non isochronicity of the center at the origin in polynomial hamiltonian systems with even degree nonlinearities abstract in 2002 x jarque and j villadelprat proved that no center in a planar polynomial hamiltonian system of degree 4 is isochronous and raised a question is there a planar polynomial hamiltonian system of even degree which has an isochronous center in this paper we give a criterion for non isochronicity of the center at the origin of planar polynomial hamiltonian systems moreover the orders of weak centers are determined our results answer a weak version of the question proving that there is no planar polynomial hamiltonian system with only even degree nonlinearities having an isochronous center at the origin

a criterion of integrability for perturbed nonresonant harmonic oscillators wick ordering of the perturbations in classical mechanics and invariance of the frequency spectrum we introduce an analogue to the renormalization theory of quantum fields into classical mechanics we also find an integrability criterion guaranteeing the convergence of birkhoff s series and an algorithm for modifying the hamiltonian to fix the frequency spectrum of the quasi periodic motions we point out its possible relevance to the transition to chaos

towards the homotopy of the k 2 local moore spectrum at p 2 abstract let v 0 be the mod 2 moore spectrum and let c be the supersingular elliptic curve over f 4 defined by the weierstrass equation y 2 y x 3 let f c be its formal group law and e c be the spectrum classifying the deformations of f c the group of automorphisms of f c which we denote by s c acts on e c further s c admits a surjective homomorphism to z 2 whose kernel we denote by s c 1 the cohomology of s c 1 with coefficients in e c v 0 is the e 2 term of a spectral sequence converging to the homotopy groups of e c h s c 1 v 0 a spectrum closely related to l k 2 v 0 in this paper we use the algebraic duality resolution spectral sequence to compute an associated graded for h s c 1 e c v 0 these computations rely heavily on the geometry of elliptic curves made available to us at chromatic level 2

the adams novikov e_2 term for pi_ l_2s 0 at the prime 2 in this paper we determine the e 2 term of the adams novikov spectral sequence converging to the homotopy groups pi l 2s 0 of l 2 localized sphere spectrum at the prime 2 the structure of the e 2 term indicates that the homotopy group pi i l 2s 0 is finite except for i 0 4 and 5

the period function for hamiltonian systems with homogeneous nonlinearities abstract the paper deals with hamiltonian systems with homogeneous nonlinearities we prove that such systems have no isochronous centers that the period annulus of any of its centres is either bounded or the whole plane and that the period function associated to the origin has at most one critical point

on the period function of x f x x 2 g x 0 abstract we study the period function t of a center o of the title s equation a sufficient condition for the monotonicity of t or for the isochronicity of o is given such a condition is also necessary when f and g are odd and analytic in this case a characterization of isochronous centers is given some classes of plane systems equivalent to such equation are considered including some kukles systems

period function for a class of hamiltonian systems abstract this paper studies the period function of the class of hamiltonian systems x h y y h x where h x y has the special form h x y f x g y and the origin is a non degenerate center more concretely if t h denotes the period of the periodic orbit contained in h x y h we solve the inverse problem of characterizing all systems with a given function t h we also characterize the limiting behaviour of t at infinity when the origin is a global center and apply this result to prove among other results that there are no nonlinear polynomial isochronous centers in this family

the concept of duality for measure projections of convex bodies abstract we show that an involution t on some class of functions on r n which reverses order meaning that if f g then t f t g has often a very specific form actually essentially unique it is done in this paper for the class of s concave functions for which this unique formula is derived these functions are for integer s exactly marginals of convex bodies of dimension n s this understanding is also extended and discussed for other classes of functions and represents from our point of view the abstract description of the concept of duality

fixed points of polarity type operators abstract a well known result says that the euclidean unit ball is the unique fixed point of the polarity operator this result implies that if in r n the unit ball of some norm is equal to the unit ball of the dual norm then the norm must be euclidean motivated by these results and by relatively recent results in convex analysis and convex geometry regarding various properties of order reversing operators we consider in a real hilbert space setting a more general fixed point equation in which the polarity operator is composed with a continuous invertible linear operator we show that if the linear operator is positive definite then the considered equation is uniquely solvable by an ellipsoid otherwise the equation can have several possibly infinitely many solutions or no solution at all our analysis yields a few by products of possible independent interest among them results related to coercive bilinear forms essentially a quantitative convex analytic converse to the celebrated lax milgram theorem from partial differential equations and a characterization of real hilbertian spaces

