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

sets of efficient points in a normed space abstract sets of efficient points in a normed space with respect to the distances to the points of a given compact set are geometrically characterized hull and closure properties are obtained these results are relevant to geometry of normed spaces and are mostly useful in the context of location theory

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

existence and multiplicity result for the singular toda system abstract we consider the toda system on a compact surface g u 1 2 1 h 1 e u 1 h 1 e u 1 d v g 1 2 h 2 e u 2 h 2 e u 2 d v g 1 4 j 1 j 1 j p j 1 u 2 2 2 h 2 e u 2 h 2 e u 2 d v g 1 1 h 1 e u 1 h 1 e u 1 d v g 1 4 j 1 j 2 j p j 1 where h i are smooth positive functions i are positive real parameters p j are given points on and i j are numbers greater than 1 we give existence and multiplicity results using variational and morse theoretical methods it is the first existence result when some of the i j s are allowed to be negative

matrix factorizations for reversible integer mapping reversible integer mapping is essential for lossless source coding by transformation a general matrix factorization theory for reversible integer mapping of invertible linear transforms is developed concepts of the integer factor and the elementary reversible matrix erm for integer mapping are introduced and two forms of erm triangular erm term and single row erm serm are studied we prove that there exist some approaches to factorize a matrix into terms or serms if the transform is invertible and in a finite dimensional space the advantages of the integer implementations of an invertible linear transform are i mapping integers to integers ii perfect reconstruction and iii in place calculation we find that besides a possible permutation matrix the term factorization of an n by n nonsingular matrix has at most three terms and its serm factorization has at most n 1 serms the elementary structure of erm transforms is the ladder structure an executable factorization algorithm is also presented then the computational complexity is compared and some optimization approaches are proposed the error bounds of the integer implementations are estimated as well finally three erm factorization examples of dft dct and dwt are given

structure of efficient sets for convex objectives this paper considers multiple objective problems in which each objective is the minimization of a convex function defined over a convex solution space and for which the number of objectives is greater than the dimension of the solution space such problems arise in location analysis where the solution space of possible facility sites may be represented by a plane but many location objectives may exist we show that the set of efficient solutions to such problems exhibit structures similar to convex hulls that is the solutions on the boundary of the efficient set are efficient for subproblems obtained by considering only small subsets of objectives the number of objectives in these subproblems is at most equal to the dimension of the solution space when the objectives are continuous and attain their minimum values in a bounded region over the solution space the efficient set of the subproblems completely determine an efficient set for the larger problem

on efficient solutions to multiple objective mathematical programs this note develops properties of quasi efficient solutions and explores interrelationships to the classical concept of efficiency in particular a point is a quasi efficient solution to a multiple objective mathematical program if and only if it is an optimal solution to a scalar maximum problem for some set of nonnegative weights on the objectives this result is then used to characterize the set of quasi efficient solutions as the union of efficient solutions to a multiple objective problem over all nonempty subsets of the objectives

generalized s transform the generalized s transform gst a family of reversible integer to integer transforms inspired by the s transform is proposed this family of transforms is then studied in detail by considering topics such as gst parameter calculation the effects of using different rounding operators in the gst and the relationship between the gst and the lifting scheme some examples of specific transforms in the gst family are also given in particular a new transform in this family is introduced and its practical utility is demonstrated

constant curvature surfaces of the supersymmetric ℂpn 1 sigma model constant curvature surfaces are constructed from the finite action solutions of the supersymmetric pn 1 sigma model it is shown that there is a unique holomorphic solution which leads to constant curvature surfaces the generalized veronese curve we give a general criterion to construct non holomorphic solutions of the model we extend our analysis to general supersymmetric grassmannian models

reversible integer color transform in this correspondence we introduce a systematic algorithm that can convert any 3 times 3 color transform into a reversible integer to integer transform we also discuss the ways to improve accuracy and reduce implementation complexity we derive the integer rgb to kla iv1v2 ycbcr dct yuv and yiq transforms that are optimal in 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

a general existence result for the toda system on compact surfaces in this paper we consider the following toda system of equations on a compact surface u1 2 1 h1eu1 h1eu1dvg 1 2 h2eu2 h2eu2dvg 1 u1 4 j 1m 1 j pj 1 u2 2 2 h2eu2 h2eu2dvg 1 1 h1eu1 h1eu1dvg 1 u2 4 j 1m 2 j pj 1 which is motivated by the study of models in non abelian chern simons theory here h1 h2h1 h2 are smooth positive functions 1 2 1 2 two positive parameters pipi points of the surface and 1 i 2 j 1 i 2 j non negative numbers we prove a general existence result using variational methods the same analysis applies to the following mean field equation u 1 heu heudvg 1 2 he u he udvg 1 which arises in fluid dynamics

avoidance of singular point in integer orthonormal transform for lossless coding in this correspondence the singular point sp problem peculiar to an integer orthonormal transform is discussed and compatibility of the integer transform with the corresponding real valued transform is improved to avoid the sp problem we introduce two point permutations of order and sign of signals since it satisfies the commutative law it becomes possible to reduce computational cost to find the best combination of the permutations which minimizes errors due to rounding of signals inside the integer transform

every unit matrix is a lulu abstract the four matrices l0u0l1u1 at the end of the title are triangular with ones on their main diagonals their product has determinant one following a question and theorem of toffoli we show that any matrix with determinant one can be factored in this way a transformation of the plane becomes a sequence of one dimensional shears with n2 1 free parameters

customizable triangular factorizations of matrices abstract customizable triangular factorizations of matrices find their applications in computer graphics and lossless transform coding in this paper we prove that any n n nonsingular matrix a can be factorized into 3 triangular matrices a plus where p is a permutation matrix l is a unit lower triangular matrix u is an upper triangular matrix of which the diagonal entries are customizable and can be given by all means as long as its determinant is equal to that of a up to a possible sign adjustment and s is a unit lower triangular matrix of which all but n 1 off diagonal elements are set zeros and the positions of those n 1 elements are also flexibly customizable such as a single row a single column a bidiagonal matrix or other specially patterned matrices a pseudo permutation matrix which is a simple unit upper triangular matrix with off diagonal elements being 0 1 or 1 can take the role of the permutation matrix p as well in some cases p may be the identity matrix besides plus a customizable factorization also has other alternatives lusp psul or sulp for lower s and puls ulsp pslu slup for upper s

new concentration phenomena for su 3 toda system in this paper we consider the following su 3 toda system 0 1 u1 2 2h1eu1 2h2eu2 0in b1 0 u2 2 2h2eu2 2h1eu1 0in b1 0 u1 u2 0on b1 0 where is a positive parameter which will go to zero h1 x h2 x h x is a positive smooth function and b1 0 is a unit ball in r2 in 29 the authors construct a family of solutions u1 u2 of 0 1 for h 1 such that the solutions blow up at the origin with mass 8 4 after some scaling with the limiting profile wi x i 2ewi in r2 r2 x i 2ewi where i 23 i for i 1 2 in that paper the authors ask whether there exist blow up solutions of mass 8 4 with the limiting profile w 2ew 0in r2 r2ew such that u1 is the sum of two bubbles and u2 has one bubble in this paper we prove that such solution exists

rigidity of holomorphic curves in complex grassmann manifolds in this paper it is proved that the holomorphic curves in complex grassmann manifolds are uniquely determined by its first and second fundamental forms up to rigid motion

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

convergence rate location and z2 condition for fully bubbling solutions to su n 1 toda systems abstract it is well known that the study of su n 1 toda systems is important not only to chern simons models in physics but also to the understanding of holomorphic curves harmonic sequences or harmonic maps from riemann surfaces to c p n one major goal in the study of su n 1 toda system on riemann surfaces is to completely understand the asymptotic behavior of fully bubbling solutions in this article we use a unified approach to study fully bubbling solutions to general su n 1 toda systems and we prove three major sharp estimates important for constructing bubbling solutions the closeness of blowup solutions to entire solutions the location of blowup points and a z 2 condition

factorization of unitary matrices factorization of n n unitary matrices as a product of n diagonal phase matrices interlaced with n 1 orthogonal matrices each one generated by a real vector is provided as a byproduct an explicit form for the weyl factorization of unitary matrices is given the results can be used at the parametrization of complex hadamard matrices and in finding the laplace beltrami operators on unitary groups

general solutions of the supersymmetric ℂp2 sigma model and its generalisation to ℂpn 1 a new approach for the construction of finite action solutions of the supersymmetric pn 1 sigma model is presented we show that this approach produces more non holomorphic solutions than those obtained in previous approaches we study the p2 model in detail and present its solutions in an explicit form we also show how to generalise this construction to n 3

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

classification and nondegeneracy of su n 1 toda system with singular sources we consider the following toda system open image in new window where i 1 0 is dirac measure at 0 and the coefficients a ij form the standard tri diagonal cartan matrix in this paper i we completely classify the solutions and obtain the quantization result sum j 1 n a ij int mathbb r 2 e u j dx 4 pi 2 gamma i gamma n 1 i quad forall 1 leq i leq n this generalizes the classification result by jost and wang for i 0 forall 1 leq i leq n ii we prove that if i i 1 j for all 1 i j n then any solution u i is radially symmetric w r t 0 iii we prove that the linearized equation at any solution is non degenerate these are fundamental results in order to understand the bubbling behavior of the toda system

