efficient mesh optimization using the gradient flow of the mean volume the signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron if we consider these up to translation and scaling the resulting quotient space is diffeomorphic to a sphere the mean volume function restricted to this sphere is a quality measure for volume elements we show that the gradient ascent of this map regularizes the building blocks of hybrid meshes consisting of tetrahedra hexahedra prisms pyramids and octahedra that is the optimization process converges to regular polyhedra we show that the normalized gradient flow of the mean volume yields a fast and efficient optimization scheme for the finite element method known as the geometric element transformation method furthermore we shed some light on the dynamics of this method and the resulting smoothing procedure both theoretically and experimentally

index and stability of symmetric periodic orbits in hamiltonian systems with application to figure eight orbit in this paper using the maslov index theory in symplectic geometry we build up some stability criteria for symmetric periodic orbits in a hamiltonian system which is motivated by the recent discoveries in the n body problem the key ingredient is a generalized bott type iteration formula for periodic solution in the presence of finite group action on the orbit for second order system we prove under general boundary conditions the close formula for the relationship between the morse index of an orbit in a lagrangian system and the maslov index of the fundamental solution for the corresponding orbit in its hamiltonian system counterpart and the boundary conditions cover the cases which appeared in the n body problem as an application we consider the stability problem of the celebrated figure eight orbit due to chenciner and montgomery in the planar three body problem with equal masses and we clarify the relationship between linear stability and its variational nature on various loop spaces the basic idea is as follows the variational characterization of the figure eight orbit provides information about its morse index based on its relation to the maslov index our stability criteria come into play

on the entropy function in sociotechnical systems abstract the entropy function h pj log pj pj being the probability of a system being in state j and its continuum analogue h p x log p x dx are fundamental in shannon s theory of information transfer in communication systems it is here shown that the discrete form of h also appears naturally in single lane traffic flow theory in merchandising goods flow from a whole saler through a retailer to a customer certain features of the process may be deduced from price distribution functions derived from sears roebuck and company catalogues it is found that the dispersion in logarithm of catalogue prices of a given year has remained about constant independently of the year for over 75 years from this it may be inferred that the continuum entropy function for the variable logarithm of price had inadvertently through sears roebuck policies been maximized for that firm subject to the observed dispersion

central configurations of the n body problem via equivariant morse theory in this paper we use the equivariant morse theory to give an estimate of the minimal number of central configurations in the n body problem in 3 in the case of equal masses we prove that the planar central configurations are saddle points for the potential energy from this we deduce the presence of non planar central configurations for every n 4

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

periodic solutions for asymptotically linear hamiltonian system with resonance both at infinity and origin via computation of critical groups abstract in this paper we study the existence of nontrivial periodic solutions for asymptotically linear hamiltonian system which has the same linear term both at infinity and origin by computation of critical groups at origin and infinity associated with maslov type index we establish the existence for nontrivial periodic solution of the system moreover some conditions employed for the computation of critical groups are weakened

on the construction of perfect morse functions on compact manifolds of coherent states perfect morse functions on the manifold of coherent states are effectively constructed the case of a compact connected simply connected lie group of symmetry having the same rank as the stationary group of the manifold of coherent states such that the manifold of coherent states is a kahlerian c space is considered it is proved that the set of perfect morse functions is dense in the set of energy functions for linear hamiltonians in the elements of the cartan algebra of the lie algebra of the representation of the group considered it is proved that the maximum number of orthogonal vectors on a coherent vector manifold is equal to the euler poincare characteristic of the manifold

counterexamples to some conjectures on the number of solutions of nonlinear equations on construit trois contre exemples resolvant par la negative certains problemes d analyse non lineaire

global anomalies on orbifolds we consider global anomalies for heterotic string theory formulated on orbifolds the vanishing of certain characteristic classes in group cohomology provides sufficient conditions for the absence of global anomalies for abelian orbifolds level matching implies these cohomology conditions so suffices for the absence of anomalies for nonabelian orbifolds level matching does not suffice and there are additional constraints we give some examples to illustrate these new constraints

equivariant morse theory for flows and an application to the n body problem a l aide de l indice de conley et de la cohomologie equivariante on deduit certaines inegalites de type morse pour un flot equivariant par rapport a l action d un groupe topologique compact

multiplicity results for two kinds of equivariant systems abstract two equivariant problems of the form e u f u are considered where f is a real function which is invariant under the action of a group g and using morse theory for each problem an arbitrarily great number of orbits of solutions is founded choosing e suitably small the first problem is a o 2 equivariant system of two equations which can be seen as a complex ginzburg landau equation while the second one is a system of m equations which is equivariant for the action of a finite group of real orthogonal matrices m m

periodic orbits of a one dimensional non autonomous hamiltonian system in this paper we study the properties of the periodic orbits of x vx t x 0 with x s1 and vx t x a t0 periodic potential called 1t0q the frequency of windings of an orbit in s1 we show that exists an infinite number of periodic solutions with a given we give a lower bound on the number of periodic orbits with a given period and by means of the morse theory

a simple proof of geodesical completeness for compact space times of zero curvature any riemannian metric on a compact manifold is geodesically complete by contrast it is widely known that incomplete compact lorentzian manifolds exist in this article it will be proven that a compact manifold with a smooth lorentz metric must be complete if the metric is flat everywhere this question of completeness which has significant implications for the classification of compact lorentzian space forms has remained unresolved until quite recently when it was finally established by carriere as a special case of an even more general result here a simple alternative proof of this special case of carriere s theorem is given the proof requires minimal mathematical machinery but still involves the use of some intriguing connections between the topology and global geometry of a compact flat space time

on the geometry of classifying spaces and horizontal slices we study the local properties of the moduli space of a polarized calabi yau manifold let u be a neighborhood of the moduli space then we know the universal covering space v of u is a smooth manifold suppose d is the classifying space of a polarized calabi yau manifold with the automorphism group g then we prove that the map from v to d induced by the period map is a pluriharmonic map we also give a kahler metric on v which is called the hodge metric we prove that the ricci curvature of the hodge metric is negative away from zero we also proved the nonexistence of the kahler metric on the classifying space of a calabi yau threefold which is invariant under a cocompact lattice of g 1 introduction let x be a polarized simply connected calabi yau manifold that is x is a simply connected compact kahler manifold of dimension n with zero first chern class and is akform of x such that h 2 x z in this paper we study the local properties of the moduli space of the polarized calabi yau manifold x by definition is the parameter space of the complex structures over x for the fixed polarization is a quasi projective variety by a theorem of viehweg 17 suppose x is a calabi yau manifold let n d im h 1 x tx 0

the morse index repeller attractor pairs and the connection index for semiflows on noncompact spaces abstract this paper extends the morse index theory of c c conley to semiflows on a noncompact meric space x is assumed to satisfy a hypothesis related to conditional contraction we collect background material define quasi index pairs and the morse index of a compact isolated invariant set k and prove that the morse index is a connected simple system we study repeller attractor pairs in k define index triples and prove their existence and several properties leading to the concepts of the connection index the connection map and the splitting class finally we consider paths continuous families of pairs k and study continuations of the morse and the connection indices along such paths the present paper is a sequel to the author s previous work on the homotopy index for infinite dimensional semiflows trans amer math soc 269 1982 351 382

uniqueness of critical points of solutions to the mean curvature equation with neumann and robin boundary conditions abstract in this paper we investigate the critical points of solutions to the prescribed constant mean curvature equation with neumann and robin boundary conditions respectively in a bounded smooth convex domain of r n n 2 firstly we show the non degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local chen huang s comparison technique and the geometric properties of approximate surfaces at the non degenerate critical points secondly we deduce the uniqueness and non degeneracy of the critical points of solutions in a rotationally symmetric domain of r n n 3 by the projection of higher dimensional space onto two dimensional plane

periodic solutions of dynamical systems with bounded potential where ve c2 r x rn r v t t x y t x v t x jjbxiw y is bounded and v v t x 0 as 1x1 co for example we prove that if 0 d v r x m and v r x m as 1x1 co there exists at least one t periodic solution of v moreover if v j t 0 0 under suitable assumptions on v y t 0 this solution is not identically zero we also give existence results for the forced system

comet like periodic orbits in the n body problem abstract take any nonresonant relative equilibrium solution of the n body problem then there is a periodic solution of the n 1 body problem where n of the particles remain close to the relative equilibrium solution and the remaining particle is close to a circular orbit of the kepler problem encircling the center of mass of the n particle system

finiteness of central configurations of five bodies in the plane we prove there are finitely many isometry classes of planar central configurations also called relative equilibria in the newtonian 5 body problem except perhaps if the 5 tuple of positive masses belongs to a given codimension 2 subvariety of the mass space

relative equilibria near stable and unstable hamiltonian relative equilibria for a symmetric hamiltonian system lower bounds for the number of relative equilibria surrounding stable and formally unstable relative equilibria on nearby energy levels are given

