high accuracy solutions to energy gradient flows from material science models a computational framework is presented for materials science models that come from energy gradient flows the models of interest lead to the evolution of structure involving two or more phases the framework includes higher order derivative models and vector problems solutions are considered in periodic cells and standard fourier spectral discretization in space is used implicit time stepping is used with adaptivity based on local error estimates the implicit system at every time step is solved iteratively with newton s method the resulting linear systems are solved in inner iterations with the conjugate gradient method using a novel preconditioner that is a constant coefficient version of the system taking values for the coefficients at the pure phase states solutions with high spatial and temporal accuracy are obtained the dependence of the condition number of the preconditioned system on the size of the time step and the order parameter in the model that represents the scaled width of transition layers between phases is investigated numerically and with formal asymptotics in a simple setting the asymptotic results require a conjecture on the rank of a modified square distance matrix results from a fast graphical processing unit implementation for a three dimensional model are shown a comparison to time stepping with operator splitting into convex and concave parts that guarantees energy decrease in the numerical scheme is done

convergence analysis and numerical implementation of a second order numerical scheme for the three dimensional phase field crystal equation abstract in this paper we analyze and implement a second order in time numerical scheme for the three dimensional phase field crystal pfc equation the numerical scheme was proposed in hu et al 2009 with the unique solvability and unconditional energy stability established however its convergence analysis remains open we present a detailed convergence analysis in this article in which the maximum norm estimate of the numerical solution over grid points plays an essential role moreover we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain and various three dimensional numerical results are presented including the numerical convergence test complexity test of the multigrid solver and the polycrystal growth simulation

discontinuous galerkin finite element approximation of the cahn hilliard equation with convection the paper is concerned with the construction and convergence analysis of a discontinuous galerkin finite element method for the cahn hilliard equation with convection using discontinuous piecewise polynomials of degree p geq1 and backward euler discretization in time we show that the order parameter c is approximated in the broken rm l infty rm h 1 norm with optimal order cal o h p tau the associated chemical potential w phi c gamma 2 delta c is shown to be approximated with optimal order cal o h p tau in the broken rm l 2 rm h 1 norm here phi c frac 1 4 1 c 2 2 is a quartic free energy function and gamma 0 is an interface parameter numerical results are presented with polynomials of degree p 1 2 3

an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

stable and efficient finite difference nonlinear multigrid schemes for the phase field crystal equation in this paper we present and compare two unconditionally energy stable finite difference schemes for the phase field crystal equation the first is a one step scheme based on a convex splitting of a discrete energy by wise et al s m wise c wang j s lowengrub an energy stable and convergent finite difference scheme for the phase field crystal equation siam j numer anal in press in this scheme which is first order in time and second order in space the discrete energy is non increasing for any time step the second scheme we consider is a new fully second order two step algorithm in the new scheme the discrete energy is bounded by its initial value for any time step in both methods the equations at the implicit time level are nonlinear but represent the gradients of strictly convex functions and are thus uniquely solvable regardless of time step size we solve the nonlinear equations using an efficient nonlinear multigrid method numerical simulations are presented and confirm the stability efficiency and accuracy of the schemes

an uncertainty principle for arithmetic sequences analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are well distributed in some appropriate sense in various discrepancy problems combinatorics researchers have analyzed limitations to equidistribution as have fourier analysts when working with the uncertainty principle in this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory formulating a general principle and giving several examples

an energy stable and convergent finite difference scheme for the modified phase field crystal equation we present an unconditionally energy stable finite difference scheme for the modified phase field crystal equation a generalized damped wave equation for which the usual phase field crystal equation is a special degenerate case the method is based on a convex splitting of a discrete pseudoenergy and is semi implicit the equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable regardless of time step size we present a local in time error estimate that ensures the pointwise convergence of the scheme

on limit points of the sequence of normalized prime gaps let pn denote the nth smallest prime number and let l denote the set of limit points of the sequence tppn 1 pnq logpnu 8 1 of normalized differences between consecutive primes we show that for k 9 and for any sequence of k nonnegative real numbers 1 2 k at least one of the numbers j i 1 i j k belongs to l it follows that at least 12 5 of all nonnegative real numbers belong to l

small gaps between primes we introduce a renement of the gpy sieve method for studying prime k tuples and small gaps between primes this renement avoids previous limitations of the method and allows us to show that for each k the prime k tuples conjecture holds for a positive proportion of admissible k tuples in particular lim infn pn m pn 1 for every integer m we also show that lim inf pn 1 pn 600 and if we assume the elliott halberstam conjecture that lim infn pn 1 pn 12 and lim infn pn 2 pn 600

generalized estimating equations to estimate the ordered stereotype logit model for panel data by modeling the effects of predictor variables as a multiplicative function of regression parameters being invariant over categories and category specific scalar effects the ordered stereotype logit model is a flexible regression model for ordinal response variables in this article we propose a generalized estimating equations gee approach to estimate the ordered stereotype logit model for panel data based on working covariance matrices which are not required to be correctly specified a simulation study compares the performance of gee estimators based on various working correlation matrices and working covariance matrices using local odds ratios estimation of the model is illustrated using a real world dataset the results from the simulation study suggest that gee estimation of this model is feasible in medium sized and large samples and that estimators based on local odds ratios as realized in this study tend to be less efficient compared with estimators based on a working correlation matrix for low true correlations the efficiency gains seem to be rather small and if the working covariance structure is too flexible the corresponding estimator may even be less efficient compared with the gee estimator assuming independence like for gee estimators more generally if the true correlations over time are high then a working covariance structure which is close to the true structure can lead to considerable efficiency gains compared with assuming independence

finite element approximation of the cahn hilliard equation on surfaces abstract in this paper we consider the phase separation on general surfaces by solving the nonlinear cahn hilliard equation using a finite element method a fully discrete approximation scheme is introduced and we establish a priori estimates for the discrete solution that does not rely on any knowledge of the exact solution beyond the initial time this in turn leads to convergence and optimal error estimates of the discretization scheme numerical examples are also provided to substantiate the theoretical results

an energy stable and convergent finite difference scheme for the phase field crystal equation we present an unconditionally energy stable finite difference scheme for the phase field crystal equation the method is based on a convex splitting of a discrete energy and is semi implicit the equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable regardless of time step size we present local in time error estimates that ensure the convergence of the scheme while this paper is primarily concerned with the phase field crystal equation most of the theoretical results hold for the related swift hohenberg equation as well

a weakly nonlinear energy stable scheme for the strongly anisotropic cahn hilliard equation and its convergence analysis abstract in this paper we propose and analyze a weakly nonlinear energy stable numerical scheme for the strongly anisotropic cahn hilliard model in particular a highly nonlinear and singular anisotropic surface energy makes the pde system very challenging at both the analytical and numerical levels to overcome this well known difficulty we perform a convexity analysis on the anisotropic interfacial energy and a careful estimate reveals that all its second order functional derivatives stay uniformly bounded by a global constant this subtle fact enables one to derive an energy stable numerical scheme moreover a linear approximation becomes available for the surface energy part and a detailed estimate demonstrates the corresponding energy stability its combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear energy stable scheme for the whole system in particular such an energy stability is in terms of the interfacial energy with respect to the original phase variable and no auxiliary variable needs to be introduced this has important implications for example in the case that the method needs to satisfy a maximum principle more importantly with a careful application of the global bound for the second order functional derivatives an optimal rate convergence analysis becomes available for the proposed numerical scheme which is the first such result in this area meanwhile for a cahn hilliard system with a sufficiently large degree of anisotropy a willmore or biharmonic regularization has to be introduced to make the equation well posed for such a physical model all the presented analyses are still available the unique solvability energy stability and convergence estimate can be derived in an appropriate manner in addition the fourier pseudo spectral spatial approximation is applied and all the theoretical results could be extended for the fully discrete scheme finally a few numerical results are presented which confirm the robustness and accuracy of the proposed scheme

second order convex splitting schemes for gradient flows with ehrlich schwoebel type energy application to thin film epitaxy we construct unconditionally stable unconditionally uniquely solvable and second order accurate in time schemes for gradient flows with energy of the form int omega f nabla phi bf x frac epsilon 2 2 delta phi bf x 2 d bf x the construction of the schemes involves the appropriate combination and extension of two classical ideas i appropriate convex concave decomposition of the energy functional and ii the secant method as an application we derive schemes for epitaxial growth models with slope selection f bf y frac14 bf y 2 1 2 or without slope selection f bf y frac12 ln 1 bf y 2 two types of unconditionally stable uniquely solvable second order schemes are presented the first type inherits the variational structure of the original continuous in time gradient flow while the second type does not preserve the variational structure we present numerical simulations for the case with slope selection which verify well known physical scaling laws for the long time coarsening process

phase field crystal equation with memory abstract phase field crystal models are used to describe several pattern formation phenomena like crystallization of liquid diffusion defects and glass formation the prototypical equation is obtained as the conserved gradient flow associated with a free energy functional of swift hohenberg type here we consider a variant of the phase field crystal equation proposed by p galenko et al to account for fast dynamics this version is characterized by the fact that the gradient flow depends on the past history of the particle density through a memory kernel k e e 0 being a relaxation time therefore the resulting nonlinear evolution equation is an integro differential equation of sixth order which is equipped with the physically relevant periodic boundary conditions we show that this problem generates a dissipative dynamical system on a suitable phase space moreover we prove the existence of a robust family of exponential attractors which is holder continuous with respect to e

the numerical simulation of the phase field crystal pfc and modified phase field crystal mpfc models via global and local meshless methods abstract in the current work we consider the two dimensional time dependent phase field crystal pfc and the modified phase field crystal models mpfc to obtain their numerical solutions for this purpose we apply two numerical meshless methods based on radial basis functions rbfs and also two meshless methods which are based on moving least squares method mls four techniques developed in this paper are globally radial basis functions grbfs radial basis functions pseudo spectral rbfs ps moving least squares mls and generalized moving least squares gmls approximations two methods based on rbfs are global and the other methods based on moving least squares are local procedures as is well known the meshless methods are suitable techniques for the numerical solution of partial differential equations on regular and non regular domains with different choices of grids in high dimensions applying the new methods on spatial domain and also using the semi implicit scheme for time variable yield a linear system of algebraic equations to solve this linear system the lu decomposition algorithm and command backslash in matlab software for gmls method are applied the implementation of boundary conditions in the rbfs ps is applied directly because the boundary conditions of the mentioned problems are periodic some numerical results show that the obtained simulations via four proposed methods are acceptable for approximating the solution of models investigated in the current paper moreover in the appendix of the paper a matlab code for gmls method is written

