tools in the orbit space approach to the study of invariant functions rational parametrization of strata functions which are equivariant or invariant under the transformations of a compact linear group g acting in a euclidean space n can profitably be studied as functions defined in the orbit space of the group the orbit space is the union of a finite set of strata which are semialgebraic manifolds formed by the g orbits with the same orbit type in this paper we provide a simple recipe to obtain rational parametrizations of the strata our results can be easily exploited in many physical contexts where the study of equivariant 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 theory in equivariant bifurcation theory in crystal field theory and in most areas where use is made of symmetry adapted functions a physically significant example of utilization of the recipe is given related to spontaneous polarization in chiral biaxial liquid crystals where the advantages with respect to previous heuristic approaches are shown

rotation numbers eigenvalues and the poincare birkhoff theorem using the relation between rotation numbers and eigenvalues we prove the existence of nontrivial t periodic solutions having precise oscillatory properties for a class of asymptotically linear second order differential equations

bounding the equivariant betti numbers of symmetric semi algebraic sets abstract let r be a real closed field the problem of obtaining tight bounds on the betti numbers of semi algebraic subsets of r k in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to ole nik and petrovski thom and milnor these bounds are all exponential in the number of variables k motivated by several applications in real algebraic geometry as well as in theoretical computer science where such bounds have found applications we consider in this paper the problem of bounding the equivariant betti numbers of symmetric algebraic and semi algebraic subsets of r k we obtain several asymptotically tight upper bounds in particular we prove that if s r k is a semi algebraic subset defined by a finite set of s symmetric polynomials of degree at most d then the sum of the s k equivariant betti numbers of s with coefficients in q is bounded by s k d o d unlike the classical bounds on the ordinary betti numbers of real algebraic varieties and semi algebraic sets the above bound is polynomial in k when the degrees of the defining polynomials are bounded by a constant as an application we improve the best known bound on the ordinary betti numbers of the projection of a compact algebraic set improving for any fixed degree the best previously known bound for this problem due to gabrielov vorobjov and zell

poincare birkhoff theorems in random dynamics we propose a generalization of the poincare birkhoff theorem on area preserving twist maps to area preserving twist maps f that are random with respect to an ergodic probability measure in this direction we will prove several theorems concerning existence density and type of the fixed points to this end first we introduce a randomized version of generalized generating functions and verify the correspondence between its critical points and the fixed points of f a fact which we successively exploit in order to prove the theorems the study we carry out needs to combine probabilistic techniques with methods from nonlinear pde and from differential geometry notably moser s method and conley zehnder theory our stochastic model in the periodic case coincides with the classical setting of the poincare birkhoff theorem

periodic solutions of second order equations via rotation numbers abstract we consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems we show how a rotation number approach together with the poincare birkhoff theorem and the phase plane analysis of the spiral properties allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity these systems may also be resonant in particular our approach can be used to deal with the problems without both global existence of the cauchy problems associated to the equation and the sign assumption on f the typical example is a partially superlinear second order equation our main result generalizes some classical results of jacobowitz and hartman among others

coexistence of bouncing and classical periodic solutions of generalized lazer solimini equation abstract the paper deals with the singular differential equation x g x p t where g has a weak singularity at x 0 sufficient conditions for a coexistence of two types of periodic solutions are presented the first type is a classical periodic solution which is strictly positive on r and does not reach the singularity the second type is a bouncing periodic solution which reaches the singularity at isolated points in particular we state a constant k 0 such that there exist at least two 2 periodic bouncing solutions having their maximum less than k and at least one 2 periodic classical solution having its minimum greater than k the proofs are based on the ideas of the poincare birkhoff twist map theorem and approximation principles

