common hypercyclic vectors for multiples of backward shift we prove that l 2 contains a dense set of vectors which are hypercyclic simultaneously for all multiples of the backward shift oper ator by constants of absolute value greater than 1

critical points for elliptic equations with prescribed boundary conditions this paper concerns the existence of critical points for solutions to second order elliptic equations of the form nabla cdot sigma x nabla u 0 posed on a bounded domain x with prescribed boundary conditions in spatial dimension n 2 it is known that the number of critical points where nabla u 0 is related to the number of oscillations of the boundary condition independently of the positive coefficient sigma we show that the situation is different in dimension n geq3 more precisely we obtain that for any fixed dirichlet or neumann boundary condition for u on partial x there exists an open set of smooth coefficients sigma x such that nabla u vanishes at least at one point in x by using estimates related to the laplacian with mixed boundary conditions the result is first obtained for a piecewise constant conductivity with infinite contrast a problem of independent interest a second step shows that the topology of the vector field nabla u on a subdomain is not modified for appropriate bounded sufficiently high contrast smooth coefficients sigma x these results find applications in the class of hybrid inverse problems where optimal stability estimates for parameter reconstruction are obtained in the absence of critical points our results show that for any finite number of prescribed boundary conditions there are coefficients sigma x for which the stability of the reconstructions will inevitably degrade

viscosity solutions of a degenerate parabolic elliptic system arising in the mean field theory of superconductivity in a type ii superconductor the magnetic field penetrates the superconducting body through the formation of vortices in an extreme type ii superconductor these vortices reduce to line singularities because the number of vortices is large it seems feasible to model their evolution by an averaged problem known as the mean field model of superconductivity we assume that the evolution law of an individual vortex which underlies the averaging process involves the current of the generated magnetic field as well as the curvature vector in the present paper we study a two dimensional reduction assuming all vortices to be perpendicular to a given direction since both the magnetic field h and the averaged vorticity are curl free we may represent them via a scalar magnetic potential q and a scalar stream function respectively we study existence uniqueness and asymptotic behaviour of solutions q of the resulting degenerate elliptic parabolic system with curvature taken into account or not by means of viscosity and weak solutions in addition we relate q to solutions h of the mean field equations without curvature finally we construct special solutions of the corresponding stationary equations with two or more superconducting phases

singular set of solutions of semilinear elliptic equations abstract we study the behavior of the singular set u u 0 for solutions u to the semilinear elliptic equation u f x u u x where is an open set in r n and f x u u a u p b u q where a b 0 and p q 0 we show that in dimension n 2 the singular set is a discrete set and if min p q 1 then a solution u satisfies an optimal growth p q near every non isolated point of the singular set also the same results are true in dimension n 3 under an n 1 dimensional density condition

a finite volume element formulation and error analysis for the non stationary conduction convection problem abstract the non stationary conduction convection problem including the velocity vector field and the pressure field as well as the temperature field is studied with a finite volume element fve method a fully discrete fve formulation and the error estimates between the fully discrete fve solutions and the accuracy solution are provided it is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions moreover it is shown that the fve method is feasible and efficient for finding the numerical solutions of the non stationary conduction convection problem and is one of the most effective numerical methods by comparing the results of the numerical simulations of the fve formulation with those of the numerical simulations of the finite element method and the finite difference scheme for the non stationary conduction convection problem

convergence and superconvergence analysis of a nonconforming finite element method for solving the signorini problem abstract in this paper we present the carey nonconforming finite element approximation of the variational inequality resulting from the signorini problem firstly we show that if the displacement field is of h 2 regularity the optimal convergence rate of o h can be obtained with respect to the energy norm secondly if stronger but reasonable h 5 2 regularity is available the superconvergence rate of o h 3 2 can be derived through the interpolated postprocessing technique finally numerical experiments are given which are consistent with our theoretical analysis

singularities in axisymmetric free boundaries for electrohydrodynamic equations we consider singularities in the electrohydrodynamic equations in a regime where we are allowed to neglect surface tension and assuming that the free surface is given by an injective curve and that either the fluid velocity or the electric field satisfies a certain non degeneracy condition we prove that either the fluid region or the gas region is asymptotically a cusp our proofs depend on a combination of monotonicity formulas and a non vanishing result by caffarelli and friedman as a by product of our analysis we also obtain a special solution with convex conical air phase which we believe to be new

