approximation of cahn hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with c1 elements a variational formulation and c1 finite element scheme with adaptive mesh refinement and coarsening are developed for phase separation processes described by the cahn hilliard diffuse interface model of transport in a mixture or alloy the adaptive scheme is guided by a laplacian jump indicator based on the corresponding term arising from the weak formulation of the fourth order non linear problem and is implemented in a parallel solution framework it is then applied to resolve complex evolving interfacial solution behavior for 2d and 3d simulations of the classic spinodal decomposition problem from a random initial mixture and to other phase transformation applications of interest simulation results and adaptive performance are discussed the scheme permits efficient robust multiscale resolution and interface characterization copyright 2008 john wiley sons ltd

evolutionary dynamics on degree heterogeneous graphs the evolution of two species with different fitness is investigated on degree heterogeneous graphs the population evolves either by one individual dying and being replaced by the offspring of a random neighbor voter model dynamics or by an individual giving birth to an offspring that takes over a random neighbor node invasion process dynamics the fixation probability for one species to take over a population of n individuals depends crucially on the dynamics and on the local environment starting with a single fitter mutant at a node of degree k the fixation probability is proportional to k for voter model dynamics and to 1 k for invasion process dynamics

utterance selection model of language change we present a mathematical formulation of a theory of language change the theory is evolutionary in nature and has close analogies with theories of population genetics the mathematical structure we construct similarly has correspondences with the fisher wright model of population genetics but there are significant differences the continuous time formulation of the model is expressed in terms of a fokker planck equation this equation is exactly soluble in the case of a single speaker and can be investigated analytically in the case of multiple speakers who communicate equally with all other speakers and give their utterances equal weight whilst the stationary properties of this system have much in common with the single speaker case time dependent properties are richer in the particular case where linguistic forms can become extinct we find that the presence of many speakers causes a two stage relaxation the first being a common marginal distribution that persists for a long time as a consequence of ultimate extinction being due to rare fluctuations

derived p adic heights and p adic l functions if e is an elliptic curve defined over a number field and p is a prime of good ordinary reduction for e a theorem of rubin relates the p adic height pairing on the p power selmer group of e to the first derivative of a cohomologically defined p adic l function attached to e bertolini and darmon have defined a sequence of derived p adic heights in this paper we give an alternative definition of the p adic height pairing and prove a generalization of rubin s result relating the derived heights to higher derivatives of p adic l functions we also relate degeneracies in the derived heights to the failure of the selmer group of e over a z p extension to be semi simple as an iwasawa module generalizing results of perrin riou

p adic l functions and p adic periods of modular forms let e be an elliptic curve which is defined over q and has stable reduction modulo a given prime p assuming that e is modular one can associate to e a p adic l function lp e s see mz swd a v vi mz t t for its construction in various cases this function is defined by a certain interpolation property and is analytic for sezp in this paper we will assume that e has split multiplicative reduction at p under this assumption the interpolation property implies that lp e 1 0 we will prove a formula for ep e 1 which was discovered experimentally by mazur tate and teitelbaum mz t t by tate s p adic urfiformization theory there is a p adic integer q epzp which we refer to as the tate period for e and a p adic analytic isomorphism

selmer groups over mathbb z _p d extensions consider an abelian variety a defined over a global field k and let l k be a mathbb z p d extension unramified outside a finite set of places of k with mathrm gal l k gamma let lambda gamma mathbb z p gamma denote the iwasawa algebra in this paper we study how the characteristic ideal of the lambda gamma module x l the dual p primary selmer group varies when l k is replaced by a strict intermediate mathbb z p e extension

a practically unconditionally gradient stable scheme for the n component cahn hilliard system we present a practically unconditionally gradient stable conservative nonlinear numerical scheme for the n component cahn hilliard system modeling the phase separation of an n component mixture the scheme is based on a nonlinear splitting method and is solved by an efficient and accurate nonlinear multigrid method and the scheme allows us to convert the n component cahn hilliard system into a system of n 1 binary cahn hilliard equations and significantly reduces the required computer memory and cpu time we observe that our numerical solutions are consistent with the linear stability analysis results we also demonstrate the efficiency of the proposed scheme with various numerical experiments

microscopic abrams strogatz model of language competition abstract the differential equation of abrams and strogatz for the competition between two languages is compared with agent based monte carlo simulations for fully connected networks as well as for lattices in one two and three dimensions with up to 10 9 agents in the case of socially equivalent languages agent based models and a mean field approximation give grossly different results

