interpolatory quadrature formulas on the unit circle for chebyshev weight functions in this paper interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands a comparison is made with szego quadrature formulas finally an application to the interval 1 1 is also carried out

domination by product measures we consider families of 0 1 valued random variables indexed by the vertices of countable graphs with bounded degree first we show that if these random variables satisfy the property that conditioned on what happens outside of the neighborhood of each given site the probability of seeing a 1 at this site is at least a value p which is large enough then this random field dominates a product measure with positive density moreover the density of this dominated product measure can be made arbitrarily close to 1 provided that p is close enough to 1 next we address the issue of obtaining the critical value of p defined as the threshold above which the domination by positive density product measures is assured for the graphs which have as vertices the integers and edges connecting vertices which are separated by no more than k units this critical value is shown to be 1 k k k 1 k 1 and a discontinuous transition is shown to occur similar critical values of p are found for other classes of probability measures on 0 1 z for the class of k dependent measures the critical value is again 1 k k k 1 k 1 with a discontinuous transition for the class of two block factors the critical value is shown to be 1 2 and a continuous transition is shown to take place in this case thus both the critical value and the nature of the transition are different in the two block factor and 1 dependent cases

existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth order elliptic equations on r n in this paper we study the following fourth order elliptic equation of the form 2 u u v x u 2 u 2 u h x u x r n where 2 is the biharmonic operator 0 n 6 v c r n r and h c r n r r under appropriate assumptions on v x and h x u some existence and multiplicity results for nontrivial solutions are obtained via variational methods

critical percolation and the minimal spanning tree in slabs the minimal spanning forest on d is known to consist of a single tree for d 2 and is conjectured to consist of infinitely many trees for large d in this paper we prove that there is a single tree for quasi planar graphs such as 2 0 k d 2 our method relies on generalizations of the gluing lemma of duminil copin sidoravicius and tassion a related result is that critical bernoulli percolation on a slab satisfies the box crossing property its proof is based on a new russo seymour welsh type theorem for quasi planar graphs thus at criticality the probability of an open path from 0 of diameter n decays polynomially in n this strengthens the result of duminil copin et al where the absence of an infinite cluster at criticality was first established 2017 wiley periodicals inc

a nested iterative scheme for indefinite linear systems in particulate flows abstract high fidelity large scale direct numerical simulation of particulate flows is of great value in a variety of industrial applications it is computationally intensive as it combines time integration solving nonlinear algebraic equations and the associated linear systems the finite element discretization of the coupled system of pdes on an unstructured grid using an arbitrary lagrangian eulerian moving mesh technique leads to very large nonlinear systems that are linearized by a version of newton s method the linear algebraic systems jacobians are sparse nonsymmetric and indefinite for which standard linear system solvers based on krylov subspace methods generally fail to converge without appropriate preconditioners the failure of krylov methods in production codes is currently being addressed by reducing the size of the time step this however leads to a very long simulation time and therefore is not always a viable approach in this study we design a hybrid inner outer iterative scheme for solving these indefinite systems which proves to be both efficient robust and ideally suited for parallel computing platforms even with appropriate large time steps comparisons with krylov subspace methods show the superiority of our proposed class of nested iterative schemes which is also scalable with respect to mesh size and insensitive to changes in properties of the fluid particles system

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

percolation in half spaces equality of critical densities and continuity of the percolation probability renormalization arguments are developed and applied to independent nearest neighbor percolation on various subsets of d d 2 yielding equality of the critical densities p c for a half space quarter space etc and ford 2 equality with the limit of slab critical densities continuity of the phase transition for the half space quarter space etc i e vanishing of the percolation probability p atp p c

on the local well posedness of quasilinear schrodinger equations in arbitrary space dimension abstract a class of quasilinear schrodinger equations is studied which contain strongly singular nonlinearities including derivatives of second order such equations have been derived as models of several physical phenomena the results of this paper apply to the superfluid film equation in fluid mechanics the local well posedness for smooth solutions of the cauchy problem is proved in arbitrary space dimension without any smallness assumption on the initial value this improves results in the literature which cover the one dimensional case only almost global well posedness results are obtained as well the proof combines nash moser techniques on implicit function theorems in frechet spaces with hyperbolic semigroup theory including evolution systems using the intersection of all sobolev spaces as basic function space the nondissipativity is overcome by a transformation procedure yielding the evolution operator and a priori estimates of higher order sobolev norms for the linearized problem here a certain trace has to vanish in view of apparent integrability conditions the transformations and the regularity theory are substantially different from the easier one dimensional case in particular continuity estimates are proved for the operator norms of the linear evolution operator in negative sobolev spaces

