defining functions for manifolds of matrices abstract we consider defining systems for manifolds of possibly nonsquare matrices the functions are algebraic expressions in the entries of the matrix the systems are local in the sense of being defined on the sets of a predefined cover of the manifold with open and dense sets we use bordered matrix techniques to characterize the manifold of matrices with a given rank in the case of square matrices we give conditions to express that zero is an eigenvalue with a given algebraic multiplicity and geometric multiplicity 1 in both cases the obtained systems have full linear rank hence they define the underlying manifolds in a regular way and can be used analytically to obtain local parametrizations of the manifold or numerically to set up newton systems with local quadratic convergence

error bounds for 2 regular mappings with lipschitzian derivatives and their applications we obtain local estimates of the distance to a set defined by equality constraints under assumptions which are weaker than those previously used in the literature specifically we assume that the constraints mapping has a lipschitzian derivative and satisfies a certain 2 regularity condition at the point under consideration this setting directly subsumes the classical regular case and the twice differentiable 2 regular case for which error bounds are known but it is significantly richer than either of these two cases when applied to a certain equation based reformulation of the nonlinear complementarity problem our results yield an error bound under an assumption more general than b regularity the latter appears to be the weakest assumption under which a local error bound for complementarity problems was previously available we also discuss an application of our results to the convergence rate analysis of the exterior penalty method for solving irregular problems

a liouville type theorem for schrodinger operators in this paper we prove a sufficient condition in terms of the behavior of a ground state of a symmetric critical operator p 1 such that a nonzero subsolution of a symmetric nonnegative operator p 0 is a ground state particularly if p j v j for j 0 1 are two nonnegative schrodinger operators defined on omega subseteq mathbb r d such that p 1 is critical in with a ground state the function psi nleq 0 is a subsolution of the equation p 0 u 0 in and satisfies psi leq c varphi in then p 0 is critical in and psi is its ground state in particular psi is up to a multiplicative constant the unique positive supersolution of the equation p 0 u 0 in similar results hold for general symmetric operators and also on riemannian manifolds

multiparameter local bifurcation on considere l equation l 0 u l 1 n u f 1 n u 0 sur un espace de banach reel u on etudie des solutions non triviales bifurcant a l origine

prediction of the mechanical response of the femur with uncertain elastic properties abstract a mandatory requirement for any reliable prediction of the mechanical response of bones based on quantitative computer tomography is an accurate relationship between material properties usually young s modulus e and bone density many such e relationships are available based on different experiments on femur specimens with a large spread due to uncertainties the first goal of this study is to pool and analyze the relevant available experimental data and develop a stochastic e relationship this analysis highlights that there is no experimental data available to cover the entire density range of the human femur and that some popular e relationships are based on data that contains extreme scatter while others are based on a very limited amount of information the second goal is to use the newly developed stochastic e relationship in high order finite element analyses feas for the computation of strains and displacements in two human proximal femurs mimicking in vitro experiments when compared with the experimental observations the fea predictions using the median of the stochastic e relationship follow the underlying distribution of the stochastic e relationship thus most deviations of the fea predictions from experimental observations can possibly be explained by uncertain elastic properties of the femur

convergence cones near bifurcation the performance of predictor solver continuation methods near bifurcation points is considered a general theory is developed for the convergence of two commonly used solvers namely newton s method and the chord method near singular points of a nonlinear equation this theory provides a uniform treatment of continuation for simple bifurcation multiple bifurcation and multiple limit point bifurcation problems for these types of bifurcation it is shown that there are conical domains of attraction for the above iterative methods centred on solution branches which correspond to isolated roots of the algebraic bifurcation equations

new bifurcation theorems via the second order optimality conditions we derive new sufficient conditions for bifurcation relying on subtle second order necessary optimality conditions for abnormal equality constrained optimization problems we relate these conditions to the known ones and demonstrate the cases when the new conditions are easier to verify

classification and unfoldings of degenerate hopf bifurcations abstract this paper initiates the classification up to symmetry covariant contact equivalence of perturbations of local hopf bifurcation problems which do not satisfy the classical non degeneracy conditions the only remaining hypothesis is that i should be simple eigenvalues of the linearized right hand side at criticality then the lyapunov schmidt method allows a reduction to a scalar equation g x 0 where g x g x a definition is given of the codimension of g and a complete classification is obtained for all problems with codimension 3 together with the corresponding universal unfoldings the perturbed bifurcation diagrams are given for the cases with codimension 2 and for one case with codimension 3 for this last case one of the unfolding parameters is a modal parameter such that the topological codimension equals in fact 2 formulas are given for the calculation of the taylor coefficients needed for the application of the results and finally the results are applied to two simple problems a model of glycolytic oscillations and the fitzhugh nerve equations

a geometric framework for the numerical study of singular points while bifurcation theory has developed rapidly in recent years there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems this paper presents such a mathematical framework for the numerical study of the bifurcation phenomena associated with a parameter dependent equation f z lambda 0 the presentation draws from differential geometry and singularity theory and provides a basis for various numerical methods used to detect and compute certain types of bifurcation points

on the computation of multi dimensional solution manifolds of parametrized equations a new algorithm is presented for computing vertices of a simplicial triangulation of thep dimensional solution manifold of a parametrized equationf x 0 wheref is a nonlinear mapping fromr n tor m p n m 1 an essential part of the method is a constructive algorithm for computing moving frames on the manifold that is of orthonormal bases of the tangent spaces that vary smoothly with their points of contact the triangulation algorithm uses these bases together with a chord form of the gauss newton process as corrector to compute the desired vertices the jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local triangulation patches several numerical examples show that the method is very efficient in computing triangulations even around singularities such as limit points and bifurcation points this opens up new possibilities for determining the form and special features of such solution manifolds

post critical analysis of structures with a nonlinear pre buckling state in the presence of imperfections in the context of the geometrically nonlinear analysis of structures by continuation methods a technique for the determination of the imperfection sensitivity is carried out the technique is based on galerkin s reduction of equilibrium equations in the subspace of extrapolation branches emanating from the bifurcation points the pre and post critical behaviour in the neighbourhood of bifurcation points however is well approximated provided that extrapolations have a large convergence domain to this end an efficient asymptotic representation of the primary and secondary equilibrium paths and procedures that overcome the ill conditioning of the systems defined in the critical points is used here

tangent vectors to a zero set at abnormal points we are concerned with sharp characterization of the contingent cone to the set defined by a finite number of equality constraints in the absence of classical regularity constrained qualification

an analysis of upwind schemes for the navier stokes equations we state a general result on approximation of branches of nonsingular solutions of nonlinear problems the abstract theory is then applied to the study of two mixed finite element schemes for the navier stokes equations using an upwind discretization of the convective terms

on a computational method for the second fundamental tensor and its application to bifurcation problems an algorithm is presented for the computation of the second fundamental tensorv of a riemannian submanifoldm ofr n fromv the riemann curvature tensor ofm is easily obtained moreover v has a close relation to the second derivative of certain functionals onm which in turn provides a powerful new tool for the computational determination of multiple bifurcation directions frequently in applications thed dimensional manifoldm is defined implicitly as the zero set of a submersionf onr n in this case the principal cost of the algorithm for computingv p at a given pointp m involves only the decomposition of the jacobiandf p off atp and the projection ofd d 1 neighboring points ontom by means of a local iterative process usingdf p several numerical examples are given which show the efficiency and dependability of the method

a priori error estimates of finite element solutions of parametrized strongly nonlinear boundary value problems abstract nonlinear boundary value problems with parameters are called parametrized nonlinear boundary problems this paper studies a priori error estimates of finite element solutions of second order parametrized strongly nonlinear boundary value problems in divergence form on one dimensional bounded intervals the banach space w 0 1 is chosen in formulation of the error analysis so that the nonlinear differential operators defined by the differential equations are nonlinear fredholm operators of index 1 finite element solutions are defined in a natural way and several a priori estimates are proved on regular branches and on branches around turning points in the proofs the extended implicit function theorem due to brezzi et al 1980 plays an essential role

