higher order approximation of analytic sets by topologically equivalent algebraic sets it is known that every germ of an analytic set is homeomorphic to the germ of an algebraic set in this paper we show that the homeomorphism can be chosen in such a way that the analytic and algebraic germs are tangent with any prescribed order of tangency moreover the space of arcs contained in the algebraic germ approximates the space of arcs contained in the analytic one in the sense that they are identical up to a prescribed truncation order

periodic solutions of second order equations via rotation numbers abstract we consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems we show how a rotation number approach together with the poincare birkhoff theorem and the phase plane analysis of the spiral properties allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity these systems may also be resonant in particular our approach can be used to deal with the problems without both global existence of the cauchy problems associated to the equation and the sign assumption on f the typical example is a partially superlinear second order equation our main result generalizes some classical results of jacobowitz and hartman among others

on a singular periodic ambrosetti prodi problem abstract we investigate the possibility of extending a classical multiplicity result by fabry mawhin and nkashama to a periodic problem of ambrosetti prodi type having a nonlinearity with possibly one or two singularities in the second part of the paper we study the existence of periodic rotating solutions for radially symmetric systems with nonlinearities of the same type

isogeometric collocation neumann boundary conditions and contact abstract isogeometric collocation methods have been proposed recently and their accuracy and efficiency demonstrated for elastostatics and explicit dynamics this paper addresses two important aspects in the development of the isogeometric collocation technology namely the imposition of neumann boundary conditions and the enforcement of contact constraints which are both treated within the same framework it is shown that the strong imposition of neumann boundary conditions may lead to a significant loss of accuracy in some situations in particular when non uniform meshes are used two possible remedies are proposed to restore the desired level of accuracy while keeping the computational cost virtually unchanged i e a hybrid collocation galerkin approach and an enhanced collocation ec approach a frictionless contact formulation suitable for the collocation framework is further proposed and shown to pass the contact patch test to machine precision when combined with the ec approach the formulation is shown to deliver accurate results and to perform robustly also for highly non uniform meshes for all the collocation formulations contact pressures are greater than or equal to zero pointwise in contrast with standard lagrange finite elements

the topology of real algebraic sets with isolated singularities in this paper we give a topological classification of real algebraic sets with isolated singularities showing that they are exactly smooth closed manifolds with smooth subpolyhedra crushed to points the question of which topological spaces are homeomorphic to real algebraic sets solutions of polynomial equations in euclidean space has been long studied in 1936 seifert showed that any smooth compact stably parallelizable manifold is diffeomorphic to a component of an algebraic set 121 and in 1952 nash extended this result to all smooth compact manifolds 11 in 1973 tognoli showed that any smooth compact manifold is diffeomorphic to a nonsingular algebraic set 13 so at least compact nonsingular algebraic sets are classified little has been done with singular algebraic sets however since the transversality arguments used by seifert nash and tognoli no longer apply except in some special cases one could use stability of singularities such as kuiper 71 and akbulut 1 used to show certain nonsmoothable pl manifolds are algebraic sets or one could use the projective version of seifert nash tognoli as king did 6 but one could still not hope these techniques would allow even a characterization of isolated singularities to get around this problem we take a cue from hironaka s resolution of singularities 4 the idea is to take a topological resolution of a space if it exists we can apply transversality techniques seifertnash tognoli to the resolved space and then blow down algebraically and end up with the original space as an algebraic set it seems likely that this technique allows one to classify all algebraic sets but in any case we show that it classifies all algebraic sets with isolated singularities in future papers we will use this technique to show for instance that all compact pl manifolds are homeomorphic to real algebraic sets 17 and that 2 dimensional real algebraic sets are topologically characterized as polyhedra satisfying sullivan s even local euler characteristic condition 16

on the periodic fucik spectrum and a superlinear sturm liouville equation in the first part of the paper a variational characterisation of the periodic eigenvalues the so called fucik spectrum of a semilinear positive homogeneous sturm liouville equation is given the proof relies on the s 1 invariance of the equation in the second part a nonlinear sturm liouville equation with typically an exponential nonlinearity is considered it is proved that under certain conditions this equation is solvable for arbitrary forcing terms the proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values in the fucik spectrum

resonance and rotation numbers for planar hamiltonian systems multiplicity results via the poincare birkhoff theorem abstract in the general setting of a planar first order system 0 1 u g t u u r 2 with g 0 t r 2 r 2 we study the relationships between some classical nonresonance conditions including the landesman lazer one at infinity and in the unforced case i e g t 0 0 at zero and the rotation numbers of large and small solutions of 0 1 respectively such estimates are then used to establish via the poincare birkhoff fixed point theorem new multiplicity results for t periodic solutions of unforced planar hamiltonian systems j u u h t u and unforced undamped scalar second order equations x g t x 0 in particular by means of the landesman lazer condition we obtain sharp conclusions when the system is resonant at infinity

boundedness in a piecewise linear oscillator and a variant of the small twist theorem consider the differential equation ddot x n 2 x h l x p t where n 1 2 dots is an integer p is a 2 pi periodic function and h l is the piecewise linear function h l x begin cases l text if x geq 1 lx text if x leq 1 l text if x leq 1 end cases a classical result of lazer and leach implies that this equation has a 2 pi periodic solution if and only if begin equation label ll hat p n 2l over pi end equation where hat p n 1 over 2 pi int 0 2 pi p t e int dt in this paper i prove that if p is of class c 5 then the condition ref ll is also necessary and sufficient for the boundedness of all the solutions of the equation the proof of this theorem motivates a new variant of moser s small twist theorem this variant guarantees the existence of invariant curves for certain mappings of the cylinder which have a twist that may depend on the angle 1991 mathematics subject classification 34c11 58f35

three dimensional simulation of unstable gravity driven infiltration of water into a porous medium infiltration of water in dry porous media is subject to a powerful gravity driven instability although the phenomenon of unstable infiltration is well known its description using continuum mathematical models has posed a significant challenge for several decades the classical model of water flow in the unsaturated flow the richards equation is unable to reproduce the instability here we present a computational study of a model of unsaturated flow in porous media that extends the richards equation and is capable of predicting the instability and captures the key features of gravity fingering quantitatively the extended model is based on a phase field formulation and is fourth order in space the new model poses a set of challenges for numerical discretizations such as resolution of evolving interfaces stiffness in space and time treatment of singularly perturbed equations and discretization of higher order spatial partial differential operators we develop a numerical algorithm based on isogeometric analysis a generalization of the finite element method that permits the use of globally smooth basis functions leading to a simple and efficient discretization of higher order spatial operators in variational form we illustrate the accuracy efficiency and robustness of our method with several examples in two and three dimensions in both homogeneous and strongly heterogeneous media we simulate for the first time unstable gravity driven infiltration in three dimensions and confirm that the new theory reproduces the fundamental features of water infiltration into a porous medium our results are consistent with classical experimental observations that demonstrate a transition from stable to unstable fronts depending on the infiltration flux

existence and multiplicity results for periodic solutions of semilinear duffing equations we obtain existence an multiplicity of harmonic and subharmonic solutions to the periodically forced nonlinear duffing s equation x g x p t in the semi linear case 0 a g x x b for x large we introduce new conditions of nonresonance with respect to the spectrum of the associated linear differential operator x x and show by some nonlinear counterexamples that the results are sharp

a nurbs based immersed methodology for fluid structure interaction abstract we introduce an isogeometric immersed and fully implicit formulation for fluid structure interaction fsi the method focuses on viscous incompressible flows and nonlinear hyperelastic incompressible solids which are a common case in various fields such as for example biomechanics in our fsi method we utilize an eulerian mesh on the whole domain and a lagrangian mesh on the solid domain the lagrangian mesh is arbitrarily located on top of the eulerian mesh in a non conforming fashion due to the formulation of our problem based on the immersed finite element method ifem we do not need mesh update or remeshing algorithms the fluid structure interface is the boundary of the lagrangian mesh but cuts arbitrarily the eulerian mesh the generalized method is used for time discretization and nurbs based isogeometric analysis is employed for the spatial discretization on both fluid and solid domains the information transfer between the two meshes is carried out using the nurbs functions which avoids the use of the so called discretized delta functions the higher order and especially the higher continuity of nurbs functions allow us to deal with severe mesh distortion in the lagrangian mesh in comparison with classical c 0 linear piecewise functions as we prove numerically our numerical solutions attain good agreement with theoretical data for free falling objects in two and three dimensions which confirms the feasibility of our methodology

on approximating submanifolds by algebraic sets and a solution to the nash conjecture in n nash showed that any compact smooth submanifold m of ir is e isotopic to a union of connected components of a real algebraic variety if n 2 dim m e isotopic just means that there are arbitrarily small smooth isotopies to components of real algebraic varieties for smaller n however he was not able to get such a strong result but conjectured it was still true in w a proof of this conjecture was given which unfortunately had a gap later i and t2 gave correct proofs in the case where n is bigger than roughly 3 dim m 2 among other results in this paper we give a proof of this conjecture of nash in particular we will show

periodic solutions of semilinear duffing equations with impulsive effects abstract in this paper we are concerned with the existence of periodic solutions for semilinear duffing equations with impulsive effects firstly for its autonomous equation any motion of the solution is same as the motion of the corresponding equation without impulses until it meets the first impulse time under the influence of impulses these two motions are likely to be quite different we introduce some reasonable assumptions on the impulsive functions to control these differences such that the information valid for the equation without impulses can always be used for the impulsive one basing on poincare birkhoff twist theorem we prove the existence of infinitely many periodic solutions secondly as for the nonautonomous equation where the autonomous case is taken as an auxiliary one we find the relation between the solutions of these two equations and then obtain the existence of infinitely many periodic solutions also by poincare birkhoff twist theorem lastly an example with special impulses satisfying the above assumptions is given

liquid vapor phase transition thermomechanical theory entropy stable numerical formulation and boiling simulations abstract we develop a new continuum mechanics modeling framework for liquid vapor flows with particular focus on the van der waals fluid by invoking microforce theory the coleman noll procedure is generalized to derive consistent constitutive relations in the presence of non local effects a new thermodynamically consistent algorithm for the van der waals model is designed using functional entropy variables and a new temporal scheme employing a family of new quadrature rules we show that the resulting fully discrete scheme is unconditionally stable in entropy and second order time accurate isogeometric analysis is utilized for spatial discretization the analytical properties of the formulation are corroborated by benchmark problems three sets of application problems are simulated to demonstrate the capability of the model and the algorithm our methodology provides a particularly useful predictive tool for boiling flows

large amplitude subharmonic oscillations for scalar second order differential equations with asymmetric nonlinearities abstract we provide sufficient conditions for the existence of subharmonic solutions for equations whose nonlinearity may have a different qualitative behaviour to the right and to the left of the origin the proofs are based on variational methods

periodic solutions of duffing s equations with superquadratic potential abstract this paper is devoted to the study of harmonic and subharmonic solutions for the second order scalar nonlinear duffing s equation x g x p t x x where g and p are continuous functions with p bounded and periodic in the first variable and g satisfying the assumption g x sign x as x among other results we prove the existence of infinitely many harmonic and subharmonic solutions of any order p p t and if the potential g x of g x satisfies certain conditions of superquadratic growth at the new existence results can be applied to situations in which the more classical superlinear growth condition g x x as x is not satisfied in this manner various preceding theorems are improved and sharpened see the introduction for more details proofs are based on a generalized version of the poincare birkhoff twist theorem due to w ding

periodic solutions of some resonant hamiltonian systems abstract based on the galerkin approximation and the morse inequality an existence theorem of periodic solutions of the hamiltonian system j z h t z is given where h t z is asymptotically linear and resonant at infinity existence of periodic solutions of the resonant asymmetric p laplacian equation is also discussed

continuous rational maps into spheres let x be a compact nonsingular real algebraic variety we prove that if a continuous map from x into the unit p sphere is homotopic to a continuous rational map then under certain assumptions it can be approximated in the compact open topology by continuous rational maps as a byproduct we also obtain some results on approximation of smooth submanifolds by nonsingular subvarieties

multiple solutions for a dirichlet problem with jumping nonlinearities ii on considere l existence des solutions pour le probleme x t g x t p t c t 0 1 x 0 x 1 0 ou p t est une fonction continue et c est une constante on suppose que g est de classe c 1 et que g est positive et strictement croissante sur 0

an isogeometric collocation method using superconvergent points abstract we develop an iga collocation method modified by collocating at points other than the standard greville abscissae the method is related to orthogonal collocation used for solving differential equations and to the superconvergence theory therefore we refer to this method as super collocation iga sc by carefully choosing the collocation points it can be seen that the iga sc converges in the first derivative energy norms at rates similar to that of the galerkin solution this is different from the collocation at greville abscissae iga c where the convergence in energy norm for odd polynomial degrees is typically suboptimal the method is tested on 1d 2d and 3d numerical examples in which it is compared to iga c and galerkin s method iga g the comparison includes a detailed cost vs accuracy analysis which shows an improved efficiency of the proposed method in particular for odd polynomial degrees