weighted matrix ordering and parallel banded preconditioners for iterative linear system solvers the emergence of multicore architectures and highly scalable platforms motivates the development of novel algorithms and techniques that emphasize concurrency and are tolerant of deep memory hierarchies as opposed to minimizing raw flop counts while direct solvers are reliable they are often slow and memory intensive for large problems iterative solvers on the other hand are more efficient but in the absence of robust preconditioners lack reliability while preconditioners based on incomplete factorizations whenever they exist are effective for many problems their parallel scalability is generally limited in this paper we advocate the use of banded preconditioners instead and introduce a reordering strategy that enables their extraction in contrast to traditional bandwidth reduction techniques our reordering strategy takes into account the magnitude of the matrix entries bringing the heaviest elements closer to the diagonal thus enabling the use of banded preconditioners when used with effective banded solvers in our case the spike solver we show that banded preconditioners i are more robust compared to the broad class of incomplete factorization based preconditioners ii deliver higher processor performance resulting in faster time to solution and iii scale to larger parallel configurations we demonstrate these results experimentally on a large class of problems selected from diverse application domains

uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation for independent translation invariant irreducible percolation models it is proved that the infinite cluster when it exists must be unique the proof is based on the convexity or almost convexity and differentiability of the mean number of clusters per site which is the percolation analogue of the free energy the analysis applies to both site and bond models in arbitrary dimension including long range bond percolation in particular uniqueness is valid at the critical point of one dimensional 1 x y 2 models in spite of the discontinuity of the percolation density there corollaries of uniqueness and its proof are continuity of the connectivity functions and except possibly at the critical point of the percolation density related to differentiability of the free energy are inequalities which bound the specific heat critical exponent in terms of the mean cluster size exponent and the critical cluster size distribution exponent e g 1 2 1 1

regular article direct simulations of 2d fluid particle flows in biperiodic domains we propose a method to simulate the motion of 2d rigid particles in a viscous incompressible fluid within the arbitrary lagrangian eulerian framework momentum equations for both the fluid and the particles are discretized and a coupled variational formulation is established by introducing an appropriate finite element approximation a symmetric linear system is obtained this system is solved by an inexact uzawa algorithm the main interest of such simulations lies in the average behaviour of a high number of particles we therefore introduced a biperiodic formulation of the problem which makes it possible to represent many body mixtures at a reasonable computational cost in order to model realistic situations an extra term must be added to the pressure this extra term is shown to be the lagrange multiplier associated with the vertical volume conservation constraint we developed an appropriate unstructured mesh generator so that the biperiodicity of the fields can be treated in a strong sense the question of particle contact is addressed and a simple technique to overcome numerical problems is proposed the influence of periodic lengths is investigated through different simulations the same case is simulated for different sizes of the window and the behaviour of some space averaged quantities is studied

multiple solutions for quasilinear elliptic equations with a finite potential well abstract we consider multiplicity of solutions for a class of quasilinear problems which has received considerable attention in the past including the so called modified nonlinear schrodinger equations by combining a new variational approach via q laplacian regularization and the compactness arguments from 4 we establish infinitely many bound state solutions for the quasilinear schrodinger type equations extending the earlier work of 4 for semilinear equations

multiple positive solutions to a class of modified nonlinear schrodinger equations abstract this paper is concerned with the existence and multiplicity of positive solutions of the following equation u u 2 u 2 u a x u p 2 u k x u where u h 1 r we will prove that the modified term 2 u 2 u is helpful to find multiple positive solutions in the case of the nonlinearity with sign changing coefficient a x surprisingly we do not need a sign condition r n a x e 1 p d x 0 which has been proven to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with sign changing nonlinearity alama and tarantello 1993 1

