hybrid systems with time scales and impulses abstract in this paper we describe hybrid systems introducing special time scales and impulsive effects so as to provide a broader perspective for hybrid systems

a class of nonholonomic kinematic constraints in elasticity we propose a first example of a simple classical field theory with nonholonomic constraints our model is a straightforward modification of a cosserat rod based on a mechanical analogy we argue that the constraint forces should be modelled in a special way and we show how such a procedure can be naturally implemented in the framework of geometric field theory finally we derive the equations of motion and we propose a geometric integration scheme for the dynamics of a simplified model

degree of mapping for nonlinear mappings of monotone type abstract a classical degree function is constructed for pseudomonotone mappings from a reflexive banach space to its dual using galerkin approximations this generalizes the leray schauder degree when the banach space is a hilbert space and yields a flexible analytical tool for the study of nonlinear elliptic problems of higher order in divergence form

multiple equilibria of elastic strings under central forces highly singular nonlinear boundary value problems of the bernoullis in this paper we study the multiplicity of equilibrium states of nonlinearly elastic strings under central forces this problem generalizes that studied by joh bernoulli in 1728 for an inextensible string the theory of elastic strings was developed by jas bernoulli in 1691 1704 cf 111 it is technically more difficult than the related problem of the elastic catenary studied in 2 4 81 because it does not have a convenient set of integrals and because the central force field may well be infinite at its center the complications due to this singularity are magnified by the requirement that the compressive contact force in the string must become infinite where the local ratio of deformed to natural length becomes 0 we find that these problems have a multiplicity of both regular and singular solutions with the analysis of the latter requiring a careful extension of the governing laws of mechanics to handle infinite forces in the rest of this section we formulate the governing equations here we pay special attention to questions of regularity which cannot be handled routinely in virtue of the singularities in the equations for a large class of problems we obtain detailed global information about the qualitative behavior of all regular solutions in section 2 we give a full characterization of singular solutions in section 3 we study purely radial solutions by means

two global several parameter bifurcation theorems and their applications abstract in this paper we investigate the structure of the nontrivial solution set for the following nonlinear operator equation u l u h u u r m x where m is a positive integer x is a banach space l x x is a positively homogeneous compact operator and h r m x x is compact with h o u near u 0 uniformly on bounded sets we obtain some results involving unilateral global bifurcation two examples of p laplacian problem with jumping nonlinearity and nonlocal boundary value problem are given to demonstrate how the theory can be applied

parameter ranges for the existence of solutions whose state components have specified nodal structure in coupled multiparameter systems of nonlinear sturm liouville boundary value problems

an isogeometric implicit g1 mixed finite element for kirchhoff space rods abstract in the paper we present a multi patch isogeometric mixed element with implicit geometric continuity at the ends for the analysis of non polar kirchhoff space rods rotations are introduced at the ends of the element analogously to the hermitian interpolation performing a change of basis for the configuration space a mixed method in which the internal forces are interpolated by means of an l 2 projection leading to a b formulation is used to cope with the membrane flexural locking that affects the solution the efficiency of the method in avoiding locking and instabilities is investigated the formulation adopted has the advantage with respect to other formulations that a non full global stiffness matrix is obtained this approach appears very appealing for efficient analyses of general assemblies of 3d non polar elements

nonlinear isogeometric spatial bernoulli beam abstract a new element formulation of a geometrically nonlinear spatial curved beam assuming bernoulli theory including torsion without warping is proposed the element formulation is derived directly from the 3d continuum by means of nonlinear kinematics thus accounting for large displacements the geometric description of the proposed element is adapted from the spatial rod of greco and cuomo 2013 and extended to a nonlinear element formulation the proposed formulation can handle arbitrary orientations of the cross section along the beam in this publication nurbs are used as basis functions for discretization since they can easily provide the required c 1 continuity between elements the presented element formulation has four degrees of freedom three for displacements and one for the rotation around the center line in order to prove the accuracy of the developed spatial bernoulli beam several numerical examples are presented and compared to analytic solutions and other element formulations