on the number of minimal two spheres of small area in manifolds with curvature bounded above a beautiful application of the morse theory of closed geodesics is provided by klingenberg s k 1 study of the number of short smooth closed geodesics in sufficiently pinched spheres together with subsequent contributions in this direction by many authors a particularly striking result see ballmann thorbergsson and ziller btz states that a generic metric on a compact simply connected riemannian manifold whose sectional curvatures satisfy 1 4 k a 1

birkhoff minimax principle for minimal surfaces with a free boundary 1 1 x 0 in b 1 2 xu xv 0 xu xv in b 1 3 x c s 1 4 xn w tx w s w c here n is the outer normal to b subscripts denote partial derivatives and are the euclidean norm and scalar product in rn 1 and ty s for y s denotes the tangent space to s at y for brevity problem 1 1 1 4 will be referred to as problem p s a solution of p s is a critical point of the following dirichlet integral

a morse complex for infinite dimensional manifolds part i abstract in this paper and in the forthcoming part ii we introduce a morse complex for a class of functions f defined on an infinite dimensional hilbert manifold m possibly having critical points of infinite morse index and co index the idea is to consider an infinite dimensional subbundle or more generally an essential subbundle of the tangent bundle of m suitably related with the gradient flow of f this part i deals with the following questions about the intersection w of the unstable manifold of a critical point x and the stable manifold of another critical point y finite dimensionality of w possibility that different components of w have different dimension orientability of w and coherence in the choice of an orientation compactness of the closure of w classification up to topological conjugacy of the gradient flow on the closure of w in the case dim w 2

symmetry breaking patterns and extended morse theory an indirect topological method is considered which constrains the possible breaking patterns of a lie group symmetry for higgs landau systems the application of the method and its efficiency are discussed

poincare s closed geodesic on a convex surface we present a new proof for the existence of a simple closed geodesic on a convex surface m this result is due originally to poincare the proof uses the 2k dimensional riemannian manifold m briefly of piecewise geodesic closed curves on m with a fixed number k of corners k chosen sufficiently large in a we consider a submanifold 0 formed by those elements of a which are simple regular and divide m into two parts of equal total curvature 2 the main burden of the proof is to show that the energy integral e restricted to 0 assumes its infimum at the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on m

unstable geodesics and topological field theory a topological field theory is used to study the cohomology of mapping space the cohomology is identified with the becchi rouet stora tyutin cohomology realizing the physical hilbert space and the coboundary operator given by the calculations of tunneling between the perturbative vacua the method is illustrated by a simple example

dynamical systems with effective like potentials on etudie l existence des solutions t periodiques pour des systemes hamiltoniens d ordre 2 de la forme x v x ou v r n 0 r est de classe c 2 et v x quand x 0

morse theory and relative equilibria in the planar n vortex problem morse theoretical ideas are applied to the study of relative equilibria in the planar n vortex problem for the case of positive circulations we prove that the morse index of a critical point of the hamiltonian restricted to a level surface of the angular impulse is equal to the number of pairs of real eigenvalues of the corresponding relative equilibrium periodic solution the morse inequalities are then used to prove the instability of some families of relative equilibria in the four vortex problem with two pairs of equal vorticities we also show that for positive circulations relative equilibria cannot accumulate on the collision set

on the homotopy index for infinite dimensional semiflows absrac r in this paper we consider semiflows whose solution operator is eventually a conditional a contraction such semiflows include solutions of retarded and neutral functional differential equations of parabolic and certain other classes of partial differential equations we prove existence of nonsmooth isolating blocks and index pairs for such semiflows via the construction of special lyapunov functionals we show that index pairs enjoy all the properties needed to define the notion of a homotopy index thus generalizing earlier results of conley 2 finally using a result of maii 9 we prove that under additional smoothness assumptions on the semiflow the homotopy index is essentially a finite dimensional concept this gives a formal justification of the applicability of waiewski s principle to infinite dimensional problems several examples illustrate the theory

morse inequalities for r constructible sheaves this note aims at generalizing of classical morse inequalities for betti numbers of compact manifolds cf lo 1 2 in this paper we work with r constructible sheaves instead and encounter the tight relation between morse theory and microlocal analysis of sheaves see witten 111 hellter and sjiistrand s 61 and henniart 7 for another approach to morse inequalities via microlocal analysis and also goresky and macpherson 3 4 who introduced the stratified morse theory this paper may be considered as a variation on kashiwara s index theorem 8 and in fact our proof is a slight modification of his