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

harmonic maps of the two sphere into a complex grassmann manifold ii where qt is a complex valued one form defined up to a factor of absolute value 1 this form qp defines a complex structure on m for x e m the space f x has an orthogonal space f x of dimension n k we denote by f x and f x their corresponding projective spaces of dimensions k 1 and n k 1

degree counting and shadow system for toda system of rank two one bubbling abstract we initiate the program for computing the leray schauder topological degree for toda systems of rank two this program still contains a lot of challenging problems for analysts as the first step we prove that if a sequence of solutions u 1 k u 2 k blows up then one of h j e u j k m h j e u j k d v g j 1 2 tends to a sum of dirac measures this is so called the phenomena of weak concentration our purposes in this article are i to introduce the shadow system due to the bubbling phenomena when one of parameters i crosses 4 and j 4 n where 1 i j 2 ii to show how to calculate the topological degree of toda systems by computing the topological degree of the general shadow systems iii to calculate the topological degree of the shadow system for one point blow up we believe that the degree counting formula for the shadow system would be useful in other problems

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

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

a new algorithm for solving planar multiobjective location problems involving the manhattan norm abstract this paper is devoted to the study of unconstrained planar multiobjective location problems where distances between points are defined by means of the manhattan norm we characterize the nonessential objectives and by eliminating them we develop an effective algorithm for generating the whole set of efficient solutions as the union of a special family of rectangles and line segments we prove the correctness of this algorithm analyze its complexity and present illustrative computational results obtained by a matlab based implementation

almost every unit matrix is a ulu abstract we call an n n matrix a shear if it is triangular with all l s on the diagonal and a unit matrix if it has unit determinant earlier we had shown that for n 3 every orthogonal matrix except for degenerate cases when one of the euler angles equals can be written in the form u 0 lu 1 where the u are upper shears and l is a lower shear then strang showed that for any n every unit matrix can be written as l 0 u 0 l 1 u 1 here we show that every unit matrix except for a subset of measure zero can be decomposed into the product of just three shears u 0 lu 1 and we present a canonical form for this decomposition on the residual subset such a decomposition is still possible up to a sign if one is allowed to suitably prepermute the rows of the matrix

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

integer fast fourier transform a concept of integer fast fourier transform intfft for approximating the discrete fourier transform is introduced unlike the fixed point fast fourier transform fxpfft the new transform has the properties that it is an integer to integer mapping is power adaptable and is reversible the lifting scheme is used to approximate complex multiplications appearing in the fft lattice structures where the dynamic range of the lifting coefficients can be controlled by proper choices of lifting factorizations split radix fft is used to illustrate the approach for the case of 2 sup n point fft in which case an upper bound of the minimal dynamic range of the internal nodes which is required by the reversibility of the transform is presented and confirmed by a simulation the transform can be implemented by using only bit shifts and additions but no multiplication a method for minimizing the number of additions required is presented while preserving the reversibility the intfft is shown experimentally to yield the same accuracy as the fxpfft when their coefficients are quantized to a certain number of bits complexity of the intfft is shown to be much lower than that of the fxpfft in terms of the numbers of additions and shifts finally they are applied to noise reduction applications where the intfft provides significantly improvement over the fxpfft at low power and maintains similar results at high power

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

products of three triangular matrices abstract we prove that any square matrix over a field is the product of at most three triangular matrices we also give explicit lul factorizations for all 2 2 and 3 3 matrices over fields

finding efficient solutions for rectilinear distance location problems efficiently abstract given n planar existing facility locations a planar new facility location x is called efficient if there is no other location y for which the rectilinear distance between y and each existing facility is at least as small as between x and each existing facility and strictly less for at least one existing facility rectilinear distances are typically used to measure travel distances between points via rectilinear aisles or street networks we first present a simple arrow algorithm based entirely on geometrical analysis that constructs all efficient locations we then present a row algorithm which is of order n log n that constructs all efficient locations and establish that no alternative algorithm can be of a lower order

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

nonessential objectives within network approaches for mcdm in gal and hanne eur j oper res 119 1999 373 the problem of using several methods to solve a multiple criteria decision making mcdm problem with linear objective functions after dropping nonessential objectives is analyzed it turned out that the solution does not need be the same when using various methods for solving the system containing the nonessential objectives or not in this paper we consider the application of network approaches for multicriteria decision making such as neural networks and an approach for combining mcdm methods called mcdm networks we discuss questions of comparing the results obtained with several methods as applied to the problem with or without nonessential objectives especially we argue for considering redundancies such as nonessential objectives as a native feature in complex information processing in contrast to previous results on nonessential objectives the current paper focuses on discrete mcdm problems which are also denoted as multiple attribute decision making madm