symmetry principles for s linear homogeneous and nonhomogeneous equations on presente des principes generaux qui demontre que des equations multiparametres homogenes et non homogenes sont symetriques par rapport a l existence de branches de solutions

local geometric invariants of integrable evolution equations the integrable hierarchy of commuting vector fields for the localized induction equation of 3d hydrodynamics and its associated recursion operator are used to generate families of integrable evolution equations which preserve local geometric invariants of the evolving curve or swept out surface

discrete time lagrangian mechanics on lie groups with an application to the lagrange top we develop the theory of discrete time lagrangian mechanics on lie groups originated in the work of veselov and moser and the theory of lagrangian reduction in the discrete time setting the results thus obtained are applied to the investigation of an integrable time discretization of a famous integrable system of classical mechanics the lagrange top we recall the derivation of the euler poinsot equations of motion both in the frame moving with the body and in the rest frame the latter ones being less widely known we find a discrete time lagrange function turning into the known continuous time lagrangian in the continuous limit and elaborate both descriptions of the resulting discrete time system namely in the body frame and in the rest frame this system naturally inherits poisson properties of the continuous time system the integrals of motion being deformed the discrete time lax representations are also found kirchhoff s kinetic analogy between elastic curves and motions of the lagrange top is also generalised to the discrete context

on the dimension and the index of the solution set of nonlinear equations we study the covering dimension and the index of the solution set to multiparameter nonlinear and semilinear operator equations involving fredholm maps of positive index the classes of maps under consideration are pseudo a proper and either approximation essential or equivariant approximation essential applications are given to semilinear elliptic bvp s

finite volume approach for the instationary cosserat rod model describing the spinning of viscous jets the spinning of slender viscous jets can be asymptotically described by one dimensional models that consist of systems of partial and ordinary differential equations whereas well established string models only possess solutions for certain choices of parameters and configurations the more sophisticated rod model is not limited by restrictions it can be considered as an regularized string model but containing the slenderness ratio in the equations complicates its numerical treatment we develop numerical schemes for fixed or enlarging time dependent domains using a finite volume approach in space with mixed central up and down winded differences and stiffly accurate radau methods for the time integration for the first time results of instationary simulations for a fixed or growing jet in a rotational spinning process are presented for arbitrary parameter ranges

finite dimensional approximation of bifurcation problems in presence of symmetries we give here a brief summary of the results of 4 14 the aim of our work was not only the finite dimensional approximation of the specific problem studied in 3 but we also wanted to show how to adapt to discrete problems and how to use in this case some classical mathematical tools e g the splitting lemma of gromoll meyer magnus the singularity theory etc such considerations are developed in 5 in particular we prove in 5 that in most cases e g in the examples given in 6 7 9 10 the discrete bifurcation problem behaves as an imperfect bifurcation problem and we improve the methods of 12 another way to use universal unfoldings in the approxima tion of bifurcation problems is suggested by 2

stability of nonlinearly elastic rods this paper describes previously unknown stabilities and instabilities of planar equilibrium configurations of a nonlinearly elastic rod that is buckled under the action of a dead load the governing equations are derived from variational principles including ones of isoperimetric type properties of stability are accordingly determined by study of the second variation stabilities to deformations both in the plane and out of the plane are considered among the newly discovered properties are secondary bifurcation from the first buckled mode marked differences between stability to two dimensional and to three dimensional variations and the stabilizing influence of resistance to twist in the isoperimetric examples the analysis makes crucial use of a novel device to account for the dependence of the second variation on constraints

a priori bounds and existence of positive solutions for semilinear elliptic systems abstract we provide a priori l bounds for classical positive solutions of semilinear elliptic systems in bounded convex domains when the nonlinearities are below the power functions v p and u q for any p q lying on the critical sobolev hyperbola our proof combines moving planes method and rellich pohozaev type identities for systems our analysis widens the known ranges of nonlinearities for which classical positive solutions of semilinear elliptic systems are a priori bounded using these a priori bounds and local and global bifurcation techniques we prove the existence of positive solutions for a corresponding parametrized semilinear elliptic system

lagrangian aspects of the kirchhoff elastic rod the relation between euler s planar elastic curves and vortex filaments evolving by the localized induction equation lie of hydrodynamics was discovered by hasimoto in 1971 basic facts about an integrable case of kirchhoff elastic rods are described here which amplify the connection between the variational problem for rods and the soliton equation lie in particular it is shown that the centerline of the kirchhoff rod is an equilibrium for a linear combination of the first three conserved hamiltonians in the lie hierarchy

on the generalized spectrum for second order elliptic systems description qualitative et quantitative detaillee du spectre generalise reel pour les systemes elliptiques du second ordre application de la theorie des perturbations lineaires des methodes de prolongement et de la caracterisation variationnelle des valeurs propres de courant hilbert

global several parameter bifurcation and continuation theorems a unified approach via complementing maps we use the concept of complementing map in order to present a unified approach to the continuation principle global implicit function theorem and global bifurcation results for compact vector fields depending on several parameters

multi peak solutions for super critical elliptic problems in domains with small holes abstract this paper deals with the slightly super critical elliptic problem u u n 2 n 2 e in u 0 in u 0 on 1 where e 0 is a small parameter and r n is a bounded domain with smooth boundary assuming that the domain exhibits k sufficiently small holes multiple solutions are constructed by gluing double spike patterns located near each of the holes

hamiltonian structures and stability for rigid bodies with flexible attachments the dynamics of a rigid body with flexible attachments is studied a general framework for problems of this type is established in the context of poisson manifolds and reduction a simple model for a rigid body with an attached linear extensible shear beam is worked out for illustration second the energy casimir method for proving nonlinear stability is recalled and specific stability criteria for our model example are worked out the poisson structure and stability results take into account vibrations of the string rotations of the rigid body their coupling at the point of attachment and centrifugal and coriolis forces

environmental bias and elastic curves on surfaces the behavior of an elastic curve bound to a surface will reflect the geometry of its environment this may occur in an obvious way the curve may deform freely along directions tangent to the surface but not along the surface normal however even if the energy itself is symmetric in the curve s geodesic and normal curvatures which control these modes very distinct roles are played by the two if the elastic curve binds preferentially on one side or is itself assembled on the surface not only would one expect the bending moduli associated with the two modes to differ binding along specific directions reflected in spontaneous values of these curvatures may be favored the shape equations describing the equilibrium states of a surface curve described by an elastic energy accommodating environmental factors will be identified by adapting the method of lagrange multipliers to the darboux frame associated with the curve the forces transmitted to the surface along the surface normal will be determined features associated with a number of different energies both of physical relevance and of mathematical interest are described the conservation laws associated with trajectories on surface geometries exhibiting continuous symmetries are also examined

behaviour of the extensible elastica solution the general form of the virtual work expression for the large strain euler bernoulli beam theory is derived using the nominal strain biot s tensor from the equilibrium equations derived from the virtual work expression it turns out that a linear relation between biot s stress tensor and the biot nominal strain tensor forms the differential equation used to derive the elastica solution moreover in the differential equation one additional term enters which is related to the extensibility of the beam axis as a special application the well known problem of an axially loaded beam is analysed due to the extensibility of the beam axis it is shown that the buckling load of the extensible elastica solution depends on the slenderness and it is of interest that for small slenderness the bifurcation point becomes unstable this means the bifurcation point changes from being supercritical which always hold for the inextensible case i e the classical elastica solution to being a subcritical point in addition higher order singularities are found as well as nonbifurcating isolated branches

complementing maps continuation and global bifurcation we state and indicate some of the consequences of a theorem whose sole assumption is the nonvanishing of the leray schauder degree of a compact vector field and whose conclusions yield multidimensional existence continuation and bifurcation results

stationary solutions of chemotaxis systems the keller segel model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance chemotaxis this paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response

rotation type solutions for spatially periodic hamiltonian systems along with solutions of 0 2 and in general for o l the hamiltonian h also called the energy is conserved this one dimensional situation can be studied in the phase space that is r x s since the sine function is periodic when the energy his prescribed and bigger than 1 then system 0 2 has two periodic solutions these periodic solutions however are not homotopic to a constant and they are usually called rotation type solutions it is the purpose of this paper to generalize this situation to equation 0 1 spatially periodic hamiltonian systems have been studied by several authors the nonautonomous case of 0 1 has been considered in l 7 for the existence of periodic solutions see also references therein the papers of liu s and the author 3 are specially related to this work there the existence of periodic solutions is established through a generalized saddle point theorem the hypotheses on h assumed in 3 will be considered here

a practical solution to the multivariate behrens fisher problem summary a practical solution to the multivariate behrens fisher problem is proposed the new procedure is constructed using the geometry of confidence ellipsoids for the two mean vectors and the condition under which two ellipsoids have no points in common the new procedure is another extension of the welch approximate degrees of freedom solution as is yao s 1965 procedure the new procedure is compared with yao s procedure in terms of the level of significance and power using monte carlo simulation

on the covering dimension of the set of solutions of some nonlinear equations we prove an abstract theorem whose sole hypothesis is that the degree of a certain map is nonzero and whose conclusions imply sharp multidimensional continuation results applications are given to nonlinear partial differential equations

nonorientability of the index bundle and several parameter bifurcation the universal parameter space for sturm liouville boundary value problems is a two torus we show that the index bundle of the universal family of linear s l boundary value problems is non orientable using this we find a one dimensional set of bifurcation points for nonlinear bvp of sturm liouville type

temporal dimension and transition out of chaos abstract a real time process could evolve governed by difference differential equations we introduce this possibility of discrete continuous evolution in the well known unidimensional logistic equation searching for new behaviour an extension to general hybrid equations is proposed

on the geometric flow of kirchhoff elastic rods recently rod theory has been applied to the mathematical modeling of bacterial fibers and biopolymers e g dna to study their mechanical properties and shapes e g supercoiling in static rod theory an elastic rod in equilibrium is the critical point of an elastic energy this induces a natural question of how to find elasticae in this paper we focus on how to find the critical points by means of gradient flows we relate a geometric function of curves to the isotropic kirchhoff elastic energy of rods so that the generalized elastic curves are the centerlines of elastic rods in equilibrium thus the variational problem for rods is formulated in curve geometry this problem turns out to be a generalization of curve straightening flows which induce nonlinear fourth order evolution equations we establish the long time existence of length preserving gradient flow for the geometric energy furthermore by studying the asymptotic behavior we show that the limit curves are the centerlines of the ki

two way manova with unequal cell sizes and unequal cell covariance matrices in this article we propose and study an approximate hotelling t2 aht test for heteroscedastic two way manova the aht test is shown to be invariant under affine transformations different choices of the contrast matrix used to define the same hypothesis and different labeling schemes of the cell mean vectors we demonstrate via intensive simulations that the aht test generally performs well and outperforms two existing approaches in terms of size and power an extension of the aht test for heteroscedastic multi way manova is briefly described a dataset from a smoking cessation trial is analyzed to illustrate the methodologies this article has supplementary material online in a single archive

the intricate global structure of buckled states of compressible columns singularity theory and global multiparameter bifurcation theory are combined to show that nonbifurcating buckled states of compressible nonlinearly elastic columns lie on branches having detailed qualitative properties these properties distinguish the branches globally together with a delicate asymptotic analysis they are used to show that the global disposition of solution branches both bifurcating and not is strikingly different from that for incompressible columns

eulerian formulation of elastic rods in numerous biological medical and engineering applications elastic rods are constrained to deform inside or around tube like surfaces to solve efficiently this class of problems the equations governing the deflection of elastic rods are reformulated within the eulerian framework of this generic tubular constraint defined as a perfectly stiff normal ringed surface this reformulation hinges on describing the rod deformed configuration by means of its relative position with respect to a reference curve defined as the axis or spine curve of the constraint and on restating the rod local equilibrium in terms of the curvilinear coordinate parametrizing this curve associated with a segmentation strategy which partitions the global problem into a sequence of rod segments either in continuous contact with the constraint or free of contact except for their extremities this re parametrization not only trivializes the detection of new contacts but also transforms these free boundary problems into classic two points boundary value problems and suppresses the isoperimetric constraints resulting from the imposition of the rod position at the extremities of each rod segment

on the dynamics of rods in the theory of kirchhoff and clebsch we discuss here the dynamical equations of a theory of elastic rods that is due to kmcm ioff 1 2 and cl bsc 3 4 this properly invariant theory is applicable to motions in which the strains relative to an undistorted configuration remain small although rotations may be large it is constructed to be a first order theory i e a theory that is complete to within an error of order two in an appropriate dimensionless measure of thickness curvature twist and extension in a first order theory of thin rods one can treat the rod as inextensible and we do so here at the outset thus at each time t the arc length parameter s for the axial curve t is employed as a material coordinate i e a parameter whose value at a material point is constant in time and not only the resultant of the shearing forces on a cross section but also the tension in the rod are reactive quantities not given by constitutive equations consider for a moment a rod that is naturally prismatic and dynamically symmetric i e a rod that in an undistorted stress free configuration is a cylinder whose directrix although not necessarily a circle bounds a figure with equal principal moments of inertia a motion of the rod is said to be planar and twist flee i e a motion of pure flexure if it is such that t lies at all times in a fixed plane which contains a principal axis of inertia of each cross section for such a motion we employ a fixed cartesian coordinate system on and for consistency with a discussion of more general motions to be given later in this paper we call the abscissa z and the ordinate x we may write fz s t fx s t for the z and x components of the resultant force f at time t on the cross section with arc length coordinate s the motion of the rod may be described by giving the z x coordinates of the points on t as functions of s and t with o s t the counterclockwise angle from the z axis to the tangent of t at s we have

hybrid systems on time scales in this paper we study hybrid dynamic systems on time scales using lyapunov like functions we obtain sufficient conditions for the practical stability of hybrid dynamic systems on time scales

atomic to continuum passage for nanotubes a discrete saint venant principle and error estimates we consider general infinite nanotubes of atoms in mathbb r 3 where each atom interacts with all the others through a two body potential at the equilibrium the positions of the atoms satisfy a euler lagrange equation when there are no exterior forces and for a suitable geometry a particular family of nanotubes is the set of perfect nanotubes at the equilibrium when exterior forces are applied on the nanotube we compare the nanotube to nanotubes of the previous family in part i of the paper this quantitative comparison is formulated in our first main result as a discrete saint venant principle as a corollary we also give a liouville classification result our saint venant principle can be derived for a large class of potentials including the lennard jones potential when the perfect nanotubes at the equilibrium are stable the approach is designed to be applicable to nanotubes that can have general shapes like for instance carbon nanotubes or dna under the oversimplified assumption that all the atoms are identical in part ii of the paper we derive from our saint venant principle a macroscopic mechanical model for general nanotubes we prove that every solution is well approximated by the solution of a continuum model involving stretching and twisting but no bending we establish error estimates between the discrete and the continuous solution more precisely we give two error estimates one at the microscopic level and one at the macroscopic level

on the connectivity properties of the solution set of parametrized families of compact vector fields a useful tool in the study of global continua of solutions of nonlinear partial differential equations is a connectivity result on the fixed points set of a l parameter family of maps that goes back to leray and schauder 171 and it was proved in its full generality by browder b in describing it we shall restrict ourselves to the case of parametrized families of compact vector fields defined in open subsets of banach spaces

double spike solutions for a critical inhomogeneous elliptic problem in domains with small holes in this paper we construct solutions which develop two negative spikes as varepsilon to0 for the problem delta u u 4 n 2 u varepsilon f x in varomega u 0 on partial varomega where varomega subset mathbb r n is a bounded smooth domain exhibiting a small hole with f geq0 f not equiv0 this result extends a recent work of clapp et al in the sense that no symmetry assumptions on the domain are required

on the behrens fisher problem a globally convergent algorithm and a finite sample study of the wald lr and lm tests in this paper we provide a provably convergent algorithm for the multivariate gaussian maximum likelihood version of the behrens fisher problem our work builds upon a formulation of the log likelihood function proposed by buot and richards 5 instead of focusing on the first order optimality conditions the algorithm aims directly for the maximization of the log likelihood function itself to achieve a global solution convergence proof and complexity estimates are provided for the algorithm computational experiments illustrate the applicability of such methods to high dimensional data we also discuss how to extend the proposed methodology to a broader class of problems we establish a systematic algebraic relation between the wald likelihood ratio and lagrangian multiplier test w lr lm in the context of the behrens fisher problem moreover we use our algorithm to computationally investigate the finite sample size and power of the wald likelihood ratio and lagrange multiplier tests which previously were only available through asymptotic results the methods developed here are applicable to much higher dimensional settings than the ones available in the literature this allows us to better capture the role of high dimensionality on the actual size and power of the tests for finite samples

the influence of elasticity on analysis the classic heritage etude des domaines suivants le catenaire solutions doubles et le calcul variationnel l elastique fonctions elliptiques analyse qualitative solutions multiples geometrie differentielle des courbes gauches systemes en petites vibrations nombres propres et modes simples equations differentielles lineaires fonctions de bessel polynomes de laguerre transformees integrales oscillations resonantes equations aux derives partielles concept de solution discontinuites la definition moderne d une fonction base fonctionnelle singularites solutions generalisees plaques la courbure gaussienne theorie tridimensionnelle deformation des regions deformation et rotation locale tenseurs nombres et vecteurs propres decomposition polaire resolution spectrale isotropie et hemitropie groupes torsion d un cylindre solution principale d un systeme d equations differentielles et son etat d approximation d autres solutions

invariant isogeometric formulations for three dimensional kirchhoff rods abstract in the isogeometric formulation for three dimensional kirchhoff rods the invariance is required such that rigid body modes associated with translations and infinitesimal rotations can be exactly represented by enforcing the invariance in the isogeometric formulation of the rod the equilibrium of forces and moments at fixed ends is preserved and the performance is improved considerably for elements of finite sizes we recognize that not all existing isogeometric formulations are invariant for general geometry of rods though an alternative formulation based on the frenet serret frame and rotation angle representation was shown to be invariant it does not work for rods with vanishing curvature while its accuracy is affected by the low continuity of the angle of twist used motivated by this two new invariant isogeometric formulations are developed through different interpolations for the angle of twist one uses the derivatives of the nurbs functions as the basis for the angle of twist interpolation and the other uses a mixed interpolation incorporating both the lagrange and nurbs functions several representative numerical examples were prepared to verify the accuracy and robustness of the two new formulations particularly for rods of variable curvatures simulated by elements of finite or large sizes the proposed formulations are demonstrated to outperform the two existing ones mentioned in predicting the displacements end reactions and stress resultants

global properties of buckled states of plates that can suffer thickness changes this paper treats the axisymmetric buckling of nonlinearly elastic cosserat plates which can suffer thickness changes as well as flexure midplane extension and shear the governing equations are accordingly quite complicated nevertheless it is shown that all solutions bifurcating or not have a simple detailed nodal structure that distinguishes branches globally

on the convex case in the positone problem for elliptic systems on etudie l existence et la dependance par rapport au parametre des solutions du systeme l u f x u dans u o sur u o dans 1 k u u 1 u k r est un domaine borne lisse et l est un operateur differentiel uniformement elliptique a coefficients continus de holder

global higher bifurcations in coupled systems of nonlinear eigenvalue problems on etablit des conditions pour l existence d une bifurcation n aire globale de dimension elevee dans des systemes generaux de problemes aux valeurs propres non lineaires multiparametre qui conservent la structure de couplage des modeles de lotka volterra diffusifs etat stationnaire

structure and dimension of global branches of solutions to multiparameter nonlinear equations this paper is concerned with the topological dimension of global branches of solutions appearing in different problems of nonlinear analysis in particular multiparameter including infinite dimensional continuation and bifurca tion problems by considering an extension of the notion of essential maps defined on sets and using elementary point set topology we are able to unify and extend in a selfcontained fashion most of the recent results on such problems our theory applies whenever any generalized degree theory with the boundary dependence property may be used but with no need of a gebraic structures our applications to continuation and bifutcation follow from the nontriviality of a local invariant in the stable homotopy group of a sphere and give information on the local dimension and behavior of the sets of solutions of bifurcation points and of continuation points

a homogeneity condition guaranteeing bifurcation in multiparameter nonlinear eigenvalue problems on montre qu un point de bifurcation de multiplicite impaire n est jamais isole dans l ensemble des points de bifurcation on introduit une notion de variete de codimension 1 et on utilise ce concept et une theorie de perturbation pour decrire completement l ensemble des points de bifurcation proches d un point de multiplicite un sous l hypothese que l application a est continue

a global bifurcation index for set valued perturbations of fredholm operators abstract in the paper the homotopy invariant detecting global bifurcations of solutions to multi parameter equations involving general set valued perturbations of linear fredholm operators of non negative index is introduced some applications to the existence problems for differential inclusions are provided

singular global bifurcation problems for the buckling of anisotropic plates this paper treats a variety of unexpected pathologies that arise in the global bifurcation analysis of axisymmetrie buckled states of anisotropic plates the geometrically exact plate theory used accounts for flexure extension and shear the nonlinear constitutive functions have very general form as a consequence of the anisotropy the trivial solution may depend discontinuously on the load parameter accordingly the equations for the bifurcation problem have the same character so that bifurcating branches of solutions become disconnected as the load parameter crosses values at which discontinuities occur the anisotropy furthermore implies that the governing equations have a singular behaviour much worse than that for isotropic plates consequently a variety of novel constructions are required to demonstrate the validity of the essential results upon which global bifurcation theory stands these results include the compactness of certain operators and the uniqueness of solutions of initial value problems for singular ordinary differential equations it is shown that in regions of solution parameter space in which the equations depend continuously on the load parameter there exist connected global branches of solution pairs that have detailed nodal properties inherited from eigenfunctions of the linearized problem moreover these nodal properties are preserved across gaps occurring where discontinuities occur the methodology used to show this result actually supports constructive methods for finding disconnected branches

nonlinear dynamical theory of the elastica the dynamical behaviour of a slender rod is analyzed here in terms of a generalization of euler s elastica theory the model includes a linear stress strain relation but nonlinear geometric terms properties of the rod may vary along its length and various boundary conditions are considered a rotational inertia term that is neglected in many theories is retained and is essential to the analysis by use of the equivalence of an energy and a sobolev norm and by reformulation of the equations as a semilinear system global existence of solutions is proved for any smooth initial data equilibrium solutions that are stable in the static sense of minimizing the potential energy are then proved to be stable in the dynamic sense due to liapounov

invariant submanifold flows given a lie group acting on a manifold our aim is to analyze the evolution of differential invariants under invariant submanifold flows the constructions are based on the equivariant method of moving frames and the induced invariant variational bicomplex applications to integrable soliton dynamics and to the evolution of differential invariant signatures used in equivalence problems and object recognition and symmetry detection in images are discussed

helices for mathematical modelling of proteins nucleic acids and polymers abstract helices occur in modelling a range of structures including ropes fibers polymers and biopolymers in a recent paper with thamwattana and hill the authors derived euler lagrange equations for modelling protein structure where the energy being minimised was assumed to depend only on the curvature and torsion of the protein backbone space curve and their first derivatives such a model is applicable to helices occurring in other scenarios and in this article the author considers more generally which energies will yield helices as solutions of their corresponding euler lagrange equations he finds in particular classes of energy for which all circular helices are solutions and an energy depending on curvature and its derivative which generates conical helices as solutions of the euler lagrange equations also included are some new results for energies depending only on curvature extending previous investigations by feoli et al a feoli v v nesterenko g scarpetta functionals linear in curvature and statistics of helical proteins nuclear phys b 705 2005 577 592

a justification of nonlinear properly invariant plate theories a single asymptotic derivation of three classical nonlinear plate theories is presented in a setting which preserves the frame invariance properties of three dimensional finite elasticity by a successive scaling of the external loading on the three dimensional body the nonlinear membrane theory the nonlinear inextensional theory and the von karman equations are derived as the leading order terms in the asymptotic expansion of finite elasticity the governing equations of the nonlinear inextensional theory are of particular interest where 1 plane strain kinematics and plane stress constitutive equations are derived simultaneously from the asymptotic analysis 2 the theory can be phrased as a minimization problem over the space of isometric deformations of a surface and 3 the local equilibrium equations are identical to those arising in the one director cosserat shell model furthermore it can be concluded that with a regular single scale asymptotic expansion it is not possible to obtain a system of plate equations in which finite membrane strain and finite bending strain occur simultaneously in the leading order term of an asymptotic analysis

l 2 flow of elastic curves with clamped boundary conditions abstract in this paper we investigate the l 2 flow of elastic curves with clamped boundary conditions in n dimensional euclidean spaces the l 2 flow corresponds to a fourth order parabolic equation in the case of closed curves the long time existence of solutions of this evolution equation has been derived in the literature we extend this result to the case of open i e non closed curves with clamped boundary conditions

spatial chaos of 3 d elastica with the kirchhoff gyrostat analogy using melnikov integrals summary the kirchhoff kinetic analogy from which the similarity between the governing equations for the static spatial equilibrium of a 3 d elastica and those for the temporal dynamics of a rigid body is constituted is revisited the melnikov integrals for detecting chaos cannot be easily formed for an elastica we shall modify the previous procedure of using the melnikov integrals for detecting temporal chaos for a gyrostat to solve the spatial chaos problem of an elastica one way to find the disturbed hamiltonian equations of the equilibrium equations of the elastica are by means of deprit s variables using the melnikov integrals the spatially chaotic deformation patterns can be resulted from the homoclinic transversal intersections of the stable and unstable manifolds at a saddle point in the poincare map when the elastica is disturbed by constant stress resultants the melnikov integrals are integrated for detecting homoclinic intersections the equations governing the evolution of the stress couples and the stress resultants are numerically integrated using the fourth runge kutta algorithms to crosscheck the analytical results the phase portraits of the poincare sections are created on a number of hyper planes to demonstrate the chaotic patterns of the stationary spatial deformations along the centreline of the elastica under weightless conditions the bounded non periodic solutions to the equilibrium of the elastica clearly show the existence of spatially chaotic deformation patterns the findings in this paper are useful in many structures in which the elastica are important examples are the mathematical modelling of the dna super coiling submarine cables and the design of the tethered satellites copyright 2004 john wiley sons ltd

on coupled multiparameter nonlinear elliptic systems on considere le systeme de problemes non lineaires aux valeurs limites de dirichlet lu x f u x v x lv x g u x v x x domaine borne de r n l est un operateur aux derivees partielles lineaire fortement et uniformement elliptique et sont des parametres reels f g r 2 r des applications continues telles que f 0 0 0 g 0 0 on presente une analyse detaillee de l ensemble des solutions du point de vue de la theorie de la bifurcation a plusieurs parametres

the influence of elasticity on analysis modern developments on examine plusieurs domaines de l analyse auxquels l elasticite a apporte une contribution importante questions de connectivite de la theorie globale des bifurcations particularites de l analyse non lineaire calcul des variations convergence faible inegalites variationnelles autres contributions

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

on the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains abstract we study a diffusive logistic equation with nonlinear boundary conditions the equation arises as a model for a population that grows logistically inside a patch and crosses the patch boundary at a rate that depends on the population density specifically the rate at which the population crosses the boundary is assumed to decrease as the density of the population increases the model is motivated by empirical work on the glanville fritillary butterfly we derive local and global bifurcation results which show that the model can have multiple equilibria and in some parameter ranges can support allee effects the analysis leads to eigenvalue problems with nonstandard boundary conditions