a least squares petrov galerkin finite element method for the stationary navier stokes equations in this paper a galerkin least squares type finite element method is proposed and analyzed for the stationary navier stokes equations the method is consistent and stable for any combination of discrete velocity and pressure spaces without requiring a babuska brezzi stability condition the existence uniqueness and convergence at optimal rate of the discrete solution is proved in the case of sufficient viscosity or small data

hypercyclic weighted shifts an operator t acting on a hilbert space is hypercyclic if for some vector x in the space the orbit tnx n 0 is dense in this paper we characterize hypercyclic weighted shifts in terms of their weight sequences and identify the direct sums of hypercyclic weighted shifts which are also hypercyclic as a consequence we show within the class of weighted shifts that multi hypercyclic shifts and direct sums of fixed hypercyclic shifts are both hypercyclic for general hypercyclic operators the corresponding questions were posed by d a herrero and they still remain open using a different technique we prove that i t is hypercycic whenever t is a unilateral backward weighted shift thus answering in more generality a question recently posed by k c chan and j h shapiro

a reduced order extrapolated crank nicolson finite spectral element method based on pod for the 2d non stationary boussinesq equations abstract in this paper we mainly utilize proper orthogonal decomposition pod to reduce the order for the coefficient vector of the classical crank nicolson finite spectral element ccnfse method of the two dimensional 2d non stationary boussinesq equations about vorticity stream functions so that the reduced order method maintains all the advantages of the ccnfse method toward this end we first establish a ccnfse format with the second order time accuracy for the two dimensional 2d non stationary boussinesq equations about vorticity stream functions and analyze the existence stability and convergence of the ccnfse solutions and then by pod we establish a reduced order extrapolated crank nicolson finite spectral element roecnfse method and analyze the existence stability and convergence of the roecnfse solutions as well as offer the flowchart for seeking roecnfse solutions finally we use two sets of numerical experiments to validate that the numerical computational consequences are accorded with the theoretical ones such that the effectiveness and feasibility of the roecnfse method are further verified both theory and method in this paper is new and completely different from the existing reduced order methods

nodal sets of thin curved layers abstract this paper is concerned with the location of nodal sets of eigenfunctions of the dirichlet laplacian in thin tubular neighbourhoods of hypersurfaces of the euclidean space of arbitrary dimension in the limit when the radius of the neighbourhood tends to zero it is known that spectral properties of the laplacian are approximated well by an effective schrodinger operator on the hypersurface with a potential expressed solely in terms of principal curvatures by applying techniques of elliptic partial differential equations we strengthen the known perturbation results to get a convergence of eigenfunctions in holder spaces this enables us in particular to conclude that every nodal set has a non empty intersection with the boundary of the tubular neighbourhood

nonconforming mixed finite element approximation to the stationary navier stokes equations on anisotropic meshes abstract the main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary navier stokes equations under anisotropic meshes that is the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure the convergence analysis is presented and the optimal error estimates both in a broken h 1 norm for the velocity and in an l 2 norm for the pressure are derived on anisotropic meshes

genericity of wild holomorphic functions and common hypercyclic vectors abstract let t be the translation operator by in the space of entire functions h c defined by t f z f z we prove that there is a residual set g of entire functions such that for every f g and every c 0 the sequence t n f is dense in h c that is g is a residual set of common hypercyclic vectors functions for the family t c 0 also we prove similar results for many families of operators as multiples of differential operator multiples of backward shift weighted backward shifts

superconvergent analysis of a nonconforming mixed finite element method for non stationary conduction convection problem abstract in this paper a backward euler mixed finite element mfe approximation scheme is presented for the non stationary conduction convection problem in which the constrained nonconforming rotated cnr q 1 element is used to approximate the velocity u the temperature t and the q 0 element to the pressure p based on the characters of these elements and some special skills i e mean value skill and a new transforming skill with respect to the superclose results of order o h 2 for u t in the broken h 1 norm and p in l 2 norm are deduced respectively here h is the subdivision parameter and the time step furthermore the global superconvergent results are obtained through the interpolation postprocessing technique finally numerical results are provided to confirm the theoretical analysis

a least squares galerkin petrov nonconforming mixed finite element method for the stationary conduction convection problem abstract in this paper a least squares galerkin petrov nonconforming mixed finite element method lsgpnmfm is proposed and analyzed for the stationary conduction convection problem we use p 2 nonconforming as approximation space for the velocity the linear element for the pressure space and the quadratic element for the temperature space the mixed finite element spaces x h and m h need not satisfy inf sup condition the existence uniqueness and convergence of the discrete solution are presented and error estimates of optimal order are derived in the case of sufficient viscosity

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

hypercyclic operators that commute with the bergman backward shift the backward shift b on the bergman space of the unit disc is known to be hypercyclic meaning it has a dense orbit here we ask which operators that commute with b inherit its hypercyclicity we show that the problem reduces to the study of operators of the form 9p b where so is a holomorphic self map of the unit disc that multiplies the dirichlet space into itself and that the question of hypercyclicity for such an operator depends on how freely 9p z is allowed to approach the unit circle as lz s 1

asymptotic behavior of the solutions of linear and quasilinear elliptic equations on ℝ ℕ we investigate the relationship between the decay at infinity of the right hand side f and solutions u of an equation lu f when l is a second order elliptic operator on r n it is shown that when l is fredholm u inherits the type of decay of f for instance exponential or power like in particular the generalized eigenfunctions associated with all the fredholm eigenvalues of l isolated or not decay exponentially no use is made of spectral theory the result is next extended when l is replaced by a fredholm quasilinear operator various generalizations to other unbounded domains higher order operators or elliptic systems are possible and briefly alluded to but not discussed in detail

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

multiplicative multitask feature learning we investigate a general framework of multiplicative multitask feature learning which decomposes individual task s model parameters into a multiplication of two components one of the components is used across all tasks and the other component is task specific several previous methods can be proved to be special cases of our framework we study the theoretical properties of this framework when different regularization conditions are applied to the two decomposed components we prove that this framework is mathematically equivalent to the widely used multitask feature learning methods that are based on a joint regularization of all model parameters but with a more general form of regularizers further an analytical formula is derived for the across task component as related to the task specific component for all these regularizers leading to a better understanding of the shrinkage effects of different regularizers study of this framework motivates new multitask learning algorithms we propose two new learning formulations by varying the parameters in the proposed framework an effcient blockwise coordinate descent algorithm is developed suitable for solving the entire family of formulations with rigorous convergence analysis simulation studies have identified the statistical properties of data that would be in favor of the new formulations extensive empirical studies on various classification and regression benchmark data sets have revealed the relative advantages of the two new formulations by comparing with the state of the art which provides instructive insights into the feature learning problem with multiple tasks

on the scaling limits at zeros of solutions of parabolic equations abstract we are concerned with the local scaling limits near zero points of solutions of general parabolic equations we show that at each zero point of finite vanishing order the solution is asymptotic to a homogeneous polynomial solution of the osculating equation with constant coefficients this provides useful information about the local structure of zero sets of solutions in particular we give upper bounds for hausdorff dimensions of zero sets of weak solutions our analysis combines a homothety scaling argument with the theory of nonautonomous dynamical systems

discontinuous approximation of viscous two phase flow in heterogeneous porous media runge kutta discontinuous galerkin rkdg and discontinuous finite volume element dfve methods are applied to a coupled flow transport problem describing the immiscible displacement of a viscous incompressible fluid in a non homogeneous porous medium the model problem consists of nonlinear pressure velocity equations assuming brinkman flow coupled to a nonlinear hyperbolic equation governing the mass balance saturation equation the mass conservation properties inherent to finite volume based methods motivate a dfve scheme for the approximation of the brinkman flow in combination with a rkdg method for the spatio temporal discretization of the saturation equation the stability of the uncoupled schemes for the flow and for the saturation equations is analyzed and several numerical experiments illustrate the robustness of the numerical method

hypercyclic and cyclic vectors abstract let x denote an arbitrary separable banach space over the field of complex numbers and b x the banach algebra of all bounded linear operators on x we prove the following results 1 an element of the space x is hypercyclic supercyclic for all positive powers tn of an operator t in b x if it is hypercyclic supercyclic for t 2 under some condition on the spectrum of the adjoint of a cyclic operator the set of all cyclic vectors of the operator is dense this result extends to any cyclic commutative subset of b x 3 under a mild condition on the spectrum of a cyclic operator t the set of all separating vectors for the commutant t of t is dense this also extends to any cyclic commutative subset of b x 4 a slightly stronger version of a theorem of k f clancey and d d rogers on cyclic vectors finally we define and discuss hereditarily hypercyclic operators