coherence thresholds in models of language change and evolution the effects of noise dynamics and network of interactions a simple model of language evolution proposed by komarova niyogi and nowak is characterized by a payoff in communicative function and by an error in learning that measure the accuracy in language acquisition the time scale for language change is generational and the model s equations in the mean field approximation are a particular case of the replicator mutator equations of evolutionary dynamics in well mixed populations this model exhibits a critical coherence threshold i e a minimal accuracy in the learning process is required to maintain linguistic coherence in this work we analyze in detail the effects of different fitness based dynamics driving linguistic coherence and of the network of interactions on the nature of the coherence threshold by performing numerical simulations and theoretical analyses of three different models of language change in finite populations with two types of structure fully connected networks and regular random graphs we find that although the threshold of the original replicator mutator evolutionary model is robust with respect to the structure of the network of contacts the coherence threshold of related fitness driven models may be strongly affected by this feature

systems of cahn hilliard equations the phase separation of alloys with two or more components is studied with emphasis on more than two components particular attention is given to differences between multicomponent and binary alloys specific topics of the paper include equilibrium theory aspects of the dynamics and numerical simulations in the equilibrium theory it is found that there is an enriched equilibrium structure that allows for multiple coexisting phases and the presence of triple points in the solution dynamic results include the characterization of the spinodal region and of the concentration variations that lead to spinodal decomposition unlike the binary theory not all composition variations lead to separation and the compositions are not restricted to the convex hull of the equilibrium concentrations linear analysis is used to predict that a pseudo binary will initially result from spinodal decomposition numerical simulations of the dynamics for a ternary alloy verify this initially but more than two phases often

multivariable varphi gamma modules and locally analytic vectors let k be a finite extension of qp and let gk gal q p k there is a very useful classification of p adic representations of gk in terms of cyclotomic modules cyclotomic means that gal k k where k is the cyclotomic extension of k one particularly convenient feature of the cyclotomic theory is the fact that the module attached to any p adic representation is overconvergent questions pertaining to the p adic local langlands correspondence lead us to ask for a generalization of the theory of modules with the cyclotomic extension replaced by an infinitely ramified p adic lie extension k k it is not clear what shape such a generalization should have in general even in the case where we have such a generalization namely the case of a lubin tate extension most modules fail to be overconvergent in this article we develop an approach that gives a solution to both problems at the same time by considering the locally analytic vectors for the action of inside some big modules defined using fontaine s rings of periods we show that in the cyclotomic case we recover the usual overconvergent modules in the lubin tate case we can prove as an application of our theory a folklore conjecture in the field stating that modules attached to f analytic representations are overconvergent

a parametric finite element method for fourth order geometric evolution equations we present a finite element approximation of motion by minus the laplacian of curvature and related flows the proposed scheme covers both the closed curve case and the case of curves that are connected via triple junctions on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches it turns out that the new scheme has very good properties with respect to area conservation and the equidistribution of mesh points we state also an extension of our scheme to willmore flow of curves and discuss possible further generalizations

comparison of voter and glauber ordering dynamics on networks we study numerically the ordering process of two very simple dynamical models for a two state variable on several topologies with increasing levels of heterogeneity in the degree distribution we find that the zero temperature glauber dynamics for the ising model may get trapped in sets of partially ordered metastable states even for finite system size and this becomes more probable as the size increases voter dynamics instead always converges to full order on finite networks even if this does not occur via coherent growth of domains the time needed for order to be reached diverges with the system size in both cases the ordering process is rather insensitive to the variation of the degreee distribution from sharply peaked to scale free

efficient numerical algorithm for multiphase field simulations phase field models have emerged as a successful class of models in a wide variety of applications in computational materials science multiphase field theories as a subclass of phase field theories have been especially useful for studying nucleation and growth in polycrystalline materials in theory an infinite number of phase field variables are required to represent grain orientations in a rotationally invariant free energy however limitations on available computational time and memory have restricted the number of phase field variables used in the simulations we present an approach by which the time and memory requirements are drastically reduced relative to standard algorithms the proposed algorithm allows us the use of an unlimited number of phase field variables to perform simulations without the associated burden on computational time or memory we present the algorithm in the context of coalescence free grain growth