the concept of duality in convex analysis and the characterization of the legendre transform in the main theorem of this paper we show that any involution on the class of lower semi continuous convex functions which is order reversing must be up to linear terms the well known legendre transform

zone diagrams in euclidean spaces and in other normed spaces zone diagrams are a variation on the classical concept of voronoi diagrams given n sites in a metric space that compete for territory the zone diagram is an equilibrium state in the competition formally it is defined as a fixed point of a certain dominance map asano matousek and tokuyama proved the existence and uniqueness of a zone diagram for point sites in the euclidean plane and reem and reich showed existence for two arbitrary sites in an arbitrary metric space we establish existence and uniqueness for n disjoint compact sites in a euclidean space of arbitrary finite dimension and more generally in a finite dimensional normed space with a smooth and rotund norm the proof is considerably simpler than that of asano et al we also provide an example of non uniqueness for a norm that is rotund but not smooth finally we prove existence and uniqueness for two point sites in the plane with a smooth but not necessarily rotund norm

correction and linearization of resonant vector fields and diffeomorphisms we extend the classical siegel brjuno russmann linearization theorem to the resonant case by showing that under a d brjuno s diophantine condition any resonant local analytic vector field resp diffeomorphism possesses a well defined correction which 1 depends on the chart but in any given chart is unique 2 consists solely of resonant terms and 3 has the property that when substracted from the vector field resp when factored out of the diffeomorphism the vector field or diffeomorphism thus corrected becomes analytically linearizable with a privileged or canonical linearizing map moreover in spite of the small denominators and contrary to a hitherto prevalent opinion the correction s analyticity can be established by pure combinatorics without any analysis

isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials in this work we study isochronous centers of two dimensional autonomous system in the plane with linear part of center type and nonlinear part given by homogeneous polynomials of fifth degree a complete classification of the necessary conditions for the time reversible systems of this class is given in order to have an isochronous center at the origin an open problem is stated for the sufficient conditions moreover we find two nonreversible isochronous families from the center cases known all the computations in order to obtain necessary conditions for such isochronous centers are given in polar coordinates and we give a proof of the isochronicity of these systems by using different methods

isochronicity for several classes of hamiltonian systems abstract in this paper we study isochronous centers of analytic hamiltonian systems giving special attention to the polynomial case we first revisit the potential systems and we show the connection between isochronicity and involutions we then study a more general system namely the ones associated to hamiltonians of the form h x y a x b x y c x y 2 as an application we classify the cubic polynomial hamiltonian isochronous centers and we give examples of nontrivial and nonglobal polynomial hamiltonian isochronous centers

bifurcation of critical periods for plane vector fields on etudie un probleme de bifurcation dans des familles de champs de vecteurs analytiques plans qui ont un centre non degenere a l origine pour toutes les valeurs d un parametre r n on definit la fonction periode p on determine les points critiques de cette fonction pres du centre avec comme parametre de bifurcation

the distance trisector curve abstract given points p and q in the plane we are interested in separating them by two curves c 1 and c 2 such that every point of c 1 has equal distance to p and to c 2 and every point of c 2 has equal distance to c 1 and to q we show by elementary geometric means that such c 1 and c 2 exist and are unique moreover for p 0 1 and q 0 1 c 1 is the graph of a function f r r c 2 is the graph of f and f is convex and analytic i e given by a convergent power series at a neighborhood of every point we conjecture that f is not expressible by elementary functions and in particular not algebraic we provide an algorithm that given x r and e 0 computes an approximation to f x with error at most e in time polynomial in log 1 x e the separation of two points by two trisector curves considered here is a special two point case of a new kind of voronoi diagram which we call the zone diagram and which we investigate in a companion paper

weak centers and bifurcation of critical periods in reversible cubic systems abstract in this paper we investigate nonhomogeneous cubic differential systems with reversibility which guarantees that the systems have a center at the origin we apply the ritt wu method to process algebraic equations and inequalities using maple v 3 on a computer we give an inductive algorithm for computing the period coefficient polynomials we find the structure of solutions of systems of algebraic equations corresponding to isochronous centers and weak centers of every finite order and we derive conditions on the parameters under which the origin is an isochronous center or a weak center of finite order we show that with appropriate perturbations local bifurcation of critical periods will occur from weak centers of finite order and isochronous centers

the homotopy groups of se 2 at p 5 revisited abstract we present a new technique for analyzing the v 0 bockstein spectral sequence studied by shimomura and yabe employing this technique we derive a conceptually simpler presentation of the homotopy groups of the e 2 local sphere at primes p 5 we identify and correct some errors in the original shimomura yabe calculation we deduce the related k 2 local homotopy groups and discuss their manifestation of gross hopkins duality

a characterization of duality through section projection correspondence in the finite dimensional setting in this paper we show that the well known duality operation in the context of convex bodies in rn is completely characterized by its property of interchanging sections with projections our results are compared to results by boroczky schneider and artstein milman who showed that in many cases the property of order reversing is sufficient to determine a duality operation up to obvious linear modifications in fact we provide another result that recovers a known characterization of duality by the property of order reversing and up to a mild condition also a characterization of duality by interchanging sections by projections

linearization of isochronous centers abstract in this paper we study isochronous centers of polynomial systems we first discuss isochronous centers of quadratic systems cubic symmetric systems and reduced kukles system all these systems have rational first integrals we give a unified proof of the isochronicity of these systems by constructing algebraic linearizing changes of coordinates we then study two other classes of systems with isochronous centers namely the class of complex systems z ip z and the class of cubic systems symmetric with respect to a line and satisfying 1 both classes consist of darboux integrable systems we discuss their geometric properties and construct the linearizing changes of coordinates we show that the class of polynomial isochronous systems carries a very rich geometry finally we discuss the geometry of the linearizing changes of coordinates in the complex plane

isochronous centers of polynomial hamiltonian systems and a conjecture of jarque and villadelprat abstract we study the conjecture of jarque and villadelprat stating that every center of a planar polynomial hamiltonian system of even degree is nonisochronous this conjecture has already been proved for quadratic and quartic systems using the correction of a vector field to characterize isochronicity and explicit computations of this quantity for polynomial vector fields we are able to describe a very large class of nonisochronous hamiltonian systems of even arbitrarily large degree

isochronicity and linearizability of planar polynomial hamiltonian systems in this paper we study isochronicity and linearizability of planar polynomial hamiltonian systems first we prove a theorem which supports a negative answer to the following open question stated by jarque and villadelprat in 15 do there exist planar polynomial hamiltonian systems of even degree having an isochronous center additionally we obtain some conditions for linearizability of complex cubic hamiltonian systems

order preserving and order reversing operators on the class of convex functions in banach spaces abstract a remarkable result by s artstein avidan and v milman states that up to pre composition with affine operators addition of affine functionals and multiplication by positive scalars the only fully order preserving mapping acting on the class of lower semicontinuous proper convex functions defined on r n is the identity operator and the only fully order reversing one acting on the same set is the fenchel conjugation here fully order preserving reversing mappings are understood to be those which preserve reverse the pointwise order among convex functions are invertible and such that their inverses also preserve reverse such order in this paper we establish a suitable extension of these results to order preserving and order reversing operators acting on the class of lower semicontinuous proper convex functions defined on arbitrary infinite dimensional banach spaces

zone diagrams existence uniqueness and algorithmic challenge a zone diagram is a new variation of the classical notion of the voronoi diagram given points sites mathbf p 1 ldots mathbf p n in the plane each mathbf p i is assigned a region r i but in contrast to the ordinary voronoi diagrams the union of the r i has a nonempty complement the neutral zone the defining property is that each r i consists of all mathbf x in mathbb r 2 that lie closer nonstrictly to mathbf p i than to the union of all the other r j j ne i thus the zone diagram is defined implicitly by a fixed point property and neither its existence nor its uniqueness seem obvious we establish existence using a general fixed point result a consequence of schauder s theorem or kakutani s theorem this proof should generalize easily to related settings say higher dimensions then we prove uniqueness of the zone diagram as well as convergence of a natural iterative algorithm for computing it by a geometric argument which also relies on a result for the case of two sites in an earlier paper many challenging questions remain open

nonexistence of isochronous centers in planar polynomial hamiltonian systems of degree four abstract in this paper we study centers of planar polynomial hamiltonian systems and we are interested in the isochronous ones we prove that every center of a polynomial hamiltonian system of degree four that is with its homogeneous part of degree four not identically zero is nonisochronous the proof uses the geometric properties of the period annulus and it requires the study of the hamiltonian systems associated to a hamiltonian function of the form h x y a x b x y c x y 2 d x y 3