a fully discrete stabilized mixed finite volume element formulation for the non stationary conduction convection problem abstract at first a semi discrete formulation with respect to time for the non stationary conduction convection problem is recalled then a fully discrete stabilized mixed finite volume element smfve formulation based on two local gauss integrals and parameter free is established directly from the semi discrete formulation with respect to time for the non stationary conduction convection problem following this the error estimates for the fully discrete smfve solutions are derived by means of the standard mixed finite element method finally some numerical experiments are presented illustrating that the numerical errors are consistent with theoretical results the degrees of freedom of the fully discrete smfve formulation are far fewer than those of the classical finite volume element fve formulation without any stabilization and its numerical solutions are more stable than those of the classical fve formulation without any stabilization thus validating that the fully discrete smfve formulation is feasible and efficient for finding the numerical solutions of the non stationary conduction convection problem

the nodal set of solutions to some elliptic problems sublinear equations and unstable two phase membrane problem abstract we are concerned with the nodal set of solutions to equations of the form u u q 1 u q 1 in b 1 where 0 q 1 2 b 1 b 1 0 is the unit ball in r n n 2 and u max u 0 u max u 0 are the positive and the negative part of u respectively this class includes the unstable two phase membrane problem q 1 as well as sublinear equations for 1 q 2 we prove the following main results a the finiteness of the vanishing order at every point and the complete characterization of the order spectrum b a weak non degeneracy property c regularity of the nodal set of any solution the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having hausdorff dimension at most n 2 locally finite when n 2 d a partial stratification theorem ultimately the main features of the nodal set are strictly related with those of the solutions to linear or superlinear equations with two remarkable differences first of all the admissible vanishing orders can not exceed the critical value 2 2 q at threshold we find a multiplicity of homogeneous solutions yielding the non validity of any estimate of the n 1 dimensional measure of the nodal set of a solution in terms of the vanishing order as a byproduct we also prove the strong unique continuation property for the unstable obstacle problem corresponding to the case 0 the proofs are based on monotonicity formulae for a 2 parameter family of weiss type functionals blow up arguments and the classification of homogeneous solutions

existence of hypercyclic operators on topological vector spaces the present paper introduces a very simple but very useful notion of the so called quasi extension ofl1 operators and proves that a large class of topological vector spaces admit continuous hypercyclic operators in particular it answers in the affirmative a question of s rolewicz posed in 1969 whether or not every infinite dimensional separable banach space admits a continuous hypercyclic operator

dynamics of lotka volterra competition systems with large interaction abstract we show that under certain additional hypothesis two population competing species models in bounded domains with diffusion and large interaction have simple dynamics in particular the solutions approach stationary states as t tends to infinity

nonlinear galerkin methods and mixed finite elements two grid algorithms for the navier stokes equations a nonlinear galerkin method using mixed finite elements is presented for the two dimensional incompressible navier stokes equations the scheme is based on two finite element spaces x h and x h for the approximation of the velocity defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h and one finite element space m h for the approximation of the pressure nonlinearity and time dependence are both treated on the coarse space we prove that the difference between the new nonlinear galerkin method and the standard galerkin solution is of the order of h 2 both in velocity h 1 omega and pressure l 2 omega norm we also discuss a penalized version of our algorithm which enjoys similar properties

proper orthogonal decomposition based reduced order stabilized mixed finite volume element extrapolating model for the nonstationary incompressible boussinesq equations abstract in this study we employ a proper orthogonal decomposition pod method to establish a pod based reduced order stabilized mixed finite volume element smfve extrapolating model with very few degrees of freedom for the nonstationary incompressible boussinesq equations we provide error estimates of the pod based reduced order smfve solutions and the algorithm implementation for the pod based reduced order smfve extrapolating model in addition we present some numerical experiments that verify the validity and reliability of the pod based reduced order smfve extrapolating model