a topological version of the poincare birkhoff theorem with two fixed points the main result of this paper gives a topological property satisfied by any homeomorphism of the annulus mathbb a mathbb s 1 times 1 1 isotopic to the identity and with at most one fixed point this generalizes the classical poincare birkhoff theorem because this property certainly does not hold for an area preserving homeomorphism h of mathbb a with the usual boundary twist condition we also have two corollaries of this result the first one shows in particular that the boundary twist assumption may be weakened by demanding that the homeomorphism h has a lift h to the strip tilde mathbb a mathbb r times 1 1 possessing both a forward orbit unbounded on the right and a forward orbit unbounded on the left as a second corollary we get a new proof of a version of the conley zehnder theorem in mathbb a if a homeomorphism of mathbb a isotopic to the identity preserves the area and has mean rotation zero then it possesses two fixed points

universality in orbit spaces of compact linear groups if p1 x p q x is a minimal integrity basis of the ideal of polynomial invariants of a compact coregular linear groupg the orbit map open image in new window yields a diffeomorphic image open image in new window of the orbit spacer n g starting from this fact we point out some properties which are common to the orbit space of all the compact coregular linear groups of transformations ofr n in particular we show that a contravariant metric matrix hat p p can be defined in the interior of open image in new window as a polynomial function of p1 p q we prove that the matrix hat p p which characterizes the set open image in new window as it is positive semi definite only forp open image in new window can be determined as a solution of a canonical differential equation which for every compact coregular linear group depends only on the numberq and on the degrees of the elements of the minimal integrity bases this allows to determine all the isomorphism classes of the orbit spaces of the compact coregular linear groups through a determination of the equivalence classes of the corresponding matrices hat p p forq 3 orbit spaces with dimensions 3 the solutions hat p p of the canonical equation are explicitly determined and the number of their equivalence classes is shown to be finite it is also shown that with a convenient choice of the minimal integrity basis the polynomial matrix elements of hat p p have only integer coefficients arguments are given in favour of the conjecture that our conclusions hold true for all values ofq our results are relevant and lead to universality properties in the physics of spontaneous symmetry breaking

removing index 0 fixed points for area preserving maps of two manifolds using the method of free modifications developed by m brown and extended to area preserving homeomorphisms we prove the following fixed point removal theorem theorem let h m m be an orientation preserving area preserving homeomorphism of an orientable two manifold m having an isolated fixed point p of index 0 given any open neighborhood n of p such that n fix h p there exists an area preserving homeomorphism h such that i h h on m n and ii h is fixed point free on n two applications of this theorem are the second fixed point for the topological version of the conley zehnder theorem on the two torus and a new proof of the second fixed point for the poincare birkhoff fixed point theorem

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

noncommutative rational functions invariant under the action of a finite solvable group abstract this paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in d generators these invariant skew fields are always finitely generated which contrasts with the free algebra case for abelian groups or solvable groups g with a well behaved representation theory it is shown that the invariant skew fields are free on g d 1 1 generators finally positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented

orbit spaces of low dimensional representations of simple compact connected lie groups and extrema of a group invariant scalar potential orbit spaces of low dimensional representations of classical and exceptional lie groups are constructed and tabulated we observe that the orbit spaces of some single irreducible representations adjoints second rank symmetric and antisymmetric tensors of classical lie groups and the defining representations of f4 and e6 are warped polyhedrons with locally more protrudent boundaries corresponding to higher level little groups the orbit spaces of two irreducible representations have different shapes we observe that dimension and concavity of different strata are not sharply distinguished we explain that the observed orbit space structure implies that a physical system tends to retain as much symmetry as possible in a symmetry breaking process in appendix a we interpret our method of minimization in the orbit space in terms of conventional language and show how to find all the extrema in the representation space of a general group invariant scalar potential monotonic in the orbit space we also present the criterion to tell whether an extremum is a local minimum or maximum or an inflection point in appendix b we show that the minimization problem can always be reduced to a two dimensional one in the case of the most general higgs potential for a single irreducible representation and to a three dimensional one in the case of an even degree higgs potential for two irreducible representations we explain that the absolute minimum condition prompts the boundary conditions enough to determine the representation vector

