good filtrations and strong f regularity of the ring of u p invariants abstract let k be an algebraically closed field of positive characteristic g a reductive group over k and v a finite dimensional g module let p be a parabolic subgroup of g and u p its unipotent radical we prove that if s sym v has a good filtration then s u p is strongly f regular

structure formation and the morphology diagram of possible structures in two dimensional diffusional growth the morphology diagram of possible structures in two dimensional diffusional growth is given in the parameter space of undercooling ensuremath delta versus anisotropy of surface tension ensuremath epsilon the building block of the dendritic structure is a dendrite with parabolic tip and the basic element of the seaweed structure is a doublon the transition between these structures shows a jump in the growth velocity we also describe the structures and velocities of fractal dendrites and doublons destroyed by noise we introduce a renormalized capillary length and density of the solid phase and use scaling arguments to describe the fractal dendrites and doublons the resulting scaling exponents for the growth velocity and the different length scales are expressed in terms of the fractal dimensions for surface and bulk of these fractal structures all the considered structures are compact on length scales larger than the diffusion length and they show fractal behavior on intermediate length scales between the diffusion length and a small size cutoff which depends on the strength of noise textcopyright 1996 the american physical society

pattern formation in three dimensional dendritic growth abstract the entire solution for three dimensional 3d nonaxisymmetric dendrites is described analytically its construction involves the 3d selection theory for the tip of the dendrite plus matching of the tail to this tip both intermediate and final asymptotics of the tail shape are given this shape which deviates strongly from the ivantsov paraboloid is in qualitative agreement with experimental observations we consider the time dependent behavior of sidebranching deformations taking into account the actual nonaxisymmetric shape of the needle crystal it is found that the amplitude of the deformation grows faster than for the axisymmetric paraboloid shape of the needle we argue that this effect resolves the puzzle that experimentally observed side branches have much larger amplitudes that can be explained by thermal noise in the framework of the axisymmetric approach the coarsening behavior of sidebranches in the nonlinear regime is shortly discussed

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

strong and weak f regularity are equivalent for graded rings it is shown that the tight closure of a submodule in a artinian module is the same as its finitistic tight closure when the modules are graded over a finitely generated inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i graded ring over a perfect field as a corollary it is deduced that for such a graded ring strong and weak f regularity are equivalent as another application the following conjecture of hochster and huneke is proved let r m be a finitely generated inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i graded ring over a field with unique homogeneous maximal ideal m then r is weakly f regular if and only if r m is weakly f regular

reconstruction of radial dirac operators we study the inverse spectral problem of reconstructing the potential of radial dirac operators acting in the unit ball of r3 for each one dimensional partial dirac operator corresponding to a nonzero angular momentum we give a complete description of the spectral data eigenvalues and suitably defined norming constants prove existence and uniqueness of solutions to the inverse problem and present the reconstruction algorithm

simplicity of rings of differential operators in prime characteristic let w be a finite dimensional representation of a linearly reductive group g over a field k motivated by their work on classical rings of invariants levasseur and stafford asked whether the ring of invariants under g of the symmetric algebra of w has a simple ring of differential operators in this paper we show that this is true in prime characteristic indeed if r is a graded subring of a polynomial ring over a perfect field of characteristic p 0 and if the inclusion r hookrightarrow s splits then d k r is a simple ring in the last section of the paper we discuss how one might try to deduce the characteristic zero case from this result as yet however this is a subtle problem and the answer to the question of levasseur and stafford remains open in characteristic zero http www luc ac be research algebra 1991 mathematics subject classification 16s32 16g60 13a35

multiscale finite difference diffusion monte carlo method for simulating dendritic solidification abstract we present a novel hybrid computational method to simulate accurately dendritic solidification in the low undercooling limit where the dendrite tip radius is one or more orders of magnitude smaller than the characteristic spatial scale of variation of the surrounding thermal or solutal diffusion field the first key feature of this method is an efficient multiscale diffusion monte carlo dmc algorithm which allows off lattice random walkers to take longer and concomitantly rarer steps with increasing distance away from the solid liquid interface as a result the computational cost of evolving the large scale diffusion field becomes insignificant when compared to that of calculating the interface evolution the second key feature is that random walks are only permitted outside of a thin liquid layer surrounding the interface inside this layer and in the solid the diffusion equation is solved using a standard finite difference algorithm that is interfaced with the dmc algorithm using the local conservation law for the diffusing quantity here we combine this algorithm with a previously developed phase field formulation of the interface dynamics and demonstrate that it can accurately simulate three dimensional dendritic growth in a previously unreachable range of low undercoolings that is of direct experimental relevance

good filtrations of symmetric algebras and strong f regularity of invariant subrings in this paper we prove that a linear action of a reductive group on a polynomial ring with good filtrations over a field of characteristic p 0 yields a strongly f regular in particular cohen macaulay invariant subring the strongly f regular property of some known examples of invariant subrings such as the coordinate rings of schubert varieties in grassmannians are recovered a similar result over a field of characteristic zero is also proved

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal