a refined conjecture of mazur tate type for heegner points in mt1 mazur and tate present a refined conjecture of birch and swinnerton dyer type for a modular elliptic curvee this conjecture relates congruences for certain integral homology cycles one c the modular symbols to the arithmetic ofe overq in this paper we formulate an analogous conjecture fore over a suitable imaginary quadratic fieldk in which the role of the modular symbols is played by heegner points a large part of this conjecture can be proved thanks to the ideas of kolyvagin on the euler system of heegner points in effect the main result of this paper can be viewed as a generalization of kolyvagin s result relating the structure of the selmer group ofe overk to the heegner points defined in the mordell weil groups ofe over ring class fields ofk an explicit application of our method to the galois module structure of heegner points is given in sect 2 2

critical dynamics and trees abstract we show that the dependence of the critical perturbation of hamiltonian systems on the frequency v is related to the statistics of large planar trees using a complex map for illustration a critical function k v is given by the radii of convergence of power series whose coefficients are represented as sums over trees a preliminary approximation based on this relation leads to an expression for the critical function as the solution of a transcendental equation which appears to provide a reasonable approximation to the true function and to have the right structure near the dense set of singularities

critical functions for complex analytic maps critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area preserving maps the author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps by means of the majorant series method some rigorous estimates are given for complex area preserving maps like the semistandard map and the modulated singular map in particular the author makes use of the brjuno function to interpolate critical maps and proves that the convergence of the brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number the author also discusses the optimality of the rigorous bounds obtained

mixed fractional heston model and the pricing of american options this paper presents a fractional version of the heston model in which the volatility brownian and price brownian are replaced by mixed fractional brownian motions with hurst parameter h 34 1 so that the model exhibits a long range dependence then the existence and uniqueness of solution of mixed fractional heston model are discussed as well as the error of an euler scheme applied on this model finally some numerical illustrations are given in the last section by computing american put option prices

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

on the existence and uniqueness of the solution to the double heston model equation and valuing lookback option abstract in this work we study the existence and uniqueness of the solution to the stochastic differential equation of the double heston model which is defined by two independent variance processes with non lipschitz diffusions besides we present a monte carlo algorithm based on the euler discretization method to price the lookback options under this model

heegner points on mumford tate curves 1 shimura curves 417 2 heegner points 429 3 the regulator term 441 4 the conjecture 445 5 applications and re nements 447

derived heights and generalized mazur tate regulators 2 algebraic preliminaries 4 2 1 notations and conventions 4 2 2 duality 5 2 2 1 local duality 5 2 2 2 global duality 6 2 3 assumptions 8 2 4 preliminary calculations 10

critical kam circles and the brjuno function the authors consider a complex hamiltonian map and show how to approximate the critical value of the perturbation parameter at which a given kam circle disappears by means of a function the brjuno function which only depends on the continued fraction expansion of the rotation number

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

hamiltonian maps in the complex plane abstract the standard map is a nonintegrable discrete time analog of the vertical pendulum detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency the functional dependence of the coordinate q on the canonical angle variable is analtyically continued into the complex plane where natural boundaries are found at constant absolute values of im the boundaries represent the appearance of chaotic motion in the complex plane when the domain of analyticity shrinks to zero the kam invariant curve is destroyed two independent numerical methods with fourier analysis in the angle variable were used one based on a variation annihilation method and the other on a double expansion the results were further checked by direct solution of the complex equations of motion the numerically simpler but intrinsically complex semipendulum and semistandard map are also studied we conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or kam surfaces of general nonintegrable systems with analytic hamiltonians

bond pricing under mixed generalized cir model with mixed wishart volatility process this paper proposes an extended version of the coxingersollross cir model with stochastic volatility and a pricing method on zero coupon bond under this model in this version we replace the standard brownian motion process with a semi martingale process named the mixed fractional brownian motion mfbm process which is a linear combination of a fractional brownian motion fbm and a standard brownian motion we assume that the part of the volatility process follows a mixed wishart process which defines by the square of the matrix valued mfbm process in order to evaluate the price of the zero coupon bond under the proposed model we use monte carlo simulation method the computed values of the zero coupon bond compare with the other interest rate models

breakup of invariant tori for the four dimensional semi standard map abstract we compute the domain of existence of two dimensional invariant tori with fixed frequency vectors for a four dimensional complex symplectic map the map is a generalization of the semi standard map studied by greene and percival it has three parameters a 1 and a 2 representing the strength of the kicks in each degree of freedom and the coupling the domain of existence of a torus in a 1 a 2 is shown to be complete and log convex for fixed k a 1 a 2 explicit bounds on the domain for fixed k are obtained numerical results show that quadratic irrationals can be more robust than the cubic irrational the spiral mean

a method for accurate stability bounds in a small denominator problem the author considers the problem of obtaining realistic lower bounds for the siegel radius recent advances of the analysis of siegel disks allows one to give a very accurate numerical algorithm based on rigorous results he finds that for non quadratic polynomial maps the maximal siegel radius might correspond to rotation numbers different from the golden mean

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

on the passage from local to global in number theory would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase in number theory the distinction local versus global with various connotations has found a home in almost every part of mathematics local problems being often a stepping stone to the more difficult global problems to illustrate what the earthy geometric terms local and global signify in number theory let us consider any diophantine equation say

on the kolyvagin cup product we define a new cohomological operation which we call the kolyvagin cup product that is a generalization of the derivative operator introduced by kolyvagin in his work on euler systems we show some of the basic properties of this operation we also define a higher dimensional derivative in certain cases and a dual operation which we call the kolyvagin cap product and which generalizes a computation of rubin

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the frequency

recent developments in the theory of hamiltonian systems area preserving mappings of an annulus occur as poincare mappings of hamiltonian systems they were studied extensively by g d birkhoff recently aubry and mather investigated the subclass of so called monotone twist mappings for which they constructed independently closed invariant cantor sets their work led to important new results their theory is discussed and related to hamiltonian systems satisfying a legendre condition the connection of this theory with the stability problem kam theory and in particular the disintegration of invariant tori is discussed various constructions of mather sets are explained in which minimal solutions of variational problems play a central role this theory has a close relation to the differential geometric investigations by morse and hedlund on geodesics on two dimensional surfaces