euler characteristics and elliptic curves abstract let e be a modular elliptic curve over without complex multiplication let p be a prime number where e has good ordinary reduction and let f be the field obtained by adjoining to all p power division points on e write g for the galois group of f over assume that the complex l series of e over does not vanish at s 1 if p 5 we make a precise conjecture about the value of the g euler characteristic of the selmer group of e over f if one makes a standard conjecture about the behavior of this selmer group as a module over the iwasawa algebra we are able to prove our conjecture the crucial local calculations in the proof depend on recent joint work of the first author with r greenberg

some remarks on the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication we establish several results towards the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields namely i the existence of an appropriate p adic l function building on works of hida and perrin riou ii the basic structure theory of the dual selmer group following works of coates hachimori venjakob et al and iii the implications of dihedral or anticyclotomic main conjectures with basechange the result of i is deduced from the construction of hida and perrin riou which in particular is seen to give a bounded distribution the result of ii allows us to deduce a corank formula for the p primary part of the tate shafarevich group of an elliptic curve in the z p 2 extension of an imaginary quadratic field finally iii allows us to deduce a criterion for one divisibility of the two variable main conjecture in terms of specializations to cyclotomic characters following a suggestion of greenberg as well as a refinement via basechange c 2011 elsevier inc all rights reserved

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

diffuse interface model for incompressible two phase flows with large density ratios we investigate the applicability of an incompressible diffuse interface model for two phase incompressible fluid flows with large viscosity and density contrasts diffuse interface models have been used previously primarily for density matched fluids and it remains unclear to what extent such models can be used for fluids of different density thereby potentially limiting the application of these models in this paper the convective cahn hilliard equation and the condition that the velocity field is divergence free are derived from the conservation law of mass of binary mixtures in a straightforward way for fluids with large density and viscosity ratios differences in the equations of motion with a previously derived quasi incompressible model are shown to result from the respective assumptions made regarding the relationship between the diffuse fluxes of two species the convergence properties of the model are investigated for cases with large density ratio quantitative comparisons are made with results from previous studies to validate the model and its numerical implementation tests show that the variation in volume during the computation is of the order of machine accuracy which is consistent with our use of a conservative discretization scheme finite volume methods for the cahn hilliard equation results of the method are compared with previous work for the change in topology of rising bubbles and rayleigh taylor instability additional results are presented for head on droplet collision and the onset of droplet entrainment in stratified flows

arithmetic of formal groups and applications i universal norm subgroups the theory of the norm map for the formal multiplicative group is well known as local class field theory in 1972 mazur 14 in his generalization of iwasawa theory to abelian varieties over number fields made quite clear that it is absolutely crucial for such a global theory first to understand the norm map for the formal groups attached to abelian varieties he achieved this understanding in case of formal groups of multiplicative type by using the theory of proalgebraic groups later on lubin and rosen 12 gave a completely elementary approach to mazur s results in the meantime hazewinkel 8 had settled the case of one dimensional commutative formal lie groups his method consists in an explicit and very complicated study of the properties of the power series coefficients of the logarithm of a formal group law by means of higher ramification theory in the same spirit vvedenskij 18 and konovalov 10 reproved and slightly extended hazewinkel s results let k q be a finite extension with ring of integers r and residue class field and koo k be a ramified zv extension with ring of integers r and f gal koo k let i r denote a smooth connected commutative formal r group of finite dimension d i e a commutative formal lie group if k k is the intermediate layer of degree p in koo k r its ring of integers and g gal k k its galois group then we know from the description of ff as a formal lie group that

a class of stable spectral methods for the cahn hilliard equation in this work the initial boundary value problem of two dimensional cahn hilliard equation is considered a class of fully discrete dissipative fourier spectral schemes are proposed moreover semi implicit prediction correction schemes are presented the numerical simulations are performed to demonstrate the effectiveness of the proposed schemes

voter model dynamics in complex networks role of dimensionality disorder and degree distribution we analyze the ordering dynamics of the voter model in different classes of complex networks we observe that whether the voter dynamics orders the system depends on the effective dimensionality of the interaction networks we also find that when there is no ordering in the system the average survival time of metastable states in finite networks decreases with network disorder and degree heterogeneity the existence of hubs i e highly connected nodes in the network modifies the linear system size scaling law of the survival time the size of an ordered domain is sensitive to the network disorder and the average degree decreasing with both however it seems not to depend on network size and on the heterogeneity of the degree distribution