resonance and rotation numbers for planar hamiltonian systems multiplicity results via the poincare birkhoff theorem abstract in the general setting of a planar first order system 0 1 u g t u u r 2 with g 0 t r 2 r 2 we study the relationships between some classical nonresonance conditions including the landesman lazer one at infinity and in the unforced case i e g t 0 0 at zero and the rotation numbers of large and small solutions of 0 1 respectively such estimates are then used to establish via the poincare birkhoff fixed point theorem new multiplicity results for t periodic solutions of unforced planar hamiltonian systems j u u h t u and unforced undamped scalar second order equations x g t x 0 in particular by means of the landesman lazer condition we obtain sharp conclusions when the system is resonant at infinity

analyzing weighted ell_1 minimization for sparse recovery with nonuniform sparse models in this paper we introduce a nonuniform sparsity model and analyze the performance of an optimized weighted l1 minimization over that sparsity model in particular we focus on a model where the entries of the unknown vector fall into two sets with entries of each set having a specific probability of being nonzero we propose a weighted l1 minimization recovery algorithm and analyze its performance using a grassmann angle approach we compute explicitly the relationship between the system parameters the weights the number of measurements the size of the two sets the probabilities of being nonzero so that when i i d random gaussian measurement matrices are used the weighted l1 minimization recovers a randomly selected signal drawn from the considered sparsity model with overwhelming probability as the problem dimension increases this allows us to compute the optimal weights we demonstrate through rigorous analysis and simulations that for the case when the support of the signal can be divided into two different subclasses with unequal sparsity fractions the weighted l1 minimization outperforms the regular l1 minimization substantially we also generalize our results to signal vectors with an arbitrary number of subclasses for sparsity

nonautonomous nonlinear odes nonresonance conditions and rotation numbers abstract we deal with the t periodic problem associated with a nonlinear scalar differential equation like 1 x f t x 0 where for x 0 and x the nonlinearity f is assumed behave linearly with a time dependent coefficient we prove that the landesman lazer conditions at zero and at infinity possess a rotational effect on small and large in the phase plane solutions of 1 as a consequence we are able to generalize previous multiplicity results in the resonant case i e when the linearizations at zero and or at infinity are resonant through the use of the poincare birkhoff fixed point theorem

finite energy foliations of tight three spheres and hamiltonian dynamics surfaces of sections are a classical tool in the study of 3 dimensional dynamical systems their use goes back to the work of poincare and birkhoff in the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions such a system is established for nondegenerate reeb flows on the tight 3 sphere by means of pseudoholomorphic curves the applications cover the nondegenerate geodesic flows on t1s2 rp3 via its double covering s3 and also nondegenerate hamiltonian systems in ir4 restricted to sphere like energy surfaces of contact type

quotients and invariants of as sets equipped with a finite group action using the geometric quotient of a real algebraic set by the action of a finite group g we construct invariants of g affine real algebraic varieties with respect to equivariant homeomorphisms with algebraic graph including additive invariants with values in z the construction requires to consider the wider category of as sets

adaptive rate reconstruction of time varying signals with application in compressive foreground extraction we propose and analyze an online algorithm for reconstructing a sequence of signals from a limited number of linear measurements the signals are assumed sparse with unknown support and evolve over time according to a generic nonlinear dynamical model our algorithm based on recent theoretical results for ell 1 ell 1 minimization is recursive and computes the number of measurements to be taken at each time on the fly as an example we apply the algorithm to online compressive video foreground extraction a problem stated as follows given a set of measurements of a sequence of images with a static background simultaneously reconstruct each image while separating its foreground from the background the performance of our method is illustrated on sequences of real images we observe that it allows a dramatic reduction in the number of measurements or reconstruction error with respect to state of the art compressive background subtraction schemes