detectability of critical points of smooth functionals from their finite dimensional approximations abstract given a critical point of a c 2 functional on a separable hilbert space we obtain sufficient conditions for it to be detectable i e visible from finite dimensional rayleigh ritz galerkin rrg approximations while examples show that even nondegenerate critical points are without any further restriction not visible we single out relevant classes of smooth functionals e g the hamiltonian action on the loop space or the functionals associated with boundary value problems for some semilinear elliptic equations such that their nondegenerate critical points are visible from their rrg approximations

symmetric trajectories for the 2n body problem with equal masses we consider the problem of 2n bodies of equal masses in mathbb r 3 for the newtonian like weak force potential r and we prove the existence of a family of collision free nonplanar and nonhomographic symmetric solutions that are periodic modulo rotations in addition the rotation number with respect to the vertical axis ranges in a suitable interval these solutions have the hip hop symmetry a generalization of that introduced in 19 for the case of many bodies and taking account of a topological constraint the argument exploits the variational structure of the problem and is based on the minimization of lagrangian action on a given class of paths

upper bounds on the length of the shortest closed geodesic on simply connected manifolds the subject of this paper is upper bounds on the length of the shortest closed geodesic on simply connected manifolds with non trivial second homology group we will give three estimates the first estimate will explicitly depend on volume and the upper bound for the sectional curvature the second estimate will depend on diameter a positive lower bound for the volume and on the possibly negative lower bound on sectional curvature the third estimate will depend on diameter on a possibly negative lower bound for the sectional curvature and on a lower bound for the simply connectedness radius the technique that we develop in order to obtain the last result will also enable us to estimate the homotopy distance between any two closed curves on compact simply connected manifolds of sectional curvature bounded from below and diameter bounded from above more precisely let c be a constant such that any metric ball of radius leq c is simply connected there exists a homotopy connecting any two closed curves such that the length of the trajectory of the points during this homotopy has an upper bound in terms of the lower bound of the curvature the upper bound of diameter and c

the topology of reduces phase spaces of the motion of vortices on a sphere abstract the motion of point vortices in a perfect incompressible fluid on a two dimensional sphere can be represented by a hamiltonian flow which is invariant under the action of the special orthogonal group so 3 following the reduction method of marsden and weinstein this motion can be described by hamiltonian flows on reduced phase spaces some topological invariants of these reduced phase spaces are calculated for suitable vorticities the results obtained can be used to give lower bounds on numbers of relative equilibria of the original system

equivariant morse theory for starshaped hamiltonian systems soit une hypersurface en forme d etoile dans r 2n le probleme de trouver des caracteristiques fermees de peut se reduire classiquement a un probleme variationnel ceci conduit a etudier une fonctionnelle s 1 equivariante sur un espace de hilbert

on the geodesical connectedness for a class of semi riemannian manifolds we prove a variational principle for geodesics on a semi riemannian manifold m of arbitrary index k and possessing k linearly independent killing vector fields that generate a timelike distribution on m using such a principle and a suitable completeness condition for m we prove some existence and multiplicity results for geodesics joining two fixed points of m

topology and phase transitions the case of the short range spherical model we characterize the topology of the phase space of the berlin kac spherical model in the context of the so called topological hypothesis for spins lying in hypercubic lattices of dimension d for zero external field we are able to characterize the topology exactly up to homology we find that even though there is a continuum of changes in the topology of the corresponding manifolds for d 3 there are abrupt discontinuities in some topological functions that could be good candidates to associate with the phase transitions that occur at the thermodynamic level we show however that these changes do not coincide with the phase transitions and conversely that no topological discontinuity can be associated to the points where the phase transitions take place at variance with what happens in the mean field version of this same model we show that these abrupt topological changes are accessible thermodynamically we conclude that even in short range systems the topological mechanism does not seem to be responsible for the triggering of a phase transition we also analyze the case of spins connected to a macroscopic number of but not all neighbors and find that similar to the results found for the fully connected version in this case the topological hypothesis seems to hold the phase transition coincides with an accumulation point of the topological changes present in configuration space the question of the ensemble equivalence in the short range spherical model is also considered

