l1 contraction and uniqueness for quasilinear elliptic parabolic equations abstract we prove the l 1 contraction principle and uniqueness of solutions for quasilinear elliptic parabolic equations of the form formula where b is monotone nondecreasing and continuous we assume only that u is a weak solution of finite energy in particular we do not suppose that the distributional derivative t b u is a bounded borel measure or a locally integrable function

the capelli identity the double commutant theorem and multiplicity free actions 0 the capelli identity cal 3 w p 39 is one of the most celebrated and useful formulas of classical invariant theory w d cl z the double commutant theorem w p 91 is likewise a basic result in the general theory of associative algebras both play key roles in weyl s book the classical groups the main purpose of this paper is to demonstrate a close connection between the two in the context of multiplicity free actions ekj of groups on vector spaces the focus of our discussion will be the structure of the differential operators which commute with a multiplicity free action applications include a new derivation of some formulas of shimura shl and rubenthaler and schiffmann rs for b functions associated to hermitian symmetric spaces and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin we also give a detailed discussion of certain aspects of multiplicity free representations here is an overview of the contents of the paper in sect 1 we review the classical capelli identities we observe how their existence is predicted by a double commutant result 1 6 and 1 8 and we show how they can be used to compute b functions we remark that capelli understood that his operators generate the center of the algebra generated by the polarization operators see b1 p 77 also capelli s motivation in introducing his operators was to explain a formula of cayley essentially the computation of a b function capelli s point of view was remarkably modern and structuralist in certain ways more modern even than that of weyl sections 2 through 9 provide a conceptual context for understanding the capelli identities as a feature of multiplicity free actions they contain a general discussion of the structure of g the algebra of polynomial coefficient differential operators which commute with a given group g of linear transformations the results of the discussion are summarized in theorem 9 1 which says in particular that the polynomial coefficient differential operators commuting with a

hele shaw type problems with dynamical boundary conditions abstract in this paper we study the nonlinear evolution equation of hele shaw type with dynamical boundary conditions that is the equation u t w f where u h w and h is the heaviside function with boundary condition x w t w k w g where denotes the outward normal vector of the fixed boundary of the domain we prove existence uniqueness and some qualitative properties of the solution

compact families of piecewise constant functions in lp 0 t b abstract a strong compactness result in the spirit of the lions aubin simon lemma is proven for piecewise constant functions in time u with values in a banach space the main feature of our result is that it is sufficient to verify one uniform estimate for the time shifts u u instead of all time shifts u u h for h 0 as required in simon s compactness theorem this simplifies significantly the application of the rothe method in the existence analysis of parabolic problems

a nonlinear time compactness result and applications to discretization of degenerate parabolic elliptic pdes abstract we propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems it is based on the original exploitation of a result related to compensated compactness rather than on a classical estimate on the space and time translates in the spirit of simon 49 our approach allows to handle various numerical discretizations both in the space variables and in the time variable in particular we can cope quite easily with variable time steps and with multistep time differentiation methods like e g the backward differentiation formula of order 2 bdf2 scheme we illustrate our approach by proving the convergence of a two point flux finite volume in space and bdf2 in time approximation of the porous medium equation

bernstein sato polynomials for maximal minors and sub maximal pfaffians abstract we determine the bernstein sato polynomials for the ideal of maximal minors of a generic m n matrix as well as for that of sub maximal pfaffians of a generic skew symmetric matrix of odd size as a corollary we obtain that the strong monodromy conjecture holds in these two cases

doubly nonlinear evolution equations of second order existence and fully discrete approximation abstract existence of solutions for a class of doubly nonlinear evolution equations of second order is proven by studying a full discretization the discretization combines a time stepping on a non uniform time grid which generalizes the well known stormer verlet scheme with an internal approximation scheme the linear operator acting on the zero order term is supposed to induce an inner product whereas the nonlinear time dependent operator acting on the first order time derivative is assumed to be hemicontinuous monotone and coercive up to some additive shift and to fulfill a certain growth condition the analysis also extends to the case of additional nonlinear perturbations of both the operators provided the perturbations satisfy a certain growth and a local holder type continuity condition a priori estimates are then derived in abstract fractional sobolev spaces convergence in a weak sense is shown for piecewise polynomial prolongations in time of the fully discrete solutions under suitable requirements on the sequence of time grids

finite volume methods for degenerate chemotaxis model a finite volume method for solving the degenerate chemotaxis model is presented along with numerical examples this model consists of a degenerate parabolic convection diffusion pde for the density of the cell population coupled to a parabolic pde for the chemoattractant concentration it is shown that discrete solutions exist and the scheme converges

arcs on determinantal varieties we study arc spaces and jet schemes of generic determinantal varieties using the natural group action we decompose the arc spaces into orbits and analyze their structure this allows us to compute the number of irreducible components of jet schemes log canonical thresholds and topological zeta functions

fast and robust numerical solution of the richards equation in homogeneous soil we derive and analyze a solver friendly finite element discretization of a time discrete richards equation based on kirchhoff transformation it can be interpreted as a classical finite element discretization in physical variables with nonstandard quadrature points our approach allows for nonlinear outflow or seepage boundary conditions of signorini type we show convergence of the saturation and in the nondegenerate case of the discrete physical pressure the associated discrete algebraic problems can be formulated as discrete convex minimization problems and therefore can be solved efficiently by monotone multigrid methods in numerical examples for two and three space dimensions we observe l 2 convergence rates of order mathcal o h 2 and h 1 convergence rates of order mathcal o h as well as robust convergence behavior of the multigrid method with respect to extreme choices of soil parameters

assessing the performance of the generalized propensity score for estimating the effect of quantitative or continuous exposures on binary outcomes propensity score methods are increasingly being used to estimate the effects of treatments and exposures when using observational data the propensity score was initially developed for use with binary exposures the generalized propensity score gps is an extension of the propensity score for use with quantitative or continuous exposures eg dose or quantity of medication income or years of education we used monte carlo simulations to examine the performance of different methods of using the gps to estimate the effect of continuous exposures on binary outcomes we examined covariate adjustment using the gps and weighting using weights based on the inverse of the gps we examined both the use of ordinary least squares to estimate the propensity function and the use of the covariate balancing propensity score algorithm the use of methods based on the gps was compared with the use of g computation all methods resulted in essentially unbiased estimation of the population dose response function however gps based weighting tended to result in estimates that displayed greater variability and had higher mean squared error when the magnitude of confounding was strong of the methods based on the gps covariate adjustment using the gps tended to result in estimates with lower variability and mean squared error when the magnitude of confounding was strong we illustrate the application of these methods by estimating the effect of average neighborhood income on the probability of death within 1 year of hospitalization for an acute myocardial infarction

local cohomology with support in ideals of maximal minors abstract suppose that k is a field of characteristic zero x is an r s matrix of indeterminates where r s and r k x is the polynomial ring over k in the entries of x we study the local cohomology modules h i i r where i is the ideal of r generated by the maximal minors of x we identify the indices i for which these modules vanish compute h i i r at the highest nonvanishing index i r s r 1 and characterize all nonzero ones as submodules of certain indecomposable injective modules these results are consequences of more general theorems regarding linearly reductive groups acting on local cohomology modules of polynomial rings

convergence of finite volume approximations for a nonlinear elliptic parabolic problem a continuous approach we study the approximation by finite volume methods of the model parabolic elliptic problem b v t rm div dv p 2 dv on 0 t times omega subset r times r d with an initial condition and the homogeneous dirichlet boundary condition because of the nonlinearity in the elliptic term a careful choice of the gradient approximation is needed we prove the convergence of discrete solutions to the solution of the continuous problem as the discretization step h tends to 0 under the main hypotheses that the approximation of the operator rm div dv p 2 dv provided by the finite volume scheme is still monotone and coercive and that the gradient approximation is exact on the affine functions of x in om an example of such a scheme is given for a class of two dimensional meshes dual to triangular meshes in particular for structured rectangular and hexagonal meshes the proof uses the rewriting of the discrete problem under a continuous form this permits us to directly apply the alt luckhaus variational techniques which are known for the continuous case

poles of the complex zeta function of a plane curve abstract we study the poles and residues of the complex zeta function f s of a plane curve we prove that most non rupture divisors do not contribute to poles of f s or roots of the bernstein sato polynomial b f s of f for plane branches we give an optimal set of candidates for the poles of f s from the rupture divisors and the characteristic sequence of f we prove that for generic plane branches f g e n all the candidates are poles of f g e n s as a consequence we prove yano s conjecture for any number of characteristic exponents if the eigenvalues of the monodromy of f are different

local cohomology with support in ideals of maximal minors and sub maximal pfaffians abstract we compute the gl equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix as well as of those with support in the ideal of 2 n 2 n pfaffians of a 2 n 1 2 n 1 generic skew symmetric matrix as an application we characterize the cohen macaulay modules of covariants for the action of the special linear group sl g on g m the main tool we develop is a method for computing certain ext modules based on the geometric technique for computing syzygies and on matlis duality

uniform in time convergence of numerical methods for non linear degenerate parabolic equations gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations conforming and non conforming finite element mixed finite element and finite volume methods we show here that this framework can be applied to a family of degenerate non linear parabolic equations which contain in particular the richards stefan s and leray lions models and we prove a uniform in time strong in space convergence result for the gradient scheme approximations of these equations in order to establish this convergence we develop several discrete compactness tools for numerical approximations of parabolic models including a discontinuous ascoli arzela theorem and a uniform in time weak in space discrete aubin simon theorem the model s degeneracies which occur both in the time and space derivatives also requires us to develop a discrete compensated compactness result

convergence of a finite volume scheme for nonlinear degenerate parabolic equations one approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u t rm div mathbf q f u delta phi u 0 by a piecewise constant function u mathcal d using a discretization mathcal d in space and time and a finite volume scheme the convergence of u mathcal d to u is shown as the size of the space and time steps tend to zero in a first step estimates on u mathcal d are used to prove the convergence up to a subsequence of u mathcal d to a measure valued entropy solution called here an entropy process solution a result of uniqueness of the entropy process solution is proved yielding the strong convergence of u mathcal d to it u some on a model equation are shown

regularization of outflow problems in unsaturated porous media with dry regions we study a porous medium with saturated unsaturated and dry regions described by richards equation for the saturation s and the pressure p due to a degenerate permeability coefficient k x s and a degenerate capillary pressure function pc x s the equations may be of elliptic parabolic or of ode type we construct a parabolic regularization of the equations and find conditions that guarantee the convergence of the parabolic solutions to a solution of the degenerate system an example shows that the convergence fails in general our approach provides an existence result for the outflow problem in the case of x dependent coefficients and a method for a numerical approximation

well posedness results for triply nonlinear degenerate parabolic equations we study well posedness of triply nonlinear degenerate elliptic parabolic hyperbolic problems of the kind b u t div a u u u f u t 0 u0 in a bounded domain with homogeneous dirichlet boundary conditions the nonlinearities b and are supposed to be continuous non decreasing and the nonlinearity a falls within the leray lions framework some restrictions are imposed on the dependence of a u u on u and also on the set where degenerates a model case is a u u f b u u u k u a0 u with a nonlinearity which is strictly increasing except on a locally finite number of segments and the nonlinearity a0 which is of the leray lions kind we are interested in existence uniqueness and stability of l entropy solutions for the parabolic hyperbolic equation b id we obtain a general continuous dependence result on data u0 f and nonlinearities b a similar result is shown for the degenerate elliptic problem which corresponds to the case of b 0 and general non decreasing surjective existence uniqueness and continuous dependence on data u0 f are shown in more generality for instance the assumptions b r r and the continuity of b 1 permit to achieve the well posedness result for bounded entropy solutions of this triply nonlinear evolution problem