on structure sensitivity of degree based topological indices one of the general requirements for any topological index ti is that similar molecules have near lying ti values which is referred to as smoothness curiously however smoothness of topological indices was until now never examined and never quantified we now propose a pertinent mathematical model for this property and undertake a comparative study of the structure sensitivity of 12 degree based topological indices by using trees

a stable and conservative finite difference scheme for the cahn hilliard equation we propose a stable and conservative finite difference scheme to solve numerically the cahn hilliard equation which describes a phase separation phenomenon numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill posed problem we design a new difference scheme based on a general strategy proposed recently by furihata and mori the new scheme inherits characteristic properties the conservation of mass and the decrease of the total energy from the equation the decrease of the total energy implies boundedness of discretized sobolev norm of the solution this in turn implies by discretized sobolev s lemma boundedness of max norm of the solution and hence the stability of the solution an error estimate for the solution is obtained and the order is o delta x 2 delta t 2 numerical examples demonstrate the effectiveness 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

on topological properties of the line graphs of subdivision graphs of certain nanostructures in the study of qsar qspr topological indices such as shultz index generalized randic index zagreb index general sum connectivity index atom bond connectivity abc index and geometric arithmetic ga index are exploited to estimate the bioactivity of chemical compounds a topological index attaches a chemical structure with a numeric number there are numerous applications of graph theory in this field of research in this paper we computed generalized randic general zagreb general sum connectivity abc ga abc4 and ga5 indices of the line graphs of 2d lattice nanotube and nanotorus of tuc4c8p q by using the concept of subdivision

propagation of singularities along broken characteristics abstract this paper contributes to the analysis of propagation of singularities for semiconcave solutions of hamilton jacobi equations under certain conditions we establish the existence and uniqueness of certain broken characteristics termed strong characteristics if u is a viscosity solution of a hamilton jacobi equation then strong characteristics carry the singularities of u

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

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

singular gradient flow of the distance function and homotopy equivalence let m be a riemannian manifold and let varomega be a bounded open subset of m it is well known that significant information about the geometry of varomega is encoded into the properties of the distance d partial varomega from the boundary of varomega here we show that the generalized gradient flow associated with the distance preserves singularities that is if x 0 is a singular point of d partial varomega then the generalized characteristic starting at x 0 stays singular for all times as an application we deduce that the singular set of d partial varomega has the same homotopy type as varomega

a second order accurate linearized difference scheme for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is of fourth order in space in this paper a linearized finite difference scheme is derived by the method of reduction of order it is proved that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete l2 norm the coefficient matrix of the difference system is symmetric and positive definite so many well known iterative methods e g gauss seidel sor can be used to solve the system

propagation of singularities for solutions of hamilton jacobi equations abstract we consider the viscosity solution of the cauchy problem for a class of hamilton jacobi equations and we show that the points of the c 1 singular support of such a function propagate along the generalized characteristics for all the times

generation of singularities from the initial datum for hamilton jacobi equations abstract we study the generation of singularities from the initial datum for a solution of the cauchy problem for a class of hamilton jacobi equations of evolution for such equations we give conditions for the existence of singular generalized characteristics starting at the initial time from a given point of the domain depending on the properties of the proximal subdifferential of the initial datum in a neighbourhood of that point

more estimates for eigenvalues and singular values abstract there is an interesting relation between the angle that a matrix forms with the identity and its eigenvalues then we show for symmetric matrices that some information about the matrix can be known if the angle is computed a lower bound for the maximum eigenvalue is obtained for symmetric positive semidefinite matrices this inequality and the upper bound for eigenvalues given in 5 provide us an interval in which must lie the maximum eigenvalue finally some results from 5 and the first part of this paper are generalized for the non symmetric case

energy stable predictor corrector schemes for the cahn hilliard equation abstract in this paper we construct a new class of predictor corrector time stepping schemes for the cahn hilliard equation which are linear second order accurate in time unconditionally energy stable and uniquely solvable then we present the stability and error estimates of the semi discrete numerical schemes for solving the cahn hilliard equation with general nonlinear bulk potentials the semi discrete scheme is further discretized using the compact central finite difference method several numerical examples are shown to verify the theoretical results in particular the numerical simulations show that the predictor corrector schemes reach the second order convergence rate at relatively larger time step sizes than the classical linear schemes the numerical strategies and theoretical tools developed in this article could be readily applied to study other phase field models or models that can be cast as gradient flow problems

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

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

propagation of singularities for solutions of nonlinear first order partial differential equations few results are available in the mathematical literature for studying the structure of the singular set of a weak solution u of f x u du 0 this paper provides new techniques to analyse such a set when u is semiconcave and f is a nonlinear convex function with respect to p the main objective achieved here is a classification of the singularities of u that propagate along lipschitz arcs such a propagation phenomenon is also described by means of a generalized characteristics inclusion

upper bound on the coarsening rate for an epitaxial growth model we study a specific example of energy driven coarsening in two space dimensions the energy is u 2 1 u 2 2 the evolution is the fourth order pde representing steepest descent this equation has been proposed as a model of epitaxial growth for systems with slope selection numerical simulations and heuristic arguments indicate that the standard deviation of u grows like t1 3 and the energy per unit area decays like t 1 3 we prove a weak one sided version of the latter statement the time averaged energy per unit area decays no faster than t 1 3 our argument follows a strategy introduced by kohn and otto in the context of phase separation combining i a dissipation relation ii an isoperimetric inequality and iii an ode lemma the interpolation inequality is new and rather subtle our proof is by contradiction relying on recent compactness results for the aviles giga energy 2003 wiley periodicals inc

topological indices of the line graph of subdivision graphs and their schur bounds in this paper we investigate topological indices of the line graphs of the tadpole graphs wheel graphs and ladder graphs using the subdivision concept which generalize the results of ranjini et al 2011 in addition we present the schur bound for the general sum connectivity co index

excess action and broken characteristics for hamilton jacobi equations abstract we study propagation of singularities for hamilton jacobi equations s t h t x s 0 t x 0 r n by means of the excess lagrangian action and a related class of characteristics in a sense the excess action gauges how far a curve x t is from being action minimizing for a given viscosity solution s t x of the hamilton jacobi equation broken characteristics are defined as curves along which the excess action grows at the slowest pace possible in particular we demonstrate that broken characteristics carry the singularities of the viscosity solution

on linear schemes for a cahn hilliard diffuse interface model numerical schemes to approximate the cahn hilliard equation have been widely studied in recent times due to its connection with many physically motivated problems in this work we propose two type of linear schemes based on different ways to approximate the double well potential term the first idea developed in the paper allows us to design a linear numerical scheme which is optimal from the numerical dissipation point of view meanwhile the second one allows us to design unconditionally energy stable linear schemes for a modified energy we present first and second order in time linear schemes to approximate the ch problem detailing their advantages over other linear schemes that have been previously introduced in the literature furthermore we compare all the schemes through several computational experiments

unconditionally energy stable time stepping scheme for cahn morral equation an unconditionally energy stable time stepping scheme is introduced to solve cahn morral like equations in the present study it is constructed based on the combination of david eyre s time stepping scheme and schur complement approach although the presented method is general and independent of the choice of homogeneous free energy density function term logarithmic and polynomial energy functions are specifically considered in this paper the method is applied to study the spinodal decomposition in multi component systems and optimal space tiling problems a penalization strategy is developed in the case of later problem to avoid trivial solutions extensive numerical experiments demonstrate the success and performance of the presented method according to the numerical results the method is convergent and energy stable independent of the choice of time stepsize its matlab implementation is included in the appendix for the numerical evaluation of algorithm and reproduction of the presented results extension of eyre s convex concave splitting scheme to multiphase systems efficient solution of spinodal decomposition in multi component systems efficient solution of least perimeter periodic space partitioning problem developing a penalization strategy to avoid trivial solutions presentation of matlab implementation of the introduced algorithm

linear first and second order unconditionally energy stable numerical schemes for the phase field model of homopolymer blends abstract in this paper we develop a series of efficient numerical schemes to solve the phase field model for homopolymer blends the governing system is derived from the energetic variational approach of a total free energy that consists of a nonlinear logarithmic flory huggins potential and a gradient entropy with a concentration dependent de gennes type coefficient the main challenging issue to solve this kind of models numerically is about the time marching problem i e how to develop suitable temporal discretizations for the nonlinear terms in order to preserve the energy stability at the discrete level we solve this issue in this paper by developing the first and second order temporal approximation schemes based on the invariant energy quadratization method where all nonlinear terms are treated semi explicitly consequently the resulting numerical schemes lead to a symmetric positive definite linear system to be solved at each time step the unconditional energy stabilities are further proved various numerical simulations of 2d and 3d are presented to demonstrate the stability and the accuracy of the proposed schemes