dynamics of book sales endogenous versus exogenous shocks in complex networks we present an extensive study of the foreshock and aftershock signatures accompanying peaks of book sales the time series of book sales are derived from the ranking system of amazon com we present two independent ways of classifying peaks one based on the acceleration pattern of sales and the other based on the exponent of the relaxation they are found to be consistent and reveal the coexistence of two types of sales peaks exogenous peaks occur abruptly and are followed by a power law relaxation while endogenous sale peaks occur after a progressively accelerating power law growth followed by an approximately symmetrical power law relaxation which is slower than for exogenous peaks we develop a simple epidemic model of buyers connected within a network of acquaintances which propagates rumors and opinions on books the comparison between the predictions of the model and the empirical data confirms the validity of the model and suggests in addition that social networks have evolved to converge very close to criticality here in the sense of critical branching processes of opinion spreading we test in detail the evidence for a power law distribution of book sales and confirm a previous indirect study suggesting that the fraction of books density distribution p s of sales s is a power law p s ensuremath sim 1 s 1 ensuremath mu with ensuremath mu ensuremath approx 2

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

global topological properties of the hopf bifurcation we study the homotopical and homological properties of the attractors evolving from a generalized hopf bifurcation we consider the lorenz equations for parameter values near the hopf bifurcation and study a natural morse decomposition of the global attractor calculating the cech homotopy type of the lorenz attractor the shape indexes of the morse sets and the morse equation of the decomposition

on central configurations this paper concerns an old problem which arose in celestial mechanics and has since taken on a life of its own the probem of central configurations of the n body problem

isosystolic inequalities for optical hypersurfaces abstract we explore a natural generalization of systolic geometry to finsler metrics and optical hypersurfaces with special emphasis on its relation to the mahler conjecture and the geometry of numbers in particular we show that if an optical hypersurface of contact type in the cotangent bundle of the 2 dimensional torus encloses a volume v then it carries a periodic characteristic whose action is at most v 3 this result is deduced from an interesting dual version of minkowski s lattice point theorem if the origin is the unique integer point in the interior of a planar convex body the area of its dual body is at least 3 2

the growth of business firms theoretical framework and empirical evidence we introduce a model of proportional growth to explain the distribution pg g of business firm growth rates the model predicts that pg g is exponential in the central part and depicts an asymptotic power law behavior in the tails with an exponent 3 because of data limitations previous studies in this field have been focusing exclusively on the laplace shape of the body of the distribution in this article we test the model at different levels of aggregation in the economy from products to firms to countries and we find that the predictions of the model agree with empirical growth distributions and size variance relationships

convex hamiltonian energy surfaces and their periodic trajectories in this paper we introduce symplectic invariants for convex hamiltonian energy surfaces and their periodic trajectories and show that these quentities satisfy several nontrivial relations in particular we show that they can be used to prove multiplicity results for the number of periodic trajectories

morse and melnikov functions for nls pde s the theory of the focusing nls equation under periodic boundary conditions together with the floquet spectral theory of its associated zakharov shabat linear operator hat l is developed in sufficient detail for later use in studies of perturbations of the nls equation counting lemmas for the non selfadjoint operator hat l are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite discd in the complex eigenvalue plane the radius of the discd is controlled by theh 1 norm of the potential overrightarrow q for this integrable nls hamiltonian system unstable tori are identified and backlund transformations are then used to construct global representations of their stable and unstable manifolds whiskered tori for the nls pde the floquet discriminant delta lambda overrightarrow q is used to introduce a natural sequence of nls constants of motion f j overrightarrow q equiv delta lambda lambda j c overrightarrow q overrightarrow q where denotes thej th critical point of the floquet discriminant a taylor series expansion of the constants f j overrightarrow q with explicit representations of the first and second variations is then used to study neighborhoods of the whiskered tori in particular critical tori with hyperbolic structure are identified through the first and second variations of f j overrightarrow q which themselves are expressed in terms of quadratic products of eigenfunctions of hat l the second variation permits identification within the disc d of important bifurcations in the spectral configurations of the operator hat l the constant f j overrightarrow q as the height of the floquet discriminant over the critical point admits a natural interpretation as a morse function for nls isospectral level sets this morse interpretation is studied in some detail it is valid globally for the infinite tail f j overrightarrow q j n which is associated with critical points outside the discd within this disc the interpretation is only valid locally with the same obstruction to its global validity as to a global ordering of the spectrum nevertheless this local morse theory together with the backlund representations of the whiskered tori produces extremely clear pictures of the stratification of nls invariant sets near these whiskered tori pictures which are useful in the study of perturbations of nls finally a natural connection is noted between the constants f j overrightarrow q of the integrable theory and melnikov functions for the theory of perturbations of the nls equation this connection generates a simple but general representations of the melnikov functions