convergence of finite volume schemes for poisson s equation on nonuniform meshes an error analysis of a family of cell center finite volume schemes for poisson s equation on cartesian product nonuniform meshes in two dimensions is presented optimal order error estimates are derived in the discrete h 1 norm under minimum smoothness requirements on the exact solution and without any additional assumption on the regularity of the mesh on quasi uniform meshes analogous estimates are obtained in the maximum norm

multi task learning for classification with dirichlet process priors consider the problem of learning logistic regression models for multiple classification tasks where the training data set for each task is not drawn from the same statistical distribution in such a multi task learning mtl scenario it is necessary to identify groups of similar tasks that should be learned jointly relying on a dirichlet process dp based statistical model to learn the extent of similarity between classification tasks we develop computationally efficient algorithms for two different forms of the mtl problem first we consider a symmetric multi task learning smtl situation in which classifiers for multiple tasks are learned jointly using a variational bayesian vb algorithm second we consider an asymmetric multi task learning amtl formulation in which the posterior density function from the smtl model parameters from previous tasks is used as a prior for a new task this approach has the significant advantage of not requiring storage and use of all previous data from prior tasks the amtl formulation is solved with a simple markov chain monte carlo mcmc construction experimental results on two real life mtl problems indicate that the proposed algorithms a automatically identify subgroups of related tasks whose training data appear to be drawn from similar distributions and b are more accurate than simpler approaches such as single task learning pooling of data across all tasks and simplified approximations to dp

a defect correction method for unsteady conduction convection problems ii time discretization in this paper a fully discrete defect correction mixed finite element method mfem for solving the non stationary conduction convection problems in two dimension which is leaded by combining the back euler time discretization with the two step defect correction in space is presented in this method we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect correction technique the stability and the error analysis are derived the theory analysis shows that our method is stable and has a good convergence property some numerical results are also given which show that this method is highly efficient for the unsteady conduction convection problems

sums of hypercyclic operators abstract every bounded operator on a complex infinite dimensional separable hilbert space can be written as the sum of two hypercyclic operators and also as the sum of two chaotic operators

on nodal sets for dirac and laplace operators we prove that the nodal set zero set of a solution of a generalized dirac equation on a riemannian manifold has codimension 2 at least if the underlying manifold is a surface then the nodal set is discrete we obtain a quick proof of the fact that the nodal set of an eigenfunction for the laplace beltrami operator on a riemannian manifold consists of a smooth hypersurface and a singular set of lower dimension we also see that the nodal set of a harmonic differential form on a closed manifold has codimension 2 at least a fact which is not true if the manifold is not closed examples show that all bounds are optimal

on random frequent universality abstract we give a probabilistic construction of frequently hypercyclic or universal entire functions using birkhoff s ergodic theorem we apply the same ideas to construct random frequently universal vectors for the polynomials of the weighted backward shift on the classical real or complex l p space

analysis of the cell centred finite volume method for the diffusion equation abstract this paper presents a rigorous theoretical analysis of the cell centred finite volume method for poisson s equation we review the traditional taylor series expansion technique which suggests that the cell centred method is inconsistent on nonuniform grids which is not confirmed by numerical experiments we then present an analysis of the method which confirms that the solution error does indeed reduce as the cell size is reduced this is supported by numerical calculations

partial regularity of the zero set of solutions of linear and superlinear elliptic equations on considere l ensemble zero u 0 des solutions de u f x u u dans ou est un ensemble ouvert dans r m et f x u u a u b u ou a 0 b 0 et 1 1

error analysis of a fully discrete finite element variational multiscale method for the natural convection problem abstract for the natural convection problem we propose a new projection based finite element variational multiscale method by defining the stabilization terms via two local gauss integrations at the element level based on the implicit backward euler and implicit crank nicolson schemes for temporal discretization and stabilized mixed finite element spatial discretization we establish two numerical schemes for the natural convection problem unconditional stabilities of the two numerical schemes are proved we derive error bounds of the fully discrete solution which are first and second order in time respectively the optimal error estimates in space could be achieved for velocity and temperature in the h 1 semi norm and for pressure in the l 2 norm with the proper choosing of stabilized parameters however the error estimates in space are suboptimal for velocity and temperature in the l 2 norm the derived theoretical results are supported by two numerical examples

