hard lefschetz theorem for valuations complex integral geometry and unitarily invariant valuations we obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets a version of the hard lefschetz theorem using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations it implies new integral geometric formulas for real submanifolds in hermitian spaces generalizing the classical kinematic formulas in euclidean

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

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

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

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

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

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

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

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

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 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

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

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 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

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

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

valuations on manifolds and rumin cohomology smooth valuations on manifolds are studied by establishing a link with the rumin de rham complex of the co sphere bundle several operations on differential forms induce operations on smooth valuations signature operator rumin laplace operator euler verdier involution and derivation operator as an application alesker s hard lefschetz theorem for even translation invariant valuations on a finite dimensional euclidean space is generalized to all translation invariant valuations the proof uses kahler identities the rumin de rham complex and spectral geometry

projection bodies in complex vector spaces the space of minkowski valuations on an m dimensional complex vector space which are continuous translation invariant and contravariant under the complex special linear group is explicitly described each valuation with these properties is shown to satisfy geometric inequalities of the brunn minkowski aleksandrov fenchel and minkowski type

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

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

contact measures in isotropic spaces abstract we revisit the contact measures introduced by firey and further developed by schneider and teufel from the perspective of the theory of valuations on manifolds this reveals a link between the kinematic formulas for area measures studied by wannerer and the integral geometry of curved isotropic spaces as an application we find explicitly the kinematic formula for the surface area measure in hermitian space

stability results for impulsive functional differential equations with infinite delays abstract this paper studies the stability problems for a class of impulsive functional differential equations with infinite delays of the form x t f t x t t x t k j k x t k k 1 2 by using the liapunov functions and razumikhin technique some new razumikhin type theorems on stability are obtained

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

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

classification of invariant valuations on the quaternionic plane abstract we describe the orbit space of the action of the group sp 2 sp 1 on the real grassmann manifolds gr k h 2 in terms of certain quaternionic matrices of moore rank not larger than 2 we then give a complete classification of valuations on the quaternionic plane h 2 which are invariant under the action of the group sp 2 sp 1

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