global bifurcation of periodic solutions in nonlinear evolution problems with periodic forcing abstract this paper discusses the global bifurcation of 2 periodic solutions of u t p x u f t u with a homogeneous dirichlet boundary condition where p x is linear elliptic and the nonlinearity f is 2 periodic in t the main differences from existing theories devoted to this type of problem can roughly be summarized as follows i the bifurcation analysis makes no use of evolution operators or related concepts poincare maps floquet multipliers etc ii the bifurcation nonbifurcation points are characterized through an associated stationary problem iii the functional setting allows for nonlinearities f exhibiting time discontinuities among other things the results include various partial generalizations of the bifurcation from the principal eigenvalue theorem which unlike the classical version do not require linear parameter dependence

improved minimal augmentation procedure for the direct computation of critical points this paper presents a new numerical procedure for the direct computation of critical points for elastic beam structures undergoing large displacements and rotations compared to the approach described by wriggers et al comput methods appl mech engrg 70 1988 329 int j numer methods engrg 30 1990 155 two main modifications are introduced first following eriksson comput methods appl mech engrg 114 1994 77 comput methods appl mech engrg 156 1998 45 comput methods appl mech engrg 179 1999 265 int j struct stability dynam 1 1 2001 the condition of criticality is expressed by a scalar equation instead of a vectorial one next the present procedure does not use exclusively the extended system obtained from the equilibrium equations and the criticality condition but also introduces intermediate iterations based purely on equilibrium equations under load or displacement control eight numerical examples presenting bifurcation and limit points are used in order to compare the performances of this new method and the one presented in comput methods appl mech engrg 70 1988 329 int j numer methods engrg 30 1990 155

computation of cusp singularities for operator equations and their discretizations we discuss the direct calculation of cusp singularities as solutions of a minimally augmented defining system which is nonsingular under the canonical cusp conditions the underlying operator equation may have any finite fredholm index and bounds on the discretization error are derived for the case of projection methods

finite dimensional approximation of non linear problems 3 simple bifurcation points note univ pavia cnr ist analisi numer i 27100 pavia italy ecole polytech f 91128 palaiseau france ecole polytech ctr math appl cnrs era 747 f 91128 palaiseau france brezzi f univ pavia ist mat appl i 27100 pavia italy isi document delivery no mn015times cited 75cited reference count 20 reference asn article 1981 001 record created on 2006 08 24 modified on 2017 05 12

second order necessary conditions for nonlinear optimization problems with abstract constraints the degenerate case in this paper we derive second order necessary conditions for optimality for an optimization problem with abstract constraints in banach spaces results for the nondegenerate case derived earlier h gfrerer siam j control optim 45 2006 pp 972 997 are extended to the degenerate case for the mathematical programming problem where the constraints are given by equality and finitely many inequality constraints our approach applies to the degenerate case when the equality constraints are not regular our results appear to be new even in this special case our second order necessary conditions are contained in the gap between the standard necessary and sufficient conditions where the only difference is the change from a nonstrict to a strict inequality our results are formulated in such a way to be applicable also to vector optimization problems

on criticality and ground states of second order elliptic equations ii abstract in this article we examine a necessary and sufficient condition for the existence of a positive solution of the equation p tw u 0 in r n where p is a second order alliptic operator which is critical in w c 0 and t 0 is small enough

bifurcation at multiple eigenvalues and stability of bifurcating solutions if k is a bounded linear operator from the real banach space u into the real banach space v and u r v has the value zero at 0 0 the existence and linear stability of the equilibrium solutions of the dynamical system k dudt u which are close to the origin in u r are studied it is assumed that u 0 0 u v is a freholm operator of index zero the only restriction on the dimension of the null space of u 0 0 and the order of vanishing at 0 0 of restricted to the null space of d 0 0 u r v is that they both be finite positive integers the main result gives conditions under which the equation which determines the equilibrium solutions in a neighborhood of the origin also determines the stability of these equilibrium solutions

on green functions of second order elliptic operators on riemannian manifolds the critical case abstract let p be a second order linear elliptic operator with real coefficients which is defined on a noncompact and connected riemannian manifold m it is well known that the equation p u 0 in m admits a positive supersolution which is not a solution if and only if p admits a unique positive minimal green function on m and in this case p is said to be subcritical in m if p does not admit a positive green function but admits a global positive super solution then such a solution is called a ground state of p in m and p is said to be critical in m we prove for a critical operator p in m the existence of a green function which is dominated above by the ground state of p away from the singularity moreover in a certain class of green functions such a green function is unique up to an addition of a product of the ground states of p and p under some further assumptions we describe the behavior at infinity of such a green function this result extends and sharpens the celebrated result of p li and l f tam concerning the existence of a symmetric green function for the laplace beltrami operator on a smooth and complete riemannian manifold m

