the separating variety for the basic representations of the additive group abstract for a group g acting on an affine variety x the separating variety is the closed subvariety of x x encoding which points of x are separated by invariants we concentrate on the indecomposable rational linear representations v n of dimension n 1 of the additive group of a field of characteristic zero and decompose the separating variety into the union of irreducible components we show that if n is odd divisible by four or equal to two the closure of the graph of the action which has dimension n 2 is the only component of the separating variety in the remaining cases there is a second irreducible component of dimension n 1 we conclude that in these cases there are no polynomial separating algebras

computation of invariants for reductive groups abstract we will give an algorithm for computing generators of the invariant ring for a given representation of a linearly reductive group the algorithm basically consists of a single grobner basis computation we will also show a connection between some open conjectures in commutative algebra and finding good degree bounds for generating invariants

special functions arising in the study of semi linear equations in circular domains rayleigh functions are defined by the formula s l n n 1 1 l n n 2 l where l 1 2 3 l n n 0 are zeros of the bessel function j n x and n 1 2 3 is the number of the zero these functions appear in the classical problems of vibrating circular membranes heat conduction in cylinders and diffraction through circular apertures in the present paper it is shown that a new family of special functions convolutions of rayleigh functions with respect to the bessel index 1 r l m p k p k m q s 1 1 l p q 2 l1 l k s 2 lforl 1 2 m 0 1 2 arises in constructing solutions of semi linear evolution equations in circular domains see also v varlamov convolution of rayleigh functions with respect to the bessel index j math anal appl 306 2005 413 424 as an example of its application a forced cahn hilliard equation is considered in a unit disc with homogeneous boundary and initial conditions construction of its global in time solutions involves the use of r 1 m and r 2 m a general representation of r l m is deduced and on the basis of that a particular result for r 2 m is obtained convenient for computing its asymptotics as m the latter issue is important for establishing a function space to which a solution of the corresponding problem belongs

simulating the kinematics of completely faceted surfaces we fully generalize a previously developed computational geometry tool 1 to perform large scale simulations of arbitrary 2 1d faceted surfaces z h x y the method is an explicit front tracking scheme that uses a compact three component facet edge junction storage mode because it naturally mirrors the intrinsic surface structure this scheme allows both rapid simulation of large ensembles and easy extraction of geometrical statistics to do so it must overcome the barrier of detecting and resolving a wide variety of topological changes that occur during surface evolution however while the variety of topological events is larger than in the case of 2d cellular networks it is still limited and our main result is a comprehensive algorithm performing these changes in the code

adaptive localized replay an efficient integration scheme for accurate simulation of coarsening dynamical systems coarsening dynamical systems cds of ordinary differential equations odes have as a defining feature the presence of discontinuities in the governing equations due to the removal of vanishing elements from the system simple commonly used integration strategies for odes neglect these discontinuities resulting in large errors in contrast available methods to resolve discontinuities exhibit total running times that scale like o n 2 as the number of elements n the resulting need to choose between either speed or accuracy frustrates attempts to gather well converged statistical data from large ensembles of elements a common goal in studies of coarsening in this work we introduce a numerical scheme called adaptive localized replay alr which overcomes this dilemma provided the underlying system has sparse neighbor dependence by analyzing the system after each timestep and then selectively re simulating only those small parts of the system that were affected by discontinuities we attain high order accuracy while maintaining running times of o n using a reference implementation of our algorithm applied to a sample problem from the field of faceted surface evolution we obtain convergence and runtime results for the alr method which compares favorably with existing methods

transient and self similar dynamics in thin film coarsening abstract we study coarsening in a simplified model of one dimensional thin films of viscous fluids on hydrophobic substrates lubrication theory shows that such films are unstable and dewet to form droplets that then aggregate over long timescales the masses and positions of the droplets can be described by a coarsening dynamical system cds consisting of odes and deletion rules we develop discrete and continuous mean field models that reproduce the well known n t o t 2 5 long time statistical power law for the number of drops a lifshitz slyozov wagner type lsw continuous model predicts the self similar distribution of drop masses matching with histograms produced by cds simulations and the discrete mean field model we also describe the distribution of drops following homogeneous versus heterogeneous dewetting and use these as initial conditions for the cds simulations that yield characteristic staircasing transients transients can also include recurring spike formation behavior in the mass distribution for idealized initial conditions we show that the transient dynamics can span the full coarsening process bypassing the power law regime entirely

wigner measures and codimension two crossings this article gives a semiclassical description of nucleonic propagation through codimension two crossings of electronic energy levels codimension two crossings are the simplest energy level crossings which affect the born oppenheimer approximation in the zeroth order term the model we study is a two level schrodinger equation with a laplacian as kinetic operator and a matrix valued linear potential whose eigenvalues cross if the two nucleonic coordinates equal zero we discuss the case of well localized initial data and obtain a description of the wavefunction s two scaled wigner measure and of the weak limit of its position density which is valid globally in time

on the convective cahn hilliard equation with degenerate mobility abstract in this paper we study the existence of weak solutions for the convective cahn hilliard equation with degenerate mobility based on the schauder type estimates we establish the global existence of classical solutions for regularized problems after establishing some necessary uniform estimates on the approximate solutions we prove the existence of weak solutions the nonnegativity and the finite speed of propagation of perturbations of solutions are also discussed

solving the regularized strongly anisotropic cahn hilliard equation by an adaptive nonlinear multigrid method abstract we present efficient second order accurate and adaptive finite difference methods to solve the regularized strongly anisotropic cahn hilliard equation in 2d and 3d when the surface energy anisotropy is sufficiently strong there are missing orientations in the equilibrium level curves of the diffuse interface solutions corresponding to those missing from the sharp interface wulff shape and the anisotropic cahn hilliard equation becomes ill posed to regularize the equation a higher order derivative term is added to the energy this leads to a sixth order nonlinear parabolic equation for the order parameter an implicit time discretization is used to remove the high order time step stability constraints dynamic block structured cartesian mesh refinement is used to highly resolve narrow interfacial layers a multilevel nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level one of the keys to the success of the method is the treatment of the anisotropic term this term is discretized in conservation form in space and is discretized fully implicitly in time numerical simulations are presented that confirm the accuracy efficiency and stability of the scheme we study the dynamics of interfaces under strong anisotropy and compare near equilibrium diffuse interface solutions to the sharp interface wulff shapes in 2d and 3d we also simulate large scale coarsening of a corrugated surface in 3d evolving by anisotropic surface diffusion we show the emergence of long range order during coarsening and an interesting mechanism of ordered coarsening

global solutions and smoothing effects for semi linear evolution equations in circular domains abstract the paper deals with global in time solutions of semi linear evolution equations in circular domains these solutions are constructed by means of the series of eigenfunctions of the laplace operator in the corresponding region a forced cahn hilliard type equation in a unit disc is considered as an example the current work focuses on revealing the mechanism of nonlinear smoothing i e on tracing the influence of smoothness of the source term on the regularity of solutions of the nonlinear mixed problem to this end convolutions of rayleigh functions with respect to the bessel index are employed the study of this new family of special functions was initiated in v varlamov convolution of rayleigh functions with respect to the bessel index j math anal appl 306 2005 413 424 v varlamov convolutions of rayleigh functions and their application to semi linear equations in circular domains j math anal appl 327 2007 1461 1478 in order to reveal the effect of nonlinear smoothing anisotropic sobolev spaces h s are introduced they are based on the sobolev spaces h s weighted by tangential derivatives so that the index is responsible for the smoothness in and s is the usual sobolev index global in time solutions of the mixed problem in question are constructed and additional smoothness with respect to the angular coordinate is established in the anisotropic sobolev spaces the above mentioned special functions are essentially used for this purpose

surface evolution of elastically stressed films under deposition by a diffuse interface model we consider the heteroepitaxial growth of thin films by numerical simulations within a diffuse interface model the model is applicable to describe the self organization of nanostructures the influence of strain surface energies and kinetics on the surface evolution is considered a matched asymptotic analysis shows the formal convergence of an anisotropic viscous cahn hilliard model to a general surface evolution equation the system is solved by adaptive finite elements in three dimensions and in special cases compared with sharp interface models