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

superconvergence in the generalized finite element method in this paper we address the problem of the existence of superconvergence points of approximate solutions obtained from the generalized finite element method gfem of a neumann elliptic boundary value problem gfem is a galerkin method that uses non polynomial shape functions and was developed in babuska et al in siam j numer anal 31 945 981 1994 babuska et al in int j numer meth eng 40 727 758 1997 melenk and babuska in comput methods appl mech eng 139 289 314 1996 in particular we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non linear equations this system does not depend on the solution of the boundary value problem for approximate solutions with second derivatives we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non linear equations we note that smooth generalized finite element approximation is easy to construct

a new fixed point theorem and periodic solutions of nonconservative weakly coupled systems abstract in this paper we prove a new fixed point theorem for the coupling of twist conditions and poincare bohl type conditions as its applications we prove the existence of periodic solutions for various nonconservative mixed type weakly coupled systems

a multiplicity result for periodic solutions of second order differential equations with a singularity abstract by the use of the poincare birkhoff fixed point theorem we prove a multiplicity result for periodic solutions of a second order differential equation where the nonlinearity exhibits a singularity of repulsive type at the origin and has linear growth at infinity our main theorem is related to previous results by rebelo 1996 1997 4 5 and rebelo and zanolin 1996 6 7 in connection with a problem raised by del pino et al 1992 1

subharmonic solutions with prescribed minimal period for nonautonomous hamiltonian systems on etablit un resultat de multiplicite pour une certaine classe de systemes hamiltoniens periodiques en temps

algebraic realization of manifolds with group actions abstract let g be a compact lie group and m a closed smooth g manifold with some restrictions on g or on the isotropy groups of the action we show that m is algebraically realized i e there exists a non singular real algebraic g variety which is equivariantly diffeomorphic to m

accurate efficient and iso geometrically flexible collocation methods for phase field models we propose new collocation methods for phase field models our algorithms are based on isogeometric analysis a new technology that makes use of functions from computational geometry such as for example non uniform rational b splines nurbs nurbs exhibit excellent approximability and controllable global smoothness and can represent exactly most geometries encapsulated in computer aided design cad models these attributes permitted us to derive accurate efficient and geometrically flexible collocation methods for phase field models the performance of our method is demonstrated by several numerical examples of phase separation modeled by the cahn hilliard equation we feel that our method successfully combines the geometrical flexibility of finite elements with the accuracy and simplicity of pseudo spectral collocation methods and is a viable alternative to classical collocation methods

periodic solutions of superlinear impulsive differential equations a geometric approach abstract a geometric method is introduced to study superlinear second order differential equations with impulsive effects basing on a reference continuous polar lifting of a planar orientation preserving homeomorphism we prove via the poincare birkhoff twist theorem the existence of infinitely many periodic solutions of conservative superlinear second order equations with the finite twist impulsive terms we also prove by developing a new twist fixed point theorem the existence of periodic solutions for non conservative superlinear second order equations with degenerate impulsive terms

the rotation number approach to the periodic fuc caron ik spectrum abstract in this paper we study the fucik spectrum of the problem q t x q t x 0 with the 2 periodic boundary condition where q t are 2 periodic after introducing a rotation number function for we prove using the hamiltonian structure and the positive homogeneity of that for any positive integer n the two boundary curves of the domain 1 n 2 in the plane are fucik curves of the result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional fucik spectrum with the dirichlet condition in particular it remains open if the fucik spectrum of is composed of only these curves

functional entropy variables a new methodology for deriving thermodynamically consistent algorithms for complex fluids with particular reference to the isothermal navier stokes korteweg equations we propose a new methodology for the numerical solution of the isothermal navier stokes korteweg equations our methodology is based on a semi discrete galerkin method invoking functional entropy variables a generalization of classical entropy variables and a new time integration scheme we show that the resulting fully discrete scheme is unconditionally stable in energy second order time accurate and mass conservative we utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions various problems are simulated to show the capability of the method our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non convex hyperbolic systems of conservation laws

the exact number of solutions for a class of ordinary differential equations through morse index computation we consider a class of second order ordinary differential equations with jumping nonlinearities that cross the first k eigenvalues under zero boundary conditions and by means of a morse index computation show the existence of exactly 2 k solutions

isogeometric collocation for large deformation elasticity and frictional contact problems abstract isogeometric collocation methods have been recently proposed as an alternative to standard galerkin approaches as they provide a significant reduction in computational cost for higher order discretizations in this work we explore the application of isogeometric collocation to large deformation elasticity and frictional contact problems we first derive the non linear governing equations for the elasticity problem with finite deformation kinematics and provide details on their consistent linearization some numerical examples demonstrate the performance of collocation in its basic and enhanced versions differing by the enforcement of neumann boundary conditions for problems with strong singularities enhanced collocation is shown to outperform basic collocation and to lead to a spatial convergence behavior very similar to galerkin whereas for weaker or no singularities enhanced and basic collocation may give very similar results a large deformation contact formulation is subsequently developed and tested in the frictional setting where collocation confirms the excellent performance already obtained for the frictionless case finally it is shown that the contact formulation in the collocation framework passes the contact patch test to machine precision in a three dimensional setting with arbitrarily inclined non matching discretizations thus outperforming most of the available contact formulations and all those with pointwise enforcement of the contact constraints

multiplicity results for an ode problem with even nonlinearity on considere le probleme u g u h dans u 0 sur ou r n est un ouvert borne a frontiere lisse en se restreignant au cas u u 2 t sin x dans 0 u 0 u 0 etant donne n 0 n il existe t nd 0 tel que pour tout t t nd 0 l equation consideree a au moins n 0 solutions

algebraic cycles and approximation theorems in real algebraic geometry let af be a compact c manifold a theorem of nash tognoli asserts that m has an algebraic model that is m is diffeomorphic to a nonsingular real algebraic set x let fv afx z 2 denote the subgroup of hk x z 2 of the cohomology classes determined by algebraic cycles of codi mension k on x assuming that m is connected orientable and dim m 5 we prove in this paper that a subgroup g of h2 m z 2 is isomorphic to h x z 2 for some algebraic model x of m if and only if w2 tm is in g and each element of g is of the form w2k for some real vector bundle over m where w2 stands for the second stiefel whitney class a result of this type was previously known for subgroups g of hx m z 2

existence uniqueness and stability of oscillations in differential equations with asymmetric nonlinearities we give conditions for the existence uniqueness and asymptotic stability of periodic solutions of a second order differential equation with piecewise linear restoring and 27r periodic forcing where the range of the derivative of the restoring term possibly contains the square of an integer with suitable restrictions on the restoring and forcing in the undamped case we give a necessary and sufficient condition

multiple periodic solutions of scalar second order differential equations abstract we prove multiplicity of periodic solutions for a scalar second order differential equation with an asymmetric nonlinearity thus generalizing previous results by lazer and mckenna 1987 1 and del pino manasevich and murua 1992 2 the main improvement lies in the fact that we do not require any differentiability condition on the nonlinearity the proof is based on the use of the poincare birkhoff fixed point theorem

the variational collocation method we propose the variational collocation method for the numerical solution of partial differential equations the conceptual basis is the establishment of a direct connection between the galerkin method and the classical collocation methods with the perspective of achieving the accuracy of the former with a computational cost of one point evaluation per degree of freedom as in the latter variational collocation requires a discrete space constructed by smooth and pointwise non negative basis functions which makes the approach immediately applicable to isogeometric analysis and some meshfree methods in this paper we concentrate on isogeometric analysis and demonstrate that there exists a set of points such that collocation of the strong form at these points produces the galerkin solution exactly we provide an estimate of these points and show that applying isogeometric collocation at the estimated points completely solves the well known odd even discrepancy in the order of spatial convergence we demonstrate the potential of variational collocation with examples of linear and non linear elasticity as well as kirchhoff plates

remarks on rate constitutive equations for finite deformation problems computational implications it is explicitly shown that if the spatial elasticity tensor of an elastic material is taken as isotropic for all possible configurations then its coefficients cannot be constants they must depend nontrivially on the jacobian determinant of the deformation gradient moreover the assumption typically made for computational purposes that its coefficients remain constant for all possible configurations is incompatible with elasticity it is further shown that an assumption widely used in the computational literature in the context of finite deformation plasticity namely relating an objective stress rate to the rate of deformation tensor through a fourth rank constant isotropic tensor is also incompatible with elasticity thus furnishing an example of an hypoelastic material which is not elastic