real analyticity of solutions of hamilton s equation il existe des constantes positives cn et sn dependant seulement de la dimension n de la variete m telles que pour une metrique initiale g 0 avec r g 0 1 l equation de hamilton a une solution jusqu a t sn et l estimation suivante est vraie pour la metrique solution en t o t sn k 0 t k k 1 12 k r 2 c n

counting rational points on hypersurfaces for any n geq 2 let f in mathbb z x 1 ldots x n be a form of degree d geq 2 which produces a geometrically irreducible hypersurface in mathbb p n 1 this paper is concerned with the number n f b of rational points on f 0 which have height at most b for any varepsilon 0 we establish the estimate n f b o b n 2 varepsilon whenever either n leq 5 or the hypersurface is not a union of lines here the implied constant depends at most upon d n and varepsilon

counting rational points on algebraic varieties for any n 2 let z pn be a geometrically integral algebraic variety of degree d this article is concerned with the number nz b of q rational points on z which have height at most b for any e 0 we establish the estimate nz b od e n bdimz e provided that d e as indicated the implied constant depends at most on d e and n

on the fundamental groups of manifolds of positive sectional curvature let m be an n manifold with sectional curvature 0 6 k 1 the main result asserts that the fundamental group of m has a finite normal cyclic subgroup with index less than w n 8 oc a constant depending on n and 8

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

amplitude variation with offset inversion using the reflectivity method abstractmost existing amplitude variation with offset avo inversion methods are based on the zoeppritz s equation or its approximations these methods assume that the amplitude of seismic data depends only on the reflection coefficients which means that the wave propagation effects such as geometric spreading attenuation transmission loss and multiples have been fully corrected or attenuated before inversion however these requirements are very strict and can hardly be satisfied under a 1d assumption reflectivity method based inversions are able to handle transmission losses and internal multiples applications of these inversions however are still time consuming and complex in computation of differential seismograms we have evaluated an inversion methodology based on the vectorized reflectivity method in which the differential seismograms can be calculated from analytical expressions it is computationally efficient a modification is implemented to transform the inversion from the interce

explicit uniform estimation of rational points ii hypersurface coverings by using the slope method we obtain an explicit uniform estimation for the density of rational points in an arithmetic projective variety with given degree and dimension embedded in a given arithmetic projective space

the number of integral points on arcs and ovals integral lattice points and that the exponent and constant are best possible however swinnerton dyer 10 showed that the preceding result can be substantially improved if we start with a fixed c strictly convex arc and consider the number of lattice points on t the dilation of by a factor t t 1 this of course is the same as asking for rational points mn n n on as n in fact swinnerton dyer proves a bound of type t zz c e t 3 5 e

integral points on elliptic curves and 3 torsion in class groups we give new bounds for the number of integral points on elliptic curves the method may be said to interpolate between approaches via diophantine techniques bp hbr and methods based on quasi orthogonality in the mordell weil lattice sil6 gs he we apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the 3 torsion part of the class group of a quadratic field the same ideas can be used to count rational points on curves of higher genus

the density of rational points on curves and surfaces let c be an irreducible projective curve of degree d in mathbb p 3 defined over overline mathbb q it is shown that c has o varepsilon d b 2 d varepsilon rational points of height at most b for any varepsilon 0 uniformly for all curves c this result extends an estimate of bombieri and pila duke math j 59 1989 337 357 to projective curves for a projective surface s in mathbb p 3 of degree d ge 3 it is shown that there are o varepsilon d b 2 varepsilon rational points of height at most b of which at most o varepsilon d b 52 27 varepsilon do not lie on a rational line in s for non singular surfaces one may reduce the exponent to 4 3 16 9d for d 4 or 5 or max 1 3 sqrt d 2 d 1 for d ge 6 even for the surface x 1 d x 2 d x 3 d x 4 d this last result improves on the previous best known as a further application it is shown that almost all integers represented by an irreducible binary form f x y in mathbb z x y have essentially only one such representation this extends a result of hooley j reine angew math 226 1967 30 87 which concerned cubic forms only the results are not restricted to projective surfaces and as an application of other results in the paper it is shown that x 1 x 2 x 3 in mathbb n 3 x 1 d x 2 d x 3 d n ll varepsilon d n theta d varepsilon with theta frac 2 sqrt d frac 2 d 1 when d ge 8 this provides the first non trivial bound for the number of representations as a sum of three d th powers

the almost cyclicity of the fundamental groups of positively curved manifolds recall that a pure f structure is a kind of generalized torus action the main result asserts that if a compact positively curved manifold mn admits an invariant pure f structure such that each orbit has positive dimension then the fundamental group has a finite cyclic subgroup with index less than wn a constant depending only on n as an application we conclude that for all 0 1 the fundamental group of a pinched n manifold either has a cyclic subgroup with index less than wn or has order less than w n a constant depending only on n and in particular this substantially improves the main result in ro1

rational points of bounded height on projective surfaces we give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces

the density of rational points on non singular hypersurfaces ii for any integers d n geq 2 let x subset mathbb p n be a non singular hypersurface of degree d that is defined over the rational numbers the main result in this paper is a proof that the number of rational points on x which have height at most b is o b n 1 varepsilon for any varepsilon 0 the implied constant in this estimate depends at most upon d varepsilon and n

on the density of rational and integral points on algebraic varieties let x n be a projective geometrically integral variety over of dimension r and degree d 4 suppose that there are only finitely many r 1 planes over on x the main result of this paper is a proof of the fact that the number n x b of rational points on xwhich have height at most b satisfies for any 0 the implied constant depends at most on d n and

deforming the metric on complete riemannian manifolds soit m g ij x une variete de riemann a n dimensions complete non compacte de tenseur de complexe riemannien r ijkl satisfaisant r ijkl 2 k 0 sur m ou 0 0 dependant seulement de n et de k 0 telle que l equation d evolution g ij x t t 2r ij x t sur m g ij x o g ij x x m ait une solution lisse g ij x t 0 pour un temps court 0 t t u k 0 et satisfait l estimation suivante pour un entier m 0 il existe des constantes c m 0 dependant seulement de n m et k 0 telles que sup x m v m r ijkl x t 2 c m t m 0 t t u k 0

nilpotent structures and invariant metrics on collapsed manifolds let mn be a complete riemannian manifold of bounded curvature say iki 0 we put mn w n c u fn c where 1n e consists of those points at which the injectivity radius of the exponential map is e the complementary set n c is called the e collapsed part of mn if x e 4 n r e then the metric ball bx r is quasi isometric with small distortion to the flat ball b0 r in the euclidean space rn after slightly

collapsing riemannian manifolds to ones of lower dimensions on considere l effondrement d une suite de varietes de riemann mi vers un espace de dimension inferieure x on etudie le cas ou l espace limite x est une variete de riemann et les courbures sectionnelles de mi sont bornees

explicit uniform estimation of rational points i estimation of heights by using the slope method in arakelov geometry we study the complexity of the singular locus of an arithmetic projective variety and explicit estimations of the arithmetic hilbert samuel function