these algorithms we present representative calculations for reduced free rigid body motion a conservative ode and a discretization of micromagnetics a dissipative pde we emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions

on robust parameter estimation in finite time without persistence of excitation the problem of adaptive estimation of constant parameters in the linear regressor model is studied without the hypothesis that regressor is persistently excited first the initial vector estimation problem is transformed to a series of the scalar ones using the method of dynamic regressor extension and mixing second several adaptive estimation algorithms are proposed for the scalar scenario in such a case if the regressor is nullified asymptotically or in finite time then the problem of estimation is also posed on a finite interval of time robustness of the proposed algorithms with respect to measurement noise and exogenous disturbances is analyzed the efficiency of the designed estimators is demonstrated in numeric experiments for academic examples

brief strict lyapunov functions for time varying systems uniformly asymptotically stable periodic time varying systems for which is known a lyapunov function with a derivative along the trajectories non positive and negative definite in the state variable on non empty open intervals of the time are considered for these systems strict lyapunov functions are constructed

on the preservation of invariants by explicit runge kutta methods a new strategy to preserve invariants in the numerical integration of initial value problems with explicit runge kutta methods is presented it is proved that this technique retains the order of the original method has an easy and cheap implementation and can be used in adaptive runge kutta codes some numerical experiments with the classical code of dormand and prince dopri5 4 based on a pair of embedded methods with orders 5 and 4 are presented to show the behavior of the new method for several problems which possess invariants

motzkin monoids and partial brauer monoids we study the partial brauer monoid and its planar submonoid the motzkin monoid we conduct a thorough investigation of the structure of both monoids providing information on normal forms green s relations regularity ideals idempotent generation minimal idempotent generating sets and so on we obtain necessary and sufficient conditions under which the ideals of these monoids are idempotent generated we find formulae for the rank smallest size of a generating set of each ideal and for the idempotent rank smallest size of an idempotent generating set of the idempotent generated subsemigroup of each ideal in particular when an ideal is idempotent generated the rank and idempotent rank are equal along the way we obtain a number of results of independent interest and we demonstrate the utility of the semigroup theoretic approach by applying our results to obtain new proofs of some important representation theoretic results concerning the corresponding diagram algebras the partial or rook brauer algebra and motzkin algebra