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

sums of squares and moment problems in equivariant situations we begin a systematic study of positivity and moment problems in an equivariant setting given a reductive group g over r acting on an affine r variety v we consider the induced dual action on the coordinate ring r v and on the linear dual space of r v in this setting given an invariant closed semialgebraic subset k of v r we study the problem of representation of invariant nonnegative polynomials on k by invariant sums of squares and the closely related problem of representation of invariant linear functionals on r v by invariant measures supported on k to this end we analyse the relation between quadratic modules of r v and associated quadratic modules of the finitely generated subring r v g of invariant polynomials we apply our results to investigate the finite solvability of an equivariant version of the multidimensional k moment problem most of our results are specific to the case where the group g r is compact

large amplitude subharmonic oscillations for scalar second order differential equations with asymmetric nonlinearities abstract we provide sufficient conditions for the existence of subharmonic solutions for equations whose nonlinearity may have a different qualitative behaviour to the right and to the left of the origin the proofs are based on variational methods

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

inequalities defining orbit spaces the orbit space of a representation of a compact lie group has a natural semialgebraic structure we describe explicit ways of finding the inequalities defining this structure and we give some applications

on the arnold conjecture for weakly monotone symplectic manifolds we show the arnold conjecture concerning symplectic fixed points in the case that the symplectic manifold is weakly monotone and all the fixed points are non degenerate in particular the conjecture is true in dimension 2 4 6 if all the fixed points are non degenerate

fast and accurate algorithms for re weighted ell _ 1 norm minimization to recover a sparse signal from an underdetermined system we often solve a constrained l1 norm minimization problem in many cases the signal sparsity and recovery performance can be further improved by replacing the l1 norm with a weighted l1 norm without prior information about the signal s nonzero elements the procedure for selecting weights is iterative in nature common approaches update the weights at every iteration using the solution of a weighted l1 problem from the previous iteration this paper presents two homotopy based algorithms that efficiently solve reweighted l1 problems first we present an algorithm that quickly updates the solution of a weighted l1 problem as the weights change since the solution changes only slightly with small changes in weights we develop a homotopy algorithm that replaces old weights with new ones in a small number of computationally inexpensive steps second we propose an algorithm that solves a weighted l1 problem by adaptively selecting weights while estimating the signal this algorithm integrates the reweighting into every step along the homotopy path by changing the weights according to changes in the solution and its support allowing us to achieve a high quality signal reconstruction by solving a single homotopy problem we compare both algorithms performance in terms of reconstruction accuracy and computational complexity against state of the art solvers and show that our methods have smaller computational cost we also show that the adaptive selection of the weights inside the homotopy often yields reconstructions of higher quality

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

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

computational and statistical tradeoffs via convex relaxation modern massive datasets create a fundamental problem at the intersection of the computational and statistical sciences how to provide guarantees on the quality of statistical inference given bounds on computational resources such as time or space our approach to this problem is to define a notion of algorithmic weakening in which a hierarchy of algorithms is ordered by both computational efficiency and statistical efficiency allowing the growing strength of the data at scale to be traded off against the need for sophisticated processing we illustrate this approach in the setting of denoising problems using convex relaxation as the core inferential tool hierarchies of convex relaxations have been widely used in theoretical computer science to yield tractable approximation algorithms to many computationally intractable tasks in the current paper we show how to endow such hierarchies with a statistical characterization and thereby obtain concrete tradeoffs relating algorithmic runtime to amount of data

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

periodic solutions of duffing s equations with superquadratic potential abstract this paper is devoted to the study of harmonic and subharmonic solutions for the second order scalar nonlinear duffing s equation x g x p t x x where g and p are continuous functions with p bounded and periodic in the first variable and g satisfying the assumption g x sign x as x among other results we prove the existence of infinitely many harmonic and subharmonic solutions of any order p p t and if the potential g x of g x satisfies certain conditions of superquadratic growth at the new existence results can be applied to situations in which the more classical superlinear growth condition g x x as x is not satisfied in this manner various preceding theorems are improved and sharpened see the introduction for more details proofs are based on a generalized version of the poincare birkhoff twist theorem due to w ding

