on the stochastic kuramoto sivanshinsky equation in this article we study the solution of the kuramoto sivashinsky equation on a bounded interval subject to a random forcing term we show that a unique solution to the equation exists for all time and depends continuously on the initial data

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

an objective 3d large deformation finite element formulation for geometrically exact curved kirchhoff rods the objective of this work is the development of a new finite element formulation for beams according to the kirchhoff theory of thin rods which includes the deformation states of axial tension torsion and bending the proposed formulation accounts for large deformations in three dimensional problem settings consisting of slender prismatic beams with arbitrarily curved initial geometries and arbitrary cross section shapes like in geometrically exact reissner theories the derived deformation measures are geometrically exact in the sense that they are consistent with the strong and weak form of the balance equations and the beam geometry for any configuration and for arbitrarily large translations rotations and strains a novel orthogonal interpolation strategy is applied to the triad field representing the cross section orientation in order to fulfill the kirchhoff constraint of vanishing shear strains in a strong sense and to preserve the objectivity of the spatially discretized problem the continuity requirements resulting from the corresponding weak form are fulfilled by a cubic hermite interpolation thus leading to a c1 continuous representation of the beam centerline fundamental properties such as objectivity path independence consistency and accuracy of the developed beam element are verified by means of suitable numerical examples

mixed finite element methods for elastic rods of arbitrary geometry the boundary value problem for rods having arbitrary geometry and subjected to arbitrary loading is studied within the context of the small strain theory the basic assumptions underlying the rod kinematics are those corresponding to the timoshenko hypotheses in the plane rectilinear case that is plane sections normal to the line of centroids in the undeformed state remain plane but not necessarily normal the problem is formulated in both the standard and mixed variational forms and after establishing the existence and uniqueness of solutions to these equivalent problems the corresponding discrete problems are studied finite element approximations of the mixed problem are shown to be stable and convergent it is shown that the equivalence between the mixed problem and the standard problem with selective reduced integration holds only for the case of rods having constant curvature and torsion though the results of numerical experiments are presented these confirm the convergent behaviour of the mixed problem

helicity and the calugareanu invariant the helicity of a localized solenoidal vector field i e the integrated scalar product of the field and its vector potential is known to be a conserved quantity under frozen field distortion of the ambient medium in this paper we present a number of results concerning the helicity of linked and knotted flux tubes particularly as regards the topological interpretation of helicity in terms of the gauss linking number and its limiting form the c lug reanu invariant the helicity of a single knotted flux tube is shown to be intimately related to the c lug reanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given helicity is decomposed into writhe and twist contributions the writhe contribution involving the gauss integral for definition see equation 4 8 which admits interpretation in terms of the sum of signed crossings of the knot averaged over all projections part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as intrinsic twist of the field lines in the flux tube around the knot see equations 5 13 and 5 15 the generic behaviour associated with the deformation of the knot through a configuration with points of inflexion points at which the curvature vanishes is analysed and the role of the twist parameter is discussed the derivation of the c lug reanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological 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

curve shortening flow coupled to lateral diffusion we present and analyze a semi discrete finite element scheme for a system consisting of a geometric evolution equation for a curve and a parabolic equation on the evolving curve more precisely curve shortening flow with a forcing term that depends on a field defined on the curve is coupled with a diffusion equation for that field the scheme is based on ideas of dziuk siam j numer anal 36 6 1808 1830 1999 for the curve shortening flow and dziuk and elliott ima j numer anal 27 2 262 292 2007 for the parabolic equation on the moving curve additional estimates are required in order to show convergence most notably with respect to the length element while in dziuk siam j numer anal 36 6 1808 1830 1999 an estimate of its error was sufficient we here also need to estimate the time derivative of the error which arises from the diffusion equation numerical simulation results support the theoretical findings

some global dynamical properties of the kuramoto sivashinsky equations nonlinear stability and attractors abstract the kuramoto sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films they are characterized by the coexistence of coherent spatial structures with temporal chaos we investigate some global dynamical properties including nonlinear stability we demonstrate their low modal behavior in terms of determining modes and that the fractal dimension of all attractors is bounded by a universal constant times l 13 8 where l is a dimensionless pattern cell size in the one dimensional case such equations are indeed a paradigm of low dimensional behavior for infinite dimensional systems

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

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

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

nonlinear dynamics of filaments ii nonlinear analysis abstract the linear stability analysis of the full time dependent kirchhoff equations for elastic filaments gives precise information about possible dynamical instabilities the associated dispersion relations derived in the preceding paper provides the selection mechanism for the shapes selected by highly unstable filaments here we perform a nonlinear analysis and derive new amplitude equations which describe the dynamics above the instability threshold the straight filament is studied in detail and the motion is shown to be described by a pair of nonlinear klein gordon equations which couple the local deformation amplitude to the twist density of particular interest is the effect of boundary conditions on the instability threshold it is shown that with suitable choice of boundary conditions the threshold of instability is delayed we also show the existence of pulse like and front like traveling wave solutions

elastic stability of dna configurations i general theory results are presented in the theory of the elastic rod model for dna among which are criteria enabling one to determine whether a calculated equilibrium configuration of a dna segment is stable in the sense that it gives a local minimum to the sum of the segment s elastic energy and the potential of forces acting on it the derived stability criteria are applicable to plasmids and to linear segments subject to strong anchoring end conditions their utility is illustrated with an example from the theory of configurations of the extranucleosomal loop of a dna miniplasmid in a mononucleosome with emphasis placed on the influence that nicking and ligation on one hand and changes in the ratio of elastic coefficients on the other have on the stability of equilibrium configurations in that example the configurations studied are calculated using an extension of the method of explicit solutions to cases in which the elastic rod modeling a dna segment is considered impenetrable and hence excluded volume effects and forces arising from self contact are taken into account

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

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

euler lagrange equations for nonlinearly elastic rods with self contact we derive the euler lagrange equations for nonlinearly elastic rods with self contact the excluded volume constraint is formulated in terms of an upper bound on the global curvature of the centre line this condition is shown to guarantee the global injectivity of the deformation of the elastic rod topological constraints such as a prescribed knot and link class to model knotting and supercoiling phenomena as observed e g in dna molecules are included by using the notion of isotopy and gaussian linking number the bound on the global curvature as a nonsmooth side condition requires the use of clarke s generalized gradients to obtain the explicit structure of the contact forces which appear naturally as lagrange multipliers in the euler lagrange equations transversality conditions are discussed and higher regularity for the strains moments the centre line and the directors is shown

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

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

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

diophantine approximation and cantor sets we provide an explicit construction of elements of the middle third cantor set with any prescribed irrationality exponent this answers a question posed by kurt mahler

a discrete geometric approach for simulating the dynamics of thin viscous threads we present a numerical model for the dynamics of thin viscous threads based on a discrete lagrangian formulation of the smooth equations the model makes use of a condensed set of coordinates called the centerline spin representation the kinematic constraints linking the centerline s tangent to the orientation of the material frame is used to eliminate two out of three degrees of freedom associated with rotations based on a description of twist inspired from discrete differential geometry and from variational principles we build a full fledged discrete viscous thread model which includes in particular a discrete representation of the internal viscous stress consistency of the discrete model with the classical smooth equations for thin threads is established formally our numerical method is validated against reference solutions for steady coiling the method makes it possible to simulate the unsteady behavior of thin viscous threads in a robust and efficient way including the combined effects of inertia stretching bending twisting large rotations and surface tension

on global solutions and blow up for kuramoto sivashinsky type models and well posed burnett equations abstract the initial boundary value problem ibvp and the cauchy problem for the kuramoto sivashinsky equation v t v x x x x v x x 1 2 v 2 x and other related 2 m th order semilinear parabolic partial differential equations in one dimension and in r n are considered global existence and blow up as well as l bounds are reviewed by using i classic tools of interpolation theory and galerkin methods ii eigenfunction and nonlinear capacity methods iii henry s version of weighted gronwall s inequalities iv two types of scaling blow up arguments for the ibvps existence of global solutions is proved for both dirichlet and navier boundary conditions for some related 2 m th order pdes in r n r uniform boundedness of global solutions of the cauchy problem are established as another related application the well posed burnett type equations v t v v p m v div v 0 in r n r m 1 are considered for m 1 these are the classic navier stokes equations as a simple illustration it is shown that a uniform l p r n bound on locally sufficiently smooth v x t for p n 2 m 1 implies a uniform l r n bound hence the solutions do not blow up for m 1 and n 3 this gives p 3 which reflects the famous leray prodi serrin ladyzhenskaya regularity results l p q criteria and re derives kato s class of unique mild solutions in r n truly bounded classic l 2 solutions are shown to exist in dimensions n 2 2 m 1

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

error analysis for the elastic flow of parametrized curves we analyze a semidiscrete numerical scheme for approximating the evolution of parametric curves by elastic flow in r n the fourth order equation is split into two coupled second order problems which are approximated by linear finite elements we prove error bounds for the resulting scheme and present numerical test calculations that confirm our analysis

