rational points on k3 surfaces a new canonical height a fundamental tenet of diophantine geometry is that the geometric properties of an algebraic variety should determine its basic arithmetic properties this is certainly true for curves where the sign of the euler characteristic of c determines whether the set of rational points on c is finite x c 0 for higher dimensional varieties there are some precise conjectures due to bombieri lang and vojta 15 which predict when the rational points on a variety should be finite or degenerate i e not zariski dense and some conjectures of manin et al 2 5 on the distribution of rational points in those cases when they are zariski dense but except for abelian varieties their subvarieties and some fano varieties varieties for which the anticanonical bundle is ample there are very few general theorems in this paper we will study the rational points on a certain class of k 3 surfaces defined over a number field k the moduli space of marked algebraic k3 surfaces is a countable union of 19 dimensional quasi projective varieties we are going to look at the 18 dimensional family studied by wehler 1 17 wehler s family consists of k3 surfaces s whose automorphism group aut s contains a subgroup d isomorphic to the free product 7 2 2 of two cyclic groups of order 2 we will use the geometric information provided by this infinite automorphism group to study the k rat ional points on s for any point pes we can look at the orbit of p under the action of d

existence of primitive divisors of lucas and lehmer numbers we prove that for n 30 every n th lucas and lehmer number has a primitive divisor this allows us to list all lucas and lehmer numbers without a primitive divisor

positive periodic solutions of first order delay differential equations with impulses using global bifurcation techniques we show the existence and multiplicity of positive periodic solutions for a class of first order delay differential equation with impulses

equidistribution quantitative des points de petite hauteur sur la droite projective we introduce a new class of adelic heights on the projective line we estimate their essential minimum and prove a result of equidistribution at every place for points of small height with estimates on the speed of convergence to each rational function r in one variable and defined over a number field k is associated a normalized height on the algebraic closure of k we show that these dynamically defined heights are adelic in our sense and deduce from this equidistribution results for preimages of points under r at every place of k our approach follows that of bilu and relies on potential theory in the complex plane as well as in the berkovich space associated to the projective line over open image in new window for each prime p

existence of positive solutions for first order discrete periodic boundary value problems with delay abstract let t 3 be a positive integer and t 0 1 2 t 1 we prove the existence of positive periodic solutions of the nonlinear discrete boundary value problem u t a t g u t u t b t f u t t t t u 0 u t our approach is based upon the global bifurcation techniques

sequential regularization methods for nonlinear higher index daes sequential regularization methods relate to a combination of stabilization methods and the usual penalty method for differential equations with algebraic equality constraints this paper extends an earlier work siam j numer anal 33 1996 pp 1921 1940 to nonlinear problems and to differential algebraic equations daes with an index higher than 2 rather than having one winning method this is a class of methods from which a number of variants are singled out as being particularly effective methods in certain circumstances we propose sequential regularization methods for index 2 and index 3 daes both with and without constraint singularities in the case of no constraint singularity we prove convergence results numerical experiments confirm our theoretical predictions and demonstrate the viability of the proposed methods the examples include constrained multibody systems

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

computing in the jacobian of a hyperelliptic curve in this paper we present algorithms suitable for computer use for computation in the jacobian of a hyperelliptic curve we present a reduction algorithm which is asymptotically faster than that of gauss when the genus g is very large

a fully explicit method for incompressible flow computation abstract a new formulation of the navier stokes equations is introduced to solve incompressible flow problems it keeps the benefits of the penalty method that is velocity and pressure can be obtained separately and no pressure poisson equation is involved unlike the penalty method the formulation is more stable or less stiff and then explicit time integration can be applied for easy implementation no linear or nonlinear system need be solved in the method in the case that a large number of time steps are needed a parallelization based on domain decomposition is applied to reduce the computational time with the explicit time integration the parallel implementation and its message passing are very simple as well

p adic dynamics the quadratic map overp adic numbers is studied in some detail we prove that near almost all indifferent fixed points it is topologically conjugate to a quasiperiodic linear map we also establish the existence of chaotic behavior and describe it using symbolic dynamics

mesures et equidistribution sur les espaces de berkovich the proof by ullmo and zhang of bogomolov s conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for small points in the associated complex abelian variety we study the analogous equidistribution property at p adic places our results can be conveniently stated within the framework of the analytic spaces defined by berkovich the first one is valid in any dimension but is restricted to algebraic metrics the second one is valid for curves but allows for more general metrics in particular to the normalized heights with respect to dynamical systems

