harmonic maps into the orthogonal group and null curves we find algebraic parametrizations of extended solutions of harmonic maps of finite uniton number from a surface to the orthogonal group mathrm o n in terms of free holomorphic data which lead to formulae for all such harmonic maps our work reveals an interesting correspondence between certain harmonic maps and the free weierstrass representation of null curves and minimal surfaces in 3 and 4 space

totally real minimal tori in in this paper we show that all totally real minimal tori in open image in new window correspond to doubly periodic finite gap solutions of the tzitzeica equation open image in new window using the results on the tzitzeica equation in integrable systems theory we describe explicitly these tori by prym theta functions

surfaces immersed in lie algebras obtained from the sigma models we study some geometrical aspects of two dimensional orientable surfaces arising from the study of sigma models to this aim we employ an identification of with the lie algebra by means of which we construct a generalized weierstrass formula for immersion of such surfaces the structural elements of the surface like its moving frame the gauss weingarten and the gauss codazzi ricci equations are expressed in terms of the solution of the model defining it further the first and second fundamental forms the gaussian curvature the mean curvature vector the willmore functional and the topological charge of surfaces are expressed in terms of this solution we present detailed implementation of these results for surfaces immersed in and lie algebras

invariant recurrence relations for cp n 1 models in this paper we present invariant recurrence relations for the completely integrable cp n 1 euclidean sigma model in two dimensions defined on the riemann sphere s 2 when its action functional is finite we determine the links between successive projection operators wavefunctions of the linear spectral problem and immersion functions of surfaces in the su n algebra together with the outlines of the proofs our formulation preserves the conformal and scaling invariance of these quantities certain geometrical aspects of these relations are described we also discuss the singularities of the meromorphic solutions of the cp n 1 model and show that they do not affect the invariant quantities we illustrate the construction procedure through the examples of the cp 2 and cp 3 models

synchronization in an array of output coupled boolean networks with time delay this brief presents an analytical study of synchronization in an array of coupled deterministic boolean networks bns with time delay two kinds of models are considered in one model the outputs contain time delay while in another one the outputs do not one restriction in this brief is that the state delay and output delay are restricted to be equal by referring to the algebraic representations of logical dynamics and using the techniques of semitensor product of matrices some necessary and sufficient conditions are derived for the synchronization of delay coupled bns examples including a practical epigenetic example are given for illustration

on the energy spectrum and parameter spaces of classicalcp n models it is known that a large class of smooth solutions ofcpn models can be constructed starting from holomorphic maps of an algebraic curve into complex projective spaces here we apply results from algebraic geometry to describe the energy spectrum and the parameter spaces for such models

strings harmonic maps and hyperbolic systems abstract we review a number of cases where strings and hyperbolic systems arise in the study of harmonic maps of pseudo riemannian manifolds

a finiteness theorem for harmonic maps into hilbert grassmannians in this article we demonstrate that every harmonic map from a closed riemannian manifold into a hilbert grassmannian has image contained within a finite dimensional grassmannian

lie groups and twistor spaces on considere un espace symetrique de riemann simplement connexe m la connexion de levi civita et le fibre torseur metrique 7b f m j m dont les fibres 7b f x m pour x m est constitue de structures complexes orthogonales

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

equivariant harmonic maps into homogeneous spaces this paper is about harmonic maps from closed riemann surfaces into homogeneous spaces such as flag manifolds and loop groups it contains the construction of a family of new examples of harmonic maps from t2 s1 s1 into f n or u n that are not holomorphic with respect to any almost complex structure on f n or u n where f n is the quotient of u n by any maximal torus and u n consists of f s1 u n smooth such that f 1 i

a genetic algorithm approach to optimize a multi products epq model with discrete delivery orders and constrained space the economic production quantity epq is one of the most applicable model in production and inventory control environments like other classical production and inventory control models it is derived based upon assumptions that cause limited real world applications continuous and constant rate delivery of orders infinite availability of warehouse spaces and applicability on a single product are some of these assumptions in order to make the epq model more applicable to real world production and inventory control problems in this paper we expand this model by assuming that the orders may be delivered discretely in the form multiple pallets in addition we may have more than one product along with warehouse space limitation under these conditions we formulate the problem as a non linear integer programming model and propose a genetic algorithm to solve it at the end we present a numerical example to demonstrate the application of the proposed methodology

the space of almost complex 2 spheres in the 6 sphere the complex dimension of the space of linearly full almost complex 2 spheres of area 4 d in the round 6 sphere is calculated to be d 8 explicit examples of these objects are constructed for every integer value of the degree d 6 d 6 7 furthermore it is shown that when d 6 this space is isomorphic to the group g2 c and when d 7 this space is empty we also show that the dimension of the space of non linearly full almost complex 2 spheres of area 4 d in the round 6 sphere is 2d 5

on algebraic problems behind the brouwer degree of equivariant maps abstract given a finite group g and two unitary g representations v and w possible restrictions on topological degrees of equivariant maps between representation spheres s v and s w are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in s v denoted by v effective applications of these congruences is limited by answers to the following questions i under which conditions is v 1 and ii does there exist an equivariant map with the degree easy to calculate in the present paper we address both questions we show that v 1 for each irreducible non trivial c g module if and only if g is solvable this provides a new solvability criterion for finite groups for non solvable groups we use 2 transitive actions to construct complex representations with non trivial characteristic regarding the second question we suggest a class of norton algebras without 2 nilpotents giving rise to equivariant quadratic maps which admit an explicit formula for the degree

monopoles non linear σ models and two fold loop spaces in this paper we study the topology of hat m k the moduli spaces ofsu 2 monopoles associated with the yang mills higgs and bogomol nyi equations and m k non linear models from quantum field theory beautiful work of donaldson 18 19 hitchin 24 25 and taubes 37 39 40 shows that gauge equivalence classes of monopoles correspond to based rational self maps of the riemann sphere similarly the non linear models we consider here are based harmonic maps from the riemann sphere to complex projectivem space in seminal work segal 35 studied m k the space of based rational maps from the riemann sphere to complex projectivem space of a fixed degreek any element of m k is clearly an element of k2cp m the space of all based continuous maps from the riemann sphere to complex projectivem space of a fixed degreek and this assignment gives rise to the natural inclusion of m k in k2cp m segal showed that these natural inclusions are homotopy equivalences through dimensionk 2m 1 as the topology of the two fold loop space 2cp m is well understood segal s result gives a very efficient way to explicitly determine the low dimensional topology of m k thus iterated loop spaces have much to say about the topology of monopoles and non linear models

controllability of probabilistic boolean control networks based on transition probability matrices in this paper we propose a new approach to investigate the controllability and reachability of probabilistic boolean control networks pbcns with forbidden states we first give a simple algebraic formula for the transition probability between two states in a given number of time step while avoiding a set of forbidden states then we construct the controllability matrix based on a new operator and some necessary and sufficient conditions are obtained for the controllability and reachability of pbcns a numerical example is given to illustrate the efficiency of the obtained results

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

classification of conformal minimal immersions of constant curvature from s 2 to hp 2 in this paper we study geometry of conformal minimal two spheres immersed in quaternionic projective spaces we firstly use bahy el dien and wood s results to obtain some characterizations of the harmonic sequences generated by conformal minimal immersions from s 2 to the quaternionic projective space hp 2 then we give a classification theorem of linearly full totally unramified conformal minimal immersions of constant curvature from s 2 to the quaternionic projective space hp 2

ensemble control of finite dimensional time varying linear systems in this article we investigate the problem of simultaneously steering an uncountable family of finite dimensional time varying linear systems with the same control signal this class of control problems motivates further research in the new subject of control theory called ensemble control a notion coming from the study of complex spin dynamics in nuclear magnetic resonance spectroscopy and imaging we derive the necessary and sufficient controllability conditions and an accompanying analytical optimal control law for an ensemble of finite dimensional time varying linear systems we show that ensemble controllability is in connection with singular values of the operator characterizing the system dynamics in addition the problem of optimal ensemble control of harmonic oscillators is studied to demonstrate the controllability results we show that the optimal solutions are pertinent to the study of time frequency limited signals and prolate spheroidal wave functions a systematic study of ensemble control systems has immediate applications to dynamical systems with parameter uncertainty as well as to wide ranging areas such as neuroscience and quantum control the work in ensemble control will foster further developments in mathematical control and systems theory

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

on minimal surfaces in a kahler manifold of constant holomorphic sectional curvature this paper studies minimal surfaces in kshler manifolds of constant holomorphic sectional curvature using the technique of the moving frame in particular we provide a classification of the minimal two spheres in cpst complex projective sf space equipped with the fubini study metric this classification can be described as follows to each holomorphic curve in c pt1 classically there is associated a particular framing of c l l called the frenet frame each element of the frenet frame induces a minimal surface in cp the classification theorem states that all minimal surfaces of topological type of the two sphere occur in this manner the theorem is proved using holomorphic differentials that occur naturally on minimal surfaces in kshler manifolds of constant holomorphic sectional curvature together with the riemann roch theorem

the classification of the surfaces with parallel mean curvature vector in two dimensional complex space forms in this paper we find new surfaces with parallel mean curvature vectors in two dimensional complex space forms that are not of constant kahler angles and then classify all of them the surface can be isometrically deformed preserving both the length of the mean curvature vector and the kahler angle from this classification the deformations appear only when the curvature of the ambiant space is nonpositive

tournaments flags and harmonic maps twistor space constructions have shown that under quite general hypotheses harmonic maps of riemann surfaces into riemannian manifolds can be represented by holomorphic curves with respect to almost complex structures which are often non integrable see for example 19 17 5 for maps into complex grassmannians the almost complex structures are invariant ones on flag manifolds and can be identified with tournaments i e digraphs in which any two nodes are joined by exactly one oriented edge elementary properties of tournaments are summarised in sect i the basic references are moon 14 and reid and beineke 18 emphasis is given to certain types of tournament which are interpreted in sect 2 in terms of the homogeneous structure of complex flag manifolds holomorphic curves in such flag manifolds are described in sect 3 by collections of subbundles of a trivial bundle over a riemann surface whose second fundamental forms define a tournament this approach relies on techniques of burstall and wood 6 and gives rise naturally to harmonic maps into grassmannians when the riemann surface has genus zero a result from algebraic geometry namely the birkhoff grothendieck decomposition theorem for holomorphic vector bundles fits into the scheme and allows one to reconstruct a tournament from a harmonic map into a grassmannian in sect 4 we use such techniques to show that harmonic maps of tl2p 1 into a complex grassmannian can be built up successively from holomorphic data this last result extends results of burstall and wood 6 and is closely related to recent work of uhlenbeck 21 and wolfson 23

stability in yang mills theories at a solution of the yang mills equations ons4 or the yang mills higgs equation on 3 the hessian of the action functional defines a natural second order elliptic operator the number of negative eigenvalues of this operator is bounded below by a multiple of the relevant topological charge the proof of this assertion requires a relative index theorem for dirac type operators on n n 3

optimal control of linear systems with parameter uncertainty a class of infinite horizon regulator problems is formulated for families of time invariant linear systems with parameter uncertainty under a regularity assumption the optimal linear state feedback control is shown to exist and is defined via a positive definite solution of a family of riccati type algebraic equations the solvability of these equations is equivalent to the stabilizability of the family of linear systems by a constant linear feedback

sequences of willmore surfaces in this paper we develop the theory of willmore sequences for willmore surfaces in the 4 sphere we show that under appropriate conditions this sequence has to terminate in this case the willmore surface either is the twistor projection of a holomorphic curve into mathbb c mathbb p 3 or the inversion of a minimal surface with planar ends in mathbb r 4 these results give a unified explanation of previous work on the characterization of willmore spheres and willmore tori with non trivial normal bundles by various authors

the topology of rational functions and divisors of surfaces segal studied the homotopy types of the spaces of holomorphic functions of the 2 sphere s 2 of closed surfaces of higher genus and of the spaces of divisors of these surfaces we continue segal s program by describing the entire stable homotopy types of these spaces in terms of the homotopy types of more familiar spaces

quantization of curvature of harmonic two spheres in grassmann manifolds various pinching theorems for curvature of minimal two spheres in grassmann manifolds have been proved in particular we show that when the curvature is large then the minimal map from s2 into g m n must be either holomorphic or antiholomorphic also minimal two spheres of curvature k 2 in g 2 4 have been classified 0 introduction it is known that starting from a harmonic map f from a riemann surface into a complex grassmann manifold one can define sequences of harmonic maps by using the 0 transform and the 0 transform these sequences are called harmonic sequences cw when f is holomorphic the sequence generated by f via the 0 transform is called a pseudoholomorphic sequence and the corresponding maps are called pseudoholomorphic maps in 1980 a m din and w j zakrzewski dz and then in 1983 j eells and j c wood ec showed that all harmonic maps from s2 to the complex projective space cpf are in fact pseudoholomorphic recently f burstall j rawnsley and s salamon announced in brs that all stable harmonic maps from s2 into irreducible hermitian symmetric spaces are either holomorphic or antiholomorphic it is also known that when the target space is a general complex grassmann manifold holomorphic curves do not generate all harmonic maps cw ra on the other hand m rigoli showed in r that for a linearly full nonsingular holomorphic curve from s2 into cpn if its curvature k 4 n then k 4 n is constant then j bolton et al bjrw generalized this result to pseudoholomorphic maps and showed that for a linearly full isometric pseudoholomorphic map of position r from s2 into cpn if its curvature k 4 2r n r n then k 41 2r n r n is constant in this paper we prove various pinching theorems of curvature for harmonic maps and show that for any nonsingular harmonic map from s2 into a complex grassmann manifold if the curvature of its induced metric is large then received by the editors january 18 1988 and in revised form april 11 1988 1980 mathematics subject classification 1985 revision primary 53a10 53c55

the construction of harmonic maps into complex grassmannians on developpe une technique d analyse des applications harmoniques d une surface de riemann dans un grassmannien complexe en utilisant des diagrammes qui etablissent plusieurs resultats

rigidity of pseudo holomorphic curves of constant curvature in grassmann manifolds rigidity of minimal immersions of constant curvature in harmonic sequences generated by holomorphic curves in grassmann manifolds is studied in this paper by lifting them to holomorphic curves in certain projective spaces we prove that for such curves the curvature must be positive and that all such simply connected curves in cp are generated by veronese curves thus generalizing calabi s counterpart for holomorphic curves in cp we also classify all holomorphic curves from the riemann sphere into g 2 4 whose curvature is equal to 2 into two families which illustrates pseudo holomorphic curves of positive constant curvature in g m n are in general not unitarily equivalent constracting to the fact that generic isometric complex submanifolds in a kaehler manifold are congruent where f x dfi for j 0 1 2 and f m c7 w n are harmonic maps one defines mj x 0 if f is anti holomorphic we call fr a pseudo holomorphic curve of position r generated by when is holomorphic where is called the directrix of fr such curves with the induced metrics from the associated grassmannians form a class of minimal immersions when special ized to g l n i cp a pseudo holomorphic curve of position r is just the r th position vector of a frenet frame of a holomorphic curve projected into cp the importance of such maps comes from the fact that any har monic map from the riemann sphere or any harmonic map of nonzero degree from a torus into cp is pseudo holomorphic 7 furthermore in 6 these curves play a central role when m s defined in the sequence 0 1 are all equal and f s are orthogonal called frenet harmonic sequences for the study of

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

invariant recurrence relations for mathbb c p n 1 models in this paper we present invariant recurrence relations for the completely integrable euclidean sigma model in two dimensions defined on the riemann sphere s2 when its action functional is finite we determine the links between successive projection operators wavefunctions of the linear spectral problem and immersion functions of surfaces in the algebra together with the outlines of the proofs our formulation preserves the conformal and scaling invariance of these quantities certain geometrical aspects of these relations are described we also discuss the singularities of the meromorphic solutions of the model and show that they do not affect the invariant quantities we illustrate the construction procedure through the examples of the and models