robust parallel smoothing for multigrid via sparse approximate inverses sparse approximate inverses are considered as smoothers for multigrid they are based on the spai algorithm m j grote and t huckle siam j sci comput 18 1997 pp 838 853 which constructs a sparse approximate inverse m of a matrix a by minimizing i ma in the frobenius norm this yields a new hierarchy of smoothers spai 0 spai 1 spai varepsilon advantages of spai smoothers over classical smoothers are inherent parallelism possible local adaptivity and improved robustness the simplest smoother spai 0 is based on a diagonal matrix m it is shown to satisfy the smoothing property for symmetric positive definite problems numerical experiments show that spai 0 smoothing is usually preferable to damped jacobi smoothing for the spai 1 smoother the sparsity pattern of m is that of a its performance is typically comparable to that of gauss seidel smoothing however both the computation and the application of the smoother remain inherently parallel in more difficult situations where the simpler spai 0 and spai 1 smoothers are not adequate the spai varepsilon smoother provides a natural procedure for improvement where needed numerical examples illustrate the usefulness of spai smoothing

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

existence of soliton solutions for a quasilinear schrodinger equation involving critical exponent in rn abstract mountain pass in a suitable orlicz space is employed to prove the existence of soliton solutions for a quasilinear schrodinger equation involving critical exponent in r n these equations contain strongly singular nonlinearities which include derivatives of the second order such equations have been studied as models of several physical phenomena the nonlinearity here corresponds to the superfluid film equation in plasma physics

quasilinear elliptic equations with critical growth via perturbation method abstract we consider a class of quasilinear schrodinger equations which include the modified nonlinear schrodinger equations a new perturbation approach is used to treat the critical exponent case giving new existence results

distributed lagrange multiplier methods for incompressible viscous flow around moving rigid bodies abstract in this article we discuss the application of a distributed lagrange multiplier based fictitious domain method to the numerical simulation of incompressible viscous flow modelled by the navier stokes equations around moving bodies we suppose the rigid bodies motion known a priori the solution method combines finite element approximations time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method numerical experiment results for two dimensional flow around a moving disk are presented

infinitely many small solutions for a modified nonlinear schrodinger equations abstract in the present paper we study the modified nonlinear schrodinger equations mnse without any growth condition on the nonlinear term we obtain the existence of infinitely many small solutions for mnse by a dual approach

bifurcations for quasilinear schrodinger equations i abstract this paper is concerned with bifurcation problems for quasilinear schrodinger type equations such as the modified nonlinear schrodinger equations which have appeared as several models from mathematical physics we give a new approach for dealing with bifurcation problems of nonlinear potential operators the novelty of our approach lies in establishing multiple bifurcations without making a reduction to finite dimensional problems the finite dimensional reduction procedures that have been used in the past work for semilinear problems but are not suitable for quasilinear problems we give a unified approach requiring much less regularity of the problems as applications we provide multiple bifurcation results for the modified nonlinear schrodinger equations

existence of positive solutions for a class of quasilinear schrodinger equations on rn abstract in this paper we study the following quasilinear schrodinger equation of the form u v x u 1 2 u 2 u u p 1 u u q 1 u x r n where n 3 2 2 n q p 2 2 n n 2 1 r n under appropriate assumptions on v x we establish the existence of ground state solutions by a minimization argument

absence of infinite cluster for critical bernoulli percolation on slabs we prove that for bernoulli percolation on a graph 2 0 k k 0 there is no infinite cluster at criticality almost surely the proof extends to finite range bernoulli percolation models on 2 that are invariant under 2 rotation and reflection 2016 wiley periodicals inc

discontinuity of the percolation density in one dimensional 1 x y 2 percolation models we consider one dimensional percolation models for which the occupation probability of a bond kx y has a slow power decay as a function of the bond s length for independent models and with suitable reformulations also for more general classes of models it is shown that i no percolation is possible if for short bondskx y p 1 this dichotomy resembles one for the magnetization in 1 x y 2 ising models which was first proposed by thouless and further supported by the renormalization group flow equations of anderson yuval and hamann the proofs of the above percolation phenomena involve rigorous renormalization type arguments of a different sort

preconditioning for the steady state navier stokes equations with low viscosity we introduce a preconditioner for the linearized navier stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero for constant coefficient problems with periodic boundary conditions we show that the preconditioning yields a system with a single eigenvalue equal to 1 so that performance is independent of both viscosity and mesh size for other boundary conditions we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon we also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration with virtually no degradation of performance

efficient preconditioning of the linearized navier stokes equations for incompressible flow we outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of navier stokes equations we describe a very general strategy for preconditioning that has two basic building blocks a multigrid v cycle for the scalar convection diffusion operator and a multigrid v cycle for a pressure poisson operator we present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid robust with respect to the time step and only deteriorates very slowly as the reynolds number is increased