prime factors of dynamical sequences let f t a q t have degree d f 2 for a given rational number x0 define xn 1 f xn for each nf 0 if this sequence is not eventually periodic and if f does not lie in one of two explicitly determined a ne conjugacy classes of rational functions then xn 1 xn has a primitive prime factor in its numerator for all su ciently large n the same result holds for the exceptional maps provided that one looks for primitive prime fac tors in the denominator of xn 1 xn hence the result for each rational function f of degree at least 2 implies a new proof that there are infinitely many primes the question of primi tive prime factors of xn dxn is also discussed for d uniformly bounded

an explicit theory of heights we consider the problem of explicitly determining the naive height constants for jacobians of hyperelliptic curves for genus 1 it is impractical to apply hilbert s nullstellensatz directly to the defining equations of the duplication law we indicate how this technical difficulty can be overcome by use of isogenies the height constants are computed in detail for the jacobian of an arbitrary curve of genus 2 and we apply the technique to compute generators of j q the mordell weil group for a selection of rank 1 examples

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

the group law on the jacobian of a curve of genus 2 an explicit description is given of the group law on the jacobian of a curve c of genus 2 the kummer surface provides a useful intermediary stage bilinear forms relating to the kummer surface imply that the global group law may be given projectively by biquadratic forms defined over the same ring as the coefficients of c it is not assumed that c has a rational weierstrass point and the theory presented applies over an arbitrary ground field

the dynamical mordell lang problem for etale maps we prove a dynamical version of the mordell lang conjecture for etale endomorphisms of quasiprojective varieties we use p adic methods inspired by the work of skolem mahler and lech combined with methods from algebraic geometry as special cases of our result we obtain a new proof of the classical mordell lang conjecture for cyclic subgroups of a semiabelian variety and we also answer positively a question of keeler rogalski stafford for critically dense sequences of closed points of a noetherian integral scheme

some current cfd issues relevant to the incompressible navier stokes equations abstract the goals of this paper are to carefully define a particular class of well set incompressible navier stokes problems in the continuum partial differential equation pde setting and to discuss some relevant and sometimes poorly understood issues related to these well posed pde problems both in the continuum world and in its computer counterpart emphasized are three common formulations used to generate computer codes the primitive variable velocity pressure formulation and two derived variable formulations pressure poisson equation and vorticity transport equation

global attractivity for certain impulsive delay differential equations abstract in this paper we study the global attractivity of zero solution of the following impulsive delay differential equation x t a t x t p t f x t t 0 t t k x t k x t k b k x t k k n some sufficient conditions to guarantee every solution of the equation converges to zero are obtained as its applications we apply our results to some well known impulsive delay ecological equations

a new algorithm for factoring polynomials over finite fields in f t there are two standard methods one is due to e berlekamp the other appears to be a folk method both are described by d knuth in 3 pp 381 397 further improvements for special values of q are given by r moenck in 5j see also e berlekamp 1 note that both methods as described in the references apply only when d 1 however straightforward modifications described below allow d 1 both methods use the calculation of resultants or equivalently the solution of linear equations to reduce the problem to finding the roots of a polynomial which has all of its roots in f when p is very small probabilistic methods are used an improvement to the folk method method along with a more explicit calculation of the work required has recently been given by m rabin 6 we present here a new probabilistic method which when combined with the above algorithms avoids the need for both resultants and linear equations it leads to algorithms which are conceptually simpler than previous methods moreover it works equally well for all finite fields f regardless of the magnitude of q when used for factoring a quadratic x2 a it reduces to berlekamp s algorithm other standard algorithms for factoring quadratics are due to d h lehmer 4 and d shanks 7 our algorithm is also suitable for finding solutions of polynomial equations over finite fields

transformation techniques and fixed point theories to establish the positive solutions of second order impulsive differential equations abstract using a new method for dealing with the impulse term of second impulsive differential equations we are concerned with determining values of for which there exist positive solutions of the following boundary value problems in the form of x t f t x t 0 t j t t k x t k x t k c k x t k k 1 2 n a x 0 b x 0 a x 1 b x 1 0 1 h s x t d t where 0 is a positive parameter c k is a real sequence with c k 1 k 1 2 n the dependence of positive solution x t on the parameter is also studied i e lim x or lim x 0 the proof of our main result is based upon transformation techniques and fixed point theories this is probably the first time the existence of positive solutions of second order impulsive differential equations has been studied by applying this technique