small gaps between products of two primes let qn denote the nth number that is a product of exactly two distinct primes we prove that qn 1 qnle 6 infinitely often this sharpens an earlier result of the authors which had 26 in place of 6 more generally we prove that if is any positive integer then qn 1 qn e 1 o 1 infinitely often we also prove several other related results on the representation of numbers with exactly two prime factors by linear forms 2008 london mathematical society

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

on the size of the first factor of the class number of a cyclotomic field we show that kummer s conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field is untrue under the assumption of two well known and widely believed conjectures of analytic number theory

regression and ordered categorical variables read before the royal statistical society by professor r l plackett on behalf of the late professor anderson at a meeting organized by the research section on wednesday october 5th 1983 professor j b copas in the chair summary a general approach to regression modelling for ordered categorical response variables y is given which is equally applicable to ordered and unordered y the regressor variables xt xi x may be continuous or categorical the method is based on the logistic family which contains a hierarchy of regression models ranging from ordered to unordered models ordered properties of the former the stereotype model are established the choice between models is made empirically on the basis of model fit this is particularly important for assessed ordered categorical response variables where it is not obvious a priori whether or not the ordering is relevant to the regression relationship model simplification is investigated in terms of whether or not the response categories are distinguishable with respect to x the models are fitted iteratively using the method of maximum likelihood examples are given

the fourier collocation method for the cahn hilliard equation abstract in this paper a fourier collocation method for numerically solving cahn hilliard equations with periodic boundary conditions is developed we establish their semidiscrete and fully discrete schemes that inherit the energy dissipation property and mass conservation property from the associated continuous problem we prove existence and uniqueness of the numerical solution and derive the optimal error bounds we perform some numerical experiments which confirm our results

limitations to the equi distribution of primes iv we construct infinitely many different polynomials of given degree which take either significantly more or significantly less prime values than expected

error analysis of a mixed finite element method for the cahn hilliard equation we propose and analyze a semi discrete and a fully discrete mixed finite element method for the cahn hilliard equation ut u 1f u 0 where 0 is a small parameter error estimates which are quasi optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain in particular it is shown that all error bounds depend on only in some lower polynomial order for small the cruxes of our analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of alikakos and fusco 2 and chen 15 to prove a discrete counterpart of it for a linearized cahn hilliard operator to handle the nonlinear term on a stretched time grid the ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the hele shaw mullins sekerka problem as 0 in 29

an energy conserving second order numerical scheme for nonlinear hyperbolic equation with an exponential nonlinear term we present a second order accurate numerical scheme for a nonlinear hyperbolic equation with an exponential nonlinear term the solution to such an equation is proven to have a conservative nonlinear energy due to the special nature of the nonlinear term the positivity is proven to be preserved under a periodic boundary condition for the solution for the numerical scheme a highly nonlinear fractional term is involved for the theoretical justification of the energy stability we propose a linear iteration algorithm to solve this nonlinear numerical scheme a theoretical analysis shows a contraction mapping property of such a linear iteration under a trivial constraint for the time step we also provide a detailed convergence analysis for the second order scheme in the 0 t norm such an error estimate in the maximum norm can be obtained by performing a higher order consistency analysis using asymptotic expansions for the numerical solution as a result instead of the standard comparison between the exact and numerical solutions an error estimate between the numerical solution and the constructed approximate solution yields an o t 3 h 4 convergence in 0 t 2 norm which leads to the necessary error estimate using the inverse inequality under a standard constraint t c h a numerical accuracy check is given and some numerical simulation results are also presented

stability analysis of large time stepping methods for epitaxial growth models numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy mbe require very large time simulation and therefore large time steps become necessary the main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit explicit approach to this end an extra term which is consistent with the order of the time discretization is added to stabilize the numerical schemes then the stability properties of the resulting schemes are established rigorously numerical experiments are carried out to support the theoretical claims the proposed methods are also applied to simulate the mbe models with large solution times the power laws for the coarsening process are obtained and are compared with previously published results

a nonstiff adaptive mesh refinement based method for the cahn hilliard equation we present a nonstiff fully adaptive mesh refinement based method for the cahn hilliard equation the method is based on a semi implicit splitting in which linear leading order terms are extracted and discretized implicitly combined with a robust adaptive spatial discretization the fully discretized equation is written as a system which is efficiently solved on composite adaptive grids using the linear multigrid method without any constraint on the time step size we demonstrate the efficacy of the method with numerical examples both the transient stage and the steady state solutions of spinodal decompositions are captured accurately with the proposed adaptive strategy employing this approach we also identify several stationary solutions of that decomposition on the 2d torus

energy stable and efficient finite difference nonlinear multigrid schemes for the modified phase field crystal equation in this paper we present two unconditionally energy stable finite difference schemes for the modified phase field crystal mpfc equation a sixth order nonlinear damped wave equation of which the purely parabolic phase field crystal pfc model can be viewed as a special case the first is a convex splitting scheme based on an appropriate decomposition of the discrete energy and is first order accurate in time and second order accurate in space the second is a new fully second order scheme that also respects the convex splitting of the energy both schemes are nonlinear but may be formulated from the gradients of strictly convex coercive functionals thus both are uniquely solvable regardless of the time and space step sizes the schemes are solved by efficient nonlinear multigrid methods numerical results are presented demonstrating the accuracy energy stability efficiency and practical utility of the schemes in particular we show that our multigrid solvers enjoy optimal or nearly optimal complexity in the solution of the nonlinear schemes

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

missing exposure data in stereotype regression model application to matched case control study with disease subclassification with advances in modern medicine and clinical diagnosis case control data with characterization of finer subtypes of cases are often available in matched case control studies missingness in exposure values often leads to deletion of entire stratum and thus entails a significant loss in information when subtypes of cases are treated as categorical outcomes the data are further stratified and deletion of observations becomes even more expensive in terms of precision of the category specific odds ratio parameters especially using the multinomial logit model the stereotype regression model for categorical responses lies intermediate between the proportional odds and the multinomial or baseline category logit model the use of this class of models has been limited as the structure of the model implies certain inferential challenges with nonidentifiability and nonlinearity in the parameters we illustrate how to handle missing data in matched case control studies with finer disease subclassification within the cases under a stereotype regression model we present both monte carlo based full bayesian approach and expectation conditional maximization algorithm for the estimation of model parameters in the presence of a completely general missingness mechanism we illustrate our methods by using data from an ongoing matched case control study of colorectal cancer simulation results are presented under various missing data mechanisms and departures from modeling assumptions

spectral approximation of pattern forming nonlinear evolution equations with double well potentials of quadratic growth this paper is concerned with the analysis of a numerical algorithm for the approximate solution of a class of nonlinear evolution problems that arise as l 2 gradient flow for the modica mortola regularization of the functional v bv t d 1 1 e v 2 td v 1 2 k d k v k 2 here is the interfacial energy per unit length or unit area t d is the flat torus in d and is a nonnegative fourier multiplier that is continuous on d symmetric in the sense that for all d and that decays to zero at infinity such functionals feature in mathematical models of pattern formation in micromagnetics and models of diblock copolymers the resulting evolution equation is discretized by a fourier spectral method with respect to the spatial variables and a modified crank nicolson scheme in time optimal order a priori bounds are derived on the global error in the l 0 t l 2 t d norm

a conservative numerical method for the cahn hilliard equation with dirichlet boundary conditions in complex domains in this paper we present a conservative numerical method for the cahn hilliard equation with dirichlet boundary conditions in complex domains the method uses an unconditionally gradient stable nonlinear splitting numerical scheme to remove the high order time step stability constraints the continuous problem has the conservation of mass and we prove the conservative property of the proposed discrete scheme in complex domains we describe the implementation of the proposed numerical scheme in detail the resulting system of discrete equations is solved by a nonlinear multigrid method we demonstrate the accuracy and robustness of the proposed dirichlet boundary formulation using various numerical experiments we numerically show the total energy decrease and the unconditionally gradient stability in particular the numerical results indicate the potential usefulness of the proposed method for accurately calculating biological membrane dynamics in confined domains

second order schemes and time step adaptivity for allen cahn and cahn hilliard models abstract in this paper we focus on efficient second order in time approximations of the allen cahn and cahn hilliard equations first of all we present the equations generic second order schemes based on a mid point approximation of the diffusion term and some schemes already introduced in the literature then we propose new ways of deriving second order in time approximations of the potential term starting from the main schemes introduced in guillen gonzalez and tierra 2013 yielding to new second order schemes for these schemes and other second order schemes previously introduced in the literature we study the constraints on the physical and discrete parameters that can appear to assure the energy stability unique solvability and in the case of nonlinear schemes the convergence of newton s method to the nonlinear schemes moreover in order to save computational cost we have developed a new adaptive time stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step finally we compare the behaviour of the schemes and the effectiveness of the adaptive time stepping algorithm through several computational experiments

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

residue classes containing an unexpected number of primes we fix a non zero integer a and consider arithmetic progressions a bmod q with q varying over a given range we show that for certain specific values of a the arithmetic progressions a bmod q contain on average significantly fewer primes than expected

unconditionally stable methods for gradient flow using convex splitting runge kutta scheme abstract we propose a convex splitting runge kutta csrk scheme which provides a simple unified framework to solve a gradient flow in an unconditionally gradient stable manner the key feature of the scheme is a combination of a convex splitting method and a specially designed multi stage two additive runge kutta method our methods are high order accurate in time and assure the gradient energy stability for any time step size we provide detailed proof of the unconditional energy stability and present issues on the practical implementations we demonstrate the accuracy and stability of the proposed methods using numerical experiments of the cahn hilliard equation

error analysis of a mixed finite element method for a cahn hilliard hele shaw system we present and analyze a mixed finite element numerical scheme for the cahn hilliard hele shaw equation a modified cahn hilliard equation coupled with the darcy flow law this numerical scheme was first reported in feng and wise siam j numer anal 50 1320 1343 2012 with the weak convergence to a weak solution proven in this article we provide an optimal rate error analysis a convex splitting approach is taken in the temporal discretization which in turn leads to the unique solvability and unconditional energy stability instead of the more standard ell infty 0 t l 2 cap ell 2 0 t h 2 l 0 t l2 l2 0 t h2 error estimate we perform a discrete ell infty 0 t h 1 cap ell 2 0 t h 3 l 0 t h1 l2 0 t h3 error estimate for the phase variable through an l 2 l2 inner product with the numerical error function associated with the chemical potential as a result an unconditional convergence for the time step tau in terms of the spatial resolution h is derived the nonlinear analysis is accomplished with the help of a discrete gagliardo nirenberg type inequality in the finite element space gotten by introducing a discrete laplacian delta h h of the numerical solution such that delta h phi in s h h sh for every phi in s h sh where s h sh is the finite element space

optimal rate convergence analysis of a second order scheme for a thin film model with slope selection abstract in this paper an energy stable second order mixed finite element scheme is proposed and analyzed for the thin film epitaxial growth model with slope selection we employ second order backward differentiation formula bdf scheme with a second order stabilized term which guarantees the long time energy stability to approximate the continuous model in terms of the convergence analysis the key difficulty to derive an optimal rate spatial estimate is associated with the appearance of the gradient operator in the nonlinear terms which may lead to a loss of optimal accuracy order to overcome this well known difficulty we make use of some auxiliary techniques over triangular elements and obtain an optimal convergence rate o h q 1 t 2 in comparison with o h q t 2 rate from a standard projection estimate furthermore we use an efficient preconditioned steepest descent psd solver for the numerical implementation a few numerical examples are presented to validate the stability and convergence

error analysis of a mixed finite element method for the molecular beam epitaxy model this paper investigates the error analysis of a mixed finite element method with crank nicolson time stepping for simulating the molecular beam epitaxy mbe model the fourth order differential equation of the mbe model is replaced by a system of equations consisting of one nonlinear parabolic equation and an elliptic equation then a mixed finite element method requiring only continuous elements is proposed to approximate the resulting system it is proved that the semidiscrete and fully discrete versions of the numerical schemes satisfy the nonlinearity energy stability property which is important in the numerical implementation moreover detailed analysis is provided to obtain the convergence rate numerical experiments are carried out to validate the theoretical results

a semi analytical fourier spectral method for the swift hohenberg equation abstract the swift hohenberg sh equation has been widely used as a model for the study of pattern formation the sh equation is a fourth order nonlinear partial differential equation and cannot generally be solved analytically therefore computer simulations play an essential role in understanding of nonequilibrium processing the aim of this paper is to present first and second order semi analytical fourier spectral methods as an accurate and efficient approach for solving the sh equation the methods are based on the operator splitting method and are to split the sh equation into linear and nonlinear subequations the linear and nonlinear subequations have closed form solutions in the fourier and physical spaces respectively the methods are simple to implement and computationally cheap to achieve high order time accuracy numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods

an energy stable linear diffusive crank nicolson scheme for the cahn hilliard gradient flow abstract we propose and analyze a linearly stabilized semi implicit diffusive crank nicolson scheme for the cahn hilliard gradient flow in this scheme the nonlinear bulk force is treated explicitly with two second order stabilization terms this treatment leads to linear elliptic system with constant coefficients and provable discrete energy dissipation rigorous error analysis is carried out for the fully discrete scheme when the time step size and the space step size are small enough second order accuracy in time is obtained with a prefactor controlled by some lower degree polynomial of 1 e here e is the thickness of the interface numerical results together with an adaptive time stepping are presented to verify the accuracy and efficiency of the proposed scheme

multiphase allen cahn and cahn hilliard models and their discretizations with the effect of pairwise surface tensions in this paper the mathematical properties and numerical discretizations of multiphase models that simulate the phase separation of an n component mixture are studied for the general choice of phase variables the unisolvent property of the coefficient matrix involved in the n phase models based on the pairwise surface tensions is established moreover the symmetric positive definite property of the coefficient matrix on an n 1 n 1 dimensional hyperplane which is of fundamental importance to the well posedness of the models can be proved equivalent to some physical condition for pairwise surface tensions the n phase allen cahn and n phase cahn hilliard equations can then be derived from the free energy functional a natural property is that the resulting dynamics of concentrations are independent of phase variables chosen finite element discretizations for n phase models can be obtained as a natural extension of the existing discretizations for the two phase model the discrete energy law of the numerical schemes can be proved and numerically observed under some restrictions pertaining to time step size numerical experiments including the spinodal decomposition and the evolution of triple junctions are described in order to investigate the effect of pairwise surface tensions

