harmonic maps and periodic toda systems harmonic maps from the point of view of periodic toda systems are considered the correspondence between certain harmonic immersions of surfaces and solutions to periodic toda systems related to nonexceptional simple lie algebras is constructed

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

on a connection between the pascal vandermonde and stirling matrices ii this article is an extension of the work presented in on a connection between the pascal vandermonde and stirling matrices i applied mathematics and computation 2003 to appear it gives a full matrix t t for which t 1 is exactly the lower triangular stochastic matrix t given in on a connection between the pascal vandermonde and stirling matrices i applied mathematics and computation 2003 to appear a new strategy to generate the elements of the matrix t t 1 is investigated a new conjecture and three combinatorial identities are also given

a multilevel approach to single and multiobjective aerodynamic optimization an optimization platform with multilevel structure which is capable of efficiently solving design optimization problems in aerodynamics is proposed the multilevel structure relies on a two way regular exchange of information between successive search levels each level can be associated with a different evaluation software different problem parameterization and or different search tool for the minimization of the same objective function s the combination of some or all of the aforementioned strategies is possible although beyond the scope of this paper the basic optimization tool which is associated with at least one of the levels is a metamodel assisted evolutionary algorithm candidate solutions that have previously been examined are memorized and serve to train radial basis function networks operating as local surrogate evaluation models to handle multiobjective optimization problems with different search techniques at each level we hybridize an evolutionary algorithm that computes the front of pareto optimal solutions at the lower level s with a gradient based method at the upper level for the purpose of refinement through local search in aerodynamic optimization the adjoint method is used to compute the gradient of the objective function for the gradient method to apply to a front of solutions rather than a single individual an approximation of the spea 2 utility function gradient needs to be devised the multilevel platform is demonstrated on mathematical problems as well as the design of isolated and compressor cascade airfoils

explicit inverse of a generalized vandermonde matrix in this paper the author gives an explicit closed form expression for the nxn inverse matrix v g k n 1 of the generalized nxn vandermonde matrix v g k n by using the elementary symmetric functions symbolic and numerical results are presented

response surface approximation of pareto optimal front in multi objective optimization a systematic approach is presented to approximate the pareto optimal front pof by a response surface approximation the data for the pof is obtained by multi objective evolutionary algorithm improvements to address drift in the pof are also presented the approximated pof can help visualize and quantify trade offs among objectives to select compromise designs the bounds of this approximate pof are obtained using multiple convex hulls the proposed approach is applied to study trade offs among objectives of a rocket injector design problem where performance and life objectives compete the pof is approximated using a quintic polynomial the compromise region quantifies trade offs among objectives 2006 elsevier b v all rights reserved

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

pseudo holomorphic curves in complex grassmann manifolds it is proved that the kahler angle of the pseudo holomorphic sphere of constant curvature in complex grassmannians is constant at the same time we also prove several pinching theorems for the curvature and the kahler angle of the pseudo holomorphic spheres in complex grassmannians with non degenerate associated harmonic sequence

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

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

the numerical rank of krylov matrices abstract in this paper we expose interesting features of krylov matrices and vandermonde matrices let s be a large symmetric matrix of order n let x be a real n vector and let k l denote the related n l krylov matrix the question considered in this paper is how the numerical rank of k l grows as l increases the key for answering this question lies in the close link between k l and the n l vandermonde matrix which is generated by the eigenvalues of s analysis of large vandermonde matrices shows that the numerical rank is expected to remain much smaller than l the proof is based on partition theorems and clustering theorems the basic tool is a new matrix equality the vandermonde pascal toeplitz equality the actual numerical rank of a vandermonde or krylov matrix depends on the distribution of the eigenvalues but often the rank is remarkable small numerical experiments illustrate these points the observation that the numerical rank of krylov matrices stays small has important practical consequences

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

evolutionary optimization of computationally expensive problems via surrogate modeling we present a parallel evolutionary optimization algorithm that leverages surrogate models for solving computationally expensive design problems with general constraints on a limited computational budget the essential backbone of our framework is an evolutionary algorithm coupled with a feasible sequential quadratic programming solver in the spirit of lamarckian learning we employ a trust region approach for interleaving use of exact modelsfortheobjectiveandconstraintfunctionswithcomputationallycheapsurrogatemodelsduringlocalsearch in contrastto earlier work we construct local surrogatemodels using radial basis functionsmotivated by theprinciple of transductive inference further the present approach retains the intrinsic parallelism of evolutionary algorithms and can hence be readily implemented on grid computing infrastructures experimental results are presented for some benchmark test functions and an aerodynamic wing design problem to demonstrate that our algorithm converges to good designs on a limited computational budget

the approximation power of moving least squares a general method for near best approximations to functionals on r d using scattered data information is discussed the method is actually the moving least squares method presented by the backus gilbert approach it is shown that the method works very well for interpolation smoothing and derivatives approximations for the interpolation problem this approach gives mclain s method the method is near best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation the interpolation approximation in r d is shown to be a c function and an approximation order result is proven for quasi uniform sets of data points

notes on particular symmetric polynomials with applications this paper presents some applications using several properties of three important symmetric polynomials elementary symmetric polynomials complete symmetric polynomials and the power sum symmetric polynomials the applications includes a simple proof of el mikkawy conjecture in m e a el mikkawy appl math comput 146 2003 759 769 and a very easy proof of the newton girard formula in addition a generalization of stirling numbers is obtained

symmetric bernoulli matrix and its cholesky decomposition abstract this study associates the symmetric bernoulli matrix with various combinatorial matrices such as pascal vandermonde and stirling matrices of the first and second kind including catalan numbers by way of dual matrices of the matrix a recurrence relation related to the entries of the cholesky factor of a symmetric positive definite matrix is presented which sheds new light on examining positive definiteness and cholesky s method of a symmetric matrix this relation is applied to give a necessary and sufficient condition for the positive definiteness of a symmetric matrix which leads to a slightly modified form of the outer product formulation for cholesky s method of a symmetric matrix

stirling matrix via pascal matrix abstract the pascal type matrices obtained from the stirling numbers of the first kind s n k and of the second kind s n k are studied respectively it is shown that these matrices can be factorized by the pascal matrices also the ldu factorization of a vandermonde matrix of the form v n x x 1 x n 1 for any real number x is obtained furthermore some well known combinatorial identities are obtained from the matrix representation of the stirling numbers and these matrices are generalized in one or two variables

conditioning and accurate computations with pascal matrices a result on the ill conditioning of pascal matrices is proved however it is shown that bidiagonal factorizations of pascal matrices can be applied to perform computations with high relative accuracy

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

passive vibration suppression of flexible space structures via optimal geometric redesign acomputationalframework is presented for the design of large flexible space structures with nonperiodicgeometries to achieve passive vibration suppression the present system combines an approximationmodel management framework ammf developed for evolutionary optimization algorithms eas with reduced basis approximate dynamic reanalysis techniques a coevolutionary genetic search strategy is employed to ensure that design changes during the optimization iterations lead to low rank perturbations of the structural system matrices for which thereduced basis methods give high quality approximations the k means algorithm is employed for cluster analysis of the population of designs to determine design points at which exact analysis should be carried out the fitness of the designs in an ea generation is then approximated using reduced basis models constructed around the points where exact analysis is carried out results are presented for the optimal design of a two dimensional cantilevered space structure to achieve passive vibration suppression it is shown that significant vibration isolation of the order of 50 db over a 100 hz bandwidth can be achieved further it is demonstrated that the ammf can potentially arrive at a better design compared to conventional approaches when a constraint is imposed on the computational budget available for optimization

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

aerodynamic shape design using evolutionary algorithms and new gradient assisted metamodels aerodynamic shape design and optimization problems based on evolutionary algorithms and surrogate evaluation tools i e the so called metamodels have recently found widespread use using metamodels trained either separately from or during the optimization loop a considerable reduction in the overall computing cost can be achieved to support metamodel based evolutionary algorithms a class of new metamodels which utilize both known responses and response gradients for their training is proposed the new gradient assisted metamodels are extensions of standard multi layer perceptrons and radial basis function networks to demonstrate the prediction capabilities of the proposed metamodels and investigate different implementation modes within search algorithms along with the relevant cpu cost a number of 2d and 3d aerodynamic shape namely airfoils and turbomachinery blades design problems are analyzed single and two objective problems aiming at designing shapes that reproduce known pressure distributions at specific operating points are considered the exact evaluation tool is a numerical solver of the compressible fluid flow equations the necessary gradient of the objective function is obtained by formulating and numerically solving adjoint equations