a predicted sequential regularization method for index 2 hessenberg daes the sequential regularization method srm is a dynamic iterative method for the numerical solution of higher index differential algebraic equations daes the srm has the advantage of being based on a regularized problem that is less stiff than those produced by standard regularization methods consequently nonstiff integrators may be used making the srm a competitive alternative to popular integrators in past work the number of srm iterations was taken to be roughly equal to the order of the numerical method used in each dynamic iteration in this paper we propose a predicted srm psrm that reduces the number of iterations in each dynamic iteration to one we give a new error analysis for explicit runge kutta methods applied to linear index 2 hessenberg daes with or without singularities we also give numerical examples to confirm the predicted convergence rates for the psrm extrapolation formulas and methods based on the differential part of the daes serve as a predictor and the srm iteration serves as a corrector implementation of higher order schemes for the psrm makes use of continuous extensions of runge kutta methods in particular we give a prediction scheme for the algebraic variable at intermediate stage points that suppresses order reduction in the differential variable near a singularity moreover the srm psrm provides new insight into operator splitting and fast convergence rates for waveform relaxation

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

an iterative perturbation method for the pressure equation in the simulation of miscible displacement in porous media the miscible displacement problem in porous media is modeled by a nonlinear coupled system of two partial differential equations the pressure velocity equation and the concentration equation an iterative perturbation procedure is proposed and analyzed for the pressure velocity equation which is capable of producing as accurate a velocity approximation as the mixed finite element method and which requires the solution of symmetric positive definite linear systems only the velocity variable is involved in the linear systems and the pressure variable is obtained by substitution trivially applying perturbation methods can only give an error o epsilon while our iterative scheme can improve the error to o epsilon m at the m th iteration level where epsilon is a small positive number thus the convergence rate of our iterative procedure is o epsilon and consequently a small number of iterations is required theoretical convergence analysis and numerical experiments are presented to show the efficiency and accuracy of our method

ergodic properties of rational mappings with large topological degree let x be a projective manifold and f x x a rational mapping with large topological degree dt ek 1 f the k 1 th dynamical degree of f we give an elementary construction of a probability measure ef such that d n t fn ef for every smooth probability measure on x we show that every quasiplurisubharmonic function is ef integrable in particular ef does not charge either points of indeterminacy or pluripolar sets hence ef is f invariant with constant jacobian f ef dt ef we then establish the main ergodic properties of ef it is mixing with positive lyapunov exponents preimages of hmost h points as well as repelling periodic points are equidistributed with respect to ef moreover when dimc x 3 or when x is complex homogeneous ef is the unique measure of maximal entropy

existence and global attractivity of positive periodic solution for an impulsive lasota wazewska model abstract sufficient conditions are obtained for the existence and global attractivity of periodic positive solution of an impulsive lasota wazewska model for the survival of red blood cells

geometry of p adic siegel discs abstract we survey recent advances in the study of regular motions over p adic fields show its varied connections with dynamics and number theory and illustrate its significance to an important class of discrete dynamical systems we also show that mappings supporting quasi periodic motions can be naturally interpreted as flows with p adic time

global structure of positive solutions for nonlocal boundary value problems involving integral conditions abstract in this paper we consider the nonlinear eigenvalue problems u h t f u 0 0 t 1 u 0 0 u 1 0 1 u s d a s where 0 1 u s d a s is a stieltjes integral with a nondecreasing and a t is not a constant on 0 1 h c 0 1 0 and h t 0 on any subinterval of 0 1 f c 0 0 and f s 0 for s 0 and f 0 f 0 f 0 lim s 0 f s s f lim s f s s we investigate the global structure of positive solutions by using global bifurcation techniques

a combinatorial li yau inequality and rational points on curves we present a method to control gonality of nonarchimedean curves based on graph theory let k denote a complete nonarchimedean valued field we first prove a lower bound for the gonality of a curve over the algebraic closure of k in terms of the minimal degree of a class of graph maps namely one should minimize over all so called finite harmonic graph morphisms to trees that originate from any refinement of the dual graph of the stable model of the curve next comes our main result we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the laplacian and some volume of the original graph this can be seen as a substitute for graphs of the li yau inequality from differential geometry although we also prove that the strict analogue of the original inequality fails for general graphs finally we apply the results to give a lower bound for the gonality of arbitrary drinfeld modular curves over finite fields and for general congruence subgroups vargamma of vargamma 1 that is linear in the index vargamma 1 vargamma with a constant that only depends on the residue field degree and the degree of the chosen infinite place this is a function field analogue of a theorem of abramovich for classical modular curves we present applications to uniform boundedness of torsion of rank two drinfeld modules that improve upon existing results and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of papikian