the lax oleinik semi group a hamiltonian point of view the weak kam theory was developed by fathi in order to study the dynamics of convex hamiltonian systems it somehow makes a bridge between viscosity solutions of the hamilton jacobi equation and mather invariant sets of hamiltonian systems although this was fully understood only a posteriori these theories converge under the hypothesis of convexity and the richness of applications mostly comes from this remarkable convergence in the present course we provide an elementary exposition of some of the basic concepts of weak kam theory in a companion lecture albert fathi exposes the aspects of his theory which are more directly related to viscosity solutions here on the contrary we focus on dynamical applications even if we also discuss some viscosity aspects to underline the connections with fathi s lecture the fundamental reference on weak kam theory is the still unpublished book of albert fathi textit weak kam theorem in lagrangian dynamics although we do not offer new results our exposition is original in several aspects we only work with the hamiltonian and do not rely on the lagrangian even if some proofs are directly inspired from the classical lagrangian proofs this approach is made easier by the choice of a somewhat specific setting we work on rm d and make uniform hypotheses on the hamiltonian this allows us to replace some compactness arguments by explicit estimates for the most interesting dynamical applications however the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic in space hamiltonians our exposition is centered on the cauchy problem for the hamilton jacobi equation and the lax oleinik evolution operators associated to it dynamical applications are reached by considering fixed points of these evolution operators the weak kam solutions the evolution operators can also be used for their regularizing properties this opens a second way to dynamical applications

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

on dynamics of lagrangian trajectories for hamilton jacobi equations characteristic curves of a hamilton jacobi equation can be seen as action minimizing trajectories of fluid particles however this description is valid only for smooth solutions for nonsmooth viscosity solutions which give rise to discontinuous velocity fields this picture holds only up to the moment when trajectories hit a shock and cease to minimize the lagrangian action in this paper we discuss two physically meaningful regularization procedures one corresponding to vanishing viscosity and another to weak noise limit we show that for any convex hamiltonian a viscous regularization allows us to construct a nonsmooth flow that extends particle trajectories and determines dynamics inside the shock manifolds this flow consists of integral curves of a particular effective velocity field which is uniquely defined everywhere in the flow domain and is discontinuous on shock manifolds the effective velocity field arising in the weak noise limit is generally non unique and different from the viscous one but in both cases there is a fundamental self consistency condition constraining the dynamics

on tsertsvadze s difference scheme for the kuramoto tsuzuki equation in this paper we have proved the second order convergence in l norm of the tsertsvadze s difference scheme for the kuramoto tsuzuki equation the existence uniqueness and iterative algorithm are also discussed in detail furthermore a l second order convergent linearized difference scheme is given for inhomogeneous equation all results are obtained without any restrictions on the meshsizes at last a numerical example is presented

stabilized linear semi implicit schemes for the nonlocal cahn hilliard equation abstract comparing with the well known classic cahn hilliard equation the nonlocal cahn hilliard equation is equipped with a nonlocal diffusion operator and can describe more practical phenomena for modeling phase transitions of microstructures in materials on the other hand it evidently brings more computational costs in numerical simulations thus efficient and accurate time integration schemes are highly desired in this paper we propose two energy stable linear semi implicit methods with first and second order temporal accuracies respectively for solving the nonlocal cahn hilliard equation the temporal discretization is done by using the stabilization technique with the nonlocal diffusion term treated implicitly while the spatial discretization is carried out by the fourier collocation method with fft based fast implementations the energy stabilities are rigorously established for both methods in the fully discrete sense numerical experiments are conducted for a typical case involving gaussian kernels we test the temporal convergence rates of the proposed schemes and make a comparison of the nonlocal phase transition process with the corresponding local one in addition long time simulations of the coarsening dynamics are also performed to predict the power law of the energy decay

on the zagreb indices of the line graphs of the subdivision graphs the aim of this paper is to investigate the zagreb indices of the line graphs of the tadpole graphs wheel graphs and ladder graphs using the subdivision concepts