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

an inverse function theorem in sobolev spaces and applications to quasi linear schrodinger equations abstract a nash moser type inverse function theorem in banach spaces with loss of derivatives is proved and applications are given to singular quasi linear schrodinger equations like the superfluid film equation in plasma physics based on an implicit function theorem in sobolev spaces a linearization method is introduced for the local and global well posedness of the cauchy problem for nonlinear evolution equations the technique compensates for a loss of derivatives in the linearized problem this is illustrated by an application to strongly singular quasi linear schrodinger equations where the nonlinearities include derivatives of second order the local well posedness of the cauchy problem is proved in sobolev spaces for arbitrary space dimension without assuming smallness assumptions on the initial value here the linearized problem is solved using hyperbolic semigroup theory including evolution systems

existence of positive solutions for a class of quasilinear schrodinger equations of choquard type abstract in this paper we study the following quasilinear schrodinger equation of choquard type u v x u u 2 u i u p u p 2 u x r n where n 3 0 n 2 n n p 2 n n 2 and i is a riesz potential under appropriate assumptions on v x we establish the existence of positive solutions

parallel finite element computations in fluid mechanics we provide an overview of the role of parallel finite element computations in fluid mechanics we emphasize the class of flow problems involving moving boundaries and interfaces some of the computationally most challenging flow problems such as fluid object and fluid structure interactions as well as free surface and two fluid flows belong to this class in the development of the methods targeting this class of problems the computational challenges involved need to be addressed in such a way that 3d computation of complex applications can be carried out efficiently on parallel computers this requirement has to be one of the key factors in designing various components of the overall solution approach such as solution techniques for the discretized equations and mesh update methods for handling the changes in the spatial domain occupied by the fluid this overview includes a description of how the computational challenges are addressed and how the computational schemes can be enhanced with special preconditioning techniques 2005 elsevier b v all rights reserved

existence and multiplicity of non trivial solutions for a class of modified schrodinger poisson systems abstract in the present paper we study the following modified schrodinger poisson system p u v x u u 1 2 u u 2 u p 1 u h x in r 3 u 2 in r 3 where 4 p 1 12 h l 2 r 3 and h 0 this model describes the interaction of a charge particle with an electro magnetic field under some hypotheses on v x we prove the existence of a non trivial ground state solution and two non trivial ground state solutions for the system p

the supercritical phase of percolation is well behaved we prove a general result concerning the critical probabilities of subsets of a lattice l it is a consequence of this result that the critical probability of a percolation process on l equals the limit of the critical probability of a slice of l as the thickness of the slice tends to infinity this verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation

fast nonsymmetric iterations and preconditioning for navier stokes equations discretization and linearization of the steady state navier stokes equations gives rise to a nonsymmetric indefinite linear system of equations in this paper we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization we confirm and supplement these analytic results with a series of numerical experiments indicating that krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter in addition we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner

mean field driven first order phase transitions in systems with long range interactions we consider a class of spin systems on d with vector valued spins sx that interact via the pair potentials jx ysx sy the interactions are generally spread out in the sense that the jx y s exhibit either exponential or power law fall off under the technical condition of reflection positivity and for sufficiently spread out interactions we prove that the model exhibits a first order phase transition whenever the associated mean field theory signals such a transition as a consequence e g in dimensions d 3 we can finally provide examples of the 3 state potts model with spread out exponentially decaying interactions which undergoes a first order phase transition as the temperature varies similar transitions are established in dimensions d 1 2 for power law decaying interactions and in high dimensions for next nearest neighbor couplings in addition we also investigate the limit of infinitely spread out interactions specifically we show that once the mean field theory is in a unique state then in any sequence of translation invariant gibbs states various observables converge to their mean field values and the states themselves converge to a product measure

existence of non trivial solution for a class of modified schrodinger poisson equations via perturbation method abstract in this paper by using the methods of perturbation and the mountain pass theorem we prove the existence of non trivial solution to a class of modified schrodinger poisson equations