preperiodic points for quadratic polynomials with small cycles over quadratic fields given a number field k and a polynomial f z in k z one can naturally construct a finite directed graph g f k whose vertices are the k rational preperiodic points of f with an edge alpha rightarrow beta if and only if f alpha beta the dynamical uniform boundedness conjecture of morton and silverman suggests that for fixed integers n ge 1 and d ge 2 there are only finitely many isomorphism classes of directed graphs g f k as one ranges over all number fields k of degree n and polynomials f z in k z of degree d in the case n d 1 2 poonen has given a complete classification of all directed graphs which may be realized as g f mathbb q for some quadratic polynomial f z in mathbb q z under the assumption that f does not admit rational points of large period the purpose of the present article is to continue the work begun by the author faber and krumm on the case n d 2 2 by combining the results of the previous article with a number of new results we arrive at a partial result toward a theorem like poonen s with a similar assumption on points of large period but over all quadratic extensions of mathbb q

stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients abstract we consider the following nonlinear impulsive delay differential equation with variable coefficients and forcing term x t a t x t f t x e t t 0 t t k x t k x t k i k x t k k n using new methods different from those in tang et al appl math comput 131 2002 373 nieto j comput appl math lett 15 2002 489 franco and nieto j comput appl math 88 1998 144 nieto j math anal appl 205 1997 423 he and yu j math anal appl 272 2002 67 we establish stability theorems for when e t 0 under the assumption that there is bk such that bkx2 x x ik x x2 for k n and x r theorems 2 1 2 4 the existence results for three positive periodic solutions and nonexistence results for periodic solution of when e t 0 under the assumption exp 0 t a u d u 0 t k 1 b k and bk 1 for all k theorems 3 1 3 2 and the existence results for periodic solution of at resonance which is caused by impulses i e exp 0 t a u d u 0 t k 1 b k under the assumption bk 1 for all k theorems 4 2 4 3 some results obtained improve and generalize the known theorems and some other results are new examples are presented to illustrate the main results

on the uniform boundedness conjecture for drinfeld modules in analogy with the famous theorems of mazur and merel on the torsion subgroups of elliptic curves one can formulate similar conjectures for the torsion points of drinfeld modules we prove some partial results for rank 2 drinfeld q t modules for example the uniform boundedness of the primary torsion

torsion anomalous points and families of elliptic curves we prove that there are at most finitely many complex lambda neq 0 1 such that two points on the legendre elliptic curve y 2 x x 1 x lambda with coordinates x 2 3 both have finite order this is a very special case of some conjectures on unlikely intersections in semiabelian schemes

preperiodic points of polynomials over global fields given a global field k and a polynomial defined over k of degree at least two morton and silverman conjectured in 1994 that the number of k rational preperiodic points of is bounded in terms of only the degree of k and the degree of in 1997 for quadratic polynomials over k q call and goldstine proved a bound which was exponential in s the number of primes of bad reduction of by careful analysis of the filled julia sets at each prime we present an improved bound on the order of s log s our bound applies to polynomials of any degree at least two over any global field k let k be a field and let k z be a rational function let denote the n iterate of under composition that is is the identity function and for n 1 n 1 we will study the dynamics on the projective line p k in particular we say a point x is preperiodic under if there are integers n m 0 such that x x the point y x satisfies n m y y and is said to be periodic of period n m note that x p k is preperiodic if and only if its orbit x n 0 is finite for example let k q and z z 29 16 then 5 4 1 4 7 4 forms a periodic cycle of period 3 and 5 4 1 4 7 4 and 3 4 each land on this cycle after one or two iterations in addition the point is of course fixed these nine q rational points are all preperiodic under meanwhile it is not difficult to see that no other point in p q is preperiodic by showing that the denominator of a rational preperiodic point must be 4 and that the absolute value must be less than 2 in general for any global field k any dimension n 1 and any morphism p p over k of degree at least two northcott proved in 1950 that the number of k rational preperiodic points of is finite 25 more precisely he showed that the preperiodic points form a set of bounded arithmetic height years later by analogy with the theorems of mazur 19 and merel 20 on k rational torsion of elliptic curves morton and silverman proposed the following conjecture 23 uniform boundedness conjecture morton and silverman 1994 given integers d n 1 and d 2 there is a constant d n d with the following property let k be a number field with k q d and let p p be a morphism of degree d defined over k then has at most preperiodic points in p k the analogy between preperiodic points and torsion comes from the fact that the torsion points of an elliptic curve e are precisely the preperiodic points of the multiplication bytwo map 2 e e in fact taking x coordinates the map 2 induces a rational function known to dynamicists as a lattes map p p of degree 4 whose preperiodic points date june 21 2005 revised december 21 2005 2000 mathematics subject classification primary 11g99 secondary 11d45 37f10