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

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

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

symmetry breaking and the twisted elastic ring symmetry breaking is considered in the context of a parameter dependent two point boundary value problem describing a twisted elastic ring that arises as a model of dna minicircles we explicitly determine via a perturbation expansion those representatives of a manifold of solutions to a symmetric bvp that persist when the symmetry is broken conditions generating a generic splitting are identified and some degenerate cases are studied in both the generic and degenerate cases the results of the perturbation expansion are used to improve the efficiency of existing algorithms for computing symmetry broken bifurcation diagrams

geodesic finite elements for cosserat rods we introduce geodesic finite elements as a new way to discretize the non linear configuration space of a geometrically exact cosserat rod these geodesic finite elements naturally generalize standard one dimensional finite elements to spaces of functions with values in a riemannian manifold for the special orthogonal group our approach reproduces the interpolation formulas of crisfield and jelenic geodesic finite elements are conforming and lead to objective and path independent problem formulations we introduce geodesic finite elements for general riemannian manifolds discuss the relationship between geodesic finite elements and coefficient vectors and estimate the interpolation error then we use them to find static equilibria of hyperelastic cosserat rods using the riemannian trust region algorithm of absil et al we show numerically that the discretization error depends optimally on the mesh size copyright 2009 john wiley sons ltd

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

isoperimetric conjugate points with application to the stability of dna minicircles a conjugate point test determining an index of the constrained second variation in one dimensional isoperimetric calculus of variations problems is described the test is then implemented numerically to determine stability properties of equilibria within a continuum mechanics model of dna minicircles

on sums of degrees of the partial quotients in continued fraction expansions of laurent series abstract for any formal laurent series x n v c n z n with coefficients c n lying in some given finite field let x a 0 x a 1 x a 2 x be its continued fraction expansion it is known that with respect to the haar measure almost surely the sum of degrees of partial quotients deg a 1 x deg a n x grows linearly in this note we quantify the exceptional sets of points with faster growth orders than linear ones by their hausdorff dimension which covers an earlier result by j wu

parametric approximation of isotropic and anisotropic elastic flow for closed and open curves deckelnick and dziuk math comput 78 266 645 671 2009 proved a stability bound for a continuous in time semidiscrete parametric finite element approximation of the elastic flow of closed curves in mathbb r d d geq2 we extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in barrett et al j comput phys 222 1 441 462 2007 siam j sci comput 31 1 225 253 2008 ima j numer anal 30 1 4 60 2010 in particular an equidistribution mesh property for this new approximation we obtain also a stability bound for a continuous in time semidiscrete scheme apart from the isotropic situation we also consider the case of an anisotropic elastic energy in addition to the evolution of closed curves we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions

nonlinear dynamics and loop formation in kirchhoff rods with implications to the mechanics of dna and cables the paper contributes a general dynamical formulation and numerical solution procedure for studying nonlinear and three dimensional dynamics of kirchhoff rods target applications include the dynamic formation of dna loops and supercoils as well as loops hockles in marine cables the formulation accommodates non homogeneous and non isotropic inextensible rods both with and without coupling of tension and torsion the utility of this formulation is illustrated by studying the dynamics and quasi static response of a clamped clamped rod subject to compression and or twist for slow loading rates the computed quasi static responses converge to published equilibrium solutions for a benchmark problem as loading rates increase new behaviors are observed including hysteresis in the neighborhood of equilibrium bifurcations the addition of chirality in the form of tension torsion coupling has a pronounced influence on the computed looped geometries this finding has implications for dna loops formed by dna protein binding as well as loop formation in helically wound wire and synthetic cables

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

numerical solution of a bending torsion model for elastic rods aiming at simulating elastic rods we discretize a rod model based on a general theory of hyperelasticity for inextensible and unshearable rods after reviewing this model and discussing topological effects of periodic rods we prove convergence of the discretized functionals and stability of a corresponding discrete flow our experiments numerically confirm thresholds e g for michell s instability and indicate a complex energy landscape in particular in the presence of impermeability

elastic stability of dna configurations ii supercoiled plasmids with self contact configurations of protein free dna miniplasmids are calculated with the effects of impenetrability and self contact forces taken into account by using exact solutions of kirchhoff s equations of equilibrium for elastic rods of circular cross section bifurcation diagrams are presented as graphs of excess link ensuremath delta mathcal l versus writhe w and the stability criteria derived in paper i of this series are employed in a search for regions of such diagrams that correspond to configurations that are stable in the sense that they give local minima to elastic energy primary bifurcation branches that originate at circular configurations are composed of configurations with mathrm d m symmetry m 2 3 dots among the results obtained are the following i there are configurations with mathrm c 2 symmetry forming secondary bifurcation branches which emerge from the primary branch with m 3 and bifurcation of such secondary branches gives rise to tertiary branches of configurations without symmetry ii whether or not self contact occurs a noncircular configuration in the primary branch with m 2 called branch ensuremath alpha is stable when for it the derivative d ensuremath delta mathcal l d mathcal w computed along that branch is strictly positive iii for configurations not in ensuremath alpha the condition d ensuremath delta mathcal l d mathcal w g0 is not sufficient for stability in fact each nonplanar contact free configuration that is in a branch other than ensuremath alpha is unstable a rule relating the number of points of self contact and the occurrence of intervals of such contact to the magnitude of ensuremath delta mathcal l which in paper i was found to hold for segments of dna subject to strong anchoring end conditions is here observed to hold for computed configurations of protein free miniplasmids

p attracting and p invariant sets for a class of impulsive stochastic functional differential equations in this paper a nonlinear and nonautonomous impulsive stochastic functional differential equation is considered by establishing an l operator differential inequality and stochastic analysis technique we obtain the p attracting set and p invariant set of the impulsive stochastic functional differential equation an example is also discussed to illustrate the efficiency of the obtained results

null controllability with constraints on the state for the linear stochastic kuramoto sivashinsky equation abstract this paper is addressed to study the null controllability with constraints on the state for the linear stochastic kuramoto sivashinsky equation we transform the problem into an equivalent null controllability problem with constraint on the control using a carleman inequality adapted to the constraints we prove the null controllability with constraints on the state for the linear stochastic kuramoto sivashinsky equation holds

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

a global attracting set for the kuramoto sivashinsky equation new bounds are given for the l2 norm of the solution of the kuramoto sivashinsky equation partial t u x t partial x 2 partial x 4 u x t u x t partial x u x t for initial data which are periodic with periodl there is no requirement on the antisymmetry of the initial data the result is mathop lim sup limits t to infty left u cdot t right 2 leqslant const l 8 5

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

a variational characterization of a hyperelastic rod with hard self contact abstract we consider an elastic rod modeled as a curve in space with an impenetrable surrounding tube of radius subject to a general class of boundary conditions the impossibility of self intersection is then imposed as a family of scalar constraints on the physical separation of nonlocal pairs of points on the rod thus the usual variational formulation of energy minimization is considered in a context of nonconvex nonsmooth optimization we show the existence of minimizers within suitably defined homotopy classes associated with both the centerline and the frame along the rod the principle results are then concerned with derivation of first order necessary conditions for optimality and some consequences of these for the contact forces and for regularity

a variational rod model with a singular nonlocal potential the classical theory of elastic rods does not account for the possibility that large deformations may involve distinct points along the rod occupying the same physical space we develop an elastic rod model with a pairwise repulsive potential such that if two non adjacent points along the rod are close in physical space there is an energy barrier which prevents contact while for points nearby along the rod the potential is describable classically this framework is developed to prove the existence of minimizers within each homotopy class where the idea of topological homotopy of a curve is generalized to elastic rods as framed curves finally the relevant first order optimality conditions are derived and used to investigate the regularity of minimizers

nonlinear dynamics of filaments i dynamical instabilities abstract the kirchhoff model provides a well established mathematical framework to study both computationaly and theoretically the dynamics of thin filaments within the approximations of linear elasticity theory the study of static solutions to these equations has a long history and the usual approach to describing their instabilities is to study the time dependent version of the kirchhoff model in the euler angle frame here we study the linear stability of the full time independent equations by introducing a new are length preserving perturbation scheme as an application we consider the instabilities of various stationary solutions such as the planar ring and straight rod subjected to twisting perturbations this scheme gives a direct proof of the existence of dynamical instabilities and provides the selection mechanism for the shape of unstable filaments

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

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

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

on a problem of k mahler diophantine approximation and cantor sets let k denote the middle third cantor set and mathcal a 3 n n 0 1 2 ldots given a real positive function let w mathcal a psi denote the set of real numbers x in the unit interval for which there exist infinitely many p q in mathbb z times mathcal a such that x p q q the analogue of the hausdorff measure version of the duffin schaeffer conjecture is established for w mathcal a psi cap k one of the consequences of this is that there exist very well approximable numbers other than liouville numbers in k an assertion attributed to k mahler explicit examples of irrational numbers satisfying mahler s assertion are also given

large deviation for the nonlocal kuramoto sivashinsky spde abstract in this paper we establish a large deviation principle for the weak solution to a nonlocal kuramoto sivashinsky stochastic partial differential equation with small noise perturbation the key technique is an application of the contraction principle

dynamics of a nonlocal kuramoto sivashinsky equation abstract in this paper we study the effects of a nonlocal term on the global dynamics of the kuramoto sivashinsky equation we show that the equation possesses a family of maximal attractors parameterized by the mean value of the initial data the dimension of the attractor is estimated as a function of the coefficient of the nonlocal term and the width of the periodic domain

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

dynamics for the stochastic nonlocal kuramoto sivashinsky equation dynamics for the stochastic kuramoto sivashinsky equation with a nonlocal term is studied we prove that the stochastic equation has a finite dimensional random attractor