on the accuracy of the finite volume element method for diffusion equations on composite grids this paper develops the finite volume element fve method which is similar to the so called control volume finite element method but tailored to composite grid applications fve is described for the general two dimensional diffusion equation in divergence form o h 3 2 discretization error estimates are then developed for the case of triangular elements these results are improved to o h 2 using a special modification involving rectilinear elements at the grid interface

some remarks on abstract universal series abstract we introduce a new class of universal series in the abstract setting this class lies between two already known classes of universal series by two examples we differentiate these classes these examples are the first examples in the case where all classes are non void we also prove that the class of universal series where the approximating indexes have upper density 1 is dense and g and that the class of lower frequently universal series is of the first category

a stabilized finite volume method for the evolutionary stokes darcy system abstract in this paper we propose a stabilized fully discrete finite volume method based on two local gauss integrals for a non stationary stokes darcy problem this stabilized method is free of stabilized parameters and uses the lowest equal order finite element triples p 1 p 1 p 1 for approximating the velocity pressure and hydraulic head of the stokes darcy model under a modest time step restriction in relation to physical parameters we give the stability analysis and the error estimates for the stabilized finite volume scheme by means of a relationship between finite volume and finite element approximations with the lower order elements finally a series of numerical experiments are provided to demonstrate the validity of the theoretical results

frequently hypercyclic operators we investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators t on separable complex f spaces t is frequently hypercyclic if there exists a vector x such that for every nonempty open subset u of x the set of integers n such that t n x belongs to u has positive lower density we give several criteria for frequent hypercyclicity and this leads us in particular to study linear transformations from the point of view of ergodic theory several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting

asymptotics of the nodal lines of solutions of 2 dimensional schrodinger equations on affine des resultats sur des proprietes nodales des l 2 solutions des equations de schrodinger a 2 dimensions

simultaneous variable selection we propose a new method for selecting a common subset of explanatory variables where the aim is to model several response variables the idea is a natural extension of the lasso technique proposed by tibshirani 1996 and is based on the joint residual sum of squares while constraining the parameter estimates to lie within a suitable polyhedral region the properties of the resulting convex programming problem are analyzed for the special case of an orthonormal design for the general case we develop an efficient interior point algorithm the method is illustrated on a dataset with infrared spectrometry measurements on 14 qualitatively different but correlated responses using 770 wavelengths the aim is to select a subset of the wavelengths suitable for use as predictors for as many of the responses as possible

splitting theorem poincare hopf theorem and jumping nonlinear problems abstract in this paper we establish a gromoll meyer splitting theorem and a shifting theorem for j c 2 0 e r and by using the finite dimensional approximation mollifiers and morse theory we generalize the poincare hopf theorem to j c 1 e r case by combining the poincare hopf theorem and the splitting theorem we study the existence of multiple solutions for jumping nonlinear elliptic equations

on the asymptotics of nodes ofl2 solutions of schrodinger equations in dimensions 3 let r n br wheren 3 andbr x n x r we investigate the asymptotics of real valued solutions l2 r of the schrodinger equation v e 0 wheree 0 begin gathered mathop lim inf limits r to infty r gamma int limits s r psi 2 d sigma int limits s r psi 2 d sigma 0 and hfill mathop lim inf limits r to infty ln volume d cap b r ln r geqq n 1 2 hfill end gathered results of this type are already non trivial for radial problems with satisfying non radial boundary conditions on r or for excited states of the hydrogen atom if one considers linear combinations of differentl waves

on the fucik spectrum in this paper we study the structure of the fucik spectrum of the set of points b a in r 2 for which the equation

bifurcation surfaces stemming from the fucik spectrum abstract let r n be a bounded domain with smooth boundary and g r r be a nonlinear function we prove existence of two dimensional bifurcation surfaces for the elliptic boundary value problem u a u b u g x u in u 0 where u min 0 u u max 0 u and a b r 2 is a pair of parameters we show that these two dimensional bifurcation surfaces stem from the fucik spectrum of the main difficulty in doing that comes from non smoothness of the operators u u in order to overcome this difficulty a variant implicit function theorem and an abstract two dimensional bifurcation theorem are proved these two theorems do not require smoothness of operators and the abstract two dimensional bifurcation theorem can be regarded as an extension of the well known crandall rabinowitz bifurcation theorem and therefore are of interest for their own sake