on a connection between symmetric polynomials generalized stirling numbers and the newton general divided difference interpolation polynomial this paper gives a connection between symmetric polynomials generalized stirling numbers and the newton general divided difference interpolation polynomial the generalized stirling numbers of the first and second kind denoted s n k a and s n k a respectively are given symbolically for the case n 5 as an illustrative example by using maple programming maple v r3 programming reference manual these numbers depend on n k and n distinct values a 1 a 2 a n the ordinary stirling numbers s n k and s n k may be obtained as special cases by taking a i i 1 i 1 1 n

the generalization of faulhaber s formula to sums of non integral powers abstract a formula for the sum of any positive integral power of the first n positive integers was published by johann faulhaber in the 1600s in this paper we generalize faulhaber s formula to non integral complex powers with real part greater than 1

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

a fast algorithm for solving linear systems of the pascal type abstract in this paper we present a fast algorithm of the complexity o n log n for solving the linear systems with coefficient matrix of the pascal type which is faster than the algorithm of the complexity o n 2 given in the paper m e a el mikkawy on solving linear systems of the pascal type applied mathematics and computation 136 2003 195 202 an application in solving non homogeneous differential equation with constant coefficients is given

convergence error and higher order sensitivity estimations differential equation with respect to the design variables the accuracies of the convergence error and higher order sensitivity estimation methods are verified using laplace euler and navier stokes equations the developed methods are used to improve the accuracy of the finite difference sensitivity calculations in iteratively solved problems aboundonthenormvalueofthe finite differencesensitivityerrorinthestatevariablesisminimizedwith respect to the finite difference step size the optimum finite difference step size is formulated as a function of the norm values of both convergenceerror andhigher order sensitivities thesensitivities calculated with the analytical and the finite difference methods are compared the performance of the proposed methods on the convergence of inverse design optimization is evaluated

remarks on two symmetric polynomials and some matrices by means of complete symmetric polynomials this paper gives a new proof for the vandermonde determinant formula another alternative proof for this formula is obtained via the collocation matrices it also gives a generalized relationship between the vandermonde the pascal and the stirling matrices a new approach to obtain the explicit inverse of the vandermonde matrix is investigated closed form expressions for some vandermonde related determinants are obtained

on conformal minimal immersions of constant curvature from s 2 to hp n we generalize here our general characterization of the harmonic sequence generated by conformal minimal immersions from s 2 to the quaternionic projective space hp n then we give a classification theorem of linearly full unramified conformal minimal immersions of constant curvature from s 2 to hp n under some conditions

extended symmetric pascal matrices via hypergeometric functions in this paper we give a matrix representation of the hypergeometric functions of the type 2f 1 a b c x as a result we obtain a connection between the hypergeometric functions the legendre polynomials and the delannoy numbers moreover it is shown that each entry of p n x y p n x y t can be represented by the hypergeometric functions where p n x y i j x i jy i j 2i 1j 1 is the extended generalized pascal matrix which is defined by zhang and liu linear algebra appl 271 1998 169

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

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

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

coupling dynamic mesh technique and heuristic algorithms in 3 d tapered wing design in this study onera m6 wing has been optimized by two parameters the wing section and the taper ratio by combining the recent mostly preferable popular approaches i e parallel computing and evolutionary techniques for the 3 d models developed during the optimization stages the mesh required has been generated by using dynamic mesh technique the code developed for this is robust and faster than the codes that produce mesh only by classical techniques an euler flow solver acer3d is used to obtain the flow parameters for each member from the results it is observed that the optimization process is working as expected during the optimization process the lift coefficient and the thickness ratio are tried to be maintained close to the design values determined at the beginning the taper ratio becomes smaller and converges to a certain value while the code tries to minimize the drag force additionally this study can be used as a reference for 2 d or 3 d aerodynamic body optimization by using heuristic type algorithms since all details are outlined referenced and interpreted with their advantages and disadvantages copyright 2007 john wiley sons ltd

filtrations factorizations and explicit formulae for harmonic maps we use filtrations of the grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a riemann surface and so all harmonic maps from the 2 sphere to the unitary group for a general class of factorizations by unitons we show how these specialize to give explicit formulae for such harmonic maps to each of the classical compact lie groups and their inner symmetric spaces the nonlinear model of particle physics our methods also give an explicit iwasawa decomposition of the algebraic loop group

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

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

optimization via separated representations and the canonical tensor decomposition abstract we introduce a new quadratically convergent algorithm for finding maximum absolute value entries of tensors represented in the canonical format the computational complexity of the algorithm is linear in the dimension of the tensor we show how to use this algorithm to find global maxima of non convex multivariate functions in separated form we demonstrate the performance of the new algorithms on several examples

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

evolutionary multimodal optimization using the principle of locality the principle of locality is one of the most widely used concepts in designing computing systems to explore the principle in evolutionary computation crowding differential evolution is incorporated with locality for multimodal optimization instead of generating trial vectors randomly the first method proposed takes advantage of spatial locality to generate trial vectors temporal locality is also adopted to help generate offspring in the second method proposed temporal and spatial locality are then applied together in the third method proposed numerical experiments are conducted to compare the proposed methods with the state of the art methods on benchmark functions experimental analysis is undertaken to observe the effect of locality and the synergy between temporal locality and spatial locality further experiments are also conducted on two application problems one is the varied line spacing holographic grating design problem while the other is the protein structure prediction problem the numerical results demonstrate the effectiveness of the methods proposed

symmetric polynomials pascal matrices and stirling matrices we consider lower triangular matrices consisting of symmetric polynomials and we show how to factorize and invert them since binomial coefficients and stirling numbers can be represented in terms of symmetric polynomials these results contain factorizations and inverses of pascal and stirling matrices as special cases this work generalizes that of several other authors on pascal and stirling matrices

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

on the global convergence of trust region algorithms using inexact gradient information trust region algorithms are an important class of methods that can be used to solve unconstrained optimization problems strong global convergence results are demonstrated for a class of methods where the gradient values are approximated rather than computed exactly provided they obey a simple relative error condition no requirement is made that gradients be recomputed to successively greater accuracy after unsuccessful iterations

commentary on an open question abstract in this commentary we give the triangular recurrence relations for the elements of the matrix t t i j l 1 d 1 s 1 and its inverse t 1 t i j s 2 dl 1 1 respectively which have also answered the open question put forward by el mikkawy moreover with the recurrences a computational maple program is written to generate the elements of t and t 1

decompositions of a matrix by means of its dual matrices with applications abstract we introduce the notion of dual matrices of an infinite matrix a which are defined by the dual sequences of the rows of a and naturally connected to the pascal matrix p i j i j 0 1 2 we present the cholesky decomposition of the symmetric pascal matrix by means of its dual matrix decompositions of a vandermonde matrix are used to obtain variants of the lagrange interpolation polynomial of degree n that passes through the n 1 points i q i for i 0 1 n

explicit construction of harmonic two spheres into the complex grassmannian we present an explicit description of all harmonic maps of finite uniton number from a riemann surface into a complex grassmannian namely starting from a constant map q and a collection of meromorphic functions and their derivatives we show how to algebraically construct all harmonic maps from the two sphere into a given grassmannian g p mathbb c n in this setting the uniton number depends on q and p and we obtain a sharp estimate for it

all harmonic 2 spheres in the unitary group completely explicitly we give a completely explicit formula for all harmonic maps of finite uniton number from a riemann surface to the unitary group u n in any dimension and so all harmonic maps from the 2 sphere in terms of freely chosen meromorphic functions on the surface and their derivatives using only combinations of projections and avoiding the usual bar partial problems or loop group factorizations we interpret our constructions using segal s grassmannian model giving an explicit factorization of the algebraic loop group and showing how to obtain harmonic maps into a grassmannian

solutions of euclidean σ models on noncompact grassmann manifolds a method for constructing explicit solutions of classical two dimensional euclidean models on grassmann manifolds of noncompact type and their supersymmetric extensions is presented it generalizes the holomorphic method developed by din zakrzewski and sasaki in the compact case by taking properly into account the geometry of indefinite metric spaces the solutions obtained are local in the sense that they may become singular on certain curves in the plane and the signature of the solution may change across such a curve