a posteriori error estimation for stabilized mixed approximations of the stokes equations the practical implementation of the lowest order pl p0 linear velocity constant pressure finite element method for the steady state incompressible navier stokes equations is addressed in this work three different types of a posteriori error indicator are introduced and each is shown to be equivalent to the discretization error our numerical results show that these indicators can be used to drive an adaptive refinement process which is specially tailored to create grids which conform to the requirements of the local stabilization it is also shown that the indicators provide an effective method for detecting local singularities in the flow solution

the cauchy problem for quasi linear schrodinger equations with x rn t r with x1 xn and summation convention one may think of the equation in 1 1 as a non linear schrodinger equation where the operator modeling the dispersion relation is non isotropic and depends also on the unknown function its conjugate and their gradients in the space variables under appropriate assumptions on the function f the method presented here extends to fully non linear schrodinger equations of the form

high energy solutions of modified quasilinear fourth order elliptic equations with sign changing potential abstract in the present paper we consider the following modified quasilinear fourth order elliptic equation 1 1 2 u u v x u 1 2 u 2 u f x u for x r n u x h 2 r n where 2 is the biharmonic operator v c r n r and f c r n r r n 5 are allowed to be sign changing two main theorems on the existence of nontrivial solutions and infinitely many high energy solutions for eq 1 1 are obtained via variational methods

on a class of periodic quasilinear schrodinger equations involving critical growth in r2 abstract we consider the equation u v x u k 2 u 2 u g x u u 0 x r 2 where v r 2 r and g r 2 r r are two continuous 1 periodic functions and k is a positive constant also we assume g behaves like exp u 4 as u we prove the existence of at least one weak solution u h 1 r 2 with u 2 h 1 r 2 the mountain pass in a suitable orlicz space together with the trudinger moser inequality are employed to establish this result such equations arise when one seeks for standing wave solutions for the corresponding quasilinear schrodinger equations schrodinger equations of this type have been studied as models of several physical phenomena the nonlinearity here corresponds to the superfluid film equation in plasma physics

approximate inverse preconditioners via sparse sparse iterations the standard incomplete lu ilu preconditioners often fail for general sparse indefinite matrices because they give rise to unstable factors l and u in such cases it may be attractive to approximate the inverse of the matrix directly this paper focuses on approximate inverse preconditioners based on minimizing i am f where am is the preconditioned matrix an iterative descent type method is used to approximate each column of the inverse for this approach to be efficient the iteration must be done in sparse mode i e with sparse matrix by sparse vector operations numerical dropping is applied to maintain sparsity compared to previous methods this is a natural way to determine the sparsity pattern of the approximate inverse this paper describes newton global and column oriented algorithms and discusses options for initial guesses self preconditioning and dropping strategies some limited theoretical results on the properties and convergence of approximate inverses are derived numerical tests on problems from the harwell boeing collection and the fidap fluid dynamics analysis package show the strengths and limitations of approximate inverses finally some ideas and experiments with practical variations and applications are presented

critical percolation on any nonamenable group has no infinite clusters we show that independent percolation on any cayley graph of a nonamenable group has no infinite components at the critical parameter this result was obtained by the present authors earlier as a corollary of a general study of group invariant percolation the goal here is to present a simpler self contained proof that easily extends to quasi transitive graphs with a unimodular automorphism group the key tool is a mass transport method which is a technique of averaging in nonamenable settings

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

sharp condition of global existence for second order derivative nonlinear schrodinger equations in two space dimensions this paper discusses a class of second order derivative nonlinear schrodinger equations which are used to describe the upper hybrid oscillation propagation by establishing a variational problem applying the potential well argument and the concavity method we prove that there exists a sharp condition for global existence and blow up of the solutions to the nonlinear schrodinger equation in addition we also answer the question how small are the initial data the global solutions exist

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

nodal soliton solutions for generalized quasilinear schrodinger equations this paper is concerned with constructing nodal radial solutions for generalized quasilinear schrodinger equations in rn which arise from plasma physics fluid mechanics as well as high power ultashort laser in matter for any given integer k 0 by using a change of variables and minimization argument we obtain a sign changing minimizer with k nodes of a minimization problem

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

nodal soliton solutions for quasilinear schrodinger equations with critical exponent this paper is concerned with constructing nodal radial solutions for quasilinear schrodinger equations in rn with critical growth which have appeared as several models in mathematical physics for any given integer k 0 by using a change of variables and minimization argument we obtain a sign changing minimizer with k nodes of a minimization problem since the critical exponent appears and the lower order term may change sign we should use more delicate arguments

existence and multiplicity of non trivial solutions for a class of modified schrodinger equation with non coercive potential in the present paper we study the following modified nonlinear schrodinger equation u v x u 12u u 2 u p 1u h x x r n p where n 3 4 0 under some hypotheses on v we obtain the existence and multiplicity of non trivial solutions

uniqueness and nondegeneracy of the ground state for a quasilinear schrodinger equation with a small parameter abstract we study least energy solutions of a quasilinear schrodinger equation with a small parameter we prove that the ground state is nondegenerate and unique up to translations and phase shifts using bifurcation theory

global existence of small solutions to a relativistic nonlinear schrodinger equation on etudie le probleme de cauchy associea une equation de schrodinger non lineaire modelisant une impulsion laser ultra intense et ultra courte dans un plasma les termes non lineaires nouveaux sont dus aux effets relativistes et a la force ponderomotrice nous prouvons l existence locale et l unicite de solutions petites lorsque la dimension de l espace transverse est egale a 2 ou 3 ainsi que l existence locale pour des donnees initiales de taille arbitraire en dimension un d espace transverse

interfaces in the potts model i pirogov sinai theory of the fortuin kasteleyn representation we develop a new analysis of the order disorder transition in ferromagnetic potts models for large numberq of spin states we use the pirogov sinai theory which we adapt to the fortuin kasteleyn representation of the models this theory applies in a rather direct way in our approach and leads to a system of non interacting contours with small activities as a consequence simpler and more natural techniques are found allowing us to recover previous results on the bulk properties of the model which then extend to non integer values ofq and to deal with non translation invariant boundary conditions this will be applied in a second part of this work to study the behaviour of the interfaces at the transition point

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

quadratic finite elements and incompressible viscous flows pressure stabilization methods are applied to higher order velocity finite elements for application to viscous incompressible flows both a standard pressure stabilizing petrov galerkin pspg method and a new polynomial pressure projection stabilization ppps method have been implemented and tested for various quadratic elements in two dimensions a preconditioner based on relaxing the incompressibility constraint is also tested for the iterative solution of saddle point problems arising from mixed galerkin finite element approximations to the navier stokes equations the preconditioner is demonstrated for bb stable elements with discontinuous pressure approximations in two and three dimensions

the blow up problem for a quasilinear schrodinger equation consider the blow up results for w2 2 rn solutions of a quasilinear schrodinger equation iut u u p 2u u 2 u 0 u t 0 u0 x x rn when x u0 l2 rn we show that the w2 2 rn solutions must blow up for any 0 r and some restriction on p we also show that the radial symmetric solutions in w2 2 rn must blow up at finite time without assuming x u0 l2 rn

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach

methods for 3d computation of fluid object interactions in spatially periodic flows abstract we present computational methods for 3d simulation of fluid object interactions in spatially periodic flows these methods include a stabilized space time finite element formulation for incompressible flows with spatial periodicity automatic mesh generation and update techniques for fluid object mixtures with spatial periodicity and parallel implementations the methods can be applied to uni periodic i e periodic in one direction bi periodic or tri periodic flows the methods are described here in the context of tri periodic flows with fluid object interactions and are applied to the simulation of sedimentation of particles in a fluid we present several case studies where the results obtained provide notable insight into the behavior of fluid particle mixtures during sedimentation

a smoothing property for frechet spaces a smoothing property s tfor frechet spaces is introduced generalizing the classical concept of smoothing operators which are important in the proof of nash moser inverse function theorems for frechet hilbert spaces property in standard form in the sense of d vogt is shown to be sufficient for s t for instance the spaces e k of infinitely differentiable functions in the sense of whitney have property s tfor an arbitrary compactk rn applications to extensions of whitney functions with estimates are included in a forthcoming paper an inverse function theorem will be proved for frechet spaces with properties s tand dn this applies to e k if the compactk k rnis subanalytic

emergent planarity in two dimensional ising models with finite range interactions the known pfaffian structure of the boundary spin correlations and more generally order disorder correlation functions is given a new explanation through simple topological considerations within the model s random current representation this perspective is then employed in the proof that the pfaffian structure of boundary correlations emerges asymptotically at criticality in ising models on mathbb z 2 with finite range interactions the analysis is enabled by new results on the stochastic geometry of the corresponding random currents the proven statement establishes an aspect of universality seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases