saving the logarithmic factor in the error term estimates of some congruence problems combining previous ideas from garaev and the first author we prove a general theorem to estimate the number of elements of a subset a of an abelian group g mathbb z n 1 times cdots times mathbb z n k lying in a k dimensional box in many cases this approach allow us to improve by a logarithm factor the range where it is possible to obtain an asymptotic estimate for the number of solutions of a given congruence

asymptotic stability of the stationary solution to an initial boundary value problem for the mullins equation of fourth order in the present article we first study the existence of the stationary solution to an initial boundary value problem for the mullins equation of fourth order which was proposed by mullins two dimensional motion of idealized grain boundaries j appl phys 27 900 1956 to describe the groove development due to the surface diffusion at the grain boundaries of a heated polycrystal then employing an energy method we prove that this stationary solution is asymptotically stable in a suitable norm as time goes to infinity

geodesic connectedness of multiwarped spacetimes abstract a new technique for the study of geodesic connectedness in a class of lorentzian manifolds is introduced it is based on arguments of brouwer s topological degree for the solution of functional equations it is shown to be very useful for multiwarped spacetimes which include different types of relativistic spacetimes connectedness by causal geodesics is also proved

hyperelliptic szpiro inequality we generalize the classical szpiro inequality to the case of a semistable family of hyperelliptic curves we show that for a semistable symplectic lefschetz fibration of hyperelliptic curves of genus g the number n of nonseparating vanishing cycles and the number d of singular fibers satisfy the inequality n 4g 2 d

on the discriminant of a hyperelliptic curve the minimal discriminant of a hyperelliptic curve is defined and used to generalize much of the arithmetic theory of elliptic curves over number fields this leads to a higher genus version of szpiro s conjecture analytically the discriminant is shown to be related to siegel modular forms of higher degree

second variation of zhang s λ invariant on the moduli space of curves we compute the second variation of the invariant recently introduced by s zhang on the complex moduli space mg of curves of genus g 2 using work of n kawazumi as a result we prove that 8g 4 is equal up to a constant to the invariant introduced some time ago by r hain and d reed we deduce some consequences for example we calculate the invariant for each hyperelliptic curve expressing it in terms of the petersson norm of the discriminant modular form 1 introduction recently independently s zhang 27 and n kawazumi 16 introduced a new interesting real valued function on the moduli space mg of complex curves of genus g 2 its value at a curve x m g is given as follows let h 0 x x be the space of holomorphic differentials on x equipped with the

symmetric roots and admissible pairing using the discriminant modular form and the noether formula it is possible to write the admissible self intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum over all places of a certain adelic invariant we provide a simple geometric interpretation for this invariant based on the arithmetic of symmetric roots we propose the conjecture that the invariant coincides with the invariant introduced in a recent paper by s w zhang this conjecture is true in the genus 2 case and we obtain a new proof of the bogomolov conjecture for curves of genus 2 over number fields

evolution of phase boundaries by configurational forces an initial boundary value problem modeling the evolution of phase interfaces in materials showing martensitic transformations is studied the model which is derived rigorously from a sharp interface model with phase interfaces driven by configurational forces and which generalizes that model consists of the equations of linear elasticity coupled with a nonlinear partial differential equation of hyperbolic character governing the evolution of the order parameter it is proved that in one space dimension global solutions exist for which the order parameter belongs to the space of functions of bounded variation other models for interface motion by martensitic transformations and by interface diffusion are suggested

viscosity solutions to a new phase field model with neumann boundary condition for solid solid phase transitions abstract we shall study an initial boundary value problem of a new phase field model which is a degenerate parabolic equation coupled to a linear elasticity sub system used to describe the solid solid phase transitions in elastically deformable solid materials we establish a series of approximate solutions to the initial boundary value problem and prove that the viscosity solutions to this initial boundary value problem exist in a one dimensional case

examples of torsion points on genus two curves we describe a method that sometimes determines all the torsion points lying on a curve of genus two defined over a number field and embedded in its jacobian using a weierstrass point as base point we then apply this to the examples y2 x5 x y2 x5 5x3 x and y2 y x5

jacobian nullwerte and algebraic equations abstract we present two applications of jacobian nullwerte both related with the resolution of algebraic equations of any degree we give a very simple expression of the roots of a polynomial of arbitrary degree in terms of derivatives of hyperelliptic theta functions this expression can be understood as an explicit proof of torelli s theorem in the hyperelliptic case we also give geometrical expressions of the discriminant of a polynomial both applications are based on a jacobian version of thomae s formula

on a model for phase separation in binary alloys driven by mechanical effects this work is concerned with the mathematical analysis of a system of partial differential equations modeling the effect of phase separation driven by mechanical actions in binary alloys like tin lead solders the system combines the quasistationary balance of linear momentum with a fourth order evolution equation of cahn hilliard type for the phase separation and it is highly nonlinearly coupled existence and uniqueness results are shown tel 0382505631 email bonetti dimat unipv it

modeles entiers des courbes hyperelliptiques sur un corps de valuation discrete let c be a hyperelliptic curve of genus g 1 over a discrete valuation field k in this article we study the models of c over the ring of integers ok of k to each weierstrass model that is a projective model arising from a hyperelliptic equation of c with integral coefficients one can associate a valuation of discriminant then we give a criterion for a weierstrass model to have minimal discriminant we show also that in the most cases the minimal regular model of c over ok dominates every minimal weierstrass model some classical facts concerning weierstrass models over ok of elliptic curves are generalized to hyperelliptic curves and some others are proved in this new setting soit k un corps de valuation discrete d anneau de valuation ok soit c une courbe hyperelliptique de genre g 1 sur k nous entendons par la que c est propre lisse et qu il existe un morphisme c pk de degre 2 on etudie dans ce travail les modeles de c sur ok c est a dire des schemas normaux propres et plats sur ok a fibres generiques isomorphes a c plus precisement on s interesse aux modeles de weierstrass de c sur ok ce sont des modeles provenant des equations hyperelliptiques de c a coefficients dans ok def 6 on etudie aussi la relation entre les modeles de weierstrass et le modele regulier minimal de c sur ok pour tout modele de weierstrass de c sur ok il y a une notion evidente du discriminant 4 3 le discriminant minimal de c est un invariant particulierement important dans l etude de l arithmetique de c comme en temoignent les travaux sur les courbes elliptiques d ailleurs ce travail generalise certains resultats bien connus sur les modeles de weierstrass entiers des courbes elliptiques dans les trois premiers paragraphes on rappelle la notion du discriminant et on definit les equations minimales et le discriminant minimal de c on montre de facon tout a fait elementaire que sur un anneau principal c admet une equation minimale globale prop 2 la notion du discriminant est liee au choix d une involution hyperelliptique sur c si g 2 est unique par contre si g 1 cela n est plus vrai dans 1 on etudie l ensemble des classes de conjugaison par les automorphismes interieurs de autk c des involutions hyperelliptiques de c et on donne une estimation de received by the editors august 22 1995 1991 mathematics subject classification primary 11g20 14h25 secondary 14g20

closed geodesies on compact lorentzian manifolds of splitting type in this paper we consider the problem of the existence of a spacelike closed geodesic on compact lorentzian manifolds tipler and galloway proved that under suitable topological properties of the manifold there exists a closed timelike geodesic in their proofs they use the hypothesis that the time coordinate of one timelike geodesic has derivative always different from zero this clearly fails for spacelike geodesies using variational methods and applying the relative category theory we prove the existence of a closed spacelike geodesic on a compact manifold of splitting type observe that thanks to the previous results the existence of at least two geometrically distinct closed geodesies on follows

on a class of geodesically connected lorentzian manifolds in this paper we study the geodesical connectedness of lorentzian manifolds we consider a connected manifold m m0 r where m0is a complete riemannian manifold endowed with a lorentzian metricgof splitting type we prove that under suitable hypotheses on the coefficients of the metricg mis geodesically connected

new explicit formulas for faltings delta invariant in this paper we give new explicit formulas for faltings delta invariant in terms of integrals of theta functions and we deduce an explicit lower bound for delta only in terms of the genus and an explicit upper bound for the arakelov green function in terms of delta furthermore we give a canonical extension of delta and the zhang kawazumi invariant varphi to the moduli space of indecomposable principally polarised complex abelian varieties

solutions to a model with nonuniformly parabolic terms for phase evolution driven by configurational forces we prove the existence of solutions global in time to an initial boundary value problem for a system of partial differential equations which consists of the equations of linear elasticity and a nonlinear nonuniformly parabolic equation of second order the problem models the behavior in time of materials with martensitic phase transformations this model with diffusive phase interfaces was derived from a model with sharp interfaces whose evolution is driven by configurational forces and can be considered to be a regularization of that model our existence proof which contributes to the verification of the model is only valid in one space dimension

regularity of solutions to a model for solid solid phase transitions driven by configurational forces abstract in a previous work we prove the existence of weak solutions to an initial boundary value problem with h 1 initial data for a system of partial differential equations which consists of the equations of linear elasticity and a nonlinear degenerate parabolic equation of second order this problem models the behavior in time of materials with martensitic phase transitions this model with diffusive phase interfaces was derived from a model with sharp interfaces whose evolution is driven by configurational forces and can be regarded as a regularization of that model assuming in this article the initial data is in h 2 we investigate the regularity of weak solutions that is difficult due to the gradient term which plays a role of a weight our proof in which the difficulties are caused by the weight in the principle term is only valid in one space dimension

solvability via viscosity solutions for a model of phase transitions driven by configurational forces abstract this article is concerned with an initial boundary value problem for an elliptic parabolic coupled system arising in martensitic phase transition theory of elastically deformable solid materials e g steel this model was proposed in alber and zhu 2007 4 and investigated in alber and zhu 2006 3 the existence of weak solutions which are defined in a standard way however the key technique used in alber and zhu 2006 3 is not applicable to multi dimensional problem one of the motivations of this study is to solve this multi dimensional problem and another is to investigate the sharp interface limits thus we define weak solutions in a way which is different from alber and zhu 2006 3 by using the notion of viscosity solution we do prove successfully the existence of weak solutions in this sense for one dimensional problem yet the multi dimensional problem is still open