numerical study of the cahn hilliard equation in one two and three dimensions abstract the cahn hilliard ch equation is related with a number of interesting physical phenomena like the spinodal decomposition phase separation and phase ordering dynamics on the other hand this equation is very stiff and the difficulty to solve it numerically increases with the dimensionality and therefore there are several published numerical studies in one dimension 1d dealing with different approaches and much fewer in two dimensions 2d in three dimensions 3d there are very few publications usually concentrate in some specific result without the details of the used numerical scheme we present here a stable and fast conservative finite difference scheme to solve the ch with two improvements a splitting potential into an implicit and explicit in time part and the use of free boundary conditions we show that gradient stability is achieved in one two and three dimensions with large time marching steps than normal methods

on anisotropic order parameter models for multi phase system and their sharp interface limits abstract for a general class of diffuse anisotropic multi phase order parameter or phase field models we use formally matched asymptotic expansions to determine the asymptotic limit when a small parameter related to the thickness of the interface tends to zero in the case of anisotropic allen cahn systems we obtain in the limit that the interface moves by anisotropic mean curvature flow at triple junctions a force balance holds which in the anisotropic case includes shear forces herring torque terms acting normal to the interface we further identify the singular limit of anisotropic cahn hilliard systems

a generalization of a theorem of imai and its applications to iwasawa theory it is proved that if k is a complete discrete valuation field of mixed characteristic 0 p with residue field satisfying a mild condition then any abelian variety over k with potentially good reduction has finite k k 1 p infty rational torsion subgroup this can be used to remove certain conditions assumed in some theorems in iwasawa theory

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

a diffuse interface model for axisymmetric immiscible two phase flow a diffuse interface model is considered for solving axisymmetric immiscible two phase flow with surface tension the navier stokes ns equations are modified by the addition of a continuum forcing the interface between the two fluids is considered as the half level set of a mass concentration c which is governed by the cahn hilliard ch equation a fourth order degenerate nonlinear parabolic diffusion equation in this work we develop a nonlinear multigrid method to solve the ch equation with degenerate mobility and couple this to a projection method for the incompressible ns equations the diffuse interface method can deal with topological transitions such as breakup and coalescence smoothly without ad hoc cut and connect or other artificial procedures we present results for rayleigh s capillary instability up to forming satellite drops the results agree well with the linear stability theory

the structure of selmer groups our objective in this paper is to prove a rather broad generalization of some classical theorems in iwasawa theory

a time adaptive finite volume method for the cahn hilliard and kuramoto sivashinsky equations this paper presents a complete finite volume method for the cahn hilliard and kuramoto sivashinsky type of equations the spatial discretization is high order accurate and suitable for general unstructured grids the time integration is addressed by means of implicit an implicit explicit fourth order runge kutta schemes with error control and adaptive time stepping the outcome is a practical accurate and efficient simulation tool which has been successfully applied to accuracy tests and representative simulations the use of adaptive time stepping is of paramount importance in problems governed by the cahn hilliard model an adaptive method may be several orders of magnitude more efficient than schemes using constant or heuristic time steps in addition to driving the simulations efficiently the time adaptive procedure provides a quantitative not just qualitative characterization of the rich temporal scales present in phase separation processes governed by the cahn hilliard phase field model

a second order accurate non linear difference scheme for the n component cahn hilliard system a b s t r a c t we consider a second order conservative nonlinear numerical scheme for the ncomponent cahn hilliard system modeling the phase separation of a n component mixture the scheme is based on a crank nicolson finite difference method and is solved by an efficient and accurate nonlinear multigrid method we numerically demonstrate the second order accuracy of the numerical scheme we observe that our numerical solutions are consistent with the exact solutions of linear stability analysis results we also describe numerical experiments such as the evolution of triple junctions and the spinodal decomposition in a quaternary mixture we investigate the effects of a concentration dependent mobility on phase separation

local discontinuous galerkin methods for the cahn hilliard type equations in this paper we develop local discontinuous galerkin ldg methods for the fourth order nonlinear cahn hilliard equation and system the energy stability of the ldg methods is proved for the general nonlinear case numerical examples for the cahn hilliard equation and the cahn hilliard system in one and two dimensions are presented and the numerical results illustrate the accuracy and capability of the methods