finite element methods for first order hyperbolic systems with particular emphasis on the compressible euler equations a petrov galerkin finite element formulation is presented for first order hyperbolic systems of conservation laws with particular emphasis on the compressible euler equations applications of the methodology are made to one and two dimensional steady and unsteady flows with shocks results obtained suggest the potential of the type of methods developed

dynamics for systems of screw dislocations the goal of this paper is the analytical validation of a model of cermelli and gurtin arch ration mech anal 148 1999 pp 3 52 for an evolution law for systems of screw dislocations under the assumption of antiplane shear the motion of the dislocations is restricted to a discrete set of glide directions which are properties of the material the evolution law is given by a maximal dissipation criterion leading to a system of differential inclusions short time existence uniqueness cross slip and fine cross slip of solutions are proved

singular solutions of elliptic equations and the determination of conductivity by boundary measurements abstract we improve some results on uniqueness and stability in the determination of the coefficient a in the equation i div a grad u 0 in when all possible pairs of dirichlet and neumann data on are known we also treat cases of anisotropic equations our method relies on the construction of solutions to i having an isolated singularity with prescribed asymptotic behaviour

the neumann problem of an elliptic equation with strong resonance in the present paper we consider a class of strong resonance problems for nonlinear elliptic neumann bvps and establish existence and multiplicity results for positive negative and sign changing solutions by using a morse theoretical approach

uniqueness of critical points of solutions to the mean curvature equation with neumann and robin boundary conditions abstract in this paper we investigate the critical points of solutions to the prescribed constant mean curvature equation with neumann and robin boundary conditions respectively in a bounded smooth convex domain of r n n 2 firstly we show the non degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local chen huang s comparison technique and the geometric properties of approximate surfaces at the non degenerate critical points secondly we deduce the uniqueness and non degeneracy of the critical points of solutions in a rotationally symmetric domain of r n n 3 by the projection of higher dimensional space onto two dimensional plane

numerical analysis of the unconditionally stable discontinuous galerkin schemes for the nonstationary conduction convection problem abstract this work aims at employing the discontinuous galerkin dg methods for solving the nonstationary conduction convection model which is motivated by advantages of dg methods the discontinuous element pair p k p k 1 p k in the space discretization and backward euler scheme in time are applied as well as an explicit upwind scheme based on picard iteration is introduced for the coupled term existence and uniqueness of the approximation solutions are proved along with preserving the unconditional stability of the numerical schemes the priori error estimates for velocity pressure and temperature are also derived numerical tests are provided to verify the theoretical analysis and to indicate the efficiency of the dg methods in simulating practical problems

a covolume method based on rotated bilinears for the generalized stokes problem we introduce a covolume or marker and cell mac method for approximating the generalized stokes problem on an axiparallel domain two grids are needed the primal grid made up of rectangles and the dual grid of quadrilaterals the velocity is approximated by nonconforming rotated bilinear elements with degrees of freedom at midpoints of rectangular elements and the pressure by piecewise constants the error in the velocity in the h 1 h norm and the pressure in the l2 norm are of first order provided that the exact velocity is in h2 and the exact pressure in h1

the stationary mean field model of superconductivity partial regularity of the free boundary abstract we prove a partial regularity result for a free boundary problem arising in the mean field theory of superconductivity to this end we extend a regularity result of caffarelli and friedman for the zero set of functions satisfying a bound v c v to functions only satisfying a bound of the form v c v where v x supp 0 t 1 v tx is the radially maximal function in addition we extend the unique continuation theorem of aronszajn and cordes to cover this case

a new low order least squares nonconforming characteristics mixed finite element method for burgers equation in this paper a new low order least squares nonconforming characteristics mixed finite element method mfem is considered for two dimensional burgers equation by use of two typical characters of the elements for approximating the velocity and flux variables a the consistency errors of the elements can be estimated as order o h 2 with respect to the mesh size h in the broken h 1 norm one order higher than their interpolation errors b the elements interpolation operators satisfy certain orthogonalities which lead to o h 2 order error estimates in l 2 norm and some superclose results the accuracy of corresponding variables in l 2 norm is improved by one order compared with other mfems in the existing literature it seems that the results provided herein have never been derived before