subharmonic hysteresis and period doubling bifurcations for a periodically driven laser the authors consider the laser rate equations for a periodically modulated class b laser the bifurcation diagram of the periodic solutions is analyzed both numerically and theoretically the bifurcation parameter is the amplitude of the periodic modulations concentration is on the period 1 and period 2 branches of solutions that emerge either from a limit point or from a period doubling bifurcation point the bifurcation diagrams of the driven laser are analyzed with and without dissipation it is seen that the laser oscillations become pulsating as their amplitude increases suggesting a new asymptotic analysis ofthe laser oscillations a large parameter proportional to the amplitude of the oscillations is identified andperiodic solutions using singular perturbation techniques are constructed equations for a poincare map are derived and an analytical expression of the first period doubling bifurcation is obtained

orbit spaces of compact coregular simple lie groups with 2 3 and 4 basic polynomial invariants effective tools in the analysis of invariant potentials the coregular real representations of the compact simple lie groups are classified and the equalities and inequalities defining their orbit spaces are determined in the case of 2 3 and 4 basic polynomial invariants the results are obtained making use of the classification of the complex coregular simple lie groups given by g w schwarz and of the determination of all the allowable metric matrices for q dimensional q 4 orbit spaces of compact coregular linear groups recently obtained by the present authors the results are used to determine in a rigorous way the minima of two so 10 and e6 invariant higgs potentials

a note on a modified version of the poincare birkhoff theorem abstract we show by constructing a suitable area preserving homeomorphism without fixed points that the twist condition considered at zero and at infinity in a recent modified version of the poincare birkhoff fixed point theorem cannot be interchanged

compressed sensing with prior information strategies geometry and bounds we address the problem of compressed sensing cs with prior information reconstruct a target cs signal with the aid of a similar signal that is known beforehand our prior information we integrate the additional knowledge of the similar signal into cs via ell 1 ell 1 and ell 1 ell 2 minimization we then establish bounds on the number of measurements required by these problems to successfully reconstruct the original signal our bounds and geometrical interpretations reveal that if the prior information has good enough quality ell 1 ell 1 minimization improves the performance of cs dramatically in contrast ell 1 ell 2 minimization has a performance very similar to classical cs and brings no significant benefits in addition we use the insight provided by our bounds to design practical schemes to improve prior information all our findings are illustrated with experimental results

erratum to generalizations of the poincare birkhoff theorem other results of 1 which rely on theorem 2 1 are correct as stated and follow from the result above a disk u in the covering space a is called positively returning for f if f u u and fn u t k u for some n k 0 where t is the generator of the infinite cyclic group of covering translations negatively returning disks are defined similarly the difference in the statement of the theorem above and the statement of theorem 2 1 of 1 is the additional requirement in item 3 that the positively and negatively returning disks in the covering space are lifts of disks in the annulus equivalently the positively and negatively returning disks in the covering space must be disjoint from their image under t k k 0 standard arguments show it suffices that t u u this additional hypothesis is satisfied in other results from 1 which use theorem 2 1 i do not know whether theorem 2 1 of 1 as originally stated is true or not

discriminants of cyclic homogeneous inequalities of three variables abstract in this article we study the structure of the cone of semidefinite forms it is a closed semialgebraic set but usually is not basic closed semialgebraic set a discriminant is a defining equation of an irreducible component of algebraic boundary of this cone we calculate discriminants using new tools characteristic variety and local cones a characteristic variety is a semialgebraic subset of a real projective variety on which the family of inequalities is essentially defined as linear functions local cone is a subcone of the psd cone which corresponds to a maximal ideal this theory works well for a family of polynomials which are invariant under an action of a finite group after we construct an abstract general theory we apply it to a family of cyclic homogeneous polynomials of three real variables of degree d we calculate some discriminants for d 3 4 5 and 6 and we show that this theory derives many new results

generalizations of the poincare birkhoff theorem in this article we prove some generalizations of the classical poincar birkhoff theorem on area preserving homeomorphisms of the annulus which satisfy a boundary twist condition the work of g d birkhoff on this theorem and its applications can be found in b1 b2 and chapter v of b3 a more modern treatment can be found in b n we prove a theorem for the open annulus a s1 x 0 1 since as we will see the theorem for the closed annulus can easily be obtained from this result for the open annulus however it is not immediately obvious what should be the analogue of the twist condition it turns out that the most general hypothesis and the most natural from the point of view of our proof involves the notion of positively and negatively returning disks for some lift of f to the covering space a r x 0 1 more precisely if f a a is a lift of f a a we will say that the e is a positively returning disk for f if there is an open disk u c a such that f u n u 0 and fp u n u k 0 forsome n k 0 here u k denotes the set x k t i x t e u thus u is disjoint from its image but under iteration by f returns to a positive translate of itself a negatively returning disk is defined similarly but with k 0 such that fl u n u 0 in section 2 we prove the following

when is a symplectic quotient an orbifold let k be a compact lie group of positive dimension we show that for most unitary k modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold the quotient of a unitary module of a finite group when k is connected we show that even a symplectomorphism to a linear symplectic orbifold does not exist our results yield conditions that preclude the symplectic quotient of a hamiltonian k manifold from being locally isomorphic to an orbifold as an application we determine which unitary su2 modules yield symplectic quotients that are z graded regularly symplectomorphic to a linear symplectic orbifold we similarly determine which unitary circle representations yield symplectic quotients that admit a regular diffeomorphism to a linear symplectic orbifold

a multiplicity result for periodic solutions of second order differential equations with a singularity abstract by the use of the poincare birkhoff fixed point theorem we prove a multiplicity result for periodic solutions of a second order differential equation where the nonlinearity exhibits a singularity of repulsive type at the origin and has linear growth at infinity our main theorem is related to previous results by rebelo 1996 1997 4 5 and rebelo and zanolin 1996 6 7 in connection with a problem raised by del pino et al 1992 1

free lines for homeomorphisms of the open annulus let h be a homeomorphism of the open annulus s 1 r isotopic to the identity and let h be a lift of h to the universal cover r x r without fixed point then we show that h admits a brouwer line which is a lift of a properly imbedded line joining one end to the other in the annulus or h admits a free essential simple closed curve

exploiting symmetries in sdp relaxations for polynomial optimization in this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite programming relaxations our special focus is on constrained problems especially when the symmetric group is acting on the variables in particular we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute the geometric quotient

tensor subalgebras and first fundamental theorems in invariant theory abstract let v be an n dimensional complex inner product space and let t t v t v be the mixed tensor algebra over v we characterize those subsets a of t for which there is a subgroup g of the unitary group u n such that a t g they are precisely the nondegenerate contraction closed graded subalgebras of t while the proof makes use of the first fundamental theorem for gl n c in the sense of weyl the characterization has as direct consequences first fundamental theorems for several subgroups of gl n c moreover a galois correspondence between linear algebraic subgroups of gl n c and nondegenerate contraction closed graded subalgebras of t is derived we also consider some combinatorial applications viz to self dual codes and to combinatorial parameters

on the dynamics of homology preserving homeomorphisms of the annulus we consider the homeomorphisms of the compact annulus a s 1 x 1 1 isotopic to the symmetry s a which interchanges the two boundary components we prove that if such a homeomorphism is in some sense conservative and twisted then it possesses a periodic orbit of period exactly two this can be regarded as a counterpart of the poincare birkhoff theorem in the isotopy class of sa

flat bases of invariant polynomials and p matrices of e7 and e8 let g be a compact group of linear transformations of a euclidean space v the g invariant c functions can be expressed as c functions of a finite basic set of g invariant homogeneous polynomials sometimes called an integrity basis the mathematical description of the orbit space v g depends on the integrity basis too it is realized through polynomial equations and inequalities expressing rank and positive semidefiniteness conditions of the p matrix a real symmetric matrix determined by the integrity basis the choice of the basic set of g invariant homogeneous polynomials forming an integrity basis is not unique so it is not unique the mathematical description of the orbit space too if g is an irreducible finite reflection group saito et al commun algebra 8 373 1980 characterized some special basic sets of g invariant homogeneous polynomials that they called flat they also found explicitly the flat basic sets of invariant homogeneous polynomials of all the irreducible finite reflection g

periodic motions of fluid particles induced by a prescribed vortex path in a circular domain abstract by means of a generalized version of the poincare birkhoff theorem we prove the existence and multiplicity of periodic solutions for a hamiltonian system modeling the evolution of advected particles in a two dimensional ideal fluid inside a circular domain and under the action of a point vortex with prescribed periodic trajectory

a theorem on orbit structures strata of compact linear lie groups we present a comprehensive constructive proof of a theorem characterizing the tangent space to a stratum orbit structure of the euclidean space rn seat of an orthogonal representation of a compact group g the characterization is made in terms of gradients of a complete set integrity basis of g invariant polynomials in a recent paper m abud and g sartori phys lett b 104 147 1981 the theorem which may be considered a generalization of a theorem by michel c r acad sci ser a 272 433 1971 has been shown to be effective in the determination of the equations of the strata and in the determination of natural extrema of g invariant functions

periodic solutions of superlinear impulsive differential equations a geometric approach abstract a geometric method is introduced to study superlinear second order differential equations with impulsive effects basing on a reference continuous polar lifting of a planar orientation preserving homeomorphism we prove via the poincare birkhoff twist theorem the existence of infinitely many periodic solutions of conservative superlinear second order equations with the finite twist impulsive terms we also prove by developing a new twist fixed point theorem the existence of periodic solutions for non conservative superlinear second order equations with degenerate impulsive terms

solutions of the cohomological equation for area preserving flows on compact surfaces of higher genus let f be a local homeomorphism of the plane with a fixed point z which is a locally maximal invariant set and which is neither a sink nor a source we prove that there are two integers q 1 and r 1 such that the sequence i fk z of the indices at z of the iterates of f satisfy i fk z 1 rq if k is a multiple of q and i fk z 1 otherwise as a corollary we deduce that there is no minimal homeomorphism on the infinite annulus or more generally on the two dimensional sphere minus a finite set of points we also construct for a local homeomorphism f as above a topological invariant which is a cyclically ordered set with an automorphism on it this allows us in particular to define a rotation number for f rational of denominator q

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

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

an improvement of the poincare birkhoff fixed point theorem if g is a twist homeomorphism of an annulus a in the plane which leaves at most one point in the interior of a fixed then there is an essential simple closed curve in the interior of a which meets its image in at most one point hence the annular region bounded by this simple closed curve and the inside component of the boundary of a is mapped onto either a proper subset or a proper superset of itself

destruction of invariant curves in the restricted circular planar three body problem by using comparison of action the classical principle of least action says that orbits of mechanical systems extremize action an important subclass are those orbits that minimize action in this paper we utilize this principle along with aubry mather theory to construct birkhoff regions of instability for a certain three body problem given by a hamiltonian system of 2 degrees of freedom we believe that these methods can be applied to construct instability regions for a variety of hamiltonian systems with 2 degrees of freedom the hamiltonian model we consider describes dynamics of a sun jupiter comet system and under some simplifying assumptions we show the existence of instabilities for the orbit of the comet in particular we show that a comet which starts close to an orbit in the shape of an ellipse of eccentricity e 0 66 can increase in eccentricity up to e 0 96 in the sequels to this paper we extend the result to beyond e 1 and show the existence of ejection orbits such orbits are initially well within the range of our solar system this might give an indication of why most objects rotating around the sun in our solar system have relatively low eccentricity

maslov index poincare birkhoff theorem and periodic solutions of asymptotically linear planar hamiltonian systems abstract we study the relationship between the twist condition in the poincare birkhoff fixed point theorem and the assumptions on the maslov index for asymptotically linear planar hamiltonian systems for this aim we develop a variant of the poincare birkhoff theorem which together with its classical version allows to obtain a lower bound for the number of nontrivial periodic solutions in terms of the gap between the maslov indexes associated to the linearizations of the planar system at zero and at infinity