dynamics of domain walls governed by the convective cahn hilliard equation the convective cahn hilliard cch equation ut uxx u u 3 xx d 2 u 2 x 0 has been suggested recently for the description of several physical phenomena including spinodal decomposition of driven phase separating systems in an external field instability of steps moving on a crystal surface and faceting of growing thermodynamically unstable surfaces in this paper the dynamics of domain walls kinks governed by the convective cahn hilliard equation is studied by means of asymptotic and numerical methods a special attention is paid to the dynamics of kink pairs and triplets that play crucial role in the coarsening ostwald ripening process for the driving parameter d d 0 2 3 and large distance between the kinks l 1 analytical formulas are found that describe the motion of the kink pairs and triplets the analytical formulas are in excellent agreement with the results of direct numerical simulations of cch equation they explain the logarithmically slow coarsening observed for the kink dynamics governed by cch equation when the distances between the kinks become large unlike the fast coarsening for moderate distances between the kinks at intermediate stages 2005 elsevier b v all rights reserved

separating invariants for 2 2 matrices abstract a minimal separating set is found for the algebra of matrix invariants of several 2 2 matrices over an infinite field of arbitrary characteristic

on a higher order convective cahn hilliard type equation a higher order convective cahn hilliard type equation that describes the faceting of a growing surface is considered with periodic boundary conditions by using a galerkin approach the existence

asymptotic solutions for the equilibrium crystal shape with small corner energy regularization dynamic models for facet formation often employ a regularization of the surface energy based on a corner energy term here we consider the effect of this regularization on the equilibrium shape of a solid particle in two dimensions using matched asymptotic expansions we determine the explicit solution for the corner shape in the presence of the regularization our results show that for a class of surface energy anisotropy models the regularized solution approaches the classic sharp corner results as the regularization approaches zero the results validate the use of the regularization in numerical calculations for the equilibrium problem finally a byproduct of the analysis is an exact solution for the equilibrium shape of a semi infinite wedge in the presence of the regularization

emergent parabolic scaling of nano faceting crystal growth nano faceted crystals answer the call for self assembled physico chemically tailored materials with those arising from a kinetically mediated response to free energy disequilibria thermokinetics holding the greatest promise the dynamics of slightly undercooled crystal melt interfaces possessing strongly anisotropic and curvature dependent surface energy and evolving under attachment detachment limited kinetics offer a model system for the study of thermokinetic effects the fundamental non equilibrium feature of this dynamics is explicated through our discovery of one dimensional convex and concave translating fronts solitons whose constant asymptotic angles provably deviate from the thermodynamically expected wulff angles in direct proportion to the degree of undercooling these thermokinetic solitons induce a novel emergent facet dynamics which is exactly characterized via an original geometric matched asymptotic analysis we thereby discover an emergent parabolic symmetry of its coarsening facet ensembles which naturally implies the universal scaling law l t 1 2 for the growth in time t of the characteristic length l

a regularized equation for anisotropic motion by curvature for realistic interfacial energies the equations of anisotropic motion by curvature exhibit backward parabolic behavior over portions of their domain thereby inducing phenomena such as the formation of facets and wrinkles in this paper a physically consistent regularized equation that may be used to analyze such phenomena is derived

non blow up phenomenon for the cahn hilliard equation with non constant mobility abstract in this paper we prove that the finite time blow up phenomenon might not occur for the cahn hilliard equation with non constant mobility and cubic nonlinearity which is quite different from the case of constant mobility we reveal such a phenomenon under some structure condition on the mobility

mean field theory for coarsening faceted surfaces a mean field theory is developed for the scale invariant length distributions observed during the coarsening of generic one dimensional faceted surfaces this theory closely follows the lifshitz slyozov wagner theory of ostwald ripening in two phase systems but the mechanism of coarsening in faceted surfaces requires the addition of convolution terms recalling work on particle coalescence and induces an unexpected coupling between the convolution and the rate of facet loss as a generic framework the theory concisely illustrates how the universal processes of facet removal and neighbor merger are moderated by the system specific mean field velocity describing average rates of length change for a simple example facet dynamics associated with the directional solidification of a binary alloy agreement between the predicted scaling state and that observed after direct numerical simulation of 40 000 000 facets is reasonable given the limiting assumption of noncorrelation between neighbors relaxing this assumption is a clear path forward toward improved quantitative agreement with data in the future

coarsening dynamics of a one dimensional driven cahn hilliard system we study the one dimensional cahn hilliard equation with an additional driving term representing say the effect of gravity we find that the driving field e has an asymmetric effect on the solution for a single stationary domain wall or kink the direction of the field determining whether the analytic solutions found by leung j stat phys 61 345 1990 are unique the dynamics of a kink antikink pair bubble is then studied the behavior of a bubble is dependent on the relative sizes of a characteristic length scale mathit e mathrm ensuremath 1 where e is the driving field and the separation l of the interfaces for el ensuremath gg 1 the velocities of the interfaces are negligible while in the opposite limit a traveling wave solution is found with a velocity v ensuremath propto e l for this latter case el ensuremath ll 1 a set of reduced equations describing the evolution of the domain lengths is obtained for a system with a large number of interfaces and implies a characteristic length scale growing as et 1 2 numerical results for the domain size distribution and structure factor confirm this behavior and show that the system exhibits dynamical scaling from very early times textcopyright 1996 the american physical society

separating invariants and finite reflection groups abstract a separating algebra is roughly speaking a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants in this paper we introduce a geometric notion of separating algebra this allows us to prove that only groups generated by reflections may have polynomial separating algebras and only groups generated by bireflections may have complete intersection separating algebras

invariants and separating morphisms for algebraic group actions the first part of this paper is a refinement of winkelmann s work on invariant rings and quotients of algebraic group actions on affine varieties where we take a more geometric point of view we show that the algebraic quotient x g given by the possibly not finitely generated ring of invariants is almost an algebraic variety and that the quotient morphism pi x rightarrow x g has a number of nice properties one of the main difficulties comes from the fact that the quotient morphism is not necessarily surjective these general results are then refined for actions of the additive group mathbb g a where we can say much more we get a rather explicit description of the so called plinth variety and of the separating variety which measures how much orbits are separated by invariants the most complete results are obtained for representations we also give a complete and detailed analysis of roberts famous example of a an action of mathbb g a on 7 dimensional affine space with a non finitely generated ring of invariants

coarsening dynamics of the convective cahn hilliard equation abstract we characterize the coarsening dynamics associated with a convective cahn hilliard equation cch in one space dimension first we derive a sharp interface theory through a matched asymptotic analysis two types of phase boundaries kink and anti kink arise due to the presence of convection and their motions are governed to leading order by a nearest neighbors interaction coarsening dynamical system cds theoretical predictions on cds include the characteristic length l m for coarsening exhibits the temporal power law scaling t1 2 provided l m is appropriately small with respect to the peclet length scale l p binary coalescence of phase boundaries is impossible ternary coalescence only occurs through the kink ternary interaction two kinks meet an anti kink resulting in a kink direct numerical simulations performed on both cds and cch confirm each of these predictions a linear stability analysis of cds identifies a pinching mechanism as the dominant instability which in turn leads to kink ternaries we propose a self similar period doubling pinch ansatz as a model for the coarsening process from which an analytical coarsening law for the characteristic length scale l m emerges it predicts both the scaling constant c of the t1 2 regime i e l m ct 1 2 as well as the crossover to logarithmically slow coarsening as l m crosses l p our analytical coarsening law stands in good qualitative agreement with large scale numerical simulations that have been performed on cch

a counterexample to hilbert s fourteenth problem in dimension 5 abstract the kernel of a certain triangular derivation of the polynomial ring k x 1 x 2 x 3 x 4 x 5 is shown to be non finitely generated over k a field of characteristic zero thus giving a new counterexample to hilbert s fourteenth problem in the lowest dimension to date