environmental periodicity and time delays in a food limited population model abstract sufficient conditions are obtained for the existence of a globally attracting positive periodic solution of the food limited population system modelled by the equation n t r t k t n t m k t c t r t n t m where m is a nonnegative integer and k r c are continuous positive periodic functions of period

a case of the dynamical mordell lang conjecture we prove a special case of a dynamical analogue of the classical mordell lang conjecture specifically let be a rational function with no periodic critical points other than those that are totally invariant and consider the diagonal action of on mathbb p 1 g if the coefficients of are algebraic we show that the orbit of a point outside the union of the proper preperiodic subvarieties of mathbb p 1 g has only finite intersection with any curve contained in mathbb p 1 g we also show that our result holds for indecomposable polynomials with coefficients in mathbb c our proof uses results from p adic dynamics together with an integrality argument the extension to polynomials defined over mathbb c uses the method of specialization coupled with some new results of medvedev and scanlon for describing the periodic plane curves under the action of on mathbb a 2

oscillation and stability of linear impulsive delay differential equations the main results of this paper are that the oscillation and the stability of a linear impulsive delay differential equation are equivalent respectively to the oscillation and the stability of a corresponding linear delay differential equation without impulses the results of this paper can be used to improve and derive some of the known results in the literature

algebraic stability and degree growth of monomial maps given a rational monomial map we consider the question of finding a toric variety on which it is algebraically stable we give conditions for when such variety does or does not exist we also obtain several precise estimates of the degree sequences of monomial maps

integer points on curves of genus two and their jacobians let c be a curve of genus 2 defined over a number field and e the image of c embedded into its jacobian j we show that the heights of points of j which are integral with respect to 2 6 can be effectively bounded as a result if p is a point on c and p its image under the hyperelliptic involution then the heights of points on c which are integral with respect to p and p can be effectively bounded in such a way that we can isolate the dependence on p and show that if the height of p is bigger than some bound then there are no points which are s integral with respect to p and p we relate points on c which are integral with respect to p to points on j which are integral with respect to 6 and discuss approaches toward bounding the heights of the latter

sequential regularization methods for higher index daes with constraint singularities the linear index 2 case standard stabilization techniques for higher index differential algebraic equations daes often involve elimination of the algebraic solution components this may not work well if there are singularity points where the constraint jacobian matrix becomes rank deficient this paper proposes instead a sequential regularization method srm a functional iteration procedure for solving problems with isolated singularities which have smooth differential solution components for linear index 2 daes we consider both initial value problems ivps and boundary value problems bvps the convergence of the srm is described and proved in detail we believe that this is the first convergence proof for any method for daes with this type of constraint singularities moreover the regularization parameter in our method is not necessarily very small so the srm is an important improvement over usual regularization methods various aspects of the subsequent numerical discretization of the regularized problems are discussed as well and some numerical verifications are carried out

cycles of quadratic polynomials and rational points on a genus 2 curve it has been conjectured that for n sufficiently large there are no quadratic polynomials in q z with rational periodic points of period n morton proved there were none with n 4 by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to x 1 16 whose rational points had been previously computed we prove there are none with n 5 here the relevant curve has genus 14 but it has a genus 2 quotient whose rational points we compute by performing a 2 descent on its jacobian and applying a refinement of the method of chabauty and coleman we hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves we also describe the three possible galois stable 5 cycles and show that there exist galois stable n cycles for infinitely many n furthermore we answer a question of morton by showing that the genus 14 curve and its quotient are not modular finally we mention some partial results for n 6

a sequential regularization method for time dependent incompressible navier stokes equations the objective of the paper is to present a method called the sequential regularization method srm for the nonstationary incompressible navier stokes equations from the viewpoint of regularization of differential algebraic equations daes and to provide a way to apply a dae method to partial differential algebraic equations pdaes the srm is a functional iterative procedure it is proved that its convergence rate is o epsilon m where m is the number of the srm iterations and epsilon is the regularization parameter the discretization and implementation issues of the method are considered in particular a simple explicit difference scheme is analyzed and its stability is proved under the usual step size condition of explicit schemes it appears that the srm formulation is new in the navier stokes context unlike other regularizations or pseudocompressibility methods in the navier stokes context the regularization parameter epsilon in the srm need not be very small and the regularized problem in the sequence may be essentially nonstiff in time direction for any epsilon hence the stability condition is independent of epsilon even for explicit time discretization numerical experiments are given to verify our theoretical results

computing conjugating sets and automorphism groups of rational functions abstract let and be endomorphisms of the projective line of degree at least 2 defined over a field f from a dynamical perspective a significant question is to determine whether and are conjugate or to answer the related question of whether a given rational function has a nontrivial automorphism we construct efficient algorithms for computing the set of conjugating maps resp the group of automorphisms with an emphasis on the case where f is a finite field or a number field each of our algorithms takes advantage of different dynamical structures so context e g field of definition and degree of the map determines the preferred algorithm

isotriviality is equivalent to potential good reduction for endomorphisms of pn over function fields abstract let k k c be the function field of a complete non singular curve c over an arbitrary field k the main result of this paper states that a morphism p k n p k n is isotrivial if and only if it has potential good reduction at all places v of k this generalizes results of benedetto for polynomial maps on p k 1 and baker for arbitrary rational maps on p k 1 we offer two proofs the first uses algebraic geometry and geometric invariant theory and it is new even in the case n 1 the second proof uses non archimedean analysis and dynamics and it more directly generalizes the proofs of benedetto and baker we will also give two applications the first states that an endomorphism of p k n of degree at least two is isotrivial if and only if it has an isotrivial iterate the second gives a dynamical criterion for whether after base change a locally free coherent sheaf e of rank n 1 on c decomposes as a direct sum l l of n 1 copies of the same invertible sheaf l

a semi explicit parallel solver for viscous incompressible flows abstract an incompressible viscous flow solver is parallelized using a database management program gpar developed for parallel domain decomposition the approach presented involves subdivision of the flow domain into sub domains called solution blocks and distribution of the solution blocks to network of computers the equations for each block are solved in block solvers while the information exchange between neighboring blocks is achieved via interface solvers the data structure for communication between block and interface solvers are provided by gpar the incompressible flow solver is based on the fractional step method in which the momentum equations are solved explicitly while the pressure equation is solved implicitly results are presented illustrating the performance of the algorithms with increasing number of blocks and processors

degree growth of meromorphic surface maps we study the degree growth of iterates of meromorphic self maps of compact kahler surfaces using cohomology classes on the riemann zariski space we show that the degrees grow similarly to those of mappings that are algebraically stable on some bimeromorphic model 0 introduction let x be a compact kahler surface and let f x x be a dominant meromorphic mapping fix a kahler class on x normalized by x 1 and define the degree of f with respect to to be the positive real number deg f f x f x where x denotes the intersection form on h 1 1 r x when x p2 and is the class of a line this coincides with the usual algebraic degree of f one can show that deg f n m 2 deg f deg f for all m n hence the limit 1 lim n deg f exists we refer to it as the asymptotic degree of f it follows from standard arguments see proposition 3 1 that 1 does not depend on the choice of that 1 is invariant under bimeromorphic conjugacy and that 1 2 where 2 is the topological degree of f

computing the mordell weil rank of jacobians of curves of genus two we derive the equations necessary to perform a two descent on the jacobians of curves of genus two with rational weierstrass points we compute the mordell weil rank of the jacobian of some genus two curves defined over the rationals and discuss the practicality of using this method

dynamical mordell lang conjecture for birational polynomial morphisms on mathbb a 2 we prove the dynamical mordell lang conjecture for birational polynomial morphisms on mathbb a 2

semi stable reduction implies minimality of the resultant abstract for a dynamical system on p n over a number field or a function field we show that semi stable reduction implies the minimality of the resultant we use this to show that every such dynamical system over a number field admits a globally minimal presentation

intersections of polynomial orbits and a dynamical mordell lang conjecture we prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection then the polynomials have a common iterate we also prove a special case of a conjectured dynamical analogue of the mordell lang conjecture