on the computation of manifolds of foldpoints for parameter dependent problems for parameter dependent nonlinear equations in r n a new augmentation method for computing submanifolds of foldpoints is presented the underlying theory is based on a characterization developed earlier of foldpoints on solution manifolds of parameterized equations the method allows for a locally convergent iterative computation of foldpoints of a certain class for a continuation along paths of such foldpoints and for a computation of simplicial approximations of higher dimensional submanifolds of foldpoints of the same type

evaluation of simple bifurcation points and post critical path in large finite rotation problems the main aspects of an algorithm for the exact evaluation of single bifurcation points along the nonlinear equilibrium trajectory of flexible mechanisms with large 3 d rotations are discussed a first version of this algorithm was presented elsewhere 1 in this work we extend its range of application to problems involving large finite rotations several flexible mechanisms and structures presenting bifurcation points are analyzed solutions of perfect bifurcation cases are compared to solutions of the literature which were computed by geometric perturbation

numerical liapunov schmidt spectral methods for k determined problems we describe several generalized liapunov schmidt methods in the bordered systems formulation for operator equations and collect information concerning classification of bifurcation functions we present a short outline of spectral methods in this bifurcation context and show the convergence of the discrete defining equations and nondegeneracy conditions for spectral finite element and difference methods to the counterparts of the original operator for k determined problems this result is then applied to a model for turing patterns in a two component mixture

fast divergence conforming reduced basis methods for steady navier stokes flow abstract reduced basis methods rb methods or rbms form one of the most promising techniques to deliver numerical solutions of parametrized pdes in real time with reasonable accuracy for incompressible flow problems rbms based on lbb stable velocity pressure spaces do not generally inherit the stability of the underlying high fidelity model and instead additional stabilization techniques must be introduced one way of bypassing the loss of lbb stability in the rbm is to inflate the velocity space with supremizer modes this however deteriorates the performance of the rbm in the performance critical online stage as additional dofs must be introduced to retain stability while these dofs do not effectively contribute to accuracy of the rb approximation in this work we consider a velocity only rb approximation exploiting a solenoidal velocity basis the solenoidal reduced basis emerges directly from the high fidelity velocity solutions in the offline stage by means of piola transforms the solenoidality of the velocity space is retained under geometric transformations making the proposed rb method suitable also for the investigation of geometric parameters to ensure exact solenoidality of the high fidelity velocity solutions that constitute the rb we consider approximations based on divergence conforming compatible b splines we show that the velocity only rb method leads to a significant improvement in computational efficiency in the online stage and that the pressure solution can be recovered a posteriori at negligible extra cost we illustrate the solenoidal rb approach by modeling steady two dimensional navier stokes flow around a naca0015 airfoil at various angles of attack

scanner influence on the mechanical response of qct based finite element analysis of long bones abstract patient specific qct based finite element qctfe analyses enable highly accurate quantification of bone strength we evaluated ct scanner influence on qctfe models of long bones a femur humerus and proximal femur without the head were scanned with k2hpo4 phantoms by seven ct scanners four models using typical clinical protocols qctfe models were constructed the geometrical dimensions as well as the qct values expressed in hounsfield unit hu distribution was compared principal strains at representative regions of interest rois and maximum principal strains associated with fracture risk were compared intraclass correlation coefficients iccs were calculated to evaluate strain prediction reliability for different scanners repeatability was examined by scanning the femur twice and comparing resulting qctfe models maximum difference in geometry was 2 3 hu histograms before phantom calibration showed wide variation between qct scans however bone density histogram variability was reduced after calibration and algorithmic manipulation relative standard deviation rsd in principal strains at rois was 0 9 fracture associated strain had 6 7 8 1 and 13 3 maximum rsd for the femur humerus and proximal femur respectively the difference in maximum strain location was quantification of strain differences showed mean rsd bounded by 6 in rois fracture associated strains in regular bones showed a mean rsd bounded by 8 strains were obtained within a 10 difference relative to the mean thus in a longitudinal study only changes larger than 20 in the principal strains may be significant iccs indicated high reliability of qctfe models derived from different scanners

determinacy of sufficiently differentiable maps variants of the algebraic conditions of mather are shown to be sufficient for the k determinacy of cu maps with respect to j flat contact or right cr equivalence relations where u k k the algebraic conditions are seen to be necessary

a b spline based heterogeneous modeling and analysis of proximal femur with graded element bone is a complex biological tissue and natural heterogeneous object the main objective of this study is to simulate quasi static loading of bio objects like human femur with b spline based modeling and its 3d finite element analysis with graded element b spline surface representation method is extended to represent material composition to develop heterogeneous solid model of proximal femur lagrangian graded element is used to assign inhomogeneous isotropic elastic properties in finite element model to improve the performance convergence study is carried out with finite element model in single leg stance load condition to test the feasibility of the model sensitivity of simulation is investigated to validate the model numerical results are compared with those of an experimental work for the same specimen in simple stance load condition obtained from one of the reference paper good agreement is achieved for vertical displacement and strains in most of the locations

on the discretization error of parametrized nonlinear equations many applications lead to nonlinear parameter dependent equations h y t y 0 where h y times t to y y 0 in operatorname rge h and the state space y is infinite dimensional while the parameter space t has finite dimension the case dim t 1 is of special interest in connection with continuation methods for this case a general theory is developed which provides for the existence of solution paths of a rather general class of such equations and of their finite dimensional approximations and which allows for an assessment of the error between these paths a principal tool in this analysis is the theory of nonlinear fredholm operators the results cover a more general class of operators than the mildly nonlinear mappings to which other approaches appear to be restricted

equilibrium subsets for multi parametric structural analysis the paper discusses multi parametric formulations for structural equilibrium problems by seeking subsets of equilibrium states which also fulfil extra conditions more detailed information on structural behaviour can be obtained these conditions can be related to analyses of e g critical behaviour imperfections or optimisation the paper shows a general problem setting and some interesting augmentations a solution method for the special case of one dimensional equilibrium subsets is described the algorithm is a generalisation of a path following algorithm where new methods are developed for the evaluation of the path tangent and for the isolation of special solution points examples are used to show the properties and the possibilities of the ideas

on solving nonlinear equations with simple singularities or nearly singular solutions the paper begins with a review of the convergence theory for newton s method near simple and other regular singularities followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy then several ways to accelerate the slow convergence of newton s method are described in some detail these are richardson extrapolation overrelaxation bordering by a singularity condition and appending the usual linear model by a quadratic term the last two approaches require limited second derivative information which can be obtained analytically by differencing or through secant updating numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy

finite dimensional approximation of nonlinear problems part ii limit points note univ paris 6 f 75230 paris 05 france ecole polytech ctr math appl cnrs equipe rech 747 f 91128 palaiseau france univ pavia cnr ist anal numer i 27100 pavia italy brezzi f univ pavia cnr ist math appl i 27100 pavia italy isi document delivery no lr761times cited 54cited reference count 13 reference asn article 1981 002doi 10 1007 bf01396184 record created on 2006 08 24 modified on 2017 05 12

an algebraic geometric method to calculate branches near a bifurcation point abstract systems of nonlinear algebraic equations with a parameter arises in many branches of mathematics and engineering computation of the solution curve is usually hindered by the existence of a bifurcation point where the rank of the corresponding matrix is not full and usually there is another branch curve passing it we introduce a method from algebraic geometry to find the branch curve a computational example is provided

a generalization of the implicit function theorem for mappings from rn 1 into rn and its applications abstract some regularity results about the local structure of the zero set of a mapping of n 1 real variables with values in r n are presented they all provide improved and generalized versions of the morse lemma for plane curves as well as generalizations of the implicit function theorem we describe their applications to one parameter nonlinear problems including an analysis of bifurcation phenomena at simple or multiple eigenvalues

numerical continuation methods a perspective in this historical perspective the principal numerical approaches to continuation methods are outlined in the framework of the mathematical sources that contributed to their development notably homotopy and degree theory simplicial complexes and mappings submanifolds defined by submersions and singularity and foldpoint theory

liouville properties and critical value of fully nonlinear elliptic operators abstract we prove some liouville properties for sub and supersolutions of fully nonlinear degenerate elliptic equations in the whole space our assumptions allow the coefficients of the first order terms to be large at infinity provided they have an appropriate sign as in ornstein uhlenbeck operators we give two applications the first is a stabilization property for large times of solutions to fully nonlinear parabolic equations the second is the solvability of an ergodic hamilton jacobi bellman equation that identifies a unique critical value of the operator

characterization and computation of singular points with maximum rank deficiency characterizations are given of points at which a mapping has a rank drop possibly greater than one multiple bifurcation points can then be computed without needing a full unfolding of the singularity nondegeneracy conditions are studied in detail

hopf bifurcation at multiple eigenvalues with zero eigenvalue on donne des resultats locaux relies a la bifurcation de hopf pour des equations paraboliques la partie lineaire autour du point d equilibre peut avoir des valeurs propres nulles

on the sensitivity of solutions of parameterized equations the sensitivity of a solution of a parameterized equation f z lambda 0 with respect to the parameter vector lambda is usually defined as the change of the state z in dependence of lambda in other words for any solution expressible in the form z lambda lambda with some smooth function z z lambda the sensitivity is the derivative dz lambda typically the solutions form a manifold m in the product of 1the state space and the parameter space and this sensitivity is available only at those points of m where the parameters can be used to define a local coordinate system this paper introduces a general sensitivity concept which applies at all solutions on m and which includes the earlier definition some general geometric interpretations of the new measure are presented and it is shown that the sensitivity analysis can be easily integrated into the solution process the theory also suggests the introduction of a readily computable second order sensitivity measure reflectin

application of elliptic regularity to bifurcation in stationary nonlinear schrodinger equations keywords bifurcation elliptic regularity fredholm operator morse index reference ana article 2003 003doi 10 1016 s0362 546x 02 00138 4view record in web of science record created on 2008 12 10 modified on 2016 08 08

finite element analysis for parametrized nonlinear equations around turning points nonlinear equations with parameters are called parametrized nonlinear equations in this paper a priori error estimates of finite element solutions of parametrized nonlinear elliptic equations on branches around turning points are considered existence of a finite element solution branch is shown under suitable conditions on an exact solution branch around a turning point also some error estimates of distance between exact and finite element solution branches are given it is shown that error of a parameter is much smaller than that of functions approximation of nondegenerate turning points is also considered we show that if a turning point is nondegenerate there exists a locally unique finite element nondegenerate turning point at a nondegenerate turning point an elaborate error estimate of the parameter is proved

computation of hopf bifurcation with bordered matrices hopf bifurcation for dynamical systems dot x f x lambda is characterized by rank deficiency two of a 2 omega 2 i where a f x x lambda in mathbb r n times n is the jacobian at the hopf point and pm i omega are the hopf eigenvalues replacing w 2 by nu and allowing nu leq 0 we introduce real and imaginary hopf matrices a with hopf numbers nu defining a codimension 2 manifold mathcal m h nu subset mathbb r n times n times mathbb r which can be described by two scalar equations alpha a nu 0 beta a nu 0 given by the solution of certain linear bordered systems these two scalar equations form a suitable augmentation of f x lambda 0 for the numerical computation of branches of hopf points in two parameter systems in contrast to chu govaerts and spence siam j numer anal 31 1994 pp 524 539 we suggest an a dependent bordering having certain advantages we construct a scalar test function for hopf points in the

folds on the solution manifold of a parametrized equation this paper presents a framework for studying singular points on the solution manifold of a parameter dependent nonlinear equation f z lambda 0 the approach is based on a systematic combination of general constrained mappings with the tangent map of differential geometry this framework is then used to develop a geometrically instructive and coordinate free treatment of fold points on the solution manifold the treatment includes a detailed analysis of the types of points that may occur on fold lines

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

on positive solutions of second order elliptic equations stability results and classification soit p un operateur uniformement elliptique d ordre 2 defini dans un domaine cr n soit p le cone convexe de toutes les solutions positives de l equation pu 0 dans on etudie les proprietes generales de p pour p et generaux

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

criticality and ground states for second order elliptic equations abstract let p be a second order elliptic operator in a domain r n suppose that p admits a positive minimal green function g p in it is the main purpose of this paper to prove that if w c is a small perturbation of p in then max w x 0 min w x 0 t u u g p outside a compact set in

existence of strong solutions for ito s stochastic equations via approximations given strong uniqueness for an ito s stochastic equation with discontinuous coefficients we prove that its solution can be constructed on any probability space by using for example euler s polygonal approximations stochastic equations in d and in domains in d are considered

finite element analysis of the landau de gennes minimization problem for liquid crystals this paper describes the landau de gennes free energy minimization problem for computing equilibrium configurations of the tensor order parameter field that characterizes the molecular orientational properties of liquid crystal materials analytical and numerical issues are addressed conditions guaranteeing well posedness existence regularity of the problem are given as is a nonlinear finite element convergence analysis

the theory of 2 regularity for mappings with lipschitzian derivatives and its applications to optimality conditions we study local structure of a nonlinear mapping near points where standard regularity and or smoothness assumptions need not be satisfied we introduce a new concept of 2 regularity a certain kind of second order regularity for a once differentiable mapping whose derivative is lipschitz continuous under this 2 regularity condition we obtain the representation theorem and the covering theorem i e stability with respect to right hand side perturbations under assumptions that are weaker than those previously employed in the literature for results of this type these results are further used to derive a constructive description of the tangent cone to a set defined by 2 regular equality constraints and optimality conditions for related optimization problems the class of mappings introduced and studied in the paper appears to be a convenient tool for treating complementarity structures by means of an appropriate equation based reformulation optimality conditions for mathematical programs with equivalently reformulated complementarity constraints are also discussed

efficient finite differences for parameter dependent singularities in two point boundary value problems we will consider the implementation of finite differences to the numerical computation of simple turning points cubic turning points simple bifurcation points and bifurcation points with corank 2 in b v ps for ordinary differential equations the singularity will be characterized by an equation or a set of equations which will be augmented with the original system to produce a regular problem block elimination will be employed for saving on the amount of work while richardson extrapolation will be used to produce accurate results numerical examples will be presented to show the efficiency of the above described algorithm

bifurcation theorems via second order optimality conditions abstract we present a new approach to bifurcation study that relies on the theory of second order optimality conditions for abnormal constrained optimization problems developed earlier by the first author this theory does not subsume the primal description of the feasible set in terms of tangent vectors or in any other way as a result we obtain new sufficient conditions for bifurcation which are to some extent complementary with respect to the known bifurcation theory

characterization and computation of generalized turning points a minimal set of augmenting equations is given so that general parameter dependent problems together with these equations will be nonsingular at points of simple rank deficiency applications to de

generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains using three different notions of the generalized principal eigenvalue of linear second order elliptic operators in unbounded domains we derive necessary and sufficient conditions for the validity of the maximum principle as well as for the existence of positive eigenfunctions for the dirichlet problem relations between these principal eigenvalues their simplicity and several other properties are further discussed 2015 wiley periodicals inc

solution manifolds and submanifolds of parametrized equations and their discretization errors the paper concerns solution manifolds of nonlinear parameterdependent equations 1 f u y0 involving a fredholm operatorf between infinite dimensional banach spacesx z andy and a finitedimensional parameter space differntial geometric ideas are used to discuss the connection between augmented equations and certain onedimensional submanifolds produced by numerical path tracing procedures then for arbitrary finite dimension of estimates of the error between the solution manifold of 1 and its discretizations are developed these estimates are shown to be applicable to rather general nonlinear boundaryvalue problems for partial differential equations

structure of positive solutions to δ v u 0 in rn on etudie la structure de toutes les solutions positives d une equation de schrodinger stationnaire ou le potentiel v est suppose etre une fonction a valeur reelle appartenant a l p loc r n avec p n 2 pour n 2 et p 1 pour n 1 on considere une formule de representation explicite pour des solutions positives leurs asymptotes a l infini et le probleme de dirichlet avec une condition aux limites a l infini

the approximation of generalized turning points newline by krylov subspace methods certain types of singular solutions of nonlinear parameter dependent operator equations were characterized by griewank and reddien 5 6 as regular solutions of suitable augmented systems for their numerical approximation an approach based on the use of krylov subspaces is here presented the application to boundary value problems is illustrated by numerical examples

an analytical study of bifurcation problems for equations involving fredholm mappings on considere des equations de la forme t v l v m v v d 0 ou est un sous ensemble ouvert d un espace norme pour un fixe t l et m sont des applications de la fermeture d 0 d un voisinage d 0 de l origine dans un espace de banach x dans un autre espace de banach y avec t 0 l 0 m 0 0 on etudie l existence de points de bifurcation

superlinearly convergent algorithms for solving singular equations and smooth reformulations of complementarity problems we propose a new algorithm for solving smooth nonlinear equations in the case where their solutions can be singular compared to other techniques for computing singular solutions a distinctive feature of our approach is that we do not employ second derivatives of the equation mapping in the algorithm and we do not assume their existence in the convergence analysis important examples of once but not twice differentiable equations whose solutions are inherently singular are smooth equation based reformulations of the nonlinear complementarity problems reformulations of complementarity problems serve both as illustration of and motivation for our approach and one of them we consider in detail we show that the proposed method possesses local superlinear quadratic convergence under reasonable assumptions we further demonstrate that these assumptions are in general not weaker and not stronger than regularity conditions employed in the context of other superlinearly convergent newton type algorithms for solving complementarity problems which are typically based on nonsmooth reformulations therefore our approach appears to be an interesting complement to the existing ones

matrix manifolds and the jordan structure of the bialternate matrix product the bialternate product of matrices was introduced at the end of the 19th century and recently revived as a computational tool in problems where real matrices with conjugate pairs of pure imaginary eigenvalues are important i e in stability theory and hopf bifurcation problems we give a complete description of the jordan structure of the bialternate product 2a in of an n n matrix a thus extending several partial results in the literature we use these results to obtain regular local defining systems for some manifolds of matrices which occur naturally in applications in particular for manifolds with resonant conjugate pairs of pure imaginary eigenvalues such defining systems can be used analytically to obtain local parameterizations of the manifolds or numerically to set up newton systems with local quadratic convergence we give references to explicit numerical applications and implementations in software we expect that the analysis provided in this paper can be used to further improve such implementations

applications of the blowing up construction and algebraic geometry to bifurcation problems a generalization of the morse lemma to vector valued functions is proved by a blowing up argument this is combined with a theorem from algebraic geometry on the number of real solutions of a system of homogeneous equations of even degree to yield a new bifurcation theorem bifurcation in a one or multi parameter problem is guaranteed if the leading term is of even degree it is often two and satisfies a regularity condition applications are given to nonlinear eigenvalue problems and to the hopf bifurcation

optimal hardy weight for second order elliptic operator an answer to a problem of agmon abstract for a general subcritical second order elliptic operator p in a domain r n or noncompact manifold we construct hardy weight w which is optimal in the following sense the operator p w is subcritical in for all 1 null critical in for 1 and supercritical near any neighborhood of infinity in for any 1 moreover if p is symmetric and w 0 then the spectrum and the essential spectrum of w 1 p are equal to 1 and the corresponding agmon metric is complete our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators the constructed hardy weight is given by an explicit simple formula involving two distinct positive solutions of the equation p u 0 the existence of which depends on the subcriticality of p in

periodic solutions of a forced lotka volterra equation abstract for a 2 species predator prey model the effect of a periodic forcing dependent on two parameters is studied a description is given for the local bifurcation curves of periodic solutions near a fixed family of solutions of the free equation

on the computation of multiple bifurcations with multiple parameters and symmetry this paper concerns numerical methods for the determination of bifurcation points of certain steady state multiparameter problems in the presence of symmetries the approach is based on the fact that under general conditions the solution set forms a manifold in the space of all state and parameter variables the reduced manifold with respect to some subsymmetry is introduced methods are presented for the local computation of the submanifold of bifurcation points of the same symmetry