unconditionally gradient stable computational schemes in problems of fast phase transitions equations of fast phase transitions in which the phase boundaries move with velocities comparable with the atomic diffusion speed or with the speed of local structural relaxation are analyzed these equations have a singular perturbation due to the second derivative of the order parameter with respect to time which appears due to phenomenologically introduced local nonequilibrium to develop unconditionally stable computational schemes the eyre theorem d j eyre unpublished proved for the classical equations based on hypotheses of local equilibrium is used an extension of the eyre theorem for the case of equations for fast phase transitions is given it is shown that the expansion of the free energy on contractive and expansive parts suggested by eyre for the classical equations of cahn hilliard and allen cahn is also true for the equations of fast phase transitions grid approximations of these equations lead to gradient stable algorithms with an arbitrary time step for numerical modeling ensuring monotonic nonincrease of the free energy special examples demonstrating the extended eyre theorem for fast phase transitions are considered

an energy stable hexagonal finite difference scheme for the 2d phase field crystal amplitude equations in this paper we construct an energy stable finite difference scheme for the amplitude expansion equations for the two dimensional phase field crystal pfc model the equations are formulated in a periodic hexagonal domain with respect to the reciprocal lattice vectors to achieve a provably unconditionally energy stable and solvable scheme to our knowledge this is the first such energy stable scheme for the pfc amplitude equations the convexity of each part in the amplitude equations is analyzed in both the semi discrete and fully discrete cases energy stability is based on a careful convexity analysis for the energy in both the spatially continuous and discrete cases as a result unique solvability and unconditional energy stability are available for the resulting scheme moreover we show that the scheme is point wise stable for any time and space step sizes an efficient multigrid solver is devised to solve the scheme and a few numerical experiments are presented including grain rotation and shrinkage and grain growth studies as examples of the strength and robustness of the proposed scheme and solver

on the distribution of squarefree integers in arithmetic progressions we investigate the error term of the asymptotic formula for the number of squarefree integers up to some bound and lying in some arithmetic progression in particular we prove an upper bound for its variance as a varies over mathbb z q mathbb z times which considerably improves upon earlier work of blomer

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

a differential delay equation arising from the sieve of eratosthenes consideration is given to the differential delay equation introduced by buchstab 1937 in connection with an asymptotic formula for the uncanceled terms in the sieve of eratosthenes maier 1985 used this result to show there is unexpected irreqularity in the distribution of primes in short intervals the function omega u is studied in this paper using numerical and analytical techniques the results are applied to give some numerical constants in maier s theorem

efficient energy stable numerical schemes for a phase field moving contact line model in this paper we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line the model is a coupled system that consists of incompressible navier stokes equations with a generalized navier boundary condition and cahn hilliard equation in conserved form in both schemes the projection method is used to deal with the navier stokes equations and stabilization approach is used for the non convex ginzburg landau bulk potential by some subtle explicit implicit treatments we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions an efficient spectral galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes numerical results show that the proposed schemes are very efficient and accurate we proposed two linear energy stable numerical schemes for a phase field moving contact line model discrete energy laws that are consistent to the continuous ones are proved the efficiency and fast convergence of the proposed schemes with spectral spatial approximations are numerically verified

computational studies of coarsening rates for the cahn hilliard equation with phase dependent diffusion mobility we study computationally coarsening rates of the cahn hilliard equation with a smooth double well potential and with phase dependent diffusion mobilities the latter is a feature of many materials systems and makes accurate numerical simulations challenging our numerical simulations confirm earlier theoretical predictions on the coarsening dynamics based on asymptotic analysis we demonstrate that the numerical solutions are consistent with the physical gibbs thomson effect even if the mobility is degenerate in one or both phases for the two sided degenerate mobility we report computational results showing that the coarsening rate is on the order of l c t 1 4 independent of the volume fraction of each phase for the one sided degenerate mobility that is non degenerate in the positive phase but degenerate in the negative phase we illustrate that the coarsening rate depends on the volume fraction of the positive phase for large positive volume fractions the coarsening rate is on the order of l c t 1 3 and for small positive volume fractions the coarsening rate becomes l c t 1 4

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

prediction of ordinal outcomes when the association between predictors and outcome differs between outcome levels there are a number of regression models which are widely used to predict ordinal outcomes the commonly used models assume that all predictor variables have a similar effect at all levels of the outcome variable if this is not the case for example if some variables predict susceptibility to a disease and others predict the severity of the disease then a more complex model is required one possibility is the multinomial logistic regression model which assumes that the predictor variables have different effects at all levels of the outcome variable an alternative is to use the stereotype family of regression models a one dimensional stereotype model makes the assumption that the effect of each predictor is the same at all outcome levels however it is possible to fit stereotype models with more than one dimension up to a maximum of min k 1 p where k is the number of outcome categories and p is the number of predictor variables a stereotype model of this maximum dimension is equivalent to a multinomial logistic regression model in that it will produce the same predicted values and log likelihood if there are sufficient outcome levels and or predictor variables there may be a number of stereotype models of differing dimension the method is illustrated with an example of prediction of damage to joints in rheumatoid arthritis

computations on the cahn hilliard model of solidification we begin numerical studies of interface migration on the two dimensional cahn hilliard equation our results add evidence to support the connection between this equation and the well known quasistatic model of solidification finite elements coupled with a second order time discretization scheme are used the nonlinear systems of equations that arise on each time step are solved by fixed point iteration while the linear systems are solved by preconditioned conjugate gradients

on integers free of large prime factors the number l x y of integers y has been given satisfactory estimates in the regions y exp log log x 5 3 6 in the intermediate range only very crude estimates have been obtained so far we close this gap and give an expression which approximates l x y uniformly for x y 2 within a factor 1 o log y log x log y y as an application we derive a simple formula for ti cx y fl x y where 1 c y we also prove a short interval estimate for l x y

first and second order energy stable methods for the modified phase field crystal equation abstract the phase field crystal pfc model was extended to the modified phase field crystal mpfc model which is a sixth order nonlinear damped wave equation to include not only diffusive dynamics but also elastic interactions in this paper we present temporally first and second order accurate methods for the mpfc equation which are based on an appropriate splitting of the energy for the pfc equation and we use the fourier spectral method for the spatial discretization the first and second order methods are shown analytically to be unconditionally stable with respect to the energy and pseudoenergy of the mpfc equation respectively numerical experiments are presented demonstrating the accuracy and energy stability of the proposed methods

an adaptive variational procedure for the conservative and positivity preserving allen cahn phase field model abstract in this paper we present an adaptive variational procedure for unstructured meshes to capture fluid fluid interfaces in two phase flows the two phases are modeled by the phase field finite element formulation which involves the conservative allen cahn equation coupled with the incompressible navier stokes equations the positivity preserving variational formulation is designed to maintain the bounded and stable solution of the allen cahn equation for the adaptivity procedure we consider the residual based error estimates for the underlying differential equations of the two phase system in particular the adaptive refinement coarsening is carried out by the newest vertex bisection algorithm by evaluating the residual error indicators based on the error estimates of the allen cahn equation the coarsening algorithm avoids the storage of the tree data structures for the hierarchical mesh thus providing the ease of numerical implementation furthermore the proposed nonlinear adaptive partitioned procedure aims at reducing the amount of coarsening while maintaining the convergence properties of the underlying nonlinear coupled differential equations we investigate the adaptive phase field finite element scheme through the spinodal decomposition in a complex curved geometry and the volume conserved interface motion driven by the mean curvature flow for two circles in a square domain we then assess the accuracy and efficiency of the proposed procedure by modeling the free surface motion in a sloshing tank in contrast to the non adaptive eulerian grid counterpart we demonstrate that the mesh adaptivity remarkably reduces the degrees of freedom and the computational cost by nearly half for similar accuracy the mass loss in the allen cahn equation via adaptivity process is also reduced by nearly three times compared to the non adaptive mesh finally we apply the adaptive numerical framework to solve the application of a dam breaking problem with topological changes

a convergent convex splitting scheme for the periodic nonlocal cahn hilliard equation in this paper we devise a first order in time second order in space convex splitting scheme for the periodic nonlocal cahn hilliard equation the unconditional unique solvability energy stability and ell infty 0 t ell 4 l 0 t l 4 stability of the scheme are established using the a priori stabilities we prove error estimates for our scheme in both the ell infty 0 t ell 2 l 0 t l 2 and ell infty 0 t ell infty l 0 t l norms the proofs of these estimates are notable for the fact that they do not require point wise boundedness of the numerical solution nor a global lipschitz assumption or cut off for the nonlinear term the ell 2 l 2 convergence proof requires no refinement path constraint while the one involving the ell infty l norm requires only a mild linear refinement constraint s le c h s c h the key estimates for the error analyses take full advantage of the unconditional ell infty 0 t ell 4 l 0 t l 4 stability of the numerical solution and an interpolation estimate of the form left phi right 4 le c left phi right 2 alpha left nabla h phi right 2 1 alpha alpha frac 4 d 4 d 1 2 3 4 c 2 h 2 1 4 d 4 d 1 2 3 which we establish for finite difference functions we conclude the paper with some numerical tests that confirm our theoretical predictions

an efficient algorithm for solving the phase field crystal model we present and discuss the development of an unconditionally stable algorithm used to solve the evolution equations of the phase field crystal pfc model this algorithm allows for an arbitrarily large algorithmic time step as the basis for our analysis of the accuracy of this algorithm we determine an effective time step in fourier space we then compare our calculations with a set of representative numerical results and demonstrate that this algorithm is an effective approach for the study of the pfc models yielding a time step effectively 180 times larger than the euler algorithm for a representative set of material parameters as the pfc model is just a simple example of a wide class of density functional theories we expect this method will have wide applicability to modeling systems of considerable interest to the materials modeling communities

calculation of polychotomous logistic regression parameters using individualized regressions summary the use of individualized logistic regression in which a series of separate simple logistic regression analyses are performed as a replacement for polychotomous logistic regression is studied the asymptotic relative efficiencies of the individual parameter estimates are observed to be generally high as are the efficiencies of predicted probability estimates and to a somewhat lesser extent joint tests of parameters from different regressions

variants of equidistribution in arithmetic progression and the twin prime conjecture let mathcal h h 1 ldots h k be a fixed set of k distinct non negative integers we show that bombieri vinogradov type theorems for a certain class of functions f in arithmetic progressions can be extended to the product f n mu 2 n h 1 cdots mu 2 n h k up to almost the same level of distribution as that for f as an application of this result we show equidistribution of tuples of squarefree integers in arithmetic progressions with a level of distribution up to 2 3 this generalizes a result of orr j number theory 3 474 497 1971 we also formulate arithmetic progression analogues of the chowla conjecture on correlations of the mobius function we are able to prove this for some special cases thereby generalizing a result of siebert and wolke math z 122 4 327 341 1971 we also show the relevance of such conjectures to the twin prime problem

energy stable multigrid method for local and non local hydrodynamic models for freezing abstract in this paper we present a numerical method for hydrodynamic models that arise from time dependent density functional theories of freezing the models take the form of compressible navier stokes equations whose pressure is determined by the variational derivative of a free energy which is a functional of the density field we present unconditionally energy stable and mass conserving implicit finite difference methods for the models the methods are based on a convex splitting of the free energy and that ensures that a discrete energy is non increasing for any choice of time and space step the methods are applicable to a large class of models including both local and non local free energy functionals the theoretical basis for the numerical method is presented in a general context the method is applied to problems using two specific free energy functionals one local and one non local functional a nonlinear multigrid method is used to solve the numerical method which is nonlinear at the implicit time step the non local functional which is a convolution operator is approximated using the discrete fourier transform numerical simulations that confirm the stability and accuracy of the numerical method are presented

a discontinuous galerkin method for the cahn hilliard equation a discontinuous galerkin finite element method has been developed to treat the high order spatial derivatives appearing in the cahn hilliard equation the cahn hilliard equation is a fourth order nonlinear parabolic partial differential equation originally proposed to model phase segregation of binary alloys the developed discontinuous galerkin approach avoids the need for mixed finite element methods coupled equations or interpolation functions with a high degree of continuity that have been employed in the literature to treat the fourth order spatial derivatives the variational formulation of the discontinuous galerkin method its implementation and numerical examples are presented in this communication it is also shown under what conditions the method is stable and an error estimate in an energy type norm is presented the method is evaluated by comparison with a standard finite element treatment in which the cahn hilliard equation is decomposed into two coupled partial differential equations

primes in intervals of bounded length in april 2013 yitang zhang proved the existence of a finite bound b such that there are infinitely many pairs of distinct primes which differ by no more than b this is a massive breakthrough makes the twin prime conjecture look highly plausible which can be re interpreted as the conjecture that one can take b 2 and his work helps us to better understand other delicate questions about prime numbers that had previously seemed intractable the original purpose of this talk was to discuss zhang s extraordinary work putting it in its context in analytic number theory and to sketch a proof of his theorem zhang had even proved the result with b 70 000 000 moreover a co operative team polymath8 collaborating only on line had been able to lower the value of b to 4680 not only had they been more careful in several difficult arguments in zhang s original paper they had also developed zhang s techniques to be both more powerful and to allow a much simpler proof indeed the proof of zhang s theorem that will be given in the write up of this talk is based on these developments in november inspired by zhang s extraordinary breakthrough james maynard dramatically slashed this bound to 600 by a substantially easier method both maynard and terry tao who had independently developed the same idea were able to extend their proofs to show that for any given integer m 1 there exists a bound bm such that there are infinitely many intervals of length bm containing at least m distinct primes we will also prove this much stronger result herein even showing that one can take bm e 8m if zhang s method is combined with the maynard tao set up then it appears that the bound can be further reduced to 576 if all of these techniques could be pushed to their limit then we would obtain b b2 12 so new ideas are still needed to have a feasible plan for proving the twin prime conjecture the article will be split into two parts the first half which appears here we will introduce the work of zhang polymath8 maynard and tao and explain their arguments that allow them to prove their spectacular results as we will discuss zhang s main novel contribution is an estimate for primes in relatively short arithmetic progressions the second half of this article sketches a proof of this result the bulletin article will contain full details of this extraordinary work part 1 primes in short intervals

analysis of mixed interior penalty discontinuous galerkin methods for the cahn hilliard equation and the hele shaw flow this paper proposes and analyzes two fully discrete mixed interior penalty discontinuous galerkin dg methods for the fourth order nonlinear cahn hilliard equation both methods use the backward euler method for time discretization and interior penalty dg methods for spatial discretization they differ from each other on how the nonlinear term is treated one of them is based on fully implicit time stepping and the other uses energy splitting time stepping the primary goal of the paper is to prove the convergence of the numerical interfaces of the dg methods to the interface of the hele shaw flow this is achieved by establishing error estimates that depend on epsilon 1 only in some low polynomial orders instead of exponential orders similar to x feng and a prohl numer math 74 2004 pp 47 84 the crux is to prove a discrete spectrum estimate in the discontinuous galerkin finite element space however the validity of such a result is not obvious because the dg space is not a subspa