integral points divisibility between values of polynomials and entire curves on surfaces abstract we prove some new degeneracy results for integral points and entire curves on surfaces in particular we provide the first examples to our knowledge of a simply connected smooth variety whose sets of integral points are never zariski dense some of our results are connected with divisibility problems i e the problem of describing the integral points in the plane where the values of some given polynomials in two variables divide the values of other given polynomials

scaling concepts in periodically modulated noisy systems abstract we show that scaling arguments are very useful to analyze the dynamics of periodically modulated noisy systems information about the behavior of the relevant quantities such as the signal to noise ratio upon variations of the noise level can be obtained by analyzing the symmetries and invariances of the system in this way it is possible to predict diverse physical manifestations of the cooperative behavior between noise and input signal as for instance stochastic resonance spatiotemporal stochastic resonance and stochastic multiresonance

local existence of the strong solutions for the full navier stokes poisson equations abstract we study the full navier stokes poisson equations for viscous polytropic fluids we prove the existence and uniqueness of local strong solutions for all initial values satisfying some compatibility conditions it is important that we allow the initial density to be vacuum

on radially symmetric solutions of the compressible isentropic self gravitating fluid abstract in this paper we mainly prove the global existence of weak solutions to the cauchy problem for the navier stokes system of compressible isentropic self gravitating fluids in r 3 when the cauchy data are radially symmetric it extends feireisl s existence theorem ducomet et al 2001 16 to the case 4 3 3 2 for radially symmetric initial data where is the specific heat ratio in the pressure if the total mass is less than a certain critical mass this conclusion also holds for 4 3 furthermore for the case of annular domain we point out the global existence radially symmetric strong solutions when the radially symmetric initial data satisfy the compatibility condition and the initial density need not be positive

a generalization of theorems of faltings and thue siegel roth wirsing in 1929 siegel proved a celebrated theorem on finiteness for integral solutions of certain diophantine equations this theorem applies to systems of polynomial equations which either a describe an irreducible curve whose projective closure has positive genus or b describe an irreducible curve of genus zero with at least three points at infinity for such systems siegel s theorem says that there are only finitely many solutions in the ring of integers of any given number field siegel s proof used the method of diophantine approximations as pioneered by thue in 1909 t to give an example if x y is a large integral solution of the equation x3 2y3 1 then

global solutions and stability for self gravitating isentropic gases abstract the global solution to the radial motion of self gravitating gases is studied the global approximate solutions are constructed by a generalized shock capturing method the nonlocal terms are carefully estimated the uniform estimate of the approximate solutions is obtained the convergence of the approximate solutions is proved by the compensated compactness method the global existence of weak entropy solution with arbitrarily large l initial data is then established

the failure distribution in percolation models of breakdown the probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation from this result and the stress voltage enhancement at the tip of the most critical defect in the network the probability of failure in percolation models of breakdown is calculated for defect fractions less than the percolation point this distribution is found to be of the form exponential of an exponential numerical simulations on the two dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the weibull 1951 form most often used in the fitting of breakdown data

directed recursion models for fractal growth fractals constructed by recursive processes are introduced to model growth phenomena these fractals are simultaneously directed and self similar in analogy with patterns growing under diffusion limited conditions the multifractal nature of the harmonic measure associated with laplacian interfaces is qualitatively interpreted using the models calculation of the largest singularity exponent allows one to make conclusions about the behaviour of diffusion limited aggregates

a simple two dimensional model for crack propagation simple models for crack growth which are closely related to the diffusion limited aggregation dla model have been explored using computer simulations in these models the bond breaking probabilities for bonds at the surface of a growing crack in a two dimensional triangular network of bonds and nodes are proportional to delta i eta and delta i is the bond strain the authors results indicate that the cracks generated by these models have a fractal structure and that their effective dimensionalities depend on both the bond breaking probability exponent eta and the boundary conditions which bonds are considered to be surface bonds that can be broken at the crack surface very similar results were obtained using shear and dilational strain

regularity of weak solutions of compressible isentropic self gravitating fluid abstract in this paper the regularity of weak solutions to the three dimensional compressible isentropic self gravitating viscous fluids is considered in periodic domain inspired by the idea of desjardins 1997 3 4 we get the higher integrability of density and we remove the additional assumption of l 0 t l q 0 in the work of choe and jin 2003 finally we prove the supremum of the density is finite and the infimum of the density is positive in the domain 0 t t 3 for a positive time t moreover moser type iteration scheme is used to obtain the l norm estimate of the velocity

complete bounded trajectories and attractors for compressible barotropic self gravitating fluid abstract in this paper we consider the global behavior of weak solutions of the navier stokes system of compressible barotropic self gravitating fluids in time in a bounded three dimension domain arbitrary forces under certain restrictions imposed on the adiabatic constant we prove the existence of global compact attractors

existence and non linear stability of rotating star solutions of the compressible euler poisson equations we prove the existence of rotating star solutions which are steady state solutions of the compressible isentropic euler poisson euler poisson equations in three spatial dimensions with prescribed angular momentum and total mass this problem can be formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having less symmetry than those functions considered in the classical auchmuty beals paper we prove the non linear dynamical stability of these solutions with perturbations having the same total mass and symmetry as the rotating star solution we also prove finite time stability of w 1 infty mathbb r 3 solutions where the perturbations are entropy weak solutions of the euler poisson equations finally we give a uniform in time a priori estimate for entropy weak solutions of the euler poisson equations

vector fields on the total space of hypersurfaces in the projective space and hyperbolicity the results we obtain in this article concern the hyperbolicity of very generic hypersurfaces in the 3 dimensional projective space we show that the kobayashi conjecture is true in this setting as long as the degree of the hypersurface is greater than 18

global existence for compressible navier stokes poisson equations in three and higher dimensions the compressible navier stokes poisson system is concerned in the present paper and the global existence and uniqueness of the strong solution is shown in the framework of hybrid besov spaces in three and higher dimensions

monte carlo approach to dendritic growth the connection between diffusion limited aggregation and the equations of dendritic growth is critically examined a different type of monte carlo simulation is proposed and used to construct two dimensional dendrite like patterns the wavelength occurring for short times is in good agreement with the linear stability analysis the time dependence of the characteristic wavelengths is also determined

on the hilbert property and the fundamental group of algebraic varieties in this paper we link the so called hilbert property hp for an algebraic variety over a number field with its fundamental group in a perspective which appears new the notion of hp comes from hilbert s irreducibility theorem and has important implications for instance towards the inverse galois problem we shall observe that the hp is in a sense opposite to the chevalley weil theorem this shall immediately entail the result that the hp can possibly hold only for simply connected varieties in the appropriate sense in turn this leads to new counterexamples to the hp involving enriques surfaces in this view we shall also formulate an alternative related property possibly holding in full generality and some conjectures which unify part of what is known in the topic these predict that for a variety with a zariski dense set of rational points the validity of the hp which is defined arithmetically is indeed of purely topological nature also a consequence of these conjectures would be a positive solution of the inverse galois problem in the paper we shall also prove the hp for a k3 surface related to the above enriques surface providing what appears to be the first example of a variety non rational over mathbb c for which the hp can be proved in an appendix we shall further discuss among other things the hp also for kummer surfaces all of this basically exhausts the study of the hp for surfaces with a zariski dense set of rational points

integral points on subvarieties of semiabelian varieties ii this paper proves a finiteness result for families of integral points on a semiabelian variety minus a divisor generalizing the corresponding result of faltings for abelian varieties combined with the main theorem of the first part of this paper this gives a finiteness statement for integral points on a closed subvariety of a semiabelian variety minus a divisor in addition the last two sections generalize some standard results on closed subvarieties of semiabelian varieties to the context of closed subvarieties minus divisors

generalizations of siegel s and picard s theorems we prove new theorems that are higher dimensional generalizations of the classical theorems of siegel on integral points on affine curves and of picard on holomorphic maps from c to affine curves these include results on integral points over varying number fields of bounded degree and results on kobayashi hyperbolicity we give a number of new conjectures describing from our point of view how we expect siegel s and picard s theorems to optimally generalize to higher dimensions in this article we prove new theorems that are higher dimensional generaliza tions of the classical theorems of siegel on integral points on affine curves and of picard on holomorphic maps from c to affine curves in section 2 we will give the statements of siegel s and picard s theorems and we will recall how these two theorems from such seemingly different areas of mathematics are related we will then proceed to give a number of new conjectures describing from our point of view how we expect siegel s and picard s theorems to optimally generalize to higher dimensions these include conjectures on integral points over varying number fields of bounded degree and conjectures addressing hyperbolic questions these conjectures appear to be fundamentally new we will then summarize our progress on these conjectures we have been able to get results in all dimensions with best possible results in many cases for surfaces

fourier acceleration of iterative processes in disordered systems technical details are given on how to use fourier acceleration with iterative processes such as relaxation and conjugate gradient methods these methods are often used to solve large linear systems of equations but become hopelessly slow very rapidly as the size of the set of equations to be solved increases fourier acceleration is a method designed to alleviate these problems and result in a very fast algorithm the method is explained for the jacobi relaxation and conjugate gradient methods and is applied to two models the random resistor network and the random central force network in the first model acceleration works very well in the second little is gained we discuss reasons for this we also include a discussion of stopping criteria

poincare series of resolutions of surface singularities let x spec r be a resolution of singularities of a normal surface singularity spec r with integral exceptional divisors e 1 e r we consider the poincare series g h n t n where h n r x o x n 1 e 1 n r e r we show that if r m has characteristic zero and pic 0 x is a semi abelian variety then the poincare series g is rational however we give examples to show that this series can be irrational if either of these conditions fails

fracture patterns and scaling laws abstract fracture is determined by the growth of cracks and can be described as a moving boundary problem similar to dielectric breakdown or viscous fingering the vectorial nature of the elastic medium allows however for a much larger variety of boundary conditions and growth rules than in the scalar electrostatic analogue giving rise to a number of new effects one can model this problem on a square lattice and it is found that the patterns of cracks can be fractal even without including noise due only to the interplay of anisotropy and memory the shapes of the cracks can be compared to the ones found experimentally for stress corrosion an effect that can be mimicked by a damage memory on the surface of the crack a quantity of experimental interest in the breaking characteristics force vs displacement in a model with quenched disorder and the possibility that several cracks be present in the system one finds scaling laws in the system size for the breaking characteristics just before the system breaks fully apart the distribution of local strain is multifractal this means that the volume fraction responsible for the final cracking goes to zero in the thermodynamic limit the scaling and multiscaling laws seem to be quite universal

regularity results for two dimensional flows of multiphase viscous fluids global regularity results for weak solutions of the navier stokes equations for two dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive lower bounds on the initial density as an application we deduce regularity properties for the integral curves of the corresponding velocity field finally we prove regularity results in the small for strong solutions