variational property of periodic kepler orbits in constant curvature spaces abstract we study the variational property of the periodic kepler orbits on the sphere the plane and the hyperbolic plane we first classify the orbits by the two constants of motion the energy and the angular momentum then we characterize the local variational property of the closed orbits by computing the maslov type indices finally we study the global variational property of the closed orbits we prove that the closed orbits on the hyperbolic plane minimizes the action among all loops which encircle the attracting center

length of geodesics on a two dimensional sphere let m be an arbitrary riemannian manifold diffeomorphic to s 2 let x y be two arbitrary points of m we prove that for every k 1 2 3 ldots there exist k distinct geodesics between x and y of length less than or equal to 4k 2 2k 1 d where d denotes the diameter of m to prove this result we demonstrate that for every riemannian metric on s 2 there are two not mutually exclusive possibilities either every two points can be connected by many short geodesics of index 0 or the resulting riemannian sphere can be swept out by short meridians

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

morse theory for symmetric functionals on the sphere and an application to a bifurcation problem abstract this paper uses conley s index to study the cortical points of a function f on a finite dimensional sphere in presence of a symmetry group the authors prove a theorem which leads to a lower bound on the number of critical points of f when the group is finite even if the action is not free this investigation has been motivated by the following bifurcation problem au lambda where a is a variational g equivariant operator an estimated on the number of branches bifurcating from an eigenvalue of a 0 is given author

orbit spaces of reflection groups with 2 3 and 4 basic polynomial invariants functions which are covariant or invariant under the transformations of a compact linear group can be expressed advantageously in terms of functions defined in the orbit space of the group i e as functions of a finite set of basic invariant polynomials the equalities and inequalities defining the orbit spaces of all finite coregular real linear groups most of which are crystallographic groups with at most four independent basic invariants are determined for each group g acting in the euclidean space the results are obtained through the computation of a metric matrix which is defined only in terms of the scalar products between the gradients of a set of basic polynomial invariants of g the semi positivity conditions are known to determine all the equalities and inequalities defining the orbit space of g as a semi algebraic variety in the space spanned by the variables in a recent paper the matrices for have been determined in an alternative way as solutions of a universal differential equation the present paper yields a partial but significant check on the correctness and completeness of these solutions our results can be easily exploited in many physical contexts where the study of covariant or invariant functions is important for instance in the determination of patterns of spontaneous symmetry breaking in the analysis of phase spaces and structural phase transitions landau s theory in covariant bifurcation theory in crystal field theory and in most areas of solid state theory where use is made of symmetry adapted functions

bumpy metrics and closed parametrized minimal surfaces in riemannian manifolds the purpose of this article is to study conformal harmonic maps f m where is a closed riemann surface and m is a compact riemannian manifold of dimension at least four such maps define parametrized minimal surfaces possibly with branch points we show that when the ambient manifold m is given a generic metric all prime closed parametrized minimal surfaces are free of branch points and are as morse nondegenerate as allowed by the group of automorphisms of e they are morse nondegenerate in the usual sense if has genus at least two lie on two dimensional nondegenerate critical submanifolds if s has genus one and on six dimensional nondegenerate critical submanifolds if has genus zero

collective purchase behavior toward retail price changes a b s t r a c t by analyzing a huge amount of point of sale data collected from japanese supermarkets we find power law relationships between price and sales numbers the estimated values of the exponents of these power laws depend on the category of products however they are independent of the stores thereby implying the existence of universal human purchase behavior the rate of sales numbers around these power laws are generally approximated by log normal distributions implying that there are hidden random parameters which might proportionally affect the purchase activity

minimal two spheres of low index in manifolds with positive complex sectional curvature suppose that s n is given a generic riemannian metric with sectional curvatures which satisfy a suitable pinching condition formulated in terms of complex sectional curvatures this pinching condition is satisfied by manifolds whose real sectional curvatures k r sigma satisfy begin aligned 1 2 k r sigma le 1 end aligned then the number of minimal two spheres of morse index lambda for n 2 le lambda le 2n 5 is at least p 3 lambda n 2 where p 3 k is the number of k cells in the schubert cell decomposition for g 3 mathbb r n 1

singular levels and topological invariants of morse bott integrable systems on surfaces abstract we classify up to homeomorphisms closed curves and eights of saddle points on orientable closed surfaces this classification is applied to morse bott foliations and morse bott integrable systems allowing us to define a complete invariant we state also a realization theorem based in two transformations and one